LI Zhen-wei,LI Bi-wen,LIU Wei,WANG Gan

(School of Mathematics and Statistics,Hubei Normal University,Huangshi 435002,China)

HOPF BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH NON-SELECTIVE HARVESTING AND TIME DELAY

LI Zhen-wei,LI Bi-wen,LIU Wei,WANG Gan

(School of Mathematics and Statistics,Hubei Normal University,Huangshi 435002,China)

In this paper,we mainly study the Hopf bifurcation and the stability of modifi ed predator-prey biological economic system with nonselective harvesting and time delay.By using the stability and bifurcation theory of diff erential-algebraic system,the conditions for stability of the positive equilibrium point are obtained,let time delay as bifurcation parameter,the existence of Hopf bifurcation and direction of Hopf bifurcation are obtained.We have improved the Leslie-Gower predator-prey system,make the system which we established more practical,so the conclusions are made more scientifi c.

stability;Hopf bifurcation;time delay;non-selective;predator-prey system; periodic solution

1 Introduction

In recent years,the increasingly serious problem of environmental degradation and resource shortage,made the analysis and modeling of biological systems more interested. The predator-prey system played a crucial role among the relationships between the biological population,and it naturally attracted much attention both for mathematicians and biologists,especially on predator-prey systems with or without time delay.As we know, delay differential equation models exhibit much more complicated dynamics than differential equation models without delay,see[1–12].A lot of researchers studied the dynamics of predator-prey models with harvesting and obtained many dynamic behaviors,such as stability of equilibrium,Hopf bifurcation,periodic solution,Bogdanov-Takens bifurcation, Neimark-Sacker bifurcation,and so on,see[10–15].

In[16],Lucas studied the dynamic properties of the following Leslie-Gower predatorprey system

where x and y denote prey and predator population densities at time t,respectively,a,d, and k are positive constants that represent the prey intrinsic growth rate,predator mortality rate,and the maximum value of the per capita reduction rate of x due to y,respectively.

At present,economic profit is a very important factor for merchants,governments and even every citizen,so it is necessary to research biologicaleconomic systems,which are often described by differential-algebraic equations or differentialdifference-algebraic equations.

In 1954,Gordon[13]studied the effect of the harvest effort on ecosystem from an economic perspective and proposed the following economic theory:

Net Economic Revenue(NER)=Total Revenue(TR)-Total Cost(TC).

This provides theoretical evidence for the establishment of diff erential-algebraic equation.

Based on the economic theory as mentioned above and system(1.1),Liu and Fu[12] considered the following Leslie-Gower predator-prey system

They investigated the Hopf bifurcation of the above system without considering the effect of time delay and the harvesting of predator.

As is known to all,delay differential equation models exhibit much more complicated dynamics than ordinary differential equation models,see[1–12],as was pointed by Kuang [17]that any modelof species dynamics without delays is an approximation at best.When we considered the model with non-selective harvesting,namely at the same time there are also the harvesting of predator and harvesting of the prey in the model,it will be more in line with the actualsituation of the predator-prey systems.

Motivated by the above discussion,in this paper,by choosing the time delay as a bifurcation parameter and consider the predator-prey systems with non-selective harvesting, we investigate a modifi ed Leslie-Gower predator-prey systems with non-selective harvesting and time delay described by the following system

where p1＞ 0 and p2＞ 0 are harvesting reward per unit harvesting effort for unit prey and predator,respectively;c1and c2are harvesting cost per unit harvesting effort for prey and predator,respectively;m is the economic profit per unit harvesting effort.

In this paper,we mainly discuss the effects ofeconomic profit on the dynamics ofsystem (1.3)in the region R3+={(x,y,E)|x ＞ 0,y ＞ 0,E ＞ 0)}.

For convenience,we let

where Xt=(x,y)T.

The rest ofthe paperisarranged asfollows:in Section 2,the localstability ofthe positive equilibrium points are investigated by the corresponding characteristic equation of system (1.3).In Section 3,by using the normal form and Hopf bifurcation theorem,we study the Hopfbifurcation ofthe nonnegative equilibrium depending on the parameter where we show that the positive equilibrium loses its stability and system(1.3)exhibits Hopfbifurcation in the second section.In Section 4,the theoretical result is supplied by a numerical example. Finally,this paper ends with a brief discussion.

2 Local Stability Analysis of System

In this section,we discuss the local stability of a positive equilibrium for system(1.3). Now,we try to find all possible positive equilibrium points of system(1.3).A point Y0= (x0,y0,E0)is an equilibrium point of system(1.3)if and only if Y0satisfy the following equations

From(2.1),we can easy get E0satisfy

where Based on the root and coeffi cient relationship of equation and γ3＜ 0,we can find at least one positive root E0,so system(1.3)has at least one positive equilibrium point,where r1＞ E0,r2＞ E0.

