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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


MINISTRY OF EDUCATION AND TRAINING
HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY
——————————-

NGUYEN PHUONG THUY

COMPETITIVE ECOSYSTEMS:
CONTINUOUS AND DISCRETE MODELS

Major: Mathematics
Code: 9460101

DOCTORAL DISSERTATION OF MATHEMATICS

HANOI - 2018

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

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


DECLARATION OF AUTHORSHIP
This work has been completed at the Department of Applied Mathematics,
School of Applied Mathematics and Informatics, Hanoi University of Science and
Technology, under the supervision of Dr. Nguyen Ngoc Doanh and Associate Prof.
Dr. habil. Phan Thi Ha Duong. I hereby declare that the results presented in
the thesis are new and have never been published fully or partially in any other
thesis/work.
Hanoi, October 2018
On behalf of Supervisors

PhD. Student

Dr. Nguyen Ngoc Doanh

Nguyen Phuong Thuy

iii


ACKNOWLEDGEMENTS

First of all, I would like to express my sincere gratitude to my supervisor, Dr.
Nguyen Ngoc Doanh for his patient guidance, encouragement and valuable advices
throughout my PhD research. I am very grateful to have the chance to work with
him, who is a very knowledge researcher and always being active and helpful supervisor.
I would like to give a special thank to my co-supervisor, Associate Prof. Dr.
habil. Phan Thi Ha Duong whom I admire not only for her professionalism in work
but also for her lifestyle and personality. The discussions with her are always very
valuable and inspired to my work.
I would like to express my gratitude to Prof. Dr. habil. Pham Ky Anh for his
many valuable comments. I would also like to say many thanks to the reviewers,
Prof. Dr. Ngo Dac Tan and Associate Prof. Dr. Le Van Hien for their suggestions
and input that led to the improvement of the thesis. And I would also like to thank
Prof. Dr. Pierre Auger, Dr. Didier Jouffre and Dr. Sidy Ly for their collaboration
in research.
It would have been much more difficult for me to complete this work without the
support and friendship of the members of the “Discrete Mathematics” Seminar at
the Institute of Mathematics, Vietnam Academy of Science and Technology (VAST),
the “Applied Mathematical Models in Control and Ecosystems” Seminar at Hanoi
University of Science and Technology and the “Modeling and Simulation of Complex
System” Seminar of WARM Team at MSLab, Faculty of Computer Science and
Engineering, Thuyloi University. I would also like to especially thank Tran Thi Kim
Oanh, Nguyen Thi Van, Dr. Ha Thi Ngoc Yen, Dr. Lai Hien Phuong, Dr. Pham
Van Trung, Dr. Le Chi Ngoc, Dr. Nguyen Hoang Thach, Dr. Nguyen The Vinh.
Thank you so much.
I would like to thank all the members of the Applied Mathematics Department,
School of Applied Mathematics and Informatics, Hanoi University of Science and
Technology for their encouragement and help in my work.
I would like to express my gratefulness to my beloved family, to my parents who
always encourage and help me at every stages of my personal and academic life and
have been longing to see this achievement come true. This thesis is a meaningful

gift for them. To my big sister Nguyen Phuong Giang, thank you for sharing your
experience in writing the thesis and spending time correcting mine. To my younger
iv


sister Nguyen Anh Thu, thank you for helping me improve my English speaking skill
and making me confident in presenting my results in conferences.
Last but not least, I would like to thank my beloved husband Quan Thai Ha,
who always stands beside me when things are up and down. For my lovely children,
Tra and Khang, their accompany definitely give me a strong motivation to reach to
this point.

Hanoi, October 2018

Nguyen Phuong Thuy

v


LIST OF ABBREVIATIONS

EBM : Equation-Based Model
IBM : Individual-Based Model
GBM : Graph-Based Model
LSE : Local Superior resource Exploiter
LIE : Local Inferior resource Exploiter
BPH : Brown Plant Hopper

1

LIST OF TABLES

Table 2.1

Equilibria of aggregated model (2.13) and local stability analysis . 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


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