Dynamics of a Stage-Structured Leslie-Gower Predator-Prey Model

Volume 2011, Article ID 149341,22pages doi:10.1155/2011/149341

Research Article

Dynamics of a Stage-Structured Leslie-Gower Predator-Prey Model

Hai-Feng Huo,

Xiaohong Wang,

and Carlos Castillo-Chavez

1Institute of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

2Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA

Correspondence should be addressed to Hai-Feng Huo,hfhuo@lut.cn Received 5 December 2010; Accepted 19 April 2011

Academic Editor: Oded Gottlieb

Copyrightq2011 Hai-Feng Huo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A generalized version of the Leslie-Gower predator-prey model that incorporates the prey population structure is introduced. Our results show that the inclusion ofage structure in the prey population does not alter the qualitative dynamics of the model; that is, we identify sufficient conditions for the “trapping” of the dynamics in a biological compact set—albeit the analysis is a bit more challenging. The focus is on the study of the boundedness of solutions and identification of sufficient conditions for permanence. Sufficient conditions for the local stability of the nonnegative equilibria of the model are also derived, and sufficient conditions for the global attractivity of positive equilibrium are obtained. Numerical simulations are used to illustrate our results.

1. Introduction

Lotka-Volterra predator-prey models have been extensively and deeply investigated1–5. In population biology, we are often interested in identifying potential mechanisms responsible for either fluctuations or the lack of fluctuations in predator-prey systems. If we let xt denote the density of prey and letytbe the density of predator, then the classical Lotka- Volterra predator-prey model is given by the following system:

xt

r1−c1y−b1x x, yt

−ε2ρ2x

y. 1.1

It is known that these equations can support population fluctuations when b1 0, but, because the model is not structurally stable, the results have been primarily used as

a metaphor and as an inspiration for mathematical and biological research on the mechanisms responsible for fluctuationsor their lackin predator-prey systems. The equations in system 1.1 set no upper limit on the percapita growth rate of the predator second term of Model 1.1 which of course is unrealistic. For example, for mammals, such a limit will be determined in part by physiological factorslength 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 on6, 7. 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, ifxtdenotes the prey density and ytthe predators’, then Leslie’s model is given by the following system of nonlinear differential equations:

xt

r1−c1y−b1x x, yt

r2−c2

y x

y, 1.2

where ri, ci,i 1,2, and b1 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, c2y/x. The rationale behind this term is based on the view that as the prey becomes numerous x → ∞then the percapita growth rate of the predatordy/ydtachieves its maximum r2. Conversely as the prey becomes scarce x → 0, the predator will go extinct since the percapita growth rate of the predator goes to−∞. An alternative interpretation of the Leslie-Gower model concludes that the carrying capacity of the predators’ environment is proportional to the number of prey available, that is,

yt r2

1− y

Ax

yr2

1− y

C

y, 1.3

whereAr2/c2can be interpreted as a prey predators’ conversion factor andCAxas the predators’ carrying capacityproportional to prey abundance. The Leslie-Gower termy/Ax has also been interpreted as a measure of the loss in percapita predator’s reproduction rate due to the relative abundanceper capitay/xof its “favorite” foodpreyx. Model1.2is often referred to as a semi-ratio-dependent predator-prey model8. Model1.2is different from the ratio-dependent predator-prey models in the studies by Wang et al.9and Hsu et al.10.

Scarcity of preyx could drive predators y to switch to alternative resources of food. In fact, there is an extensive literature on the evolutionary advantage of specialist versus generalist when it comes down to predators’ diet 11–16. Predator’s growth may also be limited by nutritional factors. In fact, evolutionary forces may lead to the predators to specialize on the most nutritious prey. The possibility that a predator does not depend on a single prey type is modelled here in a rather simple way, that is, through the addition of a positive constantdin the denominator. In fact,

yt r2

1− y αxd

y. 1.4

A modification of System 1.2 using a Holling-type II functional response for the prey population has led various researchers11,15to consider the following model:

xt

r1− c1y xk1 −b1x

x, yt

r2− c2y xk2

y,

1.5

wherer1 is the percapita growth rate of the preyx,b1 is a measure of the strength of prey on preyinterference competition,c1is the maximum value of the per capita reduction rate of prey x due to predator y, k1 measures the extent to which the environment provides protection to prey x k2 for predator y, r2 gives the maximal percapita growth rate of predatory, andc2has a similar meaning to that ofc1.

