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1. 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 documented 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 between predator and prey. Their work has important meaning for the population biology eld. They are the rst to study the phenomenon of species interactions by introducing simpli ed conditions that lead to solvable problems that have meaning until today. The proposed model is given by where N i 8 > > < > > : dN (t) dt dN 1 (t) dt 2 = r = r 1 2 N N 1 2 (t) (t) 1 N 1 N (t) K 1 2 1 (t) K 2 a a 12 21 N (t) K N 2 1 1 (t) K 2 (t) is the density of species i; i 2 f1; 2g, at time t. Parameters r ; ; are the growth rate and the carrying capacity of the species i; i 2 f1; 2g, respectively. Parameter a is interspeci c competitive coecient representing the negative e ect of species j on the growth of species i, i = j; i; j 2 f1; 2g. ij The ecological meaning of this model is that two species coexist only if the e ects of their competition are small. When the competing e ects of two species are large, one of the two species will be extinct. This famous principle is called the competitive exclusion principle. Today, this model is still applicable to competitions between a number of biophysical species in practice and in empirical observations [22, 63]. However, there are many other competing bio-systems, which cannot be explained by using the classic competition model of Lotka-Volterra (or the competitive exclusion principle). We present here two examples. In the rst example, Atkinson and 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 measured competing e ect is signi cant, the two species coexist. This result is contrary to the exclusion principle of the classical competition model. In the second example, Lei i and K iand Hanski [63] studied two species of parasites on the same Melitaeacinxia butter y. The results showed that the more competitive and less hostile species (Cotesia melitaearum) were not founded in some host species, while the less competitive (Hyposotherapy horticola) were found in all hosts of Melitaeacinxia. This result is also contrary to the principle of competitive exclusion. 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 that the environment is homogeneous and stable (expressed by the carrying capacities K for the specie i; i 2 f1; 2g), the behavior of the individual species is the same and the competition is expressed only by interspeci c competitive coecient a i . Meanwhile, these factors appear frequently and play a very important role. For example, the migration behavior of individual species is a very important factor for species survival [80, 104]. Individuals of the same species or of di erent species may have di erent behaviors. Aggressive behavior is also used by individuals of wild species to compete for accommodation, to ght against their partners, etc. In addition, individuals may also change their behaviors frequently according to the change of the environment as studied in [110, 111]. Therefore, the development of new models that take into account the complex environments and the behaviors of individuals has been interested by many mathematicians. Following are some recent approaches. The complex environment and individual migration behavior in competitive ecosystems. The competition process and the migration process have the same time scale or di erent time scales. Aggressive behavior of individuals in competitive system. Age structure (adult group and immature group) in the competitive system. ij
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 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 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 2.4 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 42 Figure 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 Domain (I): LIE wins, domain (II): extinction, domain (III): LSE wins and domain (IV): exclusion via priority effects 43 Figure 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 Domain (I): LIE wins, domain (II): extinction, domain (III): LSE wins and domain (IV): exclusion via priority effects 44 Figure 3.1 Species individual behavior at each simulation step 48 Figure 3.2 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 Figure 3.3 49 Evolution of the number of individuals of each species The red, blue and green curves represent respectively the evolution of Predator, Prey and Grass 50 Figure 3.4 Individual Based Model (on the left) and the corresponding Disk Graph Based Model (on the right) 54 Figure 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 56 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; E = 0.9 61 Figure 4.2 Example of the case where the inferior competitor wins globally in model 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 64 Figure 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 Figure 4.4 Example of the case where the inferior competitor wins globally in model 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 = Figure 4.5 68 Example of the case where the inferior competitor wins globally in model 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 68 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 71 Figure 4.7 A photo of BPH-the predator of rice 76 Figure 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 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 planthopper in mature stage from region to region and opposite 77 Figure 4.9 Compare the density of rice on patch between the original model 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 79 Figure 4.10 Equilibria and local stability analysis of the reduced model 87 Figure 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 92 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 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 94 INTRODUCTION 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 documented 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 between 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 dN1 (t) N1 (t) N2 (t) − a12 = r1 N1 (t) − , dt K1 K1 N2 (t) N1 (t) dN2 (t) = r2 N2 (t) − − 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 = 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 competitive exclusion principle Today, this model is still applicable to competitions between a number of biophysical species in practice and in empirical observations [22, 63] However, there are many other competing bio-systems, which cannot be explained by using the classic competition model of Lotka-Volterra (or the competitive exclusion principle) We present here two examples In the first example, Atkinson and 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 measured competing effect is significant, the two species coexist This result is contrary to the exclusion principle of the classical competition model In the second example, Lei and Hanski [63] studied two species of parasites on the same Melitaeacinxia butterfly The results showed that the more competitive and less hostile species (Cotesia melitaearum) were not founded in some host species, while the less competitive (Hyposotherapy horticola) were found in all hosts of Melitaeacinxia This result is also contrary to the principle of competitive exclusion 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 that the 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 same and the competition is expressed only by interspecific competitive coefficient aij Meanwhile, these factors appear frequently and play a very important role For example, the migration behavior of individual species is a very important factor for species survival [80, 104] Individuals of the same species or of different species may have different behaviors Aggressive behavior is also used by individuals of wild species to compete for accommodation, to fight against their partners, etc In addition, individuals may also change their behaviors frequently according to the change of the environment as studied in [110, 111] Therefore, the development of new models that take into account the complex environments and the behaviors of individuals has been interested by many mathematicians Following are some recent approaches • The complex environment and individual migration behavior in competitive ecosystems The competition process and the migration process have the same time scale or different time scales • Aggressive behavior of individuals in competitive system • Age structure (adult group and immature group) in the competitive system Objective The objective of this thesis is to develop models for analyzing the effects of the environment, the behaviors of individuals (aggressive behavior, hunting habits) and the age structure (adults and juveniles) on the two species of competitive ecosystems To reach this goal, we divide this thesis into main work packages: - Developing models analyzing the effects of complex environments and aggressive behavior 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 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 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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 ... 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;... Figure 4.2 Example of the case where the inferior competitor wins globally in model 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;... Figure 4.4 Example of the case where the inferior competitor wins globally in model 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;