A New Consideration of the Influence of Shelter on the Kinetic Behavior

of the Leslie-Gower Predator Prey System with Fear Effect

FENGDE CHEN, SIJIA LIN, SHANGMING CHEN, YANBO CHONG

College of Mathematics and Statistics

Fuzhou University

No. 2, wulongjiang Avenue, Minhou County, Fuzhou

CHINA

Abstract: - In this study, a Leslie-Gower predator-prey model that incorporates both fear effect and shelter is pre-

sented and investigated. It is assumed that predator species only capture and cause fear in prey species outside the

refuge, but have no impact on prey species inside the refuge. We demonstrate that the fear effect and the refuge

have no impact on the positive equilibrium’s existence and local stability. Next, we explore the system’s per-

sistence characteristic. By applying the Bendixson-Dulac criterion, we demonstrate that the requirement assures

the system’s permanence is enough to guarantee the global attractivity of the positive equilibrium. According to

our investigation, the birth rate of prey species and the refuge are two of the most critical factors in ensuring the

sustainable development of the system.

Key-Words: Predator; Prey; Fear effect; Refuge

Received: March 22, 2022. Revised: December 16, 2022. Accepted: January 7, 2023. Published: February 6, 2023.

1 Introduction

The study of the predator-prey system dominates the

field of biomathematics due to its universal existence.

Leslie [1, 2] provided the following predator-prey

model:

dt = (r1−a1P−b1H)H,

dt =r2−a2P

HP,

(1)

here Hand Prepresent the density of prey species

and the predator species at time t, respectively. The

approach described above permits a unique coexisting

fixed point

H∗=r1a2

a1r2+a2b1

, P ∗=r1r2

a1r2+a2b1

.(2)

By developing an appropriate Lyapunov function,

Korobeinikov [3] demonstrated that the positive equi-

librium is globally stable.

It is well known that prey species may stay in the

refuge to avoid the capture of predator species. This

decreases the likelihood of extinction as a result of

predation. Chen, Chen, and Xie [5] introduced the

Leslie-Gower predator-prey model with a prey sanc-

tuary as follows:

dt = (r1−b1H)H−a1(1 −m)HP,

dt =r2−a2P

(1−m)HP,

(3)

where m∈(0,1) is constant, and the authors as-

sumed that there is a refuge sheltering mH of the prey.

This makes (1 −m)Hof the prey accessible to the

predator. The system, as was shown by the authors,

permits a globally stable positive equilibrium. Ac-

cording to their research, the refuge has a complicated

effect on the ultimate density of predator species.

Recent research has shown that fear of predators

alters anti-predator defenses to such a degree that it

drastically reduces prey reproduction. Wang, Zanette,

and Zou [6] presented the following generic prey-

predator model, reflecting the cost of fear:

dt =ur0f(k, v)−du −au2−g(u)v,

dt =v−m+cg(u).

(4)

In this system, f(k, v)compensates for the cost of

anti-predator defense owning to fear, with f(k, v) =

1+kv being one of the viable expression. Since the pi-

oneering work of Wang, Zanette, and Zou [6], many

scholars have studied the predator-prey system with

the fear impact on the prey species, see [5]-[28] and

the references cited therein. For more work on Leslie-

Gower predator prey system, one could refer to [29]-

[32] and the references cited therein.

Exploring the dynamic behaviors of a predator-

prey system that contains both a fear impact and a

refuge is natural. Indeed, several scholars [7, 13, 16,

20] had done works on this direction.

The following Holling type III prey-predator sys-

tem with both fear effect and prey shelter was re-

searched by Xie and Zhang [13]:

dt =ax

1 + ny −bx2−α(1 −m)2x2y

β2+ (1 −m)2x2,

dt =−cy +kα(1 −m)2x2y

β2+ (1 −m)2x2.

(5)

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Their research has shown that an increase in fear lev-

els might enhance system stability by removing pe-

riodic solutions and reducing predator species abun-

dance at the coexist equilibrium.

