# A Predator-Prey Gompertz Model with Time Delay and Impulsive Perturbations on the Prey

Authors: Jianwen Jia; Chunhua Li
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

We introduce and study a Gompertz model with time delay and impulsive perturbations on the prey. By using the discrete dynamical system determined by the stroboscopic map, we obtain the sufficient conditions for the existence and global attractivity of the “predator-extinction” periodic solution. With the theory on the delay functional and impulsive differential equation, we obtain the appropriate condition for the permanence of the system.


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## Body

## 1. Introduction

It is well known that Gompertz equation [1] describes the growth law for a single species. The model reads as(1.1)dx(t)dt=rx(t)lnKx(t),
where x(t) is the density of the population, r is a positive constant called the intrinsic growth rate, the positive constant K is usually referred to as the environment carrying capacity or saturation level, and rln(K/x) denotes relative growth rate.Many systems in physics, chemistry, biology, and information science have impulsive dynamical behavior due to abrupt jumps at certain instants during the evolving processes. This dynamical behavior can be modeled by impulsive differential equations. The theory of impulsive differential systems has been developed by numerous mathematicians [2–4]. But more realistic models should include some of the past states of these systems, that is, ideally, a real system should be modeled by differential equations with time delays. Recently, the investigation of impulsive delay differential equations is beginning [5–7].In this paper, we need to consider a function(1.2)ϕ(x)=x2(t)1+mx2(t),
which is called Holling III functional response. The coefficient m is a positive constant and its biological meaning can be found in [8].We introduce the model as follows:(1.3)dx(t)dt=rx(t)lnKx(t)-x2(t)1+mx2(t)y(t),dy(t)dt=βx2(t-τ)1+mx2(t-τ)y(t-τ)-dy(t),t≠nT,n∈N,Δx(t)=-px(t),Δy(t)=0,t=nT,n∈N,
where x(t) and y(t) are densities of the prey and the predator, respectively, r is the Gomportz intrinsic growth rate of the prey in the absence of the predator, β is the conversion rate, d is the death rate of the predator, τ is mean length of the digest period, and p(0<p<1) represents impulsive harvest to preys by catching or pesticides at t=nT, n∈N.The initial conditions for system (1.3) are(1.4)(ϕ1(s),ϕ2(s))∈C+=C([-τ,0],R+2),ϕi(0)>0(i=1,2).From the biological point of view, we only consider system (1.3) in the biological meaning region D={(x,y)∣x,y≥0}.All outline of this paper is as follows. We give some basic knowledge in Section2. In Section 3, using discrete dynamical system determined by the stroboscopic map, we obtain the existence and global attractivity of the “predator-extinction” periodic solution. In Section 4, with the theory of delay and impulsive different equations, we obtain the sufficient condition for the permanence of the system. In the last section, we give the numerical simulation and discussion.


