Persistence and Stability for a Generalized Leslie-Gower Model with Stage Structure and Dispersal

and Applied Analysis 3 where x1 t , x2 t denote the immature and mature population densities, respectively. Here, α > 0 represents the per-capita birth rate; γ > 0 is the per-capita immature death rate; β > 0 is the death rate due to overcrowding, and τ is the “fixed” time to maturity; the term αe−γτx2 t − τ models the immature individuals who were born at time t − τ i.e., αx2 t − τ and survive and mature at time t. 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. A1 The prey population: the prey only lives in patch 1. For immature prey, α is birth rate, r1 is death rate, and the term αe−r1τx2 t − τ represents the number of immature prey that was born at time t−τ , which still survive at time t and are transferred from the immature stage to the mature stage at time t. For mature prey, r2 is death rate, r3 is the intraspecific competition rate of mature prey, a1 is the maximum value of the per-capita reduction rate of x2 due to y1, and k1 resp., k2 measures the extent to which environment provides protection to prey x2 resp., to the predator y1 . A2 The predator population: βi are the birth rate of predator in patch i, i 1, 2; Di is the dispersion rate of predator between two patches; r4 is death rate of predator in patch 2; a2 has a similar meaning to a1. It is assumed that predators in patch 1 do not capture immature prey, then we have the following delayed differential system: ẋ1 t αx2 t − r1x1 t − αe−r1τx2 t − τ , ẋ2 t αe−r1τx2 t − τ − r2x2 t − r3x 2 t − a1y1 t x2 t x2 t k1 , ẏ1 t ( β1 − a2y1 t x2 t k2 ) y1 t D1 ( y2 t − y1 t ) , ẏ2 t ( β2 − r4y2 t ) y2 t D2 ( y1 t − y2 t ) , 1.5 where x1 t and x2 t represent the densities of immature and mature individual prey in patch 1 at time t, yi t denote the density of predator species in patch i, i 1, 2 at time t, all parameters of 1.5 are positive constants. The initial conditions for system 1.5 take the form of xi θ Φi θ , yi θ Ψi θ , xi 0 > 0, yi 0 > 0, i 1, 2, 1.6 where Φ1 θ ,Φ2 θ ,Ψ1 θ ,Ψ2 θ ∈ C −τ, 0 , R4 0 , the Banach space of continuous function mapping the interval −τ, 0 into R4 0, where R4 0 { x1, x2, x3, x4 : xi ≥ 0, i 1, 2, 3, 4}. 4 Abstract and Applied Analysis For continuity of the initial conditions, we further require x1 0 ∫0 −τ αe1Φ2 s ds. 1.7 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 D ⊂ IntR4 0 such that every solution of system 1.5 with initial conditions 1.6 and 1.7 eventually enters and remains in the region D. Lemma 2.2. Solutions of system 1.5 with initial conditions 1.6 and 1.7 are positive for all t ≥ 0. Proof. Let x1 t , x2 t , y1 t , y2 t be a solution of system 1.5 with initial conditions 1.6 and 1.7 ; we first consider y1 t and y2 t for t ∈ 0, τ , ẏ1 t ∣ y1 0 D1y2 t > 0 for y2 > 0, ẏ2 t ∣ y2 0 D2y1 t > 0 for y1 > 0. 2.1 Thus, it follows that y1 t > 0, y2 t > 0 for t ∈ 0, τ . For t ∈ 0, τ , it follows from the second equation of system 1.5 that ẋ2 t ≥ [ −r2 − r3x2 t − a1y1 t x2 t k1 ] x2 t . 2.2 Consider the following auxiliary equation: u̇ t ≥ [ −r2 − r3u t − a1y1 t u t k1 ] u t ,


