# Dynamical Analysis for the Hybrid Network Model of Delayed Predator-Prey Gompertz Systems with Impulsive Diffusion between Two Patches

Authors: Li Xiang; Yurong Zhang; Dan Zhang; Zhichun Yang; Lingzhi Huang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, we consider a hybrid network model of delayed predator-prey Gompertz systems with impulsive diffusion between two patches, in which the patches represent nodes of the network such that the prey population interacts locally in each patch and diffusion occurs along the edges connecting the nodes. Using the discrete dynamical system determined by the stroboscopic map which has a globally stable positive fixed point, we obtain the global attractive condition of predator-extinction periodic solution for the network system. Furthermore, by employing the theory of delay functional and impulsive differential equation, we obtain sufficient condition with time delay for the permanence of the network.


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## Body

## 1. Introduction

Along with the continuous development of the network science, the mathematical models organized as networks have received considerable attention [1]–[3]. Taking epidemic models for an example, locations such as cities or urban areas can be represented as nodes of a network; individuals can be divided into different states, such as infection, susceptibility, immunity, etc. These individuals interact moving between connecting nodes [2, 3]. Furthermore, in the study of population dynamical systems, due to the universality and importance of the predator-prey relationship, the dynamics of the predator-prey system has been widely concerned. In recent decades, the dynamical behaviors of the predator-prey model defined on the network have enjoyed remarkable progress [4–8]. In [6], each node of the coupled network represents a discrete predator-prey system, and the network dynamics are investigated. In [7], Chang studied instability induced by time delay for a predator-prey model on complex networks and instability conditions were obtained via linear stability analysis of network organized systems.Since the severe competition, natural enemy, or deterioration of the patch environment, the population dispersal phenomena of biological species can often occur between patches. Therefore, the effect of spatial factors in population dynamics becomes a very hot subject [9, 10]. Concerning qualitative analysis for predator-prey models with diffusion, such as local (or global) stability of equilibria and the existence of periodic solutions, many nice results have been obtained (see also, e.g., [11–13]). Regretfully, in all of the above population dispersion systems, dispersal behavior of the populations is occurring at every time. That is, it is a continuous dispersal. In practice, it is often the case that population diffusion occurs in regular pulses. For example, when winter comes, birds will migrate between patches in search for a better environment, whereas they do not diffuse during other seasons. Thus, impulsive diffusion provides a more natural behavior phenomenon. At present, many scholars have applied the theory of impulsive differential equations to population dynamics, and many important studies have been performed [14–19]. Accordingly, it is an interesting subject to analyze the dynamic behaviors of the system by extending the predator-prey model with impulsive diffusion to the network version. In addition, in the 1825s, Benjamin Gompertz established the Gompertz function y=ke−ea−bx, which can be translated into a Gompertz differential equation dy/dx=by lnk/y (see [20, 21]). Compared with the logistic function, it has been proven to be a simple example to generate an asymmetric S-shaped curve [22]. Since then, many models have been established for biological growth by using the Gompertz function (e.g., [23, 24]). Furthermore, many species usually go through two distinct life stages, immature and mature. Considering that the immature becomes the mature need to spend units of time and the number of deaths in the juvenile period, it is essential to consider time-delay in stage-structured model. Many stage-structured predator-prey models with time delay and impulsive diffusive were investigated [25–28]. Liu [25] studied a delayed predator-prey model with impulsive perturbations and gave the predator-extinction periodic solution of the model, which is globally attractive and permanence. Jiao et al. [26] and Dhar and Jatav [27] investigated a delayed predator-prey model with impulsive diffusion and sufficient conditions of the global attractiveness of the predator-extinction periodic solution and the permanence were derived.Motivated by the above discussion, in this paper, we shall organize the patches into networks to investigate a delayed stage-structured functional response predator-prey Gomportz model with impulsive diffusion between two predators territories. We also consider the harvesting effort of the two mature predators. By employing the comparison theorem of impulsive differential equations and the global attractivity of the first order time-delay system, we will obtain some sufficient conditions on the global attractiveness of predator-extinction periodic solution and permanence of our model. The results can provide a reliable strategic basis for the protection of biological resources.The paper is organized as follows. In the next section, introduce model development. In Section3, some useful preliminaries are given. In Section 4, we give the conditions of the global attractivity for our model. In Section 5, we give the conditions of permanence for our model. Finally, discussion is given in Section 6.


