Survival Analysis of a Predator–Prey Model with Seasonal Migration of Prey Populations between Breeding and Non-Breeding Regions

Abstract

In this paper, we establish and study a novel predator–prey model that incorporates: (i) the migration of prey between breeding and non-breeding regions; (ii) the refuge effect of prey; and (iii) the reduction in prey pulse birth rate, in the form of a fear effect, in the presence of predators. Applying the Floquet theory and the comparison theorem of impulsive differential equations, we obtain the sufficient conditions for the stability of the prey-extinction periodic solution and the permanence of the system. Furthermore, we also study the case where the prey population does not migrate. Sufficient conditions for the stability of the prey-extinction periodic solution and the permanence are also established, and the threshold for extinction and permanence of the prey population is obtained. Finally, some numerical simulations are provided to verify the theoretical results. These results provide a theoretical foundation for the conservation of biodiversity.

1. Introduction

Migration, the seasonal round-trip movement of organisms between two (or more) locations, is ubiquitous. Migratory species are found over the earth, in terrestrial, aquatic, and aerial environments, including most vertebrates (fish, birds, mammals, amphibians, reptiles) and many invertebrate lineages [1]. In addition, many species closely related to humans are also migratory species, such as insect crop pests (Holland et al. [2]), fishery species (Parrish [3]), and ungulates competing with domestic livestock (Talbot [4]). To understand the biodynamics in population ecological management, the migration or diffusion of populations between patches has attracted the attention of many scholars, and proposed many bio-mathematical models. For example, Allen [5], Cohen and Murray [6], Zeng et al. [7], Cui et al. [8], Yang et al. [9], Takigawa [10], Freedman [11], Yan and Gao [12], and Zou and Wang [13] established and studied continuous diffusion population models using ordinary differential equations. When the individual number of species in two patches is not large enough, demographic fluctuations are often expected in the models, as stated in [14]. Studies of population models with random fluctuations and migrations between patches have attracted increasing attention in recent contributions. Zu et al. [15] discussed a single-species nonlinear diffusion system with stochastic perturbation. Zou et al. [16,17,18] studied a single-species population model with a protection zone, and they showed that protected areas can slow down the extinction speed of endangered species. Later, Wei and Wang [19] modified the model [16] and provided sustainable strategies for local managers to avoid the extinction of endangered species. Considering both white noise and Levy jump noise, Liu et al. [20] proposed and studied a logistic population model with migration. Some related studies [21,22] gave us a deeper understanding of migration patterns. Furthermore, many scholars use fractional order differential equations to establish corresponding mathematical models, which further promotes the development of population models [23,24].

In all of the above population models, the authors have always assumed that population migration occurs at any time and between any two patches simultaneously. However, the migration behavior of many species is affected by the environment, animal physiological behavior, and human activities, such as the seasonal migrations of some birds, insects, and fish [1,2]. Therefore, many scholars describe this migration phenomenon in the form of pulses in the process of modeling [25,26,27,28,29,30,31,32,33,34,35]. Global stability and persistence for impulsive migration models were studied by Huang and Yang [25], Wang et al. [26], Jiao et al. [27,28], Liu et al. [29], Shao [30], Liu and Yang [31], Liu et al. [32], Zhang et al. [33], and Wan and Jiang [34].

However, only a few studies have considered this case of periodic migration of the entire population between the patches, especially the predator–prey model with seasonal migration between breeding and non-breeding regions. We all know that mass migration of many animals can be seasonal as it is closely related to reproduction. For example, $Caribou$ is the species with the longest migration distance among land mammals. They migrate to the same location in the desolate tundra to breed each summer, and then trek south to the forest for winter [36]. $Magellanicpenguin$ is also a common seasonal migratory animal, they leave the breeding area from March to April per year, and return to the location from August to September [36,37]. Moreover, many birds (such as $Snowgeese$, $Arctictern$, $Swan$, and $Ciconiaboyciana$) breed at high latitudes in summer and fly to warm areas at low latitudes in winter [36,38]. In addition, synchronous reproduction is also a common natural phenomenon for many seasonal migrating animals (such as $Mongolian$$gzaelle$, $Wildebeest$ and $Snowgeese$), which give birth to their cubs at almost the same time when they arrive at the breeding place. A large number of dates show that almost all pregnant female $Caribou$ will give birth between 1 and 10 June [36]. Although, migratory animals face dangers during migration, and nature still ensures that migratory animals and migration cycles exist. However, human activities can disrupt this biological status rapidly. To better protect wild animals, many countries have established numerous nature reserves to provide shelters for endangered species. It is practically significant to study the role of nature reserves from the perspective of a mathematical model.

In view of the above things, we establish and study a novel predator–prey model that incorporates: (i) the migration of prey between breeding and non-breeding regions; (ii) the refuge effect of prey; and (iii) the reduction in prey pulse birth rate, in the form of a fear effect, in the presence of predators. The organization of this article is presented below. In Section 2, we formulate our mathematical model. In Section 3, we introduce some lemmas to discuss our main results. In Section 4, we analyze the stability of the prey-extinction periodic solution and the condition for the coexistence of prey and predator, respectively. Numerical simulation and discussions are provided in Section 5. Finally, the results are summarized in Section 6.

2. Model Formulation

Let $x\left(t\right)$ be the density of prey at time t. Let $y\left(t\right)$ be the density of predator at time t. In this paper, we assume that prey populations migrate seasonally between the non-breeding region (Patch 1) and the breeding region (Patch 2), while predator populations always live in Patch 2. Prey x lives in Patch 2 in $(n\gamma ,(n+l_1)\gamma ]$, when the environment in Patch 2 changes, the prey x will migrate from Patch 2 to Patch 1 in $((n+l_1)\gamma ,(n+l_1+l_2)\gamma ]$, when the environment in Patch 1 changes, the prey x will migrate from Patch 1 to Patch 2 in $((n+l_1+l_2+l_3)\gamma ,(n+1)\gamma ]$, where $\gamma$ denotes a period of seasonal succession. $0<l_i<1$, $i=1,2,3$ and $l_1+l_3+2l_2=1$. Figure 1 displays a diagram representing the migration process within a time-cycle.

Additional model assumptions are as follows:

(1) Prey x can be preyed upon by natural enemies and suffer unexpected losses during migration. An enormous migration team establishes a group defense effect against natural enemies. Thus, we model the mortality rate in the migration process by non-constant function ${\mu }_i\frac{1+\theta ex}{1+ex}$, where ${\mu }_i>0$ ($i=1,2$) denotes the maximal mortality, $e>0$ represents the coefficient of semi-saturation, $\theta \in (0,1)$ is the measure of the loss contrast between group and individual migrations.

(2) We assume that the prey population x has the habit of synchronous reproduction, i.e., cubs are born immediately when the prey x arrives at the breeding sites. The birth of cubs is modeled by $bx\left(t\right)$, where b is the intrinsic growth rate.

(3) To protect prey populations, we provide a refuge for prey populations and assume that prey in the refuge have an increased birth rate $\eta >1$. In addition, some experiments have shown that fear of predation can significantly reduce the reproduction of prey [39,40,41]. Motivated by the insightful works [39,40,41], we model the impulse birth of prey x by

$$x\left(t^{+}\right)=x\left(t\right)+\frac{(1-m)bx\left(t\right)}{1+cy\left(t\right)}+\eta bmx\left(t\right),t=(n+1)\gamma ,$$

where $0<m<1$ is the proportion of prey in refuge. $\frac{1}{1+cy}$ represents the cost of anti-predator defense owing to predation fear [40,41], $c>0$ refers to the level of fear. $x\left(t^{+}\right)=\underset{h\to 0^{+}}{\lim }x(t+h)$.

