Dynamical Systems, Bifurcation Analysis and Applications: Penang, Malaysia, August 6–13, 2018 [1st ed. 2019] 978-981-32-9831-6, 978-981-32-9832-3

Table of contents :
Front Matter ....Pages i-x
Front Matter ....Pages 1-1
Mathematical Modeling and Stability Analysis of Population Dynamics (Auni Aslah Mat Daud)....Pages 3-13
Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System (Tau Keong Ang, Hamizah M. Safuan, Ummu Atiqah M. Roslan, Mohd Hafiz Mohd)....Pages 15-29
Dynamics and Bifurcations in a Dynamical System of a Predator-Prey Type with Nonmonotonic Response Function and Time-Periodic Variation (Johan M. Tuwankotta, Eric Harjanto, Livia Owen)....Pages 31-49
Modeling and Experimental Data on the Dynamics of Predation of Rice Plants and Weeds by Golden Apple Snail (Pomacea Canaliculata) (Joel Addawe, Zenaida Baoanan, Rizavel Addawe)....Pages 51-65
Front Matter ....Pages 67-67
Analysis of a Discrete-Time Fractional Order SIR Epidemic Model for Childhood Diseases (Mahmoud A. M. Abdelaziz, Ahmad Izani Ismail, Farah A. Abdullah, Mohd Hafiz Mohd)....Pages 69-88
Front Matter ....Pages 89-89
A Tuberculosis Epidemic Model with Latent and Treatment Period Time Delays (Jay Michael R. Macalalag, Elvira P. De Lara-Tuprio, Timothy Robin Y. Teng)....Pages 91-115
Numerical Bifurcation and Stability Analyses of Partial Differential Equations with Applications to Competitive System in Ecology (Mohd Hafiz Mohd)....Pages 117-132
Front Matter ....Pages 133-133
Global Stability Index for an Attractor with Riddled Basin in a Two-Species Competition System (Ummu Atiqah Mohd Roslan, Mohd Tirmizi Mohd Lutfi)....Pages 135-146
Counting Closed Orbits in Discrete Dynamical Systems (Azmeer Nordin, Mohd Salmi Md Noorani, Syahida Che Dzul-Kifli)....Pages 147-171
Front Matter ....Pages 173-173
Computational Dynamical Systems Using XPPAUT (Ojonubah James Omaiye, Mohd Hafiz Mohd)....Pages 175-203
A Basic Manual for AUTO-07p in Computing Bifurcation Diagrams of a Predator-Prey Model (Livia Owen, Eric Harjanto)....Pages 205-224
Numerical Continuation and Bifurcation Analysis in a Harvested Predator-Prey Model with Time Delay using DDE-Biftool (Juancho A. Collera)....Pages 225-241

Citation preview

Springer Proceedings in Mathematics & Statistics

Mohd Hafiz Mohd Norazrizal Aswad Abdul Rahman Nur Nadiah Abd Hamid Yazariah Mohd Yatim Editors

Dynamical Systems, Bifurcation Analysis and Applications Penang, Malaysia, August 6–13, 2018

Springer Proceedings in Mathematics & Statistics Volume 295

Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

More information about this series at http://www.springer.com/series/10533

Mohd Hafiz Mohd Norazrizal Aswad Abdul Rahman Nur Nadiah Abd Hamid Yazariah Mohd Yatim •

•

•

Editors

Dynamical Systems, Bifurcation Analysis and Applications Penang, Malaysia, August 6–13, 2018

123

Editors Mohd Hafiz Mohd School of Mathematical Sciences Universiti Sains Malaysia USM, Penang, Malaysia

Norazrizal Aswad Abdul Rahman School of Mathematical Sciences Universiti Sains Malaysia USM, Penang, Malaysia

Nur Nadiah Abd Hamid School of Mathematical Sciences Universiti Sains Malaysia USM, Penang, Malaysia

Yazariah Mohd Yatim School of Mathematical Sciences Universiti Sains Malaysia USM, Penang, Malaysia

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-981-32-9831-6 ISBN 978-981-32-9832-3 (eBook) https://doi.org/10.1007/978-981-32-9832-3 Mathematics Subject Classification (2010): 70G60, 37H20, 97M10, 92B05, 97N80 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Organization

Southeast Asian Mathematical Society (SEAMS) School 2018 on Dynamical Systems and Bifurcation Analysis is organized by the School of Mathematical Sciences, Universiti Sains Malaysia.

Organizing Committee Advisor:

Chair: Secretary: Treasurer: Programme: Accommodation: Technical and Logistic:

Publicity: Secretariat: Liaison Officer: Publication:

Rosihan M. Ali Hailiza Kamarulhaili Ahmad Izani Md. Ismail Mohd Hafiz Mohd Yazariah Mohd Yatim Norshafira Ramli Nur Nadiah Abd Hamid Noor Saifurina Nana Khurizan Maisarah Haji Mohd Siti Amirah Abd Rahman Farah Aini Abdullah Siti Zulaikha Mohd Jamaludin Majid Khan Majahar Ali Hartini Ahmad Syed Mohamed Hussain Syed Osman Ong Wen Eng Md Yushalify Misro Amirah Azmi Shamani Supramaniam Syakila Ahmad Noor Atinah Ahmad Norazrizal Aswad Abdul Rahman

v

Preface

This volume collects research papers and survey articles by participants and speakers in the SEAMS School on Dynamical Systems and Bifurcation Analysis (DySBA), which was held in Penang, Malaysia, from 6 to 13 August 2018. The event was organized by the School of Mathematical Sciences, Universiti Sains Malaysia (USM), under the auspices of the Southeast Asian Mathematical Society (SEAMS), Centre International de Mathématiques Pures et Appliquées (CIMPA) and Commission for Developing Countries (CDC) of the International Mathematical Union (IMU). The SEAMS School 2018 on DySBA was designed as part of a series of mathematics study programmes that aims to provide the opportunity for an advanced learning experience in the field of dynamical systems via planned lectures, contributed talks and hands-on workshops. It also aimed to introduce research-based learning for advanced undergraduate students, postgraduate students and young academics. The main topics highlighted in SEAMS School 2018 on DySBA were differential equations and discrete dynamical systems with applications to mathematical biology. The event brought together both experts and novices in the theory and applications of dynamical systems and bifurcation analysis. The scientific objective of this school was to form a research network among ASEAN mathematicians to collaborate within the broader research community, as well as to open up new opportunities for researchers to link up and collaborate in these fields. The school also intended to foster a joint scientific collaboration between the School of Mathematical Sciences, USM and other international institutions. In general, this school was the first ever SEAMS school concerning dynamical systems and bifurcation theory in Malaysia. The successful organization of this event demonstrates the ability of the School of Mathematical Sciences, USM to participate actively in research engagement activities and scientific collaboration at the regional and international levels. This edited volume is prepared as a first step in the joint collaboration between ASEAN mathematicians and international researchers to work together in the fields of dynamical systems and bifurcation analysis. It will serve as a unified survey on the development and recent progress in research on dynamical systems and vii

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Preface

bifurcation from the Southeast Asia region. The authors contributing to the chapters in this book are also active researchers in these fields. This volume intends to give an exposure to prospective readers regarding the theoretical and practical aspects of dynamical systems and bifurcation analysis, as well as recent techniques on numerical continuation and computational methods. While other existing books in the field often give more attention on a particular topic in dynamical systems, our book focuses on the state-of-the-art research on dynamical systems and bifurcation analysis in a broader sense. This is crucial to ensure that readers can grasp different concepts in dynamical systems and understand how the techniques from differential equations, bifurcation analysis and numerical continuation have been employed in analysing the dynamical behaviours of the models under consideration. Among the modelling frameworks that have been employed in this volume are Ordinary Differential Equations (Part I), Fractional Differential Equations (Part II), Delay and Partial Differential Equations (Part III) and Discrete Dynamical Systems (Part IV). This volume also discusses the importance of bifurcation analysis and some numerical continuation packages that can be used to track both stable and unstable steady states together with bifurcation points to gain a better understanding of the dynamics of the systems; these concepts are highlighted in Part V under Computational Dynamical Systems. Special attention is also given to applications of dynamical systems and bifurcation analysis in mathematical biology and ecological problems. Several biological examples from recent research are employed as illustrations. Through these examples, it is shown how the techniques from dynamical systems and bifurcation theory have been applied to these different mathematical models in order to answer a range of biologically inspired questions encompassing topics such as species biodiversity and environmental issues. This book will be of great interest to researchers, scientists and educators who work in the fields of differential equations, dynamical systems, bifurcation theory and their applications. We would like to thank all participants, course lecturers and invited speakers for making the school a great success. Thanks to the organizing committee as well for their great efforts in conducting a well-run school. We are grateful to all of the authors for their contributions to this volume and to all of the reviewers for their timely and detailed feedback on the manuscripts. Additionally, we would like to extend our deepest gratitude to the School of Mathematical Sciences, Universiti Sains Malaysia, SEAMS, CIMPA, CDC-IMU, Division of Research and Innovation USM, Department of Higher Education, Ministry of Education Malaysia and Malaysian Mathematical Sciences Society for their valuable support and commitment to make the event successful. USM, Penang, Malaysia April 2019

Mohd Hafiz Mohd Norazrizal Aswad Abdul Rahman Nur Nadiah Abd Hamid Yazariah Mohd Yatim

Contents

Ordinary Differential Equations Mathematical Modeling and Stability Analysis of Population Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Auni Aslah Mat Daud Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tau Keong Ang, Hamizah M. Safuan, Ummu Atiqah M. Roslan and Mohd Hafiz Mohd Dynamics and Bifurcations in a Dynamical System of a Predator-Prey Type with Nonmonotonic Response Function and Time-Periodic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johan M. Tuwankotta, Eric Harjanto and Livia Owen Modeling and Experimental Data on the Dynamics of Predation of Rice Plants and Weeds by Golden Apple Snail (Pomacea Canaliculata) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joel Addawe, Zenaida Baoanan and Rizavel Addawe

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15

31

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Fractional Differential Equations Analysis of a Discrete-Time Fractional Order SIR Epidemic Model for Childhood Diseases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mahmoud A. M. Abdelaziz, Ahmad Izani Ismail, Farah A. Abdullah and Mohd Hafiz Mohd

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Delay and Partial Differential Equations A Tuberculosis Epidemic Model with Latent and Treatment Period Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jay Michael R. Macalalag, Elvira P. De Lara-Tuprio and Timothy Robin Y. Teng

91

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Contents

Numerical Bifurcation and Stability Analyses of Partial Differential Equations with Applications to Competitive System in Ecology . . . . . . . 117 Mohd Hafiz Mohd Discrete Dynamical Systems Global Stability Index for an Attractor with Riddled Basin in a Two-Species Competition System . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Ummu Atiqah Mohd Roslan and Mohd Tirmizi Mohd Lutfi Counting Closed Orbits in Discrete Dynamical Systems . . . . . . . . . . . . . 147 Azmeer Nordin, Mohd Salmi Md Noorani and Syahida Che Dzul-Kifli Computational Dynamical Systems Computational Dynamical Systems Using XPPAUT . . . . . . . . . . . . . . . . 175 Ojonubah James Omaiye and Mohd Hafiz Mohd A Basic Manual for AUTO-07p in Computing Bifurcation Diagrams of a Predator-Prey Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Livia Owen and Eric Harjanto Numerical Continuation and Bifurcation Analysis in a Harvested Predator-Prey Model with Time Delay using DDE-Biftool . . . . . . . . . . . 225 Juancho A. Collera

Ordinary Differential Equations

Mathematical Modeling and Stability Analysis of Population Dynamics Auni Aslah Mat Daud

Abstract This study provides a brief introduction of important terminologies and methodologies in the mathematical modelling and stability analysis of the population dynamics. As an example, a mathematical model of population dynamics for thyroid disorder during pregnancy is developed and analysed. The disorders are the second most common endocrine disorders among women in childbearing age, where inadequate or excessive amount of thyroid hormones are produced due to various causes. Thyroid disorders during pregnancy and postpartum can be divided into three types: hyperthyroidism, hypothyroidism and postpartum thyroiditis. They may lead to numerous complications to both mothers and foetuses, such as heart failure, pre-eclampsia, miscarriage, premature birth, and perinatal mortality. The model is described using a system of first order linear ordinary differential equations. Its stability is studied using Routh-Hurwitz criteria. It is found that the model has only one non-negative equilibrium, which is locally asymptotically stable. Keywords Mathematical model · Thyroid disorders · Equilibrium · Stability analysis · Routh Hurwitz criteria

1 Introduction A mathematical model is a representation of a real-world system. Mathematical modelling is an important mean for analysing the spread and control of diseases, determining dominant parameters and sensitivities to changes in parameter values. Mathematical modelling has at least two main objectives. Firstly, the models are A. A. Mat Daud (B) Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala, Nerus, Malaysia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_1

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beneficial in understanding the behaviour of the system under study and the identification of important parameters in the system, which can then be used to control the outcome. Secondly, a mathematical model can be employed to predict the future behaviour (such as weather) or measure the outcomes of experiments that cannot be performed in reality (too expensive, not reproducible, dangerous, impractical, or unethical). For example, it would be ethical and non-detrimental for researchers to kill half the female members of a population to measure the population recovery time if they do so using mathematical models instead of real populations. The solution of a differential equation may not be so important if the solution never appears in the practical, or is only physically realizable under exceptional circumstances. The equilibrium solutions of the model correspond to configurations in which the physical system does not change over time and only occur in the real world if they are stable. Meanwhile, unstable equilibria are not significant and unlikely to occur in reality because these equilibria can only be observed if the initial condition is exactly right while slight perturbations in the system or its physical surroundings will immediately dislodge the system far away from equilibrium. Biological models often study how the quantities—such as population size changes over time. Mathematically, changes in continuous variables can be described using differential equations. For simple and low-dimensional models, the ideal way to know the values of each variable at all times is by finding the general solution of the differential equations. However, mathematical models in biology are usually governed by differential equations which are very difficult or impossible to solve. Nevertheless, the stability analysis of a model enables us to obtain sufficient information about the behavior of the model without finding the explicit solution of the differential equations. This is important because generally, analyzing a differential equation qualitatively is much easier than finding its analytical solution. The objective of this study is to propose a mathematical model which can be employed to understand and predict how the subpopulations of patients with thyroid disorders (namely the number of normal non-pregnant women, the number of non-pregnant women with thyroid disorders, the number of pregnant or postpartum women with thyroid disorders without complications and the number of pregnant or postpartum women with thyroid disorders with complications) change over time. To achieve this objective, we will formulate a mathematical model for the population dynamics of the disease using compartmental modelling, and carry out the stability analysis of its equilibrium point. Thyroid Disorders The thyroid gland is an endocrine organ which produce thyroid hormones known as thyroxine (T4) and triiodothyronine (T3) which are important in regulating metabolism and development [1]. In pregnancy and postpartum, thyroid disorders are the second commonest endocrine disorder that may affect women in the form of hypothyroidism, hyperthyroidism and postpartum thyroiditis. Hypothyroidims is a condition where there is insufficient thyroid hormone which may be due to iodine deficiency, anti-thyroid antibodies, iatrogenic effects or congenital anomalies. On the contrary, an oversupply of thyroid hormones known as hyperthyroidism may

Mathematical Modeling and Stability Analysis of Population Dynamics

5

result from autoimmune disorder, pituitary tumour, thyroid nodule, thyroid cancer, thyroiditis, multiple pregnancy and molar pregnancy. These conditions may result in adverse effects to both mothers and foetoses such as maternal heart failure, maternal hypertension, maternal haemorrhage, preeclampsia, anaemia, placental abruption, miscarriage, stillbirth preterm delivery, intrauterine growth restriction, low birth weight, congenital malformation, foetal neurocognitive deficit and foetal thyroid dysfunction. Postpartum thyroiditis is an autoimmune inflammation of the thyroid gland after giving birth and is manifested as hyperthyroidism, hypothyroidism or hyperthyroidism followed by hypothyroidism [2]. If the thyroid dysfunction persists, the patient (depending on the thyroid condition) has higher risks of cardiovascular diseases, mental disorders, paralysis, infertility, osteoporosis, thyroid storm and myxoedema coma (see [1, 3, 4]). Thyroid disorders may happen to anyone at any age. A woman can only be pregnant during the childbearing age. A woman may acquire the disorder prior to or during her childbearing age and the condition may be treated, depending on the causes. Thyroid disorders during pregnancy and postpartum may be a chronic condition (where the disorder is present before the pregnancy or postpartum) or acquired during the pregnancy and postpartum period. As stated above, the condition may give rise to complications in some women. Similarly, the condition may be treated, depending on the causes. Once the postpartum period ends, the women may or may not continue to have the thyroid disorder, depending on the causes and treatments received (see [1–4]).

2 Methodology 2.1 Mathematical Modelling In this study, the modeling process begins with the construction of the flow diagram. It is a means of illustrating the model population, which is compartmentalized into state variables. In a flow diagram, the boxes indicate the state variables while the movement of people in the population is indicated by arrows. An ordinary differential equation refers to an instantaneous rate of change in a state variable with respect to the independent variable (time). The flows shown as arrows are calculated using the terms on the right-hand side of the equations: a flow out of a box is taken away from a state variable and is a negative term while a flow into a box is added and is a positive term, as discussed in Appendix A. Some parameters will be introduced to represent the proportionality rates of the flows.

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2.2 Stability Analysis An equilibrium point is stable if the system near to the equilibrium point will approach to that equilibrium point while a system is unstable if the system near to the equilibrium point moves away from that equilibrium point [5]. In this study, the stability analysis is conducted using Routh-Hurwitz criteria, which is discussed in detail in Appendix C.

3 Results and Discussion 3.1 Model The flow diagram which describes the population dynamics for thyroid disorders during pregnancy is shown in Fig. 1. There are four dependent variables which represent different stages of the disease, namely the number of normal non-pregnant women A(t), the number of non-pregnant women with thyroid disorders B(t), the number of pregnant or postpartum women with thyroid disorders without complications C(t) and the number of pregnant or postpartum women with thyroid disorders with complications D(t). There are nineteen constant parameters in this study, with values between 0 and 1, representing the following per capita rates or probability per unit time (except Pand Q which represent the numbers of individuals per unit time): μ : death due to causes other than thyroid disorders, τ : pregnant/postpartum women with thyroid disorders who die due to thyroid disorders complications, κ : pregnant/postpartum women with thyroid disorders who die due to thyroid disorders, ρ : non-pregnant women with thyroid disorders who die due to thyroid disorders, P : women entering the childbearing age population without thyroid disorders, Q : women entering the childbearing age population with thyroid disorders, ν A : women of subpopulation A leaving the childbearing age population, ν B : women of subpopulation B leaving the childbearing age population, νC : women of subpopulation C leaving the childbearing age population, ν D : women of subpopulation D leaving the childbearing age population, α : normal women in childbearing age having thyroid disorders, δ : non-pregnant women with thyroid disorders in childbearing age becoming pregnant with thyroid disorders, ξ : normal women in childbearing age having thyroid disorders during pregnancy/postpartum period, Υ : non-pregnant women with thyroid disorders recovering from thyroid disorders, η : pregnant/postpartum women with thyroid disorders recovering from them in nonpregnant state,

Mathematical Modeling and Stability Analysis of Population Dynamics

7

Fig. 1 Flow diagram of the population dynamics for thyroid disorders during pregnancy

β : pregnant/postpartum women with thyroid disorders who continue to have them in non-pregnant state, σ: pregnant/postpartum women with thyroid disorders having complications during pregnancy/postpartum period, ε : pregnant/postpartum women with thyroid disorders with complications recovering to become normal non-pregnant women after postpartum period, ω : pregnant/postpartum women with thyroid disorders with complications recovering from complications but continue to have thyroid disorders in non-pregnant state. The flow diagram in Fig. 1 is transformed to a system of first order linear ordinary differential equations. The mathematical model is governed by the following system of differential equations: dA = P − μA − ν A A − α A − ξ A + Υ B + ηC + εD dt dB = Q − ν B B − ρB + βC − δ B − μB + α A − Υ B + ω D dt dC = ξ A − ηC + δ B − βC − σC − κC − νC C − μC dt dD = σC − ω D − τ D − ν D D − μD − εD dt

(1) (2) (3) (4)

with initial conditions A (0) = A0 , B (0) = B 0 , C (0) = C 0 and D (0) = D 0 . Note that the terms in Eqs. (1)–(4) which involve the per capita rates are the products of probability per unit time and one of the state variables, which represents the number of individuals. Since Eqs. (1)–(4) represent interaction between individuals in the population, it makes sense to state that all the parameters involved are non-negative. It is also pertinent to show that all the state variables of the model are non-negative for all time. Hence, we have the following theorem (see the proof in Appendix D):

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Theorem 1 The solution set {A, B, C, D} of Eqs. (1)–(4) with non-negative initial conditions A (0) = A0 , B (0) = B 0 , C (0) = C 0 and D (0) = D 0 remains nonnegative for all time t > 0.

4 Stability Analysis By taking Eqs. (1)–(2) equal to zero and solve the resulting equations simultaneously we found that the equilibrium point ( A∗ , B ∗ , C ∗ , D ∗ ) as follow: M N2Q  (Mηδ + εσδ + N MΥ )2  −M N P N Mβδ + N ωσδ − M N S   A∗ = [(Mηξ+εσξ − N M R) N Mβδ + N ωσδ − M N 2 S −   M N 2 α + N Mβξ+N ωσξ (Mηδ + εσδ + N MΥ )]   −M N 2 Q − M N 2 α + N Mβξ+N ωσξ A∗   B = N Mβδ + N ωσδ − M N 2 S ∗

C∗ =

ξ A∗ + δ B ∗ N

D∗ =

σC ∗ M

where R = μ + ν A + α + ξ, S = ν B + ρ + δ + Υ + μ, N = η + β + σ + κ + νC + μ, M = ω + τ + ν D + μ + ε. Conducting the stability analysis of the equilibrium point (A∗ , B ∗ , C ∗ , D ∗ ) using Routh-Hurwitz criteria for n = 4, the equilibrium point is asymptotically stable if ai > 0, i = 1, 2, 3, 4 a1 a2 a3 > a32 + a12 a4 where ai , i = 1, 2, 3, 4 are the coefficients in the characteristic polynomial,   ∗  J − λI  = λ4 + a1 λ3 + a2 λ2 + a3 λ + a4

Mathematical Modeling and Stability Analysis of Population Dynamics

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where λ are the eigenvalues, I is the matrix identity, and J ∗ is the Jacobian matrix of the model, evaluated at the equilibrium point. The model in this study can be helpful to the policy makers in setting the aim for each parameter in order to achieve certain targets for the number of patients. The parameter values of the developed model can be estimated using the available data from the literature, in the context of Malaysia in order to forecast the number of patients in Malaysia. These parameter values can also be estimated for any other domain in the world using the corresponding prior data for that particular domain. Using appropriate estimated parameter values, one may predict the number of patients for all stages of the disease for the coming few years for any geographical domain.

5 Conclusion In this study, a mathematical model that describes the population dynamics of thyroid disorders during pregnancy has been formulated. The model was governed by a system of linear first order ordinary differential equations. The equilibrium point of the proposed model is determined and it is asymptotically stable. It is important to note that the research of equilibrium point and stability analysis in this study were found directly by hand. Therefore, in future, an interactive tool can be developed by modelers (for example, using computational tools such as Wolfram MATHEMATICA) for non-modellers to experiment scenarios and visualize the possible consequences under various actions. Acknowledgements This research is funded by the Ministry of Higher Education, Government of Malaysia under the Research Acculturation Grant Scheme (RAGS 57108). The results of the study were partially presented during The 4th International Conference on Mathematical Sciences (ICMS4) on November 15–17, 2016 at Palm Garden Hotel, Putrajaya, Malaysia and the SEAMS School on Dynamical Systems and Bifurcation Analysis (DySBA)on August 6–13, 2018 at Universiti Sains Malaysia (USM), Penang, Malaysia. Finally, I would like to express my gratitude to Salilah Saidun and Kritika Manimaran, who have contributed in this study.

6 Appendixes Appendix A: Formulating ODEs Based on a Flow Diagram The state variables are depicted by the boxes in the flow diagram while the arrows illustrate the movement of people between different states in the population. The flows shown as arrows are calculated using the terms on the right-hand side of the equations: a flow pointing out of a box is taken away from a state variable and is a negative term while a flow pointing into a box is added and is a positive term [6].

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Fig. 2 An example of a flow diagram

Fig. 3 The resulting governing ODEs (corresponding to the flow diagram in Fig. 2)

Some constant parameters are introduced adjacent to the arrows to represent the proportionality rates of the flows. An illustration of an example of a flow diagram is provided in Fig. 2, while the formulation of differential equations (the governing equations of the model) based on the flow diagram is shown in Fig. 3. where β is the birth rate, μ is the death rate, α is the transformation rate from X (t) to Y (t) and ρ is the transformation rate from X (t) to Y (t).

Appendix B: The Computation of Equilibrium Point Consider a mathematical model governed by a system of differential equations: d x1 = f 1 (x1, x2,..., xn ) dt d x2 = f 2 (x1, x2,..., xn ) dt .. . d xn = f n (x1, x2,..., xn ) dt

Mathematical Modeling and Stability Analysis of Population Dynamics

11

By definition, the equilibrium values of the model are all values that cause the variables to remain unchanged, or mathematically: n ) = 0 x1 , x f 1 ( 2, . . . , x n ) = 0 x1 , x f 2 ( 2, . . . , x .. . n ) = 0 f n ( x1 , x 2, . . . , x n ) is obtained by solving the equations above The equilibrium point ( x1 , x 2, . . . , x simultaneously or using any methods of solving system of equations in linear algebra.

Appendix C: The Stability Analysis Using Routh Hurwitz Criteria In this study, we will study a multiple variables model with continuous time. Therefore, the stability analysis is performed using the following steps: Step 1: Calculating a Jacobian matrix. ⎛ ∂f ⎜ ⎜ ⎜ J =⎜ ⎜ ⎜ ⎝

(x1 , ∂x 1 ∂ f2 (x1 , ∂x 1 1

x2 , . . . , xn ) x2 , . . . , xn )

∂ f1 (x1 , ∂x 2 ∂ f2 (x1 , ∂x 2

.. . ∂ fn (x1 , ∂x 1

x2 , . . . , xn )

x2 , . . . , xn ) · · ·

∂ fn (x1 , ∂x 2

x2 ,

x2 , . . . , xn )

⎞

⎟ x2 , . . . , xn ) ⎟ ⎟ ⎟ ⎟ .. .. ⎟ . . ⎠ . . . , xn ) · · · ∂∂xf nn (x1 , x2 , . . . , xn )

x2 , . . . , xn ) · · · .. .

∂ f1 (x1 , ∂x n ∂ f2 (x1 , ∂x n

where ∂∂xf ji (x1 , x2 . . . , xn ) is the partial derivative of f i with respect to its variable, x j (i, j = 1, 2, . . . , n). Step 2: Find the Jacobian matrix. The Jacobian matrix is evaluated at the equilibrium values, x1 , x n . A local 2, . . . , x is obtained. Then find the characteristic stability matrix, Jˆ = J |x1 = x1 , x2 = x2 , ..., xn =xn polynomial using det Jˆ − λI = 0, where I is the identity matrix, and rewrite in the following form: P (λ) = λn + a1 λ n−1 + · · · + an−1 λ+an with real coefficients ai for i = 1, 2, . . . , n.

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Step 3: Use Routh-Hurwiz criteria. The n Hurwitz matrices are defined as follow: ⎛ a1 1 ⎜ a3 a2   ⎜ a1 1 ⎜ H1 = (a1 ), H2 = , and Hn = ⎜ a5 a4 a3 a2 ⎜ .. .. ⎝ . .

0 a1 a3 .. .

0 1 a2 .. .

⎞ ··· 0 ··· 0 ⎟ ⎟ ··· 0 ⎟ ⎟ .. ⎟ ··· . ⎠

0 0 0 0 · · · an

Note that if j > n, then a j = 0. If and only if all det H j > 0 with j = 1, 2, . . . , n, then P (λ) has roots that are negative or have negative real part and hence the equilibrium point is said to be asymptotically stable. The Routh-Hurwitz criteria for polynomials of degree, n = 4 are [7]: a1 > 0, a3 > 0, a4 > 0, and a1 a 2 a3 > a32 + a12 a4 .

Appendix D: Proof of Theorem 1 Given that the initial conditions A (0) = A0 , B (0) = B 0 , C (0) = C 0 and D (0) = D0 are non-negative. It is clear from Eq. (1) that dA + [ξ + μ + ν + α]A(t) ≥ 0, dt so that,

 d  A(t)exp (ξ + μ + ν + α) t ≥ 0. dt

(A1)

Integrating (A1) with respect to t, gives A (t) ≥ A (0) exp [− (ξ + μ + ϕν + α) t] > 0,

∀t ≥ 0.

Similarly, it can be shown that B (t) > 0, C(t) > 0, D(t) > 0 for all time t > 0. This completes the proof. It is crucial to note that Eqs. (1)–(4) will be analyzed in a feasible region D given by   4 D = (A, B, C, D) ∈ R+ : A+ B +C + D = N , which can be easily verified to be positively invariant with respect to Eqs. (1)–(4), In what follows, the model is epidemiologically and mathematically well posed in D. (see [8]).

Mathematical Modeling and Stability Analysis of Population Dynamics

13

References 1. Boelaert, K., Franklyn, J.A.: Thyroid hormone in health and disease. J. Endocrinol. 187(1), 115 (2005). https://doi.org/10.1677/joe.1.06131 2. Carney, L.A., Quinlan, J.D., West, J.M.: Thyroid disease in pregnancy. Am. Family Physician 89(4), 273–278 (2014) 3. Cooper, D., McDermott, M., Wartofsky, L.: Hypothyroidism. J. Clin. Endocrinol. Metab. 89(11), E2 (2004). https://doi.org/10.1210/jcem.89.11.9990 4. Leo, S.D., Lee, S.Y., Braverman, L.E.: Hyperthyroidism. Lancet. 388(10047), 906918 (2016). https://doi.org/10.1016/S0140-6736(16)00278-6 5. Kopp, M.: Equilibria and stability analysis. Universitet Wien (2011) http://www.mabs.at/koop/ teaching/modelling2011/files/stab.pdf 6. Garnett, G.P.: An introduction to mathematical models in sexually transmitted disease epidemiology. Sex Transm Inf. 78, 712 (2002) 7. Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42, 599653 (2000). https:// doi.org/10.1137/S0036144500371907 8. Allen, L.J.S.: Introduction to Mathematics Biology. Pears Education, Slough (2006)

Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System Tau Keong Ang, Hamizah M. Safuan, Ummu Atiqah M. Roslan and Mohd Hafiz Mohd

Abstract The present paper examines a predator-prey fishery system by taking into account the toxin released by the fish which can lead to polluted system. Both predator and prey fish species obey the logistic population growth with their respective environmental carrying capacities. In the proposed model, both fish species produce toxins that contribute to mutual infection which is detrimental to each other. Different harvesting efforts are applied on the predator and prey fish populations, respectively. The equilibria existed in the model are studied together with the local stability properties. We consider the threshold conditions which trigger the bifurcation that occurred in the steady states. The global stability properties of coexistence equilibrium are studied by constructing an appropriate Lyapunov function. Bendixson-Dulac criterion is applied to rule out the existence of limit cycle in the system. From the bifurcation analysis, the dynamical behaviors of the system are observed as well as the persistence and extinction properties. It is shown that harvesting parameters are most likely to drive a fish population towards extinction compared to toxicant parameters which are less influential. Regions of optimal harvesting strategies were found to guarantee the persistence of both fisheries. Finally, the existence of a bionomic equilibrium solution has been examined with three possible cases. Keywords Fishery · Prey-predator · Harvesting · Toxicant · Bionomic equilibrium

T. K. Ang (B) · H. M. Safuan Universiti Tun Hussein Onn Malaysia, 84600 Pagoh, Johor, Malaysia e-mail: [email protected] U. A. M. Roslan Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia M. H. Mohd Universiti Sains Malaysia, 11800 Penang, Malaysia © Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_2

15

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1 Introduction Due to the developing global concerns on the management strategies to eradicate the problems of over-exploitation and over-utilization of natural resources, ecological modelling is proved to be one of the efficient and economic approaches to investigate the possible cause and effect [1, 2]. Natural resource such as fishery is an essential source of protein to human being. However, the depletion of fisheries has distressed the ecological balance and the effects can be detrimental if the resource becoming scarce and eventually extinct. In population ecology, the interaction between a species and the associated environments is one of the crucial elements to consider since the dynamical behaviors of the species is strictly dependent on the environment. The presence of toxin is claimed to have a hazardous effect on the population’s mortality rate and some of the pioneering work on the toxic in population models were studied by Chattopadhyay [3] and Samanta [4]. Both researchers proposed a two competing species model with toxin produced by two competitors. The toxin produced can be harmful to each other and the ratio of toxin is a decisive factor for the stability properties exhibited in the system. The abnormal level of toxic substances can induce critical fluctuation on the dynamics from both the biological and ecological perspectives, causing the system to become unstable. As the impacts of pollutant are getting immediate attention in population dynamics, numerous research were done to investigate the behaviors or dynamics caused by toxin in a more specific terrestrial community and also the aquatic environment. For instance, a series of research works were done by Huang et al. [5, 6] that emphasized on the dynamics of fish population in the presence of pollutants. Both research showed the changes in dynamics of fishery models in the presence of contaminated environment, as compared to the conventional fishery models. Both qualitative and quantitative approaches showed the consistent result that the concentration of toxin is the key factor for the population persistence. On the other hand, the commercial values of fishery resources had drawn the interest of researchers on the farming effects of fish population. Kar and Chaudhuri [7] explored the exploitation of fisheries by developing a competing fishery model which is influenced by toxicity while Das et al. [8] contributed his idea on harvesting activities of predator-prey fishery model. Similar research was studied by Ang et al. [9]. The research [7–9] examined the properties of equilibria in the system and found that harvesting activities can give rise to great impacts both biologically and economically to the fishery systems. The principal aim of the present research is to work on the dynamical nature of a toxin-polluted fishery model where mutual infection occurs among the predator and prey fish populations. Both the predator and prey fish species are subjected to nonlinear logistic growth rate with different harvesting efforts. Bifurcation analysis is performed with respect to the harvesting parameters to determine the influence of harvesting activities on the persistence and extinction properties. In the present work, we found that the effect of harvesting is more significant than that of toxin in inducing the state that lead to the extinction of the fish population.

Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System

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2 Predator-Prey Model For the purpose of investigating predator-prey fishery system in the presence of toxin produced by fish itself, a similar model as [9] is proposed:  dX = r1 X 1 − dt  dY = r2 Y 1 − dt

 X − γ1 X Y − a E 1 X − bX 2 Y, K1  Y + γ2 X Y − cE 2 Y − d X Y 2 . K2

(1)

Referring to model (1), the prey fish population, X (t) and predator fish population, Y (t) obey the logistic growth rates of r1 and r2 , restricted by the environmental carrying capacities of K 1 and K 2 respectively. Despite of the prey-predator interaction of parameters γ1 and γ2 on prey and predator fish respectively, parameters E 1 and E 2 indicate the respective harvesting effort exerted on prey and predator fish with the catchability coefficients of a and c respectively. In this model, both fish species generate some toxins that can affect the mortality rates of each other where the toxin coefficients on prey fish and predator fish are denoted by b and d respectively. The toxins are suspected to infect both fish species through the feeding process of predator on prey and also the direct infection from the contaminated aquatic environment.

3 Steady States, Equilibria and Stability Properties In current section, we examine the existing equilibria in system (1) which are meaningful in the biological point of views with the consideration of their stability behavγ1 Y γ2 X , y= and τ = r1 t, the iors. Introducing the set of scaled variables x = r1 r1 non-dimensional model of system (1) is dx = x (1 − αx) − x y − βx − δx 2 y, dτ dy = σ y (1 − ρy) + x y − y − μx y 2 , dτ where α =

(2)

r1 a E1 br1 r2 r1 cE 2 , β= , δ= , σ= , ρ= , = and μ = K 1 γ2 r1 γ1 γ2 r1 K 2 γ1 r1

dr1 . The equilibria or steady states of the system (2) are denoted in the form γ1 γ2 of Pi = (x ∗ , y ∗ ), where i = 1, 2, 3, 4 for the cases:

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    σ− 1−β , P3 = ,0 , P2 = 0, ρσ α where xˆ solves the quadratic equation of

  1 − β − α xˆ P4 = x, , ˆ δ xˆ + 1

P1 = (0, 0),

  (αμ + δ)xˆ 2 + αρσ + μ(β − 1) + δ(σ − ) + 1 xˆ + ρσ(β − 1) + σ − = 0. Lemma 1 Let ψ1 = r1 − a E 1 and ψ2 = r2 − cE 2 , then: (i) If ψ1 < 0 and ψ2 < 0, then trivial equilibrium P1 exists and stable. Both predator and prey fish populations are driven to extinction; (ii) If ψ1 < 0 and ψ2 > 0, the equilibrium P2 exists and stable, which indicates the extinction of prey fish species and persistence of predators; (iii) If ψ1 > 0 and ψ2 < 0 hold strictly, then system (1) possesses equilibrium P3 if |r1 ψ2 | > K 1 γ2 ψ1 holds together. This implies the extinction of predator fish species.

3.1 Local Stability By deriving the Jacobian matrix of model (2), we analyze the local stability characteristics of the steady states such that ⎛ J P1 = ⎝

−2x(α + δ y) + 1 − β − y

x(1 − δx)

y(1 − μy)

x(1 − 2μy) + σ(1 − 2ρy) −

⎞ ⎠.



 1−β 0 , that gives a 0 σ− pair of eigenvalues 1 − β and σ − in which the steady state P1 is a stable node if β > 1 and > σ; an unstable node if 1 > β and σ > while an unstable saddle if otherwise. For equilibrium P2 , the associated Jacobian matrix is The variational matrix for equilibrium P1 is J =

⎛

J P2

−σ ρσ ⎜ ⎜ =⎜   ⎝σ−

σ− 1−μ ρσ ρσ 1−β+

0 −σ

⎞ ⎟ ⎟ ⎟, ⎠

ρσ(1 − β) + − σ . Since σ > is ρσ the feasible condition for the existence of P2 , then the threshold condition in which transcritical bifurcation(TB) occurs is which yields the eigenvalue pair of − σ and

= σ [1 − ρ(1 − β)] .

(3)

Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System

19

If < σ [1 − ρ(1 − β)], equilibrium P2 is a stable node and becomes unstable if > σ [1 − ρ(1 − β)]. Jacobian matrix of equilibrium P3 is given by ⎛ J P3

⎜ ⎜ =⎜ ⎝

β−1 0

⎞ β−1 δ(β − 1) 1− ⎟ α α ⎟ ⎟, ⎠ β−1 σ− − α

α(σ − ) + 1 − β which gives the corresponding eigenvalues pair β − 1 and . With α the feasible condition, we can conclude that another threshold condition is =

ασ + 1 − β . α

(4)

ασ + 1 − β while it becomes an unstable α ασ + 1 − β saddle if < . Therefore, it can be noticed that the harvesting parameters α and β are the crucial factors to influence the dynamical behaviors of system (2), compared to other parameters such as δ and μ which indicate the toxicant effects as they do not appear in conditions (3) and (4). Since getting necessary conditions for linear stability of equilibrium P4 is literally cumbersome due to the presence of many parameters, we make use of the bifurcation analysis to determine the region where coexistence equilibrium P4 is stable. Moreover, we explain how the threshold conditions (3) and (4) induce transcritical bifurcation that causing the switch of stability, in Sect. 4. Equilibrium P3 is a stable node when >

3.2 Global Stability In this subsection, we construct an appropriate Lyapunov function to study the global stability of the coexistence equilibrium P4 . By considering the Lyapunov function V (x, y) given by

  y x + f (y − yˆ ) − yˆ ln , V (x, y) = (x − x) ˆ − xˆ ln xˆ yˆ

(5)

where f is a function comprises of spatial variables x and y to be determined in the consequent steps, the followingTheorem 1 holds. It is clear that the Lyapunov  function equals zero at equilibrium x, ˆ yˆ and is positive for other positive values of x and y.

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Theorem 1 Coexistence equilibrium P4 is globally asymptotically stable in the subdomain region of   (x − x) ˆ (y − yˆ ) + δ(x y − xˆ yˆ )   > 0. (6) (y − yˆ ) (x − x) ˆ − μ(x y − xˆ yˆ ) Proof The derivative of Lyapunov function V ,      x − xˆ d x y − yˆ dy dV = + f , dτ x dτ y dτ   = (x − x) ˆ α xˆ + yˆ + δ xˆ yˆ − αx − y − δx y   + f (y − yˆ ) σρ yˆ − xˆ + μxˆ yˆ − σρy + x − μx y ,    = − (x − x) ˆ α(x − x) ˆ + y − yˆ + δ(x y − xˆ yˆ )    + f (y − yˆ ) σρ(y − yˆ ) + xˆ − x + μ(x y − xˆ yˆ ) ,  ˆ − yˆ )(1 − f ) + f σρ(y − yˆ )2 = − α(x − x) ˆ 2 + (x − x)(y    + (x y − xˆ yˆ ) δ(x − x) ˆ + f μ(y − yˆ ) .   (x − x) ˆ (y − yˆ ) + δ(x y − xˆ yˆ )   , we have By choosing f = (y − yˆ ) (x − x) ˆ − μ(x y − xˆ yˆ )   dV = − α(x − x) ˆ 2 + f σρ(y − yˆ )2 . dτ dV Hence, if condition (6) holds, < 0 which implies that the coexistence equilibrium dτ P4 is globally asymptotically stable (Fig. 1).

3.3 Limit Cycle We apply the Bendixson-Dulac criterion to determine whether fishery system (2) possesses limit cycle. We consider the fishery system F1 (x, y) = x (1 − αx) − x y − βx − δx 2 y, F2 (x, y) = σ y (1 − ρy) + x y − y − μx y 2 ,

(7)

Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System

21

2

Predator, y

1.5

1

0.5

0 0

0.5

1

1.5

2

2.5

3

3.5

4

Prey, x

Fig. 1 Phase space trajectory of system (2) subject to different initial values

and the function of ζ(x, y) =

1 where xy

∂(ζ · F1 (x, y)) ∂(ζ · F2 (x, y)) + ∂x ∂y     ∂ 1 − αx − β ∂ σ(1 − ρy) − = − 1 − δx + + 1 − μy , ∂x y ∂y x   α σρ + +δ+μ . =− y x Since the expression is always negative for all the positive values of parameters, we can conclude that there is no periodic solution and closed orbit in the fishery model (2).

4 Bifurcation Results In the present research, the influence of harvesting activities on dynamical properties of a fishery system is the main intention. Therefore, by introducing set of hypothetical parameters of α = 0.15, β = 0.8, δ = 1.2, σ = 0.65, ρ = 0.25, = 0.6 and μ = 0.35, bifurcation diagrams with respect to the harvesting parameter i.e., harvesting rate on predator fish are obtained and shown is Fig. 2. In Fig. 2, solid lines illustrate the stable equilibria and dotted lines represent unstable equilibria. The calculated transcritical bifurcation (TB) points are = 0.6175 and = 1.983, obtained from the threshold conditions (3) and (4), respectively. Region I, II and II are the regions in between these points. In Region I, the prey-free equilibrium P2 is stable when the harvesting rate on predator is less than 0.6175. This explains

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that when the harvesting activities on predator fish is low, causing still abundance of predator left in the system which gives the opportunity for the predator to feed on the prey fish until the prey (food source) is extinct. Region II shows the stability property of equilibrium P4 (coexistence) where intermediate harvesting rate on the predator fish is considered. With moderate harvesting strategies on the predator, the system is in its ideal state where both species tolerate with each other to survive. In moderate predator fish harvesting, the prey fish still has the chance to reproduce and grow, and a fraction of them serves as a food source to the predator fish. Region III however, represents the state where the predator is harvested without any control. As a result, the predator fish is now at stake to be eliminated from the system as the harvesting activities are too frequent, leaving the prey species to remain its existence. Figure 3 shows the time series plots associated with Region I, II and III from Fig. 2 to provide a better explanation on the fish population densities over a period of time. In Fig. 3a, it can be clearly seen that the prey fish population extinct if the harvesting

Fig. 2 Steady-state diagrams of system (2) with respect to the harvesting parameter for a prey, x and b predator, y, respectively

(a) Region I

1.5

P4

Region II

Prey, x

P3

Region III

P3

1

TB ( =1.983)

P

4

TB ( =0.6175)

0.5

P2

0 0

P2

0.5

1

1.5

2

2.5

3

(b)

Predator, y

0.8

Region I

0.6

Region II

Region III

P2 P4

0.4

TB ( =0.6175)

0.2 P

0 0

TB ( =1.983)

P4

P

3

0.5

1

1.5

2

2.5

3

3

Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System

(a) 0.5 Prey, x Predator, y

Population

0.4 0.3 0.2 0.1 0 0

50

100

150

200

(b) 0.35

Population

0.3 Prey, x Predator, y

0.25 0.2 0.15 0.1 0.05 0

50

100

150

200

(c) 1.2 1 Population

Fig. 3 Time series plots of system (2) with initial conditions of (x0 , y0 ) = (0.3, 0.1) at a = 0.6 (Region I), b = 0.8 (Region II) and c = 2 (Region III), respectively

23

Prey, x Predator, y

0.8 0.6 0.4 0.2 0 0

50

100

150

200

24 Fig. 4 Steady-state diagrams of system (2) with respect to the harvesting parameter β for a prey, x and b predator, y, respectively

T. K. Ang et al.

(a)

(b)

rate of predator fish is too low ( < 0.6175). The population size of predator fish is decreasing because of the extinction of prey fish as the food supply decreases. Optimal harvesting rate, i.e., 0.6175 < < 1.983 ensures the coexistence of both fish population as illustrated in Fig. 3b. Figure 3c verifies that excessive harvesting rate on predator ( > 1.983) can lead to the elimination of predator fish population while prey fish who are free from hunting pressure can survive at a constant rate. On the other hand, by manipulating the parameter β as the bifurcation parameter, the fish population dynamics affected by the harvesting activities on prey fish can be observed. From Fig. 4, it is shown that when the harvesting activities on prey fish are relatively low (β < 0.6923), equilibrium P4 is stable indicating the coexistence of both fish populations. As the harvesting activities increase, population density of prey fish, which is the food source of predator fish, decreases causing the population density of predator fish to decrease (Region IV). Similar to Fig. 2, transcritical

Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System Fig. 5 Time series plots of system (2) with initial conditions of (x0 , y0 ) = (0.3, 0.1) at a β = 0.5 (Region IV) and (b) β = 0.7 (Region V), respectively

25

(a)

(b)

bifurcation occurs when the harvesting activities on prey fish exceed threshold limit of β = 0.6923. Equilibrium P2 is stable instead of P4 , implying the extinction of prey fish due to excessive harvesting while allowing the predator fish population to persist in the system as described in Region V. Figure 5a, b depict the time series plots corresponding to Region IV and V in Fig. 4, respectively. In Fig. 5a, the persistence of both prey and predator fish species is ensured due to moderate low level of prey fish harvesting (β = 0.5 < 0.6923). At high level of prey fish harvesting where β = 0.7 > 0.6923, the prey fish population goes extinct due to over-harvesting and only predator fish population can persist in the system as shown in Fig. 5b.