## 2. Hybrid Network Model-Organized Predator-Prey System

Aiello and Freedman [29] introduced the following stage-structured single species model:(1)x1′t=αx2t−γ1x1t−αe−γ1τx2t−τ,x2′t=αe−γ1tx2t−τ−γ2x22t,where x1t and x2t denote the immature and mature population densities, respectively, α>0 represents the birth rate, γ1>0 is the immature death rate, γ2>0 is the mature death and overcrowding rate, and τ represents the mean length of the juvenile period. The term αe−γ1τx2t−τ represents the immature populations who were born at time t−τ and survived at time t (with the immature death rate γ1) and therefore represents the transformation of immature to mature.Wang et al. [23] considered the following model:(2)x1′t=r1x1tlnk1x1,x2′t=r2x2tlnk2x2,t≠nτ,Δx1t=d1x2t−x1t,Δx2t=d2x1t−x2t,t=nτ,n∈ℕ,where xii=1,2 is the density of species in the ith patch, rii=1,2 is the intrinsic rate of natural increase of population in the ith patch, kii=1,2 denotes the carrying capacity in the ith patch, and dii=1,2 is dispersal rate in the ith patch. It is assumed here that the net exchange from the jth to ith patch is proportional to the difference xj−xi of population densities. The pulse diffusion occurs every τ period (τ is a positive constant). Here Δxit=xinτ+−xinτ−, where xinτ+ represents the density of population in the ith patch immediately after the nth diffusion pulse, and xinτ− represents the density of population in the ith patch before the nth diffusion pulse at time t=nτ. ri,ki and dii=1,2 are positive constants.According to the model formulation in the literature [15, 23, 25–27], in the following, we shall extend predator-prey model to the network analogue version. Firstly, we propose in this paper a predator-prey model on the network with the following assumptions:(A1)
The patches are created by predator territories and are represented as nodes of the network.(A2)
The prey population in different nodes has different growth rates. The prey population interacts locally in each patch and impulsively diffuses through connected nodes.(A3)
The predator population is divided into immature and mature. Immature becoming mature requires a constant time.(A4)
Mature predator in different nodes has a different conversion rate.(A5)
Immature predators only feed on mature predators and can not reproduce.(A6)
Mature predators in different nodes have different harvest efforts.We formulate the following hybrid network model of delayed predator-prey Gompertz system with impulsive diffusion between two patches:(3)x1′t=r1x1tlnk1x1t−p1x1tz1t,y1′t=α1z1t−α1e−ω1τ1z1t−τ1−ω1y1t,z1′t=α1e−ω1τ1z1t−τ1+β1p1x1tz1t−E1z1t−γ1z12t,x2′t=r2x2tlnk2x2t−p2x2tz2t,y2′t=α2z2t−α2e−γ1τ2z2t−τ2−γ1y2t,z2′t=α2e−ω2τ1z1t−τ1+β2p2x2tz2t−E2z2t−γ2z22t,t≠nτ,Δx1t=d1x2t−x1t,Δx2t=d2x1t−x2t,Δy1t=0,Δy2t=0,Δz1t=0,Δz2t=0,t=nτ,n∈ℕ,where xii=1,2 is the prey population density in the ith patch at time t, y1t and z1t are predator populations density with immature and mature in the first patch at time t, y2t and z2t are predator populations density with immature and mature in the second patch at time t; ri and ki are the Gompertz intrinsic growth rates and the carrying capacity in the ith patch, αi represents the growth rate of immature to mature predators in the ith patch; ω1 and γ1 are the immature and mature predator death rates in the first patch, ω2 and γ2 are the immature and mature predator death rates in the second patch, Eii=1,2 is the harvesting effort of the mature population in the ith patch. Further, τi,βi,di∈0,1 are the constant time to maturity, the conversion rates of predator, and dispersal rates of prey in the ith patch. The pulse diffusion occurs every τ period (τ is a positive constant).Also,pi·∈H; here H={f:R⟶Rf0=0,f′x>0 and f″x≤0 for all x>0}. Examples of functions found in the biological literature that satisfy H are as follows:F1f1x=ax, with a>0F2f2x=ax/1+bx, with a,b>0F3f3x=a1−e−bx, with a,b>0FunctionsF1 and F2 are known as Holling type functional responses. Function F3 is Ivlev type functional responses. Functions F1 and F2 also were regarded as incidence rate function. Function F1 is a double linear incidence rate function. Function F2 is saturated incidence rate function.We only consider system (3) in the biological meaning region: D=x1ty1t,z1t,x1t,y2tz2tx1t≥0,y1t≥0,z1t≥0,x2t≥0,y2t≥0,z2t≥0 and assume that solutions of system (3) satisfy the initial conditions:(4)ϕ1s,ϕ2s,ϕ3s,ϕ4s,ϕ5s,ϕ6s∈C−τ,0,ℝ+6,ϕi0>0,i=1,2,3,4,5,6.We can simplify model (3) organized by network and need to restrict our attention to the following model:(5)x1′t=r1x1tlnk1x1t−p1x1tz1t,z1′t=α1e−ω1τ1z1t−τ1+β1p1x1tz1t−E1z1t−γ1z12t,x2′t=r2x2tlnk2x2t−p2x2tz2t,z2′t=α2e−ω2τ1z1t−τ1+β2p2x2tz2t−E2z2t−γ2z12t,t≠nτ,Δx1t=d1x2t−x1t,Δx2t=d2x1t−x2t,Δz1t=0,Δz2t=0,t=nτ,n∈ℕ,with the initial condition(6)ϕ1s,ϕ3s,ϕ4s,ϕ6s∈C−τ,0,ℝ+4,ϕi0>0,i=1,3,4,6.


