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MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY ——————————- NGUYEN PHUONG THUY COMPETITIVE ECOSYSTEMS: CONTINUOUS AND DISCRETE MODELS Major: Mathematics Code: 9460101 ABSTRACT OF DOCTORAL DISSERTATION OF MATHEMATICS HANOI - 2018 The thesis is completed at Hanoi University of Science and Technology Advisors: Dr Nguyen Ngoc Doanh Assoc Prof Dr habil Phan Thi Ha Duong Reviewer 1: Reviewer 2: Reviewer 3: The thesis will be defended before approval committee at Hanoi University of Science and Technology: Time , date .month .year The thesis can be found at: Ta Quang Buu Library Vietnam National Library INTRODUCTION Motivation The growth and degradation of populations in the nature and the struggle of one species to dominate other species has been an interesting topic for a long time The application of mathematical concepts to explain these phenomena has been documented centuries ago The founders of mathematical-based modeling are Malthus (1798), Verhulse (1838), Pearl and Reed (1903), especially Lotka and Volterra whose most important results are published in the 1920s and 1930s Lotka and Volterra modeled, independently of each other, the competition between predator and prey 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 However, there are many other competing bio-systems, which cannot be explained by using the classic competition model of Lotka-Volterra The main reason for the limitation of Lotka-Volterra’s model is that there are too many assumptions in the model, such as the assumption that the environment is homogeneous and stable, the behavior of the individual species is the same and the competition is expressed only by interspecific competitive coefficient 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 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 for their partners, etc In addition, individuals may also change their behaviors frequently according to the change of the environment Therefore, the development of new models taking 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 the different time scale - Aggressive behavior of individuals in competitive system: The first ideas for modeling the aggressiveness of individuals through game theory were given by Pierre Auger and his collaborators in 2006 - Age structure (mature 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 four main work packages: - Develop models analyzing the effects of complex environments and aggressive behavior of the two competing ecosystems - Develop models analyzing the effect of age structure (adult and juvenile) to study competing ecosystems - Build disk-graph based models to study competing ecosystems - Implement and simulate experiments Research Methods • Equation-based and individual-based modeling methods are undertaken to model the reference systems at different scales and levels of complexity • Methods of dynamical systems and ordinary differential equations are dedicated to the study of the obtained mathematical models In particularly, method of aggregation of variables will be used, if it is necessary, to reduce to complexity of the models • Methods relating to graph theory are considered to investigate some generated graph models from the individual-based ones Results and applications The thesis presents different models and simulations which can be applied in theoretical as well as empirical studies in competitive ecosystems From the theoretical point of view, the author has successfully developed several models (some continues models for the case where two consumer species exploit a common resource with different competition strategies) and simulations (some discrete models for prey-predator systems: from the individual-based model to the generating graph of the individual-based model) In the application point of view, the author has presented some models which are very useful for different case studies such as thiof and octopus competition in Senegal coast (Case 1) and rice and brown plant-hopper (Case 2) The structure and results of the thesis The main part of this