## 2. Basic Knowledge

LetR+=[0,∞), R+2={x∈R2|x≥0}, Ω=intR+2, and N be the set of all nonnegative integers. Denote that f=(f1,f2)T is the map defined by the right-hand side of the first two equations of system (1.3). Let V0 = {V:R+×R+2→R+, continuous on (nT,(n+1)T]×R+2, and lim(t,z)→(nT+,x)V(t,z)=V(nT+,x) exist}.Definition 2.1.
LetV∈V0, then for (t,X)∈(nT,(n+1)T]×R+2, the upper right derivative of V(t,x) with respect to the impulsive differential system (1.3) is defined as
(2.1)D+V(t,X)=lim suph→0+1h[V(t+h,X+hf(t,x))-V(t,X)].
The solution of system (1.3) is a piecewise continuous function. The smoothness properties of f guarantee the global existence and uniqueness of the solution of system (1.3) [9].Lemma 2.2 (see [9]).
Considering the following impulsive differential inequalities:(2.2)dm(t)dt≤p(t)m(t)+q(t),t≠tk,k=1,2…,m(wk+)≤dkm(tk)+bk,t=tk,t≥t0,
where p,q∈PC[R+,R] and dk≥0, bk are constants, then(2.3)m(t)≤m(t0)∏t0<tk<tdkexp(∫t0tp(s)ds)+∑t0<tk<t(∏tk<tj<tdjexp(∫t0tp(s)ds))bk+∫t0t∏s<tk<tdkexp(∫stp(σ)dσ)q(s)ds,t≥t0.Lemma 2.3.
There exists a constantL=βrK(Ked/r-1)/d>0 such that x(t)≤K and y(t)≤L for each solution of system (1.3) with t large enough.Proof.
Define(2.4)V(t)=βx(t)+y(t+τ),
then V∈V0. Since dx/dt≤rx(t)ln(K/x(t)), dx/dt∣x=K=0; in addition, x(nT+)≤x(nT). Thus x(t)≤K for t large enough.
We calculate the upper right derivative ofV(t) along a solution of system (1.3) as
(2.5)D+V(t)+dV(t)≤βx(t)rlnKed/rx(t)=βx(t)rlnKed/r≤βrK(Ked/r-1).
So we have that(2.6)D+V(t)≤-dV(t)+M,t≠nT,n∈N,V(t+)≤V(t),t=nT,n∈N.
By Lemma 2.2, for t∈(nT,(n+1)T], we have that
(2.7)V(t)=(V(0+)-βrK(Ked/r-1)d)exp(-dt)+βrK(Ked/r-1)d,limt→∞V(t)≤βrK(Ked/r-1)d.
So, V(t) is ultimately bounded. Therefore by the definition of V(t), there exists a constant L=βrK(Ked/r-1)/d>0 such that x(t)≤K and y(t)≤L for each solution of system (1.3) with t large enough. This completes the proof.Lemma 2.4 (see [10]).
Consider a delay equation(2.8)ẋ(t)=ax(t-τ)-bx(t),
where a,b,τ are all positive constants and x(t)>0 for -τ≤t≤0. Thus one has the following.(1)
Ifa>b, then limt→∞x(t)=+∞.(2)
Ifa<b, then limt→∞x(t)=0.


## 3. “Predator-Extinction” Periodic Solution

### 3.1. Existence of a “Predator-Extinction” Periodic Solution

First, we begin analyzing the existence of a “predator-extinction” solution, in which predator is absent from the system, that is,(3.1)y(t)=0,t≥0.In this condition, we know the growth of the prey in the time-intervalnT≤t≤(n+1)T and give some basic properties of the following subsystem of (1.3):(3.2)dx(t)dt=rx(t)lnKx(t),t≠nT,n∈N,Δx(t)=-px(t),t=nT,n∈N.For system (3.2), we let ϖ(t)=lnx(t) and obtain a linear nonhomogeneous impulsive equation(3.3)dϖ(t)dt=-rϖ(t)+rlnK,t≠nT,n∈N,Δϖ(t)=ln(1-p),t=nT,n∈N.Solving the first equation of system (3.3) between pulses yields(3.4)ϖ(t)=lnK+(ϖ((n-1)T)-lnK)e-r(t-(n-1)T),(n-1)T<t≤nT.For the second equation of system (3.3), using discrete dynamical system determined by the stroboscopic map yields(3.5)ϖ(nT)=lnK+(ϖ((n-1)T)-lnK)e-rT+ln(1-p)=:f(ϖ(n-1)T),
where f(ϖ)=lnK+(ϖ-lnK)e-rT+ln(1-p). From (3.5), we can see that this difference system has an equilibrium ϖ0*=lnK+ln(1-p)/(1-e-rT), which implies that system (3.3) has a unique T-periodic solution(3.6)ϖ*(t)=lnK+(ϖ0*-lnK)e-r(t-(n-1)T),(n-1)T<t≤nT.Sincex0*=eϖ0*, then x*(t)=eϖ*(t) is the unique positive T-periodic solution of (3.2).In the following, we will prove thatx*(t) is globally asymptotically stable. By Lemma 2.3, we find that any solution of (3.2) is ultimately upper bounded, so we need only to prove that(3.7)limt→∞|ϖ(t)-ϖ*(t)|=0,
where ϖ*(t) is the periodic solution of system (3.3) Suppose that with lnx*(0)=lnx0*=ϖ*(0)=ϖ0*.(3.8)|ϖ(t)-ϖ*(t)|=|ϖ(0)-ϖ0*|e-rt≤e-(n-1)rT,
since limt→∞e-(n-1)rT=0; thus, limt→∞|ϖ(t)-ϖ*(t)|=0.Theorem 3.1.
System (3.2) has unique a positive periodic solution x*(t) which is globally asymptotically stable. That is, system (1.3) has a “predator-extinction” periodic solution (x*(t),0) for t∈(nT,(n+1)T], n∈N.


