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

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.


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. ese 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][5][6][7][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. erefore, 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][12][13]). Regretfully, in all of the above population dispersion systems, dispersal behavior of the populations is occurring at every time. at 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. us, 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][15][16][17][18][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 − e a− bx , which can be translated into a Gompertz differential equation dy/dx � by ln(k/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][26][27][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 predatorextinction periodic solution and permanence of our model. e results can provide a reliable strategic basis for the protection of biological resources. e 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.

Hybrid Network Model-Organized Predator-Prey System
Aiello and Freedman [29] introduced the following stagestructured single species model: where x 1 (t) and x 2 (t) denote the immature and mature population densities, respectively, α > 0 represents the birth rate, c 1 > 0 is the immature death rate, c 2 > 0 is the mature death and overcrowding rate, and τ represents the mean length of the juvenile period. e term αe − c 1 τ x 2 (t − τ) represents the immature populations who were born at time t − τ and survived at time t (with the immature death rate c 1 ) and therefore represents the transformation of immature to mature.
Wang et al. [23] considered the following model: where x i (i � 1, 2) is the density of species in the ith patch, r i (i � 1, 2) is the intrinsic rate of natural increase of population in the ith patch, k i (i � 1, 2) denotes the carrying capacity in the ith patch, and d i (i � 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 x j − x i of population densities. e pulse diffusion occurs every τ period (τ is a positive constant). Here represents the density of population in the ith patch immediately after the nth diffusion pulse, and x i (nτ − ) represents the density of population in the ith patch before the nth diffusion pulse at time t � nτ. r i , k i and d i (i � 1, 2) are positive constants. According to the model formulation in the literature [15,23,[25][26][27], in the following, we shall extend predatorprey model to the network analogue version. Firstly, we propose in this paper a predator-prey model on the network with the following assumptions: We formulate the following hybrid network model of delayed predator-prey Gompertz system with impulsive diffusion between two patches: 2 Discrete Dynamics in Nature and Society where x i (i � 1, 2) is the prey population density in the ith patch at time t, y 1 (t) and z 1 (t) are predator populations density with immature and mature in the first patch at time t, y 2 (t) and z 2 (t) are predator populations density with immature and mature in the second patch at time t; r i and k i are the Gompertz intrinsic growth rates and the carrying capacity in the ith patch, α i represents the growth rate of immature to mature predators in the ith patch; ω 1 and c 1 are the immature and mature predator death rates in the first patch, ω 2 and c 2 are the immature and mature predator death rates in the second patch, E i (i � 1, 2) is the harvesting effort of the mature population in the ith patch. Further, τ i , β i , d i ∈ (0, 1) are the constant time to maturity, the conversion rates of predator, and dispersal rates of prey in the ith patch. e pulse diffusion occurs every τ period (τ is a positive constant). Also, Examples of functions found in the biological literature that satisfy H are as follows: Functions (F1) 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: 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

Preliminaries
e solution x(t) � (x 1 (t), y 1 (t), z 1 (t), x 2 (t), y 2 (t), z 2 (t)) T of (3) is a piecewise continuous function x: R + ⟶ R 6 + . us, x(t) is continuous on (nτ, (n + 1)τ], for all n ∈ Z + and lim t⟶nτ + x(t) � x(nτ + ) exists. Obviously, the smoothness properties of x guarantee the global existence and uniqueness of the solution of (3) (see [30]). We assumed that we can obtain the following subsystem of (5): For simplicity, let u 1 � x 1 /k 1 , u 2 � x 2 /k 2 , k � k 2 /k 1 , 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: en, considering the last two equations of system (8), we get the following stroboscopic map of system (8): Here, (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. e 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 4 Discrete Dynamics in Nature and Society We see that F i (nτ) 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.

Extinction of the Predator
From the previous section, we know there exists a predator eradicated periodic solution (x 1 , 0, 0, x 2 , 0, 0) of system (3).
Proof. It is obvious from the global attraction of the periodic solution of (x 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 ε 0 > 0 sufficiently small such that It follows from that the first and third equations of system (5) that .

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.
In this paper, a delayed functional response predatorprey Gompertz system with impulsive diffusion between two patches defined on the network was investigated. e 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 predatorextinction 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. us, from eorems 1 and 2, we can easily guess that there must exist thresholds τ * 1 and τ * 2 . If τ 1 > τ * 1 and τ 2 > τ * 2 , then the immature and mature predator tends to be extinct; if τ 1 < τ * 1 and τ 2 < τ * 2 , 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.

Data Availability
No data or codes were generated or used during this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.