In Aziz-Alaoui17, a preliminary analysis of a Leslie-Gower modelSystem1.2is carried out. In the study by Korobeinikov18, the global stability of the unique coexisting interior equilibrium of System1.2is established. In the study by Aziz-Alaoui and Daher Okiye11, the existence and boundedness of solutionsincluding that of an attracting set are established as well as the global stability of the coexisting interior equilibrium for Model 1.5. There have been additional extensions, for example, in the study by Letellier and Asis- Alaoui13, the studies by Letellier et al.14and Upadhyay and Rai19, a Leslie-Gower type tritrophic model was introduced and analyzed numerically.

Nindjina et al. considered the following extension of Leslie-Gower modified with Holling-type II schemes and time delayτ:

xt

r1− c1y xk1 −b1x

x, yt

r2− c2yt−τ xt−τ k2

y,

1.6

that is, a single discrete delayτ > 0 is introduced as a negative feedback in the predator’s density. Some results associated with the global stability analysis of solutions to System 1.6have been obtained including the impact ofτ on the stability of positive equilibrium of System1.6. In fact, researchers found out that the time delay can have a destabilizing effect on the positive equilibrium of System1.6 15.

Most prey species have a life history that includes multiple stages juvenile and adults or immature and mature. In the study by Aiello and Freedman20, the population dynamics of a single species with two identifiable stages was modeled by the following system:

x₁t αx2t−γx1t−αe^{−γ τ}x2t−τ,

x₂t αe^{−γ τ}x2t−τ−βx²₂t, 1.7 wherex1t,x2tdenote the immature and mature population densities, respectively. Here, α > 0 represents the percapita birth rate,γ > 0 is the percapita immature death rate,β > 0

models death rate due to overcrowding andτ is the “fixed” time to maturity, and the term αe^{−γ τ}x2t−τmodels the immature individuals who were born at time t−τ i.e.,αx2t− τ and survive and mature at time t. The derivation and analysis of System 1.7 can be found in the study by Aiello and Freedman20. Several additional researchers21–23, and the references thereinhave investigated versions of the above single species model under various stage-structure assumptions.

Liu and Beretta24reintroduced the impact of predators. They studied a predator- prey model with Beddington-DeAngelis functional response and stage-structure on the predator population. These researchers found that predator and prey coexist if and only if the predator’s recruitment rate at the peak of prey abundance is larger than its death rate. If the system is permanent, that is, if for any solution xtof the system, there exist constantsM,m > 0 such thatm ≤ lim inf_{t→ ∞}xt ≤ lim sup_{t→ ∞}xt ≤ Mthen sufficiently

“large” predators’ interference not only stabilizes the system but also guarantees its stability against increases in the carrying capacity of the prey and increases in the birth rate of the adult predator. Finally, it was shownanalytically and numerically in the study by Liu and Beretta 24 that stability switches of interior equilibrium may occur as the maturation time delay increases. That is, stability may change from stable to unstable to finally stable, implying that “small” and “large” delays can be stabilizing. Song et al.25considered a ratio- dependent predator-prey system that incorporated “age” structure for the prey. Their analysis established boundedness of solutions, looked at the nature of equilibria and permanence as well as the local stability and global attractivity of the positive equilibrium of the model. Their results show that the inclusion of an “age” structure in the prey population does not change the qualitative dynamics of the model—albeit the analysis is more challenging.

A Leslie-Gower model that incorporates the prey’s stage structure is introduced here to study the combined effects of prey stage structure and within prey interference competitions.

Following Song et al.25, we assume that the immature prey cannot reproduce and the per capita birth rate of the mature prey isα >0, the per capita death rate of the immature prey is γ >0, the per capita death rate of the mature prey is proportional to the current mature prey population with a proportionality constantβ >0, and immature individuals become mature at ageτ. Predators only feed on the mature prey. Using these definitions, we formulate a modified Leslie-Gower and Holling-type II schemes with stage-structure for prey as follows:

x₁t αx2t−γx1t−αe^{−γ τ}x2t−τ, x₂t αe^{−γ τ}x2t−τ−βx₂²t−c1ytx2t

x2t k1 , yt yt

r2− c2yt x2t k2

.

1.8

The initial conditions are given by x2θ ≥ 0, continuous on θ ∈ −τ,0, and x10, x20, y0>0, whilex1t,x2t, andytdenote the densities of immature prey, mature prey and predator, respectively. Please note that our model1.8is different from the model in the study by Song et al.25which is based on standard ratio-dependent and symmetric cross term. Our model1.8includes the Leslie-Gower term. The differences between the standard ratio-dependent formulation and the Leslie-Gower formulation of the predator-prey system are listed in the following, standard ratio-dependent formulation can be interpreted as the effect of the predator-population on the prey population and the effect of the prey population

on the predator-population are both a function of the ratio between the two, however the Leslie-Gower formulation can be interpreted as the effect of the predator-population on the prey population is different from the effect of the prey population on the predator-population:

both effects are inversely proportional to thematureprey population plus a constant.