### 3.2. Global Attractivity of the “Predator-Extinction” Periodic Solution

Denote that(3.9)ℜ1=βK2d[mK2+exp(2ln(1/1-p))/(exp(rT)-1)].Theorem 3.2.
Ifℜ1<1, then the “predator-extinction” periodic solution (x*(t),0) of system (1.3) is globally attractive.Proof.
We denote that(x(t),y(t)) be any solution of system (1.3) with initial condition (1.4). From the second equation of system (1.3), we have that
(3.10)dy(t)dt≤βmy(t-τ)-dy(t).
We consider the following comparison equation:(3.11)dz(t)dt=βmz(t-τ)-dz(t).
If β/m-d<0, then ℜ1<1; according to Lemma 2.4, we obtain limt→∞z(t)=0.
Sincey(s)=z(s)=ϕ2(s)>0 for all s∈[-τ,0], by the comparison theorem in differential equation and the positivity of the solution, we have that y(t)→0 as t→∞.
In the following, we suppose thatβ/m≥d. Since ℜ1<1, we have that
(3.12)βK2exp(2ln(1-p)/(exp(rT)-1))1+mK2exp(2ln(1-p)/(exp(rT)-1))-d<0.
Because the functionx2/(1+mx2) is monotonically increasing with respect to x, we choose a sufficiently small positive constant ϵ such that
(3.13)β[Kexp(ln(1-p)/(exp(rT)-1))+ϵ]21+m[Kexp(ln(1-p)/(exp(rT)-1))+ϵ]2-d<0.
Noting thatdx(t)/dt≤rx(t)ln(K/x(t)),Δx(t)=-px(t) for nT<t≤(n+1)T, then we consider the following comparison system:
(3.14)dz(t)dt=rz(t)lnKz(t),t≠nT,n∈N,z(t+)=(1-p)z(t),t=nT,n∈N,z(0+)=x(0+).
From Section3.1, we see that
(3.15)z*(t)=exp{lnK+ln(1-p)1-e-rTe-r(t-(n-1)T)}(n-1)T<t≤nT,
which is unique a globally asymptotically stable positive T-periodic solution of system (3.14).
There exist a positive integern1 and an arbitrarily small positive constant ϵ for all t≥n1T such that
(3.16)x(t)≤z*(t)+ϵ≤exp{lnK+ln(1-p)1-e-rTe-rT}+ϵ=:σ.
From (3.16) and the second equation of system (1.3), for t>n1T+τ we have
(3.17)dy(t)dt≤βσ21+mσ2y(t-τ)-dy(t).
Considering the following comparison equation:(3.18)dz(t)dt=βσ21+mσ2z(t-τ)-dz(t),
we can see (3.13), that is, βσ2/(1+mσ2)<d; according to Lemma 2.4, we obtain limt→∞z(t)=0.
Sincey(s)=z(s)=ϕ2(s)>0 for all s∈[-τ,0], by the comparison theorem in differential equation and the positivity of the solution, we have that y(t)→0 as t→∞.
Next, we will prove thatx(t)→x*(t) as t→∞. Without loss of generality, we may assume that 0<y(t)≤ϵ for all t≥0. By the first equation of system (1.3), we have that
(3.19)dx(t)dt≥rx(t)lnKx(t)-x2(t)y(t)≥rx(t)lnKx(t)-Kx(t)ϵ≥rx(t)lnKe-Kϵ/rx(t).
Then we have thatz̃1(t)→x*(t) as ϵ→0 as t→∞, where z̃1(t) is unique a positive solution of
(3.20)dz1(t)dt=rz1(t)lnKe-Kϵ/rz1(t),t≠nT,n∈N,z1(t+)=(1-p)z1(t),t=nT,n∈N,z1(0+)=x(0+).
From Section 3.1, we obtain
(3.21)z̃1(t)=exp{ln(Ke-Kϵ/r)+ln(1-p)1-e-rTe-r(t-nT)},nT<t≤(n+1)T.
By using comparison theorem of impulsive differential equation [11], for any ϵ1>0, there exists T1>0 and such that t>T1(3.22)x(t)>z̃1(t)-ϵ1.
On the other hand, from the first equation of (1.3), we have that dx(t)/dt≤rx(t)ln(K/x(t)).
Considering the following comparison system:(3.23)dz2(t)dt=rz2(t)lnKz2(t),t≠nT,z2(t+)=(1-p)z2(t),t=nT,z2(0+)=x(0+),
we have that
(3.24)z̃2(t)=exp{lnK+ln(1-p)1-e-rTe-r(t-nT)},nT<t≤(n+1)T,(3.25)x(t)<z̃2(t)+ϵ1,
as t→∞ and z̃2(t)=x*(t). Let ϵ1→0, then it follows from (3.22) and (3.25), that we can see x*(t)-ϵ1<x(t)<x*(t)+ϵ1 for t large enough, which implies x(t)→x*(t) as t→∞. The proof is completed.Now, we give the following theorem.Theorem 3.3.
(1) If β/m<d, then the “predator-extinction” periodic solution (x*(t),0) is globally attractive.
(2) If β/m≥d and p>p*, T<T*, m>m*, then the “predator-extinction” periodic solution (x*(t),0) is globally attractive, where the critical values p*, T*, m* are listed in Table 1.Table 1
Critical values of some parameters of system (1.3) (R1<1 must be satisfied).
The conditions for global attractivity of(x*(t),0)p>p*p*=1-1/(βK2d-mK2)erT-1T<T*T*=1rln[1-ln(1-p)2ln(βK2/d-mK2)]m>m*m*=βd-1K2exp(-ln(1-p)2erT-1)


