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MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY ——————————- NGUYEN PHUONG THUY COMPETITIVE ECOSYSTEMS: CONTINUOUS AND DISCRETE MODELS DOCTORAL DISSERTATION OF MATHEMATICS HANOI - 2018 MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY ——————————- NGUYEN PHUONG THUY COMPETITIVE ECOSYSTEMS: CONTINUOUS AND DISCRETE MODELS Major: Mathematics Code: 9460101 DOCTORAL DISSERTATION OF MATHEMATICS HANOI - 2018 Contents DECLARATION OF AUTHORSHIP iii iv LIST OF ABBREVIATIONS LIST OF FIGURES ACKNOWLEDGEMENTS LIST OF TABLES INTRODUCTION LITERATURE REVIEW 10 1.1 Competition in ecology systems 10 1.2 Continuous models 11 1.3 Discrete models 13 1.4 Lyapunov’s methods and LaSalle’s invariance principle 16 1.5 Aggregation method 18 CONTINUOUS MODELS FOR COMPETITIVE SYSTEMS WITH 21 STRATEGY 2.1 Introduction on competitive systems 21 2.2 The classical competition model without individuals’ strategy 24 2.3 A model with an avoiding strategy 25 2.4 A model with an aggressive strategy 2.5 Discussion and Conclusion 39 32 DISCRETE MODELS FOR PREDATOR-PREY SYSTEMS 46 3.1 Introduction 46 3.2 Individual-based predator-prey model 47 3.3 Generating graph of the individual-based predator-prey model 50 3.4 3.3.1 Graph model for complex systems 50 3.3.2 Graph model for predator-prey system 52 3.3.3 Analysis of the generating graph 53 Conclusion and Perspectives 54 i APPLICATION: MODELING OF SOME REFERENCE ECOSYS57 TEMS 4.1 4.2 Modeling of the thiof-octopus system 57 4.1.1 Introduction 57 4.1.2 Model presentation 59 4.1.3 Analysis and Discussion 69 Modeling the brown plant-hopper system 74 4.2.1 Introduction 75 4.2.2 Modeling 4.2.3 Analysis and Discussion 79 76 CONCLUSION 95 BIBLIOGRAPHY 97 LIST OF PUBLICATIONS 107 ii DECLARATION OF AUTHORSHIP This work has been completed at the Department of Applied Mathematics, School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, under the supervision of Dr Nguyen Ngoc Doanh and Associate Prof Dr habil Phan Thi Ha Duong I hereby declare that the results presented in the thesis are new and have never been published fully or partially in any other thesis/work Hanoi, December 2018 On behalf of Supervisors PhD Student Dr Nguyen Ngoc Doanh Nguyen Phuong Thuy iii ACKNOWLEDGEMENTS First of all, I would like to express my sincere gratitude to my supervisor, Dr Nguyen Ngoc Doanh for his patient guidance, encouragement and valuable advices throughout my PhD research I am very grateful to have the chance to work with him, who is a very knowledge researcher and always being active and helpful supervisor I would like to give a special thank to my co-supervisor, Associate Prof Dr habil Phan Thi Ha Duong whom I admire not only for her professionalism in work but also for her lifestyle and personality The discussions with her are always very valuable and inspired to my work I would like to express my gratitude to Prof Dr habil Pham Ky Anh for his many valuable comments I would also like to say many thanks to the reviewers, Prof Dr Ngo Dac Tan and Associate Prof Dr Le Van Hien for their suggestions and input that led to the improvement of the thesis And I would also like to thank Prof Dr Pierre Auger, Dr Didier Jouffre and Dr Sidy Ly for their collaboration in research It would have been much more difficult for me to complete this work without the support and friendship of the members of the “Discrete Mathematics” Seminar at the Institute of Mathematics, Vietnam Academy of Science and Technology (VAST), the “Applied Mathematical Models in Control and Ecosystems” Seminar at Hanoi University of Science and Technology and the “Modeling and Simulation of Complex System” Seminar of WARM Team at MSLab, Faculty of Computer Science and Engineering, Thuyloi University I would also like to especially thank Tran Thi Kim Oanh, Nguyen Thi Van, Dr Ha Thi Ngoc Yen, Dr Lai Hien Phuong, Dr Pham Van Trung, Dr Le Chi Ngoc, Dr Nguyen Hoang Thach, Dr Nguyen The Vinh Thank you so much I would like to thank all the members of the Applied Mathematics Department, School of Applied Mathematics and Informatics, Hanoi University of Science and Technology for their encouragement and help in my work I would like to express my gratefulness to my beloved family, to my parents who always encourage and help me at every stages of my personal and academic life and have been longing to see this achievement come true This thesis is a meaningful gift for them To my big sister Nguyen Phuong Giang, thank you for sharing your experience in writing the thesis and spending time correcting mine To my younger iv sister Nguyen Anh Thu, thank you for helping me improve my English speaking skill and making me confident in presenting my results in conferences Last but not least, I would like to thank my beloved husband