From the first equation of system1.8we can see that

x1t _t

t−ταe^−γt−sx2sds, 1.9

x10 ₀

−ταe^{γ s}x2sds. 1.10

The last two equations in1.8do not containx1t. Hence, if we know the properties ofx2t then the properties ofx1tcan be easily obtained from1.8and1.9. Hence, we only need to consider the following system:

x₂t αe^{−γ τ}x2t−τ−βx₂²t−c1ytx2t x2t k1 , yt yt

r2− c2yt x2t k2

,

1.11

with initial conditionsx2θ≥0continuous onθ∈−τ,0andx20, y0>0.

The main purpose of this paper is to study the global dynamics of System1.11. The paper is organized as follows. InSection 2, we establish the conditions that determine the permanence of the system and obtain positiveness and boundedness results.Section 3focuses on the study of the local stability of the nonnegative equilibria. Section 4derives sufficient conditions for the global asymptotic stability of boundary equilibrium and for the global attractivity of positive equilibrium, and in theSection 5, these results are illustrated through simulations and their relevance is briefly discussed.

2. Permanence of Solutions

To prove the permanence of System1.11, we need the following lemma, which is a direct application of Theorem 4.9.1 in the study by Kuang26, see also Song et al.25and Liu et al.27.

Lemma 2.1. Consider the following equation:

xt axt−τ−bxt−cx²t, 2.1

wherea, b, c, τ >0 andxt>0, for−τ ≤t≤0.

iIfa > b, then limt→ ∞xt a−b/c.

iiIfa < b, then limt→ ∞xt 0.

Following the proof of Song et al.25and Liu et al.27, we can obtain the following lemma.

Lemma 2.2. Supposex2θ≥0 is continuous onθ∈−τ,0, andx20,y0>0, then the solution of System1.11satisfiesx2t,yt>0 for allt >0.

First, we establish a condition for the boundedness of the solutions of System1.11.

Theorem 2.3. Suppose x2θ ≥ 0 is continuous on θ ∈ −τ,0, and x20,y0 > 0, then the solutions of 1.11are bounded for all larget.

Proof. From the first equation of1.11, we have

x₂t≤αe^{−γ τ}x2t−τ−βx₂²t. 2.2 According toLemma 2.1and the standard comparison principle28, there exists aT1 > 0 and 1>0 such that

x2t≤ αe^{−γ τ}

β 1M1, fort > T1τ. 2.3

By the second equation of1.11and above inequality, we get

yt≤yt

r2− c2yt M1k2

, fort > T1τ. 2.4

From the comparison principle, there exists aT2> T1such that, for any sufficiently small 2, yt≤ M1k2r2

c2 2M2, fort > T2τ. 2.5

The proof is complete.

Now, we show that System1.11is permanent.

Theorem 2.4. Suppose that

αe^{−γ τ}−c1M2

k1 >0, 2.6

whereM2is defined by2.5, then System1.11is permanent.

Remark 2.5. Comparing the above permanent result with that results for model in Nindjin et al.15and model in Song et al.25, we see the inclusion of an extra term e^{−γ τ} in our permanence condition2.6; that is, the surviving probability of each immature prey becomes mature must be taken into account.

Proof. From the second equation of system1.11, we have

yt≥yt

r2−c2yt k2

. 2.7

It then follows that

t→ ∞liminfyt≥ k2r2

c2 m2>0. 2.8

Using the first equation of System1.11andTheorem 2.3, for sufficiently largeT, we have

x₂t≥αe^{−γ τ}x2t−τ−βx²₂t−c1ytx2t k1

≥αe^{−γ τ}x2t−τ−βx²₂t−c1M2x2t k1 .

2.9

ByLemma 2.1and the comparison principle, we have that

t→ ∞liminfx2t≥ αe^{−γ τ}−c1M2/k1

β m1>0. 2.10

Therefore, the above calculations andTheorem 2.3imply that there existMi,mi >0,i1,2, such that

0< m1≤ lim

t→ ∞infx2t≤ lim

t→ ∞sup x2t≤M1, 0< m2≤ lim

t→ ∞infyt≤ lim

t→ ∞sup yt≤M2.

2.11

The proof is complete.

