#### Abstract

A generalized version of the Leslie-Gower predator-prey model that incorporates the prey structure and predator dispersal in two-patch environments is introduced. The focus is on the study of the boundedness of solution, permanence, and extinction of the model. Sufficient conditions for global asymptotic stability of the positive equilibrium are derived by constructing a Lyapunov functional. Numerical simulations are also presented to illustrate our main results.

#### 1. Introduction

Lotka-Volterra predator-prey models have been extensively and deeply investigated (see monographs [1–5]). If we let denote the density of prey and let be the density of predator, then the classical Lotka-Volterra predator-prey model is given by the following system: The equations in system (1.1) set no upper limit on the per-capita growth rate of the predator (the second term of model (1.1) which is unrealistic. For example, for mammals, such a limit will be determined in part by physiological factors (length of the gestation period, the shortest interval between litters, the maximum average number of daughters per-litter, the age at which breeding first starts, and so on [6]). Leslie modeled the effect of such limitations via a predator-prey model where the “carrying capacity’’ of the predator's environment was assumed to be proportional to the number of prey. Hence, if and denote the prey and predator density, respectively, then Leslie's model is given by the following system: where and are positive constants. The first equation of system (1.2) is standard but the second is not because it contains the “so-called’’ Leslie-Gower term, namely, . The rationale behind this term is based on the view that as the prey becomes numerous then the per-capita growth rate of the predator achieves its maximum . Conversely as the prey becomes scarce we have that . That is, the predator must go extinct. Recently, the use of a Holling-type II functional for the prey has led various researchers [7, 8] to the consideration of the following model (a modification of system (1.2): where is the per-capita growth rate of the prey ; is a measure of the strength of prey (on prey) interference competition; is the maximum value of the per-capita reduction rate of due to ; measures the extent to which the environment provides protection to prey ( for ); gives the maximal per-capita growth rate of ; has a similar meaning to that of

In [9], the global stability of the unique coexisting interior equilibrium of system (1.2) is established. In [7], the existence and boundedness of solutions (including that of an attracting set) are established as well as the global stability of the coexisting interior equilibrium for model (1.3). There have been additional extensions, for example, in [10, 11] a Leslie-Gower type model with impulse was introduced and investigated.

The study of the role of dispersal in continuous-time metapopulation models is extensive (see [12–16] and the references cited therein). They show that dispersal can have a stabilizing influence on the system (see [12, 13]) and also can have a destabilizing influence on the system (see [14, 15]).

On the other hand, most prey species have a life history that includes multiple stages (juvenile and adults or immature and mature). In Aiello and Freedman [17], the population dynamics of a single species with two identifiable stages was modeled by the following system: where denote the immature and mature population densities, respectively. Here, represents the per-capita birth rate; is the per-capita immature death rate; is the death rate due to overcrowding, and is the “fixed’’ time to maturity; the term models the immature individuals who were born at time (i.e., ) and survive and mature at time . The derivation and analysis of system (1.4) can be found in [17]. More and More researchers (see [16–22] and the references cited therein) have investigated many kinds of predator-prey model under various stage-structure assumptions. In Xu et al. [16], they discussed a Lotka-Volterra-type predator-prey model with stage structure for predator and prey dispersal in two-patch environments. They obtained sufficient conditions of permanence and impermanence and global asymptotic stability of the positive equilibrium; they also discussed the local stability of the positive equilibrium. In [22], they studied a generalized version of the Leslie-Gower predator-prey model that incorporates the prey structure and obtained sufficient conditions of permanence and stability of the nonnegative equilibrium.

Motivated by the above works, in this paper we study the effects of stage structure for prey and predator dispersal on the global dynamics of modified version of the Leslie-Gower and Holling-type II predator-prey system. Following [16, 23], we assumethe following.

(*A *1) *The prey population *: the prey only lives in patch . For immature prey, is birth rate, is death rate, and the term represents the number of immature prey that was born at time , which still survive at time and are transferred from the immature stage to the mature stage at time . For mature prey, is death rate, is the intraspecific competition rate of mature prey, is the maximum value of the per-capita reduction rate of due to , and (resp., ) measures the extent to which environment provides protection to prey (resp., to the predator ).