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5 Bionomic Equilibrium The biological equilibria are discussed in the previous section. Now, we consider the bionomic equilibrium of fishery system (1) where it occurs when the total cost of harvesting fisheries equals the total revenue gained by selling the fisheries. Thus, the economic rent of the system, π, at any time takes the form of π = (s1 a X − c1 )E 1 + (s2 cY − c2 )E 2 ,

(8)

where s1 and s2 represent the selling price per unit biomass of the prey and predator fish respectively. Parameters c1 and c2 indicate the constant cost of harvesting per unit effort on prey and predator fish, respectively. In order to obtain the bionomic equilibrium, the following equations are considered:   X dX − γ1 X Y − a E 1 X − bX 2 Y = 0, = r1 X 1 − dt K1   dY Y + γ2 X Y − cE 2 Y − d X Y 2 = 0, = r2 Y 1 − dt K2 π = (s1 a X − c1 )E 1 + (s2 cY − c2 )E 2 = 0.

(9a) (9b) (9c)

Here, we consider three possible cases for the existence of bionomic equilibrium: Case I: s1 a X < c1 , the harvesting cost of prey fish is greater than the revenue causing the prey harvesting to stop, i.e. E 1 = 0. From Eq. (9c), we obtain c2 . Y = (10) s2 c Substituting Eq. (10) into Eq. (9a) yields X=

K 1 (r1 s2 c − γ1 c2 ) . r1 s2 c + K 1 bc2

(11)

Substituting Eq. (10) and (11) into Eq. (9b),  

  c2 K 1 (r1 s2 c − γ1 c2 ) 1 dc2 + γ2 − r2 1 − . E2 = c s2 cK 2 r1 s2 c + K 1 bc2 s2 c

(12)

Thus, bionomic equilibrium exists with the condition of r2 +

  r2 c2 K 1 (r1 s2 c − γ1 c2 ) dc2 > γ2 − . r1 s2 c + K 1 bc2 s2 c s2 cK 2

(13)

Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System

27

Case II: s2 cY < c2 , the harvesting cost of predator fish is greater than the revenue causing the predator harvesting to stop, i.e. E 2 = 0. From Eq. (9c), we have c1 X= . (14) s1 a Substituting Eq. (14) into Eq. (9b) to give Y =

K 2 (r2 s1 a + γ2 c1 ) . r2 s1 a + K 2 dc1

(15)

Now from Eq. (9a), 

    c1 K 2 (r2 s1 a + γ2 c1 ) 1 bc1 γ1 + r1 1 − − . E1 = a K 1 s1 a s1 ar2 + dc1 K 2 s1 a

(16)

Thus, bionomic equilibrium exists with the condition of r1 >

r1 c1 + K 1 s1 a



K 2 (r2 s1 a + γ2 c1 ) s1 ar2 + dc1 K 2

 γ1 +

bc1 s1 a

 .

(17)

Case III: c1 < s1 a X and c2 < s2 cY , both the harvesting costs of prey and predator fish are greater than the revenue, both harvesting activities will be operational, i.e. E 1 > 0 and E 2 > 0. From Eq. (9c), c2 c1 and Y = . X= (18) s1 a s2 c Substituting Eq. (18) into Eq. (9a) yields E1 =



   c1 c2 1 bc1 − r1 1 − γ1 + . a s1 a K 1 s2 c s1 a

(19)

Substituting Eq. (18) into Eq. (9b) yields E2 =



   c2 c1 1 dc2 + r2 1 − γ2 − . c s2 cK 2 s1 a s2 c

(20)

Hence, bionomic equilibrium exists with the following condition of 

r1 (s1 a K 1 − c1 ) K 2 (r2 s1 a + γ2 c1 ) , min K 1 (γ1 s1 a + bc1 ) r2 s1 a + dc1 K 2

 >

c2 . s2 c

(21)

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T. K. Ang et al.

Table 1 Bionomic equilibrium of fishery system (1) for Cases I, II, III in the presence of toxins Case Existence Bionomic equilibrium I II III

Yes. (38.848 > 0.023) Yes. (29 > 20.619) Yes. (min {10.816, 7.689} > 0.167)

(29.878, 0.167, 0, 21.569) (0.045, 7.689, 2.095, 0) (0.045, 0.167, 7.135, 0.852)

Table 2 Bionomic equilibrium of fishery system (1) for Cases I, II, III in the absence of toxins Case Existence Bionomic equilibrium I II III

Yes. (149.345 > 0.023) Yes. (29 > 28.751) Yes. (min {11.595, 11.495} > 0.167)

(98.563, 0.167, 0, 82.957) (0.045, 11.495, 0.062, 0) (0.045, 0.167, 7.143, 0.858)

5.1 Numerical Examples In this subsection, we make use of a set of hypothetical parameter values to interpret the effects of toxins released by the fish on the bionomic equilibrium in the fishery system for both Cases I, II, III. By taking the parameter values as r1 = 29, K 1 = 100, γ1 = 2.5, a = 4, b = 4, r2 = 1.5, K 2 = 11, γ2 = 1.5, c = 1.8, d = 1.5, s1 = 5, c1 = 0.9, s2 = 4 and c2 = 1.2, the bionomic equilibrium, in the form of (X, Y, E 1 , E 2 ) of the fishery system in the presence of toxins are acquired as shown in Table 1. In order to examine the bionomic equilibrium of the fishery system with the assumption that the fish populations are not affected by the toxins, we manipulate the parameter values of b = d = 0 where the results are summarized in Table 2. From Tables 1 and 2, it can be seen that the fish population densities are higher in Cases I and II when there is an absence of toxins. In Case III, both the population densities of predator and prey fish remain the same in both the presence and absence of toxins but the harvesting efforts that are applied are higher in the absence of toxins. Thus, it may be concluded that the effect of toxins is harmful to the fish populations and can reduce the fish population densities.

6 Conclusion In the present research, the behaviors of a predator-prey fishery model are investigated with the consideration of toxin produced by both fish populations. Both the fish populations are infected by the toxins and at the same time they are eliminated by different harvesting efforts. Model (2) is non-dimensionalized system and the local stabilities of the equilibria are analyzed by examining the Jacobian matrices and eigenvalues. We found that the harvesting parameters and β tend to be more

Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System

29

influential on the dynamics of stability properties if compared to toxin parameters δ and μ. This may due to the existence of harvesting parameters in threshold conditions (3) and (4). From the bifurcation regions, we conclude that inadequate harvesting on predator fisheries can drive the prey fisheries towards extinction but excessive harvesting on either prey and predator fish can eradicate the prey and predator fisheries themselves. In our work, the existence of bionomic equilibria is established by equating both X˙ = Y˙ = π = 0. Several possibilities of bionomic equilibrium solutions are discussed. In a nutshell, optimal harvesting activities are doubtlessly essential for the coexistence of both predator and prey fish populations to ensure a balanced aquatic ecosystem. Acknowledgements The research is supported by the Research Management Centre (RMC) Universiti Tun Hussein Onn Malaysia (Postgraduate Research Grant Code: U992) and Incentive Grant Scheme For Publication (U677).

References 1. Mutsert, K., Lewis, K., Milroy, S., Buszowski, J., Steenbeek, J.: Using ecosystem modeling to evaluate trade-offs in coastal management: effects of large-scale river diversions on fish and fisheries. Ecol. Modell. 360, 14–26 (2017) 2. Perissi, I., Bardi, U., Asmar, T.E., Lavacchi, A.: Dynamic patterns of overexploitation in fisheries. Ecol. Modell. 359, 285–292 (2017) 3. Chattopadhyay, J.: Effect of toxic substances on a two-species competitive system. Ecol. Modell. 84, 287–289 (1996) 4. Samanta, G.P.: A two-species competitive system under the influence of toxic substances. Appl. Math. Comput. 216, 291–299 (2010) 5. Huang, Q., Parshotam, L., Wang, H., Bampfylde, C., Lewis, M.A.: A model for the impact of contaminants on fish population dynamics. J. Theor. Biol. 334, 71–79 (2013) 6. Huang, Q., Parshotam, L., Wang, H., Bampfylde, C., Lewis, M.A.: The impact of environmental toxins on predator-prey dynamics. J. Theor. Biol. 378, 12–30 (2013) 7. Kar, T.K., Chaudhuri, K.S.: On non-selective harvesting of two competing fish species in the presence of toxicity. Ecol. Modell. 161, 125–137 (2003) 8. Das, T., Mukherjee, R.N., Chaudhuri, K.S.: Harvesting of a prey-predator fishery in the presence of toxicity. Appl. Math. Modell. 33, 2282–2292 (2009) 9. Ang, T.K., Safuan, H.M., Kavikumar, J.: Dynamical behaviours of prey-predator fishery model with harvesting affected by toxic substances. MATEMATIKA 34(1), 143–151 (2018)

Dynamics and Bifurcations in a Dynamical System of a Predator-Prey Type with Nonmonotonic Response Function and Time-Periodic Variation Johan M. Tuwankotta, Eric Harjanto and Livia Owen Abstract We study a two dimensional system of ordinary differential equations of a predator-prey type. We use the Holling type IV functional response which models the group defence mechanism. For this system we discuss the number of equilibria in the system and prove it using a geometrical approach. Using the classical Lagrange Multiplier method, we compute fold and cusp bifurcations for equilibrium in the system. As we turn on to numerics, we compute the other bifurcations for equilibrium, namely Hopf bifurcations, and homoclinic bifurcations. As for bifurcation of periodic solution we compute the Fold of Limit Cycle bifurcation. We also include timeperiodic variation in the system which translates most of the bifurcation sets for equilibria into bifurcation sets for periodic solutions. Furthermore, we found the swallowtail bifurcation for periodic solution in the system. Keywords Predator-prey · Bogdanov-Takens bifurcation · Bautin bifurcation · Cusp bifurcation · Swallowtail bifurcation

1 Introduction Predator-prey model. A system of two ordinary differential equations, i.e. x˙ = αx − βx y y˙ = δx y − γ y,

(1)

J. M. Tuwankotta (B) · E. Harjanto · L. Owen Analysis and Geometry, Faculty of Mathematics and Natural Sciences, Institut Teknologi, Bandung, Indonesia e-mail: [email protected] URL: https://personal.fmipa.itb.ac.id/theo/ L. Owen Department of Mathematics, University of Parahyangan, Bandung, Indonesia © Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_3

31

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where α, β, δ and γ are nonnegative real parameters, is known as the Predator-Prey system. This model was introduced by A. J. Lotka in 1920 for studying oscillations of the concentration of chemical substrates in a chemical reactions. Independently, a similar model was introduced by V. Volterra in 1926 to explain the observed periodic variations in the population of predator fish (and the corresponding variation in the population of prey fish) in the Adriatic Sea during World War I (see [2]). Also independently, the model was derived for epidemics by Kermak and McKendrick in 1927–1933 [20–22] for b = 0, and combustion theory by Semenov in 1935 [30]. A lot of evidences in the literature shows that the model has played an important role in the applications of mathematics in various fields. For examples, in molecular biology [1], enzyme kinetics [3], the natural capital investment on tourism industry [7, 8], economy [10], parasitology [11], multi-objective evolutionary optimization [12], atmospheric science [23], and control system [31]. Solutions of (1). The dynamics of (1) is indicated by periodic behavior where the densities of the predator as well as the prey, oscillate. This is easy to see by computing the implicit solution (or the first integral) for (1). Another way is to see that (1) can be written as:   α −β x˙ = x y  y γ . y˙ = x y δ − x Since Ω = {(x, y) | x > 0, y > 0} is invariant under the flow of (1), then x y > 0 for all time along the solution. This implies that (1) is orbitally equivalent with: α −β y γ y˙ = δ − , x

x˙ =

(2)

which is a Hamiltonian system with Hamiltonian: H = α ln(y) − β y − δx + γ ln x. This means that there exists a homeomorphism from R2 −→ R2 which maps orbit of (1) to orbit of (2). Using the fact that System (2) has a unique elliptic equilibrium in Ω, and that two-dimensional Hamiltonian flow is area-preserving, then we conclude that all solution of (2) as well as (1) is periodic. Variations and extensions of the model. In [17, 18, 26, 36] some non-linear response functions were introduced to capture a more realistic behavior in predation which has been observed in nature. In [17, 18] the generalization was done by requiring that the predation saturates as the density of the prey become very large. Thus, the response function is converging to a limit as the density of the prey goes to infinity.

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There are three types of response function which are introduced by C.S. Holling. Type I is when the predation grows linearly until it reaches a maximum and then stay constant, i.e.  mx, 0 ≤ x ≤ c P(x) = mc, c < x. For Type II, the response functions is monotonically increasing with a slope which is monotonically decreasing to zero while the predation converges to a limit, i.e. P(x) =

ax 1 + bx

This models the behaviour that when the population of the prey is small the rate of success of the predation is high. Type III functional response is variation of type II, i.e. ax 2 P(x) = , 1 + bx 2 where the rate of success of the predation is low when the density of the prey is too low. Zhu et al. [36] generalized one of the type III response functions. This is an interesting generalization as the rate of success of the predation decreases when the density of the prey become too large. This functional response function is called Holling Type IV, i.e. x , P(x) = 2 αx + βx + 1 which models the group defense mechanism. One can see that as x become very large then P(x) converges to zero. There are other types of generalizations (or variations) to the predator-prey type of dynamical systems. For example, Hirsch [14–16] introduced the effect of competition or cooperation into the system. Other type of variations is to include space-dynamics in the system [19, 34] or delay [23, 35]. In [34], a different type of mortality rate, namely the hyperbolic one, is considered. Another interesting focus on the study of Predator-Prey type of dynamical systems can be found for example in [33]. The author there look for the so-called canard solution by exploiting the presence of a small parameter in the system which appears as the coefficient of one of the time-derivative of the density of the prey population. As a consequence the system falls into the class of singularly perturbed dynamical systems. What is done in this note. In [4–6], Broer et al. studied a similar system as in [36] with an addition of a competition factor among the predators. The focus of these studies are describing the dynamics and bifurcations in the presence of groups defense mechanism. Broer et al. also include a periodic perturbation (to model seasonal

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variation) on the carrying capacity of the environment following the study of Rinaldy et al. [29]. The latter however consider a different type of response functions. This note describes the dynamics and bifurcations in a similar system to those in [4–6]. Sections 2 and 5 are reproductions from [13]. In Sect. 3, we apply an alternative approach for computing fold and cusp bifurcation. This approach has been described in [25] and also in [28]. In Sect. 4 we present the numerical bifurcation analysis using AUTO [9] for the system without periodic perturbation. This part is reproduced from [32].

2 System with Nonmonotonic Response Function and Periodic Perturbation In this section we consider a predator-prey dynamical system with nonmonotonic response function of Holling type IV, with α and β are real numbers. Let x and y denote the density of the population of prey and predator, respectively. The dynamical system can be written as: x˙ = x f (x, y, α, β), (3) y˙ = yg(x, y, α, β), with

y , αx 2 + βx + 1 x . g(x, y, α, β) = −δ − μy + αx 2 + βx + 1 f (x, y, α, β) = 1 − λ0 x −

The dependence of the system to the parameters α and β has been made explicit in order to indicate that these two parameters are the bifurcation parameters we have chosen. It is assumed that all parameters except α and β are positive valued real parameters. The growth rate of the prey has been normalized to one. In the absence of the predator, the population of the prey will grow exponentially to a limit of λ1 . On the other hand, in the absence of the prey the population of the predator will go to an extinction exponentially with the rate −δ. The parameter μ measures the competition factor among predators.

2.1 Equilibria of the System (3) √ Following [32] and also [4–6], we assume that α ≥ 0 while β satisfies −2 α < β < √ 2 α. This implies that αx 2 + βx + 1 > 0 for all x. There are several equilibria in the system. The origin of the phase-space is a saddle type equilibrium; this can be seen

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from the linearization of (3) in the vicinity of this equilibrium which has eigenvalues: 1 and −δ. The second equilibrium is (x, y) = ( λ1 , 0). The eigenvalues of the linearization of (3) in the vicinity of this equilibrium is −1 and −δ +

λ



1 α λ2

+

β λ

 +1

Thus, the equilibrium can be stable or of saddle type depending on the value of the parameter: δ, and λ. For a large value of λ this equilibrium is stable. Note that y = 0 is invariant with respect to the flow of (3). Furthermore, for large λ this equilibrium being stable implies extinction of the predator which is in a sense less interesting. For this reason, our study is focused on the situation where λ is relatively small so that this equilibrium is unstable. From the application point of view, the interesting equilibrium would be the one with x > 0 and y > 0. Following [13], we will show that there are at most three equilibria. The system of equations for the equilibria with x > 0 and y > 0 is: f (x, y, α, β) = 0, g(x, y, α, β) = 0.

(4)

The System (4) is equivalent to: y = (1 − λx)(αx 2 + βx + 1), 0 = y(δ + μy) − x(1 − λx),

(5)

since: αx 2 + βx + 1 = 0. To facilitate our analysis, let us write F1 (x) = (1 − λx)(αx 2 + βx + 1), and F2 (x, y) = y(δ + μy) − x(1 − λx). See Fig. 1 for an illustration for the graphs of F1 and F2 . Note that the graph of F2 (x, y) = 0, or equivalently:

Fig. 1 The graph with a solid-line is the ellipse (6) while the dotted-line is for y = (1 − λx)(αx 2 + βx + 1). This is one of the situation where D in (7) is positive. This figure is reproduced from [13]

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1 λ x− 2λ

2



δ +μ y+ 2μ



2

1 = 4



1 δ2 + λ μ



 .

(6)





1

 , 0 is:





1 δ , that passes through (0, 0), 0, − μδ , and λ1 , 0 . , − 2μ 2λ 1 equation F2 ( 2λ , y) = 0 for y > 0 which gives us the maximum

is an ellipse centered at

We can solve the value of y on the ellipse, i.e.

ymax

  δ μ −1 + 1 + 2 . = 2μ λδ

Now consider the slope of the tangent line to the ellipse at m=−

∂ F2 ∂x ∂ F2 ∂y

1

,0



 λ1  ,0 λ

λ

1 =− . δ

  It is clear that the graph of y = F1 (x) intersects x-axis at λ1 , 0 , and y-axis at (0, 1). Since F1  (x) = −3λαx 2 + (−2λβ + 2α)x − λ + β, we can compute the discriminant: D = 4(α − λβ)2 + 12λα(β − λ).

(7)

There are two cases that need to be considered. The case where: D ≤ 0. In this case, F1 (x) is a non increasing function. Then, if

     1 1 2 1 1 = −λ α F1 +β +1 0 and y > 0. However, if     1 1  < m, and ymax ≥ f , F1 λ 2λ then there are at the most two intersection points possible. The case where: D > 0. In this case, F1  (x) = 0 has two solutions, say x1 and x2 . Without loss of generality we assume that x1 < x2 . If

Dynamics and Bifurcations in a Dynamical System …

2 3



1 β − λ α

 > 0 and −

37

β−λ > 0, 3λα

then both x1 and x2 are positive. We can solve the equation: F2 (x2 , y) = 0 for y and call the solution ys . If F1 (x2 ) > ys , then the graph y = F1 (x) and the ellipse (5) have at the most three intersection points. If   β β−λ 2 1 − < 0 and − > 0, 3 λ α 3λα then both x1 and x2 are negative. Then the graph y = F1 (x) and the ellipse (5) have at the most one intersection point. If β−λ − < 0, 3λα then the local minima of F1 is negative while the local maxima is positive. Then the graph y = F1 (x) and the ellipse (5) have at the most two intersection points. This ends the proof.

3 Fold and Cusp Bifurcations of Equilibria: An Alternative Method for Computing Fold and Cusp Bifurcation Points The result on the number of equilibria for System (3) indicates the possibility for fold√and cusp bifurcations for equilibria. Let us omit the restriction that α ≥ 0 and √ −2 α < β < 2 α, which implies that the response function can be singular for some xs ∈ R. As a consequence, we need to choose the initial condition x◦ smaller than xs . By doing so, the equation for x˙ becomes negative as the solution approaches xs . Then x decreases away from xs . Following Harjanto et al. in [13], we fix the parameters δ = 1.1, λ = 0.01, μ = 0.1 and describe the local bifurcation diagram of equilibria of System (3) with α and β as the bifurcation parameters. Note that we can solve g(x, y, α, β) = 0 for α, i.e.: α=

−βλx 2 + (β − λ)x + 1 − y . x 2 (λx − 1)

Let us define: G(x, y, β) =

−βλx 2 + (β − λ)x + 1 − y . x 2 (λx − 1)

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Furthermore, we can define: F(x, y, β) = f (x, y, G(x, y, β), β) = 0. Then the solution of (4), is equivalent to the solution of: F(x, y, β) = 0 G(x, y, β) = α.

(8)

Suppose that there exists α0 such that for α < α0 , the System (3) has two equilibria, which collide with each other at α = α0 and vanishes afterward. Then System (3) undergoes fold bifurcation at α = α0 . However, this also means that given F(x, y, β) = 0, the function G(x, y, β) has maximum value of α0 . Thus, the value α0 can be found using the Lagrange multiplier method, i.e. solving: 

∇ F(x, y, β) = η∇G(x, y, β) F(x, y, β) = 0.

(9)

In general, solutions of System (9) are the family: {(x(β), y(β), η(β)} for some β. As is the case in the constrained optimization problem, the Lagrange multiplier η can be eliminated. However, we need to check if those solution really correspond to either maximum or minimum for the function G(x, y, β). The families of solutions of (9) have to satisfy, for all β: 2β 2 x 6 + (−600β 2 + 8β)x 5 + (59395β 2 − 9000β + 8008)x 4 +(−1879000β 2 + 1337580β − 1215600)x 3 +(−6050000β 2 − 49519500β + 45622580)x 2 +(−157300000β + 29477000)x − 290400000 = 0, and

yi = −5.5 + 0.1 −10xi 2 + 1000xi + 3025,

(10)

(11)

with xi being the solutions to (10). We proceed with constructing the fold line by the following procedure. First we fixed a value for β, for example: β0 = 0. Then Eq. (10) reduces to fourth degree polynomial, which can be solved numerically. In this case it produces four positive roots. Then we omit two of them using (11) since those solutions correspond to a negative value of y. Up to this point, we have (xi (β0 ), yi (β0 )), i = 1, 2 as a solution for (10) and (11). Then we can define: αi (β0 ) = G(xi (β0 ), yi (β0 ), β0 ), i = 1, 2.

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We proceed with increasing (or decreasing) β slightly, and solve the same Eq. (10). In general, (12) αi (β) = G(xi (β), yi (β), β), i = 1, 2. define the two fold curves in parameter space: (β, α). The point in parameter space where these two fold curves meet is where cusp bifurcation occurs. In this paper we follow different route to compute the cusp bifurcation point. First, let us define the Eq. (10) as H (x, β) = 0 which defines a curve in (x, β)-plane. Then, we compute the maximum point of H (x, β) = 0, by requiring: ∂H

dx ∂β = − ∂ H = 0. dβ ∂x given H (x, β) = 0. Following the method described above, we conclude that it occurs at (x, β) = (13.19624601, 0.55007938196) and it implies that cusp bifurcation point of system (3) occur at (β, α) = (0.55007938196, −0.004200565504) We plot the result (12) in Fig. 2 and conclude that: (10) has multiple root when β = 0.55007938196 (11 decimal place accuracy), which is in agreement with the result in [13], see Fig. 2 pp. 191. See also Fig. 3 which is a more detailed version of the bifurcation for equilibria that occur in (3).

Fig. 2 In this figure (on the left) we have plotted two fold curves for equilibria of (3). The two curves coalesce at a cusp point: (β, α) = (0.55007938196, −0.004200565504). This diagram is comparable with Fig. 2 pp. 191 in [13]. The diagram on the right is a plot of H (x, β) = 0 in (x, β) coordinate. The maximum point of x on H (x, β) = 0 occurs at (x, β) = (13.19624601, 0.55007938196) and it is equivalent with the cusp bifurcation point of System (3)

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Fig. 3 The numerical data we have used are δ = 1.1, λ = 0.01, and μ = 0.1. The two curves plotted using circled-line are the fold lines labelled by F1 and F2 . The curve F2  is a part of F2 which coalesce with the curve Hom where Homoclinic bifurcation occurs. The curves labelled by H1 and H3 are curves of Hopf bifurcation, while the point H2 is a degerate Hopf (or Bautin) bifurcation. The curve FLC indicates the fold of limit cycles bifurcation. We have indicated two regions in this diagram labelled by A and B which will be enlarged in the next two figures to get more details. This diagram is reproduced from [32]

Remark 1 The computation of fold bifurcation point by using Lagrange Multiplier Method, to our knowledge, was introduced in [27] and also in [13]. The method for computation of fold bifurcation is explained carefully for systems of two ordinary differential equations in [25], while for cusp bifurcation it is briefly explained in [28].

4 Numerical Bifurcations Analysis for System (3) For other bifurcations of equilibria in System (3), we will follow an equilibrium which is found for a particular value of the parameters, by varying the parameter α and β. Using the numerical continuation software AUTO97 [9], we present the bifurcation of equilibria for the system (3). The numerical data we have used are slightly different from those in the previous section. The parameter: δ and λ are chosen as are in the previous section, but μ = 0.06. We start with an equilibrium which are found for α = 0.002, β = 0.25. The local bifurcation diagrams of equilibrium of the system (3), using α and β as free parameters, are plotted in Fig. 3. This result is comparable with one of the case which is considered in [4–6, 13].

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As it is clear from the previous section, the system undergoes a codimension one bifurcation, i.e. fold bifurcation and a codimension two bifurcation, i.e. cusp bifurcation. The latter is a nontransversal intersection between two fold curves. We have observed that three fold curves, which are in Fig. 3 labelled by F1 , F2  , and F2 . The cusp point is the intersection between F1 and F2 . See also Fig. 4. Apart from the fold bifurcation, we have found Hopf bifurcation of equilibria. Then we follow the locus of this Hopf bifurcation by varying α and β. This defines a Hopf curve (a curve where Hopf bifurcaiton occurs) with three parts which are labeled by H1 , H2 which is a point, and H3 . It is also interesting to note that the H1 branch ends up in F1 in a Bogdanov-Takens point (see [24] for a definition and complete description of the bifurcation). This is a codimension two bifurcation. We have indicated the neighborhood of this point with a small box B in Fig. 3. This box has been zoomed and plotted in Fig. 4. Let us start in the domain B1 in Fig. 4. In this domain, we have an equilibrium with x > 0 and y > 0, which we will call E 0 ; note that this equilibrium has nothing to do with the Bogdanov-Takens bifurcation. As we enter the domain B2 by crossing the curve F1 , the System (3) undergoes a Fold bifurcation at where a degenerate equilibrium is created. Then the degenerate equilibrium breaks into two equilibria: one is stable (which we named E − ) and one is of saddle type (which we named E ± ). As we cross the curve H1 into the domain labelled by B3 , the equilibrium E − undergoes a Hopf bifurcation and a stable periodic solution is created. After the bifurcation, the equilibrium E − becomes unstable and we address a new name, which is E + . The stable periodic solution, on the other hand, grow in size and the

Fig. 4 A magnification of the domain B in Fig. 3 where the local bifurcation diagrams of equilibrium for System (3) are plotted in this figure. This diagram is reproduced from [32]

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collapses into the stable and unstable manifold of the saddle type equilibrium E ± . Thus, a homoclinic orbit is created. This occurs exactly on the curve Hom branching out from the Bogdanov-Takens point. As we enter the domain B5 , the homoclinic orbit vanishes and we are left with two equilibria: E ± and E + . These two equilibria collide with each other into a degenerate equilibrium and then vanishes, at another fold bifurcation on F1 . Remark 2 Orbit homoclinic to a degenerate equilibrium. From Fig. 4 we see that the curve where the homoclinic bifurcation occurs, i.e. Hom is branching out of the Bogdanov-Takens bifurcation and it extends and collides with the F2 -curve. We named the part of F2 which coalesce with Hom as F2  . On this curve, the equilibria E 0 and E ± undergoes a fold bifurcation and a degenerate equilibrium (E d ) is created. Note that we can now follow the stable periodic solution created through Hopf bifurcation; this periodic solution exists in B3 domain. Let us follow this periodic solution into the domain B4 crossing the curve F2 . Let us now enter the domain B5 ; we know that from the previous explanation on Bogdanov-Takens bifurcation as we enter the domain, the periodic solution disappear through a homoclinic bifurcation. Similarly, as we enter the domain B5 from B4 , a homoclinic orbit to a degenerate equilibrium is created. Remark 3 Bautin bifurcation. In Fig. 3 we have indicated another box labelled by A. We have zoomed into this domain and plotted the situation in Fig. 5. Apart from the existing bifurcation curve, we have a curve labelled by FLC which stands for

Fig. 5 A magnification of the domain A in Fig. 3 where the local bifurcation diagrams of equilibrium for System (3) are plotted in this figure. This diagram is reproduced from [32]

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43

Fold of Limit Cycles. On this curve, a degenerate periodic solution is created and branch out as two periodic solutions: one stable and one unstable. The curve FLC meet the Hopf curve at the popint H2 . The point H2 is also known as the codimension two bifurcation point named Bautin bifurcation, see [24] pp. 307–314 Remark 4 Transcritical bifurcation. There is also a transcritical bifurcation in System (3) which occurs for one of the equilibrium on the x-axis. All of our analysis valid as we stay in one side of the transcritical bifurcation curve. Remark 5 A complete description and phase-portrait of System (3) in each part of the parameter-space is described in [32].

5 Time Periodic Perturbation Let us consider a small, time-dependent perturbation to the carrying capacity, by introducing: (13) λ = λ0 (1 + ε sin(ωt)), with λ0 = 0.01. Similar with those in [6, 13, 29, 32], this perturbation is introduced to include seasonal time variation, for example: of the volume of a pond. It is always more convenient to work with a system of autonomous differential equations. For this, we attach to (3) a two-dimensional system of ordinary differential equations, i.e.: u˙ = u + ωv − u(u 2 + v 2 ), v˙ = −ωu + v − v(u 2 + v 2 ). With initial conditions u(0) = 0 and v(0) = 1, we have: (u, v) = (sin(ωt), cos (ωt)) is an asymptotically stable solution of this system. The extended system reads: x˙ y˙ u˙ v˙

  y , = x 1 − λ0 (1 + εu)x − αx 2 +  βx + 1)  x , = y −δ − μy + αx 2 + βx + 1 2 2 = u + ωv − u(u + v ), = −ωu + v − v(u 2 + v 2 ).

(14)

The original system can be achieved by restricting ourselves to the initial condition: (x◦ , y◦ , 0, 1) and projecting the solution of (14) to (x, y)−plane. For δ = 1.1, λ0 = 0.01, μ = 0.1, and ω = 1, while α = 0.002, β = 0.25, ε = 0, we found a periodic solution: (x, y, u, v) = (1.8161, 1.43411, cos t, sin t).

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Keeping all parameters except ε fixed, we follow this periodic solution as we increases the value of ε to 0.07. Then we let the parameters α and β vary as we follow the periodic solution at ε = 0.07.

5.1 Fold, Cusp and Torus Bifurcations of the Periodic Solution Using α and β as free parameters we follow the periodic solution at ε = 0.07. These results are plotted in Fig. 6. Evidently, the cusp bifurcation point persists under the time-periodic perturbation: the point in (α, β)-plane which is labeled by Cusp 1. One could compare Fig. 6 with the one for ε = 0 in Fig. 3. We note that the Hopf bifurcation curve in Fig. 3 becomes a torus bifurcations curve labelled by Torus 1 in Fig. 6. This is due to the fact that the periodic perturbation introduce an S 1 -action to the periodic solution which produces an invariant torus (recall that S 1 × S 1 = T2 . On the torus (which persists only in a small neighborhood of the torus curve) one would have either periodic or quasi-periodic motions. Different mechanism for vanishings two fold points. It is interesting to note that we have found another two cusp bifurcations of periodic solutions which are labelled

Fig. 6 The top left diagram is a local two parameters bifurcation diagrams for periodic solution of System (14) at ε = 0.07. One should compare this diagram with Fig. 3. The stared line represents the graph of α = 41 β 2 . On the top-right diagram, a magnification of the small rectangle in the topleft diagram is plotted. The L1 slicing is at β = 0.248, L2 at β = 0.248653, and L3 at β = 0.249. These diagrams are reproduced from [13]

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by the Cusp2 and Cusp3 points, see Fig. 6. We have plotted a small box near the point labelled as Cusp 2 and magnify the domain; the result is plotted in Fig. 6, on the right hand side plot. Let us define a small domain in (α, β, ε)-space by the Cartesian product of intervals in α, β and ε, i.e.: [−0.06, −0.03] × [0.2, 0.3] × [−0.5, 0.5]. The previously mentioned small box near the point labelled as Cusp 2 Fig. 6, can be seen as a section of this domain for ε = 0.07. On this section, we have introduced three lines, i.e. L1 = {(α, 0.248)}, L2 = {(α, 0.248653)}, and L3 = {(α, 0.249)}, for −0.04 ≤ α ≤ 0. We follow the periodic solution as we vary α along each line. The periodic solution is represented by: maxt x(t). These results are plotted in Fig. 6 as the three diagrams in the second row. Let us consider a continuation of the periodic solution as we vary α along the line L1. The periodic solution undergoes four Fold bifurcations which are labelled F1, F2, F3, and F4. The diagram for maxt x(t) against α is plotted on the most left bottom diagram in Fig. 6. On the line L2, the continuation of the periodic solution shows that two fold points F1 and F4 coalesce (see Fig. 6 the middle-bottom diagram) but not through cusp bifurcation at Cusp 2 nor Cusp 3. As we follow the periodic solution through the line L3, the coalescing fold points F1 and F4 disappear. Furthermore, we have two disconnected branches of periodic solutions in the (α, maxt x(t))-plane (see Fig. 6 the right-bottom diagram). The consequence of this is the following. If one would have followed the periodic solution on the lower branch

Fig. 7 Four sections of the codimension-three Swallowtail bifurcation in (14), for β = 0.29, β = 0.27, β = 0.25 and β = 0.23. This diagram is reproduced from [13]

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in that diagram by increasing α along the line L3, then one would have gone from α = −0.0415 to α = −0.035 without detecting any of the fold bifurcations. Those bifurcations are found by performing two parameter continuation of the point F2 and F3. These findings lead us to look for a co-dimension three bifurcation, the so-called Swallowtail bifurcation.

5.2 Swallowtail Bifurcation of a Periodic Solution By varying three parameters: ε, α, and β, we found a codimension three bifurcation: the swallowtail bifurcation. However, in our case it is different from the standard

Fig. 8 A section of the Swallowtail bifurcations sets in Fig. 7, i.e. for β = 0.25, is reproduced. On it we have made five sections which are labelled by: K1 (for ε = 0.073), K2 (for ε = 0.0704), K3 (for ε = 0.07), K4 (for ε = 0.068935), and K5 (for ε = 0.06). The plot max x of the periodic solution against α as we follow the periodic solution along the lines: K1 and K2 are plotted on the diagrams in the second row, respectively, while the diagrams in the third rows are for K j, j = 3, 4, 5. These diagrams are reproduced from [13]

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Swallowtail bifurcation due to a symmetry with respect to ε = 0 (see Fig. 7). This is understandable since −ε sin t is similar with ε sin (t − π). Thus, we are looking only at a different phase of the time-perturbation. The Swallowtail bifurcation sets of a periodic solution of (14), projected to the (α, ε)-plane, are plotted for various values of β, i.e. 0.29, 0.27, 0.25, and 0.23. To understand the dynamics, let us choose β = 0.25. We have made five slices on the (α, ε)-plane; these slices are labeled by K1 (for ε = 0.073), K2 (for ε = 0.0704), K3 (for ε = 0.07), K4 (for ε = 0.068935), and K5 (for ε = 0.06). As is done previously, we have chosen maxt x to represent the periodic solution. In Fig. 8 we have plotted the continuation of the periodic solution of (14) as we vary α, following the lines: K1, K2, . . ., and K5. On the lines: K3, K4, and K5, the diagram of the periodic solution against the parameter α are similar with the one we have observed in Fig. 6. In addition to that, on the lines K1 and K2 we have observed the situation where the fold points: F1 and F2 coalesce and disappear, through a cusp bifurcation. Remark 6 We have also detected the existence of positive time strange attractor and negative time strange attractor for System (14), see [32] for details. This is done by studying the stroboscopic map for System (14).

6 Concluding Remarks We have seen that the dynamics of a two-dimensional predator-prey type of systems of ordinary differential equations, is very rich. Here, we have used the Holling type IV functional response to model the group defense mechanism, and also competition among predators. It is interesting to note that, apart from the common bifurcations such as: fold and Hopf bifurcation, we have observed three codimension two bifurcations, namely: cusp, Bogdanov-Takens, and Bautin bifurcation. These bifurcations are three out of four on the list of codimension two bifurcations in [24]. Thus, studying this type of predator-prey system is really instructive as it provides us with real examples for some exotic bifurcations. As we add time-periodic variation into the system, then in general nondegenerate equilibrium in the system becomes periodic solution. We remark that most of the bifurcation for the equilibrium persists as we turn on this bifurcation. In addition to that, we have discussed the occurrence of Swallowtail bifurcation for periodic solution. This bifurcation seems to occur only when time-periodic perturbation is presence. Whether this is true or not is still a subject of our investigation. Since the system consists of many parameters, there might be a possibility for finding Swallowtail bifurcation which is of codimension three, related to the codimension two cusp bifurcation. Another codimension three points which we have observed here is the cuspBogdanov-Takens bifurcation. A preliminary observation has shown that as we vary

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another parameter, namely μ, we can bring the locus of the Bogdanov-Takens bifurcation to coalesce with the locus of cusp bifurcation. This too is a subject of future investigation. Acknowledgements J.M. Tuwankotta research is supported by Riset KK B, Institut Teknologi Bandung (2019).

References 1. Balagaddé, F.K., Song, H., Ozaki, J., Collins, C.H., Barnet, M., Arnold, F.H., Quake, S.R., You, L.: A synthetic escherichia coli predator-prey ecosystem. Mol. Syst. Biol. 4, 187 (2008) 2. Berryman, A.A.: The orgins and evolution of predator-prey theory. Ecology 73, 1530–1535 (1992) 3. Briggs, G.E., Haldane, J.B.S.: A note on the kinetics of enzyme action. Biochem. J. 19, 338 (1925) 4. Broer, H., Naudot, V., Roussarie, R., Saleh, K.: Bifurcations of a predator-prey model with non-monotonic response function. Comptes Rendus Mathématique 341, 601–604 (2005) 5. Broer, H., Naudot, V., Roussarie, R., Saleh, K.: A predator-prey model with non-monotonic response function. Regul. Chaotic Dyn. 11, 155–165 (2006) 6. Broer, H., Saleh, K., Naudot, V., Roussarie, R.: Dynamics of a predator-prey model with nonmonotonic response function. Discret. Contin. Dyn. Syst. A 18, 221–251 (2007) 7. Cai, Z., Wang, Q., Liu, G.: Modeling the natural capital investment on tourism industry using a predator-prey model. In: Advances in Computer Science and its Applications, Springer, pp. 751–756 (2014) 8. Casagrandi, R., Rinaldi, S.: A theoretical approach to tourism sustainability. Conserv. Ecol. 6 (2002) 9. Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X., et al.: Continuation and Bifurcation Software for Ordinary Differential Equations (with Homcont), AUTO97. Concordia University, Canada (1997) 10. Feinstein, C.H., Dobb, M.: Socialism, capitalism & economic growth, CUP Archive (1967) 11. Fenton, A., Perkins, S.E.: Applying predator-prey theory to modelling immune-mediated, within-host interspecific parasite interactions. Parasitology 137, 1027–1038 (2010) 12. Grimme, C., Lepping, J.: Integrating niching into the predator-prey model using epsilonconstraints. In: Proceedings of the 13th Annual Conference Companion on Genetic and Evolutionary Computation, ACM, pp. 109–110 (2011) 13. Harjanto, E., Tuwankotta, J.: Bifurcation of periodic solution in a predator-prey type of systems with non-monotonic response function and periodic perturbation. Int. J. Non Linear Mech. 85, 188–196 (2016) 14. Hirsch, M.W.: Systems of differential equations which are competitive or cooperative: I. limit sets. SIAM J. Math. Anal. 13, 167–179 (1982) 15. Hirsch, M.W.: Systems of differential equations that are competitive or cooperative II: convergence almost everywhere. SIAM J. Math. Anal. 16, 423–439 (1985) 16. Hirsch, M.W.: Systems of differential equations which are competitive or cooperative: III. competing species. Nonlinearity 1, 51 (1988) 17. Holling, C.S.: The components of predation as revealed by a study of small-mammal predation of the european pine sawfly. Can. Entomol. 91, 293–320 (1959) 18. Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385–398 (1959) 19. Huang, Y., Diekmann, O.: Predator migration in response to prey density: what are the consequences? J. Math. Biol. 43, 561–581 (2001)

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20. Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics. II—the problem of endemicity. Proc. R. Soc. Lond. A, 138, 55–83 (1932) 21. Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics. III—further studies of the problem of endemicity. Proc. R. Soc. Lond. A 141, 94–122 (1933) 22. Kermark, M., Mckendrick, A.: Contributions to the mathematical theory of epidemics. part I. Proc. R. Soc. A 115, 700–721 (1927) 23. Koren, I., Feingold, G.: Aerosol-cloud-precipitation system as a predator-prey problem. In: Proceedings of the National Academy of Sciences (2011) 24. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112. Springer Science & Business Media (2013) 25. Marwan, M., Tuwankotta, J.M., Harjanto, E.: Application of lagrange multiplier method for computing fold bifurcation point in a two-prey one predator dynamical system. J. Indones. Math. Soc. 24, 7–19 (2018) 26. Nagano, S., Maeda, Y.: Phase transitions in predator-prey systems. Phys. Rev. E 85, 011915 (2012) 27. Owen, L., Tuwankotta, J.: Bogdanov-takens bifurcations in three coupled oscillators system with energy preserving nonlinearity. J. Indones. Math. Soc. 18, 73–83 (2012) 28. Owen, L., Tuwankotta, J.: Computation of cusp bifurcation point in a two-prey one predator model using lagrange multiplier method, in Proceedings of the International Conference on Applied Physics and Mathematics 2019. Chulalongkorn University, Bangkok, Thailand (2019) 29. Rinaldi, S., Muratori, S., Kuznetsov, Y.: Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities. Bull. Math. Biol. 55, 15–35 (1993) 30. Semenov, N.F.: II. Chemical Kinetics and Chain Reactions (1935) 31. Sharma, A., Singh, N.: Object detection in image using predator-prey optimization. Signal Image Proc. 2 (2011) 32. Tuwankotta, J., Harjanto, E.: Strange attractors in a predator-prey type of systems with nonmonotonic response function and periodic perturbation. J. Comput. Dyn. (2019) 33. Verhulst, F.: The hunt for canards in population dynamics: a predator-prey system. Int. J. Non Linear Mech. 67, 371–377 (2014) 34. Zhang, T., Xing, Y., Zang, H., Han, M.: Spatio-temporal dynamics of a reaction-diffusion system for a predator-prey model with hyperbolic mortality. Nonlinear Dyn. 78, 265–277 (2014) 35. Zhang, T., Zang, H.: Delay-induced turing instability in reaction-diffusion equations. Phys. Rev. E 90, 052908 (2014) 36. Zhu, H., Campbell, S.A., Wolkowicz, G.S.: Bifurcation analysis of a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 63, 636–682 (2003)

Modeling and Experimental Data on the Dynamics of Predation of Rice Plants and Weeds by Golden Apple Snail (Pomacea Canaliculata) Joel Addawe, Zenaida Baoanan and Rizavel Addawe

Abstract Golden Apple Snail(GAS) (Pomacea canaliculata), popularly known in the Philippines as “golden kuhol” is considered a serious invader in a paddy ecosystem. Rice farmers consider it as a notorious invasive species and a serious pest in several rice farms. In this study, we model the predation of rice plants and weeds by GAS. We formulate three ordinary differential equations to model the simplified dynamics of apple snails, rice plants and weeds in the presence of harvesting on snails. We then investigate the mathematical features of the model and analyze the stability of equilibria. Actual death and harvesting rates on GAS gathered from actual field experiments conducted on the enclosed rice paddies are used for the parameters in the numerical simulations to demonstrate the potential effect of snail harvesting. Keywords Golden apple snail · Equilibrium states · Stability

1 Introduction Food webs are networks of interacting species that feed on each other. In a food web, we consider the predator-prey relationship. A predator is an organism that eats another organism. The prey is the organism which the predator eats. The study on the dynamics of several interacting species was developed considerably in the first half of the twentieth century. The growth and decline of population in nature and the interaction between species have been the subjects of many researchers for many years. In the study of Malthus [14], modeling effort was performed on J. Addawe (B) · R. Addawe Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio, Baguio, Philippines e-mail: [email protected] URL: http://www.upb.edu.ph/jmaddawe Z. Baoanan Department of Biology, College of Science, University of the Philippines Baguio, Baguio, Philippines

© Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_4

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population dynamics for single species denoted by the equation ddtP = kt, where the solution to this equation is an exponential function P(t) = P0 ekt , where P(t) is the population size of a particular species under study at time t, P0 is the population size at time t = 0 and k is the rate of reproduction per unit time. This model of Malthus was applied to several problems. Since this model was observed not to be realistic in time when population is seen to decelerate in the presence of limited resources like food, the model of Verhults proposed a more realistic model for the study of population dynamics denoted by ddtP = r P(1 − P). Several mathematical models consider population dynamics with harvesting either on the predator or on the prey population and is shown to have interesting results. In [5, 17, 21], the authors examined the possible existence of steady states along with the local and global stability on the dynamics of a prey-predator while studies of [8, 11, 20] are on the predator-prey interaction with harvesting in the prey population. In [10], the authors investigated a predator-prey model with Holling type of predation and harvesting of mature predators. From the several papers [7, 9, 10], it is evident that using the harvesting effort can serve as control to exponential growth of certain species and to break the usual behavior of the system and direct it to a desired steady state. In an ecosystem on rice paddies without weeds, we can see the behaviour of the snails and rice plants according to the Lotka-Voltera model [12]. In a normal rice paddy without the presence of GAS, weeds grow together with the rice plants and compete with the nutrients present on the ground. To eliminate the competition between the rice plants and weeds for nutrients, farmers need to remove weeds manually from the paddy field as it grows with the newly planted rice plants. According to the study of [2, 15], GAS was introduced in the Philippines as a dietary food due to its enriched protein content. GAS eventually became a problem for rice farmers when it eventually escape from confined environment into creeks, rivers, irrigation system and into rice paddies. The introduction of GAS into the rice paddies was observed to control growth of weeds; however, it was eventually considered as a major pest in all rice ecosystem as they damage rice plant seedlings. Several control methods such as chemical, biological, cultural, and manual are practiced in most rice paddies. No single method is sufficient to control GAS, but one of the widely practiced method of controlling GAS in the Philippines is the manual handpicking of snails from the rice paddies [2]. Due to abundance of GAS on rice paddies, farmers use molluscicide, a chemical control for GAS that feeds on newly planted rice plants. In this study, we remove the use of molluscicide and look into the dynamics of the model when we allow natural predation of GAS on both rice plants and weeds with harvesting of snails.