Zhang, Cai, Fu, and Wang [16] conducted research

on the Holling type II prey-predator system described

below, which includes both the fear effect and the prey

shelter:

dt =αx

1 + Ky −bx2−β(1 −m)xy

β+ (1 −m)x,

dt =−γy +cβ(1 −m)xy

β+ (1 −m)x.

(6)

The authors performed comprehensive research of the

aforementioned system and received comprehensive

results.

With the aim of finding out the combined effect of

prey refuge, fear effect, and Allee effect, the following

predator-prey model was examined by Huang, Zhu,

and Li [7]:

dt =rv1−v

kv−θ01

1 + ky

−a(1 −η)uv,

dt =aα(1 −η)uv −m0v.

(7)

They demonstrated how the system’s dynamic be-

haviors might become more complex via boosting

the prey shelter or Allee effect or the fear effect,

which does not change the density of prey but might

reduce the density of predator species.

Firdiansyah [20] employed a Leslie-Gower

predator-prey model with Beddington-DeAngelis

functional response to examine the influence of fear;

the model had the following form:

dt =r1x

1 + Ky −bx −px2

−α(1 −m)xy

a+b(1 −m)x+cy ,

dt =yr2−βy

(1 −m)x+γ.

(8)

The author demonstrated that an increase in fear

might reduce the population density of both species.

However in the event of a constant fear rate, the prey

refuge is beneficial to the survival of both species.

It brings to our attention that in the systems (5)-

(8), they all assumed that the prey species staying in

the refuge also has a fear effect to the predator species.

However, we argued that for those in the refuge, since

predators could not find them, predator species cer-

tainly have no influence on them, no matter the di-

rect killing or the anti-predator behaviors to reduce

the birth rate.

To this day, no academic disputes this assertion.

We believe providing a more appropriate model and

examining the system’s dynamic behaviour is prefer-

able. In fact, we shall investigate the dynamic char-

acteristics of the subsequent model.

dt =r11mH +r11(1 −m)H

1 + kP

−r12H−b1H2−a1(1 −m)HP,

dt =r2−a2

(1 −m)HP,

(9)

now, The birth rate of the prey species is r11, and the

prey species’ fear effect is 1

1+kP . We assume that the

predator only impacts prey outside the refuge and has

no influence on those prey species that remain inside

the refuge.

One could easily see that if m= 0,k= 0, i.e.,

without considering the influence of refuge and fear

effect, system (1.9) will degenerate to the following

system

dt = (r11 −r12)H−b1H2−a1HP,

dt =r2−a2

HP,

(10)

then, if r11 > r12, system (10) is equivalent to system

(1).

Also, if only restrict k= 0 in the system (9), then

the model will degenerate to

dt = (r11 −r12)H−b1H2

−a1(1 −m)HP,

dt =r2−a2

(1 −m)HP,

(11)

then, if r11 > r12, system (11) is equivalent to system

(3).

The purpose of this study is to thoroughly analyze

the dynamical behavior of the system (9) and to give

a positive answer on how m, k affects the dynamical

behavior of the system. The essay’s remaining sec-

tions are grouped as follows: In the next part, we will

examine the presence of equilibrium states and their

local stability properties. In Sections 3 and 4, respec-

tively, the properties of permanence and global stabil-

ity were examined. The impact of the fear effect and

shelter is then covered in Section 5. We describe the

related modeling and findings in Section 6 to demon-

strate the key distinction between our model and the

model with prey species experiencing the fear effect

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in all cases. The fundamental distinction between our

model and the one with all prey suffering due to the

fear effect is then shown. To demonstrate the plausi-

bility of the key conclusions, numerical simulations

are undertaken. Discussion followed the paper’s con-

clusion.

2 Existence of equilibria and their

local stability property

In this part, we will investigate the existence of equi-

librium states and their local stability quality.

Theorem 2.1.If r11 > r12 is true, then system (9) ad-

mits two nonnegative equilibria: E1r11 −r12

,0

and E2(H∗, P ∗),where

H∗=

−A2+qA2

2−4A1A3

2A1

P∗=r2(1 −m)H∗

(12)

here

A1=k(a1(−1 + m)2r2+a2b1)(1 −m)r2>0,

A2=((mr11 −r12)k+a1(−1 + m))(−1 + m)r2

+a2b1a2,

A3=−a2

2(r11 −r12)<0.