According to the above assumptions, we propose the following predator–prey model with seasonal migration of the prey population between breeding and non-breeding regions:

$$\left\{\begin{array}{c}\frac{dx\left(t\right)}{dt}=\left\{\begin{array}{c}-x\left(t\right)(d_1+a_{1}x\left(t\right)+\beta (1-m)y\left(t\right)),t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ -{\mu }_1\frac{1+\theta ex\left(t\right)}{1+ex\left(t\right)}x\left(t\right),t\in ((n+l_1)\gamma ,(n+l_1+l_2)\gamma ],\hfill \\ -x\left(t\right)(d_2+a_{2}x\left(t\right)),t\in ((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ],\hfill \\ -{\mu }_2\frac{1+\theta ex\left(t\right)}{1+ex\left(t\right)}x\left(t\right),t\in ((n+l_1+l_2+l_3)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.\hfill \\ \frac{dy\left(t\right)}{dt}=\left\{\begin{array}{c}y\left(t\right)(r_1-b_{1}y\left(t\right)+\kappa \beta (1-m)x\left(t\right)),t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ y\left(t\right)(r_2-b_{2}y\left(t\right)),t\in ((n+l_1)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.\hfill \\ x\left(t^{+}\right)=x\left(t\right)+\frac{(1-m)bx\left(t\right)}{1+cy\left(t\right)}+\eta bmx\left(t\right),t=(n+1)\gamma ,\hfill \end{array}\right.$$

(1)

where $r_1$ and $r_2$ are the growth rates of predator y in the interval $(n\gamma ,(n+l_1)\gamma ]$ and $((n+l_1)\gamma ,(n+1)\gamma ]$, respectively. $a_1$ and $a_2$ represent the intra-specific competition coefficients of population x in the interval $(n\gamma ,(n+l_1)\gamma ]$ and $((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ]$, respectively. $b_1$ and $b_2$ denote the intra-specific competition coefficients of population y in the interval $(n\gamma ,(n+l_1)\gamma ]$ and $((n+l_1)\gamma ,(n+1)\gamma ]$, respectively. $d_1$ and $d_2$ denote the death rates of prey x in the interval $(n\gamma ,(n+l_1)\gamma ]$ and $((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ]$, respectively. $\beta$ denotes the predation rate. $\kappa$ denotes the rate of conversion of nutrients into the reproduction rate of predator.

3. Preliminaries

Let $\psi \left(t\right)=\left(x\right(t),y(t\left)\right)$ be the solution of Equation (1). Obviously, $\psi :R_{+}\to R_{+}^2$ is a piecewise continuous function where $R_{+}=\{x:x\ge 0\}$ and $R_{+}^2=\{(x,y)\in R^2:x\ge 0,y\ge 0\}$. In light of [42], the uniqueness and global existence of solution of Equation (1) is guaranteed by the smoothness properties of $g={(g_1,g_2)}^T$, which represents the mapping defined by the right-hand side of Equation (1).

If $x=0$, we can obtain a prey-free subsystem as follows:

$$\frac{dy\left(t\right)}{dt}=\left\{\begin{array}{c}\begin{array}{c}y\left(t\right)[r_1-b_{1}y\left(t\right)],\hfill \end{array}t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ \begin{array}{c}y\left(t\right)[r_2-b_{2}y\left(t\right)],\hfill \end{array}t\in ((n+l_1)\gamma ,(n+1)\gamma ].\hfill \end{array}\right.$$

(2)

The analytic solution of system (2) is given by

$$y\left(t\right)=\left\{\begin{array}{c}\frac{r_{1}y\left(n\gamma \right)e^{r_1(t-n\gamma )}}{r_1+b_{1}y\left(n\gamma \right)(e^{r_1(t-n\gamma )}-1)},t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ \frac{r_{2}y\left((n+l_1)\gamma \right)e^{r_2(t-(n+l_1)\gamma )}}{r_2+b_{2}y\left((n+l_1)\gamma \right)(e^{r_2(t-(n+l_1)\gamma )}-1)},t\in ((n+l_1)\gamma ,(n+1)\gamma ].\hfill \end{array}\right.$$

(3)

Then, the stroboscopic map of (3) is

$$y\left((n+1)\gamma \right)=\frac{y\left(n\gamma \right)}{e^{-(r_1{l}_1+r_2(1-l_1))\gamma }+Ay\left(n\gamma \right)},$$

where $A=\frac{b_1}{r_1}{e}^{-r_2(1-l_1)\gamma }(1-e^{-r_1{l}_1\gamma })+\frac{b_2}{r_2}(1-e^{-r_2(1-l_1)\gamma }).$ Denote $y_n=y\left(n\gamma \right)$, then the difference equation can be derived as follows:

$$y_{n+1}=\frac{y_n}{e^{-(r_1{l}_1+r_2(1-l_1))\gamma }+A{y}_n}=h\left(y_n\right).$$

(4)

Clearly, Equation (4) has two fixed points $y^0=0$ and $y^{*}=\frac{1-e^{-(r_1{l}_1+r_2(1-l_1))\gamma }}{A}$.

Lemma 1. 

The trivial fixed point of Equation (4) is unstable, and the positive fixed point $y^{*}$ is globally asymptotically stable.

Proof. 

From (4), we obtain

$${\left|\frac{\partial h\left(y\right)}{\partial y}\right|}_{y=0}=e^{(r_1{l}_1+r_2(1-l_1))\gamma }>1,{\left|\frac{\partial h\left(y\right)}{\partial y}\right|}_{y=y^{*}}=e^{-(r_1{l}_1+r_2(1-l_1))\gamma }<1,$$

thus, the trivial fixed point is unstable and the positive fixed point $y^{*}$ of (4) is locally asymptotically stable. And because

$$\left|\frac{1}{y_n}-\frac{1}{y^{*}}\right|=\left|\frac{1}{y_{n-1}}-\frac{1}{y^{*}}\right|e^{-(r_1(l_1+2l_2)+r_2{l}_3)\gamma }\to 0asn\to +\infty ,$$

and $y_n$ is bounded, then $|y_n-y^{*}|\to 0$ as $n\to +\infty$, which implies that the fixed point $y^{*}$ is globally asymptotically stable. □

Similar to [43], we can get the following lemma.

Lemma 2. 

System (3) has a unique positive γ-periodic solution $\overline{y\left(t\right)}$, and the periodic solution $\overline{y\left(t\right)}$ is globally asymptotically stable, where

$$\overline{y\left(t\right)}=\left\{\begin{array}{c}\frac{r_1{y}^{*}{e}^{r_1(t-n\gamma )}}{r_1+b_1{y}^{*}(e^{r_1(t-n\gamma )}-1)},t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ \frac{r_2{y}_1^{*}{e}^{r_2(t-(n+l_1+l_2)\gamma )}}{r_2+b_2{y}_1^{*}(e^{r_2(t-(n+l_1+l_2)\gamma )}-1)},t\in ((n+l_1)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.$$

and

$$\overline{y\left(0^{+}\right)}=y^{*},y_1^{*}=\frac{r_1{y}^{*}{e}^{r_1{l}_1\gamma }}{r_1+b_1{y}^{*}(e^{r_1{l}_1\gamma }-1)},A=\frac{b_1}{r_1}{e}^{-r_2(1-l_1)\gamma }(1-e^{-r_1{l}_1\gamma })+\frac{b_2}{r_2}(1-e^{-r_2(1-l_1)\gamma }).$$

Next, we consider the following auxiliary system:

$$\left\{\begin{array}{c}\frac{d\omega \left(t\right)}{dt}=\left\{\begin{array}{c}-d_1\omega \left(t\right)-a_1{\omega }^2\left(t\right),t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ -\theta {\mu }_1\omega \left(t\right),t\in ((n+l_1)\gamma ,(n+l_1+l_2)\gamma ],\hfill \\ -d_2\omega \left(t\right)-a_2{\omega }^2\left(t\right),t\in ((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ],\hfill \\ -\theta {\mu }_2\omega \left(t\right),t\in ((n+l_1+l_2+l_3)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.\hfill \\ \omega \left(t^{+}\right)=[1+((1-m)+\eta m)b]\omega \left(t\right),t=(n+1)\gamma ,n\in Z_{+}.\hfill \end{array}\right.$$

(5)

$\mathbf{Assumption}$: $h_0=(1+((1-m)+\eta m)b)e^{-(d_1{l}_1+\theta {\mu }_1{l}_2+d_2{l}_3+\theta {\mu }_2{l}_2)\gamma }$.

Lemma 3. 

(i) If $h_0\le 1$, we have $\underset{t\to +\infty }{\lim }\omega \left(t\right)=0$.

(ii) If $h_0>1$, system (5) has a unique globally asymptotically stable positive periodic solution $\overline{\omega \left(t\right)}$, where

$$\overline{\omega \left(t\right)}=\left\{\begin{array}{cc}& \frac{d_2{\omega }^{*}{e}^{-d_1(t-n\gamma )}}{d_1+a_1{\omega }^{*}(1-e^{-d_1(t-n\gamma )})},t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ & {\omega }_1^{*}{e}^{-{\mu }_1\theta (t-(n+l_1)\gamma )},t\in ((n+l_1)\gamma ,(n+l_1+l_2)\gamma ],\hfill \\ & \frac{d_2{\omega }_2^{*}{e}^{-d_2(t-(n+l_1+l_2)\gamma )}}{d_2+a_2{\omega }_2^{*}(1-e^{-d_1(t-(n+l_1+l_2)\gamma )})},t\in ((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ],\hfill \\ & {\omega }_3^{*}{e}^{-{\mu }_2\theta (t-(n+l_1+l_2+l_3)\gamma )},t\in ((n+l_1+l_2+l_3)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.$$

and

$${\omega }^{*}=\frac{h_0-1}{B},{\omega }_1^{*}=\frac{d_1{\omega }^{*}{e}^{-d_1{l}_1\gamma }}{d_1+a_1{\omega }^{*}(1-e^{-d_1{l}_1\gamma })},{\omega }_2^{*}={\omega }_1^{*}{e}^{-{\mu }_1\theta l_2\gamma )},{\omega }_3^{*}=\frac{d_2{\omega }_2^{*}{e}^{-d_2{l}_3\gamma }}{d_2+a_2{\omega }_2^{*}(1-e^{-d_2{l}_3\gamma })},$$

$$B=\frac{a_2}{d_2}{e}^{-(d_1{l}_1+{\mu }_1\theta l_2)\gamma }(1-e^{-d_2{l}_3\gamma })+\frac{a_1}{d_1}(1-e^{-d_1{l}_1\gamma }).$$

Proof. 