3. Analysis of Equilibria

System1.11has the following nonnegative equilibria:

E0 0,0, E1 αe^{−γ τ}

β ,0

, E2

0,k2r2

c2

, E3

x^∗₂, y^∗

, 3.1

where

x^∗₂

αe^{−γ τ}−βk1−c1r2/c2

αe^{−γ τ}−βk1−c1r2/c2

2−4βc1r2/c2k2−αe^{−γ τ}k1

2β ,

y^∗ k2r2r2x^∗₂ c2 .

3.2

We see that the positive equilibriumE3exists if αe^{−γ τ}> c1k2r2

c2k1 . 3.3

The characteristic equation at equilibriumE0is

λ−αe^{−γ τ}e^−λτ

λ−r2 0, 3.4 and, consequently, since it has a positive eigenvalueλr2,E0is unstable.

The characteristic equation at equilibriumE1is given by the transcendental equation

λ−

−2αe^{−γ τ}αe^{−γ τ}e^−λτ

λ−r2 0. 3.5 Again,λr2is a positive eigenvalue, soE1is also unstable.

The analysis of the stability ofE2 requires a little more work. We have the following results.

Theorem 3.1. Let

Ê0 c2k1

c1k2r2αe^{−γ τ}, 3.6

then equilibriumE2is iunstable if^Ê0>1,

iilinearly neutrally stable if^Ê₀ 1, iiilocally asymptotically stable if^Ê₀<1.

Proof. i The characteristic equation of equilibriumE2is given by

λc1k2r2

c2k1 −αe^{−γ τ}e^−λτ

λr2 0, 3.7

clearly, one characteristic root isλ−r2<0, others are the roots of Fλ λc1k2r2

c2k1 −αe^{−γ τ}e^−λτ 0. 3.8

Assume that^Ê₀ > 1, therefore< αe^{−γ τ} thenF0 < 0 andF∞ ∞. HenceFλhas at least one positive root andE2is unstable.

iiSince^Ê₀1, that is,c1k2r2/c2k1αe^{−γ τ},F0 0, soλ0 is a root ofFλ 0. As Fλ ταe^{−γ τ}e^−λτ1, we haveF0>0. The rootλ0 is simple. If other roots are of form aiω, for someaandωinR, they satisfy

aαe^{−γ τ}₂

ω² αe^{−γ τ}₂

e^−2aτ. 3.9

Then, we must havea ≤ 0; that is, all other roots have nonpositive real parts. HenceE2 is linearly neutrally stable.

iiiIf^Ê₀<1, thenc1k2r2/c2k1> αe^{−γ τ}. Assume that there exists an eigenvalueλwith Reλ≥0, then we have

Reλ−c1k2r2

c2k1 αe^{−γ τ}e^−ReλτcosτImλ

≤αe^{−γ τ}e^−Reλτ−c1k2r2

c2k1 <0.

3.10

It is a contradiction, so Reλ <0. This shows that all roots ofFλ 0 must have negative real parts, hence, the equilibriumE2is locally asymptotically stable.

The proof of the theorem is complete.

Remark 3.2. Note that when the predator reaches its steady statey k2r2/c2in the absence of prey,αe^{−γ τ}can be interpreted as the per capita recruitment rate of prey andc1k2r2/c2k1 c1y/k 1approximates the per capita death rate of the prey. Therefore,^Ê0αe^{−γ τ}c2k1/c1k2r2is the basic demographic number of prey when the predator’s population size reaches its steady stateyin the absence of preyx. When^Ê0>1, the population size of prey will increase, thus E2is unstable. Similarly we can interpretiiandiiiinTheorem 3.1.

Remark 3.3. The sufficient condition given by2.6for the permanence of System1.11can be rewritten in the following form

Ê0>1αe^{−γ τ} k2β

. p0. 3.11

So a “large” basic demographic number ^Ê0 > p0 > 1 for the prey when the predator’s population size reaches its steady state in the absence of prey can guarantee the permanence of System1.11.

Now, we consider the local stability of the interior equilibriumE3 x₂^∗, y^∗. Recall there existsE3when3.3holds, that is, whenτis in the intervalI 0, τ^∗, where

τ^∗ 1

γ ln c2αk1

c1r2k2. 3.12

The characteristic equation atE3is

Dλ, τ λ²

r22βx^∗₂ c1k1y^∗ k1x₂^∗₂

λr2

2βx₂^∗ c1k1y^∗ k1x^∗₂₂

c1c2x^∗₂y^∗2 k1x^∗₂

k2x^∗₂₂

r2−λ− 2c2y^∗ k2x^∗₂

αe^{−γ τ}e^−λτ0.