thesis is divided into four chapters: Chapter 1: presents the concept of competition in ecology system as well as the approaches to study competing ecosystem including continues models and discrete models The useful tools, Lyapunov’s methods, LaSalle’s invariance principle and aggregated method, are also introduced briefly in this chapter Chapter 2: presents some continues models for the case where two consumer species exploit a common resource with different competition strategies Chapter 3: presents some discrete models for prey-predator systems: from the individual-based model to the generating graph of the individual-based model Chapter 4: presents the modeling of two ecology systems: the thiofoctopus system and the rice-brown plant hopper system Chapter LITERATURE REVIEW 1.1 Competition in ecology systems Competition plays an important role in ecological communities If the competitors are of the same species then the competition is called intraspecific competition Intraspecific competition can be for nest sites, mates, or food Intraspecific competition typically leads to decreased rates of resource intake per individual, and thus to decreased rates of individual growth or development If the competitors are of different species, it is called interspecific competition Under these conditions, the birth and death rates of one population affect these rates of the second population While intraspecific competition results in regulation of the specie’s population, interspecific competition can result in one species dominating the other, even to the point where the second species will go extinct 1.2 Continuous models Equation-based modeling has a long history in population ecology It has been used as a powerful tool which allows to make prediction about possible global emerging properties of the system in a long-term In order to describe the dynamics of ecological systems, equation-based models (EBM) often use a set of differential equations, difference equations, partial differential equations, stochastic differential equations 1.3 Discrete models Individual-based models Individual-Based Model (IBM) is a kind of computational models It simulates the actions and interactions of autonomous agents (both individual or collective entities such as organizations or groups) with a view to assess their effects on the whole of the system The system consists a finite set of elements Each element is represented by an individual, provided with attributes and local processes The dynamics of the model is generated by the interactions that occur between these individual processes These models can be used to test how changes in individual behaviors will affect the emerging overall behavior of the system Disk-graph based models A disk graph-based model is a system in which each element is represented by a circle whose size depends on a specific property of the element Each circle is then considered as a vertex, and the interaction between two elements is represented by an edge between their vertices This kind of model allows taking spatial relationships into account when modeling a system In geometric graph theory, a disk graph (DG) is simply the graph of intersection of a family of circles in the Euclidean plane Hence, graphs can be used to represent a variety of processes or states of a system: interactions, proximity, relationships between individuals, populations, events, etc 1.4 Lyapunov’s methods and LaSalle invariance principle 1.5 Aggregation method Aggregation of variables method was proposed by Pierre Auger in 2008 The considered models belong to a class of autonomous system of ordinary differential equations with two time scales can be expressed in the following form: dn = f (n) + s(n) (1.1) dτ with n ∈ Rm , where maps f and s represent the fast and slow dynamics, respectively, and is the small positive parameter measuring the time scales ratio when it is possible To perform its approximate aggregation, system (1.1) is firstly converted into slow-fast form by means of an appropriate change of variables n ∈ Rm → (x, y) ∈ Rm−k × Rk :    dx = F (x, y) + S(x, y), dτ (1.2) dy   = G(x, y), dτ where F, S, G are sufficiently smooth functions, x represents the fast variables and y represents the slow variables The aggregation method now consists in different steps: - Step 1: Taking = in the first equation of slow-fast form (1.2), i.e dx = F (x, y) For constant y, finding the asymptotically stable equilibrium dτ x∗ (y) of this system - Step 2: Substituting x∗ (y) into the second equation of slow-fast form (1.2), obtaining the aggregated system: dy = G(x∗ (y), y), (1.3) dt where t = τ represents the slow time variable - Step 3: Checking the two conditions: (H1) the system (1.3) is structurally stable and (H2) is small enough, which ensures that the asymptotic behavior of the system (1.2) can be studied through the system (1.3) Chapter CONTINUOUS MODELS FOR COMPETITION SYSTEMS WITH STRATEGY 2.1 Introduction on competition systems 2.2 The classical competition model without individual’s strategy The classical competition model is given as follows:    dR = [γ(R) − a1 C1 R − a2 C2 R]   dt       dC1 = [−d1 C1 + a1 e1 RC1 ]  dt          dC2 = [−d2 C2 + a2 e2 RC2 ] , dt (2.1) where the function γ(R) describes the resource growth When the resource is biotic, we have γ(R) = rR(1 − R/K), where r and K are the growth rate and the carrying capacity of the resource respectively And γ(R) = r(S − R) when the resource is abiotic, where r is the resource turnover rate and S is the supply concentration of the resource, which is akin to the resource carrying capacity Parameter di is the natural death rate of consumer i, represents the capture rate of consumer i on the resource and ei is the parameter related to consumer i recruitment as a consequence of consumer-resource interaction, i ∈ {1, 2} The condition for asymmetric competition, C1 for the LSE and C2 for the LIE, is given by d1 d2 < R∗ , a1 e1 a2 e2 , (2.2) where R∗ is the equilibrium level of the resource when the two consumers are absent R∗ = K when the resource is biotic and R∗ = S when it is abiotic 2.3 A model with an avoiding strategy We assume that the migration is faster than the demography and the competition We re-used the classical model (2.1) in the competitive patch and added new terms for mortality in the non-competition patch and for the migration between two patches to describe the model In this case, the model is given as follows:  dR   = ε γ(R) − a1 C1 R − a2 C2C R   dτ         dC1     dτ = ε[−d1 C1 + a1 e1 RC1 ] (2.3)    dC 2C   = (kC2N − (αC1 + α0 )C2C ) + ε[−d2C C2C + a2 e2 RC2C ]   dτ          dC2N = ((αC1 + α0 )C2C − kC2N ) − εd2N C2N , dτ where the new parameters were used in order to adapt to the model: C2C (resp C2N ) and d2C (resp d2N ) are the density and mortality rate of LIE in the competitive (resp non-competitive) patch; k is the per capita emigration rate from the non-competitive patch to the competitive patch, and αC1 + α0 represents the density dependent migration from the competitive patch to the noncompetitive patch Here α represents the strength of density-dependence in migration, i.e if there are too many LSE individuals in the competitive patch then LIE individuals are more likely to leave this patch to the non-competitive patch In the critical case when α = 0, the migration is density-independent with the per capita emigration rate α0 The parameter ε represents the ratio between two time scales t = ετ , t is the slow time scale and τ is the fast one The condition for the asymmetric competition (2.2) now becomes: d1 d2C < R∗ , a1 e1 a2 e2 (2.4) Model reduction Using the aggregation method, the system (2.3) comes into the following reduced model  dR a2 k   = γ(R) − a1 RC1 − RC2   dt H(C  1)       dC1 (2.5) = C1 [−d1 + a1 e1 R]  dt        dC2 C2   = − kd2C + d2N (αC1 + α0 ) + a2 e2 kR  dt H(C1 ) where C2 = C2C + C2N Global stability of the reduced model (2.5) When R∗ < R2+ = (kd2C + α0 d2N )/a2 e2 k, the equilibria are (R∗ , 0, 0), (0, 0, 0) (for the case of biotic resource) and (R1+ , C1+ , 0) where R1+ = d1 /a1 e1 , C1+ = γ(R1+ )/a1 R1+ which is positive since R1+ < R∗ (condition (2.4)) When R∗ > R2+ , the equilibria are (R∗ , 0, 