(*A *2) *The predator population *: are the birth rate of predator in patch , ; is the dispersion rate of predator between two patches; is death rate of predator in patch 2; has a similar meaning to . It is assumed that predators in patch 1 do not capture immature prey, then we have the following delayed differential system:
where and represent the densities of immature and mature individual prey in patch at time , denote the density of predator species in patch at time , all parameters of (1.5) are positive constants.

The initial conditions for system (1.5) take the form of where the Banach space of continuous function mapping the interval into , where

For continuity of the initial conditions, we further require

The paper is organized as follows. In Section 2, we will discuss the uniform persistence of system (1.5). In Section 3, we are concerned with the global stability of a positive equilibrium of system (1.5) by constructing Lyapunov functional and also present two numerical simulations to illustrate our main results.

#### 2. Uniform Persistence

In this section, we will discuss the uniform persistence of system (1.5) with initial conditions (1.6) and (1.7).

*Definition 2.1. *System (1.5) is said to be uniformly persistent if there exists a compact region such that every solution of system (1.5) with initial conditions (1.6) and (1.7) eventually enters and remains in the region .

Lemma 2.2. *Solutions of system (1.5) with initial conditions (1.6) and (1.7) are positive for all *

*Proof. *Let be a solution of system (1.5) with initial conditions (1.6) and (1.7); we first consider and for
Thus, it follows that for

For it follows from the second equation of system (1.5) that

Consider the following auxiliary equation:
For ; thus,

In a similar way, we consider the intervals Thus, for all

By (1.7) and the first equation of (1.5) we can obtain that
Therefore the positivity of for follows, this completes the proof.

In order to discuss the uniform persistence, we need the following result from [24].

Lemma 2.3. *Consider the following equation:
**
where and are positive constants, for We have the following:*(i)*if , then *(ii)*if , then *

Lemma 2.4. *Let be a solution of system (1.5) with initial conditions (1.6) and (1.7). Then there exists a such that
**
where is a constant and
*

*Proof. *Suppose to be any positive solution of system (1.5) with initial conditions (1.6) and (1.7). It follows from the second equation of system (1.5) that for

Consider the following auxiliary equation:
By Lemma 2.3 we obtain that
Using comparison principle, it follows that
Therefore, for sufficiently small there is a such that if
Setting it then follows (2.4) and (2.12) that, for
We define
where

It follows from (2.14) that
Therefore, there exists a and
Such that if This completes the proof.

Theorem 2.5. *System (1.5) with initial conditions (1.6) and (1.7) is uniformly persistent provided that*()* where is defined by (2.7). *

*Proof. *Suppose to be any positive solution of system (1.5) with initial conditions (1.6) and (1.7). It follows from the second equation of system (1.5) that for
Consider the following auxiliary equation:
By Lemma 2.3, we obtain that
According to comparison principle it follows that
Therefore, for sufficiently small there is a such that if
By the third and forth equation of system (1.5), we have
Consider the following auxiliary equation:
Define
Using a similar argument in the proof of [25, Lemma??2.1] we obtain
Therefore, for sufficiently small there is a such that if
Setting then by (2.4), we have
This completes the proof.

We now state a result on the extinction of the mature and immature prey.

Theorem 2.6. *The mature and immature prey population will go to extinction if holds*()*.*

*Remark 2.7. *From the we know that if the death rate of mature prey is more than the product of birth rate of immature prey and the surviving probability of each immature prey becomes mature then the mature and immature prey population will go to extinction.

*Proof. *Suppose to be any positive solution of system (1.5) with initial conditions (1.6) and (1.7). It follows from the second equation of system (1.5) that there is a
Consider the following auxiliary equation:
By Lemma 2.3, we derived from (2.29) and that
A standard comparison argument shows that
Therefore, there is a such that if Thus, we derive from (2.4) that for
We therefore obtain that
This completes the proof.