(13)

Proof. The existence of nonnegative equilibria of the

system (9) is determined by the following equations:

r11mH +r11(1 −m)H

1 + kP −r12H

r−b1H2−a1(1 −m)HP = 0,

r2−a2

(1 −m)HP= 0.

(14)

It follows directly from the second equation of (14)

that

P= 0 or P =r2(1 −m)H

.(15)

Incorporating P= 0 into the first equation, if r11 >

r12, the system admits a nonnegative boundary equi-

librium E1r11 −r12

,0. By substituting P=

r2(1 −m)H

into the first equation and simplifying,

we get the equation

A1H2+A2H+A3= 0,(16)

where A1, A2and A3are specified by (13). (16)

allows a unique positive solution H∗, consequently,

system (9) permits a single positive equilibrium

E2(H∗, P ∗)due to the presence of H∗.

Theorem 2.1 has now been proved.

Theorem 2.2. Assume that r11 > r12 holds,

then E2(H∗, P ∗)is locally asymptotically stable and

E1r11 −r12

,0is unstable.

Proof. The system’s Jacobian matrix (9) can be rep-

resented as

J(H, P ) = A11 A12

A21 A22 ,(17)

where

A11 =r11m+r11 (1 −m)

kP + 1 −r12

−2b1H−a1(1 −m)P,

A12 =−r11 (1 −m)Hk

(kP + 1)2−a1(1 −m)H,

A21 =P2a2

(1 −m)H2,

A22 =r2−2a2P

(1 −m)H.

(18)

The variational matrix’s characteristic equation is

λ2−tr(J)λ+det(J) = 0.(19)

As long as tr(J)<0and det(J)>0, which in-

dicates that both eigenvalues have negative real com-

ponents, the asymptotic stability of an equilibrium so-

lution for a continuous-time system is satisfied.

At the equilibrium E1r11 −r12

,0, the Jacobian

matrix is represented by

JE1r11 −r12

,0

= r12 −r11 B

0r2!,

(20)

where B=(−1 + m) (r11 −r12) (r11k+a1)

Hence, it has one positive characteristic root λ1=r2,

consequently, E1r11 −r12

,0is unstable.

Noting that E2(H∗, P ∗)satisfies the equation

r11m+r11(1 −m)

1 + kP ∗−r12

−b1H∗−a1(1 −m)P∗= 0,

r2−a2

P∗

(1 −m)H∗= 0.

(21)

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By applying (21), at the equilibrium E2(H∗, P ∗), we

have

JE2(H∗, P ∗)

= −H∗b1C1

C2C3!.

(22)

where

C1=−a1H∗(1 −m)−r11 (1 −m)H∗k

(kP ∗+ 1)2,

C2=a2(P∗)2

(1 −m)(H∗)2,

C3=−a2P∗

(1 −m)H∗.

Then we have

DetJE2(H∗, P ∗)

=−H∗b1a2P∗

(1 −m)H∗

+a1H∗(1 −m) + r11 (1 −m)H∗k

(kP ∗+ 1)2×

a2(P∗)2

(1 −m)(H∗)2

>0,

and

T rJE2(H∗, P ∗)=−H∗b1−a2P∗

(1 −m)H∗<0.

Consequently, E2(H∗, P ∗)is locally asymptotically

stable.

The proof of Theorem 2.2 is now complete.

3 Permanence

By means of permanence, we imply that the positive

solution of the system has a positive upper and lower

bound after a sufficient amount of time, and that these

bounds are independent of the solution. If the system

(9) is permanent, predator and prey species will co-

habit on a long-term basis.

Set

def

r2(1 −m)r11 −r12

.(23)

Theorem 3.1. Assuming that

r11m+r11(1 −m)

1 + kP1

> r12 +a1(1 −m)P1(24)

holds, system (9) is permanent.