The analytic solution of system (5) is

$$\omega \left(t\right)=\left\{\begin{array}{c}\frac{d_1\omega \left(n{\gamma }^{+}\right)e^{-d_1(t-n\gamma )}}{d_1+a_1\omega \left(n{\gamma }^{+}\right)(1-e^{-d_1(t-n\gamma )})},t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ \omega \left((n+l_1)\gamma \right)e^{-{\mu }_1\theta (t-(n+l_1)\gamma )},t\in ((n+l_1)\gamma ,(n+l_1+l_2)\gamma ],\hfill \\ \frac{d_2\omega \left((n+l_1+l_2)\gamma \right)e^{-d_2(t-(n+l_1+l_2)\gamma )}}{d_2+a_2\omega \left((n+l_1+l_2)\gamma \right)(1-e^{-d_2(t-(n+l_1+l_2)\gamma )})},\hfill \\ t\in ((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ],\hfill \\ \omega \left((n+l_1+l_2+l_3)\gamma \right)e^{-{\mu }_1\theta (t-(n+l_1+l_2+l_3)\gamma )},t\in ((n+l_1+l_2+l_3)\gamma ,(n+1)\gamma ].\hfill \end{array}\right.$$

(6)

Then, the stroboscopic map of (6) is presented by

$$\omega \left((n+1){\gamma }^{+}\right)=\frac{h_0\omega \left(n{\gamma }^{+}\right)}{1+B\omega \left(n{\gamma }^{+}\right)},$$

where $B=\frac{a_2}{d_2}{e}^{-(d_1{l}_1+{\mu }_1\theta l_2)\gamma }(1-e^{-d_2{l}_3\gamma })+\frac{a_1}{d_1}(1-e^{-d_1{l}_1\gamma }).$ Denote ${\omega }_n=\omega \left(n{\gamma }^{+}\right)$, we can obtain the following difference equation:

$${\omega }_{n+1}=\frac{h_0{\omega }_n}{1+B{\omega }_n}=g\left({\omega }_n\right).$$

(7)

Obviously, function $g\left(\omega \right)$ is a strictly increasing function and $\frac{g\left(\omega \right)}{\omega }$ is a strictly decreasing function.

(i) When $h_0\le 1$, it follows from (7) that ${\omega }_{n+1}<h_0{\omega }_n\le {\omega }_n,$ thus, $\left\{{\omega }_n\right\}$ is a strictly decreasing and bounded sequence, this implies that $\left\{{\omega }_n\right\}$ converges to ${\omega }^0\ge 0$. From (7), we obtain that ${\omega }^0=g\left({\omega }^0\right)$, and calculate that ${\omega }^0=0$, which also implies that $\underset{t\to +\infty }{\lim }\omega \left(t\right)=0$ when $h_0\le 1$.

(ii) When $h_0>1$, Equation (7) has two fixed points ${\omega }^0=0$ and ${\omega }^{*}=\frac{h_0-1}{B}$. By calculation, we have

$${\left|\frac{\partial g\left(\omega \right)}{\partial \omega }\right|}_{\omega =0}=h_0>1,{\left|\frac{\partial g\left(\omega \right)}{\partial \omega }\right|}_{\omega ={\omega }^{*}}=\frac{1}{h_0}<1,$$

thus, the trivial fixed point is unstable, and the positive fixed point ${\omega }^{*}$ is locally asymptotically stable. Moreover,

$$\left|\frac{1}{{\omega }_n}-\frac{1}{{\omega }^{*}}\right|=\frac{1}{h_0}\left|\frac{1}{{\omega }_{n-1}}-\frac{1}{{\omega }^{*}}\right|\to 0asn\to +\infty ,$$

and ${\omega }_n$ is bounded, then $|{\omega }_n-{\omega }^{*}|\to 0$ as $n\to +\infty$, which means that the fixed point ${\omega }^{*}$ is globally asymptotically stable. Similar to [43], system (5) has a globally asymptotically stable positive periodic solution

$$\overline{\omega \left(t\right)}=\left\{\begin{array}{cc}& \frac{d_2{\omega }^{*}{e}^{-d_1(t-n\gamma )}}{d_1+a_1{\omega }^{*}(1-e^{-d_1(t-n\gamma )})},t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ & {\omega }_1^{*}{e}^{-{\mu }_1\theta (t-(n+l_1)\gamma )},t\in ((n+l_1)\gamma ,(n+l_1+l_2)\gamma ],\hfill \\ & \frac{d_2{\omega }_2^{*}{e}^{-d_2(t-(n+l_1+l_2)\gamma )}}{d_2+a_2{\omega }_2^{*}(1-e^{-d_1(t-(n+l_1+l_2)\gamma )})},t\in ((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ],\hfill \\ & {\omega }_3^{*}{e}^{-{\mu }_2\theta (t-(n+l_1+l_2+l_3)\gamma )},t\in ((n+l_1+l_2+l_3)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.$$

and

$${\omega }_1^{*}=\frac{d_1{\omega }^{*}{e}^{-d_1{l}_1\gamma }}{d_1+a_1{\omega }^{*}(1-e^{-d_1{l}_1\gamma })},{\omega }_2^{*}={\omega }_1^{*}{e}^{-{\mu }_1\theta l_2\gamma },{\omega }_3^{*}=\frac{d_2{\omega }_2^{*}{e}^{-d_2{l}_3\gamma }}{d_2+a_2{\omega }_2^{*}(1-e^{-d_2{l}_3\gamma })}.$$

The proof is complete. □

Remark 1. 

From Lemma 3 and the comparison theorem, we obtain that $x\left(t\right)\le \omega \left(t\right)\le \max \{0,{\omega }^{*}\}$$+ϵ$ for $ϵ>0$ small enough and t large enough.

Lemma 4. 

Let $\overline{\omega \left(t\right)}$ be the positive periodic solution of system (5), then

$${\int }_{l_1\gamma }^{(l_1+l_2)\gamma }{\mu }_1\frac{1+\theta e\overline{\omega \left(t\right)}}{1+e\overline{\omega \left(t\right)}}dt={\mu }_1{l}_2\gamma +\frac{1-\theta }{\theta }\ln \left(\frac{1+e{\omega }_1^{*}\exp (-{\mu }_1\theta l_2\gamma )}{1+e{\omega }_1^{*}}\right),$$

$${\int }_{(l_1+l_2+l_3)\gamma }^{\gamma }{\mu }_2\frac{1+\theta e\overline{\omega \left(t\right)}}{1+e\overline{\omega \left(t\right)}}dt={\mu }_2{l}_2\gamma +\frac{1-\theta }{\theta }\ln \left(\frac{1+e{\omega }_3^{*}\exp (-{\mu }_2\theta l_2\gamma )}{1+e{\omega }_3^{*}}\right).$$

Proof. 

From Lemma 3, we obtain that

$$\begin{array}{cc}\hfill {\int }_{l_1\gamma }^{(l_1+l_2)\gamma }{\mu }_1\frac{1+\theta e\overline{\omega \left(t\right)}}{1+e\overline{\omega \left(t\right)}}dt& ={\int }_{l_1\gamma }^{(l_1+l_2)\gamma }{\mu }_1-\frac{{\mu }_1(1-\theta )e\overline{\omega \left(t\right)}}{1+e\overline{\omega \left(t\right)}}dt\hfill \\ & ={\mu }_1{l}_2\gamma +\frac{1-\theta }{\theta }{\int }_{l_1\gamma }^{(l_1+l_2)\gamma }\frac{d(e\overline{\omega \left(t\right)}+1)}{1+e\overline{\omega \left(t\right)}}\hfill \\ & ={\mu }_1{l}_2\gamma +\frac{1-\theta }{\theta }\ln \left(\frac{1+e{\omega }_1^{*}\exp (-{\mu }_1\theta l_2\gamma )}{1+e{\omega }_1^{*}}\right).\hfill \end{array}$$

Similarly,

$${\int }_{(l_1+l_2+l_3)\gamma }^{\gamma }{\mu }_2\frac{1+\theta e\overline{\omega \left(t\right)}}{1+e\overline{\omega \left(t\right)}}dt={\mu }_2{l}_2\gamma +\frac{1-\theta }{\theta }\ln \left(\frac{1+e{\omega }_3^{*}\exp (-{\mu }_2\theta l_2\gamma )}{1+e{\omega }_3^{*}}\right).$$

The proof is complete. □

Lemma 5. 

