An analysis of some models of prey-predator interaction
Abstract: Biological models of basic prey-predator interaction have been studied. This consisted, at
first, in analyzing the basic models of population dynamics such as the Malthus model, the Verhulst
model, the Gompertz model and the model with Allee effect ; then, in a second step, to analyze the
Lotka-Volterra model and its models improved by taking into account certain important hypotheses such
as competition and/or cooperation between species, existence of refuge for prey and migration of species.
For each population evolution model presented, a numerical illustration was made for its verification.
Key-Words: Difference equations, Population dynamics, Malthus model, Verhulst model, Gompertz
model, Allee Effect, Lotka-Volterra predator-prey model, Stability, competition and cooperation.
1 Introduction
Population dynamics is concerned with the fluc-
tuation over time of the number of individuals
within a population of living beings and also
makes it possible to understand the environmen-
tal influences on the numbers. There are several
criteria that determine the evolution of a given
population. We have for example: Environmen-
tal constraints (i.e., abundance or scarcity of re-
sources, quality of resources, ecological changes,
etc.) Reproduction (fertility or not of individuals,
etc.) which guarantees the survival of the species
in the absence of constant interactions with other
populations in the same environment and/or in-
teractions with the environment.
The diversity of terrestrial or marine species,
whether plants, animals, fungi and microorgan-
isms, is generally subdivided into three levels:
genetic diversity, which is the variability of
genes within a same species or a given popu-
lation. We also talk about intraspecific diver-
sity, which is characterized by the difference
between two individuals of the same species
or subspecies;
Ecosystem diversity which corresponds to the
diversity of ecosystems present on earth, the
interactions of natural populations and their
physical environment.
- Specific diversity, also called interspecific di-
versity, which corresponds to the diversity of
species.
Throughout this article, we are interested in
specific diversity. Thus, several problems on the
preservation of the environment, fauna, flora and
why not that of the survival of the human species,
will result from it since humanity draws its re-
sources there.
The modern foundations of population dy-
namics were laid in an early version of the book,
Essay on the Principle of Population, [1]. This
version opened the ”ways” to the modern study
of population dynamics. His idea was to assume
that: “if a population is not checked, it grows
geometrically”. Later, in 1838, Fran¸cois Verhulst
proposed an alternative model to that of Malthus
by introducing a process of self-regulation or
intra-specific competition, [2]. From 1926, Lotka
and Volterra were to be the pioneers in the study
of the dynamics of several interacting species, [3],
[4]. Later, several contributions to the study of
population growth rates and their interactions
emerged.
Our work is organized as follows. In section
2, we present basic models of population dynam-
ics. The models of Malthus and Verhulst which
the first biological models were the subject of
an analysis ; then we presented the Gompertz
model and the model with Allee effect which take
into account the possibility of an extinction of
species. Then in sections 3 and 4, the Lotka-
Volterra model and the models based on it are
respectively the object of special attention.
THIERRY BI BOUA LAGUI, MOUHAMADOU DOSSO, GOSSOUHON SITIONON
UFR Mathématiques et Informatique
University Université Félix Houphouët-Boigny
22 BP 582 Abidjan 22,
COTE D'IVOIRE
Received: April 16, 2023. Revised: December 2, 2023. Accepted: January 12, 2024. Published: March 11, 2024.
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
93
Volume 21, 2024
2 A few basic models
We start with four basic models of population
change over time. These are the Malthus’, Ver-
hulst’s and Gompertz’s models and the model
with Allee effect.
2.1 Malthus’s model
The Malthus model is one of the first models of
population dynamics. In [1], the author dealt
with the subject of the evolution of the human
population by adopting a relatively simple ap-
proach using the following assumptions :
x(t) is the size of a population at time t;
the increase in this population is proportional
to its size and the length of the time interval
;
the size of the population is represented by
its mean.
These hypotheses are therefore translated into
the following linear differential equation, [5]
dx(t)
dt =rx(t) (1)
where x(t)
dt denotes the population change for a
time interval and ris a constant factor of pro-
portionality which represents the coefficient of in-
crease or growth rate. By integration of equation
(1), we obtain as solution
∀t≥t0, x(t) = x0er(t−t0)(2)
with x0=x(t0)∈R+the size of the population
at the initial time t∈R+, which implies the ex-
ponential growth of this population for a certain
given initial size. The pace that this evolution
will follow depends on the values taken by the in-
trinsic growth rate r of the population, also called
the Malthusian rate. Thus, we have the following
variations :
When r < 0, the evolution of the population
is negative, The size of the population de-
creases until the extinction of the species in
question.
When r= 0, no variation in population size
can be observed. The population remains
constant.
When r > 0, the increase in the population
will be exponential, leading to an infinite de-
velopment of the species.
Figure 1: Malthus growth model
Figure 1 summarizes the different variations in
population size as a function of the values of r.
We take as initial data in the Figure x0= 10
and different values of growth rate with r∈
{−0.4; −0.5; −0.7}in blue and r∈ {0.4; 0.5; 0.6}
in red. This law applies well to microbial popu-
lations, it can be used to model the beginning of
the growth of bacteria for example and it there-
fore remains valid as long as the host environment
can contain the density of the population which
occupies it. However, note that the Malthusian
law has limitations. We present some of them :
(i) The fact that the population increases in-
finitely is not biologically satisfying ;
(ii) The prediction of the evolution over a long
time by the model is problematic, since it
does not take into account the fact that the
host environment of the population could be
saturated.
