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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


Contents
DECLARATION OF AUTHORSHIP . . . . . . . . . . . . . . . . . . .

iii

. . . . . . . . . . . . . . . . . . . . . . . . .

iv

LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . .

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

ACKNOWLEDGEMENTS

LIST OF TABLES

INTRODUCTION

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
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

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

3.4

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


4. 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

1


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 1 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

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]. . . 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 = 1 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

3


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 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Figure 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. . . . 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

4


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.

Figure 4.5

68

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. . . . . . . . . . . . . . . . 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 1 to region 2 and opposite. . . 77

Figure 4.9

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 = 0.7; 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 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. . . . . . . . . . . . . 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

5


INTRODUCTION

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 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) 1 −
,

dt
K1
K1
N2 (t)
N1 (t)
dN2 (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 = 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
6


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.
2. 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 4 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.
7


- Implementation and simulation experiments.
3. Research methods
To reach the goal of the thesis, the following methods will be possibly considered:
• Equation-based and individual-based modeling methods are undertaken to
model the reference systems at different time-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
the complexity of the models.
• Methods relating to graph theory are considered to investigate some generated
graph models from the individual-based ones.
4. Results and applications
The thesis presents different models and simulations which can be applied in theoretical as well as empirical study 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 competitive strategies) and simulations (some discrete models for preypredator 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 (Case 1) Thiof
and Octopus Competition in Senegal Coast and (Case 2) Brown Hopper-Plant and
Rice.
5. 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 systems as well as the
approaches to study competing ecosystems including continuous 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 continuous models for the case where two consumer
species exploit a common resource with different competitive 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.
8


• Chapter 4 presents the modeling of two ecology systems: the brown plant
hopper system and the thiof-octopus system.
The main content of the thesis is based on the articles listed in “List of published works of thesis”.
These results have been presented in
- International Workshop on Selected Problems in Optimization and Control
Theory (4/2/2015-7/2/2015) at Institute for Advanced Study in Mathematics (VIASM).
- The 14th Conference on “Scientific Optimization and Computation”, 4/2016,
Ba Vi.
- Seminar “Discrete Mathematics” at the Institute of Mathematics, Vietnamese
Academy of Science and Technology and Vietnam Institute for Advanced Study in
Mathematics (VIASM).
- Seminar “Applied Mathematical Models in Control and Ecosystems”, Hanoi

University of Science and Technology.
- Seminar on Modeling and Simulation of Complex System of WARM Team,
MSLab, Faculty of Computer Science and Engineering, Thuyloi University.

9


Chapter 1
LITERATURE REVIEW
In this chapter, we present some background knowledge. Section 1.1 presents the
concept of competition in ecology systems. The approaches for studying competing
ecosystems such as continuous models and discrete models are presented in Section
1.2 and 1.3. We also present some useful tools for our work as Lyapunov’s methods
and LaSalle’s invariance principle in Section 1.4. Finally, Section 1.5 introduces
briefly the method of aggregation of variables to reduce the complexity of systems.

1.1

Competition in ecology systems

The definition of competition, following the one presented in [23], is given as follow: “Competition is an interaction between individuals, brought about by a shared
requirement for a resource, and leading to a reduction in the survivorship, growth
and/or reproduction of at least some of the competing individuals concerned”. In
fact, all organisms require resources to grow, reproduce, and survive. Animals require food and water; plants require soil nutrients (e.g. nitrogen), light, and water;
etc. They are affected by the environment in which they live, and by the resources
that they obtain. No organism lives in isolation, without any interaction with the
others. Each, for at least part of its life, is a member of a population composed of
individuals of its own species. Organisms, however, cannot acquire a resource when
other organisms consume or defend that resource. Therefore, competitors reduce
each others growth, reproduction, or survival [22].

