THE STABILITY OF A PREDATOR-PREY MODEL WITH FEAR EFFECT IN PREY AND SQUARE ROOT FUNCTIONAL RESPONSE∗

Ying Huang,Zhong Li

(College of Mathematics and Computer Science,Fuzhou University,Fuzhou 350108,Fujian,PR China)

Abstract

Keywords predator-prey;fear effect;stability;Hopf bifurcation

1 Introduction

Based on some experimental studies,[1,2]showed that the cost of fear can change the anti-predator defences,thus can greatly reduce the reproduction of prey.Hence,Wang et al.[3]proposed a predator-prey model incorporating the cost of fear into prey reproduction as follows

wherer0is the birth rate of prey;dis the natural death rate of prey;kis the level of fear.They studied the stability and Hopf bifurcation of the system,and showed that the existence of Hopf bifurcation is different from that of model (1.1) without fear effect.Duan et al.[4]discussed a diffusive predator-prey model by incorporating the fear effect into prey,and found that time delay makes the dynamics behaviour of the predator-prey system more complicated.Zhang et al.[5]investigated the stability of a predator-prey system with prey refuge and fear effect.Xiao and Li[6]showed that the fear effect has no influence on the stability of system.Pal et al.[7]studied the stability of a predator-prey model with fear effect in prey and hunting cooperation,and showed two different types of bi-stabilities behaviour.

Different from the model in[3],Sasmal[8]proposed the following predator-prey model with fear effect and Allee effect in prey

where 0<θ

Braza[9]considered a predator-prey model with a modified Lotka-Volterra interaction term,which is proportional to the square root of the prey population.Since the square root term,the dynamic behavior of the origin is more subtle.N.Fakhry and R.Naji[10]investigated the following predator-prey system with fear effect and square root function response

whereris the growth rate of the prey;αis the death rate of the predator in the absence of prey;βis the conversion rate of prey to predator;kis the level of fear.It is easy to deduce that there exist a trivial equilibriumE0(0,0) and a boundary equilibriumE1(1,0).Ifβ>α,there exists a positive equilibriumE∗(x∗,y∗) ,where

They obtained the following conclusions.

Proposition A(1)E0is a saddle point.

(2)If β<α,then the trivial equilibrium E1(1,0)is locally asymptotically stable.

(3)If,then the positive equilibrium E∗is locally asymptoticallystable.

Theorem A(1)Assume that the trivial equilibrium E1is locally asymptoticallystable.If,then E1is globally asymptotically stable.

(2)Assume that the trivial equilibrium E∗is locally asymptotically stable.If,then E∗is globally asymptotically stable.

(3)If,then system(1.3)undergoes a Hopf bifurcationat E∗.

Here,we put forward to the following several interesting questions:

(1) Due to the square root functional response,the origin cannot be linearized.WhetherE0in system (1.3) is still a saddle point?To answer this question,using the blow-up method,we give the singularity of the origin.

(2) Whenβ=α,that is there exists a zero eigenvalue ofE1,[10]didn’t discuss the stability ofE1.This fact requires us to investigate the stability of the trivial equilibriumE1ifβ=α.

(3) Since the origin is a singular equilibrium,whether the equilibriaE1andE∗are globally asymptotically stable or not?The conditions for the Hopf bifurcation are too complicated.Hence,we expect to give some simple conditions to guarantee the Hopf bifurcation.

The organization of this paper is as follows:In Section 2,the stability of the equilibria of system (1.3) is investigated.In Section 3,the impact of fear effect and numerical simulations are given.Finally,a briefly conclusion is illustrated in the last section.

2 Main Results

It follows from the N.Fakhry and R.Naji[10]that every positive solution of the system (1.3) is positively invariant and uniformly bounded.In this section,we study the stability and Hopf bifurcation of system (1.3) ,and obtain the following theorems.

Theorem 2.1There exists a separatrix curve near the trivial equilibrium E0(0,0)in system(1.3),such that above the separatrix curve is the attracted parabolic field,below the separatrix curve is the hyperbolic field.

ProofSince the singularity of matrixJE0,the stability ofE0fails to be studied by using Jacobian matrix.So,we introduce the following transformations

The Jacobian matrix calculated at the origin is reduced to a zero matrix,that is

Obviously, (0,0) is a nonhyperbolic equilibrium.Therefore,we use the blow-up method to eliminate the singularity of the origin and study the dynamics of this blow-up equilibrium point.The transformations is defined by

System (2.2) has two equilibria points on the non-negativey-axis: (0,0) and (0,2α+r).The Jacobin matrix of system (2.2) at (0,0) is calculated as

Then the eigenvaluesλ1=r>0 andλ2=−2α−r<0,that is the boundary equilibrium (0,0) is a saddle point.

The Jacobin matrix of system (2.2) at (0,2α+r) is calculated as

Hence,the eigenvaluesλ1=−2α<0 andλ2=2α+r>0.Then the boundary equilibrium (0,2α+r) is a saddle point.

Therefore,the linex=0,including the equilibria points (0,0) and (0,2α+r) ,collapses to the originE0(0,0) in system (1.3).Notice that we omitted the blow-up in they-direction.Whenx=0,it follows from system (1.3) that we obtain

Hence,there exists a separatrix curve near the trivial equilibriumE0(0,0) in system (1.3) ,such that above the separatrix curve is the attracted parabolic field,below the separatrix curve is the hyperbolic field.This completes the proof of Theorem 2.1.