2 Predation Model of GAS with Harvesting in Rice Paddies In this study, we are concerned with a snail predator that predates on two prey (rice plants and weeds) system. We use the Lotka-Volterra model to understand how the farm works in the artificial ecosystem of the paddy. Modelling the growth rate of rice

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53

plants and weeds with snails based on Lotka-Voltera. The setup considers natural harvesting on the snails that eat rice plants and weeds. Also, we consider that there is competition among the snails for their survival as their population increases. In this paper, the basic system is based on the classical predator-prey system [5]. Let us assume the following: p = population density of the rice plants at time t, w = population density of the weeds at time t, s = population density of the snails at time t.

(1)

Keeping this in view, the model becomes dp = r1 p − a1 ps d t¯ dw = r2 w − a2 ws d t¯ ds = ν1 ps + ν2 ws − γs d t¯

(2)

where r1 and r2 are the intrinsic rate for the rice plants and weeds, respectively, ν1 = f 1 a1 : reproduction rate of snails per plant eaten, f 1 : the rate at which snails turn plants into offspring, a1 : predation rate coefficient (the search rate or attack efficiency of snails on rice plants), ν2 = f 2 a2 : reproduction rate of snails per weed eaten, f 2 : the rate at which snails turn weeds into offspring, a2 : predation rate coefficient (the search rate or attack efficiency of snails on weeds), and γ is the snails mortality rate. In understanding the predation dynamics, we examine solutions at equilibrium analytically: =0 r1 p − a1 ps =0 r2 w − a2 ws (3) ν1 ps + ν2 ws − γs = 0 In this section, we propose a system with the following assumptions: 1. In the absence of the predating snails, each of the rice plants and weeds grows logistically. 2. The effect of the predation is to reduce the rice plants and weeds by a term proportional to the (rice plants, weeds) and snail populations. 3. The rice plants and weeds do not help each other against the snail predation. 4. In the absence of the rice plants and weeds for food, the snail’s death rate results in inverse decay. 5. The harvesting rate on the snails is constant. For the proposed model, the model is based on the classical prey dependent predator system,

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dp = r1 p(1 − Kp1 ) − aps, d t¯ dw = r2 w(1 − Kw2 ) − aws, d t¯ ds = ν1 ps + ν2 ws − C(s)s − g1 s − g2 s 2 d t¯

(4)

where r1 , r2 are the intrinsic growth rates of the rice plants and weeds, respectively in the absence of predators; K 1 and K 2 are the environmental carrying capacity of the rice plants and weeds, respectively; a = a1 = a2 is the maximal relative increase of predation on rice plants or weeds are equal (predation coefficient or the consumption rate of rice plants and weeds, respectively; ν1 , ν2 are the conversion factors; C(s) is the harvesting rate on snails; g1 is the death rate of the snails; and, g2 is the density dependent death rate of the snails due to self limitation or the influence of predation. We consider that the self limitation can occur if there are other factors (aside from food) which causes limiting at high population densities (That is, limited space, snail predators, etc.). The aim of the study is to look into the dynamical behavior with harvesting on snails to obtain reasonable range for coexistence of both rice plants, weeds and the snail predators. In general, there are different types of harvesting functions that can be studied. But in this study, we only consider harvesting C(s) = H , a suitable constant. Now, to avoid mathematical complexity, we nondimensionalize the system (4). The population density of rice plants is scaled against its carrying capacity, K 1 . Similarly, the population density of weeds is scaled against its carrying capacity K 2 . The population density for the snail population is scaled against its predation capacity on the rice plants, r1aK 1 . Without loss of generality, the new model becomes: dx = αx(1 − x) − x z, dt dy = β y(1 − y) − yz, dt dz = δx z + yz − hz − γz 2 dt where x = ν2 K 2 α , r1

p , K1

(5)

w , α = K11 , z = αas , t¯ = rα1 t, β = αrr12 , γ = ga2 , δ = νr11 ,  = K2 r1 (H +g1 )α are all positive constants and the system with initial condir1

y=

and h = tions, x(0) ≥ 0, y(0) ≥ 0, z(0) ≥ 0.

3 Analysis and Dynamics of the Model In this section, we analyse the stability of equilibria of the system (5). First, we state and prove the theory of differential inequality.

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Lemma 1 ([1]) Let σ be a differentiable function satisfying the differential inequality σ  (t) ≤ K σ(t), a ≤ t ≤ b

(6)

σ(t) ≤ σ(a)e K (t−a) , f or a ≤ t ≤ b.

(7)

where K is constant. Then

Proof Multiply both sides of (6) by e−K t and transpose each term to the other side getting dtd {σ(t)e−K t } = e−K t [σ  (t) − K σ(t)] ≤ 0. The function σ(t)e−K t thus have a negative or zero derivative and so is nonincreasing on the interval. Therefore, σ(t)e−K t ≤ σ(a)e−K a .

3.1 Boundedness of the Model Boundedness of the model system guarantees its validity. 3 Theorem 1 All solutions of the system (5) that starts in R+ are uniformly bounded.

Proof Let x(0) ≥ 0, y(0) ≥ 0, z(0) ≥ 0 be any solution with positive initial conditions. We first define the function σ = δx + y + z. Hence, the time derivative gives dx dy dz dσ =δ + + dt dt dt dt = δαx(1 − x) − δx z + β y(1 − y) − yz + δx z + yz − hz − γz 2 = δαx(1 − x) + β y(1 − y) − hz − γz 2 . We then take any μ > 0, and add the term μσ on both sides of the equation. Now, we have dσ + μσ = δαx(1 − x) + β y(1 − y) − hz − γz 2 + μ(δx + y + z) dt   α+μ 2 α+μ 2 μ ) + δα( ) = −δα x 2 − (1 + )x + ( α 2α 2α   β+μ 2 μ β+μ 2 −β y 2 − (1 + )y + ( ) + +β( ) β 2β 2β   h−μ 2 h−μ 2 h−μ )z + ( ) + γ( ) −γ z 2 − ( γ 2γ 2γ       α+μ 2 β+μ 2 h−μ 2 ≤ δα + β +γ . 2α 2β 2γ

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Taking ω = δα

 α+μ 2 2α

+ β



β+μ 2β

2

+γ



h−μ 2γ

2

, we get dσ + μσ ≤ ω. Apply-

ing differential inequality Lemma 1, we obtain 0 ≤ σ(x, y, z) ≤ ωμ + ω(x(0),y(0),z(0)) . eμt And for t → ∞, 0 ≤ σ ≤ ωμ . 3 = Thus, the solution to the system (5) is bounded in the region Ω = R+ ω 3 {(x, y, z) ∈ R+ : 0 ≤ σ ≤ μ + κ, for κ > 0}.

3.2 Existence of Interior Equilibria By solving the equations: dx dt dy dt dz dt

= αx(1 − x) − x z = 0 = β y(1 − y) − yz = 0, = δx z + yz − hz − γz 2 = 0

(8)

for the model system (5), we get several equilibrium solutions (x, y, z). On the bound3 , we have the  equilibria E 0 = (0, 0, ary of R+  0), E 1 = (1, 0, 0),

, E 5 = 0, βγ+δ E 2 = (0, 1, 0), E 3 = (1, 1, 0), E 4 = αγ+h , 0, α(δ−h) , β(−h) . αγ+δ αγ+δ βγ+ βγ+ In the following lemma, we mentioned the equilibria at the boundary and the stability of the system (5) and the condition of existence. Lemma 2 System (5) always have six (6) boundary equilibrium points E 0 =  αγ+h (0, 0, 0), E 1 = (1, 0, 0), E 2 = (0, 1, 0), E 3 = (1, 1, 0), E 4 = αγ+δ , 0, α(δ−h) , αγ+δ  E 5 = 0, βγ+δ , β(−h) . The boundary equilibrium point E 4 exists if δ > h and equiβγ+ βγ+ librium point E 5 exists if  > h. Proof The Jacobian matrix of the above system (5) is give by ⎛

⎞ α − 2αx − z 0 −x ⎠ 0 β − 2β y − z −y J =⎝ δz z δx + y − 2γz − h Substitution by the point E 0 = (0, 0, 0) in the Jacobian matrix, we get the eigenvalues λ = α, β, −h which has two positive eigenvalues. So the point E 0 is unstable equilibrium point. Similarly, E 1 = (1, 0, 0) has eigenvalues λ = −α, β, δ − h that gives unstable equilibrium point. For E 2 = (0, 1, 0), eigenvalues λ = α, −β,  − h, are λ = −α, −β, δ +  − h, hence also unstable. For E 3 = (1, 1, 0), eigenvalues  αγ+h α(δ−h) stable only if δ +  < h. For E 4 = αγ+δ , 0, αγ+δ , the equilibrium point exists only if δ > h. Evaluating the Jacobian matrix at E 4 gives

Modeling and Experimental Data on the Dynamics of Predation …

JE4

57

⎛ ⎞ ) − ( α(δ−h) ) 0 −x α − 2α( αγ+h αγ+δ αγ+δ ⎠ =⎝ 0 β − 2β y − z −y δz z δx + y − 2γz − h

The characteristic polynomial corresponding to E 4 is     α(δ − h) αγ + h α(δ − h) PE4 (λ) = β − −λ α − 2α( )− −λ αγ + δ αγ + δ αγ + δ     α(δ − h) α(δ − h) αγ + h αγ + h ) − 2γ − h − λ + δ( )( ) δ( αγ + δ αγ + δ αγ + δ αγ + δ   α(δ − h) − λ [(A − λ)(B − λ) + C] = β− αγ + δ     α(δ − h) = β− − λ λ2 − (A + B)λ + (AB + C) αγ + δ where A =

−α(αγ + h) , αγ + δ

B=

−αγ(δ−h) αγ + δ

and C = (αδ)(δ − h)(αγ + h).

− h) − h) < 0 if β < α(δ and the eigenThe eigenvalues of PE4 (λ) are λ = β − α(δ αγ + δ αγ + δ 2 values of λ − (A + B)λ + (AB + C). We observe that (AB + C) > 0 and −(A + B) > 0 are both positive. By the condition of the Routh-Hurwitz [3] criteria, the equi− h) . librium point E 4 is locally asymptotically stable under the condition, β < α(δ αγ + δ  βγ + δ β( − h) Similarly, for E 5 = 0, βγ +  , βγ +  , the equilibrium point exists only if  > h. Evaluating the Jacobian matrix at E 5 gives

JE5

⎛ β( − ⎜α − ( βγ + ⎜ ⎜ ⎜ 0 =⎜ ⎜ ⎜ ⎝ β( − δ( βγ +

h )−λ 

h ) 

⎞ 0

0

β( − h) βγ + h )− −λ β − 2β( βγ +  βγ + 

β+h ( ) βγ + 

(

β( − h ) βγ + 

(

βγ + h 2γβ( − h) )− −h−λ βγ +  βγ + 

The characteristic polynomial corresponding to E 5 is    −β 2 γ − βh β( − h) −λ −λ PE5 (λ) = α − βγ +  βγ +      (βγ + h)(β)( − h) −βγ + βγh −λ + βγ +  (βγ + )2   β( − h) − λ [(A − λ)(B − λ) + C] = α− βγ +      β( − h) − λ λ2 − (A + B)λ + (AB + C) = α− βγ +  

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

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where A =

−β 2 γ−βh , βγ+

PE5 (λ) are λ = α −

B= β(−h) βγ+

−βγ+βγh βγ+

and C =

< 0 if α
0 and

−(A + B) = + h + βγ( − h)) > 0 are both positive. By the condition of the Routh-Hurwitz [3] criteria, the equilibrium point E 5 is locally asymptotically stable under the condition, α < β(−h) . βγ+ Lemma 3 The interior equilibrium point E 6 = (x ∗ , y ∗ , z ∗ ) of system (5) exists if β+2δ (2)(δ−h) , γ > β(δ+−h) , α > β, δ > h and  > h. When these conditions are sat1 > α+2h isfied, x ∗ , y ∗ and z ∗ are given by x ∗ = ∗

z =

αβ(δ+−h) . α+αβγ+βδ

α+αβγ−β(−h) , α+αβγ+βδ

y∗ =

βδ+αβγ+α(δ−h) , α+αβγ+βδ

and

Proof Evaluating the Jacobian matrix at E 6 = (x ∗ , y ∗ , z ∗ )

JE6

⎛ ⎞ α − 2αx ∗ − z ∗ 0 −x ∗ ⎠ 0 β − 2β y ∗ − z ∗ −y ∗ =⎝ ∗ ∗ ∗ ∗ ∗ δz z δx + y − 2γz − h

The characteristic polynomial corresponding to E 6 is   −β 2 δ − αβ 2 γ − 3αβδ + 3αβh −α2  − α2 βγ − αβh + αβ −λ −λ α + αβγ + βδ α + αβγ + βδ   2αδ − αβγδ − αβγ − 2αh + αβγh −λ α + αβγ + βδ    −α2  − α2 βγ − αβh + αβ (βδ)(δ +  − h) − α + αβγ + βδ α + αβγ + βδ   −β 2 δ − αβ 2 γ − 3αβδ + 3αβh −λ α + αβγ + βδ   −α2  − α2 βγ − αβh + αβ + −λ α + αβγ + βδ   (−αβ 2 δ − α2 β 2 γ − α2 β(δ − h))(δ +  − h) (α + αβγ + βδ)2     A B C AE B A D ( = ( − λ)( − λ)( − λ) − − λ) + ( − λ) w w w w w2 w w2 

PE 6 (λ) =

= λ 3 + a 2 λ2 + a 1 λ + a 0

where

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59

A = −α2  − α2 βγ − αβh + αβ B = −β 2 δ − αβ 2 γ − 3αβδ + 3αβh C = 2αδ − αβγδ − αβγ − 2αh + αβγh D = −αβ 2 δ − α2 β 2 γ − α2 β(δ − h))(δ +  − h) E = (βδ)(δ +  − h) w = α + αβγ + βδ and −A − B − C w AB + AC + BC − Dw − E A a1 = w2 E AB + ADw − ABC a0 = w3 a2 =

Note that a2 = (α2  − αβ − 2α(δ − h)) + αβγ(δ +  − h) + αβh + α2 βγ 2δ+β +αβγ(δ +  − h) + β 2 δ + 3αβ(δ − h) + αβ 2 γ > 0 only if 2h+α < 1 and δ > h. 2 c 2 2 For a0 = (b d + 3abd − 3abh + ab )(a e − abe + abh + a bc)(2aeh − 2ade + abcd + abc(d + e − h)) + abe(bd + a(d − h) + abc)(d + e − h)(a 2 − abe +abh + a 2 bc). 2(δ−h) Note that a0 > 0, if α > β, δ > h and γ > β(δ+−h) . Satisfying the conditions stated above, similarly, we can also show that a2 a1 > 0. Hence, we have shown that a2 > 0, and a0 > 0 and a2 a1 > 0. By the condition of the Routh-Hurwitz [3] criteria, the equilibrium point E 6 is locally asymptotically stable under the condition β+2δ (2)(δ−h) , γ > β(δ+−h) , α > β, δ > h and  > h. 1 > α+2h

4 Numerical Simulations Analytical studies are made more interesting and complete when numerical verifications are performed using experimental data on some parameters. In this section, we complement the numerical simulations using MATLAB R2015a with actual harvesting values performed in an enclosed rice paddy. The critical duration to manage GAS is during the first 10 days after transplanting and 21 days after direct wet seeding. Hence, we consider the first 30-days as the critical stage for the rice plants as they are not resistant from GAS predation. After this period, the rice plants are generally resistant to snail damage and snails are actually beneficial by feeding on weeds (Table 1).

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Table 1 Range and baseline parameter values for the model Parameter description Param Intrinsic growth rate of rice plants Intrinsic growth rate of weeds Environmental carrying capacity of rice plants Environmental carrying capacity of weeds maximal relative increase of predation on rice plants maximal relative increase of predation on weeds Efficiency to convert rice biomass into fertility Efficiency to convert weeds biomass into fertility Death rate of the snails Density death rate of snails due self limitations Harvesting rate of snails

r1 r2 K1 K2 a1 a2 ν1 ν2 g1 g2 H

Value

References

0.06–0.08 0.08–0.15 10.0 5.0 0.0–0.13 0.0–0.13 a1 a2 0.0025–0.0035 0.0025–0.0035 0.1–0.5 0.0–1.0

[13, 18] [19] Fixed Fixed [16] [16] [4] [4] [6, 22],∗ AFE [6, 22],∗ AFE ∗ AFE Simulated

∗ AFE—Data obtained from actual field experiments at a enclosed rice field in Nueva Viscaya, Philippines

(a)

(b)

1.6 rice plants weeds apple snail

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1

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1.4

0.6

1.2 0.5

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weeds

0.6

0.3

0.4 0.2

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rice plants

0.2 0

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(c)

Fig. 1 Population dynamics for the rice plants, weeds and apple snail with harvesting rate H = 0.1 and the resulting population ( p30 , w30 , s30 ) = (0.2508, 0.3858, 1.4881) after 30 days. a Corresponding phase plane plot for the first 30 days; b Trajectories for first 30 days for the different species co-existence; c Actual rice plants present after 30 days

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2 rice plants weeds apple snail

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apple snails

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1 0.8

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1

0.6 0.4

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0.6 1.5

2 1

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0.4

1 0

0.5

rice plants

0.2 0

5

10

15

20

25

30

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(c)

Fig. 2 Population dynamics for the rice plants, weeds and apple snails for the first 30 days with harvesting rate H = 0.2 and the resulting population ( p30 , w30 , s30 ) = (0.7304, 0.8871, 1.6696) after 30 days. a Corresponding phase plane plot for the first 30 days; b Trajectories for first 30 days for the different species co-existence; c Actual rice plants present after 30 days

In this study, we performed numerical simulations on the mathematical model based on the baseline parameter values gathered from literature and actual field experiments for the death and harvesting rates of GAS. Since farmers initially cultivate the paddy and weeds are initially removed, we assume that during transplanting the initial population density of young rice plants, weeds and snails respectively is ( p0 , w0 , s0 ) = (1.0, 0.25, 0.31), that corresponds to 10%, 5% and 5% of their corresponding carrying capacity. The initial values for the other parameter are r1 = 0.08, r2 = 0.15, a = 0.13, g1 = 0.0030, g2 = 0.0025, ν1 = r1 , ν2 = r2 , K 1 = 10, and K 2 = 5. Numerical simulations show that when the critical value H ∗ = 0.47, the population of rice plants during the 30-day period is monotonically increasing. For convenience, we only consider increment of 0.1 as the value of the actual harvesting rates on each quadrat. From the simulations, we then compare the actual results of harvesting rates H from observed density of rice plants on the actual field experiment for H = 0.1, 0.2, 0.3, 0.4 and 0.5.

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(b)

(a)

rice plants weeds apple snail

2.5

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3

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0.5 1 0 2.5 2

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0 0

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Fig. 3 Population dynamics for the rice plants, weeds and apple snails for the first 30 days with harvesting rate H = 0.3 and the resulting population ( p30 , w30 , s30 ) = (1.6402, 1.627, 1.8500) after 30 days. a Corresponding phase plane plot for the first 30 days; b Trajectories for first 30 days for the different species co-existence; c Actual rice plants present after 30 days

In Fig. 1, we show the population dynamics for the rice plants, weeds and apple snail for the first thirty days when there is minimal harvesting, or ten percent removal on the snails population. The simulation shows that the snail population increases and the rice plants and weeds can easily be wiped out in a few days. This numerical results is confirmed by the actual field experiment. The quadrat with ten percent harvesting shows that only a few of the rice plants and weeds have survived to a stage where snails cannot easily eat the leaves. For comparison, we used the simulated population values for H = 0.1 as the baseline. In Fig. 2, a harvesting rate of 0.2 on snails showed a decrease in the snail population and has improved the rice plants from 0.02508 to 0.7304 but has also increased the weeds from 0.3858 to 0.8871. We can see from the actual field experiment that there is an improvement in the rice plant density. Similarly, in Figs. 3 and 4, numerical simulations would show that when the snail harvesting is set at 0.30 and 0.40, the density of rice plants are shown to increase from 0.7304 to 1.6402 and 3.5914, respectively. For the weeds, simulations would result to an increase from 0.8871 to 1.6027 and 2.9911, respectively.

Modeling and Experimental Data on the Dynamics of Predation …

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4 rice plants weeds apple snail

3.5 1.2 3

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1 0.8 0.6 0.4

2.5

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0.2 0 4

1 3

4 3.5 2

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3

0.5

2.5 1

2 1.5 0

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0 0

5

10

15

20

25

30

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(c)

Fig. 4 Population dynamics for the rice plants, weeds and apple snails for the first 30 days with harvesting rate H = 0.4 and the resulting population ( p30 , w30 , s30 ) = (3.5914, 2.9911, 1.0467) after 30 days. a Corresponding phase plane plot for the first 30 days; b Trajectories for first 30 days for the different species co-existence; c Actual rice plants present after 30 days

In Fig. 5, we show the population dynamics at the optimal harvesting rate of H ∗ ≈ 0.5 for the given parameter values and the resulting observed population of rice plant.

5 Concluding Remarks This paper deals with snails, rice plants and weeds model subject to snail harvesting. The model with harvesting rate for snail predator has a great role in shaping the dynamics of the model. We obtained six (6) boundary equilibrium points and one (1) interior equilibrium point. These points were analysed for the stability of the model using various mathematical techniques. As snails predate on rice plants, the presence of snails is bound to affect the growth of rice plants and weeds. The critical duration to manage GAS is the first 30 days after transplanting of young rice plants since after this period, the rice plants are generally resistant to snail damage and snails are actually beneficial by feeding on weeds. Simulations would show that we can increase population density of rice plants by increasing snail harvesting. But,

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5 rice plants weeds apple snail

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0.15 0.1

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5 4

2

0.5

3 1

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2 0

1

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0 0

5

10

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20

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30

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(c)

Fig. 5 Population dynamics for the rice plants, weeds and apple snails for the first 30 days with harvesting rate H = 0.5 and the resulting population ( p30 , w30 , s30 ) = (4.9703, 3.8963, 0.1452) after 30 days. a Corresponding phase plane plot for the first 30 days; b Trajectories for first 30 days for the different species co-existence; c Actual rice plants present after 30 days

it can be shown that the weeds population also increases and tends to approach density, same as that of the rice plants. Though we have a decrease in the snail density, the effort to decrease the weeds has to be considered and to be addressed. We have observed that harvesting rate of 0.5 was sufficient to control GAS density and maintain monotonically increasing population for the rice plants during the critical duration where rice plants are not resistant on GAS predation. The simulations and the actual field observations for the first 30 days have shown consistent results. We have shown that, it is necessary to control the presence of snails by introducing some regulatory mechanism like harvesting of snails during the initial stage of rice plant growth, and we do not have to harvest and eliminate the snails totally in order to have a natural control on the increase of weeds. Acknowledgements This study was funded by Cordillera Studies Center and JA received research load credit from University of the Philippines Baguio.

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References 1. Birkhoff, G., Rota, G.C.: Ordinary Differential Equations. Wiley, USA (1989) 2. Cagauan, A.G., Joshi, R.C.: Golden apple snail pomocea spp. in the philippines: review on levels of infestation, control methods, utilization and future research directions. J. Agric. Life Sci. 37(2), 7–32 (2003) 3. Edelstein-Keshet, L.: Mathematical Models in Biology. Random House Inc, New York (1988) 4. Estoy Jr., G.F., Yusa, Y., Wada, T., Sakurai, H., Tsuchida, K.: Size and age at first copulation and spawning of the apple snail, pomacea canaliculata (gastropoda: Ampullariidae). Appl. Entornol. Zool. 37(1), 199–205 (2002) 5. Freedman, H.I., Waltman, P.: Persistence in models of three interacting predator-prey populations. Math. Biosci. 68, 213–231 (1984) 6. Guo, J., Martin, P.R., Zhang, C., Zhang, J.: Predation risk affects growth and reproduction of an invasive snail and its lethal effect depends on prey size. PLoS ONE 12(11), 1–14 (2017) 7. Kar, T.K., Chakraborty, K.: Effort dynamics in a prey-predator model with harvesting. Int. J. Inf. Syst. Sci. 6(3), 318–332 (2010) 8. Kar, T.K., Chaudhuri, K.S.: Regulation of a prey-predator fishery by taxation: a dynamics reaction model. Ecol. Model. 11(2), 173–187 (2003) 9. Kar, T.K., Matsuda, H.: Controllability of a harvested prey-predator system with time delay. J. Biol. Syst. 14(2), 243–254 (2006) 10. Kar, T.K., Matsuda, H.: Global dynamics and controllability of a harvested prey-predator systems with holling type iii functional response. Nonlinear Anal. Hybrid Syst. 1, 59–67 (2007) 11. Kar, T.K., Misra, S.: Influence of prey reserve in a prey-predator fishery. Nonlinear Anal. 65, 1725–1735 (2006) 12. Lotka, A.J.: Elements of Physical Biology. Williams and Wilkins Co, Baltimore, USA (1925) 13. Mahajan, G., Ramesha, M., Chauhan, B.S.: Response of rice genotypes to weed competition in dry direct-seeded rice in india. Sci. World J. 2014, 8 (2014) 14. Malthus, T.R.: An essay on principle of population and summary view of the principle of population. Penguin, England (1798) 15. Mochida, O.: Pomocea snails in the philippines. Int. Rice Res. Newsl. 12(4), 48–49 (1987) 16. Morrison, W.E., Hay, M.E.: Feeding and growth of native, invasive and non-invasive alien apple snails (ampullariidae) in the united states: invasives eat more and grow more. Biol Invasions 13, 945–955 (2011) 17. Moya-Larano, J.: Genetic variation, predator-prey interactions and food web structure. Phil. Trans. R. Soc. B 366, 1425–1437 (2011) 18. Na, S., Hong, S.Y., Kim, Y., Lee, K., Jang, S.: Estimating leaf area index of paddy rice from rapideye imagery to assess evapotranspiration in korean paddy fields. Korean J. Soil Sci. Fert. 46(4), 245–252 (2013) 19. Sen, A., Sarkar, M., Begum, M., Zaman, F., Ray, S.: Effect of spacing and weeding regime on the growth of brri dhan56. Int. J. Exp. Agric. 4, 20–29 (2014) 20. Walters, C., Christensen, V., Fulton, B., Smith, A., Hilborn, R.: Predictions from simple predator-prey theory about impacts of harvesting forage fishes. Ecol. Model. 337, 272–280 (2016) 21. Wang, S., Ma, Z.: Stabiity analysis: the effect of prey refuge in predator-prey model. Ecol. Model. 247, 95–97 (2012) 22. Yoshida, K., Hoshikawa, K., Wada, T., Yusa, Y.: Life cycle of the apple snail pomocea canaliculata (caenogastropoda:ampullariidae) inhibiting japanese paddy fields. Appl. Entomol. Zool. 44(3), 465–474 (2009)

Fractional Differential Equations

Analysis of a Discrete-Time Fractional Order SIR Epidemic Model for Childhood Diseases Mahmoud A. M. Abdelaziz, Ahmad Izani Ismail, Farah A. Abdullah and Mohd Hafiz Mohd

Abstract In this paper, a discrete-time fractional order SIR epidemic model for a childhood disease with constant vaccination program is investigated. The local asymptotic stability and bifurcation of the equilibrium points are analyzed using basic reproduction number. Flip and Neimark-Sacker (N-S) bifurcations are investigated for endemic equilibrium point and numerical simulations are carried out to illustrate the dynamical behaviors of the model. Chaos phenomenon is observed through numerical simulation inside the flip and N-S bifurcation regions. Results of the numerical simulations support the theoretical analysis. Keywords Discrete-time SIR epidemic model with fractional-order · Childhood diseases · Vaccination rate · Basic reproduction number · Flip bifurcation · Neimark-Sacker bifurcation · Chaos

1 Introduction and Preliminaries Epidemics were and still, a major cause of deaths. The black plague in Europe in the middle of the 14th century took the lives of 75–200 million people [1]. The WHO reported that in 2009–2010 there were 3–5 million cases and 100000–130000 deaths from epidemics such as Zika in Africa alone [2]. The mathematical modeling of infectious diseases and associated epidemics can reveal insights which might not be obtainable from laboratory and field studies [3]. Mathematical modeling also M. A. M. Abdelaziz (B) · A. I. Ismail · F. A. Abdullah · M. H. Mohd School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia e-mail: [email protected] A. I. Ismail e-mail: [email protected] F. A. Abdullah e-mail: [email protected] M. H. Mohd e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_5

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complements laboratory and field studies and allows for the optimal use of resources in planning appropriate strategy to control epidemics [4]. Epidemic models of infectious diseases based on a mathematical description have long been a subject of intense study for infectious diseases [5]. It is usually the case that only significant and dominant aspects of infection are included in a particular mathematical model and the model is often being simplified [6]. In mathematical modeling, it is well known that even simple mathematical models can exhibit complicated behavior [7]. In general, childhood infectious diseases such as measles, chicken pox, and rubella are among major concerns in many countries and there is usually a vaccination program for children with regards to these diseases. Some mathematical modeling studies have focused on childhood infectious diseases, for instance [8–10]. Cui et al. [11] studied a numerical scheme for a SIR epidemic model of childhood disease with a constant vaccination strategy. For this model, the total population involved in the infection was split into three classes: a susceptible class (S), an infected class (I ) and a removed class (R) consisting of vaccinated children as well as children who recover and now have permanent immunity. It is further assumed that the natural death rates μ in these different classes remain unequal and thus the total population varies and is denoted by N (t) with N (t) = S(t) + I (t) + R(t). Individuals are born into the population at a constant birth rate . The fraction of individuals vaccinated at birth is denoted by q (with 0 < q < 1) and the rest of the individuals are assumed susceptible. A susceptible individual will move into the infected group through contact with an infected individual at a rate β. An infected individual recovers at a recovery rate r and enters the removed class. The removed class, as mentioned earlier, also contains people who have been vaccinated. The governing equations to model this epidemiological system are given by: d S(t) = (1 − q) − μS (t) − β S (t) I (t) , dt d I (t) = β S (t) I (t) − (μ + r ) I (t) , dt d R(t) = q + r I (t) − μR (t) , dt

(1.1)

subject to the initial conditions S(0) = S0 ≥ 0, I (0) = I0 ≥ 0, R(0) = R0 ≥ 0. Parameters of model (1.1) are non-negative for t  0. It follows that d Ndt(t) =     μ μ − N (t) , which has the exact solution N (t) = μ 1 − e−μt + N0 e−μt with

initial value N (0) = N0  0, which implies that lim supt→∞ N (t)  μ . The feasible region for   model (1.1) i.e.  = (S, I, R) ∈ R3+ : N (t) = S(t) + I (t) + R(t)  μ is positive invariant.

Analysis of a Discrete-Time Fractional Order SIR Epidemic Model …

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 It is obvious that the model (1.1) has a free disease equilibrium point E 0 (1−q) , 0, μ  q , get it by substituting I (t) = 0. Following [12], the basic reproduction number μ for model (1.1) denoted by 0 can be obtained. Model (1.1) can rewrite as follows: dU =F −℘ dt ⎛

where,

⎞ ⎛ ⎞ ⎛ ⎞ S (t) −β S (t) I (t) μS (t) − (1 − q) ⎠. U (t) = ⎝ I (t) ⎠ , F (t) = ⎝ β S (t) I (t) ⎠ and ℘ (t) = ⎝ (μ + r ) I (t) R (t) 0 μR (t) − q − r I (t)

Then we obtain   0 −β(1−q) μ 0 μ F= and V = . 0 μ+r 0 β(1−q) μ It is well known that the basic reproduction number is the spectral radius of F V −1 where

  0 −β(1−q) μ(μ+r ) −1 FV = , is the basic repro. Thus 0 = ρ F V −1 = β(1−q) β(1−q) μ(μ+r ) 0 μ(μ+r ) duction number for the model (1.1).   The disease free equilibrium point of (1.1) E 0 (1−q) shows the level of , 0, q μ μ susceptible, recovered, and not infected population of total individuals. So that we have the following: has only one (1) If the basic reproduction number 0  1, then  the model (1.1)  (1−q) q , 0, μ . stable equilibrium i.e. disease-free equilibrium E 0 μ (2) If the basic reproduction number  > 1, then the model (1.1) has two equilib0    (1−q) q μ+r and endemic equilibrium E , 0, , ria: disease-free equilibrium E 0 1 μ μ β  0 (μq+r )−r (1−q) μ . − 1) , β (0 β(1−q) Recently, more researchers begin to investigate the qualitative properties of fractional-order epidemic models [13, 14]. These models can show realistic behavior of infection memory of disease especially, in hereditary diseases such as childhood diseases that are inherited from one or both parents [15, 16]. In this paper, we extend model (1.1) to a fractional-order model. There are several definitions of fractional derivatives [17, 18]. We shall use the Caputo definition [19] which is often used. Definition 1.1 The fractional integral of order β ∈ R+ of the function f (t), t > 0, is defined by t )β−1 f (τ ) dτ , I β f (t) = 0 (t−τ (β) and the fractional derivative of order α ∈ (n − 1, n) of f (t) , t > 0, is defined by D α f (t) = I n−α D n f (n) (t) , α > 0, where, f (n) represents the nth−order derivative of f (t), n = [α] is the value of α rounded up to the nearest integer, I β is the β−order Riemann-Liouville integral operator and (.) is Euler’s Gamma function. The operator D α is called the “α−order Caputo differential operator”.

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The fractional-order SIR epidemic model for childhood diseases with constant vaccination can be written as follows: Dtα S(t) = (1 − q) − μS − β S I, Dtα I (t) = β S I − (μ + r ) I, Dtα R(t) = q + r I − μR,

(1.2)

where, Dtα is the Caputo fractional derivative, t > 0, and α is the fractional-order satisfying α ∈ (0, 1] . It is often necessary to discretize the continuous-time epidemic model to assess the effect of larger time steps on disease spreading. As well as known that statistical data on epidemics are collected intermittently time. So, it should be noted that epidemic models which described by discrete-time systems are expected to be more realistic to study epidemics and derive many qualitative properties than the continuous-time systems [20–24]. For that, this paper aims to study the dynamical behaviors of the discrete-time SIR epidemic model constructed from the fractional-order model (1.2). Discretization process is displayed in Sect. 2. The local stability of the equilibria are obtained in Sect. 3. Further, in Sect. 4, the existence of flip bifurcation and NeimarkSacker bifurcation are analyzed. Finally, some discussions are introduced in Sect. 5.

2 Discretization Process In [25, 26] a discretization method was introduced to discretize fractional order differential equations. This discretization method is now applied to the fractional SIR model (1.2) as follows:          t t t x − βS x I x , D S(t) = (1 − q) − μS x x x          t t t x I x − (μ + r ) I x , D α I (t) = β S x x x       t t x − μR x . D α R(t) = q + r I x x α

(2.1)

Let t ∈ [0, h) , t/ h ∈ [0, 1) . Then, Dtα S(t) = (1 − q) − μS0 − β S0 I0 , Dtα I (t) = β S0 I0 − (μ + r ) I0 , Dtα R(t) = q + r I0 − μR0 , and the solution of (2.2) reduces to

(2.2)

Analysis of a Discrete-Time Fractional Order SIR Epidemic Model …

S1 (t) = S0 + J α [(1 − q) − μS0 − β S0 I0 ] tα = S0 + [(1 − q) − μS0 − β S0 I0 ] , α (α) I1 (t) = I0 + J α [β S0 I0 − (μ + r ) I0 ] tα = I0 + [β S0 I0 − (μ + r ) I0 ] , α (α) R1 (t) = R0 + J α [q + r I0 − μR0 ] tα = R0 + [q + r I0 − μR0 ] . α (α)

73

(2.3)

Now, take t ∈ [h, 2h) , 1  t/ h < 2. Thus, we arrive at Dtα S(t) = (1 − q)  − μS1 − β S1 I1 , Dtα I (t) = β S1 I1 − (μ + r ) I1 , Dtα R(t) = q + r I1 − μR1 ,

(2.4)

which have the following solution: (t − h)α [(1 − q) − μS1 (x) − β S1 (x) I1 (x)] , α (α) (t − h)α I2 (t) = I1 (x) + [β S1 (x) I1 (x) − (μ + r ) I1 (x)] , α (α) (t − h)α R2 (t) = R1 (x) + [q + r I1 (x) − μR1 (x)] . α (α)

S2 (t) = S1 (x) +

(2.5)

Repeating the discretization process n times yields (t − nh)α [(1 − q) − μSn (nx) − β Sn (nx)In (nx)] , α (α) (t − nh)α In+1 (t) = In (nx) + [β Sn (nx)In (nx) − (μ + r ) In (nx)] , α (α) (t − nh)α Rn+1 (t) = Rn (nx) + (2.6) [q + r In (nx) − μRn (nx)] , α (α) Sn+1 (t) = Sn (nx) +

where, t ∈ [nh, (n + 1) h) . For t → (n + 1) h, system (2.6) becomes hα [(1 − q) − μSn − β Sn In ] , α (α) hα = In + [β Sn In − (μ + r ) In ] , α (α) hα = Rn + [q + r In − μRn ] . α (α)

Sn+1 = Sn + In+1 Rn+1

(2.7)

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Remark 2.1 It should be noted that if α → 1 in (2.7), then the Euler’s discretization is gained.

3 Stability of Equilibria In this section, the local stability of model (2.7) is investigated. As the first two equations of (2.7) about (Sn , In ) do not include Rn and the third equation is a linear equation of Rn , the dynamical behavior of (2.7) is the same as the dynamical behavior of the model hα [(1 − q) − μSn − β Sn In ] , α (α) hα = In + [β Sn In − (μ + r ) In ] , α (α)

Sn+1 = Sn + In+1

(3.1)

which only includes Sn and In . In the following we shall study model (3.1). The Jacobian matrix of (3.1) evaluated at any equilibrium point E ∗ (S ∗ , I ∗ ) is given by 

J E

 ∗

=

1−

hα (μ + α(α) hα βI∗ α(α)

β I ∗)

α

h β S∗ − α(α) hα 1 − α(α) (μ + r − β S ∗ )

.

(3.2)

The characteristic equation of (3.2) can be written as     λ2 − T r J E ∗ λ + Det J E ∗ = 0.

(3.3)

where  hα  2μ + r − β S ∗ + β I ∗ , α (α)  ∗  hα  Det J E = 1 − 2μ + r − β S ∗ + β I ∗ α (α) 2      hα + (μ + r ) μ + β I ∗ − μβ S ∗ . α (α)   T r J E∗ = 2 −

(3.4)

Let λ1 and λ2 be the two roots of Eq. (3.3), we give the following Lemma Lemma 3.1 ([27, 28]) (i) E ∗ is locally asymptotically stable and called a sink If |λ1 | < 1 and |λ2 | < 1. (ii) E ∗ is unstable and called a source If |λ1 | > 1 and |λ2 | > 1. (iii) E ∗ is unstable and called a saddle If |λ1 | < 1 and |λ2 | > 1, (or |λ1 | > 1 and |λ2 | < 1). (iv) E ∗ is called non-hyperbolic If |λ1 | = 1 or |λ2 | = 1.

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Then, it is found two cases to discuss the local stability at the equilibria of model (3.1).   Case I: 0  1. Model (3.1) has only the disease-free equilibrium E 0 (μ+rβ )0 , 0 . The Jacobian matrix J (E 0 ) is given as J (E 0 ) =

hα μ α(α)

1−

0

α

(μ+r )0 − h α(α) h α (μ+r )(1−0 ) 1− α(α)

.

(3.5)

α

α

h h μ and λ2 = 1 − α(α) (μ + Thus J (E 0 ) has two eigenvalues are λ1 = 1 − α(α) r ) (1 − 0 ) where 0 < α  1 and h > 0. Based on Lemma 3.1 we conclude the following

Theorem 3.2 If 0 < 1, we have 

  2α(α) α (i) E 0 is a sink if 0 < h < min α 2α(α) . , )   μ  (μ+r )(1−0  2α(α) . (ii) E 0 is a source if h > max α 2α(α) , α (μ+r μ )(1−0 ) (iii) E 0 is a saddle if 

min

α

2α (α) , μ

 α

2α (α) (μ + r ) (1 − 0 )

(iv) E 0 is non-hyperbolic if h =



 < h < max

 α

2α(α) μ

or h =

α

2α (α) , μ

 α

 α

2α (α) (μ + r ) (1 − 0 )



2α(α) . (μ+r )(1−0 )

From Theorem 3.2(iv), it is clear that one of the eigenvalues of the disease-free equilibrium E 0 is 1 and other is neither 1 nor −1 and the all parameters located in where 1 ∪ 2     2α(α) 1 = (α, h, q, , β, μ, r ) : 0 < 1, h = α (μ+r , h = α 2α(α) μ )(1−0 ) and     2α(α) . , h = α (μ+r 2 = (α, h, q, , β, μ, r ) : 0 < 1, h = α 2α(α) μ )(1−0 ) Hence, the disease-free equilibrium E 0 may admit a flip bifurcation when parameters vary in a small neighborhood of 1 or 2 .    Case II: 0 > 1. Model (3.1) has two equilibria E 0 (μ+rβ )0 , 0 and E 1 μ+r , βμ β α

α

h h μ < 1 and λ2 = 1 − α(α) (μ + r ) (1 − 0 ) (0 − 1)) . For E 0 , since λ1 = 1 − α(α) > 1, then E 0 is a saddle. For E 1 , we first state the following Lemma

Lemma 3.3 ([27, 28]) Let F (λ) = λ2 − Bλ + C. Suppose that F (1) > 0, λ1 , λ2 are the two roots of F (λ) = 0. Then, (i) |λ1 | < 1 and |λ2 | < 1 if and only if F (−1) > 0 and C < 1; (ii) |λ1 | < 1 and |λ2 | > 1 or (|λ1 | > 1 and |λ2 | < 1) if and only if F (−1) < 0; (iii) |λ1 | > 1 and |λ2 | > 1 if and only if F (−1) > 0 and C > 1; (iv) λ1 = −1 and λ2 = 1 if and only if F (−1) = 0 and B = 0, 2; (v) λ1 and λ2 are complex and |λ1 | = |λ2 | if and only if B 2 − 4C < 0 and C = 1.

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The Jacobian matrix J (E 1 ) is as follows J (E 1 ) =

h α μ0 −h α (μ+r ) α(α) α(α) α h μ(0 −1) 1 α(α)

1−

.

Since the characteristic equation of J (E 1 ) can be written as F (λ) = λ2 − T r J (E 1 ) λ + Det J (E 1 ) = 0,

(3.6)

where T r J (E 1 ) = 2 −

h α μ0 , α (α)

Det J (E 1 ) = μ (μ + r ) (0 − 1)



hα α (α)



2 − μ0

hα α (α)

 + 1.

Hence,

 α 2 h > 1. Based on Lemmas (3.1) and (3.3) the F (1) = μ (μ + r ) (0 − 1) α(α) dynamical properties of E 1 can be given as follows Theorem 3.4 If 0 > 1, we have (i) E 1 is a sink if one of the following conditions holds: (i.1)   0 and 0 < h < h 1 . (i.2)  < 0 and 0 < h < h 2 . (ii) E 1 is a source if one of the following conditions holds: (ii.1)   0 and h > h 3 . (ii.2)  < 0 and h > h 2 . (iii) E 1 is a saddle if   0 and h 1 < h < h 3 . (iv) E 1 is non-hyperbolic if one of the following conditions holds: (iv.1)   0 and h = h 1 or h = h 3 . (iv.2)  < 0 and h = h 2 . where √ μ0 − )α(α) α , h 1  ( μ(μ+r )(0 −1)  0 α(α) , h 2  α (μ+r )(0 −1)  √ μ + )α(α) α h  ( 0 3

μ(μ+r )(0 −1)

and  = (μ0 )2 − 4μ (μ + r ) (0 − 1) . If the condition Theorem 3.4(iv.1) holds, then one of the two eigenvalues of J (E 1 ) is −1 and the other is neither 1 nor −1 and leads to the following (α, h, q, , β, μ, r ) ∈ 3 ∪ 4 where 3 = {(α, h, q, , β, μ, r ) : 0 > 1, h = h 1 ,  ≥ 0} ,

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and 4 = {(α, h, q, , β, μ, r ) : 0 > 1, h = h 3 ,  ≥ 0} . If (α, h, q, , β, μ, r ) ∈ 3 and h varies in the neighborhood of h 1 or (α, h, q, , β, μ, r ) ∈ 4 and h 3 , then model (3.1) may undergo a flip bifurcation at E 1 . Also when the condition Theorem 3.4(iv.2) holds, two eigenvalues of J (E 1 ) are a pair of conjugate complex numbers with modules one and leads to (α, h, q, , β, μ, r ) ∈ 5 where 5 = {(α, h, q, , β, μ, r ) : 0 > 1, h = h 2 ,  < 0} . If (α, h, q, , β, μ, r ) ∈ 5 and h varies in the neighborhood of h 2 , then model (3.1) may undergo a Neimark-Sacker bifurcation at E 1 .

4 Bifurcation at E1 In this section, the existence of flip bifurcation and Neimark-Sacker bifurcation is analyzed at the endemic equilibrium point E 1 . The parameter h is chosen as a bifurcation parameter to study the analysis of flip and Neimark-Sacker bifurcations using the idea of bifurcation theory, center manifold theorem and normal form method in [21–23].

4.1 Flip Bifurcation   μ First, we discuss the flip bifurcation analysis around E 1 μ+r , − 1) . Consid( 0 β β ering the parameter h as a bifurcation parameter, the system (3.1) can be formulated as follows: (h + h ∗ )α [(1 − q) − μSn − β Sn In ] , α (α) (h + h ∗ )α = In + [β Sn In − (μ + r ) In ] , α (α)

Sn+1 = Sn + In+1

(4.1)

where |h ∗ | 0, α2 > 0 (or α1 < 0, α2 < 0) while the period-2 points that bifurcate from E 1 are unstable if α1 > 0, α2 < 0 (or α1 < 0, α2 > 0).

4.2 Neimark-Sacker Bifurcation   μ Finally, the Neimark-Sacker bifurcation is studied at E 1 μ+r , − 1) with ( 0 β β considering the parameter h as a bifurcation parameter. A perturbation form of (3.1) can be given as

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(h + h ∗∗ )α [(1 − q) − μSn − β Sn In ] , α (α) (h + h ∗∗ )α = In + [β Sn In − (μ + r ) In ] , α (α)

Sn+1 = Sn + In+1

(4.8)

where, |h ∗∗ | 0. Hence, the two solutions of √ α α ∗∗ ) ∗∗ ) ± 2i (h+h 4H − G 2 . (4.10) are θ1,2 = 1 + G2 (h+h α(α) α(α) Since, the parameters (h, α, b, β, μ, r, ) ∈ N S B (E 1 ) and h ∗∗ varies in a small neighborhood of h ∗∗  =0, then (4.10) has two roots are a pair of complex conjugate numbers θ1,2 with θ1,2  = 1.    √   α ∗∗ ) Since θ1,2  = q (h ∗∗ ) and θ1,2 h ∗∗ =0 = 1, then we have q (0) = 1 + G (h+h α(α)   α 2 α ∗∗ ) ∗∗ ) + H (h+h = −G. = 1. Hence, we get H (h+h α(α) α(α)   α   h h α−1 G+2H α(α) d |θ |   Consequently, dh1,2 =    2 > 0.  ∗∗ h ∗∗ =0

2(α) 1+G

hα α(α)

+H

hα α(α)

n

Further, it is required that, when h ∗∗ = 0, θ , θn = 1, (n = 1, 2, 3, 4) , which is equivalent to p (0) = −2, 0, 1, 2. Since p 2 (0) − 4q(0) < 0 and q(0) = 1, therefore p 2 (0) < 4; which indicates p(0) = ±2. It is only needed to require that p(0) = 0, 1, which leads to (4.11) G 2 = 2H, 3H. Thus, the two eigenvalues θ1,2 of origin point of (4.9) do not lie on the intersection of the unit circle with the coordinate axes when h ∗∗ = 0.