For any solution $\psi \left(t\right)$ of system (1) with $\psi \left(0\right)>0$, system (1) is bounded for t large enough.

Proof. 

It follows from Remark 1 that for any $ϵ>0$, there always exist a $n_0\in Z_{+}$ such that $0\le x\left(t\right)<\max \{{\omega }^{*},0\}+ϵ$ for $t\ge n_0\gamma$. Denote $\tilde{r}=\max \{r_1,r_2+\max \{{\omega }^{*},0\}+ϵ\}$ and $\tilde{b}=\min \{b_1,b_2\}$. From (1), we have

$$\begin{array}{cc}& \frac{dy\left(t\right)}{dt}\le y\left(t\right)(\tilde{r}-\tilde{b}y\left(t\right)),t\ge n_0\gamma ,\hfill \end{array}$$

then there must be a $M_1>0$ such that $y\left(t\right)<M_1$ for t large enough. □

4. Main Results

Denote

$$h_1=\left(1+\frac{(1-m)b}{1+c{y}^{*}}+\eta mb\right)e^{-d_1{l}_1\gamma -{\mu }_1{l}_2\gamma -d_2{l}_3\gamma -{\mu }_2{l}_2\gamma -{\int }_0^{l_1\gamma }\beta (1-m)\overline{y\left(s\right)}ds}$$

and

$$\begin{array}{cc}\hfill h_2& =(1+\frac{(1-m)b}{1+c{y}^{*}}+\eta mb)\exp \{-(d_1{l}_1+d_2{l}_3)\gamma -{\mu }_1{\int }_{l_1\gamma }^{(l_1+l_2)\gamma }\frac{1+\theta e\overline{\omega \left(t\right)}}{1+e\overline{\omega \left(t\right)}}dt\hfill \\ & -{\mu }_2{\int }_{(l_1+l_2+l_3)\gamma }^{\gamma }\frac{1+\theta e\overline{\omega \left(t\right)}}{1+e\overline{\omega \left(t\right)}}dt-(1-m)\beta {\int }_0^{l_1\gamma }\overline{y\left(t\right)}dt\}.\hfill \end{array}$$

According to Lemmas 2 and 4, we can obtain that

$$\begin{array}{cc}\hfill h_1& =\left(1+\frac{(1-m)b}{1+c{y}^{*}}+\eta mb\right)e^{-(d_1{l}_1+{\mu }_1{l}_2+d_2{l}_3+{\mu }_2{l}_2)\gamma -\beta (1-m){\int }_0^{l_1\gamma }\frac{r_1{y}^{*}{e}^{r_{1}t}}{r_1+b_1{y}^{*}(e^{r_{1}t}-1)}dt}\hfill \\ & =\left(1+\frac{(1-m)b}{1+c{y}^{*}}+\eta mb\right)e^{-(d_1{l}_1+{\mu }_1{l}_2+d_2{l}_3+{\mu }_2{l}_2)\gamma -\frac{\beta (1-m)}{b_1}\ln \frac{(e^{r_1{l}_1\gamma }-1)y^{*}{b}_1+r_1}{r_1}}\hfill \\ & =\left(1+\frac{(1-m)b}{1+c{y}^{*}}+\eta mb\right){\left(1+\frac{b_1{y}^{*}(e^{r_1{l}_1\gamma }-1)}{r_1}\right)}^{\frac{-\beta (1-m)}{b_1}}{e}^{-(d_1{l}_1+({\mu }_1+{\mu }_2)l_2+d_2{l}_3)\gamma }\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill h_2& =h_1{e}^{-{\mu }_1{\int }_{l_1\gamma }^{(l_1+l_2)\gamma }\frac{1+\theta e\overline{\omega \left(t\right)}}{1+e\overline{\omega \left(t\right)}}dt-{\mu }_2{\int }_{(l_1+l_2+l_3)\gamma }^{\gamma }\frac{1+\theta e\overline{\omega \left(t\right)}}{1+e\overline{\omega \left(t\right)}}dt+{\mu }_1{l}_2\gamma +{\mu }_2{l}_2\gamma }\hfill \\ & =h_1{\left(\frac{1+e{\omega }_1^{*}}{1+e{\omega }_1^{*}\exp (-{\mu }_1\theta l_2\gamma )}\right)}^{\frac{1-\theta }{\theta }}{\left(\frac{1+e{\omega }_3^{*}}{1+e{\omega }_3^{*}\exp (-{\mu }_2\theta l_2\gamma )}\right)}^{\frac{1-\theta }{\theta }},\hfill \end{array}$$

where $y^{*}$, ${\omega }_1^{*}$ and ${\omega }_3^{*}$ are given in Lemmas 2 and 3.

4.1. Stability of Prey-Extinction Periodic Solution

Theorem 1. 

The prey-extinction periodic solution $(0,\overline{y\left(t\right)})$ of system (1) is locally asymptotically stable if $h_1<1$.

Proof. 

To prove the local asymptotic stability of periodic solution $(0,\overline{y\left(t\right)})$, we discuss the behavior of small-amplitude perturbation of the solution. Define

$$u\left(t\right)=x\left(t\right),v\left(t\right)=y\left(t\right)-\overline{y\left(t\right)}.$$

Then the linearized system of system (1) in $(n\gamma ,(n+1\left)\gamma \right]$ is obtained by

$$\left\{\begin{array}{cc}& \frac{du\left(t\right)}{dt}=-(d_1+\beta (1-m)\overline{y\left(t\right)})u\left(t\right),\hfill \\ & \frac{dv\left(t\right)}{dt}=\kappa \beta (1-m)\overline{y\left(t\right)}u\left(t\right)+(r_1-2b_1\overline{y\left(t\right)})v\left(t\right),\hfill \end{array}\right.t\in (n\gamma ,(n+l_1)\gamma ],$$

and

$$\left\{\begin{array}{cc}& \frac{du\left(t\right)}{dt}=-{\mu }_{1}u\left(t\right),\hfill \\ & \frac{dv\left(t\right)}{dt}=(r_2-2b_2\overline{y\left(t\right)})v\left(t\right),\hfill \end{array}\right.t\in ((n+l_1)\gamma ,(n+l_1+l_2)\gamma ],$$

and

$$\left\{\begin{array}{cc}& \frac{du\left(t\right)}{dt}=-d_{2}u\left(t\right),\hfill \\ & \frac{dv\left(t\right)}{dt}=(r_2-2b_2\overline{y\left(t\right)})v\left(t\right),\hfill \end{array}\right.t\in ((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ],$$

and

$$\left\{\begin{array}{cc}& \frac{du\left(t\right)}{dt}=-{\mu }_{2}u\left(t\right),\hfill \\ & \frac{dv\left(t\right)}{dt}=(r_2-2b_2\overline{y\left(t\right)})v\left(t\right),\hfill \end{array}\right.t\in ((n+l_1+l_2+l_3)\gamma ,(n+1)\gamma ],$$

The fundamental solution matrix can be calculated as follows:

$$\begin{array}{cc}\hfill {\varphi }_1\left(t\right)& =\left(\begin{array}{cc}{e}^{-{\int }_{n\gamma }^t{d}_1+\beta (1-m)\overline{y\left(s\right)}ds}& 0\\ *& e^{{\int }_{n\gamma }^t{r}_1-2b_1\overline{y\left(s\right)}ds}\end{array}\right),t\in (n\gamma ,(n+l_1)\gamma ],\hfill \end{array}$$

and

$${\varphi }_2\left(t\right)=\left(\begin{array}{cc}{e}^{-{\mu }_1(t-(n+l_1)\gamma )}& 0\\ 0& e^{{\int }_{(n+l_1)\gamma }^t{r}_2-2b_2\overline{y\left(s\right)}ds}\end{array}\right),t\in ((n+l_1)\gamma ,(n+l_1+l_2)\gamma ],$$

and

$${\varphi }_3\left(t\right)=\left(\begin{array}{cc}{e}^{-d_1(t-(n+l_1+l_2)\gamma )}& 0\\ 0& e^{{\int }_{(n+l_1+l_2)\gamma }^t{r}_2-2b_2\overline{y\left(s\right)}ds}\end{array}\right),t\in ((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ],$$

and

$$\begin{array}{cc}\hfill {\varphi }_4\left(t\right)& =\left(\begin{array}{cc}{e}^{-{\mu }_2(t-(n+l_1+l_2+l_3)\gamma )}& 0\\ 0& e^{{\int }_{(n+l_1+l_2+l_3)\gamma }^t{r}_2-2b_2\overline{y\left(s\right)}ds}\end{array}\right),t\in ((n+l_1+l_2+l_3)\gamma ,(n+1)\gamma ].\hfill \end{array}$$

The exact expression for ∗ is not provided here since it is not utilized in the subsequent calculation.