3.13

Let

Pλ, τ λ²P1τλP0τ,

Qλ, τ λQ1τ Q0τ, 3.14

where

P1τ r22βx^∗₂ c1k1y^∗ k1x^∗₂2,

P0τ r2

2βx^∗₂ c1k1y^∗ k1x^∗₂2

c1c2x^∗₂y^∗2 k1x^∗₂

k2x^∗₂2, Q1τ −αe^{−γ τ}, Q0τ −r2αe^{−γ τ}.

3.15

Then the characteristic equation atE3becomes

Dλ, τ Pλ, τ Qλ, τe^−λτ 0. 3.16

First, we will prove

P0, τ Q0, τ/0, 3.17

that is,λ0 cannot be a root of3.16for anyτ∈I.

In fact, by the definition ofx^∗₂, y^∗, we have D0, τ P0τ Q0τ P0, τ Q0, τ

r2

2βx^∗₂ c1k1y^∗ k1x^∗₂₂

c1c2x₂^∗y^∗2 k1x^∗₂

k2x^∗₂₂ −r2αe^{−γ τ} r2

2βx^∗₂ c1k1y^∗

k1x^∗₂₂ −αe^{−γ τ}

c1c2x^∗₂y^∗2 k1x^∗₂

k2x^∗₂₂ r2

⎛

⎝−βk1−c1r2

c2

αe^{−γ τ}−βk1−c1r2

c2

₂

−4β c1r2

c2 k2−αe^{−γ τ}k1

⎞⎠

>0.

3.18

Therefore,λ0 is not a root of3.16.

The characteristic equation3.16atτ 0 is

Pλ,0 Qλ,0 0, 3.19

that is,

λ² P10 Q10λP00 Q00 0. 3.20

Then,

λ²

r22βx^∗₂ c1k1y^∗ k1x^∗₂₂ −α

λr2

2βx^∗₂ c1k1y^∗

k1x^∗₂₂ c1c2x₂^∗y^∗2 k1x₂^∗

k2x^∗₂₂ −α

0.

3.21

SinceP0τ Q0τ>0 for allτ∈0, τ^∗, thenP00 Q00>0. Notice that

P10 Q10 r22βx^∗₂ c1k1y^∗

k1x₂^∗₂ −α. 3.22

IfP10Q10>0, then3.20has two solutions with negative real parts. Hence,E3is locally asymptotically stable atτ0. IfP10 Q10<0, thenE3is unstable atτ0.

To determine the local stability of the interior equilibriumE3 x^∗₂, y^∗, we proceed as follows29.

Assume thatλ±iωτ,ωτ>0 satisfy3.16, we have Piω, τ −ω²iωP1τ P0τ, PRiω, τ P0τ−ω², PIiω, τ ωP1τ,

Qiω, τ iωQ1τ Q0τ,

QRiω, τ Q0τ, QIiω, τ ωQ1τ.

3.23

The first step is to look for the positive rootsωτ>0 of

Fω, τ |Piω, τ|²− |Qiω, τ|²0 3.24

inI 0, τ^∗. Since

Fω, τ ω⁴ω²

−2P0τ P₁²τ−Q₁²τ

P₀²τ−Q²₀τ, 3.25

we have

Fω, τ ω⁴bτω²cτ 0, bτ −2P0τ P₁²τ−Q²₁τ,

cτ P₀²τ−Q²₀τ.

3.26

Depending on the signs ofbτandcτ, System3.26may have no positive real roots, or the root

ωτ 1

2

−bτ bτ²−4cτ _1/2

, τ ∈I⊆I, 3.27

or otherwise the root ω−τ

1 2

−bτ− bτ²−4cτ _1/2

, τ ∈I−⊆I, 3.28

or, as the last case, bothωτandω−τ. Note that if System3.26has no positive rootsωτ inI, then no stability switches can occur.

From the structure ofP10 Q10, a sufficient condition forE3atτ 0 to be locally asymptotically stable is given by

α−2βk1−2c1r2

c2 >0, 3.29

which impliesP10 Q10> 0. Stability switches for increasingτ inI 0, τ^∗may occur only with a pair of rootsλ±iωτ ωτreal positivethat cross the imaginary axis.

Next, we state the following theorem on the local asymptotic stability of equilibrium E3.

Theorem 3.4. The positive equilibriumE3of System1.11is locally asymptotically stable if α−2βk1−2c1r2

c2 >0, c2−2c1>0. 3.30 Remark 3.5. From3.30, we know that if the birth rate of immature preyαis sufficiently large and the maximum value of the per capita reduction rate ofxdue toyis smaller than the maximum value of the per capita reduction rate ofydue toxthen the positive equilibrium E3is locally asymptotically stable.