## 3. Preliminaries

The solutionxt=x1t,y1t,z1t,x2t,y2t,z2tT of (3) is a piecewise continuous function x:R+⟶R+6. Thus, xt is continuous on nτ,n+1τ, for all n∈Z+ and limt⟶nτ+xt=xnτ+ exists. Obviously, the smoothness properties of x guarantee the global existence and uniqueness of the solution of (3) (see [30]). We assumed that x1t,y1t,z1t,x2t,y2t,z2t≥0. If z1t=0,z2t=0, we can obtain the following subsystem of (5):(7)x1′t=r1x1tlnk1x1t,x2′t=r2x2tlnk2x2t,t≠nτ,Δx1t=d1x2t−x1t,Δx2t=d2x1t−x2t,t=nτ,n∈ℕ.For simplicity, letu1=x1/k1, u2=x2/k2, k=k2/k1, so the system (7) can be written as follows:(8)u1′t=r1u1tln1u1t,u2′t=r2u2tln1u2t,t≠nτ,Δu1t=d1ku2t−u1t,Δu2t=d21ku1t−u2t,t=nτ,n∈ℕ.Integrating and solving the first two equations of system (8) between pulses, we have the following:(9)uit=uinτ+e−rit−nτ,nτ<t≤n+1τ,i=1,2.Then, considering the last two equations of system (8), we get the following stroboscopic map of system (8):(10)u1,n+1τ=u1,nb1+d1ku2,nb2−u1,nb1,u2,n+1τ=u2,nb2+d21ku1,nb1−u2,nb2.Here, ui,n+1τ=uin+1τ+, 0<b1=e−r1τ<1, 0<b2=e−r2τ<1. Equation (10) is a difference equation. It describes the densities of the population in two patches at a pulse in terms of values at the previous pulse, in other words, stroboscopically sampling at its pulsing period. The dynamical behavior of system (10), coupled with (9), determines the dynamical behavior of system (8). To write system (8) as a map, we can define a map F:R+2⟶R+2 such that(11)F1u=u1b1+d1ku2b2−u1b1,F2u=u2b2+d21ku1b1−u2b2.We see thatFinτ describes the population densities in time nτ, and the sets of all iterations of the map F are equivalent to the set of all density sequence generated by system (10). Furthermore, we have the following.Lemma 1.
(see [23]). There exists a unique positive fixed point q=q1,q2 of F, and for every u=u1,u2>0, Fn⟶q as n⟶∞.That is, the fixed pointq=q1,q2 of F is globally stable. The trajectory of system (8) will trend to the positive periodic solution u˜1,u˜2 with a period τ, i.e.,(12)u˜it=qie−rit−nτ,nτ<t≤n+1τ,i=1,2.Then, the trajectory of system (7) will trend to the positive periodic solution x˜1,x˜2 with a period τ, i.e.,(13)x˜it=kiqie−rit−nτ,nτ<t≤n+1τ,i=1,2.Lemma 2.
There exists a constantM>0 such that x1t≤M/β1, y1t≤M, z1t≤M, x2t≤M/β2, y2t≤M, z2t≤M for each solution of (3) with all t large enough.Proof.
First, we defineVt=β1x1t+y1t+z1t+β2x2t+y2t+z2t; then, we have λ0=minω1,r1. Then, it is obvious that(14)V′t=β1r1x1tlnk1x1t+β2r2x2tlnk2x2t+α1−E1z1t+α2−E2z2t−ω1y1t−ω2y2t−γ1z12t−γ2z22t.(15)dVtdt+λ0Vt≤β1r1x1tlnk1eλ0/r1x1t+β2r2x2tlnk2eλ0/r2x2t+α1−E1+λ0z1t−γ1z12t+α2−E2+λ0z2t−γ2z22t≤β1r1k1eλ0/r1+β2r2k2eλ0/r2+α1−E1+λ024γ1+α2−E2+λ024γ2=M0.
Then, we can obtainVt≤M0/λ0+V0+−M0/λ0e−λ0t, when t⟶∞, Vt≤M0/λ0. So Vt is uniformly ultimately bounded. Hence, by the definition of Vt, there exists a constant M=M0/λ0 such that x1t≤M/β1, y1t≤M, z1t≤M, x2t≤M/β2, y2t≤M, z2t≤M for all t large enough. The proof is completed.Lemma 3.
(see [31]). Consider the following equation with delay:(16)x′t=axt−τ−bxt−cx2t.We assumed that a,b,c, and τ are positive constants. xt>0 for t∈−τ,0, we have the following:i If a>b, then limt⟶+∞xt=a−b/cii If a<b, then limt⟶+∞xt=0