0), (R2+ , 0, C2+ ), (0, 0, 0) (for the case of biotic resource) and (R1+ , C1+ , 0) where C2+ = γ(R2+ )H(0)/a2 kR2+ Theorem 2.3.1 (R1+ , C1+ , 0) is globally asymptotically stable in R3+ To summarize, in any case LSE is always the globally superior competitor Otherwise, the avoiding strategy of LIE is never successful to avoid extinction 2.4 A model with an aggressive strategy In this part, we consider the second case where LIE individuals become very aggressive so that LSE individuals have to go to a non-competitive patch The model then reads as follows:  dR   = ε γ(R) − a1 RC1C − a2 RC2   dτ         dC1C     dτ = (−(βC2 + β0 )C1C + mC1R ) + ε[−d1C C1C + Ra1 e1 C1C ] (2.6)    dC1N   = ((βC2 + β0 )C1C − mC1N ) − εd1N C1N   dτ          dC2 = ε[−d2 C2 + a2 e2 RC2 ] − εlC2 , dτ where d2 is the natural death rate of consumer 2, d1C and d1N are the natural death rates of consumer in the competitive patch and the non-competitive patch respectively The condition for the asymmetric competition becomes d1C d2 < R∗ , a1 e1 a2 e2 Model reduction  dR a1 m  = γ(R) − RC1 − a2 RC2   dt L(C  2)       dC1 C1 = [−(d1C m + d1N (βC2 + β0 )) + a1 e1 mR]  dt L(C 2)         dC2  = C2 [−(d2 + l) + a2 e2 R], dt where L(C2 ) = βC2 + β0 + m (2.7) (2.8) Table 2.1: Equilibria of aggregated model (2.8) and local stability analysis Conditions R2∗ < R1∗ 1.1 R∗ < R2∗ < R1∗ 1.2 R2∗ < R∗ < R1∗ 1.3 R2∗ < R1∗ < R∗ R1∗ < R2∗ 2.1 R∗ < R1∗ < R2∗ 2.2 R1∗ < R∗ < R2∗ 2.3 R1∗ < R∗∗ < R2∗ < R∗b 2.4 R1∗ < R2∗ < R∗∗ < R∗ 2.5 R1∗ < R2∗ < R∗ < R∗∗ unstable stable (0, 0, 0)a (0, 0, 0) (R∗ , 0, 0) (0, 0, 0) (R∗ , 0, 0) (R1∗ , C1∗ , 0) (R∗ , 0, 0) (R2∗ , 0, C2∗ ) (0, 0, 0) (0, 0, 0) (R∗ , 0, 0) (0, 0, 0) (R∗ , 0, 0) (R2∗ , 0, C2∗ ) (0, 0, 0) (R∗ , 0, 0) ˆ Cˆ1 , Cˆ2 ) (R, (0, 0, 0) (R∗ , 0, 0) ˆ Cˆ1 , Cˆ2 ) (R, (R∗ , 0, 0) (R1∗ , C1∗ , 0) (R2∗ , 0, C2∗ ) (R1∗ , C1∗ , 0) (R1∗ , C1∗ , 0) (R2∗ , 0, C2∗ ) (R1∗ , C1∗ , 0) (R2∗ , 0, C2∗ ) R1∗ = (d1C m + d1N β0 )/(a1 e1 m), C1∗ = γ(R1∗ )L(0)/a1 mR1∗ , R2∗ = (d2 + l)/(a2 e2 ), C2∗ = γ(R2∗ )/a2 R2∗ , R∗∗ = R1∗ + d1N βγ(R2∗ )/(a1 e1 ma2 R2∗ ), ˆ = R2∗ , Cˆ1 = (γ(R) ˆ − a2 R ˆ Cˆ2 )L(Cˆ2 )/a1 mR, ˆ R Cˆ2 = a1 e1 m(R2∗ − R1∗ )/d1N β a : equilibrium (0, 0, 0) appears only in the biotic resource case b : R1∗ < R∗∗ ⇔ R2∗ < R∗ individuals and two vertices are connected by an edge when the two corresponding individuals interact with each other The obtained graphs are disk graphs Some characterized properties such as maximum cliques, clustering number, distribution degree and diameter of those graphs are investigated We compare the properties of the generating graphs of individual-based predator prey models with those of some common complex system graphs We also discuss these properties in biological point of view 3.2 Individual-Based predator-prey Model We consider the dynamics of a predator-prey system living in a homogeneous environment Predator individuals exist and develop by consuming preys Meanwhile, prey individuals exist by eating grass in their living environment Environment: To simplify, we use a 2D grid environment Moreover, grass is added in the environment and being used as the resource that prey individuals can find and eat Light or dark green cells represent areas with grass The shade of green corresponds to the density of grass The darker the green is, the higher the density of grass is The white cells indicate areas without grass Species individual: Each species individual has the capacity to move, to eat and to reproduce These individuals are characterized by their level of energy An individual will die when its energy become null Individuals can gain energy by eating food In details, predator individuals eat the prey individuals while prey individuals increase their energy by eating grass Individual looses energy after each moving step When having enough energy, the individual will reproduce Individuals also looses energy for reproduction The new-born individuals appear at the same grid cell of their mother The number of new-born individuals depends on the property of each species We assume that species individuals reproduce and move stochastically