## 3.1. Existence of a “Predator-Extinction” Periodic Solution

First, we begin analyzing the existence of a “predator-extinction” solution, in which predator is absent from the system, that is,(3.1)y(t)=0,t≥0.In this condition, we know the growth of the prey in the time-intervalnT≤t≤(n+1)T and give some basic properties of the following subsystem of (1.3):(3.2)dx(t)dt=rx(t)lnKx(t),t≠nT,n∈N,Δx(t)=-px(t),t=nT,n∈N.For system (3.2), we let ϖ(t)=lnx(t) and obtain a linear nonhomogeneous impulsive equation(3.3)dϖ(t)dt=-rϖ(t)+rlnK,t≠nT,n∈N,Δϖ(t)=ln(1-p),t=nT,n∈N.Solving the first equation of system (3.3) between pulses yields(3.4)ϖ(t)=lnK+(ϖ((n-1)T)-lnK)e-r(t-(n-1)T),(n-1)T<t≤nT.For the second equation of system (3.3), using discrete dynamical system determined by the stroboscopic map yields(3.5)ϖ(nT)=lnK+(ϖ((n-1)T)-lnK)e-rT+ln(1-p)=:f(ϖ(n-1)T),
where f(ϖ)=lnK+(ϖ-lnK)e-rT+ln(1-p). From (3.5), we can see that this difference system has an equilibrium ϖ0*=lnK+ln(1-p)/(1-e-rT), which implies that system (3.3) has a unique T-periodic solution(3.6)ϖ*(t)=lnK+(ϖ0*-lnK)e-r(t-(n-1)T),(n-1)T<t≤nT.Sincex0*=eϖ0*, then x*(t)=eϖ*(t) is the unique positive T-periodic solution of (3.2).In the following, we will prove thatx*(t) is globally asymptotically stable. By Lemma 2.3, we find that any solution of (3.2) is ultimately upper bounded, so we need only to prove that(3.7)limt→∞|ϖ(t)-ϖ*(t)|=0,
where ϖ*(t) is the periodic solution of system (3.3) Suppose that with lnx*(0)=lnx0*=ϖ*(0)=ϖ0*.(3.8)|ϖ(t)-ϖ*(t)|=|ϖ(0)-ϖ0*|e-rt≤e-(n-1)rT,
since limt→∞e-(n-1)rT=0; thus, limt→∞|ϖ(t)-ϖ*(t)|=0.Theorem 3.1.
System (3.2) has unique a positive periodic solution x*(t) which is globally asymptotically stable. That is, system (1.3) has a “predator-extinction” periodic solution (x*(t),0) for t∈(nT,(n+1)T], n∈N.


