Các mô hình liên tục và rời rạc cho hệ sinh thái có yếu tố cạnh tranh (Luận án tiến sĩ)

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

MINISTRY OF EDUCATION AND TRAININGHANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

——————————-NGUYEN PHUONG THUY

COMPETITIVE ECOSYSTEMS:

CONTINUOUS AND DISCRETE MODELS

DOCTORAL DISSERTATION OF MATHEMATICS

HANOI - 2018

MINISTRY OF EDUCATION AND TRAININGHANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

DOCTORAL DISSERTATION OF MATHEMATICS

HANOI - 2018

DECLARATION OF AUTHORSHIP iii

ACKNOWLEDGEMENTS iv

LIST OF ABBREVIATIONS 1

LIST OF TABLES 2

LIST OF FIGURES 3

INTRODUCTION 6

1 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

2 CONTINUOUS MODELS FOR COMPETITIVE SYSTEMS WITH STRATEGY 21 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 32

2.5 Discussion and Conclusion 39

3 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

3.3.1 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

3.4 Conclusion and Perspectives 54

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4 APPLICATION: MODELING OF SOME REFERENCE

4.1 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

4.2 Modeling the brown plant-hopper system 74

4.2.1 Introduction 75

4.2.2 Modeling 76

4.2.3 Analysis and Discussion 79

CONCLUSION 95

BIBLIOGRAPHY 97

LIST OF PUBLICATIONS 107

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DECLARATION OF AUTHORSHIP

This work has been completed at the Department of Applied Mathematics,School of Applied Mathematics and Informatics, Hanoi University of Science andTechnology, under the supervision of Dr Nguyen Ngoc Doanh and Associate Prof

Dr habil Phan Thi Ha Duong I hereby declare that the results presented inthe thesis are new and have never been published fully or partially in any otherthesis/work

Hanoi, October 2018

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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 advicesthroughout my PhD research I am very grateful to have the chance to work withhim, 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 Dr.habil Phan Thi Ha Duong whom I admire not only for her professionalism in workbut also for her lifestyle and personality The discussions with her are always veryvaluable and inspired to my work

I would like to express my gratitude to Prof Dr habil Pham Ky Anh for hismany 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 suggestionsand input that led to the improvement of the thesis And I would also like to thankProf 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 thesupport and friendship of the members of the “Discrete Mathematics” Seminar atthe Institute of Mathematics, Vietnam Academy of Science and Technology (VAST),the “Applied Mathematical Models in Control and Ecosystems” Seminar at HanoiUniversity of Science and Technology and the “Modeling and Simulation of ComplexSystem” Seminar of WARM Team at MSLab, Faculty of Computer Science andEngineering, Thuyloi University I would also like to especially thank Tran Thi KimOanh, Nguyen Thi Van, Dr Ha Thi Ngoc Yen, Dr Lai Hien Phuong, Dr PhamVan 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 andTechnology for their encouragement and help in my work

I would like to express my gratefulness to my beloved family, to my parents whoalways encourage and help me at every stages of my personal and academic life andhave been longing to see this achievement come true This thesis is a meaningfulgift for them To my big sister Nguyen Phuong Giang, thank you for sharing yourexperience in writing the thesis and spending time correcting mine To my younger

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sister Nguyen Anh Thu, thank you for helping me improve my English speaking skilland 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 tothis point

Hanoi, October 2018

Nguyen Phuong Thuy

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LIST OF ABBREVIATIONS

EBM : Equation-Based Model

IBM : Individual-Based Model

GBM : Graph-Based Model

LSE : Local Superior resource ExploiterLIE : Local Inferior resource ExploiterBPH : Brown Plant Hopper