## 4. Extinction of the Predator

From the previous section, we know there exists a predator eradicated periodic solutionx˜1,0,0,x˜2,0,0 of system (3). In this section, we will prove that x˜1,0,0,x˜2,0,0 for the network organized system (3) is globally attractive.Theorem 1.
If(17)α1e−ω1τ1+β1p1x1∗<E1,x1∗=k1q1e−r1τ;α2e−ω2τ2+β2p2x2∗<E2,x2∗=k2q2e−r2τholds, the predator-extinction periodic solution x˜1,0,0,x˜2,0,0 of system (3) is globally attractive.Proof.
It is obvious from the global attraction of the periodic solution ofx˜1,0,0,x˜2,0,0, system (3) is equivalent to the global attraction of the periodic solution x˜1,0,x˜2,0 of system (5). From (17), we can choose ε0>0 sufficiently small such that(18)α1e−ω1τ1+β1p1x1∗+ε0<E1,α2e−ω2τ2+β2p2x2∗+ε0<E2.
It follows from that the first and third equations of system (5) that(19)x1′t≤r1x1tlnk1x1t,x2′t≤r2x2tlnk2x2t.
So we consider the following comparison impulsive differential system:(20)v1′t=r1v1tlnk1v1t,v2′t=r2v2tlnk2v2t,t≠nτ,Δv1t=d1v2t−v1t,Δv2t=d2v1t−v2t,t=nτ,n∈ℕ.
By Lemma1 and (11), we obtain the boundary periodic solution of system (20):(21)v˜it=kiqie−rit−nτ,nτ<t≤n+1τ,i=1,2,which is globally asymptotically stable. In view of the comparison theorem of the impulsive differential equation (see [30]), we have xit≤vit⟶xi˜t as t⟶∞. Then, there exist n2 with n2>n1 and t>n2 such that(22)x1t≤v1t<x1˜t+ε0,x2t≤v2t<x2˜t+ε0,nτ<t≤n+1τ,n>n2.
That is,(23)x1t≤x1˜t+ε0≤x1∗+ε0≜δ1,x2t≤x2˜t+ε0≤x2∗+ε0≜δ2,nτ<t≤n+1τ.
From the second and fourth equations of system (5), we have the following:(24)z1′t≤α1e−ω1τ1z1t−τ1−E1−β1p1δ1tz1t−γ1z12t,t>nτ+τ1,n>n2,z2′t≤α2e−ω2τ2z2t−τ2−E2−β2p2δ2tz2t−γ2z12t,t>nτ+τ2,n>n2.
Now, consider the following comparison differential system:(25)m1′t=α1e−ω1τ1m1t−τ1−E1−β1p1δ1tm1t−γ1m12t,t>nτ+τ1,n>n2,m2′t=α2e−ω2τ2m2t−τ1−E2−β2p2δ2tm2t−γ2m12t,t>nτ+τ2,n>n2.From (18), we have α1e−ω1τ1<E1−β1p1x1∗+ε0 and α2e−ω2τ2<E2−β2p2x2∗+ε0, by Lemma 3, limt⟶∞m1t=0 and limt⟶∞m2t=0. By the comparison theorem, we have limt⟶∞z1t≤limt⟶∞m1t=0 and limt⟶∞z2t≤limt⟶∞m2t=0. Because of the positivity of z1t and z2t, we know that limt⟶∞z1t=0 and limt⟶∞z2t=0. Therefore, for a small ε1>0, there exists a n3n3τ>n2τ+τ1,n3τ>n2τ+τ2 such that z1t<ε1, z2t<ε1 for all t>n3τ. From system (5), we have the following:(26)x1′t>r1x1tlnk1e−p1′0ε1/r1x1t,x2′t>r2x2tlnk2e−p1′0ε1/r2x2t,t≠nτ,Δx1t=d1x2t−x1t,Δx2t=d2x1t−x2t,t=nτ,n∈ℕ.
Here,pi•∈H and we have pixit≤pi′0xit. Consider the following comparison differential system:(27)s1′t=r1s1tlnk1e−p1′0ε1/r1s1t,s2′t=r2s2tlnk2e−p1′0ε1/r1s2t,t≠nτ,Δs1t=d1s2t−s1t,Δs2t=d2s1t−s2t,t=nτ,n∈ℕ,where s1˜t=k1e−p1′0ε1/r1q1′e−r1t−nτ and s2˜t=k2e−p2′0ε1/r2q2′e−r2t−nτ, q1′ and q2′ can be confirmed homoplastically as q1, q2. Let ε1⟶0 use the comparison theorem xit≥sit⟶xi˜t as t⟶∞; there for any ε2>0 and t large enough, there exists a n4, n>n4 such that(28)x1t>x1˜t−ε2,x2t>x2˜t−ε2.
Thus, from (23) and (28), we get x1t⟶x1˜t and x2t⟶x2˜t as t⟶∞. The proof is complete.