Process: At each simulation step, if a species individual found any food in its neighbor cells, it will capture or eat the food If there is no food in the neighbor cells, the species individuals will move stochastically to one neighbor cell Simulation implementation: We chose to use GAMA platforms to implement our models 3.3 3.3.1 Generating Graph of the IBM Graph Model for Complex Systems The main properties of some real-world complex systems such as Internet, Web, Actors, Co-author from the view of graph are summarized by Jean11 Figure 3.3: Evolution of the number of individuals of each species The red, blue and green curves represent respectively the evolution of Predator, Prey and Grass Loup Guillaume and Matthieu Latapy in 2004 These complex systems have the following common properties: - Most real-world complex systems have a low global density - Complex systems have a low average distance/diameter - The degree distribution of the graph follows a power law: pk ∼ k−α , pk is the probability of a vertex of the degree k The exponent α of the power law is generally between two and three - All these complex systems have a high clustering which seems to be independent of the size of the systems 3.3.2 Graph Model for Predator-Prey System The effected area of an individual u is the disk graph diskR (u) with radius R centered at the position of u A graph G = (V, E) of an ecology system is defined as follows: the vertex set is the set of individuals: V = {1, 2, · · · , n}; there is an edge between two individuals u and v if their effected areas intersect For each simulation iteration of the IBM and for each determined value of R, we get a corresponding graph model for the predator-prey system from the IBM 3.3.3 Analysis of the Generating Graph As most real-world complex systems have the number of edges which scales linearly with the number of vertices, the complex systems have a low density This fits well to our results in simulations (see Table 3.2) Moreover, our experimental results show that the average distance between two vertices is low This result is similar to the results of other complex systems We obtained from our experiments that the global clustering of our model is high and it seems to be independent of the size of the model This result is similar to the common properties of some complex systems The difference of our model is shown in Figure 3.6 That is the degree distribution of our model follows an exponential decrease Therefore, in our model, the number of vertices with high degree is very small The maximum clique problem is NP-complete on arbitrary graphs While a variety of algorithms have been proposed for the solution of the maximum clique problem, only a few of them have been 12 programmed and tested on graphs where the problem is difficult to solve In our work, the k-clique algorithm of Pala et.al in 2005 has been used The results in Table 3.3 proved the effectiveness of this algorithm Figure 3.5: Individual Based Model (on the left) and the corresponding Disk Graph Based Model (on the right) a) b) c) d) Figure 3.6: Distribution of degree in several simulation steps: a) at step 200, b) at step 530, c) at step 1000, d) at step 2500 3.4 Conclusion and Perspective In this chapter, we studied one of the most important ecological complex system, the predator-prey system, by combining the individual-based approach 13 Table 3.2: Some results from simulations of predator-prey system For each graph in each simulation step, n, m, density, c and d are respectively its number of vertices, numbers of links, density, clustering number and average distance n m density c d 6532 71233 1.7e-3 0.6539 7.69 6114 67031 1.8e-3 0.652 7.81 5412 61652 2.1e-3 0.6482 7.69 3032 33577 3.7e-3 0.6485 2.9 4126 31435 1.8e-3 0.6833 4.32 4514 37320 1.8e-3 0.6737 4.72 Table 3.3: Statistics about the cliques of the graphs at step of the simulation of the predator-prey competition system no of vertices no of maximum clique clique number 490 no of 4-clique no of 3-clique 24 86 and the disk graph based approach in the modeling of the system We have shown that