WANG Li-bo,XU Gui-gui

(School of Mathematical Science,Kaili University,Kaili 556011,China)

1　Introduction

Leslie[1]introduced the famous Leslie predator-prey system

where x(t),y(t)stand for the population(the density)of the prey and the predator at time t,respectively,and p(x)is the so-called predator functional response to prey.The termof the above equation is called Leslie-Gower term,which measures the loss in the predator population due to rarity(per capita y/x)of its favorite food.In case of severe scarcity,y can switch over to other populations but its growth will be limited by the fact that its most favorite food x is not available in abundance.This situation can be taken care of by adding a positive constant k to the denominator,see[2–7]and references cited therein.

It is well known that time delays play important roles in many biological dynamical systems.In general,delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause a stable equilibrium to become unstable and cause the populations to fluctuate(see[5–7]).Furthermore,the existence of periodic solutions may be changed.Naturally,more realistic and interesting models of population interactions should take into account both the seasonality of changing environment and the effects of time delay.

In recent years,Leslie-Gower model with Holling-type II was extensively studied by many scholars,many excellent results were obtained concerned with the persistent property and positive periodic solution of the system(see[18–23]and the reference therein).Because Holling-type III can describe the relationship between the predator and prey clearly.So Zhang et al.[7]studied the following system

where x(t)and y(t)represent the densities of the prey and predator population,respectively;τi≥ 0;r1,b1,a1,k1,r2,a2,and k2are positive values.Some sufficient conditions for the local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to the two delays are obtained;however,Zhang did not give sufficient conditions for the existence of positive periodic solutions and permanence.Moreover,We know that coincidence degree theory is an important method to investigate the positive periodic solutions,and some excellent results were obtained concerned with the existence of periodic solutions of the predator-prey system(see[8–14]and the references therein).

Stimulated by the above reasons,in this paper,we consider the following system:

where x(t)and y(t)represent the densities of the prey and predator population,respectively;b(t),a1(t),a2(t),k1(t),k2(t),σ(t),τi(t),i=1,2 are all positive periodic continuous functions with periodi=1,2 are ω-periodic continuous functions.In addition,we request thati=1,2,and the growth functions ri(t),i=1,2 are not necessarily positive,because the environment fluctuates randomly.Obviously,where,k2are positive constants,system(1.1)is the special case of(1.2).

To our knowledge,no such work has been done on the existence of positive periodic solutions and permanence of(1.2).Our aim in this paper is,by using the coincidence degree theory developed by Gaines and Mawhin[15],to derive a set of easily verifiable sufficient conditions for the existence of positive solutions.Then by utilizing the comparison,we obtain sufficient conditions for permanence of system(1.2).

2　Preliminaries

Let X,Z be real Banach spaces,L:DomL?X→Z be a linear mapping,and N:X→Z be a continuous mapping.The mapping L is said to be a Fredholm mapping of index zero if dim Ker L=codim ImL＜+∞and Im L is closed in Z.If L is a Fredholm mapping of index zero,then there exist continuous projectors P:X→X and Q:Z→Z such that Im P=Ker L,Ker Q=ImL=Im(I?Q).It follows that the restriction LPof L to DomL∩KerP:(I?P)X→ImL is invertible.Denote the inverse of LPby KP.The mapping N is said to be L-compact onif ? is an open bounded subset of X,QN(?)is bounded and KP(I?Q)N:→X is compact.Since ImQ is isomorphic to KerL,there exist an isomorphic J:ImL→KerL.

Lemma 2.1(Continuation theorem[15])Let ??X be an open bounded set,L be a Fredholm mapping of index zero and N be L-compact onSuppose that

(i)for each λ∈(0,1),x∈??∩DomL,Lx/=λNx;

(ii)for each x∈??∩KerL,QNx/=0;

(iii)deg{JQN,?∩KerL,0}/=0.

Then Lx=Nx has at least one solution in∩DomL.

Lemma 2.2[17]Suppose that g∈then

3　Existence of Periodic Solutions

For convenience,we denote

where f(t)is a continuous ω-periodic function.