## 5. Permanence

In this section, we will discuss the permanence of the system (3) organized by the network. To facilitate the discussion, we give the following lemma.Lemma 4.
Ifα1e−ω1τ1+β1p1q1>E1+γ1M and α2e−ω2τ2+β2p2q2>E2+γ2M, then there exist two positive constants g1 and g2 such that each positive solution of (5) satisfies z1t≥g1 and z2t≥g2 for t large enough.Proof.
From the second and fourth equations of system (5), it can be rewritten as follows:(29)z1′t=α1e−ω1τ1+β1P1x1t−E1+γ1z1tz1t−α1e−ω1τ1ddt∫t−τ1tz1udu,z2′t=α2e−ω2τ2+β2P2x2t−E2+γ2z2tz2t−α2e−ω2τ2ddt∫t−τ2tz2udu.
We can defineQ1 and Q2 as follows:(30)Q1t=z1t+α1e−ω1τ1∫t−τ1tz1udu,Q2t=z2t+α2e−ω2τ2∫t−τ2tz2udu.
Calculate the derivative ofQ1 and Q2 along the solution of (5) as follows:(31)Q1′t=α1e−ω1τ1+β1P1x1t−E1+γ1z1tz1t,Q2′t=α2e−ω2τ2+β2P2x2t−E2+γ2z2tz2t.
By using Lemma2 and combining with (30), we can obtain Q1t≤M1+α1τ1e−ω1τ1 and Q2t≤M1+α2τ2e−ω2τ2 as t⟶∞. Since α1e−ω1τ1+β1p1q1>E1+γ1M and α2e−ω2τ2+β2p2q2>E2+γ2M, we can find a sufficiently small ε>0 such that α1e−ω1τ1+β1p1q1−ε>E1+γ1M and α2e−ω2τ2+β2p2q2−ε>E2+γ2M. We suppose t0>0, such that z1t<z1∗ and z2t<z2∗ for all t>t0. It follows the first two equations of system (5) that for all t>t0,(32)x1′t>r1x1tlnk1e−p1′0z1∗/r1x1t,x2′t>r2x2tlnk2e−p1′0z2∗/r2x2t.
Here,pi•∈H, we have pixit≤pi′0xit. For all t>t0, consider the following comparison differential system:(33)l1′t=r1l1tlnk1ep1′0z2∗/r1l1tl2′t=r2l2tlnk2e−p2′0z2∗/r2l2t,t≠nτ,Δl1t=d1l2t−l1t,Δl2t=d2l1t−l2t,t=nτ,n∈ℕ.
By Lemma1, we obtain the following global asymptotically periodic unique positive solution of system (33):(34)l1˜t=k1e−p1′0z1∗/r1q1″e−r1t−nτ,l2˜t=k2e−p2′0z2∗/r2q2″e−r2t−nτ,where q1″ and q2″ can be confirmed homoplastically as q1, q2. By the comparison theorem for an impulsive differential equation, we know there exists a t1,t2t1>t0+τ1,t2>t0+τ2 such that the inequality x1t>l1t>l1˜t−ε holds for all t≥t1 and x2t>l2t>l2˜t−ε holds for all t≥t2. Let z1∗⟶0 and z2∗⟶0, then