#### 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.

#### 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 , which can be translated into a Gompertz differential equation (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 Section 3, 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:where and denote the immature and mature population densities, respectively, represents the birth rate, is the immature death rate, is the mature death and overcrowding rate, and represents the mean length of the juvenile period. The term represents the immature populations who were born at time and survived at time (with the immature death rate ) and therefore represents the transformation of immature to mature.

Wang et al. [23] considered the following model:where is the density of species in the th patch, is the intrinsic rate of natural increase of population in the th patch, denotes the carrying capacity in the th patch, and is dispersal rate in the th patch. It is assumed here that the net exchange from the th to th patch is proportional to the difference of population densities. The pulse diffusion occurs every period ( is a positive constant). Here , where represents the density of population in the th patch immediately after the th diffusion pulse, and represents the density of population in the th patch before the th diffusion pulse at time . and 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:where is the prey population density in the th patch at time , and are predator populations density with immature and mature in the first patch at time , and are predator populations density with immature and mature in the second patch at time ; and are the Gompertz intrinsic growth rates and the carrying capacity in the th patch, represents the growth rate of immature to mature predators in the th patch; and are the immature and mature predator death rates in the first patch, and are the immature and mature predator death rates in the second patch, is the harvesting effort of the mature population in the th patch. Further, are the constant time to maturity, the conversion rates of predator, and dispersal rates of prey in the th patch. The pulse diffusion occurs every period ( is a positive constant).

Also, ; here { and for all }. Examples of functions found in the biological literature that satisfy are as follows: , with , with , with

Functions and are known as Holling type functional responses. Function is Ivlev type functional responses. Functions and also were regarded as incidence rate function. Function is a double linear incidence rate function. Function is saturated incidence rate function.

We only consider system (3) in the biological meaning region: and assume that solutions of system (3) satisfy the initial conditions:

We can simplify model (3) organized by network and need to restrict our attention to the following model:with the initial condition

#### 3. Preliminaries

The solution of (3) is a piecewise continuous function . Thus, is continuous on , for all and exists. Obviously, the smoothness properties of guarantee the global existence and uniqueness of the solution of (3) (see [30]). We assumed that . If , we can obtain the following subsystem of (5):

For simplicity, let , , , so the system (7) can be written as follows:

Integrating and solving the first two equations of system (8) between pulses, we have the following:

Then, considering the last two equations of system (8), we get the following stroboscopic map of system (8):Here, , , . 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 such that

We see that describes the population densities in time , and the sets of all iterations of the map 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 of , and for every , as .*

That is, the fixed point of is globally stable. The trajectory of system (8) will trend to the positive periodic solution with a period , ,

Then, the trajectory of system (7) will trend to the positive periodic solution with a period , ,

Lemma 2. *There exists a constant such that , , , , , for each solution of (3) with all t large enough.*

*Proof. *First, we define ; then, we have . Then, it is obvious thatThen, we can obtain , when , . So is uniformly ultimately bounded. Hence, by the definition of , there exists a constant such that , , , , , for all *t* large enough. The proof is completed.

Lemma 3. *(see [31]). Consider the following equation with delay:We assumed that , and are positive constants. for , we have the following:* * If , then * * If , then *

#### 4. Extinction of the Predator

From the previous section, we know there exists a predator eradicated periodic solution of system (3). In this section, we will prove that for the network organized system (3) is globally attractive.

Theorem 1. *Ifholds, the predator-extinction periodic solution of system (3) is globally attractive.*

*Proof. *It is obvious from the global attraction of the periodic solution of , system (3) is equivalent to the global attraction of the periodic solution of system (5). From (17), we can choose sufficiently small such thatIt follows from that the first and third equations of system (5) thatSo we consider the following comparison impulsive differential system:By Lemma 1 and (11), we obtain the boundary periodic solution of system (20):which is globally asymptotically stable. In view of the comparison theorem of the impulsive differential equation (see [30]), we have as . Then, there exist with and such thatThat is,From the second and fourth equations of system (5), we have the following:Now, consider the following comparison differential system:From (18), we have and , by Lemma 3, and . By the comparison theorem, we have and . Because of the positivity of and , we know that and . Therefore, for a small , there exists a such that , for all . From system (5), we have the following:Here, and we have . Consider the following comparison differential system:where and , and can be confirmed homoplastically as , . Let use the comparison theorem as ; there for any and large enough, there exists a , such thatThus, from (23) and (28), we get and as . 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. *If and , then there exist two positive constants and such that each positive solution of (5) satisfies and for large enough.*

*Proof. *From the second and fourth equations of system (5), it can be rewritten as follows:We can define and as follows:Calculate the derivative of and along the solution of (5) as follows:By using Lemma 2 and combining with (30), we can obtain and as . Since and , we can find a sufficiently small such that and . We suppose , such that and for all . It follows the first two equations of system (5) that for all ,Here, , we have . For all , consider the following comparison differential system:By Lemma 1, we obtain the following global asymptotically periodic unique positive solution of system (33):where and can be confirmed homoplastically as , . By the comparison theorem for an impulsive differential equation, we know there exists a such that the inequality holds for all and holds for all . Let and , then holds for all and holds for all . Thus, holds for and holds for all . We make a notation as and , for convenience. So we have and . Then from (31), we can obtain the following:Let and ; we show that for all and for all . Suppose the contrary, then there are such that for , and and such that for , , and . Thus, from the third and fourth equations of system (5) imply thatThis is a contradiction. Thus, for all and for all . From (35), we have the following:which implies and as . It is a contradiction to and . Therefore, the claim is complete. By the claim, we are left to consider two cases. First, and for large enough. Second, and oscillate about and for large enough. Definewhere and . In the following, we shall show that and for are large enough; the conclusion is evident in the first case. For the second case, let and satisfy and for all and and for all , where are large enough such that for and for . Thus, and are uniformly continuous. The positive solutions of (3) are ultimately bounded and not affected by impulses. Hence, there are and are dependent on the choice of such that for all and for all . If , , our aim is obtained. Let us consider the case , , since , and , , it is clear that for all and for all . If and ; then we have that for all and for all . The same arguments can be continued and we can obtain for all and and for all . Since the interval and are arbitrarily chosen ( to be enough), we get that and for are large enough. In view of the above discussion, the choice of is independent of the positive solution of system (5), which satisfies and for sufficiently large . The proof is completed.

Theorem 2. *If and , then the system (3) is permanent.*

*Proof. *From system (3) and Lemma 2, we have the following:By the similar argument as those in the proof of Theorem 1, we have that and , and is small enough, where can be confirmed homoplastically to be . Using Lemma 2 and Lemma 4, the second and fifth equations of system (3) become as follows:It is easy to obtain and , is small enough, where