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LIST OF TABLES

Table 2.1 Equilibria of aggregated model (2.13) and local stability analysis 40Table 3.1 The statistics for several complex systems 51Table 3.2 The statistics for several steps of the simulation of the predator-

prey competition system 54Table 3.3 Statistics about the cliques of the graphs at step 1 of the simulation

of the predator-prey competition system 55Table 3.4 Statistics about the cliques of the graphs at step 530 of the simu-

lation of the predator-prey competition system 55

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LIST OF FIGURES

Figure 1.1 Principle of equation-based modeling N1 and N2 are variables

(compartments) F is the mathematical function which representsgeneral laws applied to all members of the compartments [83] 13Figure 1.2 Principle of individual-based modeling [83] 14Figure 1.3 Principle of disk graph-based modeling [83] 15Figure 2.1 Comparison of solutions of system (2.3) with their approxima-

tions through the aggregated system (2.10) for the both biotic andabiotic 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 aggregatedsystem (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= 1 and k = 1) and initial conditions R(0) = 30;

C1(0) = 20; C2C(0) = 15 and C2R(0) = 10 31Figure 2.2 Comparison of solutions of system (2.11) with their approxima-

tions through the aggregated system (2.13) for the both biotic andabiotic 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 gated system (2.13) (R, mC1/L(C2), (βC2+ β0)C1/L(C2) and

aggre-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 initialconditions R(0) = 30; C2(0) = 20; C1C(0) = 15 and C1N(0) = 10 34Figure 2.3 The outcomes of model (2.11) with the biotic resource 41

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Figure 2.4 The outcomes of model (2.11) with the abiotic resource In each

corresponding simulation, parametershave the same values as inthe case of biotic resource and the values of S and the values of

K are exactly the same 42Figure 2.5 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 main (I): LIE wins, domain (II): extinction, domain (III): LSEwins and domain (IV): exclusion via priority effects 43Figure 2.6 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 main (I): LIE wins, domain (II): extinction, domain (III): LSEwins and domain (IV): exclusion via priority effects 44Figure 3.1 Species individual behavior at each simulation step 48Figure 3.2 Distribution of individuals in several simulation steps Red, blue

Do-and green grid cells represent respectively Predator, Prey Do-andGrass individuals a) at step 10, b) at step 100, c) at step 200,d) at step 300 49Figure 3.3 Evolution of the number of individuals of each species The red,

blue and green curves represent respectively the evolution of tor, Prey and Grass 50Figure 3.4 Individual Based Model (on the left) and the corresponding Disk

Preda-Graph Based Model (on the right) 54Figure 3.5 Distribution of degree in several simulation steps: a) at step 1, b)

at step 530, c) at step 1000, d) at step 2500 56Figure 4.1 Example of the case where the inferior competitor wins globally

in model 1 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; E = 0.9 61Figure 4.2 Example of the case where the inferior competitor wins globally

in model 2 Parameters are chosen as follows r1 = 0.9; r2 =0.7; K1 = 100; K2 = 100; a12 = 0.8; a21 = 2.2; q1 = 0.6; q2 =0.3; E = 0.9; d1 = 0.2; d2= 0.4; k = 5; k = 7; m = 6; m = 0.2 64Figure 4.3 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 4.2 except for ε = 0.01 65

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Figure 4.4 Example of the case where the inferior competitor wins globally

in model 3 Parameters are chosen as follows r1 = 0.9; r2 =1.3; K1 = 100; K2 = 100; a12 = 0.8; a21 = 2.2; q1 = 0.6; q2 =0.3; E = 0.9; d1 = 0.2; d2= 0.4; k = 5; k = 7; m = 6; α = 1; α0 = 2 68Figure 4.5 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 4.4 except for ε = 0.01 68Figure 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) issaddle 71Figure 4.7 A photo of BPH-the predator of rice 76Figure 4.8 Rice and brown plant-hopper system ni is the densities of rice

respectively in patch i, i ∈ {1, 2} piA, piJ are the densities ofbrown plant-hopper in mature stage and in egg stage respectively

in patch i, i ∈ {1, 2} m, m are the dispersal rates of brown hopper in mature stage from region 1 to region 2 and opposite 77Figure 4.9 Compare the density of rice on patch 1 between the original model

plant-and the reduced one The case: rice wins globally in competition.Parameters values are chosen as follows: r1 = 0.7; r2 = 0.2;