Quan Thai Ha, who always stands beside me when things are up and down For my lovely children, Tra and Khang, their accompany definitely give me a strong motivation to reach to this point Hanoi, December 2018 Nguyen Phuong Thuy v LIST OF ABBREVIATIONS EBM : Equation-Based Model IBM : Individual-Based Model GBM : Graph-Based Model LSE : Local Superior resource Exploiter LIE : Local Inferior resource Exploiter BPH : Brown Plant Hopper LIST OF TABLES Table 2.1 Equilibria of aggregated model (2.13) and local stability analysis 40 Table 3.1 The statistics for several complex systems Table 3.2 The statistics for several steps of the simulation of the predator- 51 prey competition system 54 Table 3.3 Statistics about the cliques of the graphs at step of the simulation of the predator-prey competition system 55 Table 3.4 Statistics about the cliques of the graphs at step 530 of the simulation of the predator-prey competition system 55 LIST OF FIGURES Figure 1.1 Principle of equation-based modeling N1 and N2 are variables (compartments) F is the mathematical function which represents general laws applied to all members of the compartments [83] 13 Figure 1.2 Principle of individual-based modeling [83] 14 Figure 1.3 Principle of disk graph-based modeling [83] 15 Figure 2.1 Comparison of solutions of system (2.3) with their approximations through the aggregated system (2.10) for the both biotic and abiotic resource cases This figure shows the evolutions in time of each of the four state variables of system (2.3) (R, C1 , C1C and C1R ) and their approximations obtained from the aggregated system (2.10) (R , C1 , kC2 /H(C1 ) and (αC1 + α0 )C2 /H(C1 ) , respectively), for the same parameter values (r = 3; K = 20; S = 20; a1 = 0.8; e1 = 0.1; a2 = 0.6; e2 = 0.2; d1 = 0.4; d2C = 0.8; d2N = 0.8; α = 1.5; α0 = and k = 1) and initial conditions R(0) = 30; C1 (0) = 20; C2C (0) = 15 and C2R (0) = 10 31 Figure 2.2 Comparison of solutions of system (2.11) with their approximations through the aggregated system (2.13) for the both biotic and abiotic resource cases This figure shows the evolutions in time of each of the four state variables of system (2.11) (R, C1C , C1R and C2 ) and their approximations obtained from the aggregated system (2.13) (R, mC1 /L(C2 ), (βC2 + β0 )C1 /L(C2 ) and C2 , respectively), for the same parameter values (r = 5; K = 7; S = 7; a1 = 0.9; e1 = 0.1; a2 = 0.7; e2 = 0.2; d2 = 0.5; d1C = 0.2; d1N = 0.2; β = 5; α0 = 1, l = 0.2 and m = 0.4) and initial conditions R(0) = 30; C2 (0) = 20; C1C (0) = 15 and C1N (0) = 10 Figure 2.3 34 The outcomes of model (2.11) with the biotic resource 41 Figure 4.12: The case: rice disappears on patch Parameters values are chosen as follows: r1 = 0.7; r2 = 0.2; K = 40; a1 = 0.5; a2 = 0.7; e1 = 0.6; e2 = 0.3; α1 = 0.1; α2 = 0.2; m = 0.3; m = 0.7; d1A = 0.3; d2A = 0.5; d1J = 0.2; d2J = 0.3 Firstly, we re-write condition (4.19) of theorem as follows E1 x + E2 (1 − x) ≤ where E1 = a1 e1 K d1Jα+α − d1A and E2 = a2 e2 K d2Jα+α − d2A E1 (reps E2 ) are the functions of birth and death rates as well as the rate describing being matures of juveniles Therefore, E1 (reps E2 ) can be considered as an evolution function of BPH on patch (reps patch 2) There are several following situations: If E1 , E2 < then condition (4.19) holds In fact, in this case BPH does not grow on both patches Consequently, BPH globally gets extinct If E1 , E2 > then condition (4.19) does not hold In this case, BPH grows on both patches It is therefore survival If E1 < < E2 , then condition (4.19) is equivalent to > µ1 > E2 −E1 +E2 It means that if BPH is more likely to stay on patch (where it cannot grow), it eventually get extinct If E2 < < E1 , then condition (4.19) is equivalent to > µ2 > E1 −E2 +E1 Similarly, BPH will get extinct when it distributes more enough on patch Effects of age-structure parameters In order to study effects of age-structure parameters,we re-write condition (4.19) in the following way: G1 µ1 α1 + G2 µ2 α2 − (d1J d1A µ1 + d2J d2A µ2 ) ≤ where G1 = a1 e1 K − d1A and G2 = a2 e2 K − d2A G1 (reps G2 ) are the functions of birth and death rates That is the reason why we call G1 (reps G2 ) growth function of BPH in patch (reps patch 2) If G1 , G2 < then (∗) is satisfied This condition means 93 Figure 4.13: The case: the existence of rice and BPH on both patches Parameters values are chosen as follows: r1 = 0.3; r2 = 0.9; K = 40; a1 = 0.7; a2 = 0.1; e1 = 0.9; e2 = 0.5; α1 = 0.1; α2 = 0.1; m = 0.8; m = 0.2; d1A = 0.3; d2A = 0.5; d1J = 0.1; d2J = 0.3 that when BPH’s density decreases in both patches then BPH will die as a consequence If G1 , G2 > and (d1J d1A µ1 + d2J d2A µ2 ) ≈ then (4.19) is not satisfied This condition means that the energy that BPH gets from rice is higher than the loosing one due to death process