Global Stability of the Positive Equilibrium of a Leslie-Gower

Predator-Prey Model Incorporating Predator Cannibalism

XIAORAN LIa, QIN YUEa, FENGDE CHENb

aCollege of Finance and Mathematics

West Anhui University

Yueliang Dao, West of Yunlu Bridge, Lu An, An Hui, CHINA

bCollege of Mathematics and Statistics

Fuzhou University

No. 2, wulongjiang Avenue, Minhou County, Fuzhou

CHINA

Abstract: - A Leslie-Gower predator prey model with Holling II type cannibalism term on predator species is

proposed and studied in this paper. By constructing a suitable Lyapunov function, we show that if the positive

equilibrium exist, it is globally asymptotically stable. Our study indicates that suitable cannibalism has no influ-

ence on the persistent property of the system, however, cannibalism could reduce the final density of the predator

species and increase the final density of the prey species. Excessive cannibalism may enhance the possibility of

extinction to the predator species.

Key-Words: Leslie-Gower predator prey model; Cannibalism; Stability

Received: April 27, 2022. Revised: December 3, 2022. Accepted: December 24, 2022. Published: December 31, 2022.

1 Introduction

The aim of this paper is to investigate the dynamic be-

haviors of the following Leslie-Gower predator prey

model with predator cannibalism

dt = (r1−a1P−b1H)H,

dt =r2+c1−a2P

HP−fP 2

d+P,

(1.1)

where Hand Pare the density of prey species and

the predator species at time t, respectively. ri, i = 1,2

are the intrinsic growth rate of the prey and predator

species, respectively. r1/b1is the environment carry-

ing capacity of the prey species, fis the cannibalism

rate of predator species. c1Pis the new offsprings due

to the cannibalism. Obviously, c1< f , since it takes

depredation of a number of predator by the cannibal

to produce one new offspring. Also, it is well known

that the flow of energy decreases step by step along

the food chain, hence, generally speaking, Only 10 to

20 percent of energy can flow into the next nutritional

level, hence c1could be restrict to c1≤1

5f.

During the last decades, numerous biological

modelling were proposed and studied, see [1]-[38]

and the references cited therein. Such topics as the

influence of Allee effect([1],[2], [14], [17]-[25], the

influence of refuge ([7], [13], [37]), the influence of

stage structure ([9], [13], [32],[33],[36],[38]), the in-

fluence of harvesting ([3], [5],[16]), the influence of

commensalism or ammensalism ([10]-[12]), the influ-

ence of cannibalism ([27]-[34]) have been extensively

studied.

Recently, several scholars began to investigated

the dynamic behaviors of the cannibalism, a behavior

that consumes the same species and helps to provide

food sources, see [27]-[34] and the references cited

therein. Indeed, cannibalism often occurs in plankton,

fishes, spideres[29] and social insect populations[30].

In 2016, Basheer et al.[30] proposed the prey-

predator model with prey cannibalism as follows:

dt =u(1 + c1−u)

−uv

u+αv −cu2

u+d,

dt =δvβ−v

u,

(1.2)

where c1< c,uand vrepresent the densities of prey

and predator at time t, respectively. The parameters

c1,α,c,d,δand βare all nonnegative constants. Here

the generic cannibalism term C(u), is added in the

prey equation, and is given by

C(u) = c×u×u

u+d,

where cis the cannibalism rate. This term has a clear

gain of energy to the cannibalistic prey, and this leads

to the increase in reproduction in the prey, modeled

via adding a c1uterm to the prey equation. It seems

that this is the first time nonlinear cannibalism term

were introduced. Previously, cannibalism term were

presented in bilinear type ([29], [32]-[33]).

Recently, stimulated by the works of Basheer et

al[30], based on the traditional Lotka-Volterra preda-

tor prey system, Deng et al[31] investigated the dy-

namic behaviors of the following predator-prey model

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with cannibalism for predator:

dt =xb−αx −my,

dt =y−β+c1+nx−cy2

y+d,

(1.3)

where c1< c,xand yare the density of the prey and

predator at time t, respectively. The authors showed

that cannibalism has both positive and negative effect

on the stability of the system, it depends on the dy-

namic behaviors of the original system.

On the other hand, also stimulated by the works

of Basheer et al[30], Lin, Liu, Xie et al[34] proposed

the following Leslie-Gower predator prey model with

prey cannibalism

dt = (r1+c1−a1P−b1H)H

−fH2

d+H,

dt =r2−a2

HP,

(1.4)

where Hand Pare the density of prey species and

the predator species at time t, respectively. By apply-

ing the iterative method, the authors obtained a set of

sufficient conditions which ensure the global attrac-

tive of the positive equilibrium. One may argued that

system (1.4) seems much simple than system (1.2),

and it is no need to investigated. Indeed, Basheer et

al.[30] only investigated the local stability property of

the system (1.2), and gave no information about the

global stability property of the system. The study of

Lin, Liu, Xie et al[34] can be seen as the effort on this

direction. They tried to make some insight finding on

the cannibalism.