Proof. We only need to prove thatE3 has no stability switches asτ increases and thatE3 is stable atτ 0. Consider the roots of3.20, by the above discussion, we know if3.30holds then

P10 Q10 r22βx^∗₂ c1k1y^∗

k1x^∗₂₂ −α >0. 3.31

So the roots of3.20must have negative real parts, henceE3is stable atτ 0. Next, we prove thatE3has no stability switches asτincreases in0, τ^∗. We only need to prove that System 3.26has no positive rootsωτinI.

From3.26, we have

cτ P₀²τ−Q²₀τ P0τ Q0τP0τ−Q0τ. 3.32

We know thatP0τ Q0τ>0 and

P0τ−Q0τ r2

2βx^∗₂ c1k1y^∗

k1x^∗₂2 αe^{−γ τ}

>0. 3.33

Socτ>0.

By3.26, we also have bτ −2P0τ P₁²τ−Q²₁τ

−2

2c2y^∗ k2x^∗₂ −r2

2βx^∗₂ c1k1y^∗ k1x^∗₂₂

−2 c1c2x₂^∗y^∗2 k1x^∗₂

k2x^∗₂₂

−r2 2c2y^∗

k2x^∗₂ 2βx^∗₂ c1k1y^∗ k1x^∗₂2

₂

−α²e^{−2γ τ}

2c2y^∗ k2x₂^∗−r2

2

2βx^∗₂ c1k1y^∗ k1x₂^∗2

₂

−2 c1c2x₂^∗y^∗2 k1x^∗₂

k2x^∗₂2 −α²e^{−2γ τ}

c2y^∗ k2x₂^∗

₂

−2 c1c2x^∗₂y^∗2 k1x^∗₂

k2x^∗₂₂

2βx^∗₂ c1k1y^∗

k1x^∗₂₂ αe^{−γ τ}

2βx₂^∗ c1k1y^∗

k1x^∗₂₂ −αe^{−γ τ}

>

2βx^∗₂ c1k1y^∗

k1x^∗₂₂ αe^{−γ τ}

2βx^∗₂ c1k1y^∗

k1x₂^∗₂ −αe^{−γ τ}

c²₂y^∗2

k2x^∗₂₂ − 2c1c2y^∗2 k2x^∗₂₂

>0,

3.34 the last inequality holds because3.30and therefore we have thatbτ > 0 andcτ > 0.

HenceFω, τ/0 for allτ ∈ I 0, τ^∗, that is, there are no stability switches forτ ∈ I 0, τ^∗. The proof is complete.

4. Global Stability and Attractiveness

In this section, we establish conditions for the global stability of equilibriaE2 0, k2r2/c2 andE3 x^∗₂, y^∗of System1.11. The following theorems hold.

Theorem 4.1. Suppose that

M1k1

c1m2 αe^{−γ τ}<1, 4.1

wherem2k2r2/c2,M1αe^{−γ τ}/β 1, then the equilibriumE2 0, k2r2/c2of System1.11is globally asymptotically stable.

Remark 4.2. From4.1, we also find thatγτhas a positive effect on the extinction of prey in that a proper increase ofγτ which is defines as the “degree of stage structure” by Liu et al.

27can drive the prey into extinction, regardless of how large other coefficients were.

Remark 4.3. Inequality4.1is equivalent to

Ê0< 1 1αe^{−γ τ}/βk1

. p1. 4.2

That is, a small basic demographic number^Ê0 < p1 <1for the preywhen the predator’s population size reaches its steady state in the absence of prey can guarantee the prey’s extinctionE2is globally stable.

Proof. FromTheorem 3.1, we know thatE2 is locally asymptotically stable. Now, we only need to prove global attractiveness ofE2. By the first equation of System1.11, the proof of Theorems2.3and2.4, andx2tis nonegative, we have that

x₂t αe^{−γ τ}x2t−τ−βx²₂t−c1ytx2t x2t k1

≤αe^{−γ τ}x2t−τ−βx²₂t−c1m2x2t M1k1 .

4.3

FromLemma 2.1and4.1, we obtain that

tlim→ ∞x2t 0. 4.4

Then, there is aT0such that, fort > T0, we have− < x2t< , where is sufficiently small.

From the second equation of System1.11, we have that

yt≤yt

r2−c2yt k2

, 4.5

and, by the comparison principle, we conclude that yt≤ k2 r2

c2 , 4.6

and consequently lim_{t→ ∞}infyt≥k2r2/c2. Hence, we have that

t→ ∞limyt k2r2

c2 . 4.7

The proof is complete.