Proof. For any enough small positive constants ε >

0, set

Pε

def

r2(1 −m)r11 −r12

+ε. (25)

Inequality (24) implies that for any enough small pos-

itive constants ε > 0, the following inequalities

r11m+r11(1 −m)

1 + kP ε

> r12 +a1(1 −m)Pε

r11m+r11(1 −m)

1 + kP1

−r12 −a1(1 −m)P1

−ε > 0

(26)

holds.

By using the first equation of system (9), we have

dt =r11mH +r11(1 −m)H

1 + kP

−r12H−b1H2−a1(1 −m)HP

≤r11mH +r11(1 −m)H

1 + kP

−r12H−b1H2

≤(r11 −r12 −b1H)H,

(27)

By using Lemma 2.3 of [23] to (27), it follows that

lim sup

t→+∞

H(t)≤r11 −r12

.(28)

Hence, for ε > 0which satisfies inequality (26), there

exists a T1>0, such that

H(t)<r11 −r12

+εfor all t≥T1.(29)

For t > T1, we have, according to the second equation

of (9),

dt =r2−a2

(1 −m)HP

≤





r2−a2

(1 −m)r11 −r12

+ε





(30)

Applying Lemma 2.3 of [23] to (30) leads to

lim sup

t→+∞

P(t)≤

r2(1 −m)r11 −r12

+ε

.(31)

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Since ε > 0enough small, setting ε→0in the above

inequality, one has

lim sup

t→+∞

P(t)≤

r2(1 −m)r11 −r12

.(32)

For ε > 0which satisfies inequality (25), there exists

aT2> T1, such that for all t≥T2,

P(t)<

r2(1 −m)r11 −r12

+εdef

=Pε

1.(33)

For t > T2, the first equation of system (9) yields

dt =r11mH +r11(1 −m)H

1 + kP

−r12H−b1H2−a1(1 −m)HP

≥r11mH +r11(1 −m)H

1 + kP ε

−r12H−b1H2−a1(1 −m)HP ε

= r11m+r11(1 −m)

1 + kP ε

−r12

−a1(1 −m)Pε

1−b1H!H,

(34)

Applying Lemma 2.3 of [23] to (34) results in

lim inf

t→+∞

H(t)≥D1ε

.(35)

where

D1ε=r11m+r11(1 −m)

1 + kP ε

−r12 −a1(1 −m)Pε

Setting ε→0in the inequality shown above, we get

lim inf

t→+∞

H(t)≥D1

,(36)

where D1=r11m+r11(1 −m)

1 + kP1

−r12−a1(1−m)P1.

Hence, for ε > 0which satisfies inequality (26), there

exists a T3> T2, such that

H(t)>D1

−εfor all t≥T3.(37)

For t > T3, the second equation of (9) provides us

dt =r2−a2

(1 −m)HP

≥





r2−a2

(1 −m)D1

−ε





(38)

Applying Lemma 2.3 of [23] to (38) leads to

lim inf

t→+∞

P(t)≥

r2(1 −m)D1

−ε

.(39)

Since ε > 0enough small, setting ε→0in the above

inequality, one has

lim inf

t→+∞

P(t)≥

r2(1 −m)D1

.(40)

The equations (28), (32), (36) and (40) demonstrate

that if (24) holds, the system (9) is permanent.

Theorem 3.1 is proved.

4 Global stability

The subsequent Theorem addresses the global stabil-

ity of the positive equilibrium E2.

Theorem 4.1. Assuming

r11m+r11(1 −m)

1 + kP1

> r12 +a1(1 −m)P1(41)

holds, the positive equilibrium E2(H∗, P ∗)is glob-

ally stable.

Proof. Already, we had showed in Theorem 2.2 that

under the assumption r11 > r12, system (1) per-

mits a locally asymptotically stable positive equilib-

rium E2(H∗, P ∗). To demonstrate the attractivity of

E2, we first claim that (9) permits no limit cycles

inside the first quadrant. Set the Dulac function as

B=1

HP . Then

∂(F1B)

∂H +∂(F2B)

∂P

=E11

HP −E12

H2P−a2

(1 −m)H2

=−b1

P−a2

(1 −m)H2<0.