Competition plays an important role in ecological communities. Individuals of
the same species have very similar requirements for survival, growth and reproduction. However, the resource supply may be not enough for the demand of all
individuals. In this case, the individuals have to compete for the resource and that
leads to the resource deprivation of at least some of them. 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. It could also lead to decreases in
the amounts of stored reserves or to increased risks of predation. These may lead,
10


in turn, to decreases in survivorship and/or decreases in fecundity, which together
determine an individual’s reproductive output. The result of intraspecific competition leads to density dependent birth and death rates, i.e. as density goes up, the
birth rates drop while death rates increase. When the birth rate exceeds the death
rate, the population increases in size. When the death rate exceeds the birth rate,
the population declines. When the two rates are equal, there is no change in population size. This density therefore represents a stable equilibrium, in that all other
densities will approach it. This density is called carrying capacity of the population
and is usually denoted by K. It is called carrying capacity because it represents the
population size which the resources of the environment can just maintain (carry)
without a tendency to either increase or decrease.
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. There have
been many studies of interspecific competition between pairs (or more) of species of
all kinds. Many examples can be found in the literature [22].

1.2

Continuous models

Equation-based modeling (EBM) 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 dynamics of
ecological systems, EBMs often use a set of differential/difference equations, partial
differential equations and stochastic differential equations.
A classical and well-known equation-based model is designed by A. J. Lotka
(1932) and V. Volterra (1926) to deal with the problem of understanding and predicting the evolution of animal competing populations. The model can be used to
predict changes in the densities of species 1 and 2 over time. Then those changes
can be related to the way each species uses resources. 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 significant, one of the two species will
be extinct. This famous principle is called the competitive exclusion principle. Until
now, this competition model has been used to explain dynamics of many competitive systems in reality (see more in [22, 3, 4]). It is still applicable to competitions
between a number of biophysical species in practical and empirical observations. In
11


such classical models, individuals are assumed to be homogeneous and well mixed,
i.e. they are all treated identically. The advantage of this assumption is that these
EBMs can be handled analytically and thus be served as a basis for predicting the
dynamics of the whole system (since all of its evolution can be known in advance).
However, by using a number of assumptions (e.g. the environment is homogeneous
and stable; there is no representation of behaviors/attributes of individuals), these
models do not represent exactly the real system. For example, migration, one of
the important behaviors of individuals, is ignored in the classical Lotka-Volterra
model while it is ubiquitous in natural. The examples from the works of [6, 49, 64]
show that migration of individuals may influence competitive dynamics. The important issue is that behaviors of individuals at local level may affect the dynamics
at global level. This issue has been a central goal in theoretical and conservation
ecology. 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 first approach builds models by taking into account information about the
complex environment and the migration behaviors of individuals in competitive ecosystems. Two processes needed to be considered: the local competition
process and the migration process among patches. That leads to three following cases: (1) The competition process is faster than the migration process
[49, 76, 80, 104]; (2) The competition process and the migration process have
the same time scale [3, 4, 6, 63]; (3) The competition process is slower than
the migration process: studied by Nguyen-Ngoc D. et al. [86, 87] and some
models are based on empirical studies [77, 78].
• The second approach builds models by taking into account the aggressive
behavior of individuals in competitive systems. The first ideas for modeling
the aggressiveness of individuals through the game theory were given by Auger
P. et al. in studies [29, 51] and then in [13, 18, 74, 85].
• The third approach builds models by taking into account the age structure
(adult group and immature group) of competitive system. The age-structure
of competitive system was studied in [71, 75].
In this thesis, we focus on models based on the ideas from the first and the third
approaches.

12


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].