The linearization of the seventh equation of system (1) is

$$\left(\begin{array}{c}u\left((n+1){\gamma }^{+}\right)\\ v\left((n+1){\gamma }^{+}\right)\end{array}\right)=\left(\begin{array}{cc}1+\frac{(1-m)b}{1+c{y}^{*}}+\eta mb& 0\\ 0& 1\end{array}\right)\left(\begin{array}{c}u\left(\right(n+1\left)\gamma \right)\\ v\left(\right(n+1\left)\gamma \right)\end{array}\right).$$

According to the Floquet theory, we know that the prey-extinction periodic solution $(0,\overline{y\left(t\right)})$ is locally asymptotically stable if the absolute values of all eigenvalues of matrix

$$\begin{array}{cc}\hfill \Phi =& \left(\begin{array}{cc}1+\frac{(1-m)b}{1+c{y}^{*}}+\eta mb& 0\\ 0& 1\end{array}\right){\varphi }_1\left(l_1\gamma \right){\varphi }_2\left((l_1+l_2)\gamma \right){\varphi }_3\left((l_1+l_2+l_3)\gamma \right){\varphi }_4\left(\gamma \right)\hfill \end{array}$$

are less than 1, and the eigenvalues of matrix $\Phi$ are

$$\begin{array}{cc}\hfill {\lambda }_1=& (1+\frac{(1-m)b}{1+c{y}^{*}}+\eta mb)e^{-d_1{l}_1\gamma -{\mu }_1{l}_2\gamma -d_2{l}_3\gamma -{\mu }_2{l}_2\gamma -{\int }_0^{l_1\gamma }\beta (1-m)\overline{y\left(s\right)}ds},\hfill \end{array}$$

$$\begin{array}{c}\hfill {\lambda }_2=e^{{\int }_0^{l_1\gamma }{r}_1-2b_1\overline{y\left(s\right)}ds+{\int }_{l_1\gamma }^{\gamma }{r}_2-2b_2\overline{y\left(s\right)}ds}=e^{-(b_1{\int }_0^{l_1\gamma }\overline{y\left(s\right)}ds+b_2{\int }_{l_1\gamma }^{\gamma }\overline{y\left(s\right)}ds)}<1.\end{array}$$

We can obtain that ${\lambda }_1<1$ when $h_1<1$. Thus, the periodic solution $(0,\overline{y\left(t\right)})$ of system (1) is locally asymptotically stable when $h_1<1$. □

In the following, we study the global asymptotic stability of the periodic solution $(0,\overline{y\left(t\right)})$.

Theorem 2. 

If $h_0\le 1$ or $h_0>1$ and $h_2<1$, the prey-extinction periodic solution $(0,\overline{y\left(t\right)})$ of system (1) is globally asymptotically stable.

Proof. 

In Theorem 1, we proved the local asymptotic stability of the periodic solution $(0,\overline{y\left(t\right)})$. In the following, we simply prove that the periodic solution $(0,\overline{y\left(t\right)})$ is a global attractor.

If $h_0\le 1$, it follows from Lemma 3 that $\underset{t\to +\infty }{\lim }x\left(t\right)=0$.

If $h_0>1$, from (1), we have

$$\left\{\begin{array}{c}\frac{dy\left(t\right)}{dt}\ge y\left(t\right)[r_1-b_{1}y\left(t\right)],t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ \frac{dy\left(t\right)}{dt}=y\left(t\right)[r_2-b_{2}y\left(t\right)],t\in ((n+l_1)\gamma ,(n+1)\gamma ].\hfill \end{array}\right.$$

(8)

According to the comparison theorem, Lemmas 2 and 3, we obtain that for all $ϵ>0$, there is a $N_0\in Z_{+}$ such that

$$x\left(t\right)\le \omega \left(t\right)\le \overline{\omega \left(t\right)}+ϵ,y\left(t\right)\ge \overline{y\left(t\right)}-ϵ,$$

(9)

for $t\ge N_0\gamma$. Further, we obtain that for $t\ge N_0\gamma$,

$$\left\{\begin{array}{c}\frac{dx\left(t\right)}{dt}=-(d_1+\beta (1-m)(\overline{y\left(t\right)}-ϵ))x\left(t\right),t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ \frac{dx\left(t\right)}{dt}\le -{\mu }_1\frac{1+\theta e(\overline{\omega \left(t\right)}+ϵ)}{1+e(\overline{\omega \left(t\right)}+ϵ)}x\left(t\right),t\in ((n+l_1)\gamma ,(n+l_1+l_2)\gamma ],\hfill \\ \frac{dx\left(t\right)}{dt}\le -d_{2}x\left(t\right),t\in ((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ],\hfill \\ \frac{dx\left(t\right)}{dt}\le -{\mu }_2\frac{1+\theta e(\overline{\omega \left(t\right)}+ϵ)}{1+e(\overline{\omega \left(t\right)}+ϵ)}x\left(t\right),t\in ((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ],\hfill \\ x\left(t^{+}\right)\le (1+\frac{(1-m)b}{1+c(y^{*}-ϵ)}+\eta mb)x\left(t\right),t=(n+1)\gamma .\hfill \end{array}\right.$$

(10)

Integrating (10) from $n\gamma$ to $(n+1)\gamma$, we obtain

$$\begin{array}{c}\hfill x\left((n+1){\gamma }^{+}\right)\le \rho x\left(n{\gamma }^{+}\right),\end{array}$$

(11)

where

$$\begin{array}{cc}\hfill \rho & =(1+\frac{(1-m)b}{1+c(y^{*}-ϵ)}+\eta mb)\exp \{-(d_1{l}_1+d_2{l}_3)\gamma -{\mu }_1{\int }_{l_1\gamma }^{(l_1+l_2)\gamma }\frac{1+\theta e(\overline{\omega \left(t\right)}+ϵ)}{1+e(\overline{\omega \left(t\right)}+ϵ)}dt\hfill \\ & -{\mu }_2{\int }_{(l_1+l_2+l_3)\gamma }^{\gamma }\frac{1+\theta e(\overline{\omega \left(t\right)}+ϵ)}{1+e(\overline{\omega \left(t\right)}+ϵ)}dt-(1-m)\beta {\int }_0^{l_1\gamma }\overline{y\left(t\right)}-ϵdt\}\hfill \\ & \le (1+\frac{(1-m)b}{1+c(y^{*}-ϵ)}+\eta mb){\left(1+\frac{b_1{y}^{*}(e^{r_1{l}_1\gamma }-1)}{r_1}\right)}^{\frac{-\beta (1-m)}{b_1}}{e}^{-(d_1{l}_1+d_2{l}_3+({\mu }_1+{\mu }_2))\gamma }\hfill \\ & {\left(\frac{1+e{\omega }_1^{*}}{1+e{\omega }_1^{*}\exp (-{\mu }_1\theta l_2\gamma )}\right)}^{\frac{1-\theta }{\theta }}{\left(\frac{1+e{\omega }_3^{*}}{1+e{\omega }_3^{*}\exp (-{\mu }_2\theta l_2\gamma )}\right)}^{\frac{1-\theta }{\theta }}{e}^{({\mu }_1{l}_2+\beta l_1)\gamma ϵ}.\hfill \end{array}$$

Since $h_2<1$, we can select a $ϵ>0$ small enough such that $\rho <1$. From (11), we can imply that $x\left(t\right)\to 0$ as $t\to +\infty (n\to +\infty )$, that is, for any ${ϵ}_1>0$, there is a $N_1(>N_0)\in Z_{+}$ such that $x\left(t\right)\le {ϵ}_1$ for $t\ge N_1\gamma$. Further from (1), we have

$$\left\{\begin{array}{c}\frac{dy\left(t\right)}{dt}\le y\left(t\right)[r_1+\kappa \beta (1-m){ϵ}_1-b_{1}y\left(t\right)],t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ \frac{dy\left(t\right)}{dt}=y\left(t\right)[r_2-b_{2}y\left(t\right)],t\in ((n+l_1)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.$$