with this approach, we are able to extract more information from GBM to get deeper understanding about this ecological system such as the clicks, the local density, the global density, the average distance and the degree distribution Simulations are presented to illustrate for our results The content of this chapter is based on the paper [4] in the LIST OF PUBLICATIONS Chapter APPLICATION: MODELING OF SOME REFERENCE ECOSYSTEMS 4.1 4.1.1 Modeling of the thiof-octopus system Introduction The case of the thiof and the octopus in Senegal leads us to consider several mathematical models of two fish species competing for a common resource and that are harvested by the same fishing fleet 14 4.1.2 Model presentation Model 1: the case without refuge  n2 dn1 n1   = r1 n1 − − a12   dt K K  1    dn n n   = r2 n2 − − a21 dt K2 K2 − q1 n1 E (4.1) − q2 n2 E, where ni is the density of species i, i ∈ {1, 2} Parameters ri and Ki are the growth rate and the carrying capacity of the species i, i ∈ {1, 2} The parameter E is a constant fishing effort The parameter qi represents the capture rate of the fishing on the species i, i ∈ {1, 2} The asymmetric competition in which species is the superior competitor and species is the inferior competitor, leads to the following condition: a12 K2 a21 K1 means that species i is over exploited and/or the mortality rate is high in the refuge While the condition Ii > is related to the case where species i can invade when rare, i ∈ {1, 2} It is obvious that if species is overexploited and/or the mortality rate is high in the refuge then it gets extinct 4.1.3 Analysis and Discussion The most important result is obtained from the equilibria of the aggregated model and the local stability analysis The model shows that, in some conditions of fishing pressure, the joint dynamics of both species can reach the stable equilibrium in which the inferior competitor (octopus) wins globally and the superior competitor (thiof) goes extinct This interesting situation occurs when O1 > and O2 < In other words, the model predicts that the extinction of the superior competitor occurs when: - The fishing effort for the superior competitor (the thiof) is large enough to provoke a global negative growth rate A or else O1 > 0, i.e the superior competitor is overexploited - The fishing effort is large for the inferior competitor (the octopus) but its global growth rate M remains positive or else O2 < 0, i.e the inferior competitor is not overexploited On the contrary, the natural growth rate of the thiof r1 is smaller in comparison and the fishing pressure on this species could be large enough to provoke a global negative growth rate A for the thiof The same situation, i.e the extinction of the superior competitor, could also occur when the two global growth rates are positive A > and M > This means that in this last case, the fishing pressure is not large enough to provoke a global negative growth rate neither for the thiof nor for the octopus In that case, two supplementary conditions must be verified, I1 < and I2 > 0, which can be rewritten combining with the condition (4.2) as follows: 17 Inferior Competitor 60 40 20 0 10 15 20 Superior Competitor 25 30 Figure 4.2: Example of the model where the inferior competitor wins globally 120 Inferior Competitor 100 80 60 40 20 0 10 20 30 Superior Competitor Figure 4.4: Example of the model where the inferior competitor wins globally - I1 < and (4.2) are equivalent to K1 × K2 q1 E d1 ν2∗ − r1 r1 ν1∗ q2 E q2 µ∗2 1− − r2 r2 µ∗1 1− < a12 < where µ∗1 = m/(α0 + m), µ∗2 = − µ∗1 - I2 > and (4.2) are equivalent to  q2 E q2 µ∗2 − − r2 r2 µ∗1 K2 K2  < a21 < × K1 K1  q1 E d1 ν2∗ 1− − r1 r1 ν1∗ K1 , K2  − αd2 K1   mr2  These last inequalities signify that the negative effect of competition on the growth of the thiof (resp the octopus) exerted by the octopus (resp the thiof) 18 should be large (resp small) These last conditions are opposite to conditions (4.2) which have been assumed in order that species is the superior competitor in absence of fishing However, due to the large number of parameters, an adequate choice of