(42)

where

E11 =r11m+r11 (1 −m)

kP + 1

−r12 −2b1H−a1(1 −m)P,

E12 =r11mH +r11 (1 −m)H

kP + 1

−r12H−b1H2−a1(1 −m)HP

Now the claim follows from Dulac Theorem. Then

the claim, combined with permanence guaranteed by

Theorem 3.1 and the Bendixson-Dulac criterion, tells

us that all solutions with positive initial conditions ap-

proach E2(H∗, P ∗)as t→ ∞. The conclusion of

Theorem 4.1 is now followed.

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5 Influence of fear effect and refuge

Let us denote

F(H∗, P ∗, m, k)

=r11m+r11(1 −m)

1 + kP ∗

−r12 −b1H∗−a1(1 −m)P∗,

G(H∗, P ∗, m, k)

=r2−a2

P∗

(1 −m)H∗.

(43)

Then E2(H∗, P ∗)satisfies the equation

(F(H∗, P ∗, m, k)=0,

G(H∗, P ∗, m, k)=0.(44)

By straightforward calculation, we have

J=D(F, G)

D(H∗, P ∗)=

FH∗FH∗

GH∗GP∗

=

−b1F1

F2F3

=b1a2

(1 −m)H∗+F1F2

>0.

(45)

where

F1=−r11 (1 −m)k

(kP ∗+ 1)2−a1(1 −m),

F2=a2P∗

(1 −m) (H∗)2,

F3=−a2

(1 −m)H∗.

Using implicit function set theorem, the equation

(44) has an unique solution in the neighborhood of

E2(H∗, P ∗)

H∗=H∗(m, k), P ∗=P∗(m, k).(46)

and ∂H∗

∂k =−1

D(F, G)

D(k, P ∗),

∂P ∗

∂k =−1

D(F, G)

D(H∗, k),

∂H∗

∂m =−1

D(F, G)

D(m, P ∗),

∂P ∗

∂m =−1

D(F, G)

D(H∗, m).

(47)

By computation, we have

(1)

∂H∗

∂k =−1

r11P∗a2

(kP ∗+ 1)2H∗<0,(48)

that is, the prey density H∗decreases as kincreases;

(2)

∂P ∗

∂k =−1

r11(P∗)2a2

(kP ∗+ 1)2(H∗)2<0,(49)

that is, the predator density P∗decreases with increas-

ing k; The reason may be relying on the fact that with

the fear of predator species, the prey’s density will

decrease, and less of the food resource will finally re-

duce the density of predator species, too;

(3)

∂H∗

∂m =−1

H∗(−1 + m) (kP ∗+ 1)2>0,(50)

where

K1= 2a2 (P∗)2a1k2+1

2k2r11

+2ka1P∗+r11k+a1!P∗.

that is, the prey density, H∗, increase as mincreases.

(4)

∂P ∗

∂m =−1

a2P∗G

(−1 + m)2(H∗)2(kP ∗+ 1),(51)

where

G=b1(kP ∗+1)H∗+(−1+m)(ka11P∗+r11k+a1)P∗.

(52)

The sign of ∂P ∗

∂m depends on the sign of the term G.

(i)If

b1(kP ∗+1)H∗−(ka1P∗+r11k+a1)P∗>0,(53)

we obtain ∂P ∗

∂m <0, Thus, predator density decreases

with increasing refuge size.

(ii)If

b1(kP ∗+1)H∗−(ka1P∗+r11k+a1)P∗<0,(54)

then there exists a

m∗= 1 −b1(kP ∗+ 1)H∗

(ka1P∗+r11k+a1)P∗(55)

such that

∂P ∗

∂m >0for all 0< m < m∗,(56)

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∂P ∗

∂m <0for all 1> m > m∗,(57)

That is, for 0< m < m∗,P∗is the increasing func-

tion of m, while for msufficiently big, maybe as more

prey species remain in the refuge, predators have in-

sufficient food supplies, leading to a decrease in the

ultimate density of predator species.

6 Prey species all suffer from fear

effect

Note that in the system (9), we assume that prey

species are divided into two classes: outside the

refuge and inside the refuge. Only those outside the

refuge have a fear effect. Hence, it is natural to

compare our results with previously scholars’ works.