x1t>x1˜t−ε holds for all t≥t1 and x2t>x2˜t−ε holds for all t≥t2. Thus, x1t>q1−ε holds for t≥t1 and x2t>q2−ε holds for all t≥t2. We make a notation as σ1=q1−ε and σ2=q2−ε, for convenience. So we have α1e−ω1τ1+β1p1σ1>E1+γ1M and α2e−ω2τ2+β2p2σ2>E2+γ2M. Then from (31), we can obtain the following:(35)Q1′t>α1e−ω1τ1+β1P1x1t−E1+γ1z1tz1t,t≥t1,Q2′t>α2e−ω2τ2+β2P2x2t−E2+γ2z2tz2t,t≥t2.
Letz1¯=mint∈t1,t1+τ1z1t and z2¯=mint∈t2,t2+τ2z2t; we show that z1t≥z1¯ for all t≥t1 and z2t≥z2¯ for all t≥t2. Suppose the contrary, then there are T0,T1>0 such that z1t≥z1¯ for t∈t1,t1+τ1+T0, z1t1,t1+τ1+T0=z1¯ and z1′t1,t1+τ1+T0≤0 and such that z2t≥z2¯ for t∈t2,t2+τ2+T1, z2t2,t2+τ2+T1=z1¯, and z2′t2,t2+τ2+T1≤0. Thus, from the third and fourth equations of system (5) imply that(36)z1′t1+τ1+T0=α1e−ω1τ1z1t1+T0+β1p1x1t1+τ1+T0z1t1+τ1+T0−E1z1t1+τ1+T0−γ1z12t1+τ1+T0≥α1e−ω1τ1+β1P1σ1−E1+γ1Mz1¯>0,z2′t2+τ2+T1=α2e−ω2τ2z2t2+T1+β2p2x2t2+τ2+T1z2t2+τ2+T1−E2z2t2+τ2+T1−γ2z22t2+τ2+T1≥α2e−ω2τ2+β2P2σ2−E2+γ2Mz2¯>0.
This is a contradiction. Thus,z1t≥z1¯ for all t≥t1 and z2t≥z2¯ for all t≥t2. From (35), we have the following:(37)Q1′t≥α1e−ω1τ1+β1P1σ1−E1+γ1Mz1¯>0,t≥t1,Q2′t≥α2e−ω2τ2+β2P2σ2−E2+γ2Mz2¯>0,t≥t2,which implies Q1t⟶∞ and Q2t⟶∞ as t⟶∞. It is a contradiction to Q1t≤M1+α1τ1e−ω1τ1 and Q2t≤M1+α2τ2e−ω2τ2. Therefore, the claim is complete. By the claim, we are left to consider two cases. First, z1t≥z1∗ and z2t≥z2∗ for t large enough. Second, z1t and z2t oscillate about z1∗ and z1∗ for t large enough. Define(38)g1=minz1∗2,g1∗,g2=minz2∗2,g2∗,where g1∗=z1∗e−E1+γ1Mτ1 and g2∗=z2∗e−E2+γ2Mτ2. In the following, we shall show that z1t≥g1 and z2t≥g2 for t are large enough; the conclusion is evident in the first case. For the second case, let t1∗,t2∗>0 and ζ1,ζ2>0 satisfy z1t1∗=z1t1∗+ζ1=z1∗ and z1t<z1∗ for all t∈t1∗,t1∗+ζ1 and z2t2∗=z2t2∗+ζ2=z2∗ and z2t<z2∗ for all t∈t2∗,t2∗+ζ2, where t1∗,t2∗ are large enough such that x1t>σ1 for t1∗<t<t1∗+ζ1 and x2t>σ2 for t2∗<t<t2∗+ζ2. Thus, z1t