1.3

Discrete models

In order to study ecosystems, in this thesis, we create discrete models by coupling two approaches: individual-based model and disk-graph-based model.
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 (see Figure 1.2). 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.
We can see that IBMs seem to be natural representations of real ecological complex systems. One obvious reason for using this approach to model a real ecological
system is that individuals are building blocks of ecological systems [47, 48]. The
properties and behaviors of individuals determine the properties of the systems that
the individuals compose. In ecological systems, individuals are not identical and do
not stay the same all their life. They have to grow up, develop, acquire resources,
reproduce and interact with each other. The actions of an individual depend on its
internal and external environments and individuals change also with their actions.
13


IBMs are then particularly adapted to represent and understand the emergence of

Figure 1.2: Principle of individual-based modeling [83].

global dynamics among heterogeneous individuals sharing common environmental constraints. This approach of modeling has been applied in ecology since the
1970s/1980s. Early advocates of IBM stated that this new methodology was revolutionary and would be the right method for ecology (see, for example, [54]). Nowadays, IBMs are widely used, they are seen as one of the approaches available together
with other techniques (like EBMs). Interesting examples of IBMs in ecology can be
found in [36, 37, 46, 79, 94, 107].
A lot of simulation platforms dedicated to the implementation of IBM have been
developed for instance GAMA, Mason, Repast or NetLogo. They all have advantages and drawbacks. In our work, we chose to use GAMA [7] to implement our
models because GAMA provides a complete modeling and simulation development
environment for building spatially explicit multi-agent simulations and allows modelers to very easily implement models based on a continuous environment with explicit
geometries. Some models implemented in GAMA can be found in [8, 9, 43, 100].

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
14


Figure 1.3: Principle of disk graph-based modeling [83].
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. This approach has been interested by a lot of modelers in various fields.
For instance, models concern with bio-molecular [2, 21]; models concern with the role
of proteins and genes [45, 97]; models concern with designing effective containment
strategies for infectious diseases as in [39]. With the corpus of researches conducted
on graph theory [28, 40, 53], some interesting properties of graphs can be obtained
analytically and the level of abstraction at which they can be used is not fixed.
That allows us to represent either populations, groups or individuals in ecological
systems.
If all the circles have the same radius then the graph is called a unit disk graph
(UDG). The theory of UDG was introduced by Huson and Sen [53] and since then has
been widely developed, in many application domains, and with efficient algorithms
[28, 40, 53]. While disk graphs have been widely used in many domains, they have
not yet been popular in ecology [2, 21] although it appears that many properties of
ecological systems can be efficiently represented in this formalism.
If a modeler is interested in analyzing populations of animals or plants and their
evolution, many notions from disk graph theory will provide very interesting tools

for analysis. For instance, discovering the subgroups of individuals that have a very

15


high level of interactions and following their evolution directly refers to the notion of
“clique” in an UDG built from the interactions of the individuals in the population
[73]. Other properties studied in disk graph theory, such as “independent sets”, are
also related to notions frequently used in population ecology. Problems of finding
these properties are hard (NP-complete) problems in graph theory. But with UDG,
many efficient algorithms have been developed. For instance, [28, 40, 53] have found
a polynomial algorithm for the problem of finding the “maximal clique”.
Until now, GBM, IBM and EBM are the three prominent, remarkable modeling
approaches in ecology. Each modeling has its own strengths and weaknesses. GBM is
more likely to be used in order to model the structural relations and processes. IBM
is usually used to represent in detail the environment, the behaviors/attributes of individuals, the interactions between individuals as well as the individual-environment
interactions. On the contrary, EBM is dedicated to describe ecological systems in
a global view by using mathematical equations. Therefore, EBM is very abstract
non-reality model but is able to provide prediction about long-term behavior dynamics of systems. IBM is much more realistic but is normally very complex and is
difficult to handle an analysis of long-term behavior. Instead, a lot of simulations
are required to analyze the consequences of behaviors of the systems. Therefore,
coupling these modeling approaches in an effective way could allow us having a deep
understanding of ecology systems. Our work follows this idea. We study generating
graphs of an individual-based predator-prey model. At each time step, a graph,
called disk graphs, representing the interactions between individuals is generated.
In this graph, vertices represent individuals and two vertices are connected by an
edge if the two individuals are in interaction. We investigate some characterized
properties such as maximum cliques, clustering number, degree distribution and
diameter of those graphs.

