Global stability of Leslie-Gower Predator-prey Model with Density
Dependent Birth Rate on Prey Species and Prey Refuge
FENGDE CHEN, SIJIA LIN, SHANGMING CHEN, YANBO CHONG
College of Mathematics and Statistics
Fuzhou University
No. 2, wulongjiang Avenue, Minhou County, Fuzhou
CHINA
Abstract: - A Leslie-Gower predator prey model with density dependent birth rate on prey species and prey refuge
is proposed and studied in this paper. Sufficient condition which ensure the global stable of the positive equilibrium
is obtained. Our study indicates density dependent birth rate of prey species has negative effect on the final density
of both prey and predator species. Density dependent birth rate may lead to the Allee effect of prey species and
enhance the extinction chance of the species. Numeric simulations are carried out to show the feasibility of the
main results.
Key-Words: Leslie-Gower predator prey model; Refuge; Stability
Received: March 29, 2022. Revised: December 24, 2022. Accepted: January 16, 2023. Published: February 28, 2023.
1 Introduction
The aim of this paper is to investigate the dynamic
behaviors of the following Leslie-Gower predator
prey model with density dependent birth rate on prey
species and prey refuge
dH
dt =r11
c1+c2H−r12 −b1HH
−a1(1 −m)HP,
dP
dt =r2−a2
P
(1 −m)HP,
(1.1)
where m∈[0,1) and ai,ci, i = 1,2, b1, r11, r!2, r2
are all positive constants. where Hand Pare the den-
sity of prey species and the predator species at time
t, respectively. r11
c1+c2His the birth rate of the prey
species, r12 is the death rate of the prey species, r2
is the intrinsic growth rate of the predator species, re-
spectively.
During the past two decades, many scholars inves-
tigated the dynamic behaviors of the population mod-
elling ([1]-[40]), specially, due to its dominant impor-
tance on the nature, many scholars investigated the
dynamic behaviors of the predator prey system, see
[1]-[13], [29]-[40] and the references cited therein.
Numerous studies has been done on the Leslie-Gower
predator prey model, see [5, 8, 9, 12, 32, 33, 34,
35, 36, 37, 38, 39, 40]. There are also many schol-
ars investigated the influence of prey refuge, see
[4, 5, 7, 10, 13, 16, 20, 28].
Chen, Chen and Xie[5] proposed a Leslie-Gower
predator prey model incorporating prey refuge, which
takes the form:
dH
dt = (r1−b1H)H−a1(1 −m)HP,
dP
dt =r2−a2
P
(1 −m)HP,
(1.2)
where m∈[0,1) and ri, ai, i = 1,2, b1are all pos-
itive constants. They showed that prey refuge has
no influence on the persistent property of the sys-
tem. They also showed that increasing the prey refuge
could increase the final density of the prey species,
however, prey refuge has complex influence on the
final density of the predator species.
In system (1.2), one could easily see that without
the influence of the predator species, the prey species
takes the Logistic model
dH
dt = (r1−b1H)H. (1.3)
Here, r1is the intrinsic growth rate and b1is the den-
sity dependent coefficient. Obviously, r1=r11 −r12,
where r11 is the growth rate of the prey species, while
r12 is the death rate of the prey species. Recently,
Chen et al [6] and Zhao et al [22] argued that in some
case, the density dependent birth rate of the species is
more suitable. Now, stimulated by the work of [6, 22],
we also take the famous Beverton-Holt function as the
birth rate, then r11 in system (1.2) should be replaced
by the form r11
c1+c2xand this leads to the model (1.1).
To the best of our knowledge, model (1.1) is first time
proposed and studied.
The aim of this paper is to investigate the stabil-
ity property of the system (1.1), more precisely, we
would like to investigate the global stability of the
positive equilibrium of the system, since it indicates
the long term coexistence of the both species. We also
try to find out the influence of the density dependent
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birth rate of prey species.
In addition to this section, the rest of the paper is
arranged as follows. In next section, we will inves-
tigate the existence and local stability of the positive
equilibrium of the system (1.1). In Section 3, we will
discuss the global stability of the equilibrium by con-
structing some suitable Lyapunov function. In Sec-
tion 4, we will discuss the influence of the density de-
pendent birth rate. Numeric simulations are carried
out in Section 5 to show the feasibility of the main
results. We end this paper by a briefly discussion.