Next, we study the global attractivity of the interior equilibriumE3of System1.11.

Consider the following system:

vt a4vt−τ−a3v²t− a1vt vt a2, vt ϕt≥0, fort∈−τ,0,

v0>0,

4.8

whereai > 0,i 1,2,3,4. A similar reasoning usingLemma 2.2gives thatvt > 0 for all t≥0. From Theorem 4.9.1 in Kuang26we conclude by the following lemma.

Lemma 4.4. System4.8has a unique positive equilibrium

v^∗

a4−a2a3 a4−a2a3²4a3a2a4−a1

2a3 4.9

which is globally asymptotically stable ifa2a4−a1>0.

Finally, we have the following result.

Theorem 4.5. Suppose that

αe^{−γ τ}k1−r2

c1

k2βαe^{−γ τ} c2β

>0, β >1, αe^{−γ τ}−βk1−c1r2

c2 >0,

4.10

then the positive equilibriumE3in System1.11is globally attractive.

Remark 4.6. From4.10, we know that γτ has a negative effect on the global attractivity of positive equilibrium; that is, an increase in the value ofγτcan destroy Condition4.10.

Remark 4.7. Comparing Theorems4.1and 4.2 with Theorems 4.1 and 4.2 in Song et al.25, we also see the inclusion of an extra terme^{−γ τ}in our condition, that is, the surviving probability of each immature prey becomes mature must be taken into account.

Proof. By the first equation of System1.11, we have

x₂t≤αe^{−γ τ}x2t−τ−βx₂²t, 4.11 then byLemma 2.1and the comparison principle, for sufficiently small >0, there is aT1>0 such that

x2t< αe^{−γ τ}

β u1 4.12

fort≥T1τ. Replacing this inequality into the second equation of1.11, we have

yt≤yt

r2− c2yt u1k2

, t≥T1. 4.13

Again by the comparison principle, there is aT2 > T1τ >0 such that

yt< u1k2r2

c2 ν1, t≥T2. 4.14

Substituting4.14into the first equation of1.11, we have

x₂t≥αe^{−γ τ}x2t−τ−βx²₂t− c1ν1x2t

x2t k1. 4.15

Consider the following equation:

zt αe^{−γ τ}zt−τ−βz²t− c1ν1zt

zt k1. 4.16

From the first inequality of4.14andLemma 4.4, we see that4.16has a unique positive equilibriumz^∗ αe^{−γ τ} −βk1 αe^{−γ τ}−βk1²4βαe^{−γ τ}k1−c1ν1/2βwhich is globally asymptotically stable. Using the comparison principle, for sufficiently small > 0, we see that there is aT3> T2τsuch that

x2t> z^∗− u₁>0. 4.17

Plugging4.17into the second equation of1.11, we have that

yt≥yt

r2− c2yt u₁k2

, t≥T3. 4.18

By the comparison principle, there isT4> T3such that

yt>

u₁k2

r2

c2 − ν1, t≥T4. 4.19

Hence, we have

u₁< xt< u1, ν1< yt< ν1, t≥T4. 4.20

By replacing4.19in the first equation of1.11we see that

x₂t≤αe^{−γ τ}x2t−τ−βx²₂t− c1ν₁x2t

x2t k1. 4.21

From a similar use of the comparison principle, we conclude that there isT5> T4τsuch that x2t< z^∗₁ u2>0, t≥T5, 4.22

where z^∗₁ αe^{−γ τ} −βk1 αe^{−γ τ}−βk1²4αe^{−γ τ}k1−c1ν₁β/2β > 0 is the positive equilibrium for the equation

zt αe^{−γ τ}zt−βz²t− c1ν₁zt

zt k1. 4.23

From4.10, we have

u2 < u1. 4.24

Substituting4.22into the second equation in1.11, we have that

yt≤yt

r2− c2yt u2k2

, t≥T5. 4.25

A similar discussionas aboveimplies that for sufficiently small >0, there is aT6> T5such that

yt< u2k2r2

c2 ν2. 4.26

Sinceu2< u1, we get

ν2 < ν1. 4.27

Plugging4.26into the first equation of1.11leads to

x₂t> αe^{−γ τ}x2t−τ−βx²₂t− c1ν2x2t

x2t k1, t≥T6. 4.28 From4.10,Lemma 4.4and the comparison principle, we see that for sufficiently small >0, there is aT7> T6τsuch that

x2t> z^∗₂− u₂>0, t≥T7, 4.29

where z^∗₂ αe^{−γ τ} −βk1 αe^{−γ τ}−βk1²4αe^{−γ τ}k1−c1ν2β/2β > 0 is the positive equilibrium for the equation

zt αe^{−γ τ}zt−βz²t− c1ν2zt

zt k1. 4.30

Moreover, sinceν2< ν1we have thatu₂> u₁.