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  Next,the normal form of model (4.9) when h ∗∗ = 0 is studied. Let ω = Re θ1,2 ,  ζ = Im θ1,2 and   0 c12 . T= ω − c11 −ζ Consider the translation below 

Xn Yn





un =T υn

 .

(4.12)

Taking T−1 on both sides of (4.9), we obtain 

u n+1 υn+1



 =

ω −ζ ζ ω



un υn



 +

 f (u n , υn ) , g (u n , υn )

(4.13)

where,   f (u n , υn ) = c13 (ω − c11 ) u 2n − ζu n υn ,   1 g (u n , υn ) = [c13 (ω − c11 ) − c12 c23 ] (ω − c11 ) u 2n − ζu n υn . ζ Now, we obtain ∂2 f ∂2 f ∂2 f = 2c = −c ζ, = 0, − c , (ω ) 13 11 13 ∂u 2n ∂u n ∂υn ∂υn2 ∂3 f ∂3 f ∂3 f ∂3 f = = = = 0, ∂u 3n ∂u 2n ∂υn ∂u n ∂υn2 ∂υn3 ∂2 g 2 = (ω − c11 ) [c13 (ω − c11 ) − c12 c23 ] , ∂u 2n ζ ∂2 g ∂2 g = − [c13 (ω − c11 ) − c12 c23 ] , 2 = 0, ∂u n ∂υn ∂υn ∂3 g ∂3 g ∂3 g ∂3g = = = = 0. ∂u 3n ∂u 2n ∂υn ∂u n ∂υn2 ∂υn3 Accordingly, the nonzero discriminatory quantity  is given by 

 2   1 (1 − 2θ) θ ξ11 ξ20 − |ξ11 |2 − |ξ02 |2 + Re θξ21 = 0,  = − Re 1−θ 2 where,

(4.14)

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θ =ω + iζ,    2 1 ∂2 f ∂2 f ∂ g ∂2 g ξ11 = |(u n ,υn )=(0,0) , + + i − 4 ∂u 2n ∂υn2 ∂u 2n ∂υn2    2 1 ∂2 f ∂2 f ∂2 g ∂ g ∂2 g ∂2 f |(u n ,υn )=(0,0) , ξ20 = − +2 +i − −2 8 ∂u 2n ∂υn2 ∂u n ∂υn ∂u 2n ∂υn2 ∂u n ∂υn  2   ∂ g 1 ∂2 f ∂2 f ∂2 g ∂2 g ∂2 f ξ02 = − − 2 + i − + 2 |(u n ,υn )=(0,0) , 8 ∂u 2n ∂υn2 ∂u n ∂υn ∂u 2n ∂υn2 ∂u n ∂υn  1 ∂3 f ∂3 f ∂3 g ∂3g ξ21 = + + 2 + + 3 2 16 ∂u n ∂u n ∂υn ∂u n ∂υn ∂υn3   3 ∂ g ∂3 g ∂3 f ∂3 f |(u n ,υn )=(0,0) . i + − 2 − ∂u 3n ∂u n ∂υn2 ∂u n ∂υn ∂υn3 (4.15) From the above analysis and according to Neimark-Sacker bifurcation conditions [30], the following Theorem can be stated. Theorem 4.2 If (4.11) and (4.15) are satisfied, then the system (3.1) undergoes Neimark-Sacker bifurcation at E 1 with varying the parameter h ∗∗ in the small neighborhood of h. Furthermore, if  < 0 then an attracting invariant closed curve bifurcates from E 1 for h > h ∗∗ while if  > 0, then a repelling invariant closed curve bifurcates from E 1 for h < h ∗∗ .

5 Numerical Simulations In this section, the bifurcation diagrams and phase portraits are obtained to confirm the bifurcation analysis results discussed above. The dynamical behaviors of model (3.1) are illustrated numerically at the endemic equilibrium E 1 and they are presented through the following examples: Example 1 Choose  = 6, q = 0.45, μ = r = 0.2, β = 0.99, α = 0.98 with initial conditions (S0 , I0 ) = (0.404, 8.0479) . By calculations, one can obtain R0 ≈ 40.84, ≈  53.96 and h 1 ≈ 0.2487. Hence, model (3.1) has the endemic equilibrium E 1 0.40, 8.0479 and J (E 1 ) has two eigenvalues λ1 = −1, λ2 ≈ 0.89 i.e. |λ2 | = 1. We also obtain the discriminatory quantities ϕ1 ≈ (0.8903)−8 and ϕ2 ≈ 0.0023 which are positive. The calculations obtained ensure that E 1 undergoes a flip bifurcation and a stable period-2 points bifurcate from E 1 when parameters vary in the small neighborhood of 3 . Figure 1 shows the flip bifurcation diagram according to the values of parameters set out above when h ∈ (0.2, 0.32). From Fig. 1, it can be seen that E 1 is stable when h < h 1 and loses its stability through the flip bifurcation when h > h 1 . Also, the period-2, 4, 8 orbits start to appear when h starts to exceed the value of h 1 . From Fig. 1, it is observed that increasing h can lead to the (dis-) appearance of periodic orbit as well increases the chaotic attractors.

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Fig. 1 Flip bifurcation diagram of model (3.1) for h ∈ (0.2, 0.32) (a) α =0.98, h=0.2

(b) α =0.98, h=0.2

9

(c) α =0.98, h=0.24

8

8.05 s

n

i

n

(d) α =0.98, h=0.24

9

8.049 s

8

8.048

7

n

8.049

i

n

8.048

7 8.047

8.047

6

8.046

4

5

i

5

s, i

8.046

i

s, i

6

8.044

3

3

8.043

8.044 2

2 8.043

1 0

0

500

8.042 0.404

1000

0

0.4042 0.4044 0.4046

s

(e) α =0.98, h=0.25 8

8.041 0

500

1000

(f) α =0.98, h=0.25

(g) α =0.98, h=0.32

8.6

10

8.4

8

8.2

6

(h) α =0.98, h=0.32

n

10

in

9

6

8

3

7

i

i

4

s, i

5

s, i

0.41

11 s

7

8

4

7.8

2

7.6

0

6 5

2

4

1

3

0 −1

0.405

s

12

n

in

0.4

n

8.8 s

8.042

1

n 9

8.045

4

8.045

0

500

n

1000

7.4 −0.5

0

0.5

s

1

−2

0

500

1000

2 −5

0

n

Fig. 2 Phase portraits diagram for model (3.1) for various h corresponding to Fig. 1

5

s

10

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Fig. 3 Neimark-Sacker bifurcation diagram for model (3.1) for h ∈ (5.8, 8) (a) α =0.99, h=5

(b)α =0.99, h=5

5.5

(c) α =0.99, h=6.08

0.9998 s

5

5.5

(d) α =0.99, h=6.08

1.0002 s

n

n

i

i

0.9998

n

n

5

1.0001

4.5 4.5

0.9998 4 3.5

0.9999 3.5

0.9997

0.9998

i

i

3

s, i

s, i

1

4

0.9997

3

0.9997

2.5

2.5

0.9997

0.9996

2

2

0.9997 1.5 0.9997

1 0.5

0.9995

1.5

0.9994

1

0

500

0.9996 4.9995

1000

n

0.5

5

5.0005

0

400

s

(e) α =0.99, h=6.13

(f) α =0.99, h=6.13

s

n

600

800

1000

0.9993 4.999 4.9992 4.9994 4.9996 4.9998

(g) α =0.99, h=8.1

1.2

5.5 5

200

9

5

s

5.0002 5.0004 5.0006 5.0008

(h) α =0.99, h=8.1

1.8 s

n

n

i

i

n

n

1.15

8

1.6

7

1.4

6

1.2

4.5 1.1

4

1

2.5 2

5

1

i

3

s, i

1.05

i

s, i

3.5

4

0.8

3

0.6

2

0.4

0.95

1.5 0.9 1 0.5

1

0

500

n

1000

0.85 4.6

0

4.8

5

s

5.2

0.2

0

200

400

600

800

1000

0

3

4

5

n

Fig. 4 Phase portraits diagram for model (3.1) for various h corresponding to Fig. 3

6

s

7

8

9

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The phase portrait of E 1 is plotted in Fig. 2 for different values of h which are taken before and after the critical value h 1 . Figure 2a–d confirm the stable state of E 1 before h reaches the critical value h 1 while Fig. 2e–h confirm that the stable state of E 1 is being lost after h 1 . Example 2 Choose  = 2.727, q = 0.45, μ = 0.2, r = 0.3, β = 0.1, α = 0.99 with initial conditions (S0 , I0 ) = (4.45, 1.15) . Then R0 ≈ 1.5,  ≈ 0.11 and h 2 ≈ 6.085 and model (3.1) has the endemic equilibrium E 1 (5, 1) . Then,  the matrix J (E 1 ) possess a pair of complex conjugate eigenvalues λ1,2 with λ1,2  = 1. That makes the discriminatory quantity  ≈ −0.4846 which means that E 1 undergoes the NeimarkSacker bifurcation when parameters vary in the small neighborhood of N S (E 1 ) and an attracting invariant closed curve bifurcates from E 1 for h > h 2 . The occurrence (a) Maximal Lyapunov exponents according to flip bifurcation 0.5

0.4

0.3

Lyp

0.2

0.1

0

−0.1

−0.2 0.2

0.22

0.24

0.26

0.28

0.3

0.32

h

(b) Maximal Lyapunov exponents according to Neimark−Sacker bifurcation 0.3 0.2

Lyp

0.1 0 −0.1 −0.2 −0.3 −0.4 5.8

6

6.2

6.4

6.6

6.8

h

7

7.2

7.4

7.6

7.8

Fig. 5 a Maximal Lyapunov exponents corresponding to Fig. 1. b Maximal Lyapunov exponents corresponding to Fig. 3

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M. A. M. Abdelaziz et al.

of Neimark-Sacker bifurcation is illustrated in Fig. 3. It is clear that E 1 is stable for h < h 2 and loses its stability through Neimark-Sacker bifurcation at h = h 2 . The phase portraits corresponding to Fig. 3 are plotted in Fig. 4 for various h−values to interpret these observations. The stable equilibrium is shown in Fig. 4a– d with h-values less than h 2 while, Fig. 4e–f shows the unstable equilibrium when h is greater than h 2 . Furthermore, the stable limit cycle has appeared in Fig. 4d with h < h 2 while the attracting invariant closed curve is born with h > h 2 and can be shown in Fig. 4f. Attracting chaotic sets have also appeared when h increases further and these observations are plotted in Fig. 4g, h. Maximal Lyapunov exponents according to Figs. 1 and 3 are computed in Figs. 5a, b respectively. Figure 5 illustrates the correctness observations in Figs. 1 and 3. Negative Lyapunov exponents values in Fig. 5a indicate stable equilibrium point E 1 when h < h 1 while positive values indicate to the existence of flip bifurcation when h > h 1 . Also, it is observed from Fig. 5b that E 1 is stable when h < h 2 and loses its stability via Neimark-Sacker bifurcation when h > h 2 .

6 Conclusions In this paper, we have introduced a discrete-time SIR epidemic model (3.1) with fractional-order for childhood diseases with constant vaccination. Further, we have investigated the existence of the disease-free equilibrium E 0 and endemic equilibrium E 1 with the concept of the basic reproduction number 0 . The local stability of E 0 and E 1 are studied. The existence conditions of flip and Neimark-Sacker bifurcations are derived using the time step parameter h as a bifurcation parameter. It was also found that model (3.1) exhibits complex dynamical behaviors containing the occurrence of flip and Neimark-Sacker bifurcations, periodic orbits, attracting invariant closed curve and chaotic sets. Moreover, numerical simulations have been used to illustrate the analytical results. The findings revealed that both numerical simulation and analytical outcomes are clearly consistent. Acknowledgements The authors would like to thank the editor and the referees for their helpful comments and suggestions.

References 1. Haensch, S., Bianucci, R., Signoli, M., Rajerison, M., Schultz, M., Kacki, S., Vermunt, M., Weston, D.A., Hurst, D., Achtman, M., Carniel, E., Bramanti, B.: Distinct clones of Yersinia pestis caused the black death. PLoS Pathog 6(10), e1001134 (2010) 2. Lozano, R., Naghavi, M., Foreman, K., Lim, S., Shibuya, K., Aboyans, V., Abraham, J., Adair, T., Aggarwal, R., Ahn, S.Y., et al.: Global and regional mortality from 235 causes of death for 20 age groups in 1990 and 2010: a systematic analysis for the Global Burden of Disease Study 2010. Lancet. 380(9859), 2095–128 (2012)

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Delay and Partial Differential Equations

A Tuberculosis Epidemic Model with Latent and Treatment Period Time Delays Jay Michael R. Macalalag, Elvira P. De Lara-Tuprio and Timothy Robin Y. Teng

Abstract In this paper, a Susceptible-Exposed-Infectious-Treated (SEIT) epidemic model with two discrete time delays for the disease transmission of tuberculosis (TB) is proposed and analyzed. The first time delay τ1 represents the time of progression of an individual from the latent TB infection to the active TB disease, and the other delay τ2 corresponds to the treatment period. We begin our mathematical analysis of the model by establishing the existence, uniqueness, nonnegativity and boundedness of the solutions. We derive the basic reproductive number R0 for the model. Using LaSalle’s Invariance Principle, we determine the stability of the equilibrium points when the treatment success rate is equal to zero. We prove that if R0 < 1, then the disease-free equilibrium is globally asymptotically stable. If R0 > 1, then the disease-free equilibrium is unstable and a unique endemic equilibrium exists which is globally asymptotically stable. Numerical simulations are presented to illustrate the theoretical results. Keywords Tuberculosis · Reproductive number · Delay differential equation · Global stability · Lyapunov functional

1 Introduction Tuberculosis (TB) is an infectious disease caused by Mycobacterium tuberculosa (MTb). It is one of the top 10 causes of death worldwide. In 2017, 10 million people developed TB disease and 1.3 million died from the disease across the continents. J. M. R. Macalalag (B) Department of Mathematics, Caraga State University, Butuan City, Philippines e-mail: [email protected] J. M. R. Macalalag · E. P. De Lara-Tuprio · T. R. Y. Teng Department of Mathematics, Ateneo de Manila University, Quezon City, Philippines © Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_6

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Eight countries accounted for about 66% of the total cases worldwide with India leading the count, followed by China, Indonesia, Philippines, Pakistan, Nigeria, Bangladesh and South Africa [20]. This indicates that TB is one of the major health concerns in these countries, particularly in the Philippines. There are two TB-related conditions: (1) the latent TB infection in which the infected has no symptoms of the disease and cannot spread the bacteria to others, and (2) the active TB disease in which symptoms of the disease are apparent and the infected is capable of spreading the bacteria [2]. The disease transmission is person to person and is spread through air. The MTb is released into the air when the infectious individuals cough, sneeze, spit or talk. The probability of infection depends on the number of infectious droplet nuclei that remain suspended in the air and the duration of the exposure of a susceptible individual. Clinically, droplets of less than 5 µm diameter are already sufficient to transmit infection [11]. Hence, even a small amount of MTb inhaled can cause infection of another person. Infection may result to either latent TB infection or active TB disease. In 2017, the World Health Organization (WHO) reported that 23% of the world’s population has latent TB infection, and thus have a risk of developing active disease [20]. Some people develop TB disease soon after becoming infected (within weeks) before their immune system can fight the TB bacteria. However, some people may be infected but suppress the infection and develop symptoms years later. An individual with active TB disease may remain infectious for a long time (as long as viable MTb are present in sputum) and can remain infectious until they have undergone appropriate treatment which typically lasts for 6 to 12 months. A study in 2016 [16] revealed that TB bacteria may persist in the lungs even after patients have finished treatment and are free of clinical symptoms. Hence, the treated individuals may still be TB carriers and revert into the latent TB infection state after completing a treatment [15, 17]. Meanwhile, within a few weeks following the medication, the majority of bacteria are removed enabling the treated individuals to see resolution of symptoms and become incapable of spreading the disease [2]. The lengthy duration of treatment required for cure poses a primary problem [4]. Due to these and other related reasons, medication is often interrupted, ceased or done irregularly [14]. An observational study in 2014 [1] showed that most treatments of active TB patients are incomplete. In the case that an individual undergoing treatment does not take their TB drug medication properly, the bacteria become immune to the drug. This would make the disease difficult to cure and would often lead to treatment failure. Mathematical models can provide a useful tool in analyzing the spread of infection in a community. Compartmental models are the commonly used method to describe the spread dynamics of the disease. Several epidemiological models using ordinary differential equations for tuberculosis have been introduced and studied (see [6, 7, 10, 21, 23]). Because tuberculosis has a latency stage and this disease can be treated, the inclusion of exposed and treated compartments in an epidemic model for TB is considered by some authors (see [21, 22]). In 2015, Zhang and Feng [23] studied a Susceptible-Exposed-Infectous-Treated (SEIT) model incorporating the ideas of isolation and incomplete treatment. They concluded that increasing the treatment rate and rate of early detection of the disease will help control the spread of the disease in the community.

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In reality, most processes take time to complete. To study how these process times affect a certain system, we incorporate these into mathematical models through delay differential equations. In 2016, Li and Ma [9] incorporated a latency delay of tuberculosis in a Susceptible-Exposed-Infectious-Recovered (SEIR) model. They concluded that prolonging the latent state of the disease would eventually decrease the spread of tuberculosis in the community. In this paper, we study how the treatment duration of an individual will affect the spread dynamics of the disease in the community. We introduce in this paper an SEIT compartmental model for tuberculosis incorporating the latency and treatment period delays. We assume that the treated individual either becomes a susceptible individual if the patient has undergone a successful treatment, becomes a latent individual when most of the bacteria in the patient’s body are killed after treatment, or remains in active TB state when the treatment given for certain period of time is a failure. Moreover, while individuals undergoing treatment are still infected with the bacteria, we assume that these individuals are not infectious and cannot become infectious in the entire duration of treatment. The rest of the paper is organized as follows: in Sect. 2, we present the formulation of the mathematical model and discuss its well-posedness by showing the existence, uniqueness, nonnegativity and boundedness of the solutions under suitable initial conditions. We determine the equilibrium points of the model and the basic reproductive number R0 in Sect. 3. In Sect. 4, we use Lyapunov functional and LaSalle’s invariance principle to study the global stability of the equilibrium points when the treatment success rate is equal to zero. In Sect. 5, we perform numerical simulations to illustrate the theoretical results. Finally, we present some concluding remarks and suggestions for future works in Sect. 6.

2 The Mathematical Model The total population at time t, denoted by N (t), is divided into four compartments: the class of susceptible individuals (S), exposed or the class of latent TB infected individuals (E), infectious or the class of individuals with TB disease (I), and the class of treated individuals (T). Assume that all recruitment goes in the susceptible compartment and happens at a constant rate A. Denote the natural death rate by μ. Assume also that the number of individuals contacted by an infective per unit time, called the contact rate η of an infection, is proportional to the population size N , that is, η = k N for some k > 0. Hence, susceptible individuals are infected by individuals with active TB disease at a rate ηβN0 S I = β S I , where β = β0 k and β0 is the probability of a successful transmission of the disease when a contact occurs between a susceptible and an individual with active TB disease. In this case, a fraction c of these newly infected individuals enters the latent TB infection compartment, and the remaining fraction 1 − c moves to the active TB disease compartment. After a period of τ1 , the latent TB infected individuals develop a TB disease and will move to the infectious class.

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Fig. 1 Disease progression from the susceptible (S) compartment through the exposed (E), infectious (I) and treated (T) compartments

The infectious compartment has an additional death rate induced by the disease with constant rate coefficient . Individuals in the infectious compartment are treated with a constant rate coefficient δ. It is assumed that an individual from the treated compartment either moves back to the infectious compartment when the treatment given for a period of τ2 is a failure, to the exposed compartment when the treatment given for the period of τ2 kills most of the bacteria in the patient’s body, or to the susceptible compartment when all of the bacteria in the patient’s body are killed. Hence, the treated individuals after a period τ2 enter into the infectious compartment with a fraction m, into the latent compartment with a fraction d, and into the susceptible compartment with a fraction n so that n + d + m = 1. Also, it will be assumed that those individuals who reverted back to the latent class after being treated will no longer become infectious. Moreover, it is assumed that the individuals in the treated compartment are still infected, but are not infectious and cannot become infectious during the entire period of treatment. The above description of disease transmission (see Fig. 1) is described by the following system of differential equations S  (t) = A − μS(t) − β S(t)I (t) + nδe−μτ2 I (t − τ2 ) (1)  −μτ −μτ 1 2 E (t) = cβ S(t)I (t) − e cβ S(t − τ1 )I (t − τ1 ) − μE(t) + dδe I (t − τ2 ) (2) I  (t) = (1 − c) β S(t)I (t) + e−μτ1 cβ S(t − τ1 )I (t − τ1 ) + mδe−μτ2 I (t − τ2 ) − (μ + δ + ) I (t)

T  (t) = δ I (t) − μT (t) − e−μτ2 δ I (t − τ2 )

(3) (4)

where N (t) = S(t) + E(t) + I (t) + T (t), and S, E, I , and T are continuous realvalued functions on R. All the parameters are positive constants. Throughout this paper, the system of Eqs. (1)–(4) shall be referred to as system (SEIT). The expression e−μτ1 cβ S(t − τ1 )I (t − τ1 ) represents the introduction of infected individuals who survive the latent TB infection stage for a period τ1 . In this term, e−μτ1 represents the probability that an individual

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survives the latent period [t − τ1 , t]. The expression e−μτ2 δ I (t − τ2 ) above corresponds to the introduction of treated individuals who still live after a period τ2 of undergoing medical treatment. Here, e−μτ2 represents the probability that an individual still lives after the treatment period [t − τ2 , t]. We will now discuss the existence, uniqueness, nonnegativity and boundedness properties of the solutions to system (SEIT) under suitable initial condition. The nonnegativity implies that the human population under consideration will exist for a long time, while boundedness means that the population could not increase forever due to physical limitations brought about by many factors like the natural space and resources. By choosing a suitable initial condition, we express the implicit solution of the differential equations for E(t) and T (t) in the following lemmas: Lemma 1 The solution for E(t) of system (SEIT) is given by  E(t) =

t

cβ S(θ)I (θ)e

μ(θ−t)

 dθ +

t−τ1

when

 E(0) =

0

−τ1

t−τ2

−τ2

dδ I (θ)eμ(θ−t) dθ

cβ S(θ)I (θ)eμθ dθ.

Proof From (2), we have   eμt E  (t) + μ(E(t)) = eμt cβ S(t)I (t) − eμ(t−τ1 ) cβ S(t − τ1 )I (t − τ1 ) +eμ(t−τ2 ) dδ I (t − τ2 ).

(5)

Integrating both sides of (5) from 0 to t, we get eμt E(t) − E(0) =



t

cβ S(θ)I (θ)eμθ dθ +



t−τ1



−

dδ I (θ − τ2 )eμ(θ−τ2 ) dθ

0 0

−τ1

cβ S(θ)I (θ)eμ(θ) dθ.

Hence, if

 E(0) =

then the result follows.

t

0 −τ1

cβ S(θ)I (θ)eμ(θ) dθ, 

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Lemma2 The solution for T (t) of system (SEIT) t 0 T (t) = eμ(θ−t) δ I (θ)dθ when T (0) = eμθ δ I (θ)dθ.

is

given

by

−τ2

t−τ2

  For τ = max{τ1 , τ2 } > 0, let C = C [−τ , 0], R4 be the Banach space of con4  tinuous functions from [−τ , 0] into R4 with norm ||φ|| = ||φi ||max , where Proof The proof is similar to Lemma 1.



i=1

||φi ||max = max |φi (θ)|, for each φ = (φ1 , φ2 , φ3 , φ4 ) ∈ C. Define θ∈[−τ ,0]

C + = {φ ∈ C : φi (θ) ≥ 0, ∀i ∈ {1, 2, 3, 4}, ∀θ ∈ [−τ , 0]} . For any given continuous function X = (S, E, I, T ) : [−τ , ζ) → R4 with ζ > 0, define X t = (St , E t , It , Tt ) for each t ≥ 0 by X t (θ) = X (t + θ) = (S(t + θ), E(t + θ), I (t + θ), T (t + θ)) for all θ ∈ [−τ , 0] For system (SEIT), we will only consider solutions X = (S, E, I, T ) on [−τ , +∞) with initial condition φ = (φ1 , φ2 , φ3 , φ4 ) from the set 

+



Φ = φ ∈ C : φ1 (θ) > 0, φ2 (0) =

0

−τ1

 U (θ)dθ, φ4 (0) =

0

−τ2

 V (θ)dθ .

where U (θ) = cβφ1 (θ)φ3 (θ)eμθ and V (θ) = δφ3 (θ)eμθ . That is, for θ ∈ [−τ , 0], S(θ) = φ1 (θ) > 0, E(θ) = φ2 (θ) ≥ 0, I (θ) = φ3 (θ) ≥ 0, T (θ) = φ4 (θ) ≥ 0.

The next theorem follows immediately from the theory of retarded functional differential equations (see [5, 8]). Theorem 1 For each φ = (φ1 , φ2 , φ3 , φ4 ) ∈ Φ, the solution to system (SEIT) with initial condition φ exists and is unique. The following theorem shows that all solutions to system (SEIT) with initial conditions φ ∈ Φ remain nonnegative and bounded on [0, +∞). Theorem 2 Every solution to system (SEIT) with initial condition φ ∈ Φ is nonnegative and bounded on [0, +∞). Proof Let η = min{τ1 , τ2 }. Consider I (t) for t ∈ [0, η]. From (3), we have I  (t) ≥ (1 − c) β S(t)I (t) − I (t)(μ + δ + ).

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It then follows that 

t

I (t) ≥ I (0) exp

[(1 − c) β S(s) − (μ + δ + )] ds

0

for t ∈ [0, η]. Since I (0) ≥ 0, we then have I (t) ≥ 0 for all t ∈ [0, η]. Hence, extending these results into the intervals [η, 2η] , [2η, 3η] , . . . , we conclude that I (t) ≥ 0 for all t > 0. Consider S(t) for t ≥ 0. From (1), we have S  (t) > S(t) (−μ − β I (t)) , 

t

for which we get S(t) > S(0) exp

[−μ − β I (s)] ds . Since S(0) > 0, we con-

0

clude that S(t) > 0 for all t ≥ 0. Since I (t) ≥ 0 and S(t) > 0 for all t ≥ 0, then by Lemma 1, it follows that E(t) ≥ 0 for all t > 0. Similarly, by Lemma 2, we conclude that T (t) ≥ 0 for all t ≥ 0. To show that the solutions are bounded, consider t ≥ 0. By the above discussion, S(t) is positive and E(t), I (t), and T (t) are all nonnegative. Observe that N  (t) ≤ A A − μN (t). Thus, lim sup N (t) ≤ , that is, given ε0 > 0, there exists sufficiently μ t→∞ A large T¯ > 0 such that N (t) ≤ + ε0 for all t > T¯ . Take Δ = max{N (t) : 0 ≤ t ≤ μ   A T¯ }. Hence, N (t) ≤ max Δ, + ε0 for all t ≥ 0. Therefore, N (t) is bounded on μ [0, +∞) and the result follows.  Remark 1 If I (0) > 0, then I (t) > 0 for all t ≥ 0. This additional restriction together with S(0) > 0 will make S(t), E(t), I (t), and T (t) always positive on [0, +∞). Hereafter, we shall denote by Ω the space of all solutions to system (SEIT) with initial condition φ ∈ Φ.

3 The Basic Reproductive Number and Equilibrium Points

 0 System (SEIT) always has the disease-free equilibrium X = μA , 0, 0, 0 in Ω. Following the approach in [19], we compute the basic reproductive number R0 .

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Note that the infected compartments are E, I and T , and the only non-infected compartment is S. Let Z := (Z 1 , Z 2 , Z 3 ) = (E, I, T ) be the vector of variables for the infected compartments, and let Y := S. System (SEIT) can now be written as Z i (t) = gi (Z (t), Z (t − τ1 ), Z (t − τ2 ), Y (t), Y (t − τ1 ), Y (t − τ2 )), Y  (t) = h(Z (t), Z (t − τ1 ), Z (t − τ2 ), Y (t), Y (t − τ1 ), Y (t − τ2 )).

i ∈ {1, 2, 3}

For simplicity, let Z τ (t) = Z (t − τ ) and Y τ (t) = Y (t − τ ) for any τ > 0. Thus, the above system can be written as Z i (t) = Fi ((Z (t), Z τ1 (t), Z τ2 (t), Y (t), Y τ1 (t), Y τ2 (t)) − Vi ((Z (t), Z τ1 (t), Z τ2 (t), Y (t), Y τ1 (t), Y τ2 (t)), Y  (t) = h(Z (t), Z τ1 (t), Z τ2 (t), Y (t), Y τ1 (t), Y τ2 (t)). Note that Fi represents the rate of appearance of new infections in compartment i while Vi represents the remaining transitional terms in compartment i. Also, these Fi ’s constitute the transmission matrix F, and Vi ’s constitute the transitional matrix V. Accordingly, we have ⎡

⎤ cβ S I F = ⎣(1 − c)β S I + e−μτ1 cβ S(t − τ1 )I (t − τ1 )⎦ 0 ⎡ −μτ ⎤ e 1 cβ S(t − τ1 )I (t − τ1 ) + μE − dδe−μτ2 I (t − τ2 ) ⎦. −mδe−μτ2 I (t − τ2 ) + (μ + δ + ) I V=⎣ −δ I + μT + e−μτ2 δ I

and

0

The Jacobian matrices of F and V evaluated at X are, respectively, given by ⎡

0

⎢ F =⎢ ⎣0 0 Hence,

cβ A μ (1−c+e−μτ1 c)β A μ

0

⎤

⎡ ⎤ −μτ1 μ −dδe−μτ2 + e μcβ A 0 ⎥ ⎢ −μτ2 ⎥ 0⎥ 0⎦ . ⎦ and V = ⎣ 0 (μ + δ + ) − mδe 0 −δ + e−μτ2 δ μ 0 0

⎡

0

⎢ F V −1 = ⎢ ⎣0 0

cβ A μ((μ+δ+)−mδe−μτ2 ) (1−c+e−μτ1 c)β A μ((μ+δ+)−mδe−μτ2 )

0

0

0

⎤

⎥ 0⎥ ⎦.

The basic reproductive number R0 for system (SEIT) is the largest eigenvalue of the matrix F V −1 , that is,

A Tuberculosis Epidemic Model with Latent and Treatment Period Time Delays

R0 =

99

(1 − c + e−μτ1 c)β A . μ ((μ + δ + ) − mδe−μτ2 )

The dynamics of system (SEIT) is characterized by the quantity R0 . When ∗ R0 > 1, the model has another equilibrium point, the endemic equilibrium X = (S∗ , E ∗ , I∗ , T∗ ), where S∗ = E∗ = I∗ = = T∗ =

A (μ + δ + ) − mδe−μτ2 = , −μτ 1 c) β μR0 (1 − c + e   1  1 − e−μτ1 cβ S∗ I∗ + dδe−μτ2 I∗ , μ     β A (1 − c) + e−μτ1 c − μ (μ + δ + ) − mδe−μτ2 β [(μ + δ + ) − e−μτ2 δ [(1 − c + e−μτ1 c)n + m]]   μ (μ + δ + ) − mδe−μτ2 (R0 − 1) , and β [(μ + δ + ) − e−μτ2 δ [(1 − c + e−μτ1 c)n + m]]   1 · δ I∗ 1 − e−μτ2 . μ

(6)

(7) (8)

In epidemiology, the basic reproductive number indicates how many secondary cases a single infected individual will produce in an entirely susceptible population during its infectious period [12]. Thus, if the basic reproductive number R0 > 1, then an individual can infect more than one individual, implying the possibility of a disease outbreak which may also lead to the disease staying in the community. On the other hand, the disease will possibly die out in the community when R0 < 1. Note that R0 depends on the values of the model parameters. Hence, to achieve the ideal state where the disease eventually leaves the community, health authorities may focus on intervention strategies that will lead to values of certain parameters that yield R0 < 1. In the following remarks, we look at how the latent delay τ1 , the treatment period τ2 , and the transfer rate δ from the infectious state to the treatment state affect the value of R0 . Remark 2 For fixed parameters A, μ, δ, , β, c, d and τ2 , R0 is a decreasing function of τ1 . This is because

∂ R0 −β Ae−μτ1 c = < 0. ∂τ1 μ + δ +  − mδe−μτ2

This implies that a longer stay of the exposed individuals in the latent state results to a smaller value of R0 , eventually leading to a decrease in the rate of spread of tuberculosis in the community.

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Remark 3 For fixed parameters A, μ, δ, , β, c, d and τ1 , R0 is a decreasing function of τ2 . Again, observe that   −mδe−μτ2 β A 1 − c + e−μτ1 c ∂ R0 = 1, then system (SI) has a unique endemic equilibrium (S∗ , I∗ ) where μ S∗ and I∗ are given by (6) and (7), respectively. Denote by R0+ the set of nonnegative real numbers. For system (SI), we will consider solutions X¯ = (S, I ) on [−τ , +∞) with initial condition φ = (φ1 , φ3 ) from the set     Φ R = φ = (φ1 , φ3 ) ∈ C [−τ , 0], R20+ : φ1 (θ) > 0

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Furthermore, let Ω R be the space of all solutions to system (SI) with initial condition φ ∈ ΦR . It is important to note that for any φ = (φ1 , φ3 ) ∈ Φ R , there exists a unique solution to system (SI) with initial condition φ, which is nonnegative and bounded on [0, +∞). Due to the long medication period required in treating TB, most treatments of active TB patients are incomplete. Hence, the analysis of the model will be limited to the case when n = 0, that is, the medication given to the patient does not result to the complete removal of the bacteria in the patient’s body. Theorem 3 If R0 < 1 and n = 0, then the disease-free equilibrium (S0 , 0) of system (SI) is globally asymptotically stable in Ω R . Proof Define the functional V : Φ R → R by V (γ) = V1 (γ) + V2 (γ) + V3 (γ) + V4 (γ)

(11)

where   γ1 (0) + γ3 (0) V1 (γ) = (1 − c + e−μτ1 c) γ1 (0) − S0 − S0 ln S0  0  0 −μτ2 −μτ1 V2 (γ) = mδe γ3 (θ)dθ + e cβ γ1 (θ)γ3 (θ)dθ V3 (γ) = μe−μτ1 c



−τ2 0

−τ

−τ1

[γ1 (θ) − S0 ] dθ γ1 (θ) 2

  V4 (γ) = (1 − R0 ) μ + δ +  − mδe−μτ2



0

−τ

γ3 (θ)dθ

Certainly, V is continuous on Φ R . We can see that γ1 (0) − S0 − S0 ln γ1S(0) ≥ 0 for 0 which equality holds when γ1 (0) = S0 . Hence, V1 is nonnegative. Moreover, it easy to see that V2 and V3 are nonnegative. If R0 < 1, then V4 is nonnegative so that 0∗ 0∗ V (γ) ≥ 0. Furthermore, note that V (γ) = 0 if and only if γ = X R , where X R = 0 (S0∗ , I0∗ ) ∈ Φ R , the function identically equal to X R = (S0 , 0). Let X¯ = (S, I ) be a solution to system (SI) with initial condition φ = (φ1 , φ3 ) ∈ Φ R+ . Then X¯ t = (St , It ) ∈ Φ R for t ≥ 0, with X¯ 0 = φ, and V ( X¯ t ) = V1 ( X¯ t ) + V2 ( X¯ t ) + V3 ( X¯ t ) + V4 ( X¯ t )

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where   S(t) + I (t) V1 ( X¯ t ) = (1 − c + e−μτ1 c) S(t) − S0 − S0 ln S0  t  t V2 ( X¯ t ) = mδe−μτ2 I (θ)dθ + e−μτ1 cβ S(θ)I (θ)dθ V3 ( X¯ t ) = μe−μτ1 c



t−τ2 t

t−τ

t−τ1

[S(θ) − S0 ] dθ S(θ) 2

  V4 ( X¯ t ) = (1 − R0 ) μ + δ +  − mδe−μτ2



t

I (θ)dθ

t−τ

We now get the derivative of V along the solutions of system (SI) with initial condition φ ∈ Φ R . Considered as a function of t ∈ [0, +∞), we have  d S0 V1 ( X¯ t ) = (1 − c + e−μτ1 c) 1 − [A − μS(t) − β S(t)I (t)] + I  (t). dt S(t) (12) By substituting μS0 to A in (12), we have d (1 − c + e−μτ1 c)μ (S(t) − S0 )2 V1 ( X¯ t ) = − − (1 − c + e−μτ1 c)β S(t)I (t) dt S(t) +(1 − c + e−μτ1 c)β S0 I (t) + I  (t). Similarly, we have d V2 ( X¯ t ) = mδe−μτ2 [I (t) − I (t − τ2 )] dt + e−μτ1 cβ [S(t)I (t) − S(t − τ1 )I (t − τ1 )] ,   d [S(t) − S0 ]2 [S(t − τ ) − S0 ]2 −μτ1 ¯ V3 ( X t ) = μe − c , dt S(t) S(t − τ )   d V4 ( X¯ t ) = (1 − R0 ) μ + δ +  − mδe−μτ2 [I (t) − I (t − τ )] . dt Accordingly, we have μe−μτ1 c (S(t − τ ) − S0 )2 d (1 − c)μ (S(t) − S0 )2 V ( X¯ t ) = − − dt S(t) S(t − τ ) −(μ + δ +  − mδe−μτ2 )[1 − R0 ]I (t − τ ).

(13)

d Thus, for t ≥ 0, V  ( X¯ t ) = V ( X¯ t ) ≤ 0 if R0 < 1. dt Note that in (13), V  ( X¯ t ) = 0 if S(t) = S(t − τ ) = S0 and I (t − τ ) = 0. Conversely, if V  ( X¯ t ) = 0, then

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(μ + δ +  − mδe−μτ2 )[1 − R0 ]I (t − τ )S(t)S(t − τ ) = −[S(t) − S0 ]2 S(t − τ )(1 − c)μ − μe−μτ1 c[S(t − τ ) − S0 ]2 S(t). (14) Observe that the left-hand side expression of (14) is always nonnegative when R0 < 1 while its right-hand side expression is always nonpositive. This means that (14) holds only when − [S(t) − S0 ]2 S(t − τ )(1 − c)μ − μe−μτ1 c[S(t − τ ) − S0 ]2 S(t) = 0

(15)

[μ + δ +  − mδe−μτ2 ][1 − R0 ]I (t − τ )S(t)S(t − τ ) = 0.

(16)

and

Hence, (15) and (16) imply that S(t) = S(t − τ ) = S0 and I (t − τ ) = 0. Therefore, for all t ≥ 0, V  ( X¯ t ) = 0 if and only if S(t) = S(t − τ ) = S0 and I (t − τ ) = 0. Since V  ( X¯ t ) ≤ 0 for t ≥ 0, the function t → V ( X¯ t ), with X¯ 0 = φ, is nonin0∗ creasing on [0, +∞). Assume that φ = X R so that  := V (φ) > 0. Define G = {ψ ∈ Φ R : V (ψ) ≤ }. Then G is closed and contains X¯ t for all t ≥ 0. Furthermore, X¯ t is bounded on [−τ , 0] for all t ≥ 0. Now, if ψ ∈ G and Xˆ is the solution to system (SI) with initial condition ψ, that is, Xˆ0 = ψ, then, following a similar computation as above, we can see that V  ( Xˆ t ) ≤ 0 for all t ≥ 0. In particular, V  (ψ) ≤ 0. Moreover, for each t ≥ 0, Xˆ t is bounded and Xˆ t ∈ G . Let M be the largest invariant subset of A = {ψ ∈ cl(G) : V  (ψ) = 0}. Observe that V  (ψ) = 0 if and only if ψ1 (0) = ψ1 (−τ ) = S0 and ψ3 (−τ ) = 0. So A = {ψ ∈ cl(G) : ψ = (ψ1 , ψ3 ) where ψ1 (0) = ψ1 (−τ ) = S0 and ψ3 (−τ ) = 0}. Since M is invariant, then for each ψ ∈ M, the corresponding solution Xˆ (t) satisfies Xˆ t ∈ M ⊂ A for t ≥ 0. Hence, Xˆ t (−τ ) = Xˆ (t − τ ) = (S0 , 0) for t ≥ 0. So for t ∈ [0, τ ], we 0∗ 0∗ then get ψ = X R . Accordingly, M = {X R } = {(S0∗ , I0∗ )}. Hence, if ψ ∈ G and X is a solution to system (SI) with initial condition ψ, that is, X 0 = ψ, then by LaSalle’s Invariance Principle [8], X t converges to (S0∗ , I0∗ ) as t → +∞. Accordingly, X (t) converges to (S0 , 0) as t → +∞. By the construction of G, since φ is arbitrary, we conclude that the disease-free equilibrium (S0 , 0) is  globally asymptotically stable in Ω R . Corollary 1 If R0 < 1 and n = 0, then the disease-free equilibrium (S0 , 0, 0, 0) of system (SEIT) is globally asymptotically stable in Ω. Proof Let (S, E, I, T ) be a solution to system (SEIT) with initial condition φ ∈ Φ. Since R0 < 1, by Theorem 3, we have lim S(t) = S0 and lim I (t) = 0. By t→+∞

t→+∞

Lemmas 1 and 2, respectively, we have  lim E(t) = lim

t→+∞

t→+∞

= 0,

I (t − τ2 )e−μτ2 S(t)I (t) − S(t − τ1 )I (t − τ1 )e−μτ1 + dδ cβ μ μ



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 and

lim T (t) = δ lim

t→+∞

t→+∞

I (t) − I (t − τ2 )e−μτ2 μ

 = 0. This completes the

proof.



Theorem 4 If R0 > 1 and n = 0, then the disease-free equilibrium (S0 , 0) of system (S I ) is unstable in Ω R . Proof We linearize system (SI) at the disease-free equilibrium (S0 , 0), obtain the corresponding Jacobian matrix J at (S0 , 0), and calculate the matrix J ((S0 , 0)) − λI as   −μ − λ − βμA   0 −(μ + δ + ) + βμA 1 − c + e−τ1 (λ+μ) c + mδe−τ2 (λ+μ) − λ where λ is the eigenvalue and I is the 2 × 2 identity matrix. Accordingly, the characteristic equation at the disease-free equilibrium (S0 , 0) is given by 0 = det (J ((S0 , 0)) − λI )   βA  1 − c + e−τ1 (λ+μ) c − mδe−τ2 (λ+μ) . = (λ + μ) λ + μ + δ +  − μ (17) We prove that (S0 , 0) is unstable by showing that (17) has a positive root. Certainly, λ = −μ is a root and the other roots are determined by the equation λ=

 βA  1 − c + e−τ1 (λ+μ) c + mδe−τ2 (λ+μ) − (μ + δ + ). μ

(18)

Denote the left hand-side expression of (18) as P(λ) and the right-hand side as Q(λ). Suppose that λ is real. Note that P(0) = 0 and lim P(λ) = +∞. Furthermore, λ→+∞

observe that Q(λ) is a decreasing function of λ and

  Q(0) = (μ + δ + ) − mδe−μτ2 [R0 − 1] > 0, since R0 > 1. Consequently, the functions P and Q must intersect for some λ0 > 0. Thus, (17) has a positive root and the result follows.  Corollary 2 If R0 > 1 and n = 0, then the disease-free equilibrium (S0 , 0, 0, 0) of system (S E I T ) is unstable in Ω. Proof The result follows directly from Theorem 4.



Let Φ+ and Φ R+ be the sets of initial conditions for system (SEIT) and system (SI), respectively, such that for φ ∈ Φ and ψ ∈ Φ R , we have φ(θ) > 0 and ψ(θ) > 0. Denote further by Ω+ and Ω R+ the space of all positive solutions to system (SEIT) and system (SI) with initial conditions in Φ+ and Φ R+ , respectively.

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Theorem 5 If R0 > 1 and n = 0, then the endemic equilibrium (S∗ , I∗ ) of system (SI) is globally asymptotically stable in Ω R+ . Proof Define the functional V : Φ R+ → R by V (γ) = V1 (γ) + V2 (γ) + V3 (γ) + V4 (γ) + V5 (γ)

(19)

where V1 (γ) V2 (γ) V3 (γ) V4 (γ) V5 (γ)

 γ1 (0) = (1 − c + e c) γ1 (0) − S∗ − S∗ ln S∗ γ3 (0) = γ3 (0) − I∗ − I∗ ln I∗  0  γ3 (θ) −μτ2 γ3 (θ) − I∗ − I∗ ln dθ = mδe I∗ −τ2   0 γ1 (θ)γ3 (θ) −μτ1 γ1 (θ)γ3 (θ) − S∗ I∗ − S∗ I∗ ln dθ =e cβ S∗ I∗ −τ1  0 S∗ S∗ − 1 − ln dθ, = e−μτ1 cβ S∗ I∗ γ1 (θ) γ1 (θ) −τ −μτ1



Note that V is continuous and nonnegative on Φ R+ . Also, V (γ) = 0 if and only ∗∗ ∗∗ ∗ if γ = X R , where X R = (S∗∗ , I∗∗ ) ∈ Φ R+ , the function identically equal to X R = (S∗ , I∗ ). Let X¯ = (S, I ) be a solution to system (SI) with initial condition φ = (φ1 , φ3 ) ∈ Φ R+ . Then X¯ t = (St , It ) ∈ Φ R+ for t ≥ 0, where X¯ 0 = φ, and V ( X¯ t ) = V1 ( X¯ t ) + V2 ( X¯ t ) + V3 ( X¯ t ) + V4 ( X¯ t ) + V5 ( X¯ t ) where   S(t) V1 ( X¯ t ) = (1 − c + e−μτ1 c) S(t) − S∗ − S∗ ln S∗ I (t) V2 ( X¯ t ) = I (t) − I∗ − I∗ ln I∗  t  I (θ) I (θ) − I∗ − I∗ ln dθ V3 ( X¯ t ) = mδe−μτ2 I∗ t−τ2  t  S(θ)I (θ) S(θ)I (θ) − S∗ I∗ − S∗ I∗ ln dθ V4 ( X¯ t ) = e−μτ1 cβ S∗ I∗ t−τ1  t  S∗ S∗ − 1 − ln dθ, V5 ( X¯ t ) = e−μτ1 cβ S∗ I∗ S(θ) S(θ) t−τ We determine the derivative of V along the positive solutions of system (SI) with φ ∈ Φ R+ . Considered as a function of t, we have

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 d S∗ V1 ( X¯ t ) = (1 − c + e−μτ1 c) 1 − [A − μS(t) − β S(t)I (t)] . (20) dt S(t) By substituting μS∗ + β S∗ I∗ to A in (20), we have  S∗ d (−μ(S(t) − S∗ ) + β S∗ I∗ ) V1 ( X¯ t ) = (1 − c + e−μτ1 c) 1 − dt S(t) − (1 − c + e−μτ1 c)β S(t)I (t) + (1 − c + e−μτ1 c)β S∗ I (t).  d I∗ ¯ I  (t) = V20 − V21 , where Also, V2 ( X t ) = 1 − dt I (t) V20 = (1 − c) β S(t)I (t) + e−μτ1 cβ S(t − τ1 )I (t − τ1 ) + mδe−μτ2 I (t − τ2 ) −(μ + δ + )I (t) V21 = (1 − c)β S(t)I∗ +

mδe−μτ2 I (t − τ2 )I∗ e−μτ1 cβ S(t − τ1 )I (t − τ1 )I∗ + I (t) I (t)

−(μ + δ + )I∗ ,   d I (t − τ2 ) V3 ( X¯t ) = mδe−μτ2 I (t) − I (t − τ2 ) + I∗ ln , dt I (t)   d S(t − τ1 )I (t − τ1 ) V4 ( X¯t ) = e−μτ1 cβ S(t)I (t) − S(t − τ1 )I (t − τ1 ) + S∗ I∗ ln , dt S(t)I (t)    d S∗ S∗ S∗ S∗ V ( X¯t ) = e−μτ1 cβ S∗ I∗ − 1 − ln − − 1 − ln . dt 5 S(t) S(t) S(t − τ ) S(t − τ )

Accordingly, we get   d (S(t) − S∗ )2 S∗ V ( X¯t ) = − μ(1 − c) + (1 − c)β S∗ I∗ − (1 − c) β S∗ I∗ dt S(t) S(t)   + (1 − c + e−μτ1 c)β S∗ I (t) − (μ + δ + )I (t) + mδe−μτ2 I (t) − (1 − c)β S(t)I∗ −

e−μτ1 cβ S(t − τ1 )I (t − τ1 ) I∗ I (t)

mδe−μτ2 I (t − τ2 ) I∗ + (μ + δ + )I∗ I (t) I (t − τ2 ) S(t − τ1 )I (t − τ1 ) + e−μτ1 cβ S∗ I∗ ln + mδe−μτ2 I∗ ln I (t) S(t)I (t)   2 (S(t) − S∗ ) S∗ − μe−μτ1 c + e−μτ1 cβ S∗ I∗ − e−μτ1 cβ S∗ I∗ S(t) S(t)  S∗ S∗ + e−μτ1 cβ S∗ I∗ − 1 − ln S(t) S(t)  S∗ S ∗ − 1 − ln . − e−μτ1 cβ S∗ I∗ S(t − τ ) S(t − τ ) −

(21)

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From (10), we have   (μ + δ + )I∗ = (1 − c)β S∗ + e−μτ1 cβ S∗ + mδe−μτ2 I∗ .