BPH is therefore survival in both patches The present model takes into account a simple system of density-independence dispersal It would be interesting to consider density dependent dispersals in the model This would lead to a more complicated model and will be the subject of our future work Conclusion We have shown, in this chapter, some models for two ecological phenomena For the thiof-octopus system at the coast of Senegal, three models corresponding to three case with increasing complexity have proposed: (1) the case with refuge, (2) the case with refuge and density-independent migration and (3) the case with refuge and density-dependent migration For the rice-BPH system, a model is given by the equation-based approach We studied the local stability analysis and the global properties of their reduced models to get the knowledge about the complete model of these ecological phenomena 94 CONCLUSION Summary of contributions Competitive ecosystems have been under investigation for a long time Many models have been built to get the knowledge and to explain about these ecological phenomena in reality In this thesis, we have developed some continuous and discrete models for studying the effects of the environment, the local behaviors of individuals and the age structure of population on the competitive ecosystems both in theoretical and practical point of views The concrete results are given as follows: In term of theoretical point of view, Chapter dealt with the model with two opposite behaviors (aggressive and avoiding strategies) based on migration of individuals in a patchy (biotic and abiotic) environment revealed that under certain conditions, aggressiveness is efficient for survival of local inferior resource exploiter and even provokes global extinction of the local superior resource exploiter A new methodology of graph generating from individual-based models (a case study in predation dynamics) was proposed in Chapter A comparison with common graphs as well as the integration in term of biology point of view were reported In term of practical point of view, some effective models for two concrete ecological phenomena have been built in Chapter The competition rice-brown plant hopper model with stage structure of population showed some emerge results which support for decision makers for their management The competition thiof-octopus model coupling with fishing pressure figured out the strong increase of the fishing pressure in some areas leads to the depletion of the thiof and the invasion of its competitor, the octopus Futures works There are numerous potential research directions that we could investigate for improving the results in this thesis Here are some on which it would be nice to investigate In the current models, the behaviors of species were simple introduced by taking into account of only single competitive and non-competitive patches It would be also interesting to consider several competitive patches connected by migrations That could lead to much more complicated model but more interesting to investigate We considered, by using the discrete model, the competition of only two predatorprey species in a homogeneous environment As a perspective, we would like to consider more complex case studies of more than two species, for example, a system 95 of one 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LIST OF PUBLICATIONS Thuy Nguyen-Phuong, Doanh Nguyen-Ngoc (2016) Effects of Behavioural Strategy on the Exploitative Competition Dynamics Acta Biotheoretica, 64, pp 495-517 (SCI) Thuy Nguyen-Phuong, Doanh Nguyen-Ngoc, Pierre Auger, Sidy Ly, Didier Jouffre (2016) Can Fishing Pressure Invert the Outcome of Interspecific Competition? The Case of the Thiof and of the Octopus Along the Senegalese Coast Acta Biotheoretica, 64, pp 519-536 (SCI) Thuy Nguyen-Phuong, Oanh Tran-Thi-Kim, Doanh Nguyen-Ngoc, Effects of Fast Dispersal and Stage-Structured on Predator-Prey Dynamics: A Case Study of Brown Plant-Hopper Ecological System (accepted for publication in Vietnam Journal of Mathematical Application) Thuy Nguyen-Phuong, Doanh Nguyen-Ngoc, Duong Phan-Thi-Ha, On the Generating Graph of an Individual-Based Predator-Prey Model (submitted) 107 ... = 3; K = 20; S = 20; a1 = 0.8; e1 = 0.1; a2 = 0.6; e2 = 0.2; d1 = 0.4; d2C = 0.8; d2N = 0.8; α = 1.5; α0 = and k = 1) and initial conditions R(0) = 30; C1 (0) = 20; C2C (0) = 15 and C2R (0) =. .. values (r = 5; K = 7; S = 7; a1 = 0.9; e1 = 0.1; a2 = 0.7; e2 = 0.2; d2 = 0.5; d1C = 0.2; d1N = 0.2; β = 5; α0 = 1, l = 0.2 and m = 0.4) and initial conditions R(0) = 30; C2 (0) = 20; C1C (0) = 15... = 3; K = 20; S = 20; a1 = 0.8; e1 = 0.1; a2 = 0.6; e2 = 0.2; d1 = 0.4; d2C = 0.8; d2N = 0.8; α = 1.5; α0 = and k = 1) and initial conditions R(0) = 30; C1 (0) = 20; C2C (0) = 15 and C2R (0) =