The Influence of Nonlinear Cannibalism to Logistic Equation

FENGDE CHEN, TINGJIE ZHOU, QUN ZHU, QIANQIAN LI

College of Mathematics and Statistics

Fuzhou University

No. 2, wulongjiang Avenue, Minhou County, Fuzhou

CHINA

Abstract: - A single species model with Holling II type cannibalism term is proposed and studied in this paper.

Local and global stability property of the system are investigated. By applying the iterative method, we show that

the system always admits the unique globally asymptotically stable positive equilibrium. A threshold value R0,

which depends on the cannibalism rate and the transform rate, is obtained. Depending on R0>1,R0= 1 or

R0<1, the final density of the species will smaller or equal to or bigger than the system without cannibalism. Our

study shows that if the cannibalism rate is too large, and transform rate is too small, then R0>1and cannibalism

has negative effect on the final density of the species, which increase the extinction property of the species.

Key-Words: Single species, Cannibalism, Stability

1 Introduction

The aim of this paper is to investigate the dynamic

behaviors of the following single species model in-

corporating the nonlinear cannibalism rate

dt =x(a+c−bx)−hx2

d+x,(1.1)

where ais intrinsic rate of the species, a/bis the envi-

ronment carrying capacity, his the cannibalism rate.

C(x) = h×x×x

x+dis the the generic cannibalism

term. cx is the new offsprings due to the cannibalism.

Obviously, c < h.

During the last decades, mathematics biology be-

comes one of the important research area ([1]-[43]),

specially, many scholars investigated the dynamic be-

haviors of the ecosystem with cannibalism, see [34]-

[43] and the references cited therein. Cannibalism

often occurs in plankton[34], fishes[35], spideres[36]

and social insect populations[37]. It is a behavior that

consumes the same species and helps to provide food

sources.

In 2018, Basheer et al.[43] proposed the prey-

predator model with both predator and prey cannibal-

ism as follows:

dt =u(1 + c1−u)−uv

u+αv −cu2

u+d,

dt =δvβ−v

γu +ρv ,

(1.2)

where c1< c,uand vrepresent the densities of prey

and predator at time t, respectively. The parameters

c1,α,c,d,δand β,γ, ρ are all nonnegative constants.

Basheer et al.[43] used the Holling II type functional

response to describe the cannibalism of prey species.

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

the prey equation, and is given by

C(u) = c×u×u

u+d,(1.3)

where cis the cannibalism rate. This term has a

clear gain of energy to the cannibalistic prey. This

gain results in an increase in reproduction in the prey,

modeled via adding a c1uterm to the prey equation.

Obviously, c1< c, as it takes depredation of a num-

ber of prey by the cannibal to produce one new off-

spring. The authors of [43] tried to investigated the

local and global stability property of the equilibrium

of the system (1.2). Indeed, they used the Iterative

method to prove the global stability property of the

positive equilibrium, they first applying differential

inequality theory to the first equation of (1.2) and ob-

tained

lim sup

t→+∞

u(t)≤1 + c1.(1.3)

Hence, for ε > 0enough small, there exists a T1>0

such that

u(t)≤1 + c1+εdef

=M(1)

1.(1.4)

By using (1.4), from the second equation of (1.2), one

could obtain

lim sup

t→+∞

v(t)≤βγM(1)

1−βρ .(1.5)

Hence, for above ε > 0, there exists a T2≥T1, such

that

v(t)≤βγM(1)

1−βρ +εdef

=M(1)

2.(1.6)

Equation (1.6) together with the first equation of (1.2)

leads to

dt ≥u(1 + c1−u)−v−cu

≥uh(1 + c1−M(1)

2)−(1 + c)ui.

(1.7)

Received: May 25, 2022. Revised: February 14, 2023. Accepted: March 12, 2023. Published: April 10, 2023.

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.1

Fengde Chen, Tingjie Zhou,

Qun Zhu, Qianqian Li

E-ISSN: 2769-2477

Volume 3, 2023

Here, in (1.7), the authors had used the fact

v≤uM(1)

2(1.8)

However, from the proof of the theorem, more pre-

cisely, from (1.6), we could only obtain the fact

v(t)≤M(1)

2for all t≥T2, hence, the deduction

of (1.7) is incorrect, or at least is not strictly. Some

other similar mistakes also happened in their deduc-

tion. Hence, the conclusion of Theorem 3.3 in [43]

may not hold. One natural issue is to revisit the sta-

bility property of the system (1.2), and to give the

right conditions to ensure the stability of the positive

equilibrium. However, at present we have difficulty

in dealing with this matter. So, we try to study some

more simple model, i.e., single species model (1.1),

we hope that our study will bring some light to this

issue, and finally could solve the stability problem of

system (1.2).