Now, stimulated by the works of Basheer et

al.[30], Deng et al[31] and Lin, Liu, Xie et al[34], it

is natural to investigate the influence of predator can-

nibalism to Leslie-Gower predator prey system, this

motivated us to propose the system (1.1).

The rest of the paper is arranged as follows. In

next section, we will investigate the existence and lo-

cal stability of the equilibrium of the system (1.1). In

Section 3, we will discuss the global stability of the

equilibrium by constructing some suitable Lyapunov

function. Numeric simulations are presented in Sec-

tion 4 to show the feasibility of the main results. We

end this paper by a briefly discussion.

2 The existence and local stability of

the equilibria of system (1.1)

Concerned with the existence of the equilibria of sys-

tem (1.1), we have the following result.

Theorem 2.1.System (1.1) admits the boundary equi-

librium A(r1

b1,0); Assume that c1−f+r2>0,

then system (1.1) admits a unique positive equilibrium

B(H∗, P ∗),where

H∗=r1−a1P∗

P∗=−A2+qA2

2−4A1A3

2A1

A1= (c1−f+r2)a1+a2b1>0,

A2=d(r2+c1)a1

+ (f−c1−r2)r1+da2b1,

A3=−dr1(r2+c1)<0.

(2.1)

Proof. One could easily see that A(r1

b1,0) is the non-

negative solution of system (2.2). Next we show that

there exists a unique positive equilibrium if c1−f+

r2>0. The existence of positive equilibria of the sys-

tem (1.1) is determined by the following equations:

r1−a1P−b1H= 0,

r2+c1−a2P

H−fP

d+P= 0.

(2.2)

It follows from (2.2) that the prey isocline is l1:

r1−a1P−b1H= 0, obviously, l1is monotonically

decreases, which starts from the point (0,r1

a1)in the

P-axis to the point (r1

b1,0) in the positive H-axis.

Now let us consider the predator isocline l2:r2+

c1−a2P

H−fP

d+P= 0. l2has no definition on H= 0,

however, we could show that as H→0,P→0, in-

deed, equation r2+c1−a2P

H−fP

d+P= 0 is equiv-

alent to

a2P2+ (Hf −c1H−r2H+da2)P

−Hd(d1+r2) = 0.

(2.3)

From (2.3) and the expression of the solution of

quadratic equation, one could easily see that as H→

0, the equation has a solution P(H)→0.This means

that as H→0,l2is lies below the line l1. On the other

hand, now subs H=r1

b1to r2+c1−a2P

H−fP

d+P=

0,we could obtain

a2b1P2+ (a2b1d−c1r1+fr1−r2r1)P

−(c1+r2)r1d= 0.

(2.4)

Since a2b1>0,−(c1+r2)r1d < 0,(2.4) has a

unique positive solution P. This means that as H→

b1, line l2lies above the line l1. Above analysis shows

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that l1intersect l2at least one point. On the other

hand, l2define a function P(H), by computation, we

have

dH =Hfd +P2a2+ 2P a2d+a2d2H

(d+P)2a2P>0.

(2.5)

Hence, l2is monotonically increases. Therefore, l2

intersect l1at most one point. Above analysis shows

that l1and l2intersect only one time, consequently,

system (2.2) has unique positive solution.

From the first equation of (2.2), we have

H=r1−a1P

.(2.6)

Substituting it into the second equation of (2.2), and

simplify, we could obtain the equation

A1P2+A2P+A3= 0,(2.7)

where A1, A2, A3are defined by (2.1). (2.7) has a

unique positive solution P∗, hence, system (1.1) has

the unique positive equilibrium B(H∗, P ∗).

This ends the proof of Theorem 2.1.

Theorem 2.2. A(r1

b1,0) is unstable equilibrium, As-

sume that c1−f+r2>0, then B(H∗, P ∗)is locally

asymptotically stable.

Proof. The Jacobian matrix of the system (1.1) is cal-

culated as

J(H, P ) = 



A11 −a1H

P2a2

H2A22 

,(2.8)

where

A11 =−2Hb1−P a1+r1,

A22 =−2a2P

H+r2+c1

−2fP

d+P+f P 2

(d+P)2.

Then the Jacobian matrix of the system (1.1) about the

equilibrium A(r1

b1,0) is

J(A(r1

,0)) = −r1−a1r1

0c1+r2!.(2.9)

The eigenvalues of J(A)are λ1=−r1<0,λ2=

c1+r2>0. Thus, A(r1

b1,0) is a saddle.