(12)

for all $t\ge N_1\gamma$. According to the comparison theorem and Lemma 2, we obtain $y\left(t\right)\le z\left(t\right)$ for $t\ge N_1\gamma$, and for any sufficiently small ${ϵ}_2>0$, there is a $N_2>N_1$ such that

$$z\left(t\right)\le \overline{z\left(t\right)}+{ϵ}_2,fort\ge N_2\gamma ,$$

(13)

where $z\left(t\right)$ is the solution of the following comparison system

$$\frac{dz\left(t\right)}{dt}=\left\{\begin{array}{c}z\left(t\right)[r_1+\kappa \beta (1-m){ϵ}_1-b_{1}z\left(t\right)],t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ z\left(t\right)[r_2-b_{2}z\left(t\right)],t\in ((n+l_1)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.$$

with initial value $z\left(N_1\gamma \right)=y\left(N_1\gamma \right)$, and

$$\overline{z\left(t\right)}=\left\{\begin{array}{c}\frac{(r_1+\kappa \beta (1-m){ϵ}_1)z^{*}{e}^{(r_1+\kappa \beta (1-m){ϵ}_1)(t-n\gamma )}}{(r_1+\kappa \beta (1-m){ϵ}_1)+b_1{z}^{*}(e^{(r_1+\kappa \beta (1-m){ϵ}_1)(t-n\gamma )}-1)},t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ \frac{r_2{z}_1^{*}{e}^{r_2(t-(n+l_1+l_2)\gamma )}}{r_2+b_2{z}_1^{*}(e^{r_2(t-(n+l_1+l_2)\gamma )}-1)},t\in ((n+l_1)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.$$

and

$$\overline{z\left(N_1{\gamma }^{+}\right)}=z^{*}=\frac{1-e^{(r_1+\kappa \beta (1-m){ϵ}_1)l_1\gamma +r_2(1-l_2)\gamma }}{A_1},$$

$$z_1^{*}=\frac{(r_1+\kappa \beta (1-m){ϵ}_1)z^{*}{e}^{(r_1+\kappa \beta (1-m){ϵ}_1)l_1\gamma }}{r_1+\kappa \beta (1-m){ϵ}_1+b_1{z}^{*}(e^{(r_1+\kappa \beta (1-m){ϵ}_1)l_1\gamma }-1)},$$

$$A_1=\frac{b_1}{(r_1+\kappa \beta (1-m){ϵ}_1)}{e}^{-r_2(1-l_1)\gamma }(1-e^{-(r_1+\kappa \beta (1-m){ϵ}_1)l_1\gamma })+\frac{b_2}{r_2}(1-e^{-r_2(1-l_1)\gamma }).$$

From (9) and (13), we get

$$\overline{y\left(t\right)}-ϵ\le y\left(t\right)\le \overline{z\left(t\right)}+{ϵ}_2,fort\ge N_2\gamma .$$

Letting $ϵ\to 0$, ${ϵ}_1\to 0$, ${ϵ}_2\to 0$, we can imply that $y\left(t\right)\to \overline{y\left(t\right)}$ as $t\to +\infty .$ The proof is completed. □

4.2. Permanence

Theorem 3. 

If $h_1>1$, system (1) is permanent, that is, there are $L>0$ and $M>0$ such that $L<x\left(t\right)<M$ and $L<y\left(t\right)<M$ for t large enough.

Proof. 

It follows from Lemma 5 that there exists a $M>0$ such that $x\left(t\right)\le M$ and $y\left(t\right)\le M$ for sufficiently large t. Without loss of generality, we suppose that $x\left(t\right)\le M$ and $y\left(t\right)\le M$ for $t\ge 0$.

From (9), we have

$$y\left(t\right)\ge \overline{y\left(t\right)}-ϵ\ge \underset{t\in [0,\gamma ]}{\min }\overline{y\left(t\right)}-ϵ,fort\ge N_0\gamma .$$

Choosing $ϵ=0.5\underset{t\in [0,\gamma ]}{\min }\overline{y\left(t\right)}$, we have $y\left(t\right)\ge 0.5\underset{t\in [0,\gamma ]}{\min }\overline{y\left(t\right)}=L_2$ for $t\ge N_0\gamma$, hence, predator y is always permanent.

In the following, we prove that there is a $\underline{L_1}>0$ such that $x\left(t\right)>\underline{L_1}$ for sufficiently large t.

If $h_1>1$, by the continuity of solutions with respect to parameters, we select $L_1>0$ and ${ϵ}_1>0$ small enough such that

$$\begin{array}{cc}\hfill \sigma =& (1+\frac{(1-m)b}{1+c(Y^{*}+{ϵ}_1)}+\eta mb)e^{-(d_1+a_1{L}_1)l_1\gamma -{\mu }_1{l}_2\gamma -(d_2+a_2{L}_1)l_3\gamma -{\mu }_2{l}_2\gamma -\beta (1-m){\int }_0^{l_1\gamma }\overline{Y\left(s\right)}+{ϵ}_{1}ds}>1,\hfill \end{array}$$

where $\overline{Y\left(t\right)}$ is given (15) below.

We first prove $x\left(t\right)<L_1$ can not hold for all $t\ge 0$, if not,

$$\frac{dy\left(t\right)}{dt}\le y\left(t\right)[r_1+\kappa \beta (1-m)L_1-b_{1}y\left(t\right)],t\in (n\gamma ,(n+l_1)\gamma ].$$

Consider the following comparison system

$$\frac{dY\left(t\right)}{dt}=\left\{\begin{array}{c}Y\left(t\right)[r_1+\kappa \beta (1-m)L_1-b_{1}Y\left(t\right)],t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ Y\left(t\right)[r_2-b_{2}Y\left(t\right)],t\in ((n+l_1)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.$$

(14)

with $Y\left(0\right)=y\left(0\right)>0$. From Lemma 2, we know that System (14) has a globally asymptotically stable periodic solution

$$\overline{Y\left(t\right)}=\left\{\begin{array}{c}\frac{(r_1+\kappa \beta (1-m)L_1)Y^{*}{e}^{(r_1+\kappa \beta (1-m)L_1)(t-n\gamma )}}{(r_1+\kappa \beta (1-m)L_1)+b_1{z}^{*}(e^{(r_1+\kappa \beta (1-m)L_1)(t-n\gamma )}-1)},t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ \frac{r_2{Y}_1^{*}{e}^{r_2(t-(n+l_1+l_2)\gamma )}}{r_2+b_2{Y}_1^{*}(e^{r_2(t-(n+l_1+l_2)\gamma )}-1)},t\in ((n+l_1)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.$$

(15)

and

$$Y^{*}=\frac{1-e^{(r_1+\kappa \beta (1-m)L_1)l_1\gamma +r_2(1-l_2)\gamma }}{A_2},Y_1^{*}=\frac{(r_1+\kappa \beta (1-m)L_1)z^{*}{e}^{(r_1+\kappa \beta (1-m)L_1)l_1\gamma }}{r_1+\kappa \beta (1-m)L_1+b_1{Y}^{*}(e^{(r_1+\kappa \beta (1-m)L_1)l_1\gamma }-1)},$$

$$A_2=\frac{b_1}{(r_1+\kappa \beta (1-m)L_1)}{e}^{-r_2(1-l_1)\gamma }(1-e^{-(r_1+\kappa \beta (1-m)L_1)l_1\gamma })+\frac{b_2}{r_2}(1-e^{-r_2(1-l_1)\gamma }).$$

By the comparison theorem, we have

$$y\left(t\right)\le Y\left(t\right)\le \overline{Y\left(t\right)}+{ϵ}_{1}fort\ge n_0\gamma .$$

(16)

From (1) and (16), we derive that for $t\ge n_0\gamma$,

$$\left\{\begin{array}{c}\frac{dx\left(t\right)}{dt}=-(d_1+a_1{L}_1+\beta (1-m)(\overline{Y\left(t\right)}+{ϵ}_1))x\left(t\right),t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ \frac{dx\left(t\right)}{dt}\ge -{\mu }_{1}x\left(t\right),t\in ((n+l_1)\gamma ,(n+l_1+l_2)\gamma ],\hfill \\ \frac{dx\left(t\right)}{dt}\ge -(d_2+a_2{L}_1)x\left(t\right),t\in ((n+l_1+l_2)\gamma ,(n+l_1+l_2+l_3)\gamma ],\hfill \\ \frac{dx\left(t\right)}{dt}\ge -{\mu }_{2}x\left(t\right),t\in ((n+l_1+l_2+l_3)\gamma ,(n+1)\gamma ],\hfill \\ x\left(t^{+}\right)\ge (1+\frac{(1-m)b}{1+c(Y^{*}+{ϵ}_1)}+\eta mb)x\left(t\right),t=(n+1)\gamma .\hfill \end{array}\right.$$

(17)

Integrating (17) from $n\gamma$ to $(n+1)\gamma$, we have

$$x\left((n+1){\gamma }^{+}\right)\ge x\left(n{\gamma }^{+}\right)\sigma ,forn>n_0,$$

(18)

further,

$$x\left((n+1){\gamma }^{+}\right)\ge x\left(x\left(n_0{\gamma }^{+}\right)\right){\sigma }^{n-n_0}\to +\infty ,n\to +\infty ,$$

this contradicts the boundedness of x, thus there exists a $t_1>n_0\gamma$ such that $x\left(t_1\right)\ge L_1$.