1.4

Lyapunov’s methods and LaSalle’s invariance
principle

In this section, we present briefly Lyapunov’s methods and Lasalle’s invariance
principle which will be used in our work. Let us start with some definitions of dynamical system and stability in the sense of Lyapunov. In 1890, A.M. Lyapunov
considered the stability of typical dynamical systems described by following nonlinear ordinary differential equations
x˙ = f (x),
16

(1.1)


where x : R → Rn and the vector-valued function f : Rn → Rn is supposed to be
smooth.
Definition 1.4.1. [60]
The equilibria of the system (1.1) are zeros of the vector field given by its right-hand
side, , i.e., xe is an equlibrium of (1.1) if and only if f (xe ) = 0.

Remark: If x(t) is a solution of (1.1) with respect to the initial condition x(0) = xe
where xe is an equilibrium of (1.1) then x(t) = xe for all t.
Definition 1.4.2. [60]
The equilibrium xe of (1.1) is stable if for every given ε > 0 there is a δ = δ(ε) > 0
such that
x(0) − xe < δ ⇒ x(t) − xe < ε, ∀t ≥ 0.

(1.2)

Otherwise it is unstable.

Definition 1.4.3. [60]
The equilibrium xe of (1.1) is asymptotically stable if it is stable and
lim x(t) = xe .

t→∞

(1.3)

Definition 1.4.4. [60]
The region attraction of an equilibrium xe of (1.1) (or in some sense, the region
of asymptotic stability) is the set of all states such that the solution of (1.1) starting
at those states converges towards xe as t → ∞. If the region of attraction is the
whole state space, then xe is globally asymptotically stable.
The following is the content of Lyapunov’s methods, Lasalle’s invariant principle
and some results which are very useful in our work.
Theorem 1.4.5. [60] (Lyapunov’s first method)
Support that the dynamical system (1.1) has an equilibrium x0 and denote by A the
Jacobian matrix of f (x) evaluated at the equilibrium, A = fx (x0 ). Then x0 is stable
if all eigenvalues λ1 , λ2 , . . . , λn of A satisfy Re λi < 0.
Theorem 1.4.6. [58] (Lyapunov’s second method)
Let xe = 0 be an equilibrium state of (1.1) and D ⊂ Rn be a domain containing
xe = 0. Let V : D → R be a continuously differentiable function such that
V (0) = 0 and V (x) > 0, ∀x ∈ D\{0}.

17


If the derivative of V (x) along the trajectories of (1.1) satisfies
dV (x)
=

V˙ (x) =
dt

n

i=1

∂V (x)
x˙i =
∂xi

n

i=1

∂V (x)
fi (x) ≤ 0 in D,
∂xi

then xe = 0 is stable. Moreover, if V˙ (x) < 0 in D\{0} then xe = 0 is asymptotically
stable.
If the conditions for asymptotic stability hold globally and V (x) is radially unbounded (i.e.

x → ∞, V (x) → ∞) then, xe = 0 is globally asymptotically stable.

The function V (x) is called a Lyapunov function.
Theorem 1.4.7. [61] (LaSalle’s invariance principle)
Consider a given C 1 indefinite function V (x) the Ωc = {x : V (x) ≤ c} is a bounded
region and V˙ (x) ≤ 0 along the solutions of x˙ = f (x). Then, any solution starting
in Ωc converges to the larges invariant set M in S = {x : V˙ (x) = 0} ∩ Ωc .

Theorem 1.4.8. [98] (Existence and Uniqueness Theorem)
Consider the initial value problem x˙ = f (x), x(0) = x0 . Suppose that f is continuous
and that all its partial derivatives ∂fi /∂xj , i, j = 1, . . . , n, are continuous for x in
some open connected set D ⊂ Rn . Then for x0 ∈ D, the initial value problem has a
solution x(t) on some time interval (−τ, τ ) about t = 0, and the solution is unique.
Theorem 1.4.9. [90] (Gronwall’s Inequality)
Suppose x satisfies the following differential inequality
d
x(t) ≤ g(t)x(t) + h(t)
dt
for g continuous and h locally integrable. Then, we have that
t

exp{G(t) − G(s)}h(s)ds

x(t) ≤ x(0) exp{G(t)} +
0

for G(t) :=

t
0

g(τ )dτ .