Introduction
Lotka-Volterra predator-prey models have been extensively and deeply investigated see monographs 1-5 .If we let x t denote the density of prey and let y t be the density of predator, then the classical Lotka-Volterra predator-prey model is given by the following system: 1.1 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 2 Abstract and Applied Analysis 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 x t and y t denote the prey and predator density, respectively, then Leslie's model is given by the following system: where r i , c i , i 1, 2, and b 1 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, c 2 y/x .The rationale behind this term is based on the view that as the prey becomes numerous x → ∞ then the per-capita growth rate of the predator achieves its maximum 1/y dy/dt → r 2 .Conversely as the prey becomes scarce x → 0, we have that 1/y dy/dt → −∞.That is, the predator must go extinct.Recently, the use of a Hollingtype 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 r 1 is the per-capita growth rate of the prey x; b 1 is a measure of the strength of prey on prey interference competition; c 1 is the maximum value of the per-capita reduction rate of x due to y; k 1 measures the extent to which the environment provides protection to prey x k 2 for y ; r 2 gives the maximal per-capita growth rate of y; c 2 has a similar meaning to that of c 1 .
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: A1 The prey population: the prey only lives in patch 1.For immature prey, α is birth rate, r 1 is death rate, and the term αe −r 1 τ x 2 t − τ represents the number of immature prey that was born at time t − τ, which still survive at time t and are transferred from the immature stage to the mature stage at time t.For mature prey, r 2 is death rate, r 3 is the intraspecific competition rate of mature prey, a 1 is the maximum value of the per-capita reduction rate of x 2 due to y 1 , and k 1 resp., k 2 measures the extent to which environment provides protection to prey x 2 resp., to the predator y 1 .
A2 The predator population: β i are the birth rate of predator in patch i, i 1, 2; D i is the dispersion rate of predator between two patches; r 4 is death rate of predator in patch 2; a 2 has a similar meaning to a 1 .It is assumed that predators in patch 1 do not capture immature prey, then we have the following delayed differential system: where x 1 t and x 2 t represent the densities of immature and mature individual prey in patch 1 at time t, y i t denote the density of predator species in patch i, i 1, 2 at time t, all parameters of 1.5 are positive constants.The initial conditions for system 1.5 take the form of 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.

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 D ⊂ Int R 4 0 such that every solution of system 1.5 with initial conditions 1.6 and 1.7 eventually enters and remains in the region D. Lemma 2.2.Solutions of system 1.5 with initial conditions 1.6 and 1.7 are positive for all t ≥ 0.
Proof.Let x 1 t , x 2 t , y 1 t , y 2 t be a solution of system 1.5 with initial conditions 1.6 and 1.7 ; we first consider y 1 t and y 2 t for t ∈ 0, τ , ẏ1 t y 1 0 D 1 y 2 t > 0 for y 2 > 0, ẏ2 t y 2 0 D 2 y 1 t > 0 for y 1 > 0.
For t ∈ 0, τ , it follows from the second equation of system 1.5 that Consider the following auxiliary equation:
In order to discuss the uniform persistence, we need the following result from 24 .
Lemma 2.3.Consider the following equation: where a, b, c, and τ are positive constants, x t > 0 for t ∈ −τ, 0 .We have the following: Lemma 2.4.Let x 1 t , x 2 t , y 1 t , y 2 t be a solution of system 1.5 with initial conditions 1.6 and 1.7 .Then there exists a T 3 > 0 such that where N is a constant and

2.7
Proof.Suppose X t x 1 t , x 2 t , y 1 t , y 2 t 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 t ≥ τ, Consider the following auxiliary equation: By Lemma 2.3 we obtain that Using comparison principle, it follows that Therefore, for sufficiently small ε > 0 there is a

2.12
Setting T 2 T 1 τ, it then follows 2.4 and 2.12 that, for t ≥ T 2 ,

2.13
We define ρ t x 1 t x 2 t y 1 t y 2 t ,

2.14
where A min{r 1, r 2 }.It follows from 2.14 that Therefore, there exists a T 3 T 2 τ and This completes the proof.
Theorem 2.5.System 1.5 with initial conditions 1.6 and 1.7 is uniformly persistent provided that where N is defined by 2.7 .
Proof.Suppose X t x 1 t , x 2 t , y 1 t , y 2 t 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 t ≥ T 3 τ, 2.17 Consider the following auxiliary equation: By Lemma 2.3, we obtain that According to comparison principle it follows that Therefore, for sufficiently small ε > 0 there is a