(iii) The model also does not take into account the
availability of resources vital to the survival
of the population or at least assumes these
resources to be infinite. Which, too, is not
satisfactory.
Based on these limits, other models aimed at de-
scribing the evolution of populations over time
will introduce the notion of carrying capacity, as
we will see below.
2.2 Verhulst’s Model
Verhulst’s model is also called the logistic model,
[2]. This model takes into account the environ-
mental constraints and those related to the re-
sources in the hypotheses (i) to (iii) missing in
the evolution of the population of the biological
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
94
Volume 21, 2024
species. Hence the integration of the concept of
reception capacity, which we note K. For Ver-
hulst, a population cannot grow indefinitely with-
out encountering obstacles. It will be assumed
that the excessive density of individuals would
lead to phenomena of competition, lack of food,
unfavorable ecological conditions or simply the
destruction of natural ecological balances. From
where the differential equation below proposed in
[6].
dx(t)
dt =rx(t)1−x(t)
K(3)
where
x(t) represents the size of the population at
time t;
ris the intrinsic growth rate of the popula-
tion ;
Kis the capacity of the environment to sup-
port population growth. Beyond this limit,
the population will no longer be able to grow.
1−x(t)
Krepresents the part of the biotic
capacity still available at each instant t, that
is to say the maximum value that a given
population can reach in a given habitat.
With the following change of variable
z(t) = 1
y(t)
equation (3) reduces to
dz
dt =−rz−1
K.(4)
the solution of which is
z(t) = z0e−r(t−t0)+1
K.
Which leads to the solution of the type below
of equation (3) with the initial condition x(0) =
x0
x(t) = x0K
x0+ (K−x0)e−r(t−t0)(5)
called logistic functions. There are different cases
to describe the evolution of the population follow-
ing this model.
If 0 < x0< K then the environment can still
accommodate individuals. We will observe
a growth within this population which will
approach over the years, the limit threshold
K.
If x0=Kthen there will be no more evolu-
tion within this population.
If x0> K then intraspecific competition phe-
nomena (fight for food, etc.) will occur. This
will result in an increase in the mortality rate
and therefore a decrease in the size of the
population towards its limit K.
Figure 2 below summarizes this fact for differ-
ent values of x0
Figure 2: Verhulst’s Growth Model
Here we have taken as carrying capacity K=
10, growth rate r= 0.05 and different values
of population rate x0∈ {15; 17; 20}in blue and
x0∈ {3; 4; 5}in red. Verhulst’s model is suit-
able for several types of populations and has pro-
vided good results in some laboratory experi-
ments. Practical examples of the use of the model
to control certain populations exist. We have
the example of the elephant population of Kruger
Park.
However, Verhulst’s model has a drawback. A
drawback of the Verhulst model is that it does
not deal with the possibility of species extinc-
tion. This reality will lead scientists to perfect
the model. The examples of the models with Al-
ley effect that we will see in subsection 2.4, take
this parameter into account.
2.3 Gompertz’s model
The Gompertz model is due to Benjamin Gom-
pertz ( 1779-1865 ) a British mathematician, biol-
ogist, actuary and astronaut by self-taught train-
ing. In 1825, he tackled the subject of the aging
population in a very long article, [7] in the review
’philosophical transactions of the Royal Society of
London’. The function that he establishes makes
it possible to model a situation where a popula-
tion first grows exponentially and then ends up
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
95
Volume 21, 2024
stabilizing by approaching a certain ceiling value.
His model defines the growth of the population
according to the differential equation
dx(t)
dt =rx(t) ln K
x(t)(6)
with the parameters having the same as those of
the model above. With the following change of
variable
y(t) = ln x(t)
K,
equation (6) becomes
dy(t)
dt =−rKy(t) (7)
which leads to the solution
y(t) = y(t0)e−rK(t−t0)
Thus, with the change of variable (7), we obtain
an explicit solution of equation (6) by, [7], [8],
x(t) = Kelnx0
Ke−rK (t−t0)
(8)
It should be noted that this model admits a sin-
gle point of equilibrium which is x=Kwhich
is stable. This model evolves similarly to the lo-
gistic model (both models have the same stable
equilibrium, K). More precisely, we can see from
the Figures that the model is equivalent to the
logistic model in the neighborhood of the equi-
librium K. However, when moving away from
equilibrium K, the growth is much faster than in
the logistic model.
For this model, we take as carrying capac-
ity K= 10, growth rate r= 0.05 and the
different values of initial population rates are
x0∈ {15; 17; 20}for the curves in blue and
x0∈ {3; 4; 5}for those in red. Figure (3a) il-
lustrates the evolution of the Gompertz model
and Figure (3b) where x0={5; 35}and k= 15
gives the comparison of the Gompertz and logis-
tic models.
The red curve, which grows according to the
Gompertz model, reaches the threshold more
quickly than the other blue curve of the logistic
model curve.
As a domain of application of the Gom-
pertz model, researchers have applied it to tu-
mor growth. which led to the discussion of sev-
eral dynamic functions of the growth rate. This
Gompertz growth pattern showed cellular growth
that slows with population density and is there-
fore suitable for observing the evolution of tumor
(a) Gompertz’s Growth Model
(b) Comparison of Gompertz
and logistic models
Figure 3: Evolution of Gompertz and Logistic
models
size, [9]. The growth rate is obtained by the fol-
lowing equation
˙
N(t) = −γN(t)log N(t)
Kt > 0,(9)
N(0) = n0, γ > 0 and K > n0
where N(t) is the tumor cell concentration in the
target corganism, γindicates the net rate of tu-
mor replication and Kis the tumor carrying ca-
pacity.