Lemma 1.4.10. [91]
Let w(t, r) be a real-valued continuous function defined for t ∈ R+ , 0 ≤ r < ∞.
Let u(t) be a real-valued differential function defined for t ∈ R+ such that u (t) ≤
w(t, r(t)), for t ∈ R+ . Let r(t) be a maximal solution of r (t) = w(t, r(t)), r(0) = r0 ,
for t ∈ R+ such that u(0) ≤ r0 , then u(t) ≤ r(t), t ∈ R+ .

1.5

Aggregation method

In this section, we give a brief introduction about the aggregation method which
will help us to reduce the complexity of systems.
18


Aggregation of variables methods was well performed in [14, 15]. The considered
models belong to a class of autonomous systems of ordinary differential equations
with two time scales expressed in the following form:
dn
= f (n) + s(n),
dτ

(1.4)

with n ∈ Rm , where maps f and s represent the fast and slow dynamics, respectively,
and is a small positive parameter measuring the time scales ratio when it is possible.
To perform its approximate aggregation, system (1.4) is firstly converted into slowfast form by means of an appropriate change of variables n ∈ Rm → (x, y) ∈
Rm−k × Rk :



 dx = F (x, y) + S(x, y),
dτ
(1.5)
dy



= G(x, y),
dτ
where F, S, G are sufficiently smooth functions, x represents the fast variables and y
represents the slow variables. Finding the transformation n → (x, y) which yields the
slow-fast form (1.5) of the system (1.4) could be a difficult task and the construction
of general algorithms solving this problem is presently an active research line. On
the other hand, in some applications, for example in Auger P. et al. [16, 17], NguyenNgoc D. et al. [84, 86], the context gives a natural way to define the so-called global
variables y and thus to express the system (1.4) in a slow-fast form. The aggregation
method now consists in different steps:
• Step 1: Taking

= 0 in the first equation of slow-fast form (1.5), i.e.

dx
dτ

=

∗

F (x, y). For constant y, finding the asymptotically stable equilibrium x (y) of
this system.
• Step 2: Substituting x∗ (y) into the second equation of slow-fast form (1.5),
obtaining the aggregated system:
dy
= G(x∗ (y), y),
dt

(1.6)

where t = τ represents the slow time variable.
• Step 3: Checking the two conditions: (H1) the system (1.6) is structurally
stable and (H2)

is small enough, which ensures that the asymptotic behavior

of system (1.5) can be studied through system (1.6).
Structurally stable means dynamical systems whose phase portrait (in some domain) does not change qualitatively under all sufficiently small perturbations. The
mathematical definition of structural stability can be found in [60]. The relationship
19


between asymptotic behaviors of the two systems is an application of the classical
Tikhonov theorem. This theorem and its applications were reviewed in [106].
In this chapter, we have presented three main approaches for modeling competing
ecosystems, namely equation-based-modeling approach, individual-based-modeling
approach, disk-graph-based- modeling approach and the strength and weakness of
each approach. We also have presented briefly some useful tools for our work as
Lyapunov’s methods, LaSalle’s invariance principle and aggregation method. In
Chapter 2 and Chapter 4 we followed by presenting the equation-based-modeling
approach. Models in Chapter 2 are for general cases in competitive ecosystems,
while models in Chapter 4 are for specific cases in reality. And a model which is
built by coupling these modeling approaches, is presented in Chapter 3. All models
could allow us to have a deep and effective understanding of ecology systems.