However, since (5)-(8) incorporating the functional

response or Allee effect, We were unable to contrast

our findings with their findings directly. In this sec-

tion, we would like to study the following model.

dt =r11H

1 + kP −r12H

−b1H2−a1(1 −m)HP,

dt =r2−a2

(1 −m)HP.

(58)

We will only state the results but omit the detail proof.

Theorem 6.1.Assume that r11 > r12 holds,

then system (58) admits a boundary equilibrium

F1r11 −r12

,0and a unique positive equilibrium

F2(H∗

1, P ∗

1),where

H∗=

−B2+qB2

2−4B1B3

2B1

P∗=r2(1 −m)H∗

(59)

here

B1=kr2(1 −m)a1m2r2−2a1mr2

+a1r2+a2b1>0,

B2=a2a1m2r2−kmr12r2−2a1mr2

+kr12r2+a1r2+a2b1,

B3=−a2

2(r11 −r12)<0.

(60)

Theorem 6.2. Considering the case r11 > r12

holds, then F1r11 −r12

,0is unstable, whereas

F2(H∗

1, P ∗

1)is locally asymptotically stable.

Theorem 6.3. Assume that

r11

1 + kP1

> r12 +a1(1 −m)P1(61)

holds, where P1is defined in (23), then system (58) is

permanent.

Theorem 6.4. Assume that

r11

1 + kP1

> r12 +a1(1 −m)P1(62)

holds, then the positive equilibrium F2(H∗

1, P ∗

1)is

globally stable.

Remark 6.1. Compared with the system (9) and (58),

we found that under the assumption r11 > r12 holds,

both system admits two equilibria, and the stability

property of the equilibria is identical. However, the

situation becomes very different regarding the per-

sistent property or global stability property. Noting

that for fixed r11 and P1,r11

1 + kP1

→0as k→ ∞.

Hence, in the system (58), with the increasing fear ef-

fect, the system may not be persistent. However, in

the system (9), regardless the large of K, as long as

mtends to 1, the inequality will still be maintained.

The system could allow for a globally stable positive

equilibrium. In other words, a refuge contributes sig-

nificantly to the system’s persistence and stability.

7 Numeric simulations

Example 7.1 In system (9), let’s take the following

parameter set

r11 = 3, r12 = 1, k = 10,

b1=a2=r1=a1=r2= 1.(63)

By computation, P1= 2(1 −m),hence if

r11m+r11(1 −m)

1 + kP1

= 3m+3(1 −m)

1 + 20(1 −m)

>1 + 2(1 −m)2

=r12 +a1(1 −m)P1,

(64)

holds, the positive equilibrium E2(H∗, P ∗)is stable

on a global level. That is, for m > 0.4728930857,

system (9) provides a single globally stable pos-

itive equilibrium E2(H∗, P ∗). For m= 0.5,

system (9) allows a single positive equilibrium

E2(0.5873499784,0.2936749892).Fig.1 shows that

E2is globally asymptotically stable. Considering the

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coefficients mentioned above, then

r11

1 + kP1

1 + 10

<1 = r12

<1 + 2(1 −m)2=r12 +a1(1 −m)P1.

(65)

Therefore, the inequality (61) could not hold, and we

could not give any information about the persistent

and stability property of a system (58).

Figure 1: Global asymptotical stability of E2, the

initial conditions (H(0), P (0)) = (0.1,0.8),

(1,0.1),(1.2,0.5),(1.2,0.8),(1.2,0.1),

(0.4,0.8) and (0.2,0.8), respectively.

Example 7.2 Let us take the following parameter set

r11 = 3,

r12 =b1=a2=r1=a1=r2= 1.(66)

By computation, E2(H∗, P ∗)satisfies the equation

3m+3−3m

−k(−1 + m)H∗+ 1 −1

−H∗+ (1 −m) (−1 + m)H∗= 0,

P∗= (1 −m)H∗.

(67)

Numeric simulations (Fig. 2 and 3) show that in this

case, H∗is the increasing function of mand decreas-

ing function of k.