2 The existence and local stability of
the positive equilibrium of system
(1.1)
Concerned with the existence of the positive equilib-
rium of system (1.1), we have the following result.
Theorem 2.1.Assume that
r11 > c1r12 (2.1)
holds, then system (1.1) admits a unique positive
equilibrium B(H∗, P ∗),where
H∗=
−B2+qB2
2−4B1B3
2B1
,
P∗=r2(1 −m)H∗
a2
,
B1=c2(r2(m−1)2a1+b1a2)>0,
B2=a1c1r2(m−1)2+a2b1c1
+a2c2r12,
B3=a2(c1r12 −r11)<0.
(2.2)
Proof. The positive equilibrium of system (1.1) sat-
isfies the equation
r11
c1+c2H−r12 −b1H−a1(1 −m)P= 0,
r2−a2
P
(1 −m)H= 0.
(2.3)
From the second equation of (2.2), one has P=
r2(1 −m)H
a2
.Substituting P=r2(1 −m)H
a2
to the
first equation of (2.3) leads to
r11
c1+c2H−r12 −b1H−a1(1−m)r2(1 −m)H
a2
= 0.
(2.4)
Equation (2.4) is equivalent to
B1H2+B2H+B3= 0,(2.5)
where B1, B2, B3are defined by (2.2). (2.5) has
unique positive solution H∗, hence, under the as-
sumption (2.1) holds, system (1.1) admits a unique
positive equilibrium B(H∗, P ∗).
This ends the proof of Theorem 2.1.
Theorem 2.2. Assume that
r11 > c1r12 (2.6)
holds, B(H∗, P ∗)is locally asymptotically stable.
Proof. Under the assumption (2.6), system (1.1) ad-
mits a unique positive equilibrium B(H∗, P ∗).
The Jacobian matrix of the system (1.1) is calcu-
lated as
J(H, P )
=
A11 −a1(1 −m)H
P2a2
(1 −m)H2r2−2a2P
(1 −m)H
,
(2.7)
where
A11 =r11
c2H+c1
−r12 −b1H−a1(1 −m)P
+H −r11c2
(c2H+c1)2−b1!.
Noting that at B(H∗, P ∗),
r11
c1+c2H∗−r12 −b1H∗−a1(1 −m)P∗= 0,
r2−a2
P∗
(1 −m)H∗= 0.
(2.8)
Then the Jacobian matrix of the system (1.1) about
the equilibrium B(H∗, P ∗)is
J(B(H∗, P ∗))
= −B1−a1(1 −m)H∗
r2
P∗
H∗−r2!,
(2.9)
where
B1=H∗ r11c2
(c2H∗+c1)2+b1!.
Consequently, we have
DetJ(B(H∗, P ∗)) = r2B1+a1H∗r2
P∗
H∗>0,
and
T rJ(B(H∗, P ∗)) = −B1−r2<0.
So that both eigenvalues of J(B(H∗, P ∗)) have neg-
ative real parts, and B(H∗, P ∗)is locally asymptoti-
cally stable.
This ends the proof of Theorem 2.2.
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3 Global stability
Concerned with the global stability of the positive
equilibrium of system (1.1), we have the following
result.
Theorem 3.1. Assume that
r11 > c1r12 (3.1)
holds, B(H∗, P ∗)is globally stable.
Proof. Under the assumption (3.1) holds, sys-
tem (1.1) admits a unique positive equilibrium
B(H∗, P ∗), which satisfies the equalities
r11
c1+c2H∗−r12 −b1H∗−a1(1 −m)P∗= 0,
r2−a2
P∗
(1 −m)H∗= 0.
(3.2)
Now let us consider the following Lyapunov func-
tion:
V(H, P )
=ln H
H∗+H∗
H
+a1(1 −m)2H∗
a2ln P
P∗+P∗
P.
(3.3)
Obviously, V(H, P )is well defined and continuous
for all H, P > 0. By simple computation, we have
∂V
∂H =1
H1−H∗
H,
∂V
∂P =a1(1 −m)2H∗
a2P1−P∗
P.