parameters could be made to check all inequalities Among the two previous cases leading to the extinction of the superior competitor, we believe that the first case (A < and M > 0) is a good candidate to explain observations that are made for thiof and octopus in Senegal 4.2 4.2.1 Modeling the brown plant-hopper system Introduction Brown plant-hopper (BPH) has been recorded be the most important pests in most of the countries planting rice The biological and ecological characteristics of the BPH and rice have been studied for many years but mathematicalbased modeling has not been used to study this ecological phenomena That leads to the purpose of this study is to build a mathematical model describing a BPH-rice system 4.2.2 Modeling We consider rice and brown plant-hopper model as a predator-prey model on a two-patch environment We assume that on two given patch, the rice’s density increases in the logistic form Moreover, when the BPHs eat rice, the density of rice decrease while the density of BPH increase The BPHs will be extinct when there is no rice on the two patches We further assume that BPH individuals can move between the two patches and this migration process acts on a fast time-scale of the demography, the competition and the predation processes act on a slow time-scale on the two local patches Let ni be the density of rice respectively on patch i, i ∈ {1, 2} And piA , piJ are the densities of BPH in mature stage and in egg stage respectively on patch i, i ∈ {1, 2} Parameters r1 , r2 and K are the growth rate and the carrying capacity of rice on each rice field, respectively Parameters diA , diJ are the natural death rates of BPH in mature stage and BPH in egg stage respectively on patch i, i ∈ {1, 2} And represents the eating rates of BPH, ei is the parameter related to BPH recruitment as a consequence of predator-prey interaction, respectively on patch i, i ∈ {1, 2} We suppose that m, m are the dispersal rates of BPH in mature stage from region to region and opposite Parameter ε represents the ratio between the two time-scales t = ετ For biological reasons, we only consider the solutions (n1 , n2 , p1A , p2A , p2J ) with non-negative initial values n1 (0) ≥ 0, n2 (0) ≥ 0, p1A (0) ≥ 0, p2A (0) ≥ 0, p1J (0) ≥ 0, p2J (0) ≥ Then the completed model reads as follows: 19  n1 dn1    dτ = ε r1 n1 − K − a1 n1 p1A          n2 dn2    = ε r2 n2 − − a2 n2 p2A   dτ K         dp1A   = ε − d1A p1A + α1 p1J + mp2A − mp1A  dτ     dp2A   = ε − d2A p2A + α2 p2J + mp1A − mp2A   dτ        dp1J   = ε − d1J p1J − α1 p1J + e1 a1 n1 p1A    dτ         dp2J = ε − d p − α p + e a n p 2J 2J 2J 2 2A dτ 4.2.3 Analyses and Discussion The non-negativity and boundedness properties of the solutions of the completed model are presented in this subsection Reduced Model  n1 dn1   = n1 r1 − − a1 µ1 pA   dt K         dn2 n2   − a2 µ2 pA = n2 r2 −    dt K      dpA = − d1A µ1 + d2A µ2 pA + α1 p1J + α2 p2J   dt        dp1J   = −d1J p1J − α1 p1J + e1 a1 n1 µ1 pA     dt       dp2J  = −d2J p2J − α2 p2J + e2 a2 n2 µ2 pA dt m m where µ1 = , µ2 = and pA = p1A + p2A m+m m+m 20 Lobal Analyses of the Reduced Model There are always three equilibria on the axes: E0 (0, 0, 0, 0, 0), E1 (0, K, 0, 0, 0), and E2 (K, 0, 0, 0, 0) Equilibria on the bounded space are E3 (K, K, 0, 0, 0) which always exists, E4 (nˆ1 , 0, pˆA , pˆ1J , 0) exists under condition 1/K < 1/ˆ n1 where n ˆ1 = r1 − nˆK1 (d1A µ1 + d2A µ2 )ˆ pA (d1A µ1 + d2A µ2 )(d1J + α1 ) , pˆA = , pˆ1J = , µ1 e1 a1 α1 a1 µ1 α1 and E5 (0, n ¯ , p¯A , 0, p¯2J ) exists if 1/K < 1/¯ n2 where n ¯2 = r2 − n¯K2 (d1A µ1 + d2A µ2 )(d2J + α2 ) (d1A µ1 + d2A µ2 )¯ pA , p¯A = , p¯2J = µ2 e2 a2 α2 a2 µ2 α2 The last equilibrium is the interior one E6 (n∗1 , n∗2 , p∗A , p∗1J , p∗2J ) which exists if the following conditions are satisfied: max{ nˆ11 , n¯12 } < K < nˆ11 + n¯12 , where d1A µ1 + d2A µ2 α1 e1 a1 µ1 α2 e2 a2 µ2 − + d1J + α1 d2J + α2 K , = a1 µ1 α1 e1 a1 à1 a2 à2 e2 a2 à2 ì + × r d1J + α1 r2 d2J + α2 a µ a 1 µ2 ∗ n∗1 = K − p∗A , n∗2 = K − pA , r1 r1 e1 a1 µ1 e2 a2 µ2 p∗1J = n∗1 p∗A , p∗2J = n∗2 p∗A (d1J + α1 ) (d2J + α2 ) p∗A Stability diagram