Replacing4.22in the second equation of1.11leads to

yt≥yt

r2− c2yt u₂k2

, t≥T7. 4.31

Arguments similar to those used above guarantee the existence of aT8> T7such that

yt>

u₂k2

r2

c2 − ν2, t≥T8, 4.32

from which we get thatν2> ν1.

Repeating the above process leads to the construction of the sequencesun^∞_n1,u_n^∞_n1, νn^∞_n1,ν_n^∞_n1, andT4n>0. Fort≥T4n, we have that

0< u₁< u₂<· · ·< u_n< x2t< un<· · ·< u2< u1,

0< ν₁ < ν₂<· · ·< ν_n< yt< νn <· · ·< ν2< ν1. 4.33

Hence, the limits ofun^∞_n1,u_n^∞_n1,νn^∞_n1,ν_n^∞_n1exist. Denote that

u lim

t→ ∞un, ν lim

t→ ∞νn, u lim

t→ ∞u_n, ν lim

t→ ∞ν_n, 4.34

thenu≥u,ν≥ν. To complete the proof, we only need to showuu,νν.

By the definition ofνn andν_n, we have

ν_n

u_nk2

r2

c2 − , νn unk2r2

c2 , 4.35

thus

νn−ν_n r2

c2

un−u_n

2 . 4.36

According to the definitions ofun,u_nand4.36, we have un−u_n

αe^{−γ τ}−βk1 αe^{−γ τ}−βk1

₂ 4β

αe^{−γ τ}k1−c1ν_n 2β

−αe^{−γ τ}−βk1 αe^{−γ τ}−βk1

₂

4βαe^{−γ τ}k1−c1νn

2β 2

− 4c1

ν_n−νn

β 2β αe^{−γ τ}−βk1

₂ 4β

αe^{−γ τ}k1−c1ν_n

αe^{−γ τ}−βk1

₂

4βαe^{−γ τ}k1−c1νn

2

<− c1

ν_n−νn

αe^{−γ τ}−βk1

2 .

4.37

Letn → ∞, we have

u−u

≤ c1r2/c2 u−u

2

αe^{−γ τ}−βk1 2 , 4.38

hence

αe^{−γ τ}−βk1−c1r2

c2

u−u

≤

1αe^{−γ τ}−βk1

2 . 4.39

By4.10, we know thatαe^{−γ τ}−βk1−c1r2/c2>0 and1αe^{−γ τ}−βk1>0. Note that can be arbitrarily small, that is, letting → 0 leads to the conclusion thatuu. From4.36and lettingn → ∞, we also conclude thatνν. The proof is complete.

5. Discussion

In this paper, we consider a Leslie-Gower predator-prey type model that incorporates the prey “age” structure an extension of the ODE model in the study by Aziz-Alaoui and Daher Okiye 11. We derive the “conditional” basic demographic number ^Ê0 for the prey, that is the value of ^Ê₀ when the predator’s population size has reached its steady state in the absence of prey. We obtain sufficient conditions that ensure the boundedness of solutions as well as permanence of System1.11 ^Ê0 > p0>1. Second, we derive sufficient conditions for the local stability of nonnegative equilibria of Model 1.11. We show that E0 0,0andE1 αe^{−γ τ}/β,0are unstable,E2 0, k2r2/c2is unstable if^Ê0>1stable if

Ê0 <1, and the positive equilibriumE3exists when^Ê0>1. Finally, through the application

0 100 200 300 400 500 0

0.2 0.4 0.6 0.8 1 1.2 1.4

x2

y

Figure 1: The boundary equilibriumE2 0, k2r2/c2of System1.11is globally asymptotically stable.

0 100 200 300 400 500

0 1 2 3 4 5 6 7 8

Timet

x,y

x2

y

Figure 2: The positive equilibriumE3of System1.11is globally attractive.

of the comparison principle, sufficient conditions for the global attractivity of nonnegative equilibria are obtained. We prove thatE2is globally asymptotically stable when^Ê0< p1<1.

We conclude that the incorporation of a delay“age” structure in the preydoes not change the asymptotic behavior of the model when some restrictions are imposed on the effect of such delay. Here we provide two numerical examples to illustrate our main results.