Other more general models have been pro-
posed. We have, as an example, the famous Gom-
pertz model below, which is a growth model for
fitting real data, [10], [11].
dNG(t)
dt =NG(t)α−βlog NG(t)
K,∀t≥0,
(10)
and NG(0) = K≥0, with α, β ≥0. Note that
reparametrizations of this model were done in [11]
by Tjørve and Tjørve.
In the same continuity of research, it was
proposed and studied in [10], the growth model
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
96
Volume 21, 2024
NGP D(t) solution of differential equation (11)
which is an extension of the Gompertz model gov-
erned by equation (10).
dNGP D(t)
dt =NGP D(t)A1−1
Ab log NGP D (t)
K1+a
,
(11)
∀t≥0, NGP D(0) = K, with K > 0, A > 0, a =
0 and b > 0.
(a) The growth model
NGP D(t), solution of dif-
ferential equation (11)
(b) Tumor growth model N(t),
solution of differential equation
(9)
(c) The growth model NG(t),
solution of differential equation
(10)
Figure 4: Gompertz’s growth model extensions
According to the chronicles, we find the graph
of the Gompertz model with the extensions (9),
(10), (11). However, the models are more or less
flexible compared to that of Gompertz. Curves
(4a) and (4b) of models (10) and (11) respec-
tively, for example, present cases where the curve
can tend towards infinity, or towards a finite limit
(the carrying capacity), or even cross this limit
threshold, or even tend towards zero. Curve (4c)
of the model (9) does not cross the carrying ca-
pacity and depends largely on γand K.
For the growth model NG(t), we take as car-
rying capacity K= 100, α= 0.05 and the dif-
ferents values of β={1; 1.5; 1.1; 1000}, for the
growth model NGP D(t), we take as carrying ca-
pacity A= 1, b= 1 and the different values of
a={−0.2; −0.3; −0.5; }and for tumor growth
model N(0) = 10,
2.4 Model with Allee effect
This model is due to the American zoologist
Warder Clyde Allee (1885-1955) who made mod-
ifications to the Verhulst model. Indeed, he no-
ticed the fact that in the model of Verhulst, cer-
tain aspects that could significantly influence the
evolution of a given population have not been
taken into account. For a low density popula-
tion, it can be observed the following, [12], [13],
[14]:
It can be difficult to find a mate in sexual
species. This state of affairs induces a de-
crease in the rate of reproduction within the
population. This would imply slow or even
negative growth of the population in ques-
tion.
There is also weak intraspecific cooperation
between individuals of a small population. In
fact, the presence of many individuals pro-
motes good survival within this population.
Existence of less resistance to extreme cli-
matic conditions.
Genetic processes such as inbreeding depres-
sion and loss of genetic diversity can also in-
fluence the survival of small populations.
To materialize these hypotheses, Allee introduces
a threshold effect into the equation of the logistic
model. Below this threshold, the population is
driven to extinction. Otherwise, it grows logisti-
cally until it reaches its carrying capacity limit.
This model is expressed in the following form, [15]
:
dx(t)
dt =−rx(t)1−x(t)
Ka1−x(t)
K(12)
where x(t) represents the size of the population
at time tand ris the rate intrinsic population
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
97
Volume 21, 2024
growth as defined in the models below with r > 0
;Kis the capacity of the environment that can
support the growth of the population and Kais
the critical value below which the size of a pop-
ulation should not go down. We also say the
threshold capacity of the Allee effect, or the crit-
ical population with 0 < Ka< K.
A study of Equation (12) yields three equilib-
rium points
x∗
1= 0 ; x∗
2=K;x∗
3=Ka
where x∗
1and x∗
2are stable and the equilibrium
point x∗
3=Kais unstable. Thus, any small vari-
ation in the size of the population around the
two stable equilibrium points is compensated by
a variation in the growth rate, while a variation
in the size of the population in the vicinity of the
unstable equilibrium point is amplified by change
in growth rate. Figures 5 gives an illustration
of the evolution of a population subjected to an
Allee effect.
Figure 5: Growth model with ”Allee effect”
To obtain Figure 5, we took as limit capacity
K= 20, threshold capacity Ka= 11 and growth
rate r= 0.05. Thus for different values of the ini-
tal population rate x0∈ {8; 9.5; 12; 13; 26; 38}, we
note that when the size of the population is above
the limiting capacity Kof the environment or be-
tween the threshold capacity Kaand the limiting
capacity K, the evolution of the population fol-
lows the logistics law. On the other hand, when
this number is below the threshold capacity Ka,
the growth rate becomes negative leading to a de-
crease in numbers which itself induces a greater
decrease in the growth rate and so on. Which
leads to a very rapid extinction of the popula-
tion. There are two types of Allee effect : The
strong Allee effect and the weak Allee effect.
We speak of a strong Allee effect when below
the threshold capacity Ka, populations are
threatened with extinction.
The weak Allee effect is characterized by the
fact that the populations suffer from a lower
growth rate than if there were no Allee effect
but are however not threatened with extinc-
tion.
However, to generate the correlation between per
capita growth rate and population size, the au-
thors of [15] use
dx(t)
dt =rx(t)1−x(t)
Kx(t)−a
K(13)
with x(t) the population size, athe critical point
and Kthe force of competition which is known
as the carrying.