can be categorized depending on K A simple analysis shows 1 that if K > nˆ + n¯ then the dynamics system has equilibria: E0 , E1 , E2 , E3 and the equilibria E3 is always asymptotically stable If the equilibria E6 exists then two equilibria E4 , E5 are always unstable And E4 , E5 cannot be both stable A detail stability analysis is fully provided but is not easily categorized because there is a lot of parameters We present in Figure 4.10 an example where ones can see the whole stability picture of the model depending on K while others are fixed Precisely, the parameters are chosen as r1 = 0.3, r2 = 0.9, a1 = 0.7, a2 = 0.1, e1 = 0.9, e2 = 0.5, α1 = 0.1, α2 = 0.1, m = 0.8, m ¯ = 0.2 and K10 is the positive solution of the following equation of variable X r22 n ¯ X + r2 v + (d1A µ1 + d2A µ2 )(d2J + α2 ) e2 α2 a2 µ2 r2 X− = 0, v v where v = d1A µ1 + d2A µ2 + d2J + α2 E6 (*) means the equilibrium E6 is stable when the condition of its local stability is satisfied And in the case (**), the result of the simulations shows that the dynamical system has a limit cycle around the equilibrium E5 21 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 Figure 4.10: Equilibria and local stability analysis of the reduced model Global Analyses of the Reduced Model Theorem 4.2.1 Suppose that a1 µ1 e1 Kα1 a2 µ2 e2 Kα2 + ≤ d1A µ1 + d2A µ2 d1J + α1 d2J + α2 Then E3 (K, K, 0, 0, 0) is globally asymptotically stable on (0, +∞)5 Remark 4.2.2 The condition for locally asymptotically stability of E3 is the same as the one for its globally asymptotically stability r2 K , and n1 > where n1 , a2 µ2 pA are such that n1 ≤ lim inf t→∞ n1 (t) and pA ≤ lim inf t→∞ pA (t) Then E4 (nˆ1 , 0, pˆA , pˆ1J , 0) is globally asymptotically stable on (0, +∞)5 Theorem 4.2.3 Suppose that pA > pˆA > K K and n2 > where n1 , n2 are such 2 that n1 ≤ lim inf t→∞ n1 (t) and n2 ≤ lim inf t→∞ n2 (t) Then E6 (n∗1 , n∗2 , p∗A , p∗1J , p∗2J ) is globally asymptotically stable on (0, +∞)5 Theorem 4.2.4 Suppose that n1 > 22 Figure 4.11: The case: rice wins globally in competition Remark 4.2.5 The conditions for globally asymptotically stability of E4 (resp E6 ) are satisfied the ones for locally asymptotically stability of E4 (resp E6 ) We are interested in the result of theorem 4.2.1 where BPH gets extinct Understanding the mechanism behind this result plays an important role for the global perspectives of rice-BPH management We investigate effects of different key factors on the extinction of BPH: migration parameters and agestructure parameters Simulations are presented to illustrate for our results (see Figure 4.11) 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 The content of this chapter is based on the paper [2] and [3] in the LIST OF PUBLICATIONS 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, 23 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 ricebrown 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 predator-prey 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 of one prey and two predator species We would like also to consider more complex behaviors of individuals such as the migration behavior as well as take into account of the dynamics of the environment in the predator-prey system The present competition model about rice and brown plant hopper ecological system just took into account of the density-independent migration It would be interesting to consider density dependent migrations in the model for future study 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) ... species individuals will move stochastically to one neighbor cell Simulation implementation: We chose to use GAMA platforms to implement our models 3.3 3.3.1 Generating Graph of the IBM Graph... superior competitor in absence of fishing However, due to the large number of parameters, an adequate choice of parameters could be made to check all inequalities Among the two previous cases leading... stability picture of the model depending on K while others are fixed Precisely, the parameters are chosen as r1 = 0.3, r2 = 0.9, a1 = 0.7, a2 = 0.1, e1 = 0.9, e2 = 0.5, α1 = 0.1, α2 = 0.1, m = 0.8,