(3.4)
(3.4) shows that the positive equilibrium (H∗, P ∗)
is the only extremum of the function V(H, P )in the
positive quadrant. One could easily verifies that
lim
H→0V(H, P )
=lim
P→0V(H, P )
=lim
H→+∞
V(H, P )
=lim
P→+∞
V(H, P ) = +∞.
(3.5)
(3.4) and (3.5) show that the positive equilibrium
(H∗, P ∗)is the global minimum, that is,
V(H, P )> V (H∗, P ∗) = 1 + a1(1 −m)2H∗
a2
>0
for all H, P > 0.
Calculating the derivative of Valong the solution
of the system (1.1), by using equalities (3.2), we have
dV
dt
=1
H1−H∗
H r11
c1+c2H−r12 −b1H
−a1(1 −m)PH
+a1(1 −m)2H∗
a2P1−P∗
P×
r2−a2
P
(1 −m)HP
=H−H∗
H−r11
c1+c2H∗+b1H∗
+a1(1 −m)P∗+r11
c1+c2H
−b1H−a1(1 −m)P
+a1(1 −m)2H∗
a2P1−P∗
P×
a2
P∗
(1 −m)H∗−a2
P
(1 −m)HP
=−b1
H(H−H∗)2
+a1(1 −m)
H(H−H∗)(P∗−P)
+H−H∗
H
r11(c1+c2H∗−c1−c2H)
(c1+c2H∗)(c1+c2H)
+a1(1 −m)H∗×P−P∗
P×
P∗H−P H +P H −P H∗
H∗H
=−b1
H(H−H∗)2
+a1(1 −m)
H(H−H∗)(P∗−P)
+H−H∗
H
r11c2(H∗−H)
(c1+c2H∗)(c1+c2H)
−a1(1 −m)
P(P−P∗)2
+a1(1 −m)
H(H−H∗)(P−P∗)
=−b1
H(H−H∗)2−a1
P(P−P∗)2
−r11c2
H(c1+c2H∗)(c1+c2H)(H∗−H)2.
(3.6)
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Obviously, dV
dt <0strictly for all H, P > 0except
the positive equilibrium (H∗, P ∗), where dV
dt = 0.
Thus, V(H, P )satisfies Lyapunov’s asymptotic sta-
bility theorem, and the positive equilibrium (H∗, P ∗)
of system (1.1) is globally stable. This ends the proof
of Theorem 3.1.
4 The influence of density dependent
birth rate
From Theorem 2.1 and 3.1, it seems that c2has no in-
fluence on the existence and stability property of the
positive equilibrium. Now let us take a in-depth in-
sight on this matter.
Noting that B(H∗, P ∗)satisfies the equation Un-
der the assumption (3.1) holds, system (1.1) admits a
unique positive equilibrium B(H∗, P ∗), which satis-
fies the equalities
r11
c1+c2H∗−r12
−b1H∗−a1(1 −m)P∗= 0,
r2−a2
P∗
(1 −m)H∗= 0.
(4.1)
From the second equation of (4.1), we could obtain
P∗=r2(1 −m)H∗
a2
.(4.2)
Substituting above equality into the first equation of
(4.1), leads to
r11
c1+c2H∗−r12 −b1H∗
−a1(1 −m)r2(1 −m)H∗
a2
= 0.
(4.3)
Now let us denote
F(H∗, c2) = r11
c1+c2H∗−r12 −b1H∗
−a1(1 −m)r2(1 −m)H∗
a2
,
then equation (4.3) can be rewrite in the form
F(H∗, c2) = 0.(4.4)
Since
∂F
∂H∗=−r11c2
(c2H+c1)2−b1
−a1(1 −m)2r2
a2
<0,
(4.5)
∂F
∂c2
=−r11H
(c2H+c1)2<0,(4.6)
from (4.4)-(4.6) and the implicit function theorem, it
immediately follows that
dH∗
dc2
=−Fc2
FH∗
<0.(4.7)
(4.7) shows that H∗is the decreasing function of c2.
From (4.2) one could easily see that P∗is also the
decreasing function of c2.
From (4.3) we could also draw an interesting find-
ing, H∗→0as c2→+∞. Otherwise, assume that
there exists a δ > 0such that H∗> δ as c2→+∞.
Then one could easily see that
r11
c1+c2H∗→0as c2→0.
Consequently, F(H∗, c2)<0, which is contradict to
equation (4.2).