(22)

Note also that ln

S(t − τ1 )I (t − τ1 ) S(t − τ1 )I (t − τ1 ) S∗ = ln + ln . S(t)I (t) S∗ I (t) S(t)

(23)

Using (22) and (23) and regrouping terms in (21), we have  (S(t) − S∗ )2 S∗ d ¯ V ( X t ) = − μ(1 − c) + (1 − c)β S∗ I∗ − (1 − c) β S∗ I∗ dt S(t) S(t)   + (1 − c + e−μτ1 c)β S∗ I (t) − (μ + δ + )I (t) + mδe−μτ2 I (t) − (1 − c)β S(t)I∗ −

e−μτ1 cβ S(t − τ1 )I (t − τ1 ) I∗ I (t)

mδe−μτ2 I (t − τ2 ) I∗ + (1 − c)β S∗ I∗ + e−μτ1 cβ S∗ I∗ I (t) I (t − τ2 ) + mδe−μτ2 I∗ + mδe−μτ2 I∗ ln I (t) )I (t − τ1 ) S(t − τ S∗ 1 + e−μτ1 cβ S∗ I∗ ln + e−μτ1 cβ S∗ I∗ ln S∗ I (t) S(t)  S∗ (S(t) − S∗ )2 −μτ1 + e−μτ1 cβ S∗ I∗ − e−μτ1 cβ S∗ I∗ c − μe S(t) S(t)  S∗ S∗ −μτ1 − 1 − ln +e cβ S∗ I∗ S(t) S(t)  S∗ S∗ −μτ1 − 1 − ln −e cβ S∗ I∗ S(t − τ ) S(t − τ )  2 (S(t) − S∗ ) = −μ(1 − c + e−μτ1 c) S(t)   S∗ S(t) − + (1 − c)β S∗ I∗ 2 − S(t) S∗   S(t − τ1 )I (t − τ1 ) S(t − τ )I (t − τ1 ) 1 −μτ1 − 1 − ln −e cβ S∗ I∗ S∗ I (t) S∗ I (t)   I (t − τ I (t − τ ) ) 2 2 − 1 − ln − mδe−μτ2 I∗ I (t) I (t)  S S∗ ∗ −μτ1 − 1 − ln (24) −e cβ S∗ I∗ S(t − τ ) S(t − τ ) −

d Thus, for t ≥ 0, V  ( X¯ t ) = V ( X¯ t ) ≤ 0 if R0 > 1. dt

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In (24), for t ≥ 0, V  ( X¯ t ) = 0 if and only if S(t) = S(t − τ ) = S∗ I (t − τ2 ) S(t − τ1 )I (t − τ1 ) = = 1. Then for t ≥ 0, S(t − τ1 ) = S∗ since and S∗ I (t) I (t) τ = max{τ1 , τ2 }. Hence, I (t) = I (t − τ1 ) = I (t − τ2 ) for t ≥ 0. From (10), we can show that   I  (t) = (1 − c + e−μτ1 c)β S∗ + mδe−μτ2 − (μ + δ + ) I (t) = 0. This implies that I (t) is constant. Since S  (t) = 0, then from (9), we have 0 = A − μS∗ − β S∗ I (t) which further implies that

 A A − μ μR0 A − μS∗

 = I (t) = = β S∗ β μRA 0

A(R0 −1) R0 βA μR0

=

μ(R0 − 1) = I∗ β

for all t ≥ 0. Thus, for all t ≥ 0, V  ( X¯ t ) = 0 if and only if S(t) = S(t − τ ) = S∗ and I (t) = I (t − τ ) = I∗ . Following a similar argument as in the proof of Theorem 3, by Lasalle’s Invariance Principle [8], we conclude that the endemic equilibrium (S∗ , I∗ ) is globally  asymptotically stable in Ω R+ . Corollary 3 If R0 > 1 and n = 0, then the endemic equilibrium (S∗ , E ∗ , I∗ , T∗ ) of system (SEIT) is globally asymptotically stable in Ω+ . Proof The proof is similar to Corollary 1.



5 Numerical Simulations In this section, we study system (SEIT) in Ω numerically to illustrate the results obtained in Sect. 4. We perform numerical simulations using MATLAB DDE23. Here, we only take constant functions as initial conditions. Simulation 1 Consider the parameter values in Table 1 with n = 0, A = 15, δ = and τ2 = 43 . Hence, R0 = 0.9938 and the disease0.95, c = 0.9, d = 0.66, τ1 = 35 6 free equilibrium is (750, 0, 0, 0). Also, we take the following initial conditions: (a) (φ1 , φ2 , φ3 , φ4 ) = (100, 76, 40, 290); (b) (φ1 , φ2 , φ3 , φ4 ) = (1500, 1766, 600, 4700) ; (c) (φ1 , φ2 , φ3 , φ4 ) = (3000, 4000, 550, 500); and (d) (φ1 , φ2 , φ3 , φ4 ) = (20500, 5700, 100, 5000). Figure 2 shows that for different initial conditions, the trajectories of the solutions converge to the (750, 0, 0, 0). This is consistent with our result that the disease-free equilbrium is globally asymptotically stable in Φ when R0 < 1. Simulation 2 Consider the same parameter values as in Simulation 1 except that we decrease the values of τ1 = 35 to τ1 = 21 , τ2 = 34 to τ2 = 21 , and δ = 0.95 to 6 δ = 0.2. Hence, R0 = 2.299 and the disease-free equilibrium is still (750, 0, 0, 0).

A Tuberculosis Epidemic Model with Latent and Treatment Period Time Delays Table 1 Parameter values of Tuberculosis transmission model Parameter Description Value μ  δ β A c n d m τ1 τ2

Natural death rate TB disease death rate Treatment rate Transmission coefficient Recruitment rate Fraction of newly infected individuals who enter the latent class Fraction of leaving treated individuals who enter the susceptible class Fraction of leaving treated individuals who enter the latent class Fraction of leaving treated individuals who enter the infectious class Latency delay Treatment period

109

Unit

Taken from

Year−1 Year−1 Year−1 Year−1 Year−1 Year−1

[3] [3] Assumed [21] Assumed Assumed

n ∈ [0, 1] Year−1

Assumed

d ∈ [0, 1] Year−1

Assumed

Year−1

Assumed

Year Year

Assumed Assumed

0.02 0.3 δ ∈ (0, 1) 0.0014 Variable c ∈ (0, 1)

m∈ [0, 1] Variable Variable

Fig. 2 (Simulation 1) The disease-free equilibrium is asymptotically stable if R0 < 1 and n = 0

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Fig. 3 (Simulation 2) The disease-free equilibrium becomes unstable and the endemic equilibrium is asymptotically stable if R0 > 1 and n = 0

By choosing 100 different initial conditions (φ1 , φ2 , φ3 , φ4 ), we see that in Fig. 3 that the trajectories of the solutions do not converge to (750, 0, 0, 0). This shows that the disease-free equilbrium becomes unstable in Φ when R0 > 1. Remark 5 In Simulation 2, we obtained R0 = 2.299 and the endemic equilibrium (S∗ , E ∗ , I∗ , T∗ ) = (326.3, 125.0, 18.5, 1.8) now exists. Observe that each of the solution curves of S, E, I, and T converges to S∗ , E ∗ , I∗ and T∗ , respectively, as shown in Fig. 3. This is consistent with our result that the endemic equilibrium is globally asymptotically stable whenever R0 > 1. Simulation 3 Consider the parameter values in Table 1 with n = 0, A = 50, δ = 0.5, c = 0.9, d = 0.66, τ1 = 21 and τ2 = 43 . Hence, R0 = 5.316 and the diseasefree equilibrium is (2500, 0, 0, 0) . Also, we now have the endemic equilibrium (470.3, 1020.3, 61.6, 22.9). Consider the following initial conditions: (a) (φ1 , φ2 , φ3 , φ4 ) = (50, 26, 10, 240); (b) (φ1 , φ2 , φ3 , φ4 ) = (100, 76, 40, 290); (c) (φ1 , φ2 , φ3 , φ4 ) = (1500, 1766, 600, 4000); and (d) (φ1 , φ2 , φ3 , φ4 ) = (2000, 150, 200, 1750). Figure 4 shows that for the above initial conditions, the trajectories of the solutions converge to (470.3, 1020.3, 61.6, 22.9). This is again consistent with the result that the endemic equilibrium is globally asymptotically stable whenever it exists in Ω+ .

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Fig. 4 (Simulation 3) The endemic equilibrium is globally asymptotically stable if R0 > 1 and n=0

Even though the previous analyses of the model were only limited to the case when n = 0, it is also interesting to visualize what happens to system (SEIT) when n = 0 as shown below. Simulation 4 Consider the parameter values in Table 1 with n = 0.1, A = 10, τ1 = 10, τ2 = 1, δ = 0.95, c = 0.1, d = 0.35, and m = 0.55. Hence, R0 = 0.907 and the disease-free equilibrium is (S0 , E 0 , I0 , T0 ) = (500, 0, 0, 0). By choosing 100 different initial conditions (φ1 , φ2 , φ3 , φ4 ), we see that each of the solution curves of S, E, I, and T converges to S0 , E 0 , I0 and T0 , respectively, as shown in Fig. 5. Simulation 5 Consider the parameter values in Table 1 with n = 0.35, A = 15, τ1 = 21 , τ2 = 21 , δ = 0.2, c = 0.9, d = 0.5, and m = 0.15. Hence, R0 = 2.122 and the endemic equilibrium is (S∗ , E ∗ , I∗ , T∗ ) = (353.4, 96.4, 18.6, 1.8). Again, by choosing 100 different initial conditions (φ1 , φ2 , φ3 , φ4 ), we see that each of the solution curves of S, E, I, and T converges to S∗ , E ∗ , I∗ and T∗ , respectively, as illustrated in Fig. 6.

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Fig. 5 (Simulation 4) The disease-free equilibrium is asymptotically stable if R0 < 1 and n = 0

6 Concluding Remarks and Further Studies Time delays are unavoidable in many engineering and biological processes ([8, 18]), and these can also be observed in epidemic models of infectious diseases. In this paper, we considered an SEIT model for the transmission of tuberculosis, where the population of humans is completely described by the interaction of susceptible, exposed (with latent TB infection), infectious (with active TB disease) and treated individuals. There are two time delays that are incorporated in the model: one represents the length of time for an individual exposed to the bacteria to become infectious, and the other delay corresponds to the treatment period. We then studied how these delays influence the dynamics of the entire system. The existence and uniqueness, nonnegativity and boundedness of the solutions were shown in Sect. 2. In Sect. 3, we defined the basic reproductive number R0 and showed that this value plays a vital role in the dynamics of disease transmission in the community. We studied the relationship between R0 and the parameters τ1 , τ2 and δ. In particular, we showed that R0 is a decreasing function of τ1 , τ2 and δ, which may be helpful in determining possible avenues of intervention to reduce the rate of spread of the disease in the community. In Sect. 4, when the treatment success rate is equal to zero, we established the global stability of the equilibrium points through Lyapunov functionals

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Fig. 6 (Simulation 5) The endemic equilibrium is asymptotically stable if R0 > 1 and n = 0

and the LaSalle’s Invariance Principle. If R0 < 1, then all solutions converge to the disease-free equilibrium, which implies that the disease will eventually be eliminated from the community. If R0 > 1, then all positive solutions converge to the endemic equilibrium, which implies that the disease will stay in the community. Numerical simulations to illustrate the theoretical results were presented in Sect. 5. This paper confirms the conclusion in [9] that extending the latent period of tuberculosis can reduce the rate of spread of the disease in the community. Furthermore, we have shown not only that increasing the rate of undergoing treatment (δ) will help control the spread of the disease [23], but also that increasing the length of stay (τ2 ) in the treatment phase can slow down the spread of disease in the community. The stability of the equilibrium points is studied under the assumption that there is no chance of successfully curing an infected individual from the disease (n = 0). From the numerical simulations when n = 0, it was shown that the solutions converge to the disease-free equilibrium when R0 < 1, and the solutions converge to the endemic equilibrium when R0 > 1. Hence, it is also interesting to analyze qualitatively the global stability of the equilibrium points when this particular assumption is relaxed. We leave this as a future work. Furthermore, tuberculosis may be transmitted from a mother with active TB disease to her child through birth [13]. Hence, introducing vertical transmission in the model is a possible extension of this work.

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References 1. Babiarz, K.S., Suen, S.C., Goldhaber-Fiebert, J.D.: Tuberculosis treatment discontinuation and symptom persistence: an observational study of Bihar, India’s public care system covering >100,000,000 inhabitants. BMC Public Health 14(418) (2014). https://doi.org/10.1186/14712458-14-418 2. Centers for Disease Control and Prevention (2018). Tuberculosis. https://www.cdc.gov/tb/ topic/treatment/tbdisease.htm 3. Colijn, C., Cohen, T., Murray, M.: Emergent heterogeneity in declining tuberculosis epidemics. J. Theor. Biol. 247, 765–774 (2007). https://doi.org/10.1016/j.jtbi.2007.04.015 4. Connolly, L.E., Edelstein, P.H., Ramakrishnan, L.: Why is long-term therapy required to cure tuberculosis? PLoS Med. 4(3), e120 (2007). https://doi.org/10.1371/journal.pmed.0040120 5. Driver, R.: Ordinary and Delay Differential Equations. Springer, New York (1977) 6. Feng, Z., Castillo-Chavez, C., Capurro, A.F.: A model for Tuberculosis with Exogenous Reinfection. Theor. Popul. Biol. 57, 235–247 (2000). https://doi.org/10.1006/tpbi.2000.1451 7. Feng, Z., Iannelli, M., Milner, F.: A two-strain tuberculosis model with age of infection. SIAM J. Appl. Math. 62, 1634–1656 (2002). https://doi.org/10.1137/S003613990038205X 8. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press Inc., California (1993) 9. Li, J., Ma, M.: Impact of prevention in a tuberculosis model with latent delay. Adv. Diff. Equ. (2016). https://doi.org/10.1186/s13662-016-0934-z 10. Liu, L.J., Zhang, T.L.: Global stability for a tuberculosis model. Math. Comput. Model. 54, 836–845 (2011). https://doi.org/10.1016/j.mcm.2011.03.033 11. Madkour, M.M.: Tuberculosis. Springer, Berlin, Heidelberg (2004) 12. Martcheva, M.: An Introduction to Mathematical Epidemiology. Springer, US (2015) 13. Mittal, H., Das, S., Faridi, M.M.: Management of newborn infant born to mother suffering from tuberculosis: current recommendations and gaps in knowledge. Indian J. Med. Res. 140(1), 32– 9 (2014) 14. Mittal, C.C., Gupta, S.: Noncompliance to DOTS: how it can be decreased. Indian J. Commun. Med. 36, 27–30 (2011). https://doi.org/10.4103/0970-0218.80789 15. Neyrolles, O., Hernandez-Pando, R., Pietri-Rouxel, F., et al.: Is Adipose Tissue a Place for Mycobacterium tuberculosis Persistence? PLOS One 1(1), e43 (2016). https://doi.org/10.1371/ journal.pone.0000043 16. NIH/National Institute of Allergy and Infectious Diseases: TB can persist in lungs despite treatment, researchers find. Science Daily. http://www.sciencedaily.com/releases/2016/09/ 160906131454.htm 17. Public Library of Science: Tuberculosis Bacillus hides from immune system in host’s fat cells. ScienceDaily. ScienceDaily, 24 December (2006). https://www.sciencedaily.com/releases/ 2006/12/061221074735.htm 18. Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer (2011) 19. Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002). https://doi.org/10.1016/S0025-5564(02)00108-6 20. World Health Organization: Global Tuberculosis Report 2018. World Health Organization, Geneva (2018) 21. Xu, Y., Huang, G., Zhao, Z.: Dynamics of a tuberculosis model with treatment and selfcure. In: Tan, H. (Eds.) Informatics in Control, Automation and Robotics, Lecture Notes in Electrical Engineering, Springer, Berlin, Heidelberg 132 (2015) https://doi.org/10.1007/9783-642-25899-2_66.

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Numerical Bifurcation and Stability Analyses of Partial Differential Equations with Applications to Competitive System in Ecology Mohd Hafiz Mohd

Abstract Bifurcation analysis is a powerful technique for investigating the dynamical behaviours of nonlinear systems. While this approach has been employed extensively in analysing ordinary-differential equations and other deterministic models, the use of bifurcation analysis in studying the dynamics of partial differential equations (PDE) is yet limited. This chapter illustrates the progress on how numerical bifurcation and stability analyses can be used in understanding the overall dynamics of a PDE system under consideration. By considering an ecological example of competitive system with environmental suitability and spatial diffusion terms, distinct behaviours of the model e.g. alternative stable states, multi-species coexistence and extinction phenomena are demonstrated as interspecific competition and dispersal strength change. Further investigation reveals the existence of several threshold values in ecologically-relevant parameters corresponding to distinct bifurcations (e.g. saddle-node and transcritical), which lead to different stability properties of PDE solution branches. Keywords Partial differential equations · Numerical bifurcation · Stability analysis · Ecological modelling

1 Introduction Whether it is down to climate change, invasive species problem or environmental disturbance, species ecosystems across the globe are changing. When this situation happens, some species leave their original habitats in search of a more suitable place where they can live. This process is called dispersal [1]. Apart from dispersal, different ecological forces have been demonstrated to shape species distributions across a geographical region: environmental components (i.e. abiotic factors) [2–5], M. H. Mohd (B) School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia e-mail: [email protected] URL: http://math.usm.my/hafizmohd © Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_7

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and species interactions (i.e. biotic factors such as competition) [6–8]. While the influences of competition and abiotic environments are common subjects in the literature [9–13], majority of these ecological studies concentrate on simple two-species systems like in the classical work of Lotka and Volterra [14, 15]. However, real ecosystems in nature consist of complex interactions between multiple interacting species. To better understand the joint effects species can have on one another while living in certain environments, it is crucial to consider an ecological system with more than two-interacting species to mimic the reality. Plus, the interplay of competitive interactions and dispersal process in a multi-species ecosystem with environmental heterogeneity is a complex subject (e.g. refer to [16, 17] for reviews) that needs further studies. Motivated by these observations, my approach in investigating these ecological issues is by employing a Lotka-Volterra type model consisting of multi-species competitive interactions, which remains biologically relevant and also interesting. To investigate the influences of local dispersal, I incorporate a simple diffusion term and this inclusion leads to a partial differential equations (PDE). I also add a spatial variation in carrying capacity to model changing environments, which can affect the distributions of species. By employing this competitive system, there is a twofold aim of this study. The first is to discuss some techniques in performing bifurcation analysis to examine the dynamics of a PDE system under consideration. Also, the stability analysis of this model is investigated. The second aim is to better understand how these theoretical findings relate directly to the ecological systems; consistent with this aim, I examine the important roles of biotic interactions, local dispersal and environmental components on multi-species community assemblies. To achieve these aims, I address these ecological problems by extending the deterministic systems discussed in [7, 18, 19] to model competition, environmental factors and local dispersal in multi-species ecosystems. By investigating the findings of the models under various dispersal and competition intensities, I examine the combined influences of local dispersal and species interactions on community dynamics across spatially-varying environments. Additionally, I also employ different methodologies to solve my PDE model with competitive interactions so as to study the combined influences of biotic interactions and dispersal on species biodiversity; to do this, the stability properties of different branches of PDE solutions are analysed using the methods from bifurcation theory and dynamical systems. By considering several dispersal scenarios, I construct some bifurcation diagrams of solution behaviour as the strength of competition is varied using numerical bifurcation analysis in XPPAUT package [20]. This analysis provides a comprehensive way for examining bifurcational changes in dynamics and illustrating different outcomes of species interactions. Additionally, bifurcation analysis has the capability of demonstrating the qualitative behaviours and revealing the long-term distributions of the ecological systems. It is also useful for examining the overall dynamics of the systems under consideration and for better comprehending of certain emergent phenomena such as ecological tipping (or bifurcation) points, multi-species coexistence and exclusion of species. This article is structured as follows. After discussing the multi-species competitive PDE system, I illustrate different outcomes of species interactions under various

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dispersal strength using numerical simulation methods. By using bifurcation analysis, I then discuss some theoretical insights on the interplay of competitive interactions and local dispersal on species biodiversity. I also conduct stability analysis numerically to analyse the stability properties of different steady states in this PDE model. Finally, I discuss how my theoretical results relate directly to real multi-species ecosystems.

2 A Partial Differential Equations Model with Biotic Interactions, Abiotic Environments and Dispersal By considering a system of m-species in a one-dimensional domain, 0 ≤ x ≤ 1, I study the following PDE model for the densities Ni (x, t) with x corresponds to location and t corresponds to time [18, 19]: ⎛ r i Ni ⎝ ∂ Ni = K i (x) − ∂t K i (x)

m  j=1

⎞ αi j N j ⎠ + Di

∂ 2 Ni ∂x 2

(i = 1, 2, . . . , m)

(1)

where ri indicates the natural growth rate of species, K i indicates the environmental carrying capacity, Di indicates the local dispersal strength of species i and αi j indicates the competitive effect of species j on species i. The intraspecific competition coefficients αii may effectively be set to equal 1 (i.e. competitive effect between two individuals of species i is the same) by non-dimensionalising the density of species i with its intraspecific competition coefficient αii ; this rescaling gives me the ratio of intraspecific to interspecific competition, which is represented by the remaining competition coefficients αi j . All of the parameters are assumed to be positive. In general, system (1) is a spatially varying competition model of Lotka-Volterra type [7, 19], which becomes a PDE with the inclusion of a spatial diffusion term. Without dispersal (Di = 0), the dynamics of system (1) at a particular location (or environment) x does not depend on the behaviours from other locations. It is assumed that these interacting species exert symmetrical competitive strength e.g. αi j = α ji = α. The competition is also supposed to be local, which means that species can compete with other species if they are in the same location x. Consistent with these assumptions, a two-species (e.g. m = 2) system represents the simplest model of type (1). Depending on the ratio of the carrying capacities KK 21 and the intensity of competition α (with α = 1 a necessity), biotic interactions within each location x can result in different dynamical behaviours: (i) local coexistence when α < KK 21 < α1 (meaning that depending on the magnitude of α and also ratio of carrying capacities, competitive interactions can lead to species coexistence if interspecific competitive strength is weaker than intraspecific competition i.e. when 0 < α < 1); (ii) alternative stable states when α1 < KK 21 < α (similarly, when α > 1 i.e. interspecific competitive strength is stronger than intraspecific competition, alternative stable states behaviour can occur). The analysis can also be formulated for

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multiple species (e.g. m = 3) systems and for the case of asymmetrical competitive interactions (αi j = α ji ); further details and elaborations on these theoretical results can be found in [21–24]. Note that I have also examined the outcomes of system (1) for a plausible values (or range) of α due to the uncertainty issue in choosing the intensity of competitive strength. I also examine the environmental influences on the distributions of species by incorporating a spatial element x into the carrying capacity component; in particular, K i (x) corresponds to carrying capacity of species i, which changes across environmental gradients. The component x could be employed to represent abiotic environments (e.g. elevation, ambient temperature or salinity), or this term could also be used to represent a location within a geographical region, which may vary in climatic conditions and thus affect species distributions. Apart from the influences of species interactions, how each of these interacting species responds to the environmental gradients can also shape species distributions. To demonstrate the combined effects of different ecological forces in a multi-species community, I assume that the carrying capacity of species changes according to a nonlinear function with respect to x. This is called nonlinear environmental gradients and I study these effects in a three-species model (m = 3). Though I could employ a simple function of carrying capacity that varies linearly with x (as discussed by [1]), there are certain disadvantages of this kind of assumption: (i) unbounded increase in carrying capacity as x varies; (ii) from a theoretical viewpoint, this linear function corresponds to assuming that species’ fundamental niches change through an infinite spatial distance. These problems can be removed by employing a nonlinear environmental gradient e.g. a quadratic function: 

 (x − xi )2 , 0.001 (2) K i (x) = max K i,max 1 − wi2 where xi corresponds to the spatial location at which the carrying capacity for species i attained a maximum value K i,max and wi corresponds to the fundamental niche’s width. To make sure system (1) is well defined outside each species fundamental niche, I choose K i (x) to be a small value e.g. 0.001. Notice that the parameter values used in the numerical results section are given in Table 1 (unless stated otherwise in the figure caption). These parameter values are chosen primarily because they depict interesting scenarios of competition between three-species with different climatic tolerance e.g. warm-, cold- and temperate-region species. It should be noted that these species’ fundamental niches also illustrate an example of a mid-domain effect, meaning that more species ranges overlap near the centre of a geographical region [25, 26]. This pattern of species diversity has been observed empirically; for instance, different studies of small mammals along elevational gradients observe patterns of mid-domain effect in which species diversity peaks at an intermediate region [27, 28]. To model local dispersal among neighbouring locations, the diffusion term is incorporated into the system, with parameter Di represents dispersal strength among species. For the sake of brevity, it is assumed equal strength of dispersal (Di = D) for

Numerical Bifurcation and Stability Analyses of Partial … Table 1 Parametrisation of the model Symbol Description ri K i (x) x K 1,max K 2,max K 3,max x1 x2 x3 w1 w2 w3 Di αi j

The intrinsic growth rate of species i Carrying capacity of species i given by Eq. (2) Location within a geographical region Maximum carrying capacity of species 1 Maximum carrying capacity of species 2 Maximum carrying capacity of species 3 The most favourable location for species 1 The most favourable location for species 2 The most favourable location for species 3 The width of the fundamental niche for species 1 The width of the fundamental niche for species 2 The width of the fundamental niche for species 3 Diffusion coefficient Competition coefficient (values given in figure captions)

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Parameter/Function 1 Function 0–1 3500 5000 5000 0.8 0.2 0.5 0.6 0.7 0.25 0–0.0025

all the interacting and no migration is possible when species arrives at the boundary of locations; this is done by imposing zero-flux boundary conditions in the system:  ∂ Ni  = 0. Di ∂x x=0,1

(3)

3 Results Without local dispersal (Di = 0), I found that species distribution depends on the magnitude biotic interactions α. It is discovered that local coexistence is possible (i.e. species can survive at the same location) when interspecific competitive strength is weaker than intraspecific competition (0 < α < 1). I also observe the emergence of alternative stable states with the long-term species distributions vary with initial abundances, and this situation occurs when α > 1 (i.e. interspecific competitive strength is stronger than intraspecific competition). In this case, local coexistence of species is impossible due to competitive exclusion phenomenon: only one species can persist at a particular location x through alternative stable states [1, 29]. In the following sections, I demonstrate the combined effects of biotic interactions and dispersal on the presence-absence of species. I highlight these observations using bifurcation and stability analyses, with respect to the occurrence of alternative stable states, species coexistence and the stability of multi-species compositions.

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3.1 Numerical Simulation Results of a PDE Model System (1) with zero-flux boundary conditions (3) is solved numerically using MATLAB [30] pdepe solver. In particular, the method of lines is used by pdepe solver to solve systems of PDE in one spatial dimension x and temporal variable t. In the method of lines, the spatial domain 0 ≤ x ≤ 1 is divided into a mesh with A + 1 equally spaced nodes x j = j h for j = 0, 1, . . . , A, where h = A1 is the uniform mesh size. The derivative with respect to x in system (1) is transformed to a difference equation using second-order central difference approximation: N j+1 − 2N j + N j−1 ∂2 N j = ∂x 2 h2

(4)

This gives me a system of 3(A + 1) ordinary differential equations (ODE), one for the density of each species at a set of equally distance x points. Then, the resulting system of ODE is solved till t = 1000 to ensure convergence to a steady state. I used h = 2 × 10−3 and initial conditions as indicated in each figure section. The results are insensitive to a reduction in grid spacing h and they are illustrated in Figs. 1 and 2. When Di > 0 (i.e. with the inclusion of local dispersal into the model), species distribution is affected by migration of species from different locations x. For example, Figs. 1 and 2 show the distributions of species given by the PDE system. This diagram depicts species presence-absence at α = 1.3 for weak (Fig. 1a, b), medium (Fig. 1c, d), strong (Fig. 2a, b) and stronger (Fig. 2c, d) dispersal scenarios, with two distinct initial abundances with environments favourable to species 3 (left column) and environments favourable to species 1 and 2 (right column). In general, when α > 1, alternative stable states phenomenon and the dependency on initial abundances are observed in this PDE model. Near the central region, incorporation of weak migration level into the system (1) promotes coexistence of three(Fig. 1a) or two-species (Fig. 1b) with species 3 (red) goes extinct across the spatial domain. It is observed that the occurrence of alternative stable states is still evident in this multi-species communities under moderate dispersal intensity (Fig. 1c, d), as the dynamical behaviours of system (1) change with species establishment order; in this situation, dispersal weakens exclusion (of all but one species) effects that have been observed in no-dispersal scenario and allow multiple species coexistence near the central region. The salient features between these simulation results are observed under rapid dispersal scenario. In this case, I realise that alternative stable states phenomenon, which occurs in this PDE model is eliminated by strong dispersal level. Based on Fig. 2a–d, only coexistence of two-species outcome (with species 3 absent) is possible for different initial abundances, as rapid dispersal process can lead to some species (e.g. species 3) to be displaced.

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Fig. 1 Simulation results of the PDE model under several dispersal strength: D = 0.0005 (a, b); D = 0.001 (c, d). x-axis corresponds to spatial locations and y-axis corresponds to the steady-state densities of species. The intrinsic growth rate of species i, ri = 1. Carrying capacities of species following Eq. (2) with K 1,max = 3500, K 2,max = 5000, K 3,max = 5000, x1 = 0.8, x2 = 0.2, x3 = 0.5, w1 = 0.6, w2 = 0.7 and w3 = 0.25. Left column, species densities at α = 1.3 when initial abundances favour species 3: N1 (x) = 0.1K 1 (x) , N2 (x) = 0.1K 2 (x) , N3 (x) = 0.9K 3 (x)). Right column, species densities at α = 1.3 when initial abundances favour species 1 and 2: N1 (x) = 0.9K 1 (x) , N2 (x) = 0.9K 2 (x) , N3 (x) = 0.1K 3 (x). These plots are computed by numerical simulation with MATLAB pdepe solver

3.2 One-Parameter Bifurcation Analysis of a PDE Model I employed bifurcation analysis using XPPAUT to verify my simulation findings in Figs. 1 and 2 and gain some dynamical systems insights on the overall dynamics of my PDE model. System (1) is discretised using the method of lines in which the derivative with respect to x in system (1) is transformed to a difference equation using a second-order central difference approximation (4), with constant mesh size h. The zero-flux boundary conditions (3) are coded in the systems for the end points using a finite difference approximation. This yields a system of ODE, which is solved for t = 1000 in XPP using cvode solver. From a numerical methods perspective, cvode is an example of stiff solver in the XPPAUT tool kit; in general, discretisation of PDE using the method of lines often results in a large system of ODE, which is numerically stiff; in this case, it is useful to employ this kind of solver in order to

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Fig. 2 Simulation results of the PDE model under several dispersal strength: D = 0.0015 (a, b); D = 0.002 (c, d). x-axis corresponds to spatial locations and y-axis corresponds to the steady-state densities of species. The intrinsic growth rate of species i, ri = 1. Carrying capacities of species following Eq. (2) with K 1,max = 3500, K 2,max = 5000, K 3,max = 5000, x1 = 0.8, x2 = 0.2, x3 = 0.5, w1 = 0.6, w2 = 0.7 and w3 = 0.25. Left column, species densities at α = 1.3 when initial abundances favour species 3: N1 (x) = 0.1K 1 (x) , N2 (x) = 0.1K 2 (x) , N3 (x) = 0.9K 3 (x)). Right column, species densities at α = 1.3 when initial abundances favour species 1 and 2: N1 (x) = 0.9K 1 (x) , N2 (x) = 0.9K 2 (x) , N3 (x) = 0.1K 3 (x). These plots are computed by numerical simulation with MATLAB pdepe solver

avoid numerical problems and to ensure convergence to a solution. Then, the steady state is continued in AUTO, in which I tracked stable and unstable steady states and also bifurcation points as a model parameter changes. Continuation results shown in this chapter used a maximum/minimum allowable step size of parameter, 10−1 /10−6 . I observe a qualitatively different outcomes in my simulation findings by considering various dispersal scenarios. To examine the differences observed in species distributions across spatial region x as the intensity of competition α changes, I used bifurcation analysis to track the stable and unstable steady states of the system (1). For instance, Fig. 3 depicts the steady-state density of species 3 (N3 ) at x = 0.5 with weak/moderate (A: D = 0.001), strong (B: D = 0.0015) and stronger (C and D: D = 0.002 and D = 0.0025, respectively) dispersal levels, as the intensity of competition α is changed. Similar graphs could be plotted for N1 and N2 .

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Fig. 3 The density of species 3 (N3 ) for PDE model at location x = 0.5 as the strength of biotic interactions α vary under various dispersal scenarios: a D = 0.001, b D = 0.0015, c D = 0.002, d D = 0.0025. The points B P corresponds to transcritical bifurcation and L P corresponds to saddle-node bifurcation in the PDE. Three branches of N3 : (i) three-species steady-states (middle branches); (ii) three-species steady-states, with species 3 presence (upper branches); (iii) two-species steady-states, with species 3 absence (lower branches). The black curves correspond to unstable steady-states and red curves correspond to stable steady-states. Other parameter values as in Table 1. These plots are computed by numerical continuation using XPPAUT

As competition coefficient α changes, different dynamical behaviours occur in the PDE model with weak and medium dispersal levels e.g. multi-species coexistence and alternative stable-states outcomes (Fig. 1a–d). For the values of competitive strength that have been investigated (e.g. 0.5 ≤ α ≤ 1.5), I expect there occurs a critical value of α, which leads to qualitatively distinct species distributions. By using numerical continuation techniques (Fig. 3a), I discover that this threshold competition coefficient corresponds to transcritical bifurcation point (BP). In general, there are three steady state branches: the upper branch corresponds to coexistence of threespecies and lower branch corresponds to coexistence of two-species with species 3 absent. Red (respectively, Black) curves show stable (respectively, unstable) steady state solutions. There is a middle branch corresponding to unstable steady states with three-species coexistence (black curves) separating the stable steady states of upper and lower branches. When α < B P (e.g. B P = 1.16 in Fig. 3a), I observe the steady state with three-species coexistence is stable; when α > B P, I find that

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alternative stable states emerge with the density N3 tends toward upper or lower branch of steady-state (red curves), and this situation depends on initial abundances of species. From various simulation experiments conducted, I observe qualitatively different dynamics for rapid dispersal levels; in particular, I find that alternative stable states phenomenon vanishes for relatively intense competitive interactions (e.g. α = 1.3 in Fig. 2a–d). Another important observation from my numerical simulation experiments is that there occurs threshold values for competition intensity in which alternative stable states emerge and then vanish as α increases. My continuation result (Fig. 3b) reveals that there are critical values of α in the system (1): the upper (respectively, lower) threshold L P with L P = 1.22 (respectively, B P with B P = 1.11) in Fig. 3b corresponds to saddle-node (respectively, transcritical) bifurcation and alternative stable states emerge for small set of values of competition intensity i.e. when B P < α < L P. Based on this analysis, saddle-node bifurcation is a crucial mechanism behind the disappearance of alternative stable-states in my PDE model. In the case of stronger dispersal levels, I notice that the occurrence of alternative stable states is reduced as α changes. Closer examination of the continuation results in Fig. 3c, d illustrates that the two bifurcation points, B P and L P, approach each other and, finally, coalesce (compare Fig. 3c with 3d). This situation results in the parameter regions supporting alternative stable states phenomenon to vanish. In general, the interaction of transcritical and saddle-node bifurcations is investigated extensively in different studies e.g. Kooi et al. [31], Van Voorn and Kooi [32] and Saputra et al. [33], so interested readers are referred to the aforementioned work for details explanation.

3.3 Stability Analysis of PDE Solutions I also verified that steady states obtained are stable. To do this, the derivative with respect to t in system (1) is equated with zero and the derivative with respect to x is transformed to a difference equation using a second-order central difference approximation (Eq. 4), with constant mesh size h. The zero-flux boundary conditions (3) are coded in the equations for the end points using a finite difference approximation. This leads to a system of 3(A + 1) non-linear equations, one for the density of each species at a series of uniformly spaced x points, x j = j h for j = 0, 1, . . . , A, where h = A1 . This system is solved for steady state using the MATLAB built-in function, fsolve, with initial guess is the same as initial condition indicated in figure section. To determine the stability of steady state, the Jacobian matrix is calculated numerically in fsolve and then the eigenvalues are computed using eig function. The steady state is stable if all eigenvalues of the Jacobian evaluated at the steady state have negative real parts. Generally, a PDE system possesses infinite spectrum of eigenvalues. In practice, these eigenvalues can be obtained by transforming a PDE model under consideration into a large system of nonlinear ordinary differential equations (ODE) using

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finite-difference approximation, as discussed in the previous paragraph. Then, I need to employ a mesh refinement technique in order to ensure that the approximated eigenvalues from the ODE system are in agreement with the PDE model. To do this, first I consider a coarse grid of finite-difference approximation, i.e. A is small (e.g. A = 40 points) and later I calculate the eigenvalues of the system. Then, I refine the number of finite-difference points (or meshes) by taking larger values of A (e.g. A = 50 or 100 points), which means that I add more finite-difference terms into the system. The obtained spectra of eigenvalues for different A are compared. As the number of meshes A increases, the approximated ODE eigenvalues should converge to those of the PDE system under consideration. These eigenvalues can be used to determine stability of my PDE solutions. For the model (1), I have conducted the mesh refinement analysis and interested readers are referred to [29] for further details. The main lesson learned from this

Fig. 4 Some of the spectrum of eigenvalues for the PDE model under moderate dispersal level D = 0.001. These eigenvalues spectra correspond to different solution branches in Fig. 3A at α = 1.1: a stable three-species steady-state, with species 3 presence (upper branch); c unstable twospecies steady-state, with species 3 absence (lower branch). In b (respectively, d), the eighth largest eigenvalues are plotted to demonstrate that the steady-state in the upper (respectively, lower) branch is stable (respectively, unstable). The blue dots correspond to negative eigenvalues and red dot corresponds to a positive eigenvalue. Other parameter values as in Table 1. These plots are computed using MATLAB fsolve and eig functions

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Fig. 5 Some of the spectrum of eigenvalues for the PDE model under moderate dispersal level D = 0.001. These eigenvalues spectra correspond to different solution branches in Fig. 3A at α = 1.3: a stable three-species steady-state, with species 3 presence (upper branch); c stable two-species steady-state, with species 3 absence (lower branch); e unstable three-species steady-state (middle branch). In b (respectively, d), the eighth largest eigenvalues are plotted to demonstrate that the steady-state in the upper (respectively, lower) branch is stable. In f, the eighth largest eigenvalues are also plotted to show that the steady-state in the middle branch is unstable. The blue dots correspond to negative eigenvalues and red dot corresponds to a positive eigenvalue

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analysis is that increasing the number of meshes above 40 points has no effect on the approximation of eigenvalues. As an example, in Fig. 4a, c (respectively, Fig. 5a, c, e), I illustrate some of the spectrum of eigenvalues with A = 50 points for different solution branches in my bifurcation diagram, Fig. 3a, with fixed value of competition coefficient i.e. α = 1.1 (respectively, α = 1.3). Taking a vertical slice at α = 1.1 in Fig. 3a, I observe two different steady-states: the upper (respectively, lower) branch steady-state is stable (respectively, unstable). To verify the stability properties of these steady-states, it is only necessary to compare the first few eigenvalues of the spectrum, shown in Fig. 4a, c; so, I plotted the first eighth eigenvalues for α = 1.1 and α = 1.3, and the results are shown in Fig. 4b and d, respectively. When α = 1.1, I observe that on the upper branch (Fig. 4b), the largest eigenvalue, and hence every eigenvalue is negative. So, it is confirmed that this upper branch is stable. For the lower branch (Fig. 4d), it is discovered that the largest eigenvalue is positive (red dot), while the remaining eigenvalues are negative. Closer investigation using stability analysis indicates that steady-states lying on the lower branch are unstable, particularly when α < B P. Similar analysis is performed for α > B P with Fig. 5b, d, f show the results for α = 1.3; I find both upper (Fig. 5b) and lower (Fig. 5d) steady-state branches are stable, and the middle branch is unstable (Fig. 5f). Overall, I observe that the stability analysis of PDE solution branches are in a good agreement with my bifurcation analysis results in the previous section. The results are also insensitive to a reduction in grid spacing.

4 Discussion and Ecological Implications Taken together, I find that species interactions, local dispersal and abiotic environments significantly shape species distributions in an ecological system under consideration. I have used partial differential equations to demonstrate the important roles of interactions among multiple species and local dispersal in spatially-changing environments. It is discovered that species distributions depend on both the intensity of competitive interactions and dispersal strength. The system predicts species coexistence and the emergence of alternative stable states over a substantial range of parameter values, suggesting that these results are not very sensitive to the specific choice of parameters. The occurrences of coexistence between multiple species and exclusion of species in community assemblies depend strongly on the interplay of these biotic and abiotic factors. My findings show that in no-dispersal case and in the presence of intense interspecific competition, the model demonstrates alternative stable states phenomenon. In this situation, only one species persists at any particular locality and this is in agreement with the principle of competitive exclusion [11, 12]. From a conservation perspective, we need to be aware of the significant roles of biotic interactions in shaping species distributions. For instance, it has been documented in some ecological studies that introduced species have been released to natural ecosystems for several reasons, including economic gain and food sources; in some cases, they have the ability

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to outcompete our weak native species [34]. Also, the introduction of European rabbits to Australia had resulted in the destruction of vegetation, and the decline in native species through competition [34, 35]. Due to intense competitive interactions, the introduction of starlings and house sparrows in North America has caused the population density of native bluebird to reduce significantly [34]. Based on this information, introduced species can outcompete our vulnerable native species, sometimes causing their extinction, and altering ecosystem functioning. To mitigate ecological disruption from species introductions in the future, the consequence of biotic interactions should therefore be considered when developing conservation strategies and forecasts. By incorporating dispersal into ecological systems, the impacts of competitive exclusion (which emerges due to the absence of dispersal) is reduced; this situation causes the sensitivity of alternative stable states to be weakened; as a result, dispersal promotes coexistence of multiple species in the PDE system. These findings are in parallel with several other ecological observations: dispersal can expand species range overlap (also their geographic distributions) [36], and permit survival of multiple species in otherwise unsuitable locations [37–39]; consequently, local coexistence between multiple species is possible due to dispersal [7, 40, 41] and in turn enhancing the biodiversity of species. I have also discovered different observations of local dispersal: though my findings depict the occurrence of alternative stable states are still evident in a multi-species competitive system with dispersal, I also discover that the competitive exclusion phenomenon can be softened due to dispersal impacts, and this situation allows coexistence of multiple species. To gain better understanding of the systems dynamics and investigate the different outcomes of species interactions, I also performed bifurcation analysis to track both stable and unstable steady states as a model parameter is varied. Continuation result suggests a theoretical explanation for the observed differences in species presenceabsence in my simulation results, and also in a number of different studies [7]; I find that there are threshold values for competitive strength, which correspond to transcritical and saddle-node bifurcations in the model under different dispersal strength. Based on these bifurcation points, I characterise the bifurcation scenarios that give rise to several outcomes of species interactions (e.g. stable coexistence, competitive exclusion and alternative stable states), as discussed by some ecological studies [7]. Overall, this study demonstrates the influential impacts of biotic interactions and local dispersal in determining species biodiversity across heterogeneous environments. My work provides some explanations on the possible causes of distinct species distributions observed across environmental gradients, especially from the theoretical point of views. More investigation is needed into this topic in order to better understand the effects of distinct ecological forces on multi-species community dynamics. One obvious limitation of this work is that the results may be limited to the assumed parameter values e.g. ri , K i,max , xi , wi and Di ; more work is required to investigate the dynamics of this ecological system as some of these parameters are varied. Also for future research, I aim to investigate the interaction of alternative stable states with other ecological factors such as numerous dispersal processes, different biotic

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interactions e.g. predation and also mutualism. In short, I recommend the use of partial differential equations or other mathematical modelling frameworks, together with the techniques from bifurcation theory and dynamical systems in order to get a better understanding of the overall dynamical behaviours of ecological systems under consideration. Acknowledgements Mohd Hafiz Mohd is supported by the Universiti Sains Malaysia (USM) Fundamental Research Grant Scheme (FRGS) No. 203/PMATHS/6711645.