Now,we derive the formula for determining the properties of the positive equilibrium point ofsystem(1.3).As in[13],first we consider the localparametric ψ ofthe third equation of system(1.3),which is defined as follows

where

is a smoothing mapping,that is

Then we can obtain the parametric system of system(1.3)as follows (

Noticing that g(ψ(Z(t)))=0,so we can get the linearized system of parametric system(2.3) at(0,0)as follows

From(2.4),we can obtain the characteristic equation of the linearized system of parametric system(2.2)at(0,0)as follows

By eq.(2.5),when τ=0,it is obvious that,then,two roots of eq. (2.5)has always negative teal parts,i.e.,the positive equilibrium point of system(1.3)is locally asymptotically stable.

Now,based on the above discussion,we study the local stability around the positive equilibrium point for system(1.3)and the existence of Hopf bifurcation occurring at the positive equilibrium point when τ＞ 0.

If iω is a root of eq.(2.5),and substituting iω (ω is a positive real number)into eq. (2.5),and separating the real and imaginary parts,two transcendental equations can be obtained as follows

Since sin(ω τ)2+cos(ω τ)2=1 and adding(2.6)and(2.7),we obtain

Substituting ω0into(2.6)and solving for τ,we get

Thus when τ= τn,the characteristic equation(2.5)has a pair of purely imaginary roots iω0.

Lemma 2.1Denote by λn(τ)= ηn(τ)+iωn(τ)the root of(2.5)such that ηn(τn)=0, ωn(τn)= ω0,n=0,1,2,···.Then the following transversality condition η′n(τn)is satisfied.

ProofDifferentiating eq.(2.5)with respect to τ,we obtain

Noting that

The proof is completed.

From the above analysis and[17,18],we have the following results.

Theorem 2.1(i)For system(1.3),its positive equilibrium point Y0is locally asymptotically stable for τ∈ [0,τ0)and unstable for τ＞ τ0.

(ii)System(1.3)undergoes Hopf bifurcation at the positive equilibrium point Y0for τ= τn,n=0,1,2,···.

3 Direction and the Stability of Hopf Bifurcation

In this section,we investigate the direction ofbifurcation and the stability ofbifurcation periodic orbits from the positive equilibrium point Y0of system(1.3)at τ= τ0by using the normalform approach theory and center manifold theory introduced by Hassard[15].

Now,we re-scare the time by

for simplicity,we continue to use Z said ¯Z,then the parametric system(2.3)of system (1.3)is equivalent to the following functional differential equation(FDE)system in C= C([−1,0],R2),

where Z(T)=(y1(t),y2(t))T,and Lµ:C → R,f:R × C → R are given,respectively,by

where

and φ =(φ1,φ2) ∈ C.By the Riesz representation theorem,there exists a matrix whose components are bounded variation functions θ ∈ [−1,0]such that

where

Then system(3.1)can be rewritten as

For ψ ∈ C1([0,1],(R2)∗),the adjoint operator A∗of A as

where ηTis the transpose of the matrix η.

For φ ∈ C1([−1,0],R2)and for ψ ∈ C1([0,1],(R2)∗),in order to normalize the eigenvectors of operator A and adjoint operator A∗,we define a bilinear inner product

where η(θ)= η(θ,0).It is easy to verify that A(0)and A∗are a pair of adjoint operators.

By the discussion in Section 2,we know that ±iωτ0are eigenvalues of A(0).Thus they are also eigenvalues of A∗.Next we calculate the eigenvector q(θ)of A associated to the eigenvalue iω τ0and the eigenvector q∗(s)of A∗associated to the eigenvalue −iω τ0.Then it is not diffi cult to show that

where

Moreover,〈q∗(s),q(θ)〉=1 and 〈q∗(s),¯q(θ)〉=0.

In the reminder of this section,we use the same notations as those in[15].We fi rst compute the coordinates to describe the center manifold C0at µ =0.Define

On the center manifold C0,we have

In fact,z and ¯z are local coordinates for center manifold C0in the direction of q and ¯q∗. Note that W is real if ztis real.We consider only real solutions.For the solution zt∈ C0, since µ =0 and(3.3),we have

rewrite it as

where

From(3.3)and(3.8),we have

Rewrite(3.11)as

where

Substituting the corresponding series into(3.12)and comparing the coeffi cient,we obtain

Notice that

and(3.6)we obtain

According to(3.8)and(3.9),we know that

where

By(3.7),it follows that

That is,

Comparing the coeffi cients with(3.10),it follows that

Now we compute W20(θ)and W11(θ).From(3.11)and(3.15),we have that for θ ∈ [−1,0),

Comparing the coeffi cients with(3.13),we can obtain that

Substituting the above equalities into(3.14),it follows that

Solving(3.18),we have

In what follows,we seek appropriate E and F in(3.19).From(3.11)and(3.15),we have

where

Substituting(3.19)–(3.21)into(3.14)and noting that

We obtain

It is easy to obtain E and F from(3.22)and(3.23),that is

Therefore we can compute the following quantitieswhich determine the direction of Hopf bifurcation and stability of bifurcated periodic solutions of system(1.3)at the critical value τ0.