2.21
By the third and forth equation of system 1.5 , we have

2.22
Consider the following auxiliary equation:

2.24
Using a similar argument in the proof of 25, Lemma 2.1 we obtain lim

2.25
Therefore, for sufficiently small ε > 0 there is a T 5 T 4 τ such that if t ≥ T 5 ,

2.26
Setting T 6 T 5 τ, then by 2.4 , we have

2.27
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 H2 holds Remark 2.7.From the H2 , we know that if the death rate of mature prey r 2 is more than the product of birth rate of immature prey α and the surviving probability of each immature prey becomes mature e −r 1 τ , then the mature and immature prey population will go to extinction.
Proof.Suppose X t x 1 t , x 2 t , y 1 t , y 2 t 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

2.28
Consider the following auxiliary equation: x 2 t 0.

2.32
We therefore obtain that lim t → ∞ x 1 t 0.

2.33
This completes the proof.

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 H1 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 H1 hold.Assume further that where

3.1
where , and N is defined by 2.7 .
Then the positive equilibrium E * x * 1 , x * 2 , y * 1 , y * 2 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:

3.2
Noting that E * x * 2 , y * 1 , y * 2 is a positive equilibrium of system 3.2 , we can rewrite system 3.2 as

3.4
Calculating the derivative of V 1 t along solution of system 1.5 , we have

3.6
Using the inequality ab ≤ 1/2 ka 2 1/2k b 2 , it follows from 3.6 that where parameters A, B are positive constants to be determined.Define

3.8
Setting A 2, B 1, c 3 1, then it follows from 3.7 and 3.8 that

3.10
N, n 2 , and n 3 are defined in 2.16 , 2.21 , and 2.26 , respectively.If H1 and H3 hold and ε > 0 is sufficiently small, we have A i > 0, i 1, 3. In view of Lyapunov theorem 26 , we conclude that the positive equilibrium E * x * 2 , y * 1 , y * 2 of system 3.2 is globally asymptotically stable.Thus, we have lim

3.11
Using L'Hospital's rule, it follows from 2.4 and 3.11 that

3.12
This completes the proof.

Abstract and Applied Analysis 13
It is interesting to discuss the local stability of the positive equilibrium E * x * 1 , x * 2 , y * 1 , y * 2 of system 1.5 .The characteristic equation of the positive equilibrium E * of system 1.5 is of the form λ r 1 P λ Q λ e −λτ 0, 3.13 where

3.18
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 E * .
Theorem 3.3.Suppose that system 1.5 admits a positive equilibrium 1 If Δ l 2 −3m ≤ 0, then the positive equilibrium E * of system 1.5 is locally asymptotically stable.

Two Examples
In this section, we give two examples to illustrate our main results.
integration can be carried out using standard MATLAB algorithm.Numerical simulation also confirms the fact see Figure 1 .

4.2
System 4.2 has a unique boundary equilibrium E * 0, 0, 1, 1 .It is easy to show that H2 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 .

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 H1 and H3 , 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 H2 , the immature and mature prey population will go to extinction.

3 , 2 y 6
and a 1 b 1 a 2 b 2 − a 0 b 0 > 0, then by Routh-Hurwitz Theorem the positive equilibrium E * of system 1.5 is locally asymptotically stable when τ 0. Let F y P iy 2 − Q iy ly 4 my 2 n 0, 3.16
16, x 2 t denote the immature and mature population densities, respectively.Here, α > 0 represents the per-capita birth rate; γ > 0 is the per-capita immature death rate; β > 0 is the death rate due to overcrowding, and τ is the "fixed" time to maturity; the term αe −γτ x 2 t − τ models the immature individuals who were born at time t − τ i.e., αx 2 t − τ and survive and mature at time t.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.
It is easy to show that if τ < 0.8973, then H1 and H3 hold for system 4.1 .By Theorem 2.5 we see that system 4.1 is uniformly persistent when τ < 0.8973.By Theorem 3.1 we see that the positive equilibrium of system 4.1 is globally asymptotically stable when τ 0.5.Numerical 4.2.Consider the following system: ẋ1 t 5x 2 t − x 1 t − 5e −1 x 2 t − 1 , 1 t y 2 t − y 1 t , ẏ2 t 1 − y 2 t y 2 t y 1 t − y 2 t .