Also, in this case, by computation, P∗satisfies the

following equation

3m+3−3m

kP ∗+ 1 −1

+P∗

−1 + m−(1 −m)P∗= 0.

(68)

Numeric simulation (Fig. 4 ) shows that P∗is the

decreasing function of k. Fixed k= 0.5, Figure

5 shows that there exists a m∗, such that P∗is

increasing in (0, m∗)and decreasing in (m∗,1).

Figure 2: Relationship of H∗,mand k.

Example 7.3 In system (9), let us take the following

parameter set

r11 = 3, r12 = 1, k = 10, m = 0.1,

b1=a2=r1=a1=r2= 1.(69)

By computation, P1= 2(1 −m) = 1.8,hence

r11m+r11(1 −m)

1 + kP1

= 0.3 + 2.7

1 + 18

<0.3+0.15 = 0.45

<1+1.62 = 1 + 2(1 −m)2

=r12 +a1(1 −m)P1.

(70)

According to Theorem 2.1 and 2.2, the system admits

a locally asymptotically stable positive equilibrium.

Additionally, since inequalities (24) and (41) are not

met, we have unsure about the positive equilibrium’s

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Figure 3: Relationship of H∗and k, here we

choose m=1

global stability feature. Numeric simulation (Fig. 6),

however, reveals that the positive equilibrium in this

situation is globally asymptotically stable.

8 Discussion

The fear effect of the predator on prey species is

widespread, and recently, many scholars have been

working in this direction([5]-[26]). On the other hand,

prey species could live in the shelter to reduce direct

killing of predator species. The influence of refuge re-

mains a long and important research subject of inves-

tigation in the predator-prey system. Recently, sev-

eral scholars ([14]-[16], [18], [20]-[22]) tried to com-

bine these two aspects and to propose some new mod-

eling of predator-prey system. They had made some

critical progress in this direction, but all of these stud-

ies involved that the prey species inside or outside of

the sanctuary all suffer from the effect of fear. Such

an assumption seems unreasonable since there are no

predator species within the sanctuary.

Stimulated by this fact, beginning with our ear-

lier research ([5]), we suggest the system (9). Under

the permits that r11 > r12, i.e., the birth rate of prey

species is greater than its mortality rate, we discover

the following: The system is capable of supporting

both a boundary equilibrium E1and a positive equi-

librium E2.One could easily see that E1is unstable

while E2is locally asymptotically stable (see Theo-

rem 2.2 for more detailed discussion).

It is natural to explore the property of global sta-

bility of the equilibrium as this property reflects the

Figure 4: Relationship of P∗,mand k.

long-run coexistence of the two species. In system (1)

and (2), by developing some proper Lyapunov func-

tions, the authors demonstrated the stability property

of the positive equilibrium. However, with the in-

troduction of the fear effect, the Lyapunov function

used in [3] and [5] could not be applied directly to the

system (9). In this research, we began by investigat-

ing the system’s persistence. Then, by applying the

Dulac criterion, we derived sufficient conditions to

ensure the globally asymptotical stability of positive

equilibrium. Intriguingly, the condition that ensures

the permanence of the system is sufficient to assure

the globally asymptotically stable of positive equilib-

rium, which indicates that if (24) holds, the system

could not exhibit bifurcation behaviors and could not

have a periodic solution.

The main innovation of this article is the assump-

tion that the prey population in the refuge is unaf-

fected by the fear effect. As can be seen from Section

6, such an assumption, compared to the classical hy-

pothesis, where scholars assumed that the prey popu-

lation is affected by the fear effect no matter where it

is located, and the dynamic behavior is vastly differ-

ent. Under our assumption, predator and prey popula-

tions can always coexist as long as the refuge is large

enough. In contrast, under the classical hypothesis,

prey populations tend to go extinct if the fear effect is

too large.

Example 7.3 demonstrates that Theorems 3.1 and

4.1 have space for improvement; we want to note this

at the conclusion of the study. With the methodology

used in this research, this seems implausible. This is

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Figure 5: Relationship of P∗and m, here we

choose k=1

left for further research.

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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).

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