Theorem 3.1(i)The direction of Hopfbifurcation is determined by the sign ofµ2:the Hopfbifurcation is supercritical if µ2＞ 0 and the Hopf bifurcation is subcriticalif µ2＜ 0.

(ii)The stability of bifurcated periodic solution is determined by β2:the periodic solution are stable if β2＞ 0 and unstable if β2＜ 0.

(iii)The period ofbifurcation periodic solution is determined by t2:the period increase if t2＞ 0,decrease if t2＜ 0.

4 Numerical Simulations

As an example we consider the differential-algebraic predator-prey system(1.3)with the parameters r1=1.6,r2=1.3,b=k=m=0.5,p1=7,p2=6,c1=5,c2=3,that is,

And by the discussions in Section 2 and Section 3,we determine the stability of the positive equilibrium point and Hopf bifurcation.Here,for convenience,we only discuss one of the positive equilibrium point Y0of system(4.1),and others positive equilibrium points of system(4.1)can be similar studied.we can easily get Y0=(2.0053,3.1480,0.0256),and by computing,we get ω0=0.9942, τ0=0.6473.So by Theorem 2.1,the equilibrium point Y0is asymptotically stable when τ∈ [0,τ0)=[0,0.6473)and unstable when τ＞ 0.6473.

When τ=0,we can easily show that the positive equilibrium point

is asymptotically stable.

By the theory of Hassard[15],as it is discussed in former section,we also determine the direction of Hopf bifurcation and the other properties of bifurcating periodic solution.By computing,we can obtain the following values C1(0)=0.5303 − 0.4428i, λ′(τ0)=1.6352+ 1.1431i,it follows that µ2= −0.3243 ＜ 0,β2=1.0607 ＞ 0,t2=1.2643 ＞ 0,from which and Theorem 3.1 we conclude that the Hopf bifurcation of system(4.1)occurring at τ0=0.6473 is subcriticaland the bifurcating periodic solution exists when τcross τ0to the left and the bifurcating periodic solution is unstable.

By Theorem 3.1,the positive equilibrium point Y0of system(4.1)is locally asymptotically stable when τ=0.62 ＜ τ0as is illustrated by computer simulation in Fig.1.And periodic solutions occur from Y0when τ=0.682 ＞ τ0as is illustrated by computer simulation in Fig.2.

5 Discussion

Nowadays,economic profit is a very important factor for governments,merchants,and even citizen,and the harvested biologicalresources in the predator-prey systems are usually sold as commodities in the market in order to achieve the economic interest.So modelling and qualitative analysis for bio-economic system are necessary.

Compared with most other researches on dynamics ofpredator-prey population,see[1, 5,12,18],the main contribution ofthis paper lies in the following aspect.The predator-prey system we consider incorporate delay and non-selective harvesting,which could make our model more realistic and the analysis result in this paper is more scientific.So our paper provide a new ideal and a effi cacious method for the qualitative analysis of the Hopf bifurcation of the differential-algebraic biologicaleconomic system.In addition,stage structure, diffusion effects,disease effects may be incorporated into our bio-economic system,which would make the bio-economic system exhibit much more complicated dynamics.

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一类带无选择性捕获和时滞的捕食食饵系统的Hopf分支分析

李震威,李必文,刘 炜,汪 淦
(湖北师范学院数学与统计学院,湖北 黄石 435002)

本文主要研究了一个改进的带时滞和无选择捕获函数的捕食-食饵生态经济系统的稳定性和Hopf分支. 利用微分代数系统的稳定性理论和分支理论, 得到了系统正平衡点稳定性的条件, 以及当时滞τ 作为分支参数时系统产生Hopf分支的条件. 对Leslie-Gower捕食-食饵模型进行了一定程度的完善, 使得建立的模型更符合实际情况,因此得到的结论也更加科学.

稳定性;Hopf分支; 时滞; 无选择性; 捕食食饵系统; 周期解

34D20;34K18;34C23

O29;O193

tion:34D20;34K18;34C23

A < class="emphasis_bold">Article ID:0255-7797(2017)02-0257-14

0255-7797(2017)02-0257-14

∗Received date:2014-11-16 Accepted date:2015-02-26

Foundation item:Supported by the Funding Program of Higher School Outstanding Youth Scientifi c and Technological Innovation Team in Hubei of China(T201412).

Biography:Li Zhenwei(1991–),male,born at Qianjiang,Hubei,major in ordinary diff erential equations and control theory.