Furthermore, to capture both effects (strong
and weak) in a dynamic population model, we
have equation (14) below which is analogous to
that proposed by Nagumo in [16] in the context
of active transmission of d ’an impulse along a
nerve axon, [13].
dx(t)
dt =rx(x−a)1−x
K.(14)
This model has three equilibria, x= 0, x=K
and x=a. When 0 <a<K,x= 0 and x=K
are stable, while x=ais unstable. This situation
corresponds to the strong Allee effect.
3 Lotka-Volterra Model
The system of Lotka, [17], [18] and Volterra, [4],
[19], also called prey-predator system, is one of
the most famous systems of differential equations
of its time. Its particularity is that it is the first
mathematical model of two interacting popula-
tions. It is a non-linear autonomous dynamic sys-
tem.
Lotka worked on the dynamics of autocatalytic
chemical reactions, before extending his model to
organic systems, then to the evolution of popu-
lations living in communities. As for Volterra, it
wanted to explain qualitatively the fluctuations
of fish stocks in the Adriatic Sea. Indeed, af-
ter the First World War, the zoologist Umberto
d’Anconna, made a paradoxical observation, con-
cerning the quantities of fish of different species
that were caught in the Adriatic Sea; thus dur-
ing the war the sardine fishery had decreased, the
share of these in the catches, which should have
increased, had however decreased in favor of their
predators, the sharks as shown on Figure 6.
It makes the following assumptions :
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
98
Volume 21, 2024
Figure 6: Two fish populations interacting. (Im-
age Futura-sciences)
- We consider two interacting fish species
(preys and predators).
- The predator population is assumed to feed
exclusively on the prey population.
- The population of prey has an unlimited food
source.
Under these assumptions, the model describ-
ing the evolution over time of these two popula-
tions is as follows, [20], [21] :
dx(t)
dt =x(t) (α−βy(t))
dy(t)
dt =−y(t) (δ−γx(t))
α, β, γ, δ ∈R∗
+
(15)
where x(t) represents the population density
of prey and y(t) that of predators in the envi-
ronment, dx(t)
dt and dy(t)
dt are respectively the vari-
ations of prey populations and predators, for a
variation of time. By studying the system (15),
we notice that:
- In the absence of predators the prey popu-
lation follows the Malthusian law. It grows
indefinitely. It is the same observation for
predators in the absence of prey ; but this
time, we are witnessing a rapid extinction of
this population.
- The model admits two equilibrium points :
(x∗, y∗) = (0,0) and (x∗, y∗)=(δ
γ,α
β) which
are respectively unstable and stable.
- The average of the densities with respect to
time remains constant and at equilibrium,
corresponds to (¯x, ¯y) = ( δ
γ,α
β) where ¯xand
¯yare the average density of prey and preda-
tors respectively.
- Trajectories are closed around equilibrium
point.
These different points are illustrated by the
evolution curves (7a) and the phase portrait (7b)
by taking the initial population rates x0= 3 and
y0= 1.5.
(a) Evolution of populations
(b) Field lines and phase por-
trait
Figure 7: Chronicles and phase portrait of Lotka-
Volterra’s model.
Indeed, Figure (7a) represents the evolution
of prey populations (in green color) and preda-
tors (in blue color), as a function of time, as well
as the phase portrait (bottom curve) of prey pop-
ulations ( on the abscissa) and predators (on the
ordinate). The following remarks are made on
the figures:
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
99
Volume 21, 2024
- The maximum amplitudes are constant for
the two populations.
- The period is identical for the two popula-
tions.
- The phase difference between the amplitudes
of the populations is constant.
As for Figure 7b, it represents the field lines
and the phase portrait. The plot of this field
gives a good idea of the dynamics of the system.
We come to the conclusion that, for certain prey-
predator interactions, the results obtained are ac-
tually observed in nature. The Lotka-Volterra
model has specific characteristics and serves as a
basis for other models.
However, the model does not take into account
the capacities of the real environment, it assumes
them to be unlimited. The phenomenon of in-
traspecific competition is not taken into account.
Moreover, whatever the values of the parameters,
we end up with limit cycles (see Figure 7b) while
we expect the disappearance of prey when the re-
production rate is too low compared to the pre-
dation rate.
4 Models based on the
Lotka-Volterra model
By considering the basic model of Lotka-Volterra,
several other scientists will add certain hypothe-
ses to try to describe observable phenomena in
the sense of the evolution of populations in inter-
action. In fact, this evolution responds to laws
that relate to the intrinsic organization of the
different populations. As a result, remarkable be-
haviors are developed by populations in their own
right to reach prey whenever necessary for preda-
tors or to find techniques of protection against
their ”executioners” for prey. Thus, the struggle
for survival will lead animals of different species
to associate with the aim of reaping mutual ben-
efits from this union, without living at the ex-
pense of one another. We then speak of mu-
tualism between populations where populations
adopt simultaneous behaviors tending to monop-
olize the resources of an environment. In this case
we speak of competition between living species.
This competition can be intraspecific (between
members of the same species) or interspecific (be-
tween populations of different species).
4.1 Model of competition
Competition between species induces a strug-
gle for survival. Each species will negatively im-
pact the other. This will cause the growth rate of
each species to decrease. We consider two species
in competitive interaction, [22], [23]. The differ-
ential equation describing the interaction of com-
petition between two species is given below.
,
dx
dt =α1x1−x
K1
−β1
y
K1
dy
dt =α2y1−y
K2
−β2
x
K2
(16)
where the parameters α1,α2,β1,β2,K1and K2
are all positive. The evolution of each population
follows the logistic law. In this equation,
-xand yare the proportions of populations in
interaction ;
-α1and α2are respectively the rates increase
in populations xand y.