Theorem 3.1Assumehold,where H2is defined in the proof,then system(1.2)has at least one positive ω-periodic solution.

ProofLet x(t)=ex1(t),y(t)=ex2(t),then from(1.2),we have

It is easy to see that if system(3.1)has one ω -periodic solutionthenis a positive ω-periodic solution of(1.2).Therefore,we only need to prove that(3.1)has at least one ω-periodic solution.

Take X=Z={x(t)=(x1(t),x2(t))T∈C(R,R2):x(t+ω)=x(t)}and denote

then X and Z are Banach spaces when they are endowed with the norms‖·‖.

We define operators L,P and Q as follows,respectively,

where DomL={x∈X:x(t)∈C1(R,R2)},and define N:X→Z by the form

thus L is a Fredholm mapping of index zero.Furthermore,the generalized inverse(to L)KP:ImL→KerP∩DomL has the form

Thus

and

Obviously,QN and KP(I?Q)N are continuous.Moreover,QNKp(I?Q)Nare relatively compact for any open bounded set ??X.Hence,N is L-compact on，here ? is any open bounded set in X.

Corresponding to the operator equation Lx= λNz,λ ∈ (0,1),we have

Suppose that x(t)=(x1(t),x2(t))T∈ X is an ω-periodic solution of system(3.2)for a certain λ ∈ (0,1).By integrating(3.2)over the interval[0,ω],we obtain

From(3.2)–(3.4),we obtain

and

Noting that x=(x1(t),x2(t))T∈ X.Then there exist ξi,ηi∈ [0,ω]such that

It follows from(3.3)and(3.7)that

which implies that

It follows from(3.5),(3.8)and Lemma 2.2 that,for any t∈ [0,ω],

From(3.7),(3.9)and(3.4)that

i.e.,

which together with(3.6)and Lemma 2.2 imply

In addition,from(3.3)and(3.7),we get

which implies that

then together with(3.5)and Lemma 2.2 imply

From(3.9),(3.7)and(3.4),we have

i.e.,

which,together with(3.6)and Lemma 2.2 imply

It follows from(3.9)–(3.12)that

Obviously,H0is independent of λ.

Considering the following algebraic equations

If system(3.14)has a solution or a number of solutions x?=(x?1,x?2)T,then similar arguments as those of(3.9)–(3.12)show that

Hence

Set ? ={x=(x1,x2)T∈ X:‖x‖＜ H0}.Then,Lx/= λNx for x ∈ ?? and λ ∈ (0,1),that is ? satisfies condition(i)in Lemma 2.1.

Suppose x∈ ??∩KerL with‖x‖=H0.If(3.14)has at least one solution,we obtain from(3.15)that

If system(3.14)does not have a solution,then

Thus condition(ii)in Lemma 2.1 is satisfied.

Finally in order to prove(iii)in Lemma 2.1 we define homomorphism mapping

and

where μ∈[0,1]is a parameter.We will show that if x=(x1,x2)T∈??∩KerL,x=(x1,x2)Tis a constant vector in R2with max{|x1|,|x2|}=H0,then H(x1,x2,μ)/=0.Otherwise,suppose that x=(x1,x2)T∈ R2with max{|x1|,|x2|}=H0satisfying H(x1,x2,μ)=0,that is,

Similar argument as those of(3.14),(3.15)show that

which is a contradiction.

Hence by a direct calculation,we have

So(iii)in Lemma 2.1 is satisfied.By applying Lemma 2.1,we conclude that system(1.2)has at least one positive ω-periodic solution.The proof is completed.

Remark 3.1It is notable that our result only need b(t),a1(t),a2(t),k1(t),k2(t),τi(t)i=1,2,σ(t)are all positive ω-periodic continuous functions;but ri(t)∈ C(R,R),i=1,2 are ω-periodic continuous functions,and the growth functions ri(t),i=1,2 are not necessarily positive.It is reasonable on the biology.In addition,one can easily find that time delays τi(t),i=1,2, σ(t)do not necessarily remain nonnegative.Moreover,Theorem 3.1 will remain valid for systems(1.2)if the delayed terms are replaced by the term with discrete time delays,state-dependent delays,or deviating argument.Hence,time delays of any type or the deviating argument have no effect on the existence of positive solutions.