#### 3. Global Stability

In this section, we study the global asymptotic stability of a positive equilibrium of system (1.5). By Theorem 2.5 we see that if satisfies, system (1.5) is uniformly persistent, which implies that system (1.5) must have at least one positive equilibrium. So in the following we assume that a positive equilibrium exists and denote it by

Theorem 3.1. *Let hold. Assume further that*()* where
**where is a sufficient small constant, and is defined by (2.7).**Then the positive equilibrium of system (1.5) is globally asymptotically stable.*

*Remark 3.2. *Theorem 3.1 shows that if the time delay due to maturity is sufficiently small, the positive equilibrium of system (1.5) is globally asymptotically stable.

*Proof. *We first consider the following subsystem:
Noting that is a positive equilibrium of system (3.2), we can rewrite system (3.2) as
Define
Calculating the derivative of along solution of system (1.5), we have
Setting By (3.5) we obtain
Using the inequality it follows from (3.6) that
where parameters are positive constants to be determined.

Define
Setting , then it follows from (3.7) and (3.8) that
where
and are defined in (2.16), (2.21), and (2.26), respectively.

If and hold and is sufficiently small, we have In view of Lyapunov theorem [26], we conclude that the positive equilibrium of system (3.2) is globally asymptotically stable. Thus, we have
Using L'Hospital's rule, it follows from (2.4) and (3.11) that
This completes the proof.

It is interesting to discuss the local stability of the positive equilibrium of system (1.5).

The characteristic equation of the positive equilibrium of system (1.5) is of the form where here Clearly, is a negative eigenvalue. If which implies that and then by Routh-Hurwitz Theorem the positive equilibrium of system (1.5) is locally asymptotically stable when

Let where Let and then (3.16) becomes By applying the results on the distribution of roots of (3.16) and (3.18) in [27] and [26, Theorem??4.1, page 83], we therefore derive the following results on the stability of the positive equilibrium

Theorem 3.3. *Suppose that system (1.5) admits a positive equilibrium and *(1)*If then the positive equilibrium of system (1.5) is locally asymptotically stable.*(2)*If and then there exists a positive number such that the positive equilibrium of system (1.5) is locally asymptotically stable if and is locally unstable if ; further, as increases through , bifurcates into small amplitude periodic solutions, here, *

#### 4. Two Examples

In this section, we give two examples to illustrate our main results.

*Example 4.1. *Consider the following system:
where the parameter is a positive constant.

System (4.1) has a unique positive equilibrium . It is easy to show that if , then and hold for system (4.1). By Theorem 2.5 we see that system (4.1) is uniformly persistent when . By Theorem 3.1 we see that the positive equilibrium of system (4.1) is globally asymptotically stable when . Numerical integration can be carried out using standard MATLAB algorithm. Numerical simulation also confirms the fact (see Figure 1).

*Example 4.2. *Consider the following system:

System (4.2) has a unique boundary equilibrium . It is easy to show that holds for system (4.2). By Theorem 2.6 we see that mature and immature prey population goes to extinction. Numerical integration can be carried out using standard MATLAB algorithm. Numerical simulation also confirms the fact (see Figure 2).

#### 5. Discussion

In this paper, we discussed a generalized Leslie-Gower-type predator-prey model with stage structure for prey and predator dispersal in two-patch environments. By using comparison arguments we established sufficient conditions for system (1.5) to be permanent. By constructing Lyapunov functionals, sufficient conditions are derived for the global asymptotic stability of the positive equilibrium of system (1.5). By Theorem 3.1 we see that if the birth rate of immature prey and the extent to which environment provides protection to mature prey and predator in patch 1, respectively, are high and the maximum value of the per-capita reduction rate of mature prey due to predator in patch 1 is low satisfying (*H *1) and (*H *3), the positive equilibrium of system (1.5) is globally asymptotically stable. By Theorem 2.6 we see that if the death rate of mature prey is more than the transformation rate of immatures to matures satisfying (*H*2), the immature and mature prey population will go to extinction.

#### Acknowledgments

The first author was partially supported by the Key Project of Chinese Ministry of Education (209131), the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the NSF of Bureau of Education of Gansu Province of China for Postgraduate Tutors (0803-01), the Development Program for Outstanding Young Teachers in Lanzhou University of Technology (Q200703), and the Doctor's Foundation of Lanzhou University of Technology. The second author was partially supported by the Young Teacher's Foundation of Dali University (2008X34).