Since we are interesting in the influence of den-
sity dependent birth rate, above analysis shows that
with the increasing of c2. the density of both prey
and predator are decreasing, and if c2is enough large,
the final density of prey species will approach to zero,
which increasing the extinction property of the prey
species.
5 Numeric simulations
Now let’s consider the following two examples.
Example 5.1
dH
dt =2
1 + H−1−HH
−1·(1 −0.5)HP,
dP
dt =1−1·P
(1 −0.5)HP,
(5.1)
where corresponding to system (1.1), we take r11 =
2, c1=c2=r12 =b1=a1=r2=a2= 1, m =
0.5,then,
r11 = 2 >1 = c1r12,
hence, it follows from Theorem 3.1 that the unique
positive equilibrium B(0.3689,0.1844) of system
(5.1) is globally stable. Fig. 1 and 2 support this as-
sertion.
Example 5.2
dH
dt =2
1 + c2H−1−HH
−1·(1 −0.5)HP,
dP
dt =1−1·P
(1 −0.5)HP,
(5.2)
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where all the coefficients are the same as Example
5.1, only take c2as the variable coefficients, then,
r11 = 2 >1 = c1r12,
it follows from Theorem 3.1 that the system
(5.2) always admits a unique positive equilibrium
B(H∗, P ∗), which is globally stable. Obviously, H∗
and P∗are the function of c2.In this case, H∗satisfies
the equation
2
c2H∗+ 1 −1−1.25H∗= 0.
Numeric simulation (Fig.3) shows that with the
increasing of c2,H∗is decreasing and finally H∗is
approach to zero.
Figure 1: Dynamic behaviors of the first
species in system (5.1), the initial condition
(H(0), P (0)) = (1.5,1.5),(1.5,0.3),(0.2,0.1)
and (0.4,1.5), respectively.
6 Discussion
Stimulated by the works of Chen et al[5], Chen et al[6]
and Zhao et al[22], based on the model (1.2), we fur-
ther incorporate the density dependent birth rate to the
prey species, and this result in the system (1.1).
Our study shows that under some very nature as-
sumption, more precisely, for the prey species, the
birth rate is larger than the death rate, then the sys-
tem could exits a unique positive equilibrium, which
is globally stable. Obviously, if we assume that c1=
1, c2= 0,then system (1.1) is reduced to the system
Figure 2: Dynamic behaviors of the second
species in system (5.1), the initial condition
(H(0), P (0)) = (1.5,1.5),(1.5,0.3),(0.2,0.1)
and (0.4,1.5), respectively.
considered in [5], and Theorem 3.1 is degenerate to
Theorem 2.1 in [5], it is in this sense, we generalize
the main result of Chen et al[5].
It is curiously that Theorem 2.1 and 3.1 are inde-
pendent of the coefficient c2, however, one could eas-
ily see that H∗is the implicit function of c2, our study
shows that H∗and P∗are both the decreasing func-
tion of c2.Also, H∗→0, P ∗→0as c2→+∞.It
is well known that if the amount of the species is less
than a threshold, then, many endangered species will
have Allee effect[10, 15, 23], which means that the
population size will decrease if it is too sparse, this
will enhance the possibility of the extinction of prey
species.
To sum up, by introducing the density dependent
birth rate of prey species, we show that generally
speaking, the system could still be coexist in a sta-
ble state. However, with the increasing influence of
the density dependent birth rate, the final density of
both predator and prey species will reduced, and this
may have negative effect on the long time survival of
the prey and predator species.
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Figure 3: Relationship of H∗and c2
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WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.5
Fengde Chen, Sijia Lin, Shangming Chen, Yanbo Chong
E-ISSN: 2224-2678
47
Volume 22, 2023
Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Sijia Lin, Yanbo Chong wrote the draft.
Shangming Chen carried out the simulation.
Fengde Chen proposed the problem.
Sources of funding for research
presented in a scientific article or
scientific article itself
This work is supported by the Natural Science Foun-
dation of Fujian Province(2020J01499).
Creative Commons Attribution
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/li-
censes/by/4.0/deed.en_US
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.5
Fengde Chen, Sijia Lin, Shangming Chen, Yanbo Chong
E-ISSN: 2224-2678
48
Volume 22, 2023
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