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 using the iterative method. Numeric

simulations are presented in Section 4 to show the fea-

sibility 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 x0= 0 and the unique positive equilibrium

x∗

1,where

x∗

1=a+c−h−bd +√δ

2b,

δ= (a+c−h−bd)2+ 4bd(a+c).

(2.1)

Proof. The equilibria of system (1.1) satisfies the

equation

x(a+c−bx)−hx2

d+x= 0.(2.2)

Equation (2.2) has three solution x0= 0, and

x∗

1=a+c−h−bd +√δ

2b,

x∗

2=a+c−h−bd −√δ

2b.

(2.3)

where δis defined by (2.1). Noting that

√δ=p(a+c−h−bd)2+ 4bd(a+c)

>|a+c−h−bd|,

hence

a+c−h−bd +√δ > 0,

a+c−h−bd −√δ < 0.

Therefore,

x∗

1>0, x∗

2<0.

Hence, system (1.1) admits a unique positive equilib-

rium x∗

Theorem 2.2. x0= 0 is unstable equilibrium, and x∗

is locally asymptotically stable equilibrium.

Proof. Set

F=x(a+c−bx)−hx2

d+x.(2.4)

Then

F0=a+c−2bx −2hx

d+x+hx2

(d+x)2.(2.5)

Substituting x0, x∗

1into F0leads to

F0|x=x0=a+c > 0.(2.6)

So, x0is unstable.

F0|x=x∗

1=−a−c+h(x∗

1)2

(d+x∗

1)2

=x∗

1(a+c−bx∗

d+x∗

1−a−c

< a +c−bx∗

1−a−c

=−bx∗

1<0.

(2.7)

So, x∗

1is locally asymptotically stable.

This ends the proof of Theorem 2.2.

3 Global attractivity

Concerned with the global attractivity of the positive

equilibrium, we have the following result.

Theorem 3.1. The positive equilibrium x∗

1is globally

attractive.

Proof. From (1.1) we have

dt ≤x(a+c−bx),(3.1)

Hence,

lim sup

t→+∞

x(t)≤a+c

b.(3.2)

For ε > 0enough small, without loss of generality,

we may assume that ε < 1

a+c

b+h

,it follows from

(3.2) that there exists a T1>0such that

x(t)<a+c

b+εdef

=M1.(3.3)

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.1

Fengde Chen, Tingjie Zhou,

Qun Zhu, Qianqian Li

E-ISSN: 2769-2477

Volume 3, 2023

From (1.1) we also have

dt ≥xa+c−(b+h

d)x,(3.4)

Hence,

lim inf

t→+∞

x(t)≥a+c

b+h

.(3.5)

For above ε > 0, it follows from (3.5) that there exists

aT2> T1such that

x(t)>a+c

b+h

−εdef

=m1.(3.6)

From (3.3), for t≥T2, we have

dt ≤xa+c−(b+h

d+M1

)x,(3.7)

Hence,

lim sup

t→+∞

x(t)≤a+c

b+h

d+M1

.(3.8)

For above ε > 0, it follows from (3.2) that there exists

aT3> T2such that

x(t)<a+c

b+h

d+M1

+ε

def

=M2.(3.9)

From (1.1) we also have

dt ≥xa+c−(b+h

d+m1

)x,(3.10)

Hence,

lim inf

t→+∞

x(t)≥a+c

b+h

d+m1

.(3.11)

For above ε > 0, it follows from (3.5) that there exists

aT2> T1such that

x(t)>a+c

b+h

d+m1

−ε

def

=m2.(3.12)

Repeating the above procedure, we get four se-

quences mi,Mi,i = 1,2, .... such that

Mi=a+c

b+h

d+Mi−1

+ε

mi=a+c

b+h

d+mi−1

−ε

(3.13)

From the deduction process, for t > max{T2i}, we

have

mi< x(t)< Mi.(3.14)

We claim that sequences Miis strictly decreasing,

and sequences miis strictly increasing. To proof this

claim, we will carry out by induction. Obviously, we

have

M1=a+c

b+ε > a+c

b+h

d+M1

+ε

2=M2.(3.15)

m1=a+c

b+h

−ε < a+c

b+h

d+m1

−ε

2.(3.16)

(3.15) and (3.16) show that the conclusion holds for

i= 2. Let us assume now that our claim is true for

i=k, that is,

Mk< Mk−1, mk> mk−1,(3.17)

then

d+Mk−1

d+Mk

d+mk−1

d+mk

(3.18)

And so

Mk+1 =a+c

b+h

d+Mk

+ε

k+ 1

<a+c

b+h

d+Mk−1

+ε

k=Mk,

(3.19)

mk+1 =a+c

b+h

d+mk

−ε

k+ 1

>a+c

b+h

d+mk−1

−ε

k=mk.