Theorem 2.2(1)If α<β,the boundary equilibrium E1(1,0)in system(1.3)is a saddle point.

(2)If α≥β,the boundary equilibrium E1(1,0)in system(1.3)is stable.

ProofIt follows from Proposition A that the boundary equilibriumE1(1,0) in system (1.3) is stable ifβ<α,and is a saddle point ifα<β.

Now let’s focus onα=β.Then system (1.3) can be rewritten as

The Jacobin matrix of system (2.3) atE1is calculated as

that is the eigenvaluesλ1=−r<0 andλ2=0.

We transform the equilibriumE1to the origin by making a transformation thatX=x−1,Y=y.Then we have a Taylor expansion at the origin as follows

whereP1(X,Y) andQ1(X,Y) areC∞functions of at least order third of (X,Y).

LettingX1=−rX−Y,Y1=Y,introducing a new time variableτbyτ=−rt,rewritingτast,we have

whereP2(X1,Y1) andQ2(X1,Y1) areC∞functions of at least order third of (X1,Y1).

Therefore,the coefficient of.According to Theorem 7.1 of[11],we can conclude thatE1(1,0) is an attracting saddle-node,which includes a stable parabolic sector.This completes the proof of Theorem 2.2.

Theorem 2.3(1)If,the positive equilibrium E∗(x∗,y∗)in system(1.3)is stable.

(2)If,the positive equilibrium E∗(x∗,y∗)in system(1.3)is unstable.

ProofThe Jacobin matrix of system (1.3) atE∗is calculated as

Note thatx∗<1,we obtain

and

Therefore,ifβ2−3α2>0,that is,then the boundary equilibriumE∗in system (1.3) is unstable.Ifβ2−3α2<0,that is,then the boundary equilibriumE∗in system (1.3) is stable.This completes the proof of Theorem 2.3.

Theorem 2.4If,system(1.3)undergoes a supercritical Hopf bifurca-tion at positive equilibrium E∗and exists a stable limit cycle around E∗.

ProofFrom the proof of Theorem 2.3,when,we have DetJE∗>0 and TrJE∗=0.The positive equilibriumE∗beco mes a non-hyperbolic equilibrium,and the Jacobin matrix atE∗has a pair of imaginary roots.To ensure the occurrenceof Hopf bifurcation,we must check the transversality condition for Hopf Bifurcation.By simple calculation,we have

Hence,the positive equilibriumE∗loses its stability through non-degenerate Hopf bifurcation.the limit cycle.Using,we first translatethe equilibriumE∗ofsystem

Now wewillcalculate the Lyapunovnumberl1atE∗to determinethestability of (1.3) to the origin by employing the transformationsNote that.Thus,system (1.3) in a neighborhood of the origin can be written as

where

The first Lyapunov numberl1[12]to determine the stability of limit cycle for a planar system (2.6) is given by the following formula

Obviouslyl1<0,system (1.3) undergoes a supercritical Hopf bifurcation at positive equilibriumE∗and exists a stable limit cycle aroundE∗.This completes the proof of Theorem 2.4.

3 Discussion and Numerical Simulations

In this section,we show the impacts of fear effect and square root functional response on the dynamics behavior of system (1.3).Noting that,the fear effect has no in fluence on the prey density.For convenience,let,so.By simple computation,we obtain

which implies that the density of predators is a decreasing function with respect to fear effect.Hence,the fear effect reduces the predator density,but does not affect the prey density (Figure 1).

Using blow-up method,we show thatE0is a singularity point,which is different from that of[10].That is there exists a separatrix curve near the trivial equilibriumE0(0,0) in system (1.3) ,such that above the separatrix curve is the attracted parabolic field,below the separatrix curve is the hyperbolic field.Since the singularity of the origin,the trivial equilibriumE1and the positive equilibriumE∗cannot be globally asymptotically stable.Ifα=β,we show that the trivial equilibriumE1is stable.If,it follows from Theorem 2.4 that system (1.3) undergoes a supercritical Hopf bifurcation at positive equilibriumE∗and exists a stable limit cycle aroundE∗.Hence,we give a more simpler condition than that of[10]to guarantee the Hopf bifurcation.

Figure 1:The influence of fear effect on y∗ with r=4,α=0. 8,β=1.

Letr=1,k=6,β=1.The origin is always a singular equilibrium.Hence,we show the different stability of system (1.3) with the change ofα.Ifα=1.1>β,the trivial equilibriumE0(0,0) is locally asymptotically stable (Figure 2 (a)).Ifα=β=1,the trivial equilibriumE0 (0,0) is still locally asymptotically stable (Figure 2 (b)).If,the trivial equilibriumE0(0,0) becomes unstable,and the positive equilibriumE∗(x∗,y∗) is locally asymptotically stable (Figure 2 (c)).If,the positive equilibriumE∗(x∗,y∗) becomes unstable.System (1.3) undergoes a supercritical Hopf bifurcation and exists a stable limit cycle aroundE∗(Figure 2 (d)).

Figure 2:The dynamics behaviour of predator-prey model (1.3) with r=1,k=6,β=1.