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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, October 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, October 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 ... competing ecosystems - Building disk-graph based models to study competing ecosystems Luận án đủ file: Luận án full ... 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... Figure 4.1 Example of the case where the inferior competitor wins globally in model Parameters are chosen as follows r1 = 0.7; r2 = 1.3; K1 = 100; K2 = 100; a12 = 0.8; a21 = 2.2; q1 = 0.6; q2 = 0.3;