Two cases should be considered.

$\mathbf{Case}\mathbf{a}:$ If $x\left(t\right)\ge L_1$ for all $t\ge n_0\gamma$, the proof is complete.

$\mathbf{Case}\mathbf{b}:$$x\left(t\right)$ is oscillatory about $L_1$ for t large enough. Let $\overline{t}=\underset{t\ge t_1}{\inf }\{t:x\left(t\right)<L_1\}$, then there must be $n_1\in Z_{+}$ such that $\overline{t}\in [n_1\gamma ,(n_1+1)\gamma )$ $(n_1>n_0)$. Clearly, $x\left(t\right)\ge L_1$ for $t\in [t_1,\overline{t}]$. Since $x\left(t\right)$ is a monotonically decreasing function on the interval $[n\gamma ,(n+1\left)\gamma \right)$, we obtain $x\left(\overline{t}\right)=L_1$ and $x\left({\overline{t}}^{+}\right)>L_1$ if $\overline{t}=n_1\gamma$. If $\overline{t}\ne n_1\gamma$, we get $x\left(\overline{t}\right)=L_1$.

Choose $p\in Z_{+}$ large enough such that

$$p>n_0,{(1+\frac{(1-m)b}{1+cM}+\eta mb)}^{n_0+1}{e}^{-\overline{d}(n_0+1)}{\sigma }^{p-n_0}>1,$$

where $\overline{d}=\max \{d_1+a_{1}M+\beta (1-m)M,d_2+a_{2}M,{\mu }_1,{\mu }_2\}$.

We claim that there must be a $t_2\in [(n_1+1)\gamma ,(n_1+1+p)\gamma )$ such that $x\left(t_2^{+}\right)>L_1$. Otherwise, $x\left(t\right)<L_1$ for $t\in [(n_1+1)\gamma ,(n_1+1+p)\gamma )$. Letting $Y\left((n_1+1)\gamma \right)=y\left(0\right)$, we obtain from (14)–(16) that $y\left(t\right)\le \overline{Y\left(t\right)}+{ϵ}_1$ for $t\ge (n_1+1+n_0)\gamma$. From (18), we have

$$x\left((n_1+p+1){\gamma }^{+}\right)\ge x\left((n_1+1+n_0){\gamma }^{+}\right){\sigma }^{p-n_0}.$$

Because

$$\left\{\begin{array}{c}\frac{dx\left(t\right)}{dt}\ge -(d_1+a_{1}M+\beta (1-m)M)x\left(t\right),t\in (n\gamma ,(n+l_1)\gamma ],\hfill \\ \frac{dx\left(t\right)}{dt}\ge -{\mu }_{1}x\left(t\right),t\in ((n+l_1)\gamma ,(n+l_1+l_2)\gamma ],\hfill \\ \frac{dx\left(t\right)}{dt}\ge -(d_2+a_{2}M)x\left(t\right),t\in ((n+l_1+l_2)\gamma ,(n+1+l_1+l_2+l_3)\gamma ],\hfill \\ \frac{dx\left(t\right)}{dt}\ge -{\mu }_{2}x\left(t\right),t\in ((n+1+l_1+l_2+l_3)\gamma ,(n+1)\gamma ],\hfill \end{array}\right.$$

(19)

from (19), we have

$$\left\{\begin{array}{cc}& \frac{dx\left(t\right)}{dt}\ge -\overline{d}x\left(t\right),t\in ((n+l_1+l_2+l_3)\gamma ,(n+1+l_1+l_2+l_3)\gamma ],\hfill \\ & x\left(t^{+}\right)\ge (1+\frac{(1-m)b}{1+cM}+\eta mb)x\left(t\right),t=(n+l_1+l_2+l_3)\gamma \hfill \end{array}\right.$$

Integrating above inequality on the interval $[\overline{t},n_1+1+n_0]$, we have

$$\begin{array}{cc}\hfill x\left((n_1+1+n_0)\gamma \right)& \ge x\left(\overline{t}\right){(1+\frac{(1-m)b}{1+cM}+\eta mb)}^{n_0}{e}^{-\overline{d}(n_0+1)}\hfill \\ & =L_1{(1+\frac{(1-m)b}{1+cM}+\eta mb)}^{n_0}{e}^{-\overline{d}(n_0+1)}.\hfill \end{array}$$

And

$$\begin{array}{cc}\hfill x\left((n_1+p+1){\gamma }^{+}\right)& \ge x\left((n+1+n_0){\gamma }^{+}\right){\sigma }^{p-n_0}\hfill \\ & \ge L_1{(1+\frac{(1-m)b}{1+cM}+\eta mb)}^{n_0+1}{e}^{-\overline{d}(n_0+1)}{\sigma }^{p-n_0}>L_1,\hfill \end{array}$$

which is a contradiction. Therefore, we know that there exists $t_2>(n_1+1)\gamma$ such that $x\left(t_2^{+}\right)>L_1$. We further conclude that $x\left(t\right)>e^{-d(p+1)\gamma }{L}_1=\underline{L_1}$ for $t\in (t_1,t_2]$.

We can continue the same discussion for $t\ge t_2$ since $x\left(t_2^{+}\right)>L_1$. Therefore, $x\left(t\right)\ge \underline{L_1}$ for t large enough. □

If the seasonal migration of species x is not considered, the system (1) will degenerate into the following a predator–prey system with impulsive birth, fear effects and refuge effects.

$$\left\{\begin{array}{c}\left.\begin{array}{c}\frac{dx\left(t\right)}{dt}=x\left(t\right)[-d_0-a_{0}x\left(t\right)-\beta (1-m)y\left(t\right)],\hfill \\ \frac{dy\left(t\right)}{dt}=y\left(t\right)[r_0-b_{0}y\left(t\right)+\kappa \beta (1-m)x\left(t\right)],\hfill \end{array}\right\}t\in (n\gamma ,(n+1)\gamma ],\hfill \\ \left.\begin{array}{c}x\left(t^{+}\right)=(1+\frac{(1-m)b}{1+cy\left(t\right)}+\eta mb)x\left(t\right),\hfill \\ y\left(t^{+}\right)=y\left(t\right),\hfill \end{array}\right\}t=(n+1)\gamma .\hfill \end{array}\right.$$

(20)

We assume in model (20) that $r_0,a_0,d_0,b_0,\alpha ,\beta ,\kappa ,b$ are positive constants, $0\le m\le 1$ and $\eta >1$.

It is easy to obtain that the system (20) has a prey-extinction periodic solution $(0,\frac{r_0}{b_0})$. Denote

$$h_3=(1+\frac{(1-m)b{b}_0}{b_0+c{r}_0}+\eta mb)e^{-(d_0+(1-m)\beta \frac{r_0}{b_0})\gamma }.$$

By adopting the same proof methods as Theorems 1–3, we can also derive the following results:

Theorem 4. 

(1) If $h_3<1$, the prey-extinction periodic solution $(0,\frac{r_0}{b_0})$ of System (20) is globally asymptotically stable.

(2) If $h_3>1$, system (20) is permanent.

Remark 2. 

From Theorem 4, we can easily see that $h_3$ is the threshold for the permanence and extinction of the prey population in System (20), that is, the prey x of System (20) will be extinct if $h_3<1$, and the prey x is permanent if $h_3>1$.