## 3.2. Global Attractivity of the “Predator-Extinction” Periodic Solution

Denote that(3.9)ℜ1=βK2d[mK2+exp(2ln(1/1-p))/(exp(rT)-1)].Theorem 3.2.
Ifℜ1<1, then the “predator-extinction” periodic solution (x*(t),0) of system (1.3) is globally attractive.Proof.
We denote that(x(t),y(t)) be any solution of system (1.3) with initial condition (1.4). From the second equation of system (1.3), we have that
(3.10)dy(t)dt≤βmy(t-τ)-dy(t).
We consider the following comparison equation:(3.11)dz(t)dt=βmz(t-τ)-dz(t).
If β/m-d<0, then ℜ1<1; according to Lemma 2.4, we obtain limt→∞z(t)=0.
Sincey(s)=z(s)=ϕ2(s)>0 for all s∈[-τ,0], by the comparison theorem in differential equation and the positivity of the solution, we have that y(t)→0 as t→∞.
In the following, we suppose thatβ/m≥d. Since ℜ1<1, we have that
(3.12)βK2exp(2ln(1-p)/(exp(rT)-1))1+mK2exp(2ln(1-p)/(exp(rT)-1))-d<0.
Because the functionx2/(1+mx2) is monotonically increasing with respect to x, we choose a sufficiently small positive constant ϵ such that
(3.13)β[Kexp(ln(1-p)/(exp(rT)-1))+ϵ]21+m[Kexp(ln(1-p)/(exp(rT)-1))+ϵ]2-d<0.
Noting thatdx(t)/dt≤rx(t)ln(K/x(t)),Δx(t)=-px(t) for nT<t≤(n+1)T, then we consider the following comparison system:
(3.14)dz(t)dt=rz(t)lnKz(t),t≠nT,n∈N,z(t+)=(1-p)z(t),t=nT,n∈N,z(0+)=x(0+).
From Section3.1, we see that
(3.15)z*(t)=exp{lnK+ln(1-p)1-e-rTe-r(t-(n-1)T)}(n-1)T<t≤nT,
which is unique a globally asymptotically stable positive T-periodic solution of system (3.14).
There exist a positive integern1 and an arbitrarily small positive constant ϵ for all t≥n1T such that
(3.16)x(t)≤z*(t)+ϵ≤exp{lnK+ln(1-p)1-e-rTe-rT}+ϵ=:σ.
From (3.16) and the second equation of system (1.3), for t>n1T+τ we have
(3.17)dy(t)dt≤βσ21+mσ2y(t-τ)-dy(t).
Considering the following comparison equation:(3.18)dz(t)dt=βσ21+mσ2z(t-τ)-dz(t),
we can see (3.13), that is, βσ2/(1+mσ2)<d; according to Lemma 2.4, we obtain limt→∞z(t)=0.
Sincey(s)=z(s)=ϕ2(s)>0 for all s∈[-τ,0], by the comparison theorem in differential equation and the positivity of the solution, we have that y(t)→0 as t→∞.
Next, we will prove thatx(t)→x*(t) as t→∞. Without loss of generality, we may assume that 0<y(t)≤ϵ for all t≥0. By the first equation of system (1.3), we have that
(3.19)dx(t)dt≥rx(t)lnKx(t)-x2(t)y(t)≥rx(t)lnKx(t)-Kx(t)ϵ≥rx(t)lnKe-Kϵ/rx(t).
Then we have thatz̃1(t)→x*(t) as ϵ→0 as t→∞, where z̃1(t) is unique a positive solution of
(3.20)dz1(t)dt=rz1(t)lnKe-Kϵ/rz1(t),t≠nT,n∈N,z1(t+)=(1-p)z1(t),t=nT,n∈N,z1(0+)=x(0+).
From Section 3.1, we obtain
(3.21)z̃1(t)=exp{ln(Ke-Kϵ/r)+ln(1-p)1-e-rTe-r(t-nT)},nT<t≤(n+1)T.
By using comparison theorem of impulsive differential equation [11], for any ϵ1>0, there exists T1>0 and such that t>T1(3.22)x(t)>z̃1(t)-ϵ1.
On the other hand, from the first equation of (1.3), we have that dx(t)/dt≤rx(t)ln(K/x(t)).
Considering the following comparison system:(3.23)dz2(t)dt=rz2(t)lnKz2(t),t≠nT,z2(t+)=(1-p)z2(t),t=nT,z2(0+)=x(0+),
we have that
(3.24)z̃2(t)=exp{lnK+ln(1-p)1-e-rTe-r(t-nT)},nT<t≤(n+1)T,(3.25)x(t)<z̃2(t)+ϵ1,
as t→∞ and z̃2(t)=x*(t). Let ϵ1→0, then it follows from (3.22) and (3.25), that we can see x*(t)-ϵ1<x(t)<x*(t)+ϵ1 for t large enough, which implies x(t)→x*(t) as t→∞. The proof is completed.Now, we give the following theorem.Theorem 3.3.
(1) If β/m<d, then the “predator-extinction” periodic solution (x*(t),0) is globally attractive.
(2) If β/m≥d and p>p*, T<T*, m>m*, then the “predator-extinction” periodic solution (x*(t),0) is globally attractive, where the critical values p*, T*, m* are listed in Table 1.Table 1
Critical values of some parameters of system (1.3) (R1<1 must be satisfied).
The conditions for global attractivity of(x*(t),0)p>p*p*=1-1/(βK2d-mK2)erT-1T<T*T*=1rln[1-ln(1-p)2ln(βK2/d-mK2)]m>m*m*=βd-1K2exp(-ln(1-p)2erT-1)