(3.20)

Above analysis shows that Miis strictly decreasing

sequence, miis strictly increasing sequence. Set

lim

i→+∞

Mi=M, lim

i→+∞

mi=m. (3.21)

Setting i→+∞in (3.13) leads to

M=a+c

b+h

d+M

m=a+c

b+h

d+m

(3.22)

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.1

Fengde Chen, Tingjie Zhou,

Qun Zhu, Qianqian Li

E-ISSN: 2769-2477

Volume 3, 2023

(3.22) shows that M, m are all the positive solution

of (2.2). By Theorem 2.1, (2.2) has a unique positive

solution x∗

1. Hence, we conclude that M=m=x∗

that is

lim

t→+∞

x(t) = x∗

1.(3.23)

Thus, the unique interior equilibrium x∗

1is globally

attractive. This completes the proof of Theorem 3.1.

4 The influence of cannibalism

It’s well known that the Logistic equation

dt =x(a−bx) (4.1)

admits a unique positive equilibrium x∗=a

b,which

is globally asymptotically stable. Theorem 2.1, 2.2

and 3.1 shows that system (1.1) admits a unique pos-

itive equilibrium x∗

1=a+c−h−bd +√δ

2b, which

is also globally asymptotically stable. From this, we

can draw the first conclusion:

(I) For the Logistic equation, nonlinear cannibal-

ism has no influence on the persistent property of

the system.

Next, let’s compare the big or small of x∗and x∗

Noting that

x∗

1−x∗=−a+c−h−bd +√δ

2b.(4.2)

From (4.2), by simple computation, we have

(i) If

a>h−c

cd ,(4.3)

then x∗

1> x∗;

(ii) If

a=h−c

cd ,(4.4)

then x∗

1=x∗;

(iii) If

a<h−c

cd ,(4.5)

then x∗

1< x∗.

Without loss of generality, since we are interest-

ing in the influence of cannibalism, we may assume

that a, b, d are fixed positive constants, noting that h

is the cannibalism rate and ccan be denote by trans-

form rate. We then have the following results.

(II) If cis enough small, then the inequality (4.5)

holds, in this case, the cannibalism will decrease

the final density of the species.

(III) If cis enough large, such that h−c→0, then

the inequality (4.3) holds, in this case, the cannibal-

ism will increase the final density of the species.

Remark 4.1. Set R0=a(h−c)

bcd , then R0can be

seen as the threshold parameter of the system (1.1). If

R0<1, then x∗

1> x∗; If R0>1,then x∗

1< x∗, and

if R0= 1, then cannibalism has no influence on the

final density of the species.

Finally, noting that x∗

1is the function of hand c,

we have

∂x∗

∂h =−a+c−bd −h+√δ

2b√δ<0.(4.6)

∂x∗

∂c =a+c+bd −h+√δ

2b√δ>0.(4.7)

That is

(IV) The final density of the species is the decreas-

ing function of cannibalism rate and the increasing

function of the transform rate.

5 Numeric simulations

Example 5.1. Now let’s fixed a=b=d= 1, c =

0.01,then

x∗

1(h) = 0.005 −2−h

2+1

2ph2−0.02h+ 4.0401.

Fig. 1 shows that x∗

1is the strictly decreasing function

of h, also, if his enough large, then x∗

1→0.That

is, for the fixed transform rate, if the cannibalism co-

efficient his enough large, the species may driven to

extinction, though at first sight, for the fixed cannibal-

ism rate, the system is permanent. The final density of

the species may become very small if the cannibalism

rate is enough large.

Example 5.2. Now let’s fixed a=b=d= 1, h = 2,

then

x∗

1(c) = −1 + 1

2c+1

2pc2+ 8.

Fig. 2 shows that x∗

1is the strictly increasing function

of c.