and z2t are uniformly continuous. The positive solutions of (3) are ultimately bounded and not affected by impulses. Hence, there are T1′,T2′0<T1′<τ1,0<T2′<τ2 and T1′,T2′ are dependent on the choice of t1∗,t2∗ such that z1t>z1∗/2 for all t∈t1∗,t1∗+T1′ and z2t>z2∗/2 for all t∈t2∗,t2∗+T2′. If ζ1<T1′, ζ2<T2′, our aim is obtained. Let us consider the case T1′<ζ1≤τ1, T2′<ζ2≤τ2, since z1′t>−E1+γ1Mz1t, z1t1∗=z1∗ and z2′t>−E2+γ2Mz2t, z2t1∗=z2∗, it is clear that z1t≥g1 for all t∈t1∗,t1∗+τ1 and z2t≥g2 for all t∈t2∗,t2∗+τ2. If ζ1>τ1 and ζ2>τ2; then we have that z1t≥g1 for all t∈t1∗,t1∗+τ1 and z2t≥g2 for all t∈t2∗,t2∗+τ2. The same arguments can be continued and we can obtain z1t≥g1 for all t∈t1∗+τ1,t1∗+ζ1 and z2t≥g2 and for all t∈t2∗+τ2,t2∗+ζ2. Since the interval t1∗,t1∗+ζ1 and t2∗,t2∗+ζ2 are arbitrarily chosen (t1∗,t2∗ to be enough), we get that z1t≥g1 and z2t≥g2 for t are large enough. In view of the above discussion, the choice of g1,g2 is independent of the positive solution of system (5), which satisfies z1t≥g1 and z2t≥g2 for sufficiently large t. The proof is completed.Theorem 2.
Ifα1e−ω1τ1+β1p1q1>E1+γ1M and α2e−ω2τ2+β2p2q2>E2+γ2M, then the system (3) is permanent.Proof.
From system (3) and Lemma 2, we have the following:(39)x1′t≥r1x1tlnk1e−p1′0M/nx1t,x2′t≥r2x2tlnk2e−p2′0M/r2x2t,t≠nτ,Δx1t=d1x2t−x1t,Δx2t=d2x1t−x2t,t=nτ,n∈ℕ.By the similar argument as those in the proof of Theorem 1, we have that x1t≥f1−ε and x2t≥f2−ε, and ε>0 is small enough, where(40)fi=kie−pi′0M/riq¯ie−riτ,i=1,2.q¯1andq¯2 can be confirmed homoplastically to be q1,q2. Using Lemma 2 and Lemma 4, the second and fifth equations of system (3) become as follows:(41)y1′t≥α1g1−Me−ω1τ1−ω1y1t,y2′t≥α2g2−Me−ω2τ2−ω2y2t.
It is easy to obtainy1t≥ρ1−ε and y2t≥ρ2−ε, ε>0 is small enough, where ρ1=α1g1−Me−ω1τ1/ω1−e−ω1τ1/ω1 and ρ2=α2g2−Me−ω2τ2/ω2−e−ω2τ2/ω2. Hence, by Lemma 2, Lemma 4, and the above discussion, we obtain that system (3) is permanent. The proof is completed.