## 4. Permanence

In the above section, we have proved that, whenp>p*,  T<T*, or m>m*, the “predator-extinction” periodic solution (x*(t),0) is globally attractive. But in natural world, the predator cannot be eradicated totally. In order to save resources, we need to keep the prey and predator coexisting when the prey does not bring about immense economic losses. Next, we will discuss the permanence of system (1.3).Definition 4.1.
System (1.3) is said to be uniformly persistent if there are positive constants mi(i=1,2) and a finite time T0 such that for all solutions (x(t),y(t)) with initial values x(0+)>0, y(0+)>0, x(t)≥m1, and y(t)≥m2 hold for all t>T0.Definition 4.2.
System (1.3) is said to be permanent if there exists a compact region D⊂intΩ, such that every solution of system (1.3) with initial condition (1.4) will eventually enter and remain in region D.
Denote that(4.1)ℜ2=βK2d[mK2+exp(2ln(1/1-p)/(1-exp(-rT)))].Theorem 4.3.
Ifℜ2>1, then there exist two positive constants m1 and m2 such that x(t)≥m1, y(t)≥m2 for t large enough, that is, system (1.3) is uniformly persistent.Proof.
Suppose that(x(t),y(t)) is any positive solution of system (1.3) with initial condition (1.4). From Lemma 2.3, we know that x(t)≤K, y(t)≤L.
Firstly, from the first equation of system (1.3), we know that dx(t)/dt≥rx(t)ln(Ke-KL/r/x(t)). Now, we consider the following equation:
(4.2)dz3(t)dt=rz3(t)lnKe-KL/rz3(t),t≠nT,n∈N,z3(t+)=(1-p)z3(t),t=nT,n∈N,z3(0+)=x(0+).
therefore we have that(4.3)z̃3(t)=exp{lnKe-KL/r+ln(1-p)1-e-rTe-r(t-nT)}nT<t≤(n+1)T.
There exists a ϵ>0 small enough such that, for sufficiently large t,
(4.4)x(t)>z̃3(t)-ϵ≥exp{ln(Ke-KL/r)+ln(1-p)1-e-rT}-ϵ=:m1.
Secondly, we will findm2 such that y(t)≥m2.
The second equation of system (1.3) may be rewritten as follows:
(4.5)dy(t)dt=(βx2(t)1+mx2(t)-d)y(t)-ddt∫t-τtβx2(θ)1+mx2(θ)y(θ).
Define(4.6)V(t)=y(t)+∫t-τtβx2(θ)1+mx2(θ)y(θ).
Calculating the derivative of V(t) along the solution of (1.3), it follows from (4.5) that
(4.7)dV(t)dt=βx2(t-τ)1+mx2(t-τ)y(t-τ)-dy(t)+βx2(t)1+mx2(t)y(t)-βx2(t-τ)1+mx2(t-τ)y(t-τ)=d(βdx2(t)1+mx2(t)-1)y(t).
Sinceℜ2>1, then we can deduce that
(4.8)r2K[2ln(1-p)1-e-rT+ln(K2(βd-m))]>0.
Hence, we choose two positive constants m2* and ϵ1 small enough such that
(4.9)βdζ21+mζ2>1,
where
(4.10)ζ=Kexp{-Km2*r}·exp{ln(1-p)1-e-rT}-ϵ1>0,0<m2*<r2K[2ln(1-p)1-e-rT+ln(K2(βd-m))].
In the following, we will find m2 such that y(t)≥m2. There are two cases.
(1) We claim that the inequalityy(t)<m2* cannot hold for all t>t1(t1>0); otherwise, there is a positive constant t1 such that y(t)<m2* for all t>t1.
From the first equation of system (1.3), we have that
(4.11)dx(t)dt≥rx(t)lnKe-Km2*/rx(t).
We have that x(t)≥z̃4(t), where z̃4(t) is unique a positive solution of
(4.12)dz4(t)dt=rz4(t)lnKe-Km2*/rz4(t),t≠nT,n∈N,z4(t+)=(1-p)z4(t),t=nT,n∈N,z4(0+)=x(0+).