Example 5.3. Now let’s fixed a=b=d= 1,

then we have x∗= 1, from (4.3)-(4.5), we know

that if h > 2c, then x∗

1> x∗,if h= 2c, then

x∗

1=x∗, if h < 2c, then x∗

1< x∗. Fig. 3 shows that

x∗

1(h, c)smaller or bigger than x∗, depending on the

parameters lies below or above the line h= 2c.

Now let’s fix c= 0.5,h= 0.5,1and 2, respec-

tively. Then, for h= 0.5, x∗

1> x∗, for h= 1,

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.1

Fengde Chen, Tingjie Zhou,

Qun Zhu, Qianqian Li

E-ISSN: 2769-2477

Volume 3, 2023

Figure 1: Relationship of x∗

1and h.

Figure 2: Relationship of x∗

1and c.

x∗

1=x∗, and for h= 2, x∗

1< x∗. Fig.4-6 supports

this assertion. From Fig.4-6 we could also find that

x∗

1is the decreasing function of h.

Now let’s fix h= 1,c= 0.3,0.5and 0.7,

respectively. Then, for c= 0.7, x∗

1> x∗, for c= 0.5,

x∗

1=x∗, and for c= 0.3, x∗

1< x∗. Fig.7-9 supports

this assertion. From Fig.7-9 we could also find that

x∗

1is the increasing function of c.

Figure 3: Relationship of hand c.x∗

1(h, c)

smaller or bigger than x∗, depending on the pa-

rameters lies above or below the line h= 2c.

6 Conclusion

Based on the traditional Logistic equation and the

works of Basheer et al.[42, 43], we proposed a sin-

gle species model incorporating the nonlinear canni-

balism. Already, Basheer et al.[42] incorporated the

cannibalism to the Holling-Tanner model with ratio-

dependent functional response (i.e., system (1.2)).

They showed that cannibalism in the prey cannot sta-

bilize the unstable interior equilibrium in the ODE

case, but can destabilize the stable interior equilib-

rium, leading to a stable limit cycle. In this paper,

we focus our attention to the single species model,

our study shows that the system with cannibalism al-

ways admits a unique globally asymptotically stable

equilibrium, which means that the cannibalism has

no influence on the persistent property of the system.

However, we could show that depending on the pa-

rameter regime, the final density of the species maybe

larger or smaller or equal to the final density of the the

system without cannibalism. It’s in this sense, that

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.1

Fengde Chen, Tingjie Zhou,

Qun Zhu, Qianqian Li

E-ISSN: 2769-2477

Volume 3, 2023

Figure 4: Dynamic behaviors of the system (1.1),

with a=b=d= 1, c = 0.5, h = 0.5.

Figure 5: Dynamic behaviors of the system (1.1),

with a=b=d= 1, c = 0.5, h = 1.

Figure 6: Dynamic behaviors of the system (1.1),

with a=b=d= 1, c = 0.5, h = 2.

Figure 7: Dynamic behaviors of the system (1.1),

with a=b=d= 1, h = 1, c = 0.3.

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.1

Fengde Chen, Tingjie Zhou,

Qun Zhu, Qianqian Li

E-ISSN: 2769-2477

Volume 3, 2023

Figure 8: Dynamic behaviors of the system (1.1),

with a=b=d= 1, h = 1, c = 0.5.

Figure 9: Dynamic behaviors of the system (1.1),

with a=b=d= 1, h = 1, c = 0.7.

cannibalism may have positive or negative or has no

influence on the final density of the species. Also, if

the cannibalism rate is enough large while the trans-

form rate is enough small, then the species may in-

crease its probability of the extinction in the sense that

the final density of the species may approach to zero.

We hope our findings could be applied to more

complicated situation, such as the competition model

or the mutualism model. Also, as was shown in the

introduction section, the results about the global sta-

bility property of the positive equilibrium of system

(1.2) may not right, is it possible for us to investigate

the stability property of the positive equilibrium by

using the iterative method? We leave these problem

for future investigation.

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E-ISSN: 2769-2477

Volume 3, 2023

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Qun Zhu, Qianqian Li

E-ISSN: 2769-2477

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Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

Tingjie Zhou wrote the draft.

Qun Zhu and Qianqian Li carried out the simulation.

Fengde Chen proposed the issue and revise the paper.

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).

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International , CC BY 4.0)

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censes/by/4.0/deed.en_US

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.1

Fengde Chen, Tingjie Zhou,

Qun Zhu, Qianqian Li

E-ISSN: 2769-2477

Volume 3, 2023

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.