References 1. Mohd, M.H.B.: Modelling the presence-absence of multiple species. Ph.D. Thesis, University of Canterbury, New Zealand (2016) 2. Shmida, A.V.I., Wilson, M.V.: Biological determinants of species diversity. J. Biogeogr. 1–20 (1985) 3. Pearson, R.G., Dawson, T.P.: Predicting the impacts of climate change on the distribution of species: are bioclimate envelope models useful? Global Ecol. Biogeogr. 12(5), 361–371 (2003) 4. Soberón, J.: Grinnellian and Eltonian niches and geographic distributions of species. Ecol. Lett. 10(12), 1115–1123 (2007) 5. Kearney, M., Porter, W.: Mechanistic niche modelling: combining physiological and spatial data to predict species-ranges. Ecol. Lett. 12(4), 334–350 (2009) 6. Wisz, M.S., Pottier, J., Kissling, W.D., Pellissier, L., Lenoir, J., Damgaard, C.F., Dormann, C.F., Forchhammer, M.C., Grytnes, J.A., Guisan, A.: The role of biotic interactions in shaping distributions and realised assemblages of species: implications for species distribution modelling. Biol. Rev. 88(1), 15–30 (2013) 7. Case, T.J., Holt, R.D., McPeek, M.A., Keitt, T.H.: The community context of species’ borders: ecological and evolutionary perspectives. Oikos 108(1), 28–46 (2005) 8. Gilman, S.E., Urban, M.C., Tewksbury, J., Gilchrist, G.W., Holt, R.D.: A framework for community interactions under climate change. Trends Ecol. Evol. 25(6), 325–331 (2010) 9. Darwin, C.: On the Origins of Species by Means of Natural Selection. John Murray (1859) 10. Grinnell, J.: The niche-relationships of the California thrasher. Auk 34(4), 427–433 (1917) 11. Gause, G.F.: Experimental studies on the struggle for existence. J. Exp. Biol. 9(4), 389–402 (1932) 12. Gause, G.F.: The Struggle for Existence. Dover Publications (1934) 13. Hutchinson, G.E.: Concluding Remarks: Cold Spring Harbor Symposium on Quantitative Biology, vol. 22. Cold Spring Harbor Laboratory Press, pp. 415–427 (1957) 14. Lotka, A.J.: Elements of Physical Biology. Williams and Wilkins (1925) 15. Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926) 16. Cosner, C.: Reaction-diffusion-advection models for the effects and evolution of dispersal. Discret. Contin. Dyn. Syst. A 34(5), 1701–1745 (2014) 17. Holmes, E.E., Lewis, M.A., Banks, J.E., Veit, R.R.: Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75(1), 17–29 (1994) 18. MacLean, W.P., Holt, R.D.: Distributional patterns in St. Sroix Sphaerodactylus lizards: the taxon cycle in action. Biotropica 11(3), 189–195 (1979) 19. Roughgarden, J.: Theory of population genetics and evolutionary ecology: an introduction. Macmillan Publishing (1979) 20. Ermentrout, B.: Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students. SIAM (2002) 21. Kot, M.: Elements of Mathematical Ecology. Cambridge University Press (2001)

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22. Gotelli, N.J.: A Primer of Ecology. Sinauer Associates Incorporated (1995) 23. Chesson, P.: Mechanisms of maintenance of species diversity. Annu. Rev. Ecol. Syst. 31, 343– 358 (2000) 24. Case, T.J.: An Illustrated Guide to Theoretical Ecology. Oxford University Press (2000) 25. Colwell, R.K., Rahbek, C., Gotelli, N.J.: The mid-domain effect and species richness patterns: what have we learned so far? Am. Nat. 163(3), E1–E23 (2004) 26. Colwell, R.K., Lees, D.C.: The mid-domain effect: geometric constraints on the geography of species richness. Trends Ecol. Evol. 15(2), 70–76 (2000) 27. McCain, C.M.: The mid-domain effect applied to elevational gradients: species richness of small mammals in Costa Rica. J. Biogeogr. 31(1), 19–31 (2004) 28. McCain, C.M.: Elevational gradients in diversity of small mammals. Ecology 86(2), 366–372 (2005) 29. Mohd M.H.B.: Modelling biotic interactions, dispersal effects and the stability of multi-species community compositions. In: AIP Conference Proceedings, vol. 1974, p. 020079. AIP Publishing (2018) 30. Higham, D.J., Higham, N.J.: MATLAB guide. SIAM (2016) 31. Kooi, B.W., Boer, M.P., Kooijman, S.A.L.M.: Resistance of a food chain to invasion by a top predator. Math. Biosci. 157(1), 217–236 (1999) 32. Van Voorn, G.A.K., Kooi, B.W.: Smoking epidemic eradication in a eco-epidemiological dynamical model. Ecol. Complex. 14, 180–189 (2013) 33. Saputra, K.V.I., Van Veen, L., Quispel, G.R.W.: The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Dyn. Contin. Discret. Impuls. Syst. Ser. B Math. Anal. 14, 233–250 (2010) 34. Sinclair, A.R.E., Fryxell, J.M., Caughley, G.: Wildlife Ecology. Wiley, Conservation and Management (2006) 35. Short, J., Bradshaw, S.D., Giles, J., Prince, R.I.T., Wilson, G.R.: Reintroduction of macropods (Marsupialia: Macropodoidea) in Australia a review. Biol. Conserv. 62(3), 189–204 (1992) 36. Dytham, C.: Evolved dispersal strategies at range margins. Proc. R. Soc. Lond. B Biol. Sci. 276(1661), 1407–1413 (2009) 37. Davis, A.J., Jenkinson, L.S., Lawton, J.H., Shorrocks, B., Wood, S.: Making mistakes when predicting shifts in species range in response to global warming. Nature 391(6669), 783–786 (1998) 38. Davis, A.J., Lawton, J.H., Shorrocks, B., Jenkinson, L.S.: Individualistic species responses invalidate simple physiological models of community dynamics under global environmental change. J. Anim. Ecol. 67(4), 600–612 (1998) 39. Pulliam, H.R.: On the relationship between niche and distribution. Ecol. Lett. 3(4), 349–361 (2000) 40. Amarasekare, P., Nisbet, R.M.: Spatial heterogeneity, source-sink dynamics, and the local coexistence of competing species. Am. Nat. 158(6), 572–584 (2001) 41. Lei, G., Hanski, I.: Spatial dynamics of two competing specialist parasitoids in a host metapopulation. J. Anim. Ecol. 67(3), 422–433 (1998)

Discrete Dynamical Systems

Global Stability Index for an Attractor with Riddled Basin in a Two-Species Competition System Ummu Atiqah Mohd Roslan and Mohd Tirmizi Mohd Lutfi

Abstract We consider a competition system between two-species containing riddled basin and second basin attractors. To characterize local geometry of riddled basin, we compute a global stability index for the attractor in the system. Our results show that the index varies from ∞ down to positive values within a parameter region. The changes of the index indicates that the attractor looses its stability from asymptotically stable attractor to riddled basin attractor. Thus, the stability index has a great potential to become a new study on bifurcation of dynamical system since it is able to characterize different types of geometry of basins of attraction. Keywords Riddled basin · Stability index · Dynamical system

1 Introduction: Motivation Dynamical system is a useful study on long time behaviour of an initial value problem. They are often applied to predict either for mathematical purposes or applications in real-world problems. In this paper, we focus on discrete-time dynamical system in form of: xn+1 = f (xn ), where f : X → X , xn ∈ X , f is a continuous map on metric space, X with discrete time n ∈ Z. This system is called an iterative map. The main concern in dynamical systems is the behaviour of initial conditions that approach to certain solution of the system. Apparently, a vital question may arise: how do we quantify or measure the movement of orbits of initial conditions in a system? In fact, there are many tools have been introduced to answer this question, including the well-known Lyapunov exponents. But, it is not the reason to signify the end of this study. We use a stability U. A. Mohd Roslan (B) · M. T. Mohd Lutfi Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_8

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index approach that proposed by Podvigina and Ashwin [1], to explore the quantification of the stated movement. This index applies the concept of measure theory in order to measure the orbit movement. Podvigina and Ashwin [1] originally introduced the stability index to characterize the local geometry of basin of attraction in the case of heteroclinic cycles. Subsequently, Castro and Lohse [2] developed such index to study the stability of simple heteroclinic cycles and networks. In 2015, Keller [3] developed a stability index for an attractor in a chaotically driven concave maps. A year later, Mohd Roslan and Ashwin [4] investigated the local and global stability indices for an attractor with riddled basin of a piecewise linear expanding map. The authors finally proved the stability index for a point in attractor can be formulated in terms of Lyapunov exponents of the map and the stability index for the attractor of the map. Referring to Ashwin et al. [6], Mohd Roslan [5] computed the stability index numerically for a coupled electronic-circuit system. Thus, in this study a global stability index for an attractor with riddled basin in a two-species competition system is computed based on previous example by Viana et al. [7]. The authors in [7] quantified the stability of orbits and periodic orbits on the attractor using the Lyapunov exponents. The paper is organized as follows: in Sect. 2, we review some basic concepts of attraction properties in dynamical systems with an example of the well-known logistic map. In Sect. 3, we discuss the concept of riddled basin as well as some conditions for such a basin to occur. Section 4 is devoted to the discussion on the global stability index as introduced by [4]. An algorithm for the computations of stability index for a set is also introduced in this section. In Sect. 5, we consider a two-species competition system containing riddled basin and give the main result on the global stability index for an attractor in the system numerically. Finally, we discuss and conclude our findings in the last section.

2 Basic Definitions In dynamical systems, we are interested to study invariant sets for a system. Invariant sets are set which do not change with time. Some examples of invariant sets are fixed points (i.e. equilibrium points), periodic orbits, limit cycles and attractors. More precisely, Definition 1 (Invariant set) A compact set A ⊂ X is invariant if and only if x ∈ A implies that f (x) ∈ A and f −1 (x) ∈ A, i.e. f (A) = A. This definition interprets as given any initial conditions that started in A, its whole trajectory also stay in A, so it consists of union of trajectories. Definition 2 (Fixed point) The point x0 ∈ X is a fixed point for f : X → X if f (x0 ) = x0 .

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Definition 3 (Orbit) The orbit of an initial condition x0 ∈ X is n { f n (x0 )}∞ n=0 = {x 0 , f (x 0 ), f ( f (x 0 )), ..., f (x 0 ), ...},

where f n denotes the nth iterate of f . Definition 4 (Periodic orbit) The point x0 ∈ X is a periodic point of period n for f : X → X if f n (x0 ) = x0 . Example 1 Consider the logistic map f : [0, 1] → [0, 1] given by xn+1 = f (xn ) = μxn (1 − xn ), with μ > 0. To find the fixed points or points with period one, we need to solve the following μx(1 − x) = x . Moreover, to find which gives two solutions of fixed points: x11 = 0 and x12 = μ−1 μ points of period two or period-2 orbits, it is necessary to solve f 2 (x) = f ( f (x)) = x i.e. μ(μx(1 − x))(1 − (μx(1 − x))) = x. Therefore, we have x(μ2 x(1 − x)[1 − μx(1 − x)] − 1) = 0. By factorizing the above equation, we obtain   μ−1 (−μ2 x 2 + μ2 x + μx − μ − 1) = 0. x x− μ Thus, the equation −μ2 x 2 + μ2 x + μx − μ − 1 = 0 i.e. μ2 x 2 − μ(μ + 1)x + (μ + 1) = 0 gives a pair of solutions of period two points as follows: x21 =

μ+1+

√ (μ + 1)(μ − 3) 2μ

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and x22 =

μ+1−

√ (μ + 1)(μ − 3) 2μ

for μ > 3.

2.1 Stability of Invariant Sets According to Glendinning [8], there are three fundamental concepts of stability in dynamical systems, namely Lyapunov stable, quasi-asymptotically stable and asymptotically stable. Definition 5 (Lyapunov stable) We say A is Lyapunov stable (start near stay near) if for all ε > 0 there is a δ > 0 such that d(x, A) < δ implies that d( f n (x), A) < ε for all time n > 0. Definition 6 (Quasi-asymptotically stable) We say A is quasi-asymptotically stable (tends to eventually) if there exists δ > 0 such that d(x, A) < δ implies that d( f n (x), A) → 0 as n → ∞. Definition 7 (Asymptotically stable) A is asymptotically stable (tends to directly) if and only if (a) A is Lyapunov stable, and (b) A is quasi-asymptotically stable.

2.2 Basin of Attraction According to Glendinning [8], basin of attraction is defined as the largest neighbourhood of invariant set for which all the initial conditions in the neighbourhood tend to the invariant set. Whereas in our study, basin of attraction refers to set of all initial conditions whose orbits converge to A. From now on, we let A ⊂ X be a compact invariant set, and its basin is defined as: Definition 8 The basin of attraction, denoted by B(A) is the set of points x ∈ X whose ω-limit set is contained in A, i.e. B(A) = {x ∈ X : ω(x) ⊆ A}, where ω(x) =

 m≥0

 n≥m

f n (x).

Alexander et al. [9] describes the word ‘contained in’ in the above definition as the set of initial conditions whose orbits are asymptotic to A.

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2.3 Attractor Roughly speaking, an attractor is an invariant set for which all nearby orbits of initial conditions converge [10]. To visualize the concept of attractor and its basin of attraction, we discuss the dynamics of fixed points in logistic map from previous example: Example 2 Recall that the logistic map has two fixed points x11 = 0 and x12 = μ−1 . μ These fixed points are stable if | f (x∗ )| < 1 and unstable if | f (x∗ )| > 1 where x∗ denotes the fixed point. Let us consider for the case μ > 1. For x11 , we have that )= f (0) = μ. Since μ > 1, therefore x11 is unstable. Meanwhile, for x12 , f ( μ−1 μ 2 − μ. Since 2 − μ < 1, therefore x12 is stable for 1 < μ < 3. Hence, we say that x12 is a fixed point attractor in this range. Furthermore, if any initial conditions which satisfies 0 < x0 < 1, the orbits will converge to the attractor x12 . Thus, we say that [0, 1] is the basin of attraction of the attractor x12 [11]. The convergence of initial conditions to A can also be quantified by using the concept in measure theory. This theory is highlighted as the main concern in this paper, therefore, we use the Lebesgue measure which denoted by  to measure the attractiveness of initial conditions to A, starting with the definition of weak attractor which is the main ingredient for Milnor attractor. Definition 9 (Weak attractor [12]) A compact invariant subset A ⊂ X is a weak attractor if and only if (B(A)) > 0, i.e. the basin has strictly positive Lebesgue measure. Milnor [13] introduced a stronger definition of attractor as follows: Definition 10 (Milnor attractor) A compact invariant subset A ⊂ X is a Milnor attractor if and only if (a) it is a weak attractor, and (b) for any closed proper subset A ⊆ A that is compact and invariant satisfies (B(A) \ B(A )) > 0 (also strictly positive measure). Heretofore, both weak and Milnor attractors require the basins with positive measure. However, it is possible for a basin to have full measure (i.e. basin with measure equal to 1), such that the basin contains an asymptotically stable attractor. By combining the concept of full measure basin with Definition 7, we then have the following: Definition 11 (Asymptotically stable attractor (a.s.a.)) A is a.s.a. if and only if (a) A is asymptotically stable, and (b) basin of A has full measure (B(A)) = 1.

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3 Riddled Basin of Attraction The concept of riddled basin has been introduced by Alexander et al. [9]. It is named because it is “riddled” with holes of another basin. In particular, suppose that a system has two attractors A and Ac (another attractor that is complement to A) with basins B(A) and B(Ac ) respectively. Basin B(A) is riddled by basin B(Ac ) if for every point p in B(A), a small ball of radius δ centred at p, Bδ ( p), has a positive Lebesgue measure of points belonging to basin B(Ac ), irrespective of how small δ is. This type of basin has been studied extensively by many researchers in theoretical and applications. A theoretical example by Buescu et al. [12], while the application as such in the field of biology, Viana et al. [7] has considered a competition model of two species of flour beetles. Their results stated that only one species survive. However, the occurence of extinction species is highly sensitive to initial population and environmental conditions. Recent work on riddled basin by Dai et al. [14] investigated the escaping crisis in an integrate-and-fire model of an electronic relaxation oscillator. Let Bδ (A) be δ-neighbourhood and d(·, ·) be the Hausdorff distance between two sets. Then Bδ (A) = {x ∈ X : d(x, A) < δ}. In Ashwin et al. [6], they define the riddled basin as: Definition 12 (Riddled basin) A Milnor attractor A has a riddled basin if for all x ∈ B(A) and δ > 0, then (Bδ (A) ∩ B(A))(Bδ (A) ∩ B(Ac )) > 0. In fact, (Bδ (A) ∩ B(A)) > 0 and (Bδ (A) ∩ B(Ac )) > 0 [9].

3.1 Conditions for the Existence of Riddled Basins Riddled basins occur in interesting way. In this section, we refer to Viana et al. [7] for some conditions of the occurence of riddled basin. 1. 2. 3. 4.

There exists an invariant subspace M ⊂ X . The dynamics on M has a chaotic attractor A. There is another attractor Ac not belonging to M. The attractor A is transversely stable in X , i.e. for typical orbits on the attractor, the Lyapunov exponents for infinitesimal perturbations along the directions transversal to the invariant subspace M are all negative.

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4 A New Approach of Bifurcation: The Stability Index In this section, we discuss the latest approach: the stability index. This approach has a great potential in the study of bifurcation for solving various problems in dynamical systems. The index was originally introduced for an individual point by [1], but subsequently Mohd Roslan and Ashwin [4] renamed as local stability index. In this paper, we focus on global stability index in which the index for a set in general, and attractor in particular, rather than for a single point on the attractor is computed. Thus, we adapt the following definition of global stability index based on Mohd Roslan and Ashwin [4]: Definition 13 Let A be an attractor and let δ > 0. We define the following proportions (Bδ (A) ∩ B(A)) , Σδ (A) := (Bδ (A)) and 1 − Σδ (A) :=

(Bδ (A) ∩ B(A)c ) (Bδ (A))

where B(A)c is complement of B(A). Then stability index of A is σ(A) := σ+ (A) − σ− (A), which exists when the following converge: ln(Σδ (A)) ln(1 − Σδ (A)) , σ+ (A) := lim . δ→0 δ→0 ln δ ln δ

σ− (A) := lim

We note that 0 ≤ Σδ (A) ≤ 1 and therefore σ(A) ∈ [−∞, ∞]. This means that the index may vary from −∞ to ∞.

4.1 Algorithm for Computing Stability Index Here, we introduce the steps to compute the stability index for the attractor A: 1. 2. 3. 4.

Choose a parameter in the system Choose δ-neighbourhood around A Compute the proportion Σδ (A) that converge to A Reduce the size of δ-neighbourhood

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5. Compute the proportion as δ → 0 (i) If proportion = 1, then σ(A) = ∞. (ii) If proportion = 0, then σ(A) = −∞. (iii) If 0 < proportion < 1, then σ(A) > 0. 6. Repeat Steps 1 to 5 for different values of parameter We apply the above algorithm for the computations of stability index for an attractor in a numerical example in the next section.

5 A Numerical Example with Riddled Basin In this section, we consider a discrete map of two-species competition system from Viana et al. [7] of the form xn+1 = 4xn (1 − xn ), yn+1 = yn + κyn (1 − yn ) cos(3πxn ),

(1)

where 0 ≤ κ ≤ 1 represents the strength of competition between species x and y. Notice that the first equation in (1) is in fact the usual logistic map equation. This indicates that species x obeys the logistic growth rate. This map has two attractors; one at A0 = [0, 1] × {0} and the other at A1 = [0, 1] × {1}. It has been shown in [7] that basin of A0 is riddled with basin of A1 and basin of A1 is also riddled with basin of A0 . In this paper, we characterize such a basin using the stability index approach. Figure 1 depicts the approximate basin of the map (1) for various values of κ. The black strips represents the basin of attraction of attractor A0 denoted as B(A0 ), contains the initial conditions that are attracted to A0 . Meanwhile, the orange area indicates the basin of attraction of A1 , denoted as B(A1 ), that contains the initial conditions that are attracted to A1 . Note that both basins have fractal structure and overlap on each other. By referring to Definition 13, the proportions can be computed as: Σδ (A0 ) =

area of black strips in δ-neighbourhood area of black strips and orange in δ-neighbourhood

and 1 − Σδ (A0 ) =

orange area in δ-neighbourhood , area of black strips and orange in δ-neighbourhood

where δ-neighbourhood is chosen as [0, 1] × [0, δ] since we only consider for positive region thus biologically meaningful. Figure 2 shows the result on the proportion of black strips (i.e. B(A0 )) on varying κ = 0, ..., 1.

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Fig. 1 The numerical approximation of basin of attraction for map (1) for various κ. The black strips represented by B(A0 ) is the basin such that the initial conditions attracted to A0 = [0, 1] × {0} while the orange area represented by B(A1 ) is the basin where initial conditions are attracted to A1 = [0, 1] × {1}. a For κ = 0.3, A0 is a.s.a. b For κ = 0.8, A0 is a Milnor attractor such that its basin B(A0 ) is riddled with B(A1 ). In this case, both basins have positive measure

Fig. 2 The plot of proportion of black strips (B(A0 )) versus κ = 0, ..., 1. The proportion is 1 for κ = 0, ..., 0.5 and the proportion is less than 1 for 0.5 < κ ≤ 1

5.1 Computations of Stability Index σ( A0 ) In this section, we compute the stability index for attractor A0 numerically. It is essential to know which region the riddled basin occurs for this system. We follow the steps in algorithm introduced in Sect. 4.1 to compute the stability index. Firstly, we set the parameter between the range of 0 ≤ κ ≤ 1. Then, we choose a δ-neighbourhood around A0 in which A0 = [0, 1] × {0}. Later, the proportion of points that converge

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to A0 or to A1 within the δ-neighbourhood is computed. This proportion is evaluated by shrinking the size of the δ-neighbourhood. The results show the following cases: (a) When 0 ≤ κ ≤ 0.5, the proportion Σδ (A0 ) increases and finally hits 1 as δ → 0. When Σδ (A0 ) = 1, this indicates that the δ-neighbourhood only filled with initial conditions that converge to A0 . (b) When 0.5 < κ ≤ 1, Σδ (A0 ) → 1 as δ → 0 but never attains 1. This proportion increases as more and more black strips occupy the δ-neighbourhood around A. This is the case where riddled basin occurs as there are always initial conditions that are escape to the second attractor, i.e. to A1 no matter how small the size of δ-neighbourhood is. Then σ− (A0 ) is determined from the slope of log(Σδ (A0 )) versus log(δ). Meanwhile, σ+ (A0 ) is determined by the slope of the plot of log(1 − Σδ (A0 )) versus log(δ). Then, the stability index is obtained by solving σ(A0 ) = σ+ (A0 ) − σ− (A0 ). We observe that the index varies continuously from ∞ and turns down to a positive value before continue to increase monotonically to higher positive values. The index varies with κ as follows (see Fig. 3):

Fig. 3 The numerical approximation of stability index σ(A0 ) for the attractor A0 = [0, 1] × {0} for map (1) by varying κ. Note that the value 1 on vertical axis in the graph shows the index value is ∞. When 0 ≤ κ ≤ 0.5, σ(A0 ) = ∞ since all initial conditions within δ-neighbourhood of A0 belong to B(A0 ). The index is then jumps down to a positive value (approximately 0.775) and increase afterwards. The increasing pattern means that more and more points belonging to B(A0 ). Trajectories converge to B(A0 ) are more than the trajectories that converge to B(A1 ). This is where the riddled basin occurs since there are always trajectories that escape to B(A1 ) no matter how small the δ-neighbourhood is

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(a) When 0 ≤ κ ≤ 0.5, σ(A0 ) = ∞. (b) When 0.5 < κ ≤ 1, σ(A0 ) > 0. The above result in (b) agrees with results from Mohd Roslan and Ashwin [4] and Keller [3] in which for -almost all points x, the stability index σ(A0 ) is positive when the riddled basin occurs.

6 Discussion This study investigates the local geometry of riddled basin of attraction for a twospecies competition system by using the stability index. We have computed numerically the global stability index for an attractor in the system. This system shows that riddled basin occurs within a range of parameter κ where for Lebesgue almost all points on the attractor, the stability index is positive. From the biological point of view, we know that κ represents the competition strength between the two species x and y. Our results show that for 0 ≤ κ ≤ 0.5, we have an asymptotically stable attractor meaning that since the competition strength is low, the two species coexist and live together in harmony. However, as the competition strength increases, the system becomes unstable thus the riddled basin occurs. This behaviour indicates that only one species will survive, depending on the initial population and environmental conditions. As the stability index in the current paper computed using numerical simulation, it would be of interest to study both local and global stability indices theoretically for this biological system in order to have more stronger results. For example, future work can be done to see whether there is relationship between stability index with Lyapunov exponents.

References 1. Podvigina, O., Ashwin, P.: On local attraction properties and a stability index for heteroclinic connections. Nonlinearity 24, 887–929 (2011) 2. Castro, S.B.S.D., Lohse, A.: Stability in simple heteroclinic networks in R4 . Dyn. Syst. 29(4), 451–481 (2014) 3. Keller, G.: Stability index for chaotically driven concave maps. J. Lond. Math. Soc. 29(4), 451–481 (2015) 4. Mohd Roslan, U.A., Ashwin, P.: Local and global stability indices for a riddled basin attractor of a piecewise linear map. Dyn. Syst. 31, 375–392 (2016) 5. Mohd Roslan, U.A.: Stability index for an attractor with riddled basin in dynamical systems. In: AIP Conference Proceedings, vol. 1870, pp. 0400071–0400076 (2017). https://doi.org/10. 1063/1.4995839 6. Ashwin, P., Buescu, J., Stewart, I.: From attractor to chaotic saddle: a tale of transverse instability. Nonlinearity 9, 703–737 (1996) 7. Viana, R.L., Camargo, S., Pereira, R.F., Verges, M.C., Lopes, S.R., Pinto, S.E.S.: Riddled basins in complex physical and biological systems. J. Comput. Interdiscip. Sci. 1(2), 73–82 (2009)

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8. Glendinning, P.: Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. Cambridge University Press, UK (1994) 9. Alexander, J.C., Yorke, J.A., You, Z., Kan, I.: Riddled basin. Int. J. Bifurc. Chaos 2, 795–813 (1992) 10. Devaney, R.L.: An Introduction to Chaotic Dynamical Systems, 2nd edn. Addison-Wesley, USA (1989) 11. Ott, E.: Chaos in Dynamical Systems, 2nd edn. Cambridge University Press, UK (2002) 12. Buescu, J.: Exotic Attractors: from Liapunov Stability to Riddled Basins. Birkhäuser Verlag, Switzerland (1997) 13. Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99, 177–195 (1985) 14. Dai, J., He, D.-R., Xu, X.-L., Hu, C.K.: A riddled basin escaping crisis and the universality in an integrate-and-fire circuit. Phys. A Stat. Mech. Appl. 500, 72–79 (2018). https://doi.org/10. 1016/j.physa.2018.02.047

Counting Closed Orbits in Discrete Dynamical Systems Azmeer Nordin, Mohd Salmi Md Noorani and Syahida Che Dzul-Kifli

Abstract For a discrete dynamical system, the following functions: (i) prime orbit counting function, (ii) Mertens’ orbit counting function, and (iii) Meissel’s orbit sum, describe the different aspects of the growth in the number of closed orbits of the system. These are analogous to counting functions for primes in number theory. The asymptotic behaviour of those functions can be determined by two approaches: by (i) Artin-Mazur zeta function, or (ii) number of periodic points per period. In the first approach, the analyticity and non-vanishing property of the zeta function lead to the asymptotic equivalence of the prime orbit and Mertens’ orbit counting functions. In the second approach, the estimate on the number of periodic points per period is used to obtain the order of magnitude of all those counting functions. This chapter will introduce the counting functions and demonstrate both approaches in some categories of shift spaces, such as shifts of finite type, countable state Markov shifts, Dyck shifts and Motzkin shifts. Keywords Prime orbit theorem · Mertens’ orbit theorem · Meissel’s orbit theorem · Artin-Mazur zeta function · Shift of finite type · Countable state Markov shift · Dyck shift · Motzkin shift

Notations The following notations are used in this chapter: (i) asymptotic equivalence; f (x) ∼ g(x) if and only if lim

x→∞

f (x) = 1. g(x)

A. Nordin (B) · M. S. Md. Noorani · S. C. Dzul-Kifli Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia e-mail: [email protected] M. S. Md. Noorani e-mail: [email protected] S. C. Dzul-Kifli e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_9

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(ii) little-o notation; f (x) = o(g(x)) if and only if lim

x→∞

f (x) = 0. g(x)

(iii) big-O notation; f (x) = O(g(x)) if and only if there exist k ∈ R and positive real A such that | f (x)| ≤ A · g(x) for any x ≥ k. (iv) order of magnitude; f (x)  g(x) if and only if there exist k ∈ R and positive reals A and B such that A · g(x) ≤ f (x) ≤ B · g(x) for any x ≥ k. (v) divisibility. For d, n ∈ N, d|n if and only if dn ∈ N.

1 Discrete Dynamical Systems and Closed Orbits 1.1 Closed Orbits Let X be a set and T : X → X be a map on X . The pair (X, T ) forms a discrete dynamical system with evolution of a point x ∈ X as the sequence {(T k (x)}∞ k=0 . The set γ(x) = {T k (x) | k ∈ N0 } is called the orbit of x. If T n (x) = x for some n ∈ N, then x is called a periodic point of period n. Furthermore, if T k (x) = x for any k ∈ {1, 2, . . . , n − 1}, then n is the least period of x. In this case, the orbit γ(x) is finite. It is called a closed orbit, usually denoted as τ (x). Definition 1 For a discrete dynamical system (X, T ), if x ∈ X is periodic with least period n, then the set τ (x) = {x, T (x), T 2 (x), . . . , T n−1 (x)} is called a (prime) closed orbit (or periodic orbit) of period (or length) |τ | = n. Example 1 The following demonstrate the periodic points and the closed orbits in some simple systems: (a) R under identity map T (x) = x; Any point x is periodic with least period 1. So, τ (x) = {x}.

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(b) R under T (x) = −x; Any non-zero x is periodic with least period 2. So,  τ (x) =

{0} if x = 0, {x, −x} otherwise.

(c) Unit circle S1 under doubling map T (x) = 2x for x ∈ [0, 2π); 2πk for all k ∈ {0, 1, ...2n − 2}, which are The periodic points of period n are n 2 −1 the (2n − 1)th roots of unity. However, the closed orbits of period n are difficult to be expressed explicitly. (d) [0, 1] under logistic map T (x) = 4x(1 − x). It is difficult to obtain the periodic points, and hence the closed orbits. However, it is easy to check that there is a closed orbit of period 3. By Sharkovskii’s theorem [26], this implies that there is a closed orbit for any period. Given a system, it is common to study its closed orbits. This is because some dynamical properties of a system depend on the nature of the closed orbits. For example, (i) the count of the closed orbits are invariant under topological conjugacy, so conjugate systems have same number of closed orbits per period, (ii) for a system to be Devaney chaotic [6], the union of all closed orbits must be dense, and (iii) for a system to be Pn dense [7], the union of all closed orbits of period at least n is dense for any n ∈ N. Another interesting question on closed orbits of a system is the growth of closed orbits, i.e. how much the number of closed orbits increases per period. To answer this, it is useful to introduce some counting functions for closed orbits. Remark 1 In Example 1, the systems in (c) and (d) have finite number of closed orbits per period, unlike the systems in (a) and (b). The counting functions in the next section are only defined for a system with finite number of closed orbits per period.

1.2 From Counting Primes to Counting Closed Orbits Recall that in number theory, some counting functions for primes are introduced to study the distribution of primes. These include: (i) prime counting function; It is defined as π(x) =

 prime p≤x

1.

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It counts the number of primes less than or equal to x. Hadamard [10] and Vallée Poussin [29] were the first ones to prove that π(x) ∼ (ii) Mertens’ prime counting function; It is defined as M(x) =

x . log x 



prime p≤x

 1 1− . p

The additive version is defined as 

M (x) =

prime p≤x

Mertens [21] proved that M(x) ∼

1 . p

e−γ log x

where γ is the Euler-Mascheroni constant, and M (x) = log log x + M + o(1) where M is the Meissel-Mertens constant. (iii) Meissel’s sum. It is defined as  1 m(a) = p(log p)a prime p for real a > 0. As pointed out by Lindqvist and Peetre [20], Meissel found that as a → 0+ , 1 m(a) = + M + O(a) a where M is the Meissel-Mertens constant. The proof of all results above can be found in [12]. Motivated by the idea in number theory, it is possible to define the counting functions for closed orbits. These are dynamical analogues to the counting functions for primes. These counting functions describe the distribution of closed orbits in a system. Definition 2 Let (X, T ) be a discrete dynamical system with the topological entropy h > 0. Define O(n) as the number of closed orbits τ of period |τ | = n. The counting functions for the closed orbits are defined as:

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(i) prime orbit counting function πT (x) =



1

|τ |≤x

=



O(n)

n≤x

for x ∈ N, (ii) Mertens’ orbit counting function MT (x) =



1

1−



eh|τ |   1 O(n) 1 − hn = e n≤x |τ |≤x

and its additive version MT (x) =

 |τ |≤x

=

1 eh|τ |

 O(n) n≤x

ehn

for x ∈ N, and (iii) Meissel’s sum m T (a) = =

 τ ∞  n=1

1 |τ |a eh|τ | O(n) n a ehn

for real a > 0. Remark 2 Definition 2 involves the notion of topological entropy. However, its definition is sophisticated, so it is not included here. For interested reader, the definition can be found in [1]. For some systems, the topological entropy is reduced to simpler definition. As in counting primes, it raises a question on how these counting functions behave as x → ∞. The asymptotic behaviour of the counting functions can be determined by two approaches, which is either via (i) Artin-Mazur zeta function, or (ii) number of periodic points per period. Both approaches will be discussed and demonstrated by using some examples from shift spaces.

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1.3 Shift Spaces In simple term, a shift space is a set of sequences of certain symbols equipped with a map. For a sequence in this set, the symbols appear by following some rules or restrictions. The map shifts the sequence to the left. Let A be a set of n symbols, typically taken to be A = {0, 1, . . . , n − 1}. For w = w1 w2 . . . wn ∈ An and x = . . . x−2 x−1 x0 x1 x2 . . . ∈ AZ , w is said to occur in x if there exists i ∈ Z such that xi xi+1 . . . xi+n−1 = w1 w2 . . . wn . It is denoted as w ≺ x. n Definition 3 Let A be a set of n symbols, and F ⊂ ∪∞ n=1 A . Define the set XF = {x ∈ AZ | w ⊀ x ∀w ∈ F}

and the map σ : XF → XF that maps x to σ(x) as (σ(x))i = xi+1 ∀i ∈ Z. The pair (XF , σ) is called a shift space (or shift). The set A is called the alphabet and its element a ∈ A is called a letter. The element x ∈ XF is called a point. The element w ∈ An that occurs in some x ∈ XF is called an allowed word (or word). Denote Bn (XF ) as the set of all allowed words of length n. The set L(XF ) = ∪∞ n=1 Bn (XF ) is called the language. The element w ∈ F is called a forbidden word. The map σ is called the shift map. The map σ is usually omitted in the notation of a shift space, so it is written simply as XF . Note that a forbidden word will not occur in any point in XF . Therefore, the set F acts as a form of rules or restrictions for a word to occur in a point. Example 2 The following are some examples of simple shift spaces: (a) full k-shift; Let A = {0, 1, . . . , k − 1} for some k ∈ N and F = ∅. Then, X∅ = AZ is called the full k-shift. Any w ∈ An is allowed, so it will occur in some points in X∅ . (b) golden mean shift; Let A = {0, 1} and F = {11}. Then, X{11} is called the golden mean shift. Only 11 is forbidden, so it will not occur in any point in X{11} . (c) even shift. Let A = {0, 1} and F = {102n−1 1 | n ∈ N}. Then, X{102n−1 1}n∈N is called the even shift. The forbidden words consist of an odd number of 0’s between two consecutive 1’s. Therefore, for any point in X{102n−1 1}n∈N , if there is two consecutive 1’s, then the number of 0’s between them must be even. The topological entropy of a shift space is defined based on Bn (XF ).

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Definition 4 Let XF be a shift space. Then, the topological entropy is given by h(XF ) = lim

n→∞

1 log |Bn (XF )|. n

Example 3 (Example 2 revisited) (a) For the full k-shift X∅ , it is easy to check that |Bn (X∅ )| = k n . Therefore, the topological entropy is log k. |Bn (X{11} )|. (b) For the golden mean shift X{11} , it is not straightforward to obtain  √  1+ 5 However, the topological entropy is found to be log ϕ, where ϕ = 2 is the golden ratio (see [19]). (c) For the even shift, the topological entropy is also log ϕ (see [19]). Other properties of a shift space are irreducibility and mixing. Definition 5 Let XF be a shift space. (i) XF is said to be irreducible if for any u, v ∈ L(XF ), there exists w ∈ L(XF ) such that uwv ∈ L(XF ). (ii) XF is said to be (topologically) mixing if for any u, v ∈ L(XF ), there exists N ∈ N such that for any n ≥ N , there exists w ∈ Bn (XF ) such that uwv ∈ L(XF ). By definition, mixing implies irreducibility. Example 4 (Example 2 revisited) (a) For the full k-shift X∅ , note that u0n v ∈ L(X∅ ) for any u, v ∈ L(X∅ ) and any n ∈ N. So, it is mixing, and hence irreducible. (b) For the golden mean shift X{11} , note that u0n v ∈ L(X{11} ) for any u, v ∈ L(X{11} ) and any n ∈ N. So, it is mixing, and hence irreducible. (c) The even shift is mixing, and hence irreducible based on [14]. Some shift spaces have been categorised into different types based on their construction. Examples are shifts of finite type, countable state Markov shifts (this is an extension of shift spaces over infinite alphabet), Dyck shifts and Motzkin shifts. These will be introduced later. The theory on shift spaces is originated to study general dynamical systems, but now, it has been used significantly in coding theory which involves data storage and transmission (see [19]).

2 Orbit Growth via Artin-Mazur Zeta Function Artin and Mazur [5] introduced a dynamical zeta function for a discrete dynamical system. The zeta function is a generating function for the number of periodic points per period.

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Definition 6 Let (X, T ) be a discrete dynamical system and F(n) be the number of periodic points of period n. Then, the Artin-Mazur zeta function for this system is defined as ∞   F(n) n z ζT (z) = exp n n=1 for z ∈ C. Note that ζT (z) is only defined if F(n) is finite for any n ∈ N. Furthermore, it is not guaranteed that its radius of convergence is non-zero. However, for most systems including the shift spaces discussed here, the zeta function has non-zero radius of convergence. Example 5 (Example 1(c) revisited) For the doubling map T on S1 , note that F(n) = 2n − 1. So, its Artin-Mazur zeta function is ζT (x) =

1−z . 1 − 2z

1 The radius of convergence is . 2 Regarding orbit growth, the analyticity and non-vanishing property of ζT (x) can determine the asymptotic behaviour of πT (x), MT (x) and MT (x). Theorem 1 Let (X, T ) be a discrete dynamical system, h > 0 be its topological entropy and ζT (x) be its Artin-Mazur zeta function. Suppose that (i) the radius of convergence for ζT (x) is e−h , and (ii) the function α(z) = ζT (x) · (1 − eh z) is analytic and non-zero in the region |z| < Re−h for some R > 1. Then, (a) (Prime Orbit Theorem) πT (x) ∼

eh(x+1) , (eh − 1)x

MT (x) ∼

e−γ x · α(e−h )

(b) (Mertens’ Orbit Theorem)

where γ is Euler-Mascheroni constant, and MT (x) = log x + log α(e−h ) + γ + C + o(1)

Counting Closed Orbits in Discrete Dynamical Systems

where C=

155

    1 1 log 1 − h|τ | + h|τ | e e τ

(C does converge). Outline of proof : The results in (a) and (b) follow from [24] on shifts of finite type, and [27] on Axiom A flows respectively. Both proofs do not depend on the nature of the systems, but rely on the properties of their Artin-Mazur zeta function and some combinatorial calculation. Therefore, the proofs can be generalised for any system having similar zeta function. The proofs are lengthy, so they are not included here. Example 6 (Examples 1(c) and 5 revisited) For the doubling map T on S1 , the 1 topological entropy is h = log 2. So, the radius of convergence of ζT (z) is , which 2 is indeed e−h . Furthermore, α(z) = 1 − z is analytic and non-zero in the region |z| < Re−h for any R ∈ (1, 2). By Theorem 1, πT (x) ∼ MT (x) ∼

2x+1 , x 2e−γ , x

and MT (x) = log x − log 2 + γ + C + o(1) for some constant C. Theorem 1 will be used to obtain similar results for shifts of finite type and certain countable state Markov shifts.

2.1 Example: Shifts of Finite Type Recall that the set F defines a shift space XF . However, F may not be unique; there n  may be a set F  ⊂ ∪∞ n=1 A such that F = F but XF  = XF . Example 7 Let A = {0, 1} and F = {11n | n ∈ N}. Note that any word in F begins with 11. Therefore, it is sufficient to set 11 as the only forbidden word. By taking F  = {11}, it is obtained that XF = XF  . This is indeed the golden mean shift in Example 2(b). If there is such finite F  , then the shift space is called a shift of finite type. Definition 7 Let XF be a shift space. XF is called a shift of finite type if there exists n a finite set F  ⊂ ∪∞ n=1 A such that XF  = XF .

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Example 8 (Example 2 revisited) (a) The full k-shift X∅ is a shift of finite type because F = ∅ itself is finite. (b) The golden mean shift X{11} is a shift of finite type because F = {11} itself is finite. (c) The even shift is not a shift of finite type. Suppose otherwise that such finite F  exists. Let  be the maximum length of words in F  . Note that 102−1 1 of length 2 + 1 must contain some word in F  . This is not possible since any sub-word from 102−1 1 of length at most  is allowed in the even shift. n If F is finite, then it is possible to obtain a finite set F  ⊂ ∪∞ n=1 A such that every  word in F has length exactly  for some  ∈ N. Indeed, set  as the maximum length of words in F. Define

F  = {wu | w ∈ F, u ∈ A−|w| }. It is easy to check that XF  = XF . From now on, given a shift of finite type XF , it is assumed that F is finite and contains words of length exactly . A shift of finite type XF can be expressed in form of a directed graph G(XF ) and transition matrix T (XF ). The graph G(XF ) is constructed by the following way: (i) the set of vertices are A \ F, and (ii) for each pair u, v ∈ A \ F, there is an edge from u to v if u 2 u 3 . . . u  = v1 v2 . . . v−1 . T (XF ) is a square matrix with index A \F such that its entry is given by:  T (XF )(u,v) =

Fig. 1 G (X{11} ) for the golden mean shift X{11}

1 if there exists an edge from u to v, 0 otherwise.

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Example 9 (Example 2(b) revisited) For the golden mean shift X{11} , the graph G(X{11} ) is given in Fig. 1. Using the index {00, 01, 10} in this order, the transition matrix is given by ⎛ ⎞ 110 T (X{11} ) = ⎝0 0 1⎠ . 110 Remark 3 The following are some comments on the graph representation and transition matrix of a shift of finite type: (a) T (XF ) depends on the order of the index. However, the results that follow from T (XF ) will be the same regardless of the order. (b) There is a different graph representation (and hence the transition matrix) for a shift of finite type. Again, the results that follow from any representation will be the same. In matrix theory, recall the notions of irreducibility and aperiodicity for an n × n real, non-negative matrix A as follows: (i) A is said to be irreducible if for every pair of indices i and j, there is  ∈ N such that (A )(i, j) > 0; (ii) An irreducible A is said to be aperiodic if for every pair of indices i and j there exists  ∈ N such that (Ak )(i, j) > 0 for any k ≥ . From Perron-Frobenius theorem, if a matrix A is irreducible, then there exists a positive real eigenvalue λ A (called the Perron value) such that |μ| ≤ λ A for other eigenvalue μ of A. Furthermore, if A is also aperiodic, then |μ| < λ A for other eigenvalue μ of A (see [17]). The properties of the transition matrix T (XF ) are related to the properties of the shift of finite type XF . Proposition 1 ([19]) Let XF be a shift of finite type and T (XF ) be its transition matrix. (a) XF is irreducible if and only if T (XF ) is irreducible. (b) XF is mixing if and only if T (XF ) is irreducible and aperiodic. The topological entropy and Artin-Mazur zeta function for a shift of finite type are obtained from its transition matrix. Theorem 2 ([19]) Let XF be an irreducible shift of finite type and T (XF ) be the irreducible transition matrix with Perron value λT (XF ) . Then, h(XF ) = log λT (XF ) . Theorem 3 ([19]) Let XF be a shift of finite type and T (XF ) be its transition matrix.

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Then, ζσ (z) =

1 det(I − z · T (XF ))

where I is the identity matrix. Example 10 (Example 2(b) and 9 revisited) For the golden mean shift X{11} , the transition matrix T (X{11} ) is indeed irreducible and aperiodic, so it is mixing (and hence irreducible). Furthermore, the Perron value for T (XF ) is the golden ratio ϕ, so the topological entropy is log ϕ. The Artin-Mazur zeta function is given by ζσ (z) =

1 . 1 − z − z2

Now, it is ready to focus on the asymptotic behaviour of the counting functions for shifts of finite type. This is restricted to the mixing shifts of finite type only. Theorem 4 ([24, 27]) Let XF be a mixing shift of finite type with the topological entropy h. Then, (a) (Prime Orbit Theorem) πσ (x) ∼

eh(x+1) , (eh − 1)x

Mσ (x) ∼

e−γ x · α(e−h )

(b) (Mertens’ Orbit Theorem)

where γ is Euler-Mascheroni constant and α(z) = ζσ (z) · (1 − e−h ), and Mσ (x) = log x + log α(e−h ) + γ + C + o(1) for some constant C. Proof Note that det(I − z · T (XF )) =

(1 − μz) where μ runs over the eigenvalμ

ues of the transition matrix T (XF ). Based on the product, it is possible to consider non-zero μ only. From now on, it is assumed that μ is non-zero. By Theorem 3, 1 . ζσ (z) = (1 − μz) μ

This implies that ζσ (z) has pole at μ−1 for each eigenvalue μ, including the Perron value λT (XF ) . Furthermore, since XF is mixing, |μ| < λT (XF ) for other eigenvalue μ. This implies that λ−1 T (XF ) is the closest pole to the origin. Therefore, the radius of

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−h convergence of ζσ (z) is λ−1 from Theorem 2. So, condition T (XF ) , which is indeed e (i) of Theorem 1 is satisfied. Consider α(z) = (1 − λT (XF ) z) · ζσ (z). Note that α(z) has pole at point μ−1 for each eigenvalue μ other than λT (XF ) . Since |μ| < λT (XF ) , by setting

   λT (XF ) , R ∈ 1, min μ |μ| it is obtained that α(z) is analytic in the region |z| < R · λ−1 T (XF ) . Furthermore, α(z) is non-zero everywhere. So, condition (ii) of Theorem 1 is satisfied. So, the result follows from Theorem 1.  Remark 4 There is a similar result for shifts of finite type which are irreducible, but not mixing. However, a slight modification is needed to be done on Theorem 1 and the proof is not straightforward.

