Các mô hình liên tục và rời rạc cho hệ sinh thái có yếu tố cạnh tranh luận án ti

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. iv LIST OF ABBREVIATIONS. 1 LIST OF TABLES. 2 LIST OF FIGURES .1 Competition in ecology systems .4 Lyapunov’s methods and LaSalle’s invariance principle.

CONTINUOUS MODELS FOR COMPETITIVE SYSTEMS WITH STRATEGY 21 2.1 Introduction on competitive systems .2 The classical competition model without individuals’ strategy .3 A model with an avoiding strategy .4 A model with an aggressive strategy .5 Discussion and Conclusion. DISCRETE MODELS FOR PREDATOR-PREY SYSTEMS 46 3.2 Individual-based predator-prey model .3 Generating graph of the individual-based predator-prey model .1 Graph model for complex systems .2 Graph model for predator-prey system .3 Analysis of the generating graph .4 Conclusion and Perspectives. APPLICATION: MODELING OF SOME REFERENCE ECOSYS- TEMS 57 4.1 Modeling of the thiof-octopus system .3 Analysis and Discussion .2 Modeling the brown plant-hopper system .3 Analysis and Discussion. 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. 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. 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 super- visor.

I would like to give a special thank to my co-supervisor, Associate Prof. 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.

Pham Ky Anh for his many valuable comments. I would also like to say many thanks to the reviewers, Prof. Ngo Dac Tan and Associate Prof. Le Van Hien for their suggestions and input that led to the improvement of the thesis.

And I would also like to thank Prof. 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 1 LIST OF TABLES Table 2.1 Equilibria of aggregated model (2.13) and local stability analysis .1 The statistics for several complex systems .2 The statistics for several steps of the simulation of the predator- prey competition system.3 Statistics about the cliques of the graphs at step 1 of the simulation of the predator-prey competition system.4 Statistics about the cliques of the graphs at step 530 of the simu- lation of the predator-prey competition system. 55 2 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].2 Principle of individual-based modeling [83].3 Principle of disk graph-based modeling [83].1 Comparison of solutions of system (2.3) with their approxima- tions 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 = 0.5; α0 = 1 and k = 1) and initial conditions R(0) = 30; C1 (0) = 20; C2C (0) = 15 and C2R (0) = 10.2 Comparison of solutions of system (2.11) with their approxima- tions 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 aggre- gated 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 = 0.4) and initial conditions R(0) = 30; C2 (0) = 20; C1C (0) = 15 and C1N (0) = 10.3 The outcomes of model (2.11) with the biotic resource .4 The outcomes of model (2.11) with the abiotic resource. In each corresponding simulation, parametershave the same values as in the case of biotic resource and the values of S and the values of K are exactly the same.5 The left panel is about domains of the space (l, d1N , β) for the different outcomes of model 2.13 of the abiotic resource case. Do- main (I): LIE wins, domain (II): extinction, domain (III): LSE wins and domain (IV): exclusion via priority effects.6 The left panel is about domains of the space (l, d1N , β0 ) for the different outcomes of model 2.13 of the biotic resource case.

Do- main (I): LIE wins, domain (II): extinction, domain (III): LSE wins and domain (IV): exclusion via priority effects .1 Species individual behavior at each simulation step.2 Distribution of individuals in several simulation steps. Red, blue and green grid cells represent respectively Predator, Prey and Grass individuals. a) at step 10, b) at step 100, c) at step 200, d) at step 300.3 Evolution of the number of individuals of each species. The red, blue and green curves represent respectively the evolution of Preda- tor, Prey and Grass.4 Individual Based Model (on the left) and the corresponding Disk Graph Based Model (on the right).5 Distribution of degree in several simulation steps: a) at step 1, b) at step 530, c) at step 1000, d) at step 2500.1 Example of the case where the inferior competitor wins globally in model 1.

Parameters are chosen as follows r1 = 0.2 Example of the case where the inferior competitor wins globally in model 2. Parameters are chosen as follows r1 = 0.3 Example of the case where the inferior competitor wins globally in model 2: A comparison between the aggregated model (blue dots) and the complete model (red curve). The parameters are the same as in Figure 4.4 Example of the case where the inferior competitor wins globally in model 3. Parameters are chosen as follows r1 = 0.5 Example of the case where the inferior competitor wins globally in model 3.

A comparison between the aggregated model (blue dots) and the complete model (red curve). The parameters are the same as in Figure 4.6 Two cases where there exists a strictly positive equilibrium: (a) the case where (n∗1 , n∗2 ) is stable, (b) the case where (n∗1 , n∗2 ) is saddle.7 A photo of BPH-the predator of rice.8 Rice and brown plant-hopper system. ni is the densities of rice respectively in patch i, i ∈ {1, 2}. piA , piJ are the densities of brown plant-hopper in mature stage and in egg stage respectively in patch i, i ∈ {1, 2}.

m, m are the dispersal rates of brown plant- hopper in mature stage from region 1 to region 2 and opposite.9 Compare the density of rice on patch 1 between the original model and the reduced one. The case: rice wins globally in competition. Parameters values are chosen as follows: r1 = 0.10 Equilibria and local stability analysis of the reduced model.11 The case: rice wins globally in competition. Parameters values are chosen as follows: r1 = 0.12 The case: rice disappears on patch 2.

Parameters values are chosen as follows: r1 = 0.13 The case: the existence of rice and BPH on both patches. Param- eters values are chosen as follows: r1 = 0. Motivation The growth and degradation of populations in the nature and the struggle of one species to dominate other species have been an interesting topic for a long time. The application of mathematical concepts to explain these phenomena have been doc- umented for centuries.

The founders of mathematical-based modeling are Malthus (1798), Verhulse (1838), Pearl and Reed (1903), especially Lotka and Volterra whose most important results were published in the 1920s and 1930s. Lotka and Volterra modeled, independently of each other, the competition be- tween predator and prey. Their work has important meaning for the population biology field. They are the first to study the phenomenon of species interactions by introducing simplified conditions that lead to solvable problems that have meaning until today.

The proposed model is given by    dN 1 (t) N1 (t) N2 (t) = r1 N1 (t) 1 − − a12 ,   dt K1 K1    dN2 (t) N2 (t) N1 (t) = r2 N2 (t) 1 − − a21 ,   dt K2 K2  where Ni (t) is the density of species i, i ∈ {1, 2}, at time t. Parameters ri and Ki are the growth rate and the carrying capacity of the species i, i ∈ {1, 2}, respectively. Parameter aij is interspecific competitive coefficient representing the negative effect of species j on the growth of species i, i 6= j, i, j ∈ {1, 2}. The ecological meaning of this model is that two species coexist only if the effects of their competition are small.

When the competing effects of two species are large, one of the two species will be extinct.