Noting that B(H∗, P ∗)satisfies the equation

r1−a1P∗−b1H∗= 0,

r2+c1−a2P∗

H∗−fP ∗

d+P∗= 0.

(2.10)

The Jacobian matrix of the system (1.1) about the

equilibrium B(H∗, P ∗)is

J(B(H∗, P ∗))

= −H∗b1−a1H∗

(P∗)2

(H∗)2B!,(2.11)

where B=−a2P∗

H∗−fP ∗

d+P∗+f(P∗)2

(d+ (P∗))2.

Then we have

DetJ(B(H∗, P ∗)

=H∗b1a2P∗

H∗+fP ∗

d+P∗−f(P∗)2

(d+ (P∗))2

+a1H∗a2

(P∗)2

(H∗)2>0,

and

T rJ(B(H∗, P ∗)|

=−H∗b1−dfH∗

(d+H∗)2−a2P∗

H∗

−fP ∗

d+P∗+f(P∗)2

(d+ (P∗))2<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.

3 Global stability of the positive

equilibrium

Concerned with the global stability property of the

positive equilibrium, we have the following result.

Theorem 3.1. The positive equilibrium B(H∗, P ∗)is

globally attractive provide that

c1−f+r2>0 (3.1)

holds.

Proof. B(H∗, P ∗)satisfies the equalities

r1−a1P∗−b1H∗= 0,

r2+c1−a2

P∗

H∗−fP ∗

d+P∗= 0.

(3.2)

We will adapt the idea of Chen, Chen and Xie[35]

to prove Theorem 3.1. More precisely, we construct

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the following Lyapunov function:

V(H, P ) = ln H

H∗+H∗

+a1H∗

a2ln 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

H1−H∗

H,

∂V

∂P =a1H∗

a2P1−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 + a1H∗

for all H, P > 0.

Calculating the derivative of Valong the solution

of the system (1.1), by using equalities (3.2), we have

H1−H∗

Hr1−a1P−b1HH

+a1H∗

a2P1−P∗

P·r2+c1

−a2P

H−fP

d+PP

=H−H∗

Ha1P∗+b1H∗−a1P−b1H

+a1H∗

·1−P∗

P·a2

P∗

H∗+fP ∗

d+P∗

−a2P

H−fP

d+P

=−b1

H(H−H∗)2+a1

H(H−H∗)(P∗−P)

+a1H∗·P−P∗

P·P∗H−P H +P H −P H∗

H∗H

+a1H∗

·1−P∗

P·fP ∗

d+P∗−fP

d+P

=−b1

H(H−H∗)2

+a1

H(H−H∗)(P∗−P)

−a1

P(P−P∗)2

+a1

H(H−H∗)(P−P∗)

+a1H∗

P−P∗

fd(P∗−P)

(d+P)(d+P∗)

=−b1

H(H−H∗)2−a1

P(P−P∗)2

−fd

(P∗−P)2

(d+P)(d+P∗).

(3.6)

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 cannibalism

Let

F(H∗, P ∗, c1, f) = r1−a1P∗−b1H∗,

G(H∗, P ∗, c1, f) = r2+c1−a2P∗

H∗

−fP ∗

d+P∗,

then rewrite equation (3.2) in the form

(F(H∗, P ∗, c1, f)=0,

G(H∗, P ∗, c1, f)=0.(4.1)

By simple computation, we have

J=D(F, G)

D(H∗, P ∗)

=

FH∗FP∗

GH∗GP∗

=

−b1−a1

a2P∗

(H∗)2−E1

=b1E1+a1a2P∗

(H∗)2>0.

(4.2)

where

C1=a2

H∗+f

d+P∗−fP ∗

(d+P∗)2.

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Using implicit function set theorem, the equation

(4.1) has an unique solution in the neighborhood of

B(H∗, P ∗)

H∗=H∗(c1, f), P ∗=P∗(c1, f).(4.3)

and

∂H∗

∂c1=−1

D(F,G)

D(c1,P ∗),

∂P ∗

∂c1=−1

D(F,G)

D(H∗,c1),

(4.4)

∂H∗

∂f =−1

D(F,G)

D(f,P ∗),

∂P ∗

∂f =−1

D(F,G)

D(H∗,f).