K = 40; a1 = 0.2; a2 = 0.2; e1 = 0.05; e2 = 0.05; α1 = 0.2;

α2 = 0.3; m = 0.3; m = 0.7; d1A = 0.3; d2A = 0.5; d1J = 0.2;

d2J = 0.3 79Figure 4.10 Equilibria and local stability analysis of the reduced model 87Figure 4.11 The case: rice wins globally in competition Parameters values

are chosen as follows: r1 = 0.7; r2 = 0.2; K = 40; a1 = 0.2;

a2 = 0.2; e1 = 0.05; e2 = 0.05; α1 = 0.2; α2 = 0.3; m = 0.3;

m = 0.7; d1A= 0.3; d2A = 0.5; d1J = 0.2; d2J = 0.3 92Figure 4.12 The case: rice disappears on patch 2 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 93Figure 4.13 The case: the existence of rice and BPH on both patches Param-

eters 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 94

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1 Motivation

The growth and degradation of populations in the nature and the struggle of onespecies to dominate other species have been an interesting topic for a long time Theapplication 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 whosemost important results were published in the 1920s and 1930s

Lotka and Volterra modeled, independently of each other, the competition tween predator and prey Their work has important meaning for the populationbiology field They are the first to study the phenomenon of species interactions byintroducing simplified conditions that lead to solvable problems that have meaninguntil today The proposed model is given by

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 This famous principle is called the competitiveexclusion principle Today, this model is still applicable to competitions between anumber of biophysical species in practice and in empirical observations [22, 63].However, there are many other competing bio-systems, which cannot be ex-plained by using the classic competition model of Lotka-Volterra (or the competitiveexclusion principle) We present here two examples In the first example, Atkinsonand Shorrocks [12] studied the competition of two species for having phytoplankton(food) in multiple environments Competition is noted when one of the two species

is absent, resulting in an increase the remaining species Although the measuredcompeting effect is significant, the two species coexist This result is contrary to theexclusion principle of the classical competition model In the second example, Lei

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and Hanski [63] studied two species of parasites on the same Melitaeacinxia fly The results showed that the more competitive and less hostile species (Cotesiamelitaearum) were not founded in some host species, while the less competitive (Hy-posotherapy horticola) were found in all hosts of Melitaeacinxia This result is alsocontrary to the principle of competitive exclusion.

butter-The main reason for the limitation of Lotka-Volterra’s classic competition model

is that there are too many assumptions in the model, such as the assumption thatthe environment is homogeneous and stable (expressed by the carrying capacities

Ki for the specie i, i ∈ {1, 2}), the behavior of the individual species is the sameand the competition is expressed only by interspecific competitive coefficient aij.Meanwhile, these factors appear frequently and play a very important role Forexample, the migration behavior of individual species is a very important factorfor species survival [80, 104] Individuals of the same species or of different speciesmay have different behaviors Aggressive behavior is also used by individuals ofwild species to compete for accommodation, to fight against their partners, etc Inaddition, individuals may also change their behaviors frequently according to thechange of the environment as studied in [110, 111]

Therefore, the development of new models that take into account the complexenvironments and the behaviors of individuals has been interested by many mathe-maticians Following are some recent approaches

• The complex environment and individual migration behavior in competitiveecosystems The competition process and the migration process have the sametime scale or different time scales

• Aggressive behavior of individuals in competitive system

• Age structure (adult group and immature group) in the competitive system

2 Objective

The objective of this thesis is to develop models for analyzing the effects of theenvironment, the behaviors of individuals (aggressive behavior, hunting habits)and the age structure (adults and juveniles) on the two species of competitiveecosystems To reach this goal, we divide this thesis into 4 main work packages:

- Developing models analyzing the effects of complex environments and aggressivebehavior of the two competing ecosystems

- Developing models analyzing the effect of age structure (adult and juvenile)the studied competing ecosystems

- Building disk-graph based models to study competing ecosystems

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