2.2 Example: Countable State Markov Shifts Recall that for a shift space, the alphabet A is set to be finite. However, it is possible to extend the definition for countably infinite A, usually taken to be N. A countable state Markov shift is an example of shift spaces over countably infinite alphabet. This is an extension of the concept of shifts of finite type. For this type of shift spaces, the rules or restrictions are not commonly given in form of forbidden words, but in form of a transition matrix (with each entry either 0 or 1). Definition 8 Let A = N and T be an N × N transition matrix. Define the set XT = {x ∈ AZ | T(xi ,xi+1 ) = 1 ∀i ∈ Z}. The pair (XT , σ) is called a countable state Markov shift. Intuitively, the transition matrix T indicates which letter can appear after a given letter. For any pair of letters a, b ∈ A, b can appear after a if the (a, b)-entry of T is 1. Similarly, b cannot appear after a if the (a, b)-entry of T is 0. Formally speaking, (i) ab is allowed in XT if and only if T(a,b) = 1, and (ii) ab is forbidden in XT if and only if T(a,b) = 0. (See [17] for more details.) A countable state Markov shift XT can also be represented as a directed graph G(XT ). The graph G(XT ) is constructed by the following way: (i) the set of vertices are A = N, and (ii) for each pair of letters a, b ∈ A, there is an edge from a to b if T(a,b) = 1.

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Fig. 2 G (XT ) for the renewal shift

Example 11 Sarig [25] introduced the renewal shift XT with ⎛

1 ⎜1 ⎜ ⎜0 ⎜ T = ⎜0 ⎜ ⎜0 ⎝ .. .

1 0 1 0 0 .. .

1 0 0 1 0 .. .

1 0 0 0 1 .. .

⎞ ... . . .⎟ ⎟ . . .⎟ ⎟ . . . .⎟ ⎟ ⎟ . . .⎠ .. .

Based on T , (i) for any a ∈ N, 1a is allowed in XT , and (ii) for any b ∈ N and b ≥ 2, b(b − 1) is allowed in XT . The graph G(XT ) is shown in Fig. 2. The irreducibility and aperiodicity of the transition matrix determine the irreducibility and mixing of the countable state Markov shift. Proposition 2 ([17]) Let XT be a countable state Markov shift. (a) XT is irreducible if and only if T is irreducible. (b) XT is mixing if and only if T is irreducible and aperiodic. However, since T is infinite-dimensional, it does not make sense to consider its eigenvalues. Therefore, the definition of Perron value for T is different in this case. Definition 9 Let XT be an irreducible countable state Markov shift. The Perron value for T is defined as λT = lim

n→∞

 n

(T n )(i,i)

regardless of the choice of index i ∈ N. Remark 5 The Perron value λT may not be finite. For this section, it is assumed that λT is finite. The topological entropy of a countable state Markov shift is related to the Perron value. Theorem 5 Let XT be an irreducible countable state Markov shift with Perron value λT . Then, h(XT ) = log λT .

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Example 12 (Example 11 revisited) For the renewal shift, T is irreducible and aperiodic. So, XT is mixing, and hence irreducible. For any n ∈ N, ⎛

2n−1 2n−1 2n−1 2n−1 ⎜2n−2 2n−2 2n−2 2n−2 ⎜ ⎜ .. .. .. .. ⎜ . . . . ⎜ ⎜ 2 2 2 2 ⎜ ⎜ 1 1 1 1 Tn =⎜ ⎜ 1 0 0 0 ⎜ ⎜ 0 1 0 0 ⎜ ⎜ 0 0 1 0 ⎜ ⎜ 0 0 0 1 ⎝ .. .. .. .. . . . .

⎞ ... . . .⎟ ⎟ ⎟ . . .⎟ ⎟ . . .⎟ ⎟ . . .⎟ ⎟. . . .⎟ ⎟ . . .⎟ ⎟ . . .⎟ ⎟ . . .⎟ ⎠ .. .

Therefore, (T n )(1,1) = 2n−1 . Consequently, the Perron value is 2, and hence the topological entropy is log 2. Unfortunately, there is no general closed form of the Artin-Mazur zeta function for a countable state Markov shift. This is because (i) countable state Markov shifts are a broad type of shift spaces; There is further classification, such as transient, recurrent and positive recurrent countable state Markov shifts [17]. However, the classification has no common property that is useful to determine the general form of their zeta function. (ii) infinite-dimensional matrix is difficult to be analysed; A shift of finite type is well-represented by a finite matrix, which is easy to be analysed, for example, in term of its eigenvalues. Its eigenvalues lead to its zeta function. This is not the case for a countable state Markov shift, since it is almost impossible to consider all eigenvalues of its transition matrix. (iii) the number of periodic points per period F(n) may not be finite. Shift spaces over finite alphabet, including shifts of finite type, always have finite F(n) for all n ∈ N. This is not necessarily true if the alphabet is infinite. In fact, up to date, there is no useful equivalent condition for a countable state Markov shift to have finite F(n) for all n ∈ N. Therefore, given a countable state Markov shift, its zeta function has to be obtained directly by using F(n). In case when F(n) is finite, it is obtained by F(n) = Trace(T n ). Example 13 (Examples 11 and 12 revisited) For the renewal shift, note that F(n) = 2n − 1, and hence

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ζσ (x) =

1−z . 1 − 2z

This is the same as the zeta function for the doubling map T on S1 in Example 6. Therefore, 2x+1 πσ (x) ∼ , x Mσ (x) ∼

2e−γ , x

and Mσ (x) = log x − log 2 + γ + C + o(1) for some constant C.

2.3 Discussion In this section, it has been seen that the Artin-Mazur zeta function of a system can be used to determine the asymptotic behaviour of the counting functions. Although this approach seems straightforward, some problems may arise due to the zeta function itself. These include: (i) the zeta function is difficult to be obtained; In many systems, it is difficult to obtain the number of periodic points F(n) directly to calculate its zeta function. So, the zeta function may be found by using some dynamical properties of the system and advanced combinatorial method. This may turn out difficult. (ii) the zeta function is sophisticated to be analysed. The zeta function for the system may be complicated, so analysing its analyticity and non-vanishing property may be difficult. In this section, shifts of finite type and renewal shift have rational zeta function, thus easy to be analysed. However, some systems indeed have complicated zeta function, including Dyck and Motzkin shifts.

3 Orbit Growth via Number of Periodic Points per Period Recall that in Definition 2, the counting functions are expressed in term of the number of closed orbits O(n). Based on [3, 4], O(n) is related to the number of periodic points F(n) by 1  n  F(d) μ O(n) = n d|n d

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where μ : N → {−1, 0, 1} is the Möbius function (see [12]). Using this fact, it is obtained that (i) πT (x) = (ii) MT (x) = (iii) m T (a) =

 1  n  F(d), μ n d|n d n≤x

 1  n  F(d), μ nehn d|n d n≤x

∞  n=1

1 n a+1 ehn

 n  F(d). μ d d|n

In this approach, an estimate for F(n) is obtained. The estimate is then used to obtain the order of magnitude for the counting functions. Theorem 6 Let (X, T ) be a discrete dynamical system with the topological entropy h > 0. Define F(n) to be the number of periodic points of period n. If F(n)  ehn , then (a) (Prime Orbit Theorem) πT (x) 

ehx , x

(b) (Mertens’ Orbit Theorem) MT (x)  log x, and (c) (Meissel’s Orbit Theorem) m T (a) 

1 a

as a → 0+ . Outline of proof : The results follow from [3, 4] on Dyck and Motzkin shifts. Again, those proofs do not depend on the nature of the systems, but rely on the estimate on F(n) and some combinatorial calculation. Therefore, the proofs can be generalised for any system having similar estimate of F(n). The proofs are lengthy, so they are not included here.

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Example 14 (Examples 1(c) and 5 revisited) Recall that for the doubling map on S1 , 1−z it is obtained that F(n) = 2n − 1, h = log 2 and ζT (z) = . Note that 2n−1 ≤ 1 − 2z F(n) ≤ 2n , and hence F(n)  2n . By Theorem 6, (i) πT (x) 

2x , x

(ii) MT (x)  log x, and (iii) m T (a) 

1 a

as a → 0+ . Theorem 6 will be used to obtain similar results for Dyck and Motzkin shifts.

3.1 Example: Dyck Shifts Dyck shift is a shift space such that its alphabet consists of matching symbols. The alphabet is equipped with a monoid operation. The forbidden words are defined based on the monoid operation. M Definition 10 For M > 1, let A = ∪i=1 {li , ri } be an alphabet with M symbols li ’s and M matching symbols ri ’s. Let S be a monoid (with identity 1 and zero 0) generated by A under the following operation:  1 if i = j, (i) li ◦ ri = 0 otherwise, (ii) 1 ◦ a = a ◦ 1 = a for all a ∈ A ∪ {0, 1}, and (iii) 0 ◦ a = a ◦ 0 = 0 for all a ∈ A ∪ {0, 1}. n Define the map r ed : ∪∞ n=1 A → S as

r ed(a1 a2 ...ak ) = a1 ◦ a2 ◦ ... ◦ ak . Define the set

D M = {x ∈ AZ | r ed(w) = 0 for all w ≺ x}.

The pair (D M , σ) is called the Dyck shift.

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In some literatures, li is called a left delimiter while ri is called a right delimiter. n For the Dyck shift, w ∈ ∪∞ n=1 A is forbidden if and only if r ed(w) = 0. It is not a shift of finite type. Example 15 In Dyck shift D2 , (a) r1l2 r1l1 is forbidden because r ed(r1l2 r1l1 ) = 0, and (b) l1l2 r2 l2 is allowed because r ed(l1l2 r2 l2 ) = l1l2 = 0. Remark 6 In Definition 10, the construction of the Dyck shift requires that M > 1. Otherwise, if M = 1, then the shift space is the full 2-shift, which is trivial. The topological entropy and Artin-Mazur zeta function for the Dyck shift were found by Krieger [18] and Keller [16] respectively. Theorem 7 ([18]) For the Dyck shift D M , the topological entropy is h(D M ) = log(M + 1). Theorem 8 ([16]) For the Dyck shift D M , the Artin-Mazur zeta function is √ 1 − 4M z 2 ) ζσ (z) = . √ (1 − 2M z + 1 − 4M z 2 )2 2(1 +

Note that the zeta function is not rational and seems sophisticated. Therefore, the approach via zeta function is difficult to be used in this case. The number of periodic points F(n) for the Dyck shift was found by Hamachi and Inoue [11]. Theorem 9 For the Dyck shift D M , the number of periodic points of period n is ⎧  n  i   n−1 ⎪ 2 ⎨2 (M + 1)n − i=0 if n is odd, M i F(n) =   n     2 n i n ⎪ ⎩2 (M + 1)n − i=0 M + nn M 2 if n is even. i 2

Alsharari et al [4] proved that F(n)  (M + 1)n for the Dyck shift D M . With that, Theorem 6 can be used to obtain the order of magnitude of the counting functions for the Dyck shift.

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Theorem 10 ([4]) For the Dyck shift D M , (a) (Prime Orbit Theorem) πσ (x) 

(M + 1)x , x

(b) (Mertens’ Orbit Theorem) Mσ (x)  log x, and (c) (Meissel’s Orbit Theorem) m σ (a) 

1 a

as a → 0+ .

3.2 Example: Motzkin Shift Motzkin shift is an extension to Dyck shift, but its alphabet contains additional symbols. The alphabet is equipped with similar monoid operation. N M Definition 11 For M > 1 and N ≥ 0, let A = (∪i=1 {li , ri }) ∪ (∪i=1 {n i }) be an alphabet with M symbols li ’s, M matching symbols ri ’s and N additional symbols n i ’s. Let S be a monoid (with identity 1 and zero 0) generated by A under the following operation:  1 if i = j, (i) li ◦ ri = 0 otherwise, (ii) 1 ◦ a = a ◦ 1 = a for all a ∈ A ∪ {0, 1}, (iii) 0 ◦ a = a ◦ 0 = 0 for all a ∈ A ∪ {0, 1}, (iv) n i ◦ n j = 1 for all i and j, and (v) n i ◦ a = a ◦ n i = a for all a ∈ A ∪ {0, 1} and all i. n Define the map r ed : ∪∞ n=1 A → S as

r ed(a1 a2 ...ak ) = a1 ◦ a2 ◦ ... ◦ ak . Define the set M M,N = {x ∈ AZ | r ed(w) = 0 for all w ≺ x}. The pair (M M,N , σ) is called the Motzkin shift. Similarly, li , ri and n i are called a left delimiter, a right delimiter and a neutral symbol respectively.

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Note that if N = 0, then the Motzkin shift M M,0 is the Dyck shift D M . Similarly, Motzkin shift is not a shift of finite type. Remark 7 Again, in Definition 11, it is required that M > 1 by construction. In fact, if M = 1, then the shift space is the full (N + 2)-shift, which is trivial. Example 16 For the Motzkin shift M2,2 , (a) r1l2 n 1 n 2 r1l1 is forbidden because r ed(r1l2 n 1 n 2 r1l1 ) = 0, (b) n 1l1l2 r2 l2 n 2 is allowed because r ed(n 1l1l2 r2 l2 n 2 ) = l1l2 = 0. The topological entropy and the Artin-Mazur function for the Motzkin shift were found by Inoue [13]. Theorem 11 ([13]) For the Motzkin shift M M,N , the topological entropy is h(D M ) = log(M + N + 1). Theorem 12 ([13]) For the Motzkin shift M M,N , the Artin-Mazur zeta function is ζσ (z) =



(1 − N z)2 − 4M z 2 )  . (1 − (2M + N )z + (1 − N z)2 − 4M z 2 )2 2(1 − N z +

As before, the approach via zeta function is difficult to be used in this case. The number of periodic points F(n) for the Motzkin shift was found by Inoue [13] (it is lenghty, so it is not included here). Alsharari et al [3] again proved that F(n)  (M + N + 1)n for the Motzkin shift M M,N . Based on Theorem 6, this implies the order of magnitude of the counting functions for the Motzkin shift. Theorem 13 ([3]) For the Motzkin shift M M,N , (a) (Prime Orbit Theorem) πσ (x) 

(M + N + 1)x , x

(b) (Mertens’ Orbit Theorem) Mσ (x)  log x, and (c) (Meissel’s Orbit Theorem) m σ (a)  as a → 0+ .

1 a

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3.3 Other Results In Theorem 6, the estimate F(n)  ehn only leads to the order of magnitude of the counting functions. This result produces bounds for the counting functions, but the bounds are not sharp. Pakangkapon and Ward [23] obtained sharper bounds for the counting functions by using orbit monoids. Theorem 14 ([23]) Let (X, T ) be a discrete dynamical system. Define F(n) to be the number of periodic points of period n. Suppose that there exists constants C1 > 0, h > 0 and h  < h such that 

F(n) = C1 ehn + O(eh n ). Then, (a) C1 hn O(n) = e +O n



(b) eh(x+1) +O πT (x) = x(eh − 1)



eh n n 

 ,

ehx



3

x2

,

and (c)

  1 . MT (x) = C1 log x + O x

There are many results in [23] which are not included here, including m T (a). However, those results involve the notion of orbit monoid, which is not discussed here. Regarding the Dyck shift, Akhatkulov et al [2] found better results for the asymptotic behaviour of the counting functions. Theorem 15 ([2]) For the Dyck shift D M , (a)

  n  R F(n) = 2(M + 1)n 1 + O √ n √ 2 M R= < 1, M +1

where

(b) πσ (x) =

2(M + 1)x Mx

   1 1+O , x

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(c) Mσ (x) = 2 log x + C + εx for some constant C > 0 and εx ∼ (d) m σ (a) =

1 , and x

1 +C +O a



a a+1



for a > 0. In fact, up to date, the authors have found the asymptotic equivalence of πσ (x), Mσ (x) and Mσ (x) for Dyck and Motzkin shifts. Currently, the result is under review.

3.4 Discussion In this section, it has been seen that the number of periodic points F(n) of a system can be used to determine the order of magnitude for the counting functions. However, similar to approach via zeta function, some problems may arise due to F(n) itself. For example, (a) F(n) may be difficult to be obtained for some systems, and (b) even it has been found, it may be difficult to obtain estimate for F(n) if it is sophisticated. In fact, this approach only produces results on the order of magnitude of the counting functions. This is less precise compared to the results on the asymptotic equivalence of the counting functions via zeta function.

4 Conclusion It has been seen that the asymptotic behaviour of the counting functions for closed orbits in a discrete dynamical system can be obtained via either Artin-Mazur zeta function or the number of periodic points per periods. Although this chapter demonstrates some examples from shift spaces, both approaches can be applied to other systems. Up to date, there are some results for other systems such as suspension flows [24], Axiom A flows [27, 28], toral automorphisms [15, 22, 30], some algebraic dynamical systems [8, 9] and reference therein. There should be more further work carried out regarding this research interest. Some suggestions are: (i) to obtain sharper result for the asymptotic behaviour of the counting functions, especially the Meissel’s sum m T (a),

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(ii) to obtain the asymptotic behaviour of the counting functions for other types of shift spaces, and (iii) to obtain the asymptotic behaviour of the counting functions for other types of discrete dynamical systems. Acknowledgements The authors would like to acknowledge the grants FRGS/1/2017/STG06/ UKM/01/1 and FRGS/1/2017/STG06/UKM/02/2 by Ministry of Higher Education, Malaysia, and DIP-2017-011 by Universiti Kebangsaan Malaysia for financial support in this research.

References 1. Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114(2), 309 (1965) 2. Akhatkulov, S., Noorani, M.S.M., Akhadkulov, H.: An analogue of the prime number, Mertens and Meissels theorems for closed orbits of the Dyck shift. In: AIP Conference Proceedings, vol. 1830 (2017) 3. Alsharari, F., Noorani, M.S.M., Akhadkulov, H.: Analogues of the Prime Number Theorem and Mertens Theorem for closed orbits of the Motzkin shift. Bull. Malays. Math. Sci. Soc. 40(1), 307319 (2017) 4. Alsharari, F., Noorani, M.S.M., Akhadkulov, H.: Estimates on the number of orbits of the Dyck shift. J. Inequalities Appl. 2015(1), 112 (2015) 5. Artin, M., Mazur, B.: On periodic points. Ann. Math. 81(1), 8299 (1965) 6. Devaney, R.: An Introduction of Chaotic Dynamical Systems, 2nd edn. Addison-Wesley, California (1989) 7. Dzul-Kifli, S.C., Good, C.: On devaney chaos and dense periodic points: period 3 and higher implies chaos. Am. Math. Mon. 122(8), 773780 (2015) 8. Everest, G., Miles, R., Stevens, S., Ward, T.: Orbit-counting in non-hyperbolic dynamical systems. J. fur die Reine und Angew. Math. 608, 155182 (2007) 9. Everest, G., Miles, R., Stevens, S., Ward, T.: Dirichlet series for finite combinatorial rank dynamics. Trans. Am. Math. Soc. 362, 199227 (2009) 10. Hadamard, J.: Sur la distribution des zros de la fonction ζ(s) et ses consquences arithmtiques. Bulletin de la Socit Mathmatique de France 24, 199–220 (1896) 11. Hamachi, T., Inoue, K.: Embedding of shifts of finite type into the Dyck shift. Monatshefte für Mathematik 145(2), 107129 (2005) 12. Hardy, G.H., Wright, E.M.: An Introduction to Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008) 13. Inoue, K.: The zeta function, periodic points and entropies of the Motzkin shift. ArXiv Mathematics e-prints. http://arxiv.org/abs/math/0602100 (2010). Last accessed 6 Sep 2018 14. Ismail, M.S., Dzul-Kifli, S.C.: The dynamical properties of even shift space. In: AIP Conference Proceedings vol. 1870(1) (2017) 15. Jaidee, S., Stevens, S., Ward, T.: Mertens theorem for toral automorphisms. Proc. Am. Math. Soc. 139(5), 16 (2011) 16. Keller, G.: Circular codes, loop counting, and zeta-functions. J. Comb. Theory Ser. A 56(1), 7583 (1991) 17. Kitchens, B.P.: Symbolic Dynamics: One-sided. Two-sided and Countable State Markov Shifts. Springer, Berlin (1998) 18. Krieger, W.: On the uniqueness of the equilibrium state. Math. Syst. Theory 8, 97 (1974) 19. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)

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20. Lindqvist, P., Peetre, J.: On a number-theoretic sum considered by Meissel–a historical observation. Nieuw Arch. Wisk. 15(3), 175–179 (1997) 21. Mertens, F.: Ein Beitrag zur analytischen Zahlentheorie. J. reine angew. Math. 78, 46–62 (1874) 22. Noorani, M.S.M.: Mertens theorem and closed orbits of ergodic toral automorphisms. Bull. Malaysian Math. Soc. 22, 127133 (1999) 23. Pakapongpun, A., Ward, T.: Functorial orbit counting. J. Integer Seq. 12(2), 120 (2009) 24. Parry, W., Pollicott, M.: Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics. Société Mathématique de France, France (1990) 25. Sarig, O.: Zeta functions for the renewal shift. Kyoto Univ. Math. Anal. Lab. 1404, 98–104 (2004) 26. Sharkovskii, A.N.: Coexistence of cycles of a continuous map of the line onto itself. Ukranian Math. Z. 16, 61–71 (1964) 27. Sharp, R.: An analogue of Mertens theorem for closed orbits of Axiom A flows. Boletim da Sociedade Brasileira de Matemática 21(2), 205229 (1991) 28. Sharp, R.: Prime orbit theorems with multi-dimensional constraints for Axiom A flows. Monatshefte fr Mathematik 114(34), 261304 (1992) 29. Vallée Poussin, C.J.: Recherches analytiques de la thorie des nombres premiers. Annales de la Societe Scientifique de Bruxelles 20, 183256, 281352, 363397; 21, 351–368 (1896) 30. Waddington, S.: The prime orbit theorem for quasihyperbolic toral automorphisms. Monatshefte für Mathematik 112(3), 235248 (1991)

Computational Dynamical Systems

Computational Dynamical Systems Using XPPAUT Ojonubah James Omaiye and Mohd Hafiz Mohd

Abstract This article is written as a guide for researchers on how to employ the techniques in numerical continuation and bifurcation analysis using XPPAUT. This is a free software package to solve and analyse dynamical systems numerically. The article starts with a gentle introduction to XPPAUT, how to install this software, and an overview of the numerical routines. By using ordinary differential equations as an example, readers are guided to solve for the steady-states and also perform some graphical analysis, such as phase portraits and time-series plots. Thereafter, the sections gradually increase in complexity, covering general steps in bifurcation analysis and how to produce complete bifurcation diagrams, particularly co-dimension one and co-dimension two bifurcation plots. Keywords Bifurcation analysis · XPPAUT · Dynamical systems · Ecological model

1 Introduction In this article, we employed XPPAUT package to solve examples on ordinary differential equations. The article is planned to serve as a laboratory manual, which can be used as a resource material to introduce the readers to XPPAUT tools. XPPAUT has the advantages over other packages for many reasons. Firstly, it is an open source and its operation is compatible with Unix and Window environments as well as Mac operating system by means of appropriate X-windows server. In addition, XPPAUT is the only package that is able to integrate the bifurcation package AUTO. Since XPPAUT is an open source package, it can be freely downloaded from the developer’s website: http://www.math.pitt.edu/~bard/bardware/binary/latest/xppwin.zip;

O. J. Omaiye · M. H. Mohd (B) School of Mathematical Sciences, Universiti Sains Malaysia, 11800, USM Gelugor, Penang, Malaysia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_10

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information on how to install XPPAUT and tutorials could be found in this website. Also, details of XPPAUT are described in the developer’s book [1].

2 Basics of Solving Ordinary Differential Equations 2.1 Creating the ODE File Consider a system of ordinary differential equations (ODE), as an example [2]:  x β(1 − m)yx dx = αx 1 − − , dt k 1 + a(1 − m)x dy cβ(1 − m)yx = −γ y + . dt 1 + a(1 − m)x

(1)

System (1) is an extension of the Rosenzweig-MacArthur model in [3]. The model incorporates a prey-refuge such that x and y, respectively, represent the densities of prey and predator at time, t. Also, α denotes the growth rate of the prey; k signifies the carrying capacity of prey; γ represents the death rate of the predator; β denotes the attack rate of the predator; a is the half-saturation constant; c is the prey conversion factor (i.e., for every prey consumed, a newborn predator will be produced). Given mx as the prey population, the term m represents the fraction of the prey refuge that can protect the population of prey. In this case, let m ∈ [0, 1) such that (1 − m)x becomes the available number of prey that can be consumed by the predator. If there is no prey refuge present, then m = 0 and Eq. (1) reduces to the standard form of the Rosenzweig-MacArthur model. Recall that we can solve Eq. (1) for steady-states and check for its stability by using analytical method. Instead of analytical method, we can use XPPAUT to explore the dynamical behaviours of the system of Eq. (1). To use XPPAUT for this purpose, first we create an input file, which informs the program about the variables, parameters names and the equations. The input files must have the file extension .ode which we refer to as the ODE file. Below is an example of the ODE file for Eq. (1): # preyrefuge.ode # # right hand sides dx/dt=(a1*x)*(1-(x/k))-((b*(1-m)*y*x)/(1+a*(1-m)*x)) dy/dt=-e*y+((c*b*(1-m)*x*y)/(1+a*(1-m)*x)) # In Equation (\ref{e1}), $a1=alpha$, $b=beta$, $e=gamma$ # parameters par $m=0.3,a1=10,k=100,a=0.02,e=0.05,b=0.6,c=0.02$ #

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# some initial conditions init $x=1,y=1$ # # some numeric @ dt=0.1,nout=25,total=1000,meth=cvode,tol=1e-10,tol=1e-10 @ maxstor=1000,bound=5000 @ $xp=t,yp=x,xlo=0,ylo=0,xhi=100,yhi=100$ # we are done done

The line shown with a symbol # simply indicates further comments on this coding. Both the parameter values and init statement given are discretionary, which by default are set to zero if no choice of parameter values or initial conditions are set. In order to type in the first file exactly as it appears, use a text editor such as Wordpad or Notepad, then name the file as preyrefuge.ode and save it.

2.2 Installing and Running the Program Details on XPPAUT installation procedures can be found on the website below: http://www.math.pitt.edu/~bard/xpp/installonwindows.html The following details illustrate some preliminary steps on installing XPPAUT: 1. Install X11 server, Xming using the link: http://sourceforge.net/projects/xming/ files/latest/download. – You must ensure that you run the Xming program from Start → All Programs/All Apps → Xming → Xming. Then, confirm that the Xming program is smoothly running. When “X” icon is displayed in the Windows Taskbar Notifications Area, then the Xming is perfectly installed. 2. Download XPPAUT “xppwin.zip” using the link: http://www.math.pitt.edu/ ~bard/bardware/binary/latest/xppwin.zip – Right-click on the file to extract or unzip the downloaded “xppwin.zip” file and then click to “Extract All...”, which will show a “xppall” folder. – Right-click the extracted xppall folder and Copy; then open your computer’s “C:” drive, right-click in a blank space and then Paste the folder in the root of the “C:”. – Note: Place the “xppall” folder such that the full path to the folder is “C:\ xppall”. – Open the “xppall” folder to search for “xpp - Shortcut” file, when found, rightclick on the file and then “Copy” the shortcut. Then right-click on a blank space anywhere on the Desktop and “Paste” the shortcut on the Desktop.

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3. To get started with the XPP/XPPAUT, first drag and drop the ODE file onto the “xpp - Shortcut” on your Desktop.

2.3 The Main Window As illustrated in Fig. 1, the Main Window has several features for graphics, menus, and many other features and buttons. To input commands, you may either use mouse to click on the menu items at the left column or simply tap the shortcuts on the keyboard. Both options either using full commands or keyboard shortcuts are included in this guide. However, you may later prefer to use keyboard shortcuts more often as you get used to XPPAUT. The keyboard shortcut is easy to use as you just need to press capital letters accordingly; for instance, the commands for Initialconds and phAsespace, can be represented by the capital letters I and A. The upper section of the Main Window displays typed input such as parameter values (Param), initial conditions (ICs) and equations (Eqns). Also, at the bottom most section of the Main Window is for information that concerns several things and short description of the highlighted menu items. In order to easily change your parameters and initial data, three little slider boxes with the words parameter are provided.

Fig. 1 The Main Window

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Quitting the Program

After the use of the XPPAUT, you can exit this software or programme by clicking on File, Quit, and then Yes (or simply press the keyboard shortcuts F, Q, Y, respectively).

2.4 Solving the Equations and Graphical Plotting In this section, we will solve the Eq. (1), select different initial conditions using a mouse, save the plots in different files, and create some files for printing.

2.4.1

Setting the Main Windows for Plotting

The Main Window displays a box with axes X and Y. By default, the window title is indicated as X versus T , which shows that X represents the variable on the vertical axis and T on the horizontal axis. Instead of X , you may wish to plot Y versus T . In this case, simply click on the command, select Xi versus t (i.e., click X) and then key in Y. On the basis of Eq. (1), use 0–100 along the horizontal axis and 0–100 along the vertical axis as the plotting range. You may want to change this range by clicking on Viewaxes and then click 2D (or simply press the keyboard shortcuts V and then followed by 2). A dialog box will appear and then fill it in as follows: X-axis: T Xmax: 100 Y-axis: X Ymax: 100 Xmin: 0 Xlabel: X Ymin: 0 Ylabel: T When you are through, click OK.

2.5 Computing the Solutions: Time Series Plot In order to illustrate the solutions of Eq. (1) in terms of time series plot, let us do the following steps below: 1. Launch XPPAUT by dragging and dropping ODE file onto the “xpp - Shortcut” on your Desktop and set up the view accordingly. 2. To solve and plot the graph of X versus T , press Initialconds - (G)o (or press I then G).

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Fig. 2 Time series plot for prey population density, x at time t

Figure 2 shows the time series plot for prey population density, x at time t based on the solution of Eq. (1). To plot predator population density, y on the same axis, click Graphic stuff – (A)dd curve; a New Curve window will be displayed as shown in Fig. 3. Now, adjust Y-axis section from X to Y. You may set another colour for the curve y. To do this, change the code at Color column from 0 (black) to 1 (red) and then click OK. Figure 4 is the full set of solutions for Eq. (1), which represents both prey and predator species (x and y, respectively). In order to label this graph, click Viewaxes – 2D (or simply press the keyboard shortcuts V and then followed by 2). In the 2D View window, input T against the Xlabel and X,Y against the Ylabel. When you are through, click OK (as illustrated in Fig. 5). Figure 6 shows labeled axes of final time series plot. Finally, to save this figure, click Graphic stuff - (P)ostscript and click OK at the Postscript parameters window. A new dialog box will appear; select file and click OK as in Fig. 7. The .ps file is saved in xppall folder in C-Windows as auto file.

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Fig. 3 Options to change axes, colour and line type for time series

3 Computing the Solution: Phase Portrait In order to have a better understanding of the dynamics of a given system under consideration, we can also use XPPAUT to plot the phase portrait, which illustrates the behaviour of the model from different initial conditions. To do this, the following steps can be employed to plot the phase portrait. 1. Launch XPPAUT by dragging and dropping the ODE file onto the “xpp - Shortcut” on your Desktop and set up the view accordingly. 2. First, plot graph of X versus T by clicking Initialconds - (G)o. 3. Change the axes to Y versus X and re-label the graph. To do this, click Viewaxes and then click 2D (or simply press the keyboard shortcuts V and then followed by 2). Immediately, a dialog box will be displayed for you to fill the fields as shown below: X-axis: X Xmax: 100 Y-axis: Y Ymax: 100 Xmin: 0 Xlabel: X Ymin: 0 Ylabel: Y When you are through, click OK. Figure 8 shows the output of the steps.

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Fig. 4 Time series plot for prey population densities, x and y at time t

4. A phase portrait is not complete without nullclines plotted along with it. A nullcline can be added for X and Y by clicking nullcline – (N)ew (or simply press the keyboard shortcuts N and then followed by another N). The nullclines of Eq. (1) are shown by the red and green curves as in Fig. 9. As shown in Fig. 10, the point of intersection between the red and green curves (orange circle) corresponds to two-species steady-state (i.e., coexistence of species). From the knowledge of local stability analysis, the two-species steady state is unstable. We can establish this fact using XPPAUT by conducting stability analysis numerically. This can be done by clicking on Sing pts and then select Mouse. Thereafter, click inside the orange circle as illustrated in Fig. 10. An Equilibria window will be displayed as shown in Fig. 11. Observe that on the Equilibria window, the two species steady-state (X, Y ) is (6.4935, 24.287). The summary of the stability analysis result of this steady-state is shown in CWindows in Fig. 12 (black-coloured window). As clearly shown at the bottom of this window, the eigenvalues are complex conjugate with positive real part which implies that it is an unstable spiral. 5. Having carried out the stability analysis, we can then return to our phase portrait and complete the diagram. The direction field is also important in the phase portrait. To add direction field, click Dir.field/flow then select (D)irect field and

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Fig. 5 Options to change axes of the time series plot

then Enter Grid number of your choice (Fig. 13). Most often, the default value of 16 is good to be used as Grid number. Figure 14 shows the resulting output together with the direction fields. 6. Now, it is possible to study how the trajectories of the model evolve over time from the direction fields. To illustrate these trajectories from different initial conditions, press Initialconds – m(I)ce. Thereafter, click at different locations on the quadrant to generate these trajectories. Different trajectories correspond to the different initial conditions used to solve Eq. (1). When you are done, press ESC on your keyboard. Figure 15 is the resulting output of clicking at different starting locations on the quadrant. 7. Usually, it is necessary to adjust X and Y scales as desired before saving the phase portrait. This can be done by clicking on Viewaxes and then click 2D. Immediately, a window will be displayed as in Fig. 16. Within the 2D View window, select your desired range of values for X and Y . For example, let our new range of values for X and Y be Xmax: 40 and Ymax: 40.

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Fig. 6 Time series plot showing labelled axes

8. Having chosen the new range of values as in the previous page, our phase portrait can now be saved by clicking Graphic stuff – (P)ostscript. Then, in the Save the data window, click O K and select or rename it as a new file. When you are through, click O K .

4 Bifurcation and Continuation: Co-dimension 1 Bifurcation Diagram Using XPPAUT, we can track the stable and unstable steady-states and bifurcation points of the system as some parameters are varied. To achieve this, we employ numerical continuation and bifurcation analysis to describe how solutions to the systems such as in Eq. (1) change over parameters. In this case, XPPAUT can conveniently be used for this purpose since it serves as interface to many of the features that are found in the AUTO package for continuation/bifurcation analysis. Although AUTO may be considered as the best package for continuation/bifurcation analysis, the stand-alone version of AUTO still has some problems. One of such problem is associated with coding compilation requirement of the equations in FORTRAN computer language. Thus, AUTO implemented in XPPAUT can be use as a powerful

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Fig. 7 Options to save the data in the computer directory

alternative. However, the results should not be considered as an ultimate but need to be observed with caution. Detailed discussion on bifurcation analysis can be found in [4].

4.1 General Steps for Bifurcation Analysis 1. In the first place, click Initialconds – Go and then Initialconds – Last. Carry out this step several times to ensure that the solutions achieve stable steady-state. For limit cycles, integrate over one full period only to obtain a good estimate of the period. 2. Now, click File – Auto (or simply press the keyboard shortcuts F and then A) to display the AUTO window. 3. Select your parameters to be varied using the Parameter function. 4. Choose the parameters to be plotted and the varied range using the Axes – Hilo function. Note that a co-dimension one bifurcation diagram need to be plotted first before we can proceed to a co-dimension two bifurcation analysis.

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Fig. 8 Phase portrait

5. Then, define the direction, step-size and others using the Numerics function. 6. Finally, click Run and then Steady-state or Periodic.

4.2 Coding for Bifurcation Analysis Using AUTO For co-dimension 1 bifurcation analysis in AUTO, the previous .ode file must be extended by including some AUTO numeric. Below is an example of ODE file for Eq. (1) with AUTO components: # preyrefuge.ode # # right hand sides dx/dt=(a1*x)*(1-(x/k))-((b*(1-m)*y*x)/(1+a*(1-m)*x)) dy/dt=-e*y+((c*b*(1-m)*x*y)/(1+a*(1-m)*x)) # In Eq 1, a1=alpha, b=beta, e=gamma # parameters par m=0.45,a1=10,k=100,a=0.02,e=0.09,b=0.6,c=0.02

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Fig. 9 Phase portrait with nullclines

# # some initial conditions init x=1,y=1 # # some XPP numeric @ dt=.1,nout=25,total=1000,meth=cvode,tol=1e-10,tol=1e-10 @ maxstor=1000,bound=5000 @ xp=t,yp=x,xlo=0,ylo=0,xhi=100,yhi=100 # # AUTO numeric @ autoxmin=0,autoxmax=1,autoymin=-10,autoymax=120 @ dsmax=1,ds=0.00001,dsmin=.0000001 # we are done done

The initial parameter step size for the bifurcation computation is defined by incorporating ds in the calculation. The ds sign (+ or −) is a signal for AUTO to recognise whether to vary the parameter either to the right (if positive) or the left (if negative). The ds step-size can be altered accordingly, since the first one is just a “suggestion”

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Fig. 10 Phase portrait showing the point of intersection between the red and green curves Fig. 11 Equilibrium point window

depending on the values of dsmin (the minimum step size of parameter) and dsmax (the maximum step size of parameter) stated by the user. Note that AUTO may miss out some important points for bigger ds.

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Fig. 14 Phase portrait with the direction fields

4.3 Performing Bifurcation Analysis Using AUTO Now, we will explore the bifurcation structure of Eq. (1), which is an extension of the Rosenzweig-MacArthur model incorporating a prey-refuge. To do this, we will use the previous parameters in our preyrefuge.ode file: m = 0.45, e = 0.09 and then fix the other parameters and initial conditions. After that, we can now commence the bifurcation analysis by following the steps below: 1. Launch XPPAUT by dragging and dropping the preyrefuge.ode file onto the “xpp - Shortcut” on your Desktop and set up the view accordingly. First, plot the graph of X, Y versus T as discussed earlier in time series plot to obtain Fig. 17. If we change the value of m = 4.5 to m = 0.2 and then run the simulation, some periodic solutions as illustrated in Fig. 18 will be produced. Notice that Fig. 18 is similar to Fig. 6 of our earlier analysis. Thus, we are interested to know how the change in the parameter m affects the (dis-)appearance of different steady-states and periodic solution in the Eq. (1). 2. To begin the bifurcation analysis, run the simulation again for m = 0.45. Now, Click Initialconds – Go and then click Initialconds – Last for many times. This is to ensure that the steady-state has been obtained.

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Fig. 15 Phase portrait with different initial conditions

3. Next, choose File – Auto to show the AUTO window. In the AUTO window, the parameter m must be listed as Par1 under Parameter since it is the bifurcation parameter. Figure 19 illustrates the outcome of steps 2 and 3. 4. Set up the graphics axes using Axes – Hilo (if it has not been taken care of in your code). 5. Set up the Numerics (if it has not been taken care of in your code). 6. Now, to start the plot, click Run and then Steady state. As shown in Fig. 20, the beginning of the diagram will show many marked points (+) on the curve. Usually, the curve may appear as either red or black. In this case, the red curve indicates stable steady-state and the black curve indicates unstable steady-state. 7. To extend the diagram, click Grab to enable you to see a small drop-down menu as shown in Fig. 21. This usually shows a bold cross (+) on the curve at the position labeled 1. Also, below the entire diagram are information about the plot generated. Also, m represents the parameter value at the location where the bold cross appeared. We can also see other possible solutions generated. To do this, move the cursor (i.e., the bold cross) over the solution curve generated by clicking the left or right key. As you move the cursor, you may come across bifurcation points (e.g., the transcritical bifurcation point). Such bifurcation points are usually labeled below Ty section and they denote the branch points (BP).

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Fig. 17 Time series plot with m = 4.5

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Fig. 18 Time series plot with m = 0.2

Fig. 19 Co-dimension 1 bifurcation diagram

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Fig. 20 Co-dimension 1 bifurcation diagram showing the red (stable steady-state) and black (unstable steady-state) curves

Fig. 21 Co-dimension 1 bifurcation diagram with information on bifurcation point

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Fig. 22 Co-dimension 1 bifurcation diagram: “grabbing” the point

Fig. 23 AutoNum window

8. Having finished with the checking for branch points, return the cursor to the location labeled as point (1) or simply select EP and then Enter. This means that the point (1) is “grabbed” as the starting point for another computation. Then, click Numerics (Fig. 22). Immediately, the AutoNum window will be displayed as shown in Fig. 23; sometimes with positive or negative sign at the ds. In our case, change the sign from positive to negative such that ds = − 0.00001. Then, click OK and click Run. 9. So far, Figs. 24 and 25 illustrate the basic steps to continue the solution branch to another EP which is at the left of point (1). Since ds = −0.00001, the parameter

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Fig. 24 Co-dimension 1 bifurcation diagram with ds = −0.00001

Fig. 25 AutoNum window

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m reduces as the solution branch moves towards the left. Thus, we can see in Fig. 24 as the outcome of the steps. 10. Observe that the transcritical bifurcation has occurred at the upper right in our solution diagram. Such a bifurcation happens whenever there is stability exchange between the two steady states. In our case, we have two-species steadystate (X, Y ) and single-species steady-state (K , 0). To see this phenomenon take place in our co-dimension 1 bifurcation diagram under consideration, click Grab—and then choose BP. Thereafter, click Enter and choose Numerics. An AutoNum window will automatically be displayed with ds carrying negative sign. Remove the negative sign and change to positive sign in ds section. When done, click O K as shown in Fig. 25. Then, click Run – Switch. Figure 25 leads to Fig. 26 with straight lines and curves represent single-species stable steady-state (K , 0) and two-species stable steady-state (X, Y ), respectively. Notice that there exist a threshold value of m, which corresponds to the point BP, where the straight lines and curves intersect. This point BP is the transcritical bifurcation mentioned earlier, where the single-species stable steadystate (K , 0) exchanges stability with the two-species stable steady-state (X, Y ). Thus, the black curve in the diagram corresponds to the unstable steady-state; such that if m is less than BP, only two-species steady-state is stable and if greater than BP, only single-species steady-state is stable. 11. To continue on the left region of the red curve, click Grab and move the cursor along the curve until you get to a point where HB appears under Ty (Fig. 27). That point corresponds to the Hopf bifurcation. Now, press Enter at this point

Fig. 26 Co-dimension 1 bifurcation diagram with ds = 0.00001

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Fig. 27 Co-dimension 1 bifurcation diagram showing Hopf bifurcation point

Fig. 28 Co-dimension 1 bifurcation diagram showing the green circles (stable periodic orbits)

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Fig. 29 Co-dimension 1 bifurcation diagram: optimising the plotting region

12. 13.

14.

15.

to “grab” the HB point as a new start for another computation. When you are through, click Numerics and an AutoNum window will be displayed. Change the ds to negative if otherwise and clik OK. Next, click Run to pop-up a screen with Hopf Pt label and select Periodic. Figure 28 presents the output of the steps discussed before. The green circles illustrate stable periodic orbits. On the other hand, if the circles are open, then it implies that the periodic solution is unstable. Also, the upper and lower points in the periodic solution, reveal the maximum and minimum values of X attained by the periodic solution of the plot. It is possible to observe these maximum and minimum values in our time series plot. To obtain an optimise plotting region, it is important to scale our axes before we save Fig. 28 as postscript/pdf format. Thus, click Axes – hI-lo. In the displayed AutoPlot window in Fig. 29, adjust the values to your desired range for X and Y and then click O K . Then, save the plot by clicking File – Postscript and then press OK.

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5 Bifurcation and Continuation: Co-dimension 2 Bifurcation Diagram In this section, we will perform the two-parameter bifurcation analysis, which is a more detailed bifurcation analysis of Eq. (1). Recall that the bifurcation analysis carried out earlier is simply a one-parameter bifurcation analysis. In this case, we observe that as bifurcation parameter m is varied there are appearance and disappearance of different bifurcations. To have a more detailed analysis of the dynamics of the system, we will construct bifurcation diagram in terms of two parameters. This is to enable us to investigate the effects of changing another parameter value, for example γ, on the dynamics of the Eq. (1). Note that γ corresponds to the term e in our earlier preyrefuge.ode code. Bifurcation analysis in two parameters is a continuation from one parameter bifurcation analysis. This implies that you must first start from one parameter bifurcation, and thereafter grab different bifurcation points generated to study two-parameter bifurcation curves. In general, the procedures to execute the two-parameter bifurcation analysis using XPPAUT start with selecting Axes and then Two parameters. This action will pop-up a window to enable you to select the main parameter and the second parameter. The following steps below are guides to obtain the two-parameter bifurcation diagram of Eq. (1): 1. Start up XPPAUT as discussed earlier and ensure that you obtain the oneparameter bifurcation diagram as previously shown in Fig. 28. 2. To continue Hopf bifurcation point in (γ, m)-parameter space, first click Grab – Select H B and click Enter – Axes – Two par. An AutoPlot window will be displayed; change Main Parm, Second Parm and the range values of x and y axes. Now, press O K and then click Run – Two Param. The partial blue curve shown in the Fig. 30, corresponds to Hopf bifurcation for different values of γ and m. Notice that the blue curve is not complete and to continue the curve to the left hand side of (γ, m)-space, click Numerics and if the ds sign is positive, change it to negative in the AutoNum window displayed. When you are through, click O K and then click Run – Two Param. Thus, these steps lead to a complete blue curve as shown in Fig. 31. 3. Next, we want to continue the transcritical bifurcation point in (γ, m)-parameter space. To do this, first click Axes – hI-lo and then press O K . This will take you to the one-parameter bifurcation diagram earlier shown in Fig. 28. Thereafter, click Grab – choose B P and then Enter. This will enable you to grab the transcritical bifurcation point in order to continue the point as γ and m are varied in (γ, m)parameter space. Now, click Axes – Two par and then press O K . Then, press Run – Two param. Figure 32 illustrate the outcome of these steps. The cyan curve that appeared in Fig. 32 corresponds to the transcritical bifurcation point in (γ, m)-parameter space. 4. The cyan curve is not yet complete and need to be extended. To do this, click Numerics and if the ds sign is negative, change it to positive in the AutoNum

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Fig. 30 Co-dimension 2 bifurcation diagram showing a blue curve (Hopf bifurcation)

Fig. 31 Co-dimension 2 bifurcation diagram showing a complete blue curve

window displayed. When you are through, click O K and then click Run Two Param. Thus, these steps lead to a complete cyan curve as shown in Fig. 33. 5. Now save the figure by clicking File – Postscript. Select file and then press OK.

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Fig. 32 Co-dimension 2 bifurcation diagram showing a cyan curve (transcritical bifurcation)

Fig. 33 Co-dimension 2 bifurcation diagram showing a complete cyan curve

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6 Conclusion We have discussed the techniques in numerical continuation and bifurcation analysis using XPPAUT in this article. The text is written in a simple and organised steps with adequate diagrams to aid the understanding of the contents. The guidelines provided are good attempts in assisting new users for the XPPAUT software. We hope that it will be beneficial to users in research especially to the early-stage postgraduate students. Acknowledgements Authors are supported by the Universiti Sains Malaysia (USM) Fundamental Research Grant Scheme (FRGS) No. 203/PMATHS/6711645.