If the time delayed term σ(t)vanishes,τ1(t)≡ τ1,τ2≡ τ2and k21(t)≡ k1,k2(t)≡ k2,then system(1.2)is reduced to system(1.1)which was studied by Zhang et al.in[7].Thus from Theorem 3.1,we have the following result.

Corollary 3.1Assumehold,where

Then system(1.1)has at least one positive ω-periodic solution.

Remark 3.2In[7],Zhang et al.suppose ri(t),i=1,2 are positive.From Corollary 3.1,it is easy to known that ri(t)∈C(R,R),so ri(t),i=1,2 are not necessarily positive.We improve the result of[7].

4　Permanence

Definition 4.1System(1.2)is said to be permanent if there exist positive constants T,Mi,mi,i=1,2,such that any solution(x(t),y(t))Tof(1.2)satisfies m1≤x(t)≤M1,m2≤y(t)≤M2for t≥T.

Lemma 4.1[16]If a＞ 0,b＞ 0,τ(t)≥ 0,then

(1)if y′(t) ≤ y(t)[b? ay(t? τ(t))],then there exists a constant T ＞ 0 such that

(2)if y′(t) ≥ y(t)[b? ay(t? τ(t))],then there exists a constant T and M such that y(t)＜ M for t＞ T,then for any small constant ε＞ 0,there exists a constant T?＞ T such tha

Lemma 4.2There exists positive constant T0such that the solution(x(t),y(t))of(1.2)satisfies

where

ProofIf follows from system(1.2)that

From Lemma 4.1 yield that there exists a positive constant T1such that x(t)≤M1for t≥T1.Then we get

So there exists a positive T0≥T1such that y(t)≤M2for t≥T0.

Lemma 4.3If?1＞ 0 then there exists a positive constant T?such that the solution(x(t),y(t))of system(1.2)satisfies

where ε is a small enough positive constant and

ProofIf follows from Lemma 4.2 and system(1.2)that for t≥T0,

which,together with Lemma 4.1 and Lemma 4.2,implies that there exists a positive constant T?≥T0such that x(t)≥ m1and y(t)≥ m2for t≥T?.

From Lemma 4.2 and Lemma 4.3,we can get the following result on the permanence of system(1.1).

Theorem 4.1If?1＞0,then system(1.2)is permanent.

Similar to the proofs of Lemma 4.2 and Lemma 4.3,we have

Corollary 4.1If?1＞0,then system(1.1)is permanent.

Example 1Consider the following equation

where r1(t)=3+2sin(12πt),b(t)=1? 0.1sin(12πt),a1(t)=0.5+0.1sin(12πt),r2(t)=0.8+0.2sin(12πt),and k2=1,It is easy to calculation,and all the conditions in Theorems 3.1,3.2 and 4.1 hold.So we know system(4.3)has at least one positive periodic solution and permanent(see Figures 1,2,we take x(0)=1,y(0)=5 and x(0))=4,y(0)=5).

Figure 1

Figure 2

Example 2If r1(t)=8+2sin(2πt),b(t)=2?0.1sin(2πt),a1(t)=0.5+0.1sin(2πt),k21(t)=9,r2(t)=0.8+0.2sin(2πt),τ1(t)=1,τ2(t)=0.5,σ(t)=0,a2(t)=0.3?0.1sin(2πt),and k2(t)=1,It is easy to calculation,and all the conditions in Theorems 3.1,3.2 and 4.1 hold.So we know system(4.2)has at least one positive periodic solution and permanent(see Figure 3).

Figure 3

[1]Leslie P H.Some further notes on the use of matrices in population mathematics[J].Biometrika,1948,35(3-4):213–245.