20

Chapter 2
CONTINUOUS MODELS FOR COMPETITIVE
SYSTEMS WITH STRATEGY
In this chapter, we investigate a system of two species exploiting a common resource by using continuous models. We consider both abiotic resource (i.e. resource
with a constant supply rate) and biotic resource (i.e. resource having the properties
of reproduction and self-limitation). We are interested in the asymmetric competition where a given consumer is the locally superior resource exploiter (LSE) and the
other is the locally inferior resource exploiter (LIE). We consider the interference
competition where the LIE individuals can use two opposite strategies to compete
with the LSE individuals. The first one is the avoiding strategy where LIE individuals go to a non-competitive patch to avoid the competition with LSE individuals.
The second one is the aggressive strategy where LIE individuals become very aggressive to chase their competitors to non-competitive patches. We further assume
that there is no resource in the non-competitive patch so that individuals have to
come back to the competitive patch for their survival. Another assumption is that
the migration process is faster than the demography and the competition processes.
The experimented models show that being aggressive is efficient for survival of the
LIE and even provokes global extinction of the LSE. This result does not depend on
the nature of resource. The chapter is organized as follows. Section 2.1 introduces
competitive systems with strategy. Section 2.2 gives a short introduction to the classical exploitative competition model. Section 2.3 studies the case where LIE uses
the avoiding strategy. Section 2.4 focuses on the case where LIE uses the aggressive
strategy. Section 2.5 shows some result analysis, conclusions and perspectives. The
content of this chapter is based on the paper [1] in the LIST OF PUBLICATIONS.

2.1

Introduction on competitive systems

There are two main types of competitions between species: interference competition and exploitative competition. Exploitative competition occurs indirectly
through a common limiting resource which acts as an intermediate. For example,
the use of the resource by one species depletes the amount available for others [68].

21

On the other hand, interference competition occurs directly between individuals
in the way that individuals of one species (1) interfere with the foraging, the survival, the reproduction of other species or (2) prevent the physical presence of other
species in a portion of the habitat. Mechanisms for interference competition include
pheromones, and violent behaviours extending to cannibalism [85, 93].
While a lot of investigations have focused on exploitative competition, there has
been little theory on interference competition. Almost all the studies on interference competition follows the Lotka-Volterra tradition by considering the resource
dynamics to be implicit. The resource is assumed not to be accumulated within the
system, so it can be treated as an input rather than a state variable. The interaction
between exploitative and interference competitions remains largely unexplored for
species that exploit a dynamic resource [85, 105].
In nature, individuals of the same population and of different populations are
able to use different behavioural strategies, i.e. a kind of interference competition, to
compete with others for a common resource. Some phenotypic characteristics, such
as aggressivity, can differ between populations. For instance, in urban populations
(e.g. domestic cats), individuals rarely fight, while in rural populations individuals
are more likely to be aggressive for mating and for getting access to some resources.
Individuals, as living organisms, are able to learn and to change strategy along their
lifetime according to the environmental conditions, to their age, to their physical
conditions and to the results of previous contests [48]. Behavioural plasticity allows
an individual to be more flexible and to adopt the behaviour that can maximize its
survival in the present environmental condition.
Recently, the effects of behavioural plasticity on the competitive dynamics have
attracted a lot of attentions. In [18], the impact of using two strategies (“being
aggressive” and “not being aggressive”) by predator individuals on the predation
dynamics has been investigated. In [71, 74], the authors studied the effects of aggressive behaviour on the age-structured population. In those investigations, aggressive
behaviour was represented by coupling the Hawk-Dove game theory with the population dynamics. It was assumed that the changing of strategies is faster than other
processes such as competition, predation and demography. In our previous contribution [85], we also investigated the interference competition between two consumer
species where individuals may use two opposite strategies (being aggressive or not

being aggressive) to compete with each other. The key issue is to study the effects
of two opposite strategies that may be used by the inferior competitor in order to
avoid extinction in competition with the superior competitor and to survive globally.
We showed that being aggressive is an efficient strategy for the survival of inferior
22