From Section 3.1, we have that
(4.13)z̃4(t)=exp{lnKe-Km2*/r+ln(1-p)1-e-rTe-r(t-nT)}nT<t≤(n+1)T.
By comparison theory [11], for any ϵ1>0, there exists a T1>0, for t>T1, such that
(4.14)x(t)>z̃4(t)-ϵ1≥exp{lnKe-Km2*/r+ln(1-p)1-e-rT}-ϵ1=:ζ.
From (4.7) and (4.14), we have that
(4.15)dV(t)dt≥d(βdζ2(t)1+mζ2(t)-1)y(t),t≥T1.
Denote that(4.16)yl=mint∈[T1,T1+τ]y(t).
We show thaty(t)≥yl for all t≥T1. Otherwise, there exists a nonnegative constant T2 such that y(t)≥yl, for t∈[T1,T1+τ+T2), y(T1+τ+T2)=yl and ẏ(T1+τ+T2)≤0.
Thus, from the second equation of (1.3) and (4.12), (4.14), we can see that
(4.17)dy(T1+τ+T2)dt>(βζ21+mζ2-d)yl=d(βdζ21+mζ2-1)yl>0,
which is a contradiction to y(t)≤L. Hence we get that y(t)≥yl>0 for all t≥T1. Meanwhile, we can see that dV(t)/dt>0, which implies that V(t)→∞, t→∞.
This is a contradiction toV(t)≤(1+βτK)L. Thus, for any positive constant t1, the inequality y(t)≤m2* cannot hold for all t≥t1.
(2) Ify(t)≥m2* holds for all t large enough, then our aim is obtained. Otherwise, y(t) is oscillatory about m2*. Let m2=min{m2*/2,m2*e-dτ}.
We will prove thaty(t)≥m2. There exist two positive constants t̅, χ such that y(t̅)=y(t̅+χ)=m2* and y(t)<m2* for t∈[t̅,t̅+χ].
Whent̅ is large enough, the inequality x(t)>σ holds true for t∈[t̅,t̅+χ].
Sincey(t) is continuous and bounded and does not have impulsive effort, we conclude that y(t) is uniformly continuous. There exist a constant T3 (0<T3<τ and T3 is independent to t̅) such that y(t)>m2*/2 for all t∈[t̅,t̅+T3].
Ifχ<T3, then our aim is obtained.
IfT3<χ≤τ, from the second equation of (1.3), we have that dy(t)/dt≥-dy(t) for t∈[t̅,t̅+χ], and then we have that y(t)≥m2*e-dτ for t̅<t≤t̅+χ≤t̅+τ. Therefore, y(t)≥m2 for t∈[t̅,t̅+τ].
Ifχ>τ, by the second equation of system (1.3), we have that y(t)≥m2 for t∈[t̅,t̅+τ], the same as above claim; we can obtain y(t)≥m2 for t∈[t̅+τ,t̅+χ].
Since the interval[t̅,t̅+χ] is arbitrarily chosen, we get that y(t)≥m2 for t large enough.
This proof is complete.Theorem 4.4.
Ifℜ2>1, then system (1.3) is permanent.Proof.
Suppose that(x(t),y(t)) is any solution of system (1.3) with initial condition (1.4). By Theorem 4.3, there exist positive constants m1, m2, and T′ such that x(t)≥m1, y(t)≥m2 for t≥T′. Set
(4.18)D={(x,y)∈R+2∣m1≤x(t)≤K,m2≤y(t)≤L}.
ThenD is a bounded compact region and D⊂ int Ω. By Theorem 4.3, every solution of system (1.3) with initial condition (1.4) eventually enters and remains in region D.
The proof is complete.Theorem 4.5.
Ifp<p*, T>T*, or m<m*, then system (1.3) is permanent, where the critical values p*, T*, m* are listed in Table 2.Table 2
Critical values of some parameters of system (1.3) (R2>1 must be satisfied).
The conditions for the permanence of system (1.3)p<p*p*=1-1/(βK2d-mK2)1-e-rTT>T*T*=-1rln[1+ln(1-p)2ln(βK2/d-mK2)]m<m*m*=βd-1K2exp{-ln(1-p)21-e-rT}