(4.5)

By computation, we have

D(F, G)

D(c1, P ∗)

=

Fc1FP∗

Gc1GP∗

=

0−a1

1−a2

H∗−f

d+P∗+fP ∗

(d+P∗)2



=a1>0,

D(F, G)

D(H∗, c1)

=

FH∗Fc1

GH∗Gc1

=

−b10

a2P∗

(H∗)21

=−b1<0,

D(F, G)

D(f, P ∗)

=

FfFP∗

GfGP∗

=

0−a1

−P∗

d+P∗−a2

H∗−f

d+P∗+fP ∗

(d+P∗)2



=−a1P∗

d+P∗<0,

D(F, G)

D(H∗, f)

=

FH∗Ff

GH∗Gf

=

−b10

a2P∗

(H∗)2−P∗

d+P∗



=b1P∗

P∗(d+P∗)>0,

(4.6)

thus,

∂H∗

∂c1

=−a1

J<0,

∂P ∗

∂c1

Jb1>0,

∂H∗

∂f =1

a1P∗

d+P∗>0,

∂P ∗

∂f =−1

b1P∗

P∗(d+P∗)<0.

(4.7)

Above analysis shows that both H∗is the decreas-

ing function of c1and increasing function of f, while

P∗is the increasing function of c1and the decreas-

ing function of f. Noting the fact c1< f , which can

be seen that the cannibalism has negative effect on

the final density of the predator species, while it has

the positive effect on the density of the prey species.

Such a result seems naturally, since cannibalism can

be seen as the predator has other food resource, and

this reduce the direct predating of prey species. How-

ever, cannibalism means that the predator species take

itself as food resource, this certainly has negative ef-

fect on predator species.

5 Numeric simulations

Now let’s consider the following two examples.

Example 5.1

dt =1−HH−HP,

dt =1−1·P

H+ 0.1PP

−0.5P2

1 + P,

(5.1)

where corresponding to system (1.1), we take r1=

b1=a1=r2=a2= 1, f = 0.5, c1= 0.1,then,

r2+c1−f= 0.6>0,

hence, it follows from Theorem 4.1 that the unique

positive equilibrium B(0.55,0.45) of system (5.1) is

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globally stable. Fig. 1 and 2 support this assertion.

Example 5.2

dt =1−HH−HP,

dt =1−1·P

H+ 0.2fP P

−fP2

1 + P,

(5.2)

where all the coefficients are the same as Exam-

ple 5.1, only take fas the variable coefficients, also,

choose c1= 0.2f, then, if

r2+c1−f=r2−0.8f= 1 −0.8f > 0,

i.e.,

f < 1.25,

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 f. In this case, P∗satisfies

the equation

1−P∗

1−P∗−0.8fP ∗= 0.

Numeric simulation (Fig.3) shows that with the

increasing of f,P∗is decreasing and finally P∗is

approach to zero.

6 Discussion

Recently, Deng et al [31] incorporated the Basheer

type cannibalism [30] to the traditional Lotka-Volterra

predator prey system, this led to the system (1.3).

They showed that if system (1.3) admits the positive

equilibrium, then the equilibrium is globally stable.

On the other hand, Lin et al[34] also incorporated the

Basheer type cannibalism [30] to the prey species in

Leslie-Gower predator prey system, by applying the

iterative method, they also obtained a set of sufficient

conditions which ensure the globally attractive of pos-

itive equilibrium of the system. Stimulated by their

works, we incorporating predator cannibalism to the

Leslie-Gower predator prey system, this leads to sys-

tem (1.1).

Noting that condition (3.1) is enough to ensure the

existence of the positive equilibrium of system (1.1),

and the proof of Theorem 3.1 is independent of the

condition (3.1), hence, we can draw the conclusion:

Once system (1.1) admits a unique positive equilib-

rium, it is globally stable. Such a property is simi-

lar to the traditional Leslie Gower predator prey sys-

tem[35].

Figure 1: Dynamic behaviors of the first

species in system (4.1), 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.

Figure 2: Dynamic behaviors of the second

species in system (4.1), 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.

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Figure 3: Relationship of P∗and fin system

(5.2).

Our study also indicates that the cannibalism of

predator species has negative effect on the predator

species and positive effect on the prey species, since

with the increasing of cannibalism, the final density

of predator species is reduced and the final density of

prey species is increasing.

We would like to mention here that to this day, still

seldom did scholars investigate the dynamic behav-

iors of the nonautonomous cannibalism predator prey

model, we will do some study on this direction in the

future.

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Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

Xiaoran Li carried out the computation and wrote the

draft.

Qin Yue carried out the simulation.

Fengde Chen was responsible for the proposing of

the problem.

Sources of funding for research

presented in a scientific article or

scientific article itself

Social Science Project of Anhui Provincial Depart-

ment of Education (2022AH051657).

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International , CC BY 4.0)

This article is published under the terms of the

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censes/by/4.0/deed.en_US

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DOI: 10.37394/23202.2022.21.44

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Volume 21, 2022