5. Numerical Simulations

As shown by the above theorems, it is crucial to gain a better understanding of how the refuge effect and fear effect impact the survival of migratory species. We present numerical simulations using the Matlab R2017b software to verify our theoretical results. Fixed parameters

$$\begin{array}{cc}& d_1=0.15,a_1=0.011,r_1=0.1,b_1=0.4,{\mu }_1=0.3,{\mu }_2=0.3,e=5.1,d_2=0.15,\hfill \\ & a_2=0.01,\theta =0.2,r_2=0.7,b_2=0.6,\kappa =0.8,\beta =1.2,b=2,\eta =3,\gamma =4,\hfill \\ & l_1=0.32,l_2=0.18,l_3=0.32.\hfill \end{array}$$

5.1. The Role of the Refuge Effect

First, by setting $m=0.45$ and $c=0.1$, we can calculate that

$$\begin{array}{cc}\hfill h_1& =\left(1+\frac{(1-m)b}{1+c{y}^{*}}+\eta mb\right){\left(1+\frac{b_1{y}^{*}(e^{r_1{l}_1\gamma }-1)}{r_1}\right)}^{\frac{-\beta (1-m)}{b_1}}{e}^{-(d_1{l}_1+({\mu }_1+{\mu }_2)l_2+d_2{l}_3)\gamma }\hfill \\ & =\left(1+\frac{1.1}{1+0.1\times 1.0847}+2.7\right){\left(1+\frac{0.15\times 1.0847(e^{0.128}-1)}{0.1}\right)}^{\frac{-0.66}{0.4}}{e}^{-0.4704}\hfill \\ & \approx 0.9629<1\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill h_2& =h_1{\left(\frac{1+e{\omega }_1^{*}}{1+e{\omega }_1^{*}\exp (-{\mu }_1\theta l_2\gamma )}\right)}^{\frac{1-\theta }{\theta }}{\left(\frac{1+e{\omega }_3^{*}}{1+e{\omega }_3^{*}\exp (-{\mu }_2\theta l_2\gamma )}\right)}^{\frac{1-\theta }{\theta }}\hfill \\ & \approx 0.9629{\left(\frac{1+5.1\times 34.6397}{1+5.1\times 34.6397\times e^{-0.0432}}\right)}^4{\left(\frac{1+5.1\times 19.7493}{1+5.1\times 19.7493e^{-0.0432} )}\right)}^4\hfill \\ & \approx 0.9629\times 1.1874\times 1.1866\approx 1.3567>1.\hfill \end{array}$$

It follows from Theorem 1 that the prey-extinction periodic solution $(0,\overline{y\left(t\right)})$ is locally asymptotically stable as shown in Figure 2 and Figure 3. Figure 2 reveals the phase diagrams for the survival and extinction of the prey population for different initial values $(x\left(0^{+}\right),y\left(0^{+}\right))$$\in [0,1]\times [0,1],$ where the red region in Figure 2 is a region of attraction for the periodic solution of prey-extinction. In addition, we can also observe from Figure 3 that System (1) has two steady states: the prey-extinction periodic solution where the prey population goes extinct, and a positive periodic solution where both the prey and predator populations oscillate periodically and positively.

Moreover, when we take a smaller refuge effect intensity of $m=0.3$, and other parameters remain unchanged, by calculating, we have

$$\begin{array}{cc}\hfill h_2& =h_1{\left(\frac{1+e{\omega }_1^{*}}{1+e{\omega }_1^{*}\exp (-{\mu }_1\theta l_2\gamma )}\right)}^{\frac{1-\theta }{\theta }}{\left(\frac{1+e{\omega }_3^{*}}{1+e{\omega }_3^{*}\exp (-{\mu }_2\theta l_2\gamma )}\right)}^{\frac{1-\theta }{\theta }}\hfill \\ & \approx 0.6762\times 1.1873\times 1.1864\approx 0.9526<1,\hfill \end{array}$$

System (1) has a unique globally asymptotically stable prey-extinction periodic solution as shown in Figure 4a, that is, the prey population will eventually become extinct for any initial value $(x\left(0^{+}\right),y\left(0^{+}\right))\in R_{+}^2$. If we choose a larger refuge effect intensity of $m=0.5$, the calculation yields

$$\begin{array}{cc}\hfill h_1& =\left(1+\frac{1}{1+0.1\times 1.0847}+3\right){\left(1+\frac{0.15\times 1.0847(e^{0.128}-1)}{0.1}\right)}^{\frac{-3}{2}}{e}^{-0.4704}\hfill \\ & \approx 1.0787>1\hfill \end{array}$$

from Theorem 3, we obtain that System (1) is permanent (see Figure 4b).

5.2. The Role of the Fear Effect

According to Theorems 1–3, we can confirm that the fear effect is detrimental to the permanence of the prey population in System (1). In the following, we take into account the following four cases of c: (i) $c=0.5$, (i) $c=0.8$, (iii) $c=1.2$, (iv) $c=2$. The other parameters are the same as in Figure 4b and the initial value $(x\left(0^{+}\right),y\left(0^{+}\right))=(0.1,1)$. Figure 5 shows that the density of the prey population will shift from a high-density state to a low-density state or even extinction as the value of c increases.

5.3. The Dynamic Behaviors of System (20)

In the following, we consider the dynamic behaviors of the prey–predator System (20) where the prey population does not migrate. Given the parameters of System (20):

$$d_0=0.15,a_0=0.01,r_0=0.2,b_0=0.6,\beta =1.2,\kappa =0.8,b=2,\eta =3,\gamma =4,c=0.4,$$

and then vary the value of m. We first give $m=0.3$, and calculate that

$$\begin{array}{cc}\hfill h_3& =\left(1+\frac{(1-m)b{b}_0}{b_0+c{r}_0}+\eta mb\right)e^{-(d_0+(1-m)\beta \frac{r_0}{b_0})\gamma }\hfill \\ & =\left(1+\frac{1.4}{1+0.4÷3}+1.8\right)e^{-1.72}\approx 0.7226<1.\hfill \end{array}$$

We can see from Theorem 4 that System (20) has a unique prey-extinction periodic solution $(0,\frac{1}{3})$, which is globally asymptotically stable as shown in Figure 6a. When $m=0.5$, it can be calculated that

$$\begin{array}{cc}\hfill h_3& =\left(1+\frac{1}{1+0.4÷3}+3\right)e^{-1.4}\approx 1.204>1,\hfill \end{array}$$

by Theorem 4, we obtain that System (20) is permanent (see Figure 6b).

6. Discussion

In Section 4, we obtain conditions for the local asymptotic stability and global asymptotic stability of the prey-extinction periodic solution, as well as for the persistence of the system. The numerical verification is carried out in Section 5 (see Figure 3, Figure 4 and Figure 6). From the proof of Theorem 2, it is easy to see that the condition for global asymptotic stability of the prey-extinction periodic solution is only a sufficient condition. If $h_1<1<h_2$, the prey-extinction periodic solution is locally asymptotically stable by Theorems 1 and 2. From Figure 2 and Figure 3, we observe an interesting situation that the prey population may persist when $x\left(0^{+}\right)$ is large enough and $y\left(0^{+}\right)$ is small enough. Unfortunately, for this phenomenon, we are currently limited to numerical simulations and are unable to use existing mathematical methods to give the domain of attraction for the prey-extinction periodic solution. Of course, the parameter $0<\theta <1$ plays an important role for this scenario to occur. As can be seen from the expression of $h_2$, $h_2$ decreases as $\theta$ decreases. In particular, if $\theta =1$, i.e., the effect of group migration on the population is not taken into account, then it follows from Theorems 1 and 2 that the prey-extinction periodic solution is globally asymptotically stable when $h_1<1$. Combined with Theorem 3, this leads to the conclusion that $h_1$ is a threshold for extinction and persistence of prey populations.

In Model (1), we consider the effects of the refuge effect and the fear effect on the survival of prey populations. According to the expression of $h_1$, we can conclude that the value of $h_1$ increases as m increases or c decreases. In other words, increasing the proportion of refuges or reducing the fear level of prey towards predators is favorable for the conservation of the prey population (see Figure 3, Figure 4 and Figure 5). Furthermore, in Theorem 3, we have only explored the permanence of the system (1), but as can be seen from Figure 4 and Figure 5, there exists a positive periodic solution to System (1). However, we have not yet found a good mathematical method to address the existence of the positive periodic solution, which is one of our future research directions. Future work can also modify Model (1) on the basis of model (1), such as introducing stochastic perturbations of environmental factors into Model (1) [19,22] and considering the migration of predator populations [27].

7. Conclusions

In this work, we establish and study a novel predator–prey model that incorporates: (i) the migration of prey between breeding and non-breeding areas; (ii) the refuge effect of prey; and (iii) the reduction in the prey pulse birth rate, in the form of fear effect, in the presence of predator. By applying the Floquet theory and the comparison theorem of impulsive differential equations, sufficient conditions for the stability of the prey-extinction periodic solution and the permanence of the system (1) are obtained. The study shows that the the prey refuge effect and the fear effect play a crucial role on the survival of the prey population. Furthermore, we have also investigated the case in which the prey population does not migrate (see System (20)). Sufficient conditions for the stability of the prey-extinction periodic solution and the permanence of System (20) are also established, and the threshold for extinction and permanence of the prey population is obtained.