-β1and β2respectively characterize the com-
petitive pressure exerted by the population x
on the population yand that exerted by the
population yon the population x.
-K1and K2respectively represent the carry-
ing capacities of the environment of popula-
tion xand population y.
A study of the model makes it possible to obtain
four points of equilibrium. Indeed, by making the
following time scale changes : τ=α1t,u=x
K1,
v=y
K2,φ1=β1K2
K1,φ2=β2K1
K2and ρ=α2
α1.
System (16) becomes
du
dτ =u(1 −u−φ1v)
dv
dτ =ρ(1 −v−φ2u)
(17)
of the following points of equilibrium (0,0), (1,0),
(0,1) and (u∗, v∗) = 1−φ1
1−φ1φ2
,1−φ2
1−φ1φ2.
The equilibrium point (u∗, v∗) has a biologi-
cal sense when φ1φ2= 1, u∗>0 and v∗>0.
Note also that the equilibrium point (0,0) is an
unstable node, the second equilibrium point (1,0)
is a saddle point, if φ2<1 (respectively a sta-
ble node, if φ2>1); as for the third equilibrium
point (0.1), it is a saddle point, if φ1<1 (respec-
tively a stable node, if φ1>1) and for the fourth
equilibrium point (u∗, v∗), it is a saddle point (re-
spectively a stable node) if φ1φ2<1 (respectively
if φ1φ2>1.).
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
100
Volume 21, 2024
Let x(0) and y(0) be the initial densities of the
populations assumed to be strictly positive. From
a biological point of view, we have the following
conclusions:
- If φ1<1 and φ2<1 then the densities con-
verge to the equilibrium (u∗, v∗) ;
- If φ1>1 and φ2>1 then we have a bistabil-
ity phenomenon. One of the populations dies
out while the other tends towards its carrying
capacity.
- If φ1<1 and φ2>1 then the densities con-
verge towards the point of equilibrium (1,0)
with the population xwhich dominates the
other.
- If φ1>1 and φ2<1 then the reverse of the
previous case occurs.
We illustrate the evolution of the populations
in Figure 8 by assuming that the growth rates
(α1= 0.9 and α2= 0.7), with the initial popula-
tion rates (u0= 40 and v0= 40) . In addition,
the carrying capacities (k1= 8 and k2= 5.1) are
taken into account in the context of our example
for all figures.
1) In Figure (8a), after a rapid reduction in the
respective numbers due to the competitive
pressure that each species exerts on the other,
the two species maintain their numbers con-
stant for a certain time. The considered pa-
rameters are (β1= 0.02; β2= 0.05), so that
(φ1<1; φ2<1).
2) In Figure (8b), one of the species disappears
to the benefit of the other as long as the other
species resists the competition. We have con-
sidered the parameters (β1= 1.6; β2= 0.7),
so that (φ1<1; φ2>1).
3) In Figure (8c), with the choice of parame-
ters (β1= 1.7; β2= 0.98) such that (φ1>
1; φ2>1) , the interspecific competition of
species 1 on species 2 is higher than the in-
traspecific competition of species 1. Species
2 dies out while the other tends toward its
carrying
4) The case of Figure (8d) is the opposite of
the case of (8c). We take as parameters
(β1= 1,8; β2= 1,98) which imply (φ1>
1; φ2<1). In this case, species 1 is extin-
guished while the other tends towards its car-
rying capacity
(a)
(b)
(c)
(d)
Figure 8: Chronicles of the competition model
according to the values of φ1,2
4.2 Model of cooperation
Cooperation (or mutualism) between populations
can be intraspecific or interspecific. It occurs
when the individuals of the different populations
come together for their mutual survival. This in-
teraction will allow mutualist populations to take
advantage of each other immediately for some or
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
101
Volume 21, 2024
delayed in time for others.
The Lotka-Volterra model of cooperation, pro-
posed in [24], is given by the system of differential
equations below
dx
dt =α1x1−x
K1
+µ1
y
K1
dy
dt =α2y1−y
K2
+µ2
x
K2
(18)
where µ1and µ2represent the impact of co-
operation on the size of the population xand on
that yrespectively. Note that the model of coop-
eration is opposed to the model of competition,
[25]. The evolution of each population follows
the logistic law. As in the competition model, we
also have four points of equilibrium (0,0), (1,0),
(0,1) and (¯u, ¯v) = 1 + φ1
1−φ1φ2
,1 + φ2
1−φ1φ2where
φ1φ2= 1, φ1>0 and φ2>0. A study of equi-
librium points shows that
- the point of equilibrium (0,0) is an unsta-
ble node and the points of equilibrium (1,0),
(0,1) are saddle points.
- When φ1φ2<1, the equilibrium (¯u, ¯v) is
globally asymptotically stable. Cooperation
is then weak and the two populations coexist
with constant numbers at equilibrium.
- When φ1φ2>1, the equilibrium point (¯u, ¯v)
is unstable. In this case, the cooperation is
strong and the two populations grow without
limit.
The equilibrium point (¯u, ¯v) induces a problem
of overcrowding of populations. But, what will
happen when the capacity of the respective envi-
ronment of each population is exceeded ? Won’t
the populations enter into competition ? Re-
cent studies, in [26], have highlighted this phe-
nomenon. We illustrate this in the following ex-
ample by taking as growth rates α1= 0.09 and
α2= 0.07 ; and the carrying capacities k1= 8
and k2= 5.1. We therefore obtain Figure 9 which
shows the evolution of the populations in three
different cases.
