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




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


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


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

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 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

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 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


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

(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