References 1. Ermentrout, B.: Simulating, Analysing and Animating Dynamical Systems: A Guide to XPPAUT for Research and Studiess, 1st edn. SIAM, New York (2002) 2. Kar, T.K.: Stability analysis of a prey–predator model incorporating a prey refuge. Commun. Nonlinear Sci. Numer. Simul. 10(6), 681–691 (2005) 3. Rosenzweig, M.L., MacArthur, R.H.: Graphical representation and stability conditions of predator-prey interactions. Am. Nat. 97(895), 209–223 (1963) 4. Kuznetsov, Y.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York (1998)

A Basic Manual for AUTO-07p in Computing Bifurcation Diagrams of a Predator-Prey Model Livia Owen and Eric Harjanto

Abstract This is a beginner’s manual of AUTO-07p, a continuation and bifurcation software for ordinary differential equation (ODE). AUTO package is available for Windows, Mac OS, or UNIX/Linux platform. The directory auto/07p/demos has many tutorial demos for algebraic system, ODE and partial differential equation. We focus on the continuation of solutions of system of ODE. In this manual, we will learn the main tools in AUTO by doing cusp and pp2 demos step-by-step. In the first example, we will generate one- and two-parameter bifurcation diagrams. The second example is a 2D predator-prey model with the detection of a Hopf bifurcation. We plot some orbits and time series plot. We provide two options for running AUTO, i.e., by using Unix and Python commands. Keywords AUTO · AUTO-07p · Dynamical system · Bifurcation diagram · Hopf bifurcation · Fold bifurcation

1 Introduction AUTO is a numerical software for doing continuation and solving bifurcation problems in ordinary differential equations (ODES) and partial differential equations. In 1981, the first version of AUTO is developed by Eusebius Doedel, with subsequent major contribution by several researchers, including Alan Champneys, Fabio Dercole, Thomas Fairgrieve, Yuri Kuznetsov, Bart Oldeman, Randy Paffenroth, Bjorn Sandstede, Xianjun Wang, and Chenghai Zhang. This manual is a summary of [1] and the latest version, AUTO-07p, will be used here. AUTO-07p is written in Fortran and it can be installed in Windows, Mac OS or UNIX/Linux platform. In this manual, we will follow the installation process using UNIX/Linux platform, i.e., Ubuntu. We

L. Owen (B) · E. Harjanto Analysis and Geometry Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha No 10, Bandung, Indonesia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_11

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focus on the continuation of solutions of system of ODEs: u (t) = f(u(t), p), f(., .), u(.) ∈ Rn

(1)

with an initial condition, where p denotes one or more free parameters. Two demos that are available in AUTO package will be explained using Unix commands and Python commands. The first demo is cusp demo and the other one is pp2. In the cusp demo, one- and two-parameter continuations will be computed to generate their bifurcation diagrams. While, in pp2 demo, we detect Hopf bifurcation and then some periodic solutions will be computed. For more information about bifurcation analysis, see bifurcation’s theory book [2].

2 Preparation There are several things to be noted when installing AUTO-07p: 1. Internet connection is needed to make the installation process runs smoothly since you will need to install some packages from the Ubuntu repository. 2. AUTO file: auto07p-0.9.1.tar.gz must be downloaded first from http://cmvl.cs. concordia.ca/auto. 3. Terminal will be used to install and run AUTO-07p.

2.1 Installation The details of installation are available in [1]. As mentioned earlier, here, we are going to use the terminal to install AUTO-07p: Step 1 Go to the directory that have AUTO file: auto07p-0.9.1.tar.gz. You have to unzip the file using command: gunzip auto07p-0.9.1. tar.gz, followed by tar xvfo auto07p-0.9.1.tar. You will have a new directory auto with a subdirectory auto/07p. Step 2 Change directory to auto/07p directory and install the packages enlisted below: 1. 2. 3. 4. 5. 6.

Python: python-matplotlib and ipython PLAUT04: SoQt (libsoqt-dev or libsoqt4-dev) and libsimage PLAUT: xterm GUI94: lesstif2-dev or libmotif-dev Manual: LATEX(tetex or texlive) and transfig Others: gcc, gfortran, and g++

After installing the above packages, try typing ./configure. If all the packages needed have been installed, it will tell you that AUTO has been configured. If

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not, you need to install some other packages that is mentioned and type again ./configure until it tells you that AUTO has been configured. Step 3 Type make to compile AUTO. Now, you can call AUTO package whenever you want to activate AUTO-07p. You can do this at any directory by typing source/home/.../auto/07p/cmds/ auto.env.sh.

2.2 Input and Output File There are two main input files, which are the equation-file (xxx.f90) and the constants file (c.xxx). Make sure you give the same name for these two files. We will often use the c.xxx for the continuation computation. AUTO produces a standard output and three output files. The standard output shows various types of points where some of them are shown in Table 1. Three types of output files are 1. b.xxx: “b” is for bifurcation, and this file contains the bifurcation diagram file, 2. d.xxx: “d” is for diagnostic, and this file contains the continuation step size for each continuation step, the eigenvalues of the Jacobian (see Sect. 5.4) and many more, 3. s.xxx: “s” is for solution, and this file contains the solution file.

Table 1 The various types of points Number in s file Label (3) (4) (5)

HB – LP

(6) (7) (8) (9) (−4) (−9) (−22) (−31) (−32)

BP PD TR EP UZ MX CP BT GH

Description Hopf bifurcation User-specified regular output point Fold or saddle-node bifurcation (differential equations) Branch point (differential equations) Period doubling bifurcation Torus bifurcation End point of family; normal termination Output at user-specified parameter value Abnormal termination; no convergence Cusp bifurcation on fold curve Bogdanov-Takens bifurcation on Hopf curve Generalised Hopf bifurcation on Hopf curve

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Fig. 1 Command window, tektronix and editor

Fig. 2 Starting up in the command window—cusp demo

2.3 Starting Up AUTO We work with three windows, which are the editor, command window and tektronix (see Fig. 1). The editor window basically will be used for editing xxx.f90 and c.xxx files. It can also be used to open the output files b.xxx or s.xxx. Let us start with the command window. First, we will explore the simplest demo in AUTO, i.e., cusp demo in the next section. Click the command window and type as follows (see Fig. 2). Now our main directory is auto07/demos/cusp.

3 Running AUTO Using Unix Commands Consider the normal form of cusp bifurcation: x˙ = μ + λx − x 3 . Open cusp.f90 in editor as shown in Fig. 3. Follow the steps below:

(2)

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Step 1 Input variables and parameters that will be used. Step 2 U(n) define nth variable and PAR(n) define nth parameter. In cusp.f90, x = U(1) state x is the first variable, lambda = PAR(1) state λ is the first parameter, and mu = PAR(2) state μ is the second parameter. Step 3 F(n) define the right hand side of n th equation of the system. In cusp.f90, F(1) = mu + lambda*x - x**3 state the right hand side of (2). Step 4 Initialise the parameters, PAR(1) = 1 and PAR(2) = 0. Step 5 Initialise an equilibrium that satisfy the system with parameters in Step 4. In cusp.f90, we get the equilibrium point (0, 0) by solving 0 + x − x 3 = 0.

3.1 One Parameter Continuation Now, we focus on the c.cusp file that will be used regularly (see Fig. 4). Line 1 Rename the first parameter with lambda and the second parameter with mu. If you do not need to rename the parameters, just delete this line. Line 2 Rename the first variable with x. If you do not need to rename the variable, just delete this line. Line 3 NDIM (number of dimension of the system). IPS = 1 is for computing stationary solutions of ODEs with detection of Hopf bifurcations, IPS = 2 is for computing periodic solutions (see pp2 demo). IRS = 0 is for the first run, IRS = N is for starting the computation at the previously computed solution with Label N . ILP = 1 is for detecting folds. Line 4 In ICP, type the name of the free parameter (make sure you type two parameters for a two-parameter continuation). Line 5 By default, ISW = 1, which indicates one parameter continuation. If IRS is the label of either a fold, a Hopf, a period-doubling or a torus bifurcation point, then ISW = 2. Line 8 DS = N , where N > 0 for forward continuation and N < 0 for backward continuation. Line 10 UZSTOP = { mu  : [−2.0, 2.0]} is for labelling the point when parameter mu equals to −2 and 2. The computation will stop when it has reached those values. To continue the computation, use UZR = { mu  : [−2.0, 2.0]}. Next, we go to the command window. Since the value of DS in the previous file is positive, we are now running forward continuation. Follow the steps below and refer to Fig. 5. Step 1 Type @r cusp to run forward continuation. We will get fold point (LP) at Label 2.

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Fig. 3 The cusp.f90 file

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Fig. 4 The c.cusp file (compact version of constant file)

Fig. 5 Forward continuation result of cusp demo. We get fold point or LP at Label 2

Fig. 6 DS < 0 for backward continuation

Step 2 Type @sv cuspforward to save the result as cuspforward and AUTO will produce three output files. For backward continuation, we need to change the sign of DS in c.cusp file editor. Follow the steps below: Step 1 Change the sign of DS in c.cusp file editor (see Fig. 6). Step 2 In the command window, type @r cusp. Step 3 Then, type @sv cuspbackward to save the result as cuspbackward. Step 4 Type @ap cuspbackward cuspforward to append cuspbackward to cuspforward (see Fig. 7).

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Fig. 7 Backward continuation result of cusp demo. We get LP at Label 2

3.2 Plotting the Bifurcation Diagram To plot the bifurcation diagram, type @p and followed by the file name. Here we type @p cuspforward and the tektronix window will appear. In tektronix window, follow the steps below: Enter command Type axis. Enter horizontal and vertical axis Type a pair of numbers X, Y , where X is the horizontal axis and Y is the vertical axis. Number 1 is for the free parameter μ. Here we type 1, 3 since we are plotting in (μ, x) coordinate. Enter command Type bd0 and AUTO will automatically set the range of horizontal axis and vertical axis. If you want to set the range of horizontal axis and vertical axis, type bd. Solution labels Type y if you want to put the solution labels, and then follow the rest (see Fig. 8). We will then obtain a bifurcation diagram showing a variation of parameter μ as illustrated in Fig. 9.

3.3 The Constant File for Two-Parameter Continuation For a two-parameter continuation, in the previously cuspforward file, we change IRS = 2 to start the computation from the fold point (Label 2), and ISW = 2 to run a two-parameter continuation (see Fig. 10). Now, we follow the fold bifurcation curves as μ and λ are varied. Similar procedures as in the previous section are used:

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Fig. 8 Plotting command for one parameter continuation

Fig. 9 The branch of continuation equilibria as a variation on parameter μ. Two fold bifurcation points are labelled with 2

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Fig. 10 IRS = 2 is used to start the computation from Label 2 and ISW = 2 is used to run a two-parameter continuation

Forward continuation Type @r cusp cuspforward to start the computation from Label 2 of cuspforward. Save as cuspforward2. Backward continuation Change the sign of DS in c.cusp. Type @r cusp cuspforward then save as cuspbackward2 and append to cuspforward2 (see Fig. 11). Plotting the bifurcation diagram Type @p cuspforward2 then choose 1, 4 in for the axis. We obtain the bifurcation diagram of system (3) in (μ, λ) coordinate as shown in Fig. 12, where the two fold curves end up in a cusp bifurcation point (Label 7).

4 Running AUTO Using Python Commands We have shown how to run AUTO using Unix commands in previous section. In this section, we run AUTO using Python commands. First, clear or restart the command window and follow the steps below: Step 1 Call the source by typing source/home/../auto/07p/cmds/ auto.env.sh. Step 2 Type auto to run AUTO using Python. Step 3 Type cd then mkdir cusp to create an empty work directory. Step 4 Type cd cusp then demo(‘cusp’) to the cusp demo files to the work directory. Step 5 Type demofile(‘cusp.auto’) to execute cusp demo. It is easy to follow the rest. Step 6 Execute p = plot(mu) to plot the bifurcation diagram on PyPLAUT. Choose x for the Y axes and the right diagram of Fig. 13 is obtained. Explore all icons below the diagram. Step 7 Continue the demo and a cusp point in the right diagram of Fig. 14 is obtained. We should obtain the same results as in Sect. 3 in a simpler way. The bifurcation diagram can be saved by clicking save postscript on the File menu.

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Fig. 11 Cusp bifurcation point (CP Label 7) in two parameter-continuation

Fig. 12 Bifurcation diagram of (2) in (μ, λ) coordinate. Two fold curves end up in a cusp bifurcation point labelled with 7

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Fig. 13 (Left) Running demofile(‘cusp.auto’) part 1. (Right) One parameter bifurcation diagram on PyPLAUT

Fig. 14 (Left) Running demofile(‘cusp.auto’) part 2. (Right) Two parameter bifurcation diagram on PyPLAUT

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5 Predator-Prey Model In this section, we use pp2 demo, a 2D predator-prey model with harvesting. The system is given by u 1  = p2 u 1 (1 − u 1 ) − u 1 u 2 − p1 (1 − e− p3 u 1 ), u 2  = −u 2 + p4 u 1 u 2 .

(3)

For instance, let u 1 be the density of prey fish and u 2 be the shark density. In pp2.f90, parameter p1 are varied from 0 until 1, and parameters p2 , p3 and p4 are fixed ( p2 = p4 = 3 and p3 = 5). For the first step, choose p1 = 0 and we find that one of the equilibria is (0, 0). The equation file (pp2.f90) is shown in Fig. 15.

5.1 Running pp2 Demo Using Unix Commands Let us follow the steps below: Step 1 Change the directory to pp2 demo. Type cd.. then cd pp2. Step 2 Run the c.pp2, type @r pp2. Step 3 We obtain the result as shown in Fig. 16, and then save as pp21. Step 4 Plot the pp21. We obtain the bifurcation diagram as illustrated in Fig. 17 showing a variation of parameter quota, with fold bifurcation point (Label 5) and Hopf bifurcation point (Label 9).

5.2 Periodic Solution For periodic solution, first, we need to add period as a new parameter. Then, we continue the periodic solution (Hopf bifurcation point at Label 9) in two-parameter continuation of parameter quota and period. Let us follow the steps below: Step 1 In c.pp2 file, change IPS = 2, IRS = 9 and add parameter 11 in ICP (see Fig. 18). Step 2 In command window, type @r pp2 pp21 for running pp2 from latest result. Step 3 Save the result as pp2period (see Fig. 19). Now we want to plot some closed orbit in (fish, shark) coordinate. Type @p pp2period and in tetronix, and let us follow the steps below to obtain Figs. 20 and 21. Enter command

Type 2d.

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Fig. 15 The pp2.f90 file. PAR(:4) = (/0.0,3.0,5.0,3.0/) with p1 = 0, p2 = 3, p3 = 5, p4 = 3 and U = 0.0 and the equilibrium point u 1 = u 2 = 0

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Fig. 16 The result for running c.pp2

Fig. 17 The branch of continuation equilibria as variation on parameter quota. There are fold bifurcation point labelled with 5 or quota = quota f , and Hopf bifurcation point labelled with 9. When quota < quota f , there are four equilibria, which are f ish = 0, f ish = 0.333333 and the other two equlibria collapse at quota = quota f and then vanish. Consequently, when quota < quota f , there are two equilibria, which are f ish = 0 and f ish = 0.333333

Enter labels Type some labels that you want to plot or type a fo all. Here we plot orbits for Label 11, 15, 19 and 23. Enter axes Type a pair of numbers X, Y , where X is the horizontal axis and Y is the vertical axis. Number 1 is for time parameter quota, 2 is for fish and 3 is for shark. Here we type 2 3 and get some orbits in (fish, shark) coordinate, see Fig. 20.

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Fig. 18 The c.pp2 file

Fig. 19 The command window c.pp2 file

Enter axes Type d to display the 2D plot. Time series plot To obtain the time series plot of fish population, redo the previous procedure and type 1 2 and we get Fig. 21.

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Fig. 20 Some orbits in (fish, shark) coordinate

Fig. 21 The time series plot of fish population density

5.3 Running pp2 Demo Using Python Commands In this section, we run pp2 demo using Python commands. Let us follow the steps below: Step 1 Type clear. Step 2 Type auto then we run AUTO using Python. Step 3 Type mkdir pp2 the cd pp2 to copy pp2 demo to the work directory. Step 4 Type demofile(‘pp2.auto’) to run the script pp2.auto. It is easy to follow the rest.

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Fig. 22 One-parameter bifurcation diagram of pp2 on PyPLAUT

Fig. 23 (Left) Solution curves in (fish, shark) coordinate. (Right) The time series plot of fish and shark population density

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Step 5 Type plot(‘pp2’) to plot a one-parameter bifurcation diagram as shown in Fig. 22). Type plot(‘ps’) and explore to get Fig. 23.

5.4 The Stability of Equilibrium AUTO provides the eigenvalues of the Jacobian matrix that give information about the stability of the equilibrium point in s.xxx file. It is known from the literatures (see, for example, [2]) that the Jacobian matrix of a fold point (or LP) has at least one zero eigenvalues and a Hopf point (or HB) has purely imaginary eigenvalues. To obtain these eigenvalues, open the d.pp21 file, and then search for LP and HB points. We get the results as in Fig. 24 for fold point, where the eigenvalues are

Fig. 24 The eigenvalues of Jacobian matrix of LP point in d.pp21 file are 0.233485 and −1.53415 × 10−8 ≈ 0

Fig. 25 The eigenvalue of Jacobian matrix of HB point in d.pp21 file are −2.03224 × 10−10 ± 0.604782i ≈ ±0.604782i

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0.233485 and −1.53415 × 10−8 ≈ 0, and as in Fig. 25 for Hopf point, where the eigenvalues are −2.03224 × 10−10 ± 0.604782i ≈ ±0.604782i. Acknowledgements We would like to express our sincere gratitude to SEAMS School 2018 on Dynamical Systems and Bifurcation Analysis (DySBA) committee for offering us this opportunity. Livia Owen acknowledges the financial support from The Indonesian Education Scholarship Program (LPDP), Ministry of Finance of the Republic of Indonesia.

References 1. Doedel, E.J., Champneys, A.R., Dercole, F. et al.: AUTO-07P: continuation and bifurcation software for ordinary differential equations. California Institute of Technology, Pasadena, California (2008) 2. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, vol. 112, 2nd edn. Springer, New York (1998)

Numerical Continuation and Bifurcation Analysis in a Harvested Predator-Prey Model with Time Delay using DDE-Biftool Juancho A. Collera

Abstract Time delay has been incorporated in models to reflect certain physical or biological meaning. The theory of delay differential equations (DDEs), which has seen extensive growth in the last seventy years or so, can be used to examine the effects of time delay in the dynamical behaviour of systems being considered. Numerical tools to study DDEs have played a significant role not only in illustrating theoretical results but also in discovering interesting dynamics of the model. DDE-Biftool, which is a Matlab package for numerical continuation and numerical bifurcation analysis of DDEs, is one of the most utilized and popular numerical tools for DDEs. In this paper, we present a guide to using the latest version of DDE-Biftool targeted to researchers who are new to the study of time delay systems. A short discussion of an example application, which is a harvested predator-prey model with a single discrete time delay, will be presented first. We then implement this example model in DDE-Biftool, pointing out features where beginners need to be cautious. We end with a comparison of our theoretical and numerical results. Keywords Delay differential equations · Numerical continuation · Numerical bifurcation analysis · Time delay systems

1 Introduction Time delay has been incorporated in models to reflect certain physical or biological meaning. Examples include optical feedback in laser systems [1–3], maturation age in stage structured population models [4], and delayed information in queueing models [10] just to name a few. The theory of delay differential equations (DDEs) [9, 12], J. A. Collera (B) University of the Philippines Baguio, Gov. Pack Road, 2600 Baguio, Philippines e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2019 M. H. Mohd et al. (eds.), Dynamical Systems, Bifurcation Analysis and Applications, Springer Proceedings in Mathematics & Statistics 295, https://doi.org/10.1007/978-981-32-9832-3_12

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which has seen extensive growth in the last seventy years or so, can be used to examine the effects of time delay in the dynamical behaviour of systems being considered. Numerical tools to study DDEs have played a significant role not only in illustrating theoretical results but also in discovering interesting dynamics of the model. DDEBiftool [8, 11], which is a Matlab package for numerical continuation and numerical bifurcation analysis of DDEs, is one of the most utilised and popular numerical tools for DDEs. In this paper, we present a guide to using the latest version of DDE-Biftool targeted to researchers who are new to the study of time delay systems. A short theoretical discussion of the example, which is a harvested predator-prey model with a single discrete time delay, will be given in the next section. In Sect. 3, we implement this example model in DDE-Biftool and compare the theoretical and numerical results. We conclude the paper with a summary and thoughts on using DDE-Biftool in studying time delay systems.

2 Harvested Predator-Prey Model with Time Delay We first discuss the model that we are going to use as an example for numerical continuation and numerical bifurcation analysis. The theoretical results presented here will be compared to the numerical results obtained in the succeeding section.

2.1 The Model We consider the following model, studied in [14], with a single discrete time delay parameter τ > 0 : ⎧ d x(t) ⎪ ⎪ = r x(t) − ax(t)x(t − τ ) − bx(t)y(t) − h, ⎪ ⎨ dt (1) ⎪ ⎪ dy(t) ⎪ ⎩ = cx(t)y(t) − dy(t) − k. dt Here, x(t) and y(t) are the state variables denoting, respectively, the densities of the prey and predator populations at time t. The parameter a is the ratio of the intrinsic growth rate r of the prey and the carrying capacity K for the prey population in the absence of the predation. The rate of consumption of prey by the predator is given by the parameter b while c measures the conversion of prey consumed into the predator reproduction rate. The death rate of the predator is represented by d. Both species are assumed to have economic value and are harvested. The parameters h and k denote the harvesting rates of the prey and predator populations, respectively. All parameters in system (1) are positive real numbers.

Numerical Continuation and Bifurcation Analysis … Table 1 Parameter values r a b 3.50

0.04

1.00

227

c

d

h

k

0.05

0.30

0.02

0.01

The equilibrium solutions of system (1) are solutions (x(t), y(t)) of system (1) satisfying d x(t)/dt = 0 and dy(t)/dt = 0, and hence are obtained by solving for constant values x and y in the following system of nonlinear equations: ⎧ ⎨ r x − ax 2 − bx y − h = 0, (2) ⎩ cx y − dy − k = 0. Since h > 0 and k > 0, the value of x nor the value of y can not be zero. From the equations in system (2), we get y = (r x − ax 2 − h)/bx and y = k/(cx − d). Hence, the positive equilibria of system (1) exist provided (r 2 − 4ah) > 0 and (cx − d) > 0. Example 1 Solving the nonlinear equations in system (2) with parameter values given in Table 1, we obtain the following equilibrium solutions of system (1): E 1 = (6.061458, 3.254242), E 2 = (87.432881, 0.002456).

(3) (4)

2.2 Local Stability of the Equilibrium Solutions If we let x(t) = x(t − τ ) and y(t) = y(t − τ ), then the right-hand side of system (1) can be expressed as 

   f (x, y, x, y) r x − ax x − bx y − h = . g(x, y, x, y) cx y − dy − k

(5)

The linearized system corresponding to system (1) about an equilibrium solution E ∗ = (x ∗ , y ∗ ) is given by dX(t) = AX(t) + BX(t − τ ), dt  where X(t) =

 x(t) and the matrices A and B are as follows: y(t)

(6)

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 A= 

∂ f /∂x ∂ f /∂ y ∂g/∂x ∂g/∂ y

∂ f /∂x ∂ f /∂ y B= ∂g/∂x ∂g/∂ y

 (x,y,x,y)=(x ∗ ,y ∗ ,x ∗ ,y ∗ )

,

(7)

.

(8)



(x,y,x,y)=(x ∗ ,y ∗ ,x ∗ ,y ∗ )

The characteristic equation corresponding to the linear system (6) is det(λI − A − Be−λτ ) = 0,

(9)

where I is the  2× 2 identity matrix. Equation (9) is obtained by using the ansatz x X(t) = eλt 0 to the linear system (6). If all roots of the characteristic Eq. (9) lie y0 in the open left-half plane, i.e., Re λ < 0 for all roots λ of Eq. (9), then the equilibrium E ∗ is locally asymptotically stable. Using Eqs. (7) and (8) and the functions f and g in (5), we obtain  A=

r − ax ∗ − by ∗ −bx ∗ cy ∗ cx ∗ − d



 and B =

 −ax ∗ 0 . 0 0

Thus, we can write the characteristic equation (9) as  2 λ + a1 λ + a2 + (a3 λ + a4 ) e−λτ = 0,

(10)

where a1 = −(r − ax ∗ − by ∗ ) − (cx ∗ − d), a2 = (r − ax ∗ − by ∗ )(cx ∗ − d) + bcx ∗ y ∗ , a3 = ax ∗ , and a4 = −ax ∗ (cx ∗ − d). When the time delay τ = 0, Eq. (10) reduces to the quadratic equation λ2 + (a1 + a3 )λ + (a2 + a4 ) = 0.

(11)

Both roots of equation (11) have negative real part if and only if (a1 + a3 ) > 0

and

(a2 + a4 ) > 0.

(12)

Hence, for the case τ = 0, the equilibrium E ∗ is locally asymptotically stable whenever conditions in (12) are satisfied. We want to know if E ∗ , under the conditions in (12) will become unstable as we vary the time delay parameter. Suppose that the conditions in (12) are satisfied and consider now the case where τ > 0. Initially, E ∗ is locally asymptotically stable, i.e., all roots of the characteristic equation (10) with τ = 0 lie in the open left-half plane. If one or more roots of equation (10) cross the imaginary axis and move towards the open right-half plane as τ is increased, then E ∗ will switch stability and becomes unstable. We have two possibilities: either a real root of equation (10) will cross the imaginary axis, i.e., λ = 0 is a root of equation (10) at some critical delay value, or a pair of complex conjugate roots of equation (10) cross the imaginary axis, i.e.,

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λ = ±iω is a root of equation (10) at some critical delay value where ω is a nonzero real number. If λ = 0 is a root of equation (10), then (a2 + a4 ) = 0. However, since (a2 + a4 ) > 0 from conditions in (12), then λ = 0 is not a root of the characteristic equation (10). Suppose now that Eq. (10) has a pair of purely imaginary roots λ = ±iω. Since the right-hand side of Eq. (10) is an entire function, complex roots of equation (10) come in conjugate pairs. Thus, without loss of generality, we may assume that ω > 0. Since λ = iω with ω > 0 satisfies equation (10), we have (−ω 2 + ia1 ω + a2 ) + (ia3 ω + a4 )e−iωτ = 0.

(13)

This gives the following equations: a4 cos(ωτ ) + a3 ω sin(ωτ ) = ω 2 − a2 , a4 sin(ωτ ) − a3 ω cos(ωτ ) = a1 ω,

(14) (15)

after using the Euler’s formula in Eq. (13) and then matching the real and imaginary parts on both sides of Eq. (13). We can eliminate τ by squaring each side of Eqs. (14) and (15) and then adding corresponding sides. We obtain (ω 2 − a2 )2 + a12 ω 2 = a42 + a32 ω 2 , which we can write as ω 4 + αω 2 + β = 0,

(16)

where α = a12 − 2a2 − a32 and β = a22 − a42 . If we let u = ω 2 , then Eq. (16) becomes the following quadratic equation in u: h(u) := u 2 + αu + β = 0.

(17)

If Eq. (17) does not have a positive root, then Eq. (10) cannot have purely imaginary roots. That is, the roots of the characteristic equation (10) that are in the open lefthalf plane when τ = 0 remain in the open left-half plane as the time delay parameter τ is increased. In other words, if Eq. (17) does not have a positive root, then the equilibrium E ∗ remains locally asymptotically stable for all τ > 0. Note that if the coefficients in Eq. (17) satisfy the following conditions: α>0

and

β > 0,

(18)

then both roots of equation (17) have negative real parts. That is, under the conditions in (18), Eq. (17) does not have positive roots. Therefore, the equilibrium E ∗ is locally asymptotically stable for all τ ≥ 0 whenever conditions in (12) and (18) are satisfied (cf. Theorem 4 of [14]). Lemma 1 The number of positive roots of equation (17) is determined as follows:

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≥0 0 with h(u) > 0 > 0 with h(u) = 0 > 0 with h(u) < 0

Number of Positive Roots 0 1 1 0 1 2

where (u, h(u)) is the vertex of the parabola given by the graph of the function h(u) in Eq. (17). Example 2 Using the parameter values in Table 1, we obtain α = −2.031311 and β = 0.972753 approximately. The graph of the quadratic function h(u) given in Eq. (17) is a parabola with vertex at (u, h(u)) = (1.015655, −0.058803). Since α < 0, β > 0, and h(u) < 0, by Lemma 1, Eq. (17) has two positive roots. Let us denote these positive roots by u − and u + with u − < u + . Solving equation (17), we obtain the roots u − = 0.773162 and u + = 1.258149, with corresponding ω− = 0.879297

ω+ = 1.121672,

and

(19)

which are roots of equation (16). Consequently, the characteristic equation (10) has purely imaginary roots ±iω− and ±iω+ . Remark 1 The conditions in (3.13) of [14] are in fact the three inequalities in the last row of the table in Lemma 1, i.e., α < 0,

β>0

and

h(u) < 0,

(20)

where Eq. (17) has exactly two positive roots.

2.3 Critical Delay Values Let us now determine the critical time delay values where the purely imaginary roots ±iω− and ±iω+ of Eq. (10), obtained in Example 2, will occur. Under the conditions (20), Eq. (10) has purely imaginary roots ±iω− and ±iω+ . Thus, the values ±ω− and ±ω+ satisfy Eqs.(14) and (15). We can compute for sin(ωτ ) and cos(ωτ ) from Eqs. (14) and (15) to obtain tan(ωτ ) =

ω(a3 ω 2 + a1 a4 − a2 a3 ) . (a4 − a1 a3 )ω 2 − a2 a4

The purely imaginary roots ±iω− (resp. ±iω+ ) of Eq. (10) occurs when the time delay τ = τk− (resp. τ = τk+ ) for k = 0, 1, 2, 3, . . . with τk± =



  2 + a1 a4 − a2 a3 ) ω± (a3 ω± 1 tan−1 + 2πk . 2 ω± (a4 − a1 a3 )ω± − a2 a4

(21)

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Table 2 Values of τk− and τk+ for k = 0, 1, 2, 3, 4 τ0− τ1− τ2− τ3− τ4−

τ0+ τ1+ τ2+ τ3+ τ4+

= −1.752556 = 5.393140 = 12.538836 = 19.684531 = 26.830227

= 1.3794139 = 6.9810371 = 12.582660 = 18.184284 = 23.785907

Example 3 Using Eq. (21) with the parameter values in Table 1 and corresponding ω± given in (19), we obtain the critical delay values shown in Table 2.

2.4 Transversality Conditions We saw earlier that at τ = τk− (resp. τ = τk+ ) for k = 0, 1, 2, 3, . . . , the characteristic equation (10) has purely imaginary roots ±iω− (resp. ±iω+ ). We want to know if these roots along the imaginary axis will move towards the open right-half plane or towards the open left-half plane. We address this by determining if the rate of change of the Re λ with respect to τ at the critical time delay values τ± is positive or negative, where λ = λ(τ ) is a root of the characteristic equation (10). Recall from Eq. (17), that h(u) = u 2 + αu + β. Hence, h  (u) = 2u + α. As shown in Eq. (3.15) of [14], we have

 d(Reλ)  sign  dτ τ

=

τk±

    2 = sign h  (ω± ) = sign h  (u ± )

in our notation. Since the graph of h(u) is decreasing (resp. increasing) at u = u − (resp. at u = u + ), we know that h  (u − ) < 0 (resp. h  (u + ) > 0). Therefore,

 d(Reλ)  sign  dτ τ

= τk±

= ±1.

This means that the root λ(τ ) of the characteristic equation (10) that lies on the imaginary axis when τ = τk− (resp. when τ = τk+ ) moves towards the open left-half plane (resp. towards the open right-half plane).

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3 Numerical Continuation and Bifurcation Analysis Numerical tools to study DDEs have played a significant role not only in illustrating theoretical results but also in discovering interesting dynamics of the model. DDEBiftool [8, 11], which is a Matlab package for numerical continuation and numerical bifurcation analysis of DDEs, is one of the most utilized and popular numerical tools for DDEs. It was originally developed by Engelborghs [8] as part of his PhD work at the KU Leuven under supervision of D. Roose. DDE-Biftool provides a set of capabilities that is similar to what a range of alternative tools do for ordinary differential equations (ODEs) and maps, such as Matcont [6], COCO [5] and AUTO [7]. Another tool performing a similar set of tasks for DDEs, particularly for timedependent DDEs with time-dependent delays, is Knut [13]. Aside from continuation of steady-state and periodic-orbit solutions which are typically done by varying a single parameter, DDE-Biftool can also continue bifurcations in two parameters. This includes steady-state folds, Hopf bifurcations, folds of periodic orbits, period doublings, and torus bifurcations. It can also perform normal form analysis for equilibria. DDE-Biftool is GNU Octave compatible and has a BSD licence such that it can be run completely as free software. The most recent version, DDE-Biftool v3.1.1, is maintained by J. Sieber and can be downloaded from https://sourceforge.net/projects/ddebiftool. The manual for this newest version of DDE-Biftool is provided at [11]. In this section, we illustrate the use of DDE-Biftool to perform numerical continuation and bifurcation analysis of system (1) varying the delay parameter τ . The boxed commands are the required commands and can be saved in a single m-file for convenience. addpath(’../ddebiftool/’,’../ddebiftool_utilities/’); pp_sys = @(x,p)[... p(1)*x(1,1) - p(2)*x(1,1).*x(1,2) - p(3)*x(1,1).*x(2,1) - p(6); p(4)*x(1,1).*x(2,1) - p(5)*x(2,1)-p(7)]; funcs = set_funcs(’sys_rhs’,pp_sys,’sys_tau’,@()[8]);

We start with addpath which identifies the location of the folders ddebiftool and ddebiftool_utilities containing the functions that we need for continuation and bifurcation analysis. This should be adjusted depending on where the user intends to do their computations and where the folders ddebiftool and ddebiftool_utilities were copied. Next, we encode the right-hand side of system (1) in the function named pp_sys. The state variables x1 (t), x1 (t − τ ), x2 (t), and x2 (t − τ ) are respectively denoted by x(1,1), x(1,2), x(2,1), and x(2,2). Here, the first index refers to the component while the second index refers to the delay number. For simplicity, we represent the parameters r , a, b, c, d, h, and k by p(1), p(2), p(3), p(4), p(5), p(6), and p(7), respectively.

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We then set-up the function structure funcs identifying the previously defined system right-hand side pp_sys as ’sys_rhs’. The time delay τ will be the eighth parameter in our parameter list. Thus, assigning [8] in ’sys_tau’. parbd = {’min_bound’,[0,8],’max_bound’,[8,15],’max_step’,[8,0.05]}; [br,success] = SetupStst(funcs,... ’parameter’,[3.50 0.04 1.00 0.05 0.30 0.02 0.01 0.00],... ’x’,[6.00; 3.00],’contpar’,8,’step’,0.02, parbd{:})

In parbd, we set the minimum bound, maximum bound, and maximum step size for the time delay τ , which is our main continuation and bifurcation parameter. Here, we choose τ to be from 0 to 15 because τ is non-negative and since we want to see the dynamics as we vary the time delay τ up until τ2+ = 12.582660 (see Table 2). Next, we set-up the branch of equilibria which we denote here by br. For ’parameter’, we use the parameter values from Table 1. Note that the initial value set for τ , which is eighth in the parameter list, is 0.00. For ’x’, we use the initial guess [6.00; 3.00] targeting the equilibrium E 1 with values given in Eq. (3). Our continuation parameter is τ , so ’contpar’ is 8. After running the commands above, we get the following results. br = method: [1x1 struct] parameter: [1x1 struct] point: [1x2 struct] success = 1

This means that our attempt to set-up a branch of equilibria is successful. The equilibrium branch br now contains two points. The first branch point in br has τ = 0 while the second branch point in br has τ = 0.02 since ’step’ is assigned a value 0.02. The corrected value for E 1 can be obtained by typing the following in the Command Window. >> format long; br.point(1).x ans = 6.061458241811056 3.254242134274988

The value of the time delay parameter τ for the first and second points in the equilibrium branch br are obtained by typing the following in the Command Window. >> br.point(1).parameter(8) ans = 0 >> br.point(2).parameter(8) ans = 0.020000000000000

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Fig. 1 (Left) The equilibrium branch br obtained by using the branch continuation function br_contn. (Right) The same branch br with stability information obtained by using the function br_stabl

figure(1); clf; br.method.continuation.plot = 1; [br,s,f,r] = br_contn(funcs,br,300); ylim([5,7]); set(gca,’FontSize’,20); figure(2); clf; br = br_stabl(funcs,br,0,1); [xm,ym] = df_measr(0,br); br_splot(br,xm,ym); ylim([5,7]); set(gca,’FontSize’,20);

We now continue the equilibrium branch br and then determine the stability of the continued branch. The above commands yield two plots of the equilibrium branch br as shown in Fig. 1. The plot on the left panel of Fig. 1 shows br with additional 300 points. So now br contains a total of 302 points which is achieved by using the function br_contn. Meanwhile, the plot on the right panel of Fig. 1 shows the same branch br but with stability information. This is obtained using the function br_stabl. The stable and unstable parts of the branch are in green and red, respectively, while the Hopf bifurcation points are marked with asterisks (∗). It is worth noting that in this particular example the stability switches occur at the Hopf bifurcation points. Here, there are five stability switches. The function df_measr gives the default xm and ym for the equilibrium branch br. We can check what these are by typing xm and ym in the Command Window. >> xm xm = field: ’parameter’ subfield: ’’ row: 1 col: 8 func: ’’ >> ym ym = field: ’x’ subfield: ’’

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row: 1 col: 1 func: ’’

Hence, for the plots in Fig. 1, the horizontal axis is terms of the time delay parameter τ while the vertical axis is in terms of the state variable x(t). If you want the vertical axis to be in terms of y(t) instead of x(t), you need to type in ym.row=2 in the Command Window to make the desired change. Here, we keep the vertical axis in terms x(t) for future plots. br.method.stability.minimal_real_part = -2; nunst = GetStability(br); ind_hopf = find(abs(diff(nunst))==2)

The function GetStability yields nunst which is the number of characteristic roots in the open right-half plane. Hopf bifurcation occurs when a complex conjugate pair of simple characteristic roots crosses the imaginary axis. The above set of commands gives ind_hopf which is the list of points in the equilibrium branch br where Hopf bifurcation occurs. ind_hopf = 30 110 142 251 252

The code below gives an animation showing the movement of the characteristic roots on the complex plane as the time delay parameter τ is varied. Pay attention to the value of τ when a pair of complex conjugate roots crosses the imaginary axis. for i = 1:length(br.point); clf; figure(33); hold on; plot([-0.5 0.5], [0 0], ’b’, [0 0], [-5 5], ’b’); p_splot(br.point(i)); axis([-0.20 0.20 -1.5 1.5]); tau = br.point(i).parameter(8); text(0.01,0.2,[’\tau = ’,num2str(tau,’%2.2f’)],’FontSize’,32); M(i) = getframe(gcf); end

The value of the time delay parameter τ at the branch points given in ind_hopf are obtained as follows. >> for i = 1:length(ind_hopf) critical_tau(i) = br.point(ind_hopf(i)).parameter(8); end >> critical_tau’ ans =

1.348598400000001 5.348598399999998 6.948598399999993 12.398598400000049 12.448598400000050

Comparing with values in Table 2, the first three values above are correct up the first decimal digit while the last two are not. We have to keep in mind that the above

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values are mere approximations since the maximum step size that we assigned for the time delay parameter τ in parbd is just 0.05. [br_hopf1,success] = SetupHopf(funcs,br,ind_hopf(1))

The function SetupHopf allows us to initialise the continuation of Hopf bifurcations. Here, we denote by br_hopf1 the Hopf branch which as we saw earlier occurs approximately at the point ind_hopf(1) along the equilibrium branch br. The above command yields br_hopf1 = method: [1x1 struct] parameter: [1x1 struct] point: [1x1 struct] success = 1

which means that our attempt to set-up the Hopf branch br_hopf1 is successful. At the first branch point of br_hopf1, the value of the time delay parameter τ can be obtained by typing in the following in the Command Window. >> br_hopf1.point(1).parameter(8) ans = 1.379413927096384

The above value is a correction to the initial guess 1.348598400000001 obtained from br.point(ind_hopf(1)).parameter(8). Moreover, this corrected value matches the value of τ0+ given in Table 2. [br_hopf2,success] [br_hopf3,success] [br_hopf4,success] [br_hopf5,success]

= = = =

SetupHopf(funcs,br,ind_hopf(2)) SetupHopf(funcs,br,ind_hopf(3)) SetupHopf(funcs,br,ind_hopf(4)) SetupHopf(funcs,br,ind_hopf(5))

Similarly, the correct value of τ for the next four Hopf bifurcation points are obtained by setting up Hopf branches. Here, we denote the next four Hopf branches as br_hopf2, br_hopf3, br_hopf4, and br_hopf5. The correct values of τ for the next four Hopf bifurcation points are obtained as follows. >> [br_hopf2.point(1).parameter(8);... br_hopf3.point(1).parameter(8);... br_hopf4.point(1).parameter(8);... br_hopf5.point(1).parameter(8)] ans =

5.393140023781609 6.981037144585396 12.538835708751554 12.538835707843552

These values match the values in Table 2 of τ1− , τ1+ , and τ2− , but not that of τ2+ . We remedy this by adding ’excludefreqs’,br_hopf4.point(1).omega in the previous code which removes undesired eigenvalues from consideration.

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[br_hopf5,success] = SetupHopf(funcs,br,ind_hopf(5),... ’excludefreqs’,br_hopf4.point(1).omega)

The correct value for τ2+ is now obtained by typing in the following in the Command Window. >> br_hopf5.point(1).parameter(8) ans = 12.582660362074412

Branches of periodic solutions can be obtained from the identified Hopf bifurcations. Among these branches of periodic solutions, the branch emanating from the third Hopf bifurcation shows some interesting dynamics. We focus on this branch for the rest of this section. [br_psol3,success] = SetupPsol(funcs,br,ind_hopf(3))

The function SetupPsol allows us to set-up a branch of periodic solutions. Here, we denote by br_psol3 the branch of periodic solutions that emanates from the third Hopf bifurcation which as we saw earlier occurs approximately at the point ind_hopf(3) along the equilibrium branch br. The above command yields br_psol3 = method: [1x1 struct] parameter: [1x1 struct] point: [1x2 struct] success = 1

which means that our attempt to set-up the branch of periodic solutions br_psol3 is successful. br_psol3.method.continuation.plot = 0; [br_psol3,s,f,r] = br_contn(funcs,br_psol3,60); br_psol3 = br_stabl(funcs,br_psol3,0,1); xm_psol=xm; ym_psol.field=’profile’; ym_psol.subfield=’’; ym_psol.row=1; ym_psol.col=’all’; ym_psol.func=’max’; figure(3); clf; hold on; br_splot(br,xm,ym); br_splot(br_psol3,xm_psol,ym_psol); axis([6.5 9.5 0 45]); set(gca,’FontSize’,20);

The above commands yield a plot of the periodic-solution branch br_psol3 together with the equilibrium branch br as shown in Fig. 2. Plotting the branch br_psol3 requires defining xm_psol and ym_psol. Here, xm_psol is the same as xm, i.e., the time delay parameter τ . For ym_psol, we use the maximum function taking just the maximum value of the periodic x(t).

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Fig. 2 Plot of the branch of periodic solutions br_psol3 emanating from the third Hopf bifurcation point at τ = τ2− = 6.981037144585396

nunst_psol3 = GetStability(br_psol3,’exclude_trivial’,true); ind_pd = find(abs(diff(nunst_psol3))==1,1,’first’) [per2,success] = DoublePsol(funcs,br_psol3,ind_pd)

Observe that br_psol3 is initially stable and then it becomes unstable at a point marked with (). This stability switch actually occurs at a period-doubling bifurcation. We can get the value of τ where this period-doubling bifurcation occurs by setting up the branch of period-2 solutions per2 using the function DoublePsol. The value of τ where the period-doubling bifurcation occurred is obtained by typing in the following commands in the Command Window. >> per2.point(1).parameter(8) ans = 8.464201107682122

figure(4); clf; hold on; per2.method.continuation.plot = 0; [per2,s,f,r] = br_contn(funcs,per2,45); per2 = br_stabl(funcs,per2,0,1); br_splot(br_psol3,xm_psol,ym_psol); br_splot(per2,xm_psol,ym_psol); axis([8.2 9.2 30 43]); set(gca,’FontSize’,20);

We continue the branch of period-2 solutions per2 emanating from the perioddoubling bifurcation () along the branch of periodic solutions br_psol3. Figure 3 shows the plot this continued branch together with the periodic solutions branch br_psol3. The period-2 solutions branch per2 also undergoes stability switch at a point marked with (). As before, this is also a period-doubling bifurcation. We leave it to the reader to verify that this second period-doubling bifurcation occurs at the value τ = 8.757752002502176. That is, beyond this value, we can expect a period-4

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Fig. 3 Plot of the branch of period-2 solutions per2 emanating from the period-doubling bifurcation point () along the branch of periodic solutions br_psol3 where τ = 8.464201107682122

Fig. 4 Time series plots of x(t) and y(t) for different values of the time delay parameter τ showing period-1 solutions (τ = 7.10), period-2 solutions (τ = 8.60), and period-4 solutions (τ = 8.78)

solution. Figure 4 shows the time series plots of x(t) and y(t) for different values of the time delay parameter τ showing period-1 solutions (τ = 7.10), period-2 solutions (τ = 8.60), and period-4 solutions (τ = 8.78). We end this section with an exercise for the readers to check if the period-doubling bifurcations will go on and will lead to a cascade of period-doubling bifurcations and eventually to chaos.

4 Conclusions In this paper we presented how to use DDE-Biftool to obtain and analyze branches of solutions to a system of delay differential equations. We did this by revisiting the work of Toaha and Hassan [14] on a harvested predatory-prey model with time

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delay. We first discussed this model theoretically using a more simplified approach. Then, we implement the model in DDE-Biftool to study it numerically. The values of the time delay where Hopf bifurcations occur were numerically obtained and matches the theoretical results. In addition, branches of periodic solutions were also obtained using numerical continuation. At one of the branches of periodic solutions, some interesting dynamics occurred. The occurrence of period-doubling bifurcations, which could lead to chaotic behaviour of the system, was observed. Here, we emphasized the importance of both theory and numerics in studying models. We hope that this paper served its purpose of introducing researchers to time delay systems and its implementation in DDE-Biftool which reveals more dynamical behaviour of the model being considered. Acknowledgements The author acknowledges the support of University of the Philippines Baguio, CIMPA, IMU-CDC, SEAMS, and Universiti Sains Malaysia for his participation to SEAMS School 2018 on Dynamical Systems and Bifurcation Analysis. The author also would like to thank the referees for their valuable reviews that improved the quality of this paper.

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12. Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer, New York (2011). https://doi.org/10.1007/978-1-4419-7646-8 13. Szalai, R.: Knut: a continuation and bifurcation software for delay-differential equations (version 8), Department of Engineering Mathematics, University of Bristol (2013). http://rs1909. github.io/knut/ 14. Toaha, S., Hassan, M.A.: Stability analysis of predator-prey population model with time delay and constant rate of harvesting. Punjab Univ. J. Math. 40, 37–48 (2008)