[2]Aziz-Alaoui M A.Study of a Leslie-Gower-type tritrophic population[J].Chaos Sol.Frac.,2002,14(8):1275–1293.

[3]Gakkhar S,Singh B.Dynamics of modified Leslie-Gower-type prey-predator model with seasonally varying parameters[J].Chaos,Sol.Frac.,2006,27(5):1239–1255.

[4]Zhu Y L,Wang K.Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower Holling-type II schemes[J].J.Math.Anal.Appl.,2011,384(2):400–408.

[5]Nindjin A F,Aziz-Alaoui M A.Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay[J].Nonl.Anal.RWA,2006,7(5):1104–1118.

[6]Sunita G,Anuraj S.Complex dynamics in a prey predator system with multiple delays[J].Commun.Nonl.Sci.Numer.Simul.,2012,17(2):914–929.

[7]Zhang Z Z,Yong H Z,Fu M.Hopf bifurcation in a predator-prey system with Holling type III functional response and time delays[J].J.Appl.Math.Comput.,2014,44(1-2):337–356.

[8]Wang L L,Li W T.Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional rsponse[J].J.Comput.Appl.Math.,2004,162(2):341–357.

[9]Huo H F,Li W T.Periodic solutions of delayed Leslie-Gower predator-prey models[J].Appl.Math.Comput.,2004,155(3):591–605.

[10]Wang Q,Zhou J,Wang Z J,Ding M M,Zhang H Y.Existence and attractivity of a periodic solution for a ratio-dependent Leslie system with feedback controls[J].Nonl.Anal.RWA,2011,12(1):24–33.

[11]Wang K.Existence and global asymptotic stability of positive periodic solution for a predator-prey system with mutual interference[J].Nonl.Anal.RWA,2009,10(5):2774–2783.

[12]Lu S P.On the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments[J].J.Math.Anal.Appl.,2003,280(2):321–333.

[13]Song X Y,Li Y F.Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect[J].Nonl.Anal.RWA,2008,9(1):64–79.

[14]Zhang F Q,Zheng C W.Positive periodic solutions for the neutral ratio-dependent predator-prey model[J].J.Comput.Appl.Math.,2011,61(8):2221–2226.

[15]Gaines R E,Mawhin J L.Coincidence degree and nonlinear equations[M].Berlin:Springer-Verlag,1997.

[16]Chen F D.Permanence and global stability of nonautonomous Lotka-Volterra model with predatorprey and deviating arguments[J].Appl.Math.Comput.,2006,173(2):1082–1100.

[17]Wang D S.Four positive periodic solutions of a delayed plankton allelopathy system on time scales with multiple exploited(or harvesting)terms[J].IMA J.Appl.Math.,2013,78(3):449–473.

[18]Radouane Y,Adnani F El,Alaoui H T.Limit cycle and numerical similations for small and large delays in a predator-prey model with modied Leslie-Gower and Holling-type II scheme[J].Nonl.Anal.RWA,2008,9(5):2055–2067.

[19]Radouane Y,Adnani F El,Alaoui H T.Stability of limit cycle in a predator-prey model with modied Leslie-Gower and Holling-type II schemes with time delay[J].Appl.Math.Sci.,2007,1(1-4):119–131.

[20]Li Z X.Periodic solution for a modified Leslie-Gower model with feedback control[J].Acta Math.Appl.Sin.,2015,38(1):37–52.

[21]Xie X D,Xue Y L,Chen J H,Li T T.Permanence and global attractivity of a nonautonomous modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge[J].Adv.Diff.Equa.,2016,2016(1):1–11.

[22]Yuan R,Jiang W H,Wang Y.Saddle-node-Hopf bifurcation in a modified Leslie-Gower predatorprey model with time-delay and prey harvesting[J].J.Math.Anal.Appl.,2015,422(2):1072–1090.

[23]Tang X P,Li J Y,Gao W J.Existence of positive periodic solution for an impusive Holling-II predator-prey system with time delay[J].J.Math.,2009,29(6):761–768.