1) In Figure (9a), we consider the parameters
β1= 1.8 and β2= 1.63 such that φ1φ2>1,
and the initial population rates u0= 12 and
v0= 10. Cooperation is strong in this case
and the two populations grow without limit.
(a)
(b)
(c)
Figure 9: Chronicles of the cooperation model
2) In Figure (9b), we take β1= 1.8 and β2= 0.4
so that φ1φ2<1, and the initial population
rates (u0= 12 and v0= 10). The coopera-
tion is then weak and the two populations co-
exist with constant numbers in equilibrium.
3) Figure (9c) shows a phenomenon of over-
population. The initial population rates are
(u0= 120 and v0= 100). This phenomenon
leads to the destruction of natural ecological
equilibrium. We then witness the extinction
of the species.
4.3 Model of competition and
cooperation
In this section, we want to know if we can go from
an interspecific cooperation phenomenon to an in-
terspecific competition phenomenon, depending
on the density of the species. populations. In-
deed, the possibility for a population to pass from
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
102
Volume 21, 2024
mutualism to competition according to the evolu-
tion of population density is inevitable. Too high
a density of mutualist populations could therefore
be the basis of a change in interaction for cer-
tain associations. Consequently, to rethink the
terms cooperation or competition, the authors of
[26] proposed the following coefficient for a cer-
tain population xj.
δij =bixj−cix2
j
1 + dix2
j
i, j = 1,2 and i=j. (19)
where bi,ciand diare constants depending on
the environment. Thus, for two interacting pop-
ulations x,y, we have the following system of
differential equations
dx
dt =α1x1−x
K1
+δ12
y
K1
dy
dt =α2y1−y
K2
+δ21
x
K2
(20)
putting (x, y)=(x1, x2). The other parameters
are the same as those defined in subsection 4.2.
We get equilibrium points (x∗, y∗) which
change in nature according to the variation of
the different parameters. For example, we have
the equilibrium point (K1,0) which is an unstable
saddle point when the interaction is mutualistic
and becomes stable when the populations move
towards competition, [26].
The ecologist Zhibin Zhang in ”Chinese
Academy of Sciences” worked on a model of
species coexistence by associating competition
and cooperation, [27]. The system of equations
describing this interaction is
dx
dt =α1xc1−x−a1(y−b1)2
dy
dt =α2yc2−y−a2(x−b2)2
(21)
where α1,α2,a1,a2,b1,b2,c1and c2are all pos-
itive. The coefficients α1and α2are respectively
the rates increase in populations xand y.
The hypothesis is that for 0 < x ⩽b2or
0< y ⩽b1, the associated species is mutu-
alistic otherwise the population concerned be-
comes a competitor at a higher density. When
the growth of each population is zero, we obtain
a stable equilibrium characterized by the point
(c1−x−a1(y−b1)2, c2−y−a2(x−b2)2).
(a)
(b)
(c)
Figure 10: Chronicles of the competition-
cooperation model
The evolution of populations is summarized by
the examples in Figure 10 below.
We have the different cases of figures accord-
ing to the values of the parameters. In Figure
(10a) where initial population rates are x0= 12
and y0= 15, for example, the populations are
in a situation of cooperation while Figure (10b)
with initial population rates x0= 40 and y0= 40,
presents a situation of competition. As for Fig-
ure (10c), we see the two preceding cases succeed
one another over a long period taking as initial
population rates x0= 8 and y0= 8. Indeed,
when the species are in a situation of cooperation,
the numbers increase simultaneously then reach
a threshold where they enter into competition in
this case the densities of the populations regress
until reaching a threshold where the species be-
come cooperative again.
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
103
Volume 21, 2024
4.4 Lotka-Volterra model with refuge
for prey
Consider two populations in predation interac-
tion. Suppose that a proportion ϕ(x, y) of prey
can find a shelter allowing it to escape from
predators. The differential equation system de-
scribing this interaction is given by, [28]:
dx
dt =α1x−d1y(x−ϕ(x, y))
dx
dt =−α2y+d2y(x−ϕ(x, y))
(22)
Parameters d1and d2are positive. When all the
preys are within reach of the predators, we find
the equation (15) which has two points of equi-
librium (0,0) and (x∗,y∗) with,
x∗=α2
d2
+ϕ(x, y)and y∗=α1
d1
+α1
α2
d2
d1
ϕ(x, y)
Note that the nature of each equilibrium point de-
pends on the value taken by ϕ(x, y) and also on
the initial conditions. The figures and the phase
portrait describing the behavior of the species
which interact according to system (22) are rep-
resented by Figure 11. The growth rates consid-
ered here are α1= 0.5 for prey and α2= 0.1 for
predators. We have also taken the parameters
d1= 0.1 and d2= 0.4, and the inital population
rate x0= 50 and y0= 40 ; and used the following
value for ϕ(x, y) = mx with m∈]0.1[.
We notice that, the closer the value of mis to
0, the dynamics is close to that of system (15) of
which a representation is given in Figure (11a).
However, when the value of mis close to 1, we
notice that, on Figure (11b), over time the pop-
ulations of predators tend towards extinction be-
tween 250 and 275, while the populations of prey
seem to evolve without limit. Figure (11c) shows
the phase portrait.