## 5. Numerical Simulation and Discussion

In this paper, we introduce and discuss a predator-prey system model with Holling III response functional under time delay on the predator and impulsive perturbations on the prey. From Section3, there exists a predator-extinction periodic solution of system (1.3); when ℜ1<1, the predator eradication periodic solution is globally attractive. From Section 4, when ℜ2>1, system (1.3) is permanent.In the following, we will analyze the influence of them on the dynamics of system (1.3). We consider the hypothetical set of parameter values as r=1.5, K=2, β=0.6, d=0.4, and τ=0.Figure1 is the dynamical behavior of system (1.3) with r=1.5, K=2, β=0.6, d=0.4, τ=0, p=0.5, T=0.45, and ℜ1=0.9658214856<1. (a) is the timeseries of prey population (x) for periodic oscillation; (b) is the timeseries of population (y) for extinction; (c) is the phase portrait of the prey and the predator population for global attractivity of the “predator-eradication” periodic solution.(a)(b)(c)Figure2 is the dynamical behavior of system (1.3) with r=1.5, K=2, β=0.6, d=0.4, τ=0, p=0.2, T=1, and ℜ2=2.083554740>1. (a) is the timeseries of prey population (x) for permanence; (b) is the timeseries of population (y) for permanence; (c) is the phase portrait (T-periodic solution) of the prey and the predator population of system (1.3).(a)(b)(c)By Theorems3.2 and 4.3, we know that, when ℜ1=0.9658214856<1, the “predator-eradication” periodic solution is globally attractive (Figure 1); when ℜ2=2.083554740>1, system (1.3) is permanence (Figures 2(a) and 2(b)).In Figure1(b), we know that the predator population dies ultimately; we know that, although impulsive catching is larger or the period of pulsing is shorter, we kill the prey largely, and the predator population will decrease largely. It is very difficult for it to prey on prey; the predator can die out earlier than the prey. From Figure 2, we suppose that a smaller impulsive catching rate or a longer period can cause the prey and predator populations to coexist.



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*Source: 256195-2010-02-02.xml*