## 6. Illustrative Example and Discussion

Example 1.
Consider the hybrid network model (3) of delayed predator-prey Gompertz system with impulsive diffusion between two patches, in which p1x1=1.5x1,p2x2=5x2,τ=0.5, and(42)r1=1.1,k1=0.2,α1=0.4,ω1=0.5,β1=0.3,γ1=0.2,τ1=0.8,d1=0.35;r2=1.2,k2=0.6,α2=0.5,ω2=0.4,β2=0.2,γ2=0.2,τ2=0.8,d2=0.3.
From (11), we compute the fixed point q=1.7659,0.7553.Case 1.
E1=0.6andE2=1. In Theorem 1, we have x1∗,x2∗=0.2199,0.5752 and(43)α1e−ω1τ1+β1p1x1∗−E1=−0.2329<0;α2e−ω2τ2+β2p2x2∗−E2=−0.0618<0.
It follows from Theorem1 that the predator-extinction periodic solution x˜1,0,0,x˜2,0,0 of the system (3) is globally attractive. Figure 1 shows the attractivity of the solution with the initial condition x1s=0.01,x2s=0.02,ands∈−0.8,0.Figure 1
The time response in Case1.
(a)(b)(c)(d)(e)(f)Case 2.
E1=0.01andE2=0.02. In Theorem 2, we have M=5.0261 and(44)α1e−ω1τ1+β1p1q1−E1−γ1M=0.0476>0;α2e−ω2τ2+β2p2q2−E2−γ2M=0.0931>0.
It follows from Theorem2 that system (3) is permanent. Figure 2 shows the permanent of the solution with the above initial condition.
In this paper, a delayed functional response predator-prey Gompertz system with impulsive diffusion between two patches defined on the network was investigated. The patches represent nodes of a network such that the prey population interacts locally in each patch and occurs diffusively over links connecting nodes. By extending system (3) to the network version, we analyzed that the predator-extinction solution of system (3) is globally attractive and obtained the permanence condition of system (3). We also observed that constant time delay and the growth rate of the immature predator can bring obvious effects on the dynamics of the system, and the stability and extinction (or prey and predators coexist) of the system are determined by their thresholds. Thus, from Theorems 1 and 2, we can easily guess that there must exist thresholds τ1∗andτ2∗. If τ1>τ1∗ and τ2>τ2∗, then the immature and mature predator tends to be extinct; if τ1<τ1∗ and τ2<τ2∗, then the system will be permanent. If immature predator growth rates are below (or above) their thresholds, then the predators become extinct (or prey and predators coexist) in all patches. Hence, the immature predator growth rates and the predator’s maturation times play an important role in delayed functional response predator-prey Gompertz systems with impulsive diffusion between the two patches. We hope that the results will provide a reliable tactic basis for biological resource protection. In addition, it is meaningful to generalize the results to the case of n-patch case, but it is difficult to discuss the properties of the n-dimension stroboscopic map. One may instigate this problem in the future.Figure 2
The time response in Case2.
(a)(b)(c)(d)(e)(f)



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