4.5 Lotka-Volterra model with
migration
In the model of the prey-predator relationship,
if we assume that ”the preys have the possi-
bility of moving, that they can go outside
their initial environment, either in search
of food or to escape their predators, and
that predators also have the possibility of
going outside their environment in search
of preys”; it would then be necessary to take
into account the spatial dimension of the prob-
lem. Taking this factor into account leads to a
(a)
(b)
(c)
Figure 11: Chronicles of Lotka-Volterra model
with refuge for prey
type of spatio-temporal model, which is used in
ecology or biology. The dynamics of this interac-
tion is then described by the following system of
partial differential equations, [29] :
∂u
∂t =d1∆u+ (α1−b1u−ζ1)u
∂v
∂t =d2∆v+ (α2−ζ2)v
(23)
We choose a bounded spatial domain with
boundary conditions that can be of the Neumann
type for example, in which the populations of
prey and predators to be studied are found. In
this equation we have
-uand vrespectively represent the density of
prey and predators at time tand at position
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
104
Volume 21, 2024
x;
- ∆ is the Laplacian operator; d1and d2are
the diffusion coefficients of prey and predator
respectively ;
-ζ1and ζ2denote the functional responses of
the predator to the prey (ie the variations in
the number of prey consumed per individual
and per day).
The other terms are the same as before.
We obtain four equilibrium points (0,0), (1,0),
(0,¯v) and (u∗, v∗). The nature of each of them
depends on the considered domain and the initial
conditions.
This model is suitable for example for the
study of a biological invasion, which consists of
the rapid numerical and spatial expansion of a
population outside its initial environment.
It has moreover been used recently (in France
between 2000 and 2003), to study the spread of
the horse chestnut leafminer.
Figure 12 presents the chronicle of two species
propagating in the same way in space .
(a)
(b)
Figure 12: Propagation of two prey-predator
species in time and space
The following growth rates are used: α1= 0.5
for the prey (chestnut trees) and α2= 0.1 for the
predators (miners) with u0= 500; v0= 300. The
functional response of the predator to the prey is
Holling II type. In this case we will take in system
(23),
ζ1=b2v
u+hu
and ζ2=c v
u+hv
.
Thus, the model (23) is rewritten as follows
∂u
∂t =d1
∂2u
∂x2+α1−b1u−b2v
u+huu
∂v
∂t =d2
∂2v
∂x2+α2−c v
u+hvv
(24)
The spread of predators (miners), represented
by the black color in Figure (12a), is strong from
the start but ends up fading over time under
the effect of the resistance of the prey (chestnut
trees). Figure (12b) represents a view in the (t, x)
plane.
5 Conclusion
Basic biological models of prey-predator interac-
tion have been investigated. We began by pre-
senting the first models of population dynamics
which prompted the implementation of other im-
proved models. These are the simple Malthus
model which does not take into account the ca-
pacity of the environment and the limited ex-
istence of the resources of the population, the
Verhulst model which takes into account these
two hypotheses, then the model of Gompertz on
the rate of aging and mortality of the population
and finally the model with the Allee effect which
takes into account the introduction of a thresh-
old effect in the equation of the logistic model.
Then we made an analysis of the Lotka-Volterra
model which is a model of evolution of two popu-
lations living in community. It is also called prey-
predator model.
Finally, five models based on the Lotka-
Volterra model were analyzed. These are :
the model of competition which induces com-
petition between species living in communi-
ties;
the model of cooperation (also called mutu-
alism) which takes into account the fact that
the different populations work together for
their mutual survival;
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
105
Volume 21, 2024
the model of competition and cooperation
which gives the possibility for a population
to move from mutualism to competition ac-
cording to the evolution of the density of the
population;
the Lotka-Volterra model with refuge for prey
which assumes that a proportion ϕ(x, y) of
prey can find a shelter allowing it to escape
predators ;
and the Lotka-Volterra model with migration
assuming that preys have the possibility of
going outside their initial environment, either
in search of food or to escape their predators,
as well as predators in search of preys.
During our analyses, we carried out numeri-
cal tests, thus illustrating the dynamics of the
populations according to the different models pre-
sented.
In future research, we plan to study the sys-
tems of more than three species in a natural envi-
ronment and their application in the preservation
of the ecosystem.
Acknowledgment:
The authors would like to thank all the
reviewers for their thoughtful comments and
efforts towards improving our manuscript.
References:
[1] T.R. Malthus, An Essay on the Principle of
Population, vol. 2, 6th edition, (1826).
[2] P. F. Verhulst, Notice on the law that a
population follows in its growth (Notice sur
la loi que la population poursuit dans son
accroissement). Corr. Math. Phys.,
10(1838) 113-121.
[3] J. Hofbauer, V. Hutson and W. Jansen,
Coexistence for systems governed by
difference equations of Lotka-Volterra type,
J. Math. Biol. 25 (1987) 553-570.
[4] V. Volterra, Lessons on the Mathematical
Theory of Struggle for Life (Le¸cons sur la
Th´eorie Math´ematique de la Lutte pour la
Vie), Gauthier-Villars, Paris, (1931).
[5] E. Ibarguen-Mondragon, M. Vergel-Ortega,
& S. Gomez-Vergel, Malthus Model applied
to exponential growth of Covid-19. Rebista
Boletin Redipe, 9(11)(2020) 159-164.
[6] M. Rayungsari, A. In’am, and M. Aufin,
Genetic Algorithm to Estimate Parameters
of Indonesian Population Growth Model, in
Proceedings of the International Conference
on Community Development (ICCD 2020),
vol. 477, no. Iccd. Paris, France: Atlantis
Press, 2020. ISBN 978-94-6239-253-3 pp.
426–430.
[7] B. Gompertz, On the Nature of the
Function Expressive of the Law of Human
Mortality, and on a New Mode of
Determining the Value of Life
Contingencies. Philosophical Transactions
of the Royal Society of London, Vol. 115,
(1825), pp. 513-583.
[8] N. Pavlov, G. Spasov, A. Rahnev and N.
Kyurkchiev, A New Class Of
Gompertz–Type Software Reliability
Models, International Electronic Journal of
Pure and Applied Mathematics, Vol. 12 No.
1, (2018) 43-57.
[9] A. R. Sheergojria, P. Iqbala, P. Agarwalb
and N. Ozdemire, Uncertainty-based
Gompertz growth model for tumor
population and its numerical analysis, An
International Journal of Optimization and
Control: Theories &Applications, Vol.12,
No.2 (2022) pp.137150.
[10] M. Asadi, A. Di Crescenzo, F. A. Sajadi &
S. Spina. A generalized Gompertz growth
model with applications and related
birth-death processes. Ricerche di
Matematica, (2020),1-36.
[11] K. Wu, D. Darcet, Q. Wang and D.
Sornette, Generalized logistic growth
modeling of the COVID-19 outbreak in 29
provinces in China and in the rest of the
world, (2020).
[12] A. M. Kramer, B. Dennis, A. M. Liebhold
and J. Drake. The Evidence for Allee
Effects. Population Ecology,
51(3)(2009),341-354.
[13] M. N. Kuperman, G. Abramson, Allee
effect in models of interacting species.
Chaos, Solitons Fract. (2021), 153:111512
[14] P.A. Stephene, W.J. Sutherland,
Consequence of the Allee effet for
behaviour, ecology and conservation, Trends
Ecol. Evol., vol. 14(10)(1999) 401-405.
[15] J.M. Drake, A.M. Kramer, Allee effects.
Nat Educ Knowl, 3(10):2, (2011).
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
106
Volume 21, 2024
[16] J. Nagumo, S. Arimoto and S. Yoshizawa,
Proc. IRE, 50(1962)2061–2070.
[17] A. J. Lotka, Analytical note on certain
rhythmic relations in organic systems. Proc.
Natl. Acad. Sci. 6, (1920) 410-415.
[18] A. J. Lotka, Elements of Physical Biology.
Williams &Wilkins, Baltimore, (1925).
[19] V. Volterra, Fluctuations in the abundance
of a species considered mathematically.
Nature 118 (1926)558–560. Reprinted in
L.A. Real, J.H. Brown (eds.) Foundations
of Ecology. University of Chicago Press,
(1991) pp. 283–285.
[20] T. Blaszak, Lotka-Volterra models of
Predator-Prey Relationships,
https://web.mst.edu/huwen/teaching ˜
Predator Prey Tyler Blaszak.pdf, Missouri
University of Science and Technology, (last
accessed on 2023-04-25).
[21] K. Mahtani, An epidemiological application
of the Lotka-Volterra model to predict
population dynamics of COVID-19,
Parabola. Vol. 59, Issue 1 (2023).
[22] F. J. Ayala, M. E. Gilpin And J. G.
Ehrenfeld, Competition Between Species:
Theoretical Models and Experimental
Tests, Theoretical Populatfon Biology
4,(1973) 331-356.
[23] P. Waltman, Competition Models in
Population Biology, SIAM, PA, (1983).
[24] J. Hofbauer, K. Sigmund. Evolutionary
Games and Population Dynamics,
Cambridge University Press (1998).
[25] C. Bergeron, Mathematical vision of the
equilibrium of biodiversity (Original :
Vision math´ematique sur l’´equilibre de la
biodiversit´e). Bulletin AMQ, Vol. LV, No 4,
(2015) 59.
[26] M. Hernandez. Dynamics of transitions
between population interactions : a
nonlinear interaction α-function defined.
Proceedings Royal Society of London B, 265
(1998)pp. 1433-1440.
[27] Z. Zhang, Mutualism or Cooperation
among Competitors promotes Coexistence
and competitive Ability, Ecological
Modelling, 164 (2003) 271-282.
[28] A. Okubo and S. A. Levin, Diffusion and
Ecological Problems : Modern Perspectives,
second Edition, Interdisciplinary Applied
Mathematics, vol. 14, Springer, New York,
(2001).
[29] B. I. Camara, Complexity of prey-predator
model dynamics with diffusion and
applications (Complexit´e de dynamiques de
mod`eles proie-pr´edateur avec diffusion et
applications), PHD Thesis, Universit´e du
Havre, (2009).
https;//theses.hal.science/tel-
00460361/document.
Contribution of Individual Authors to the
Creation of a Scien-
tific Article (Ghostwriting
Policy)
All authors have contributed equally to creation
of this article.
Indeed,
Thierry Bi Boua Lagui and Gossouhon
Sitionon carried out the digital simula-
tion and the interpretation of the re-
sults of the curves of the different models.
Thierry Bi Boua Lagui and
Mouhamadou Dosso wrote the article
after checking the simulation results.
Soma Ouattara has made corrections to
the English translation of the article.
Sources of Funding for Research Presented
in a Scientific Article or Scientific Article
Itself
There is no funding for this article.
Creative Common0 At-
tribution License 4.0
(Attribution 4.0 International , CC BY
4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
US
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.10
Thierry Bi Boua Lagui,
Mouhamadou Dosso, Gossouhon Sitionon
E-ISSN: 2224-2902
107
Volume 21, 2024
