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,**

^{1}**Xiaohong Wang,**

^{2}**and Carlos Castillo-Chavez**

^{2}*1**Institute of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China*

*2**Department 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 let*yt*be the density of predator, then the classical Lotka-
Volterra predator-prey model is given by the following system:

*x*^{}t

*r*1−*c*1*y*−*b*1*x*
*x,*
*y*^{}t

−ε2*ρ*2*x*

*y.* 1.1

It is known that these equations can support population fluctuations when *b*1 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, if*xt*denotes the prey density
and *yt*the predators’, then Leslie’s model is given by the following system of nonlinear
diﬀerential equations:

*x*^{}t

*r*1−*c*1*y*−*b*1*x*
*x,*
*y*^{}t

*r*2−*c*2

*y*
*x*

*y,* 1.2

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 percapita growth rate of the predatordy/ydtachieves its maximum
*r*2. 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,

*y*^{}t *r*2

1− *y*

*Ax*

*yr*2

1− *y*

*C*

*y,* 1.3

where*Ar*2*/c*2can be interpreted as a prey predators’ conversion factor and*CAx*as the
predators’ carrying capacityproportional to prey abundance. The Leslie-Gower term*y/Ax*
has also been interpreted as a measure of the loss in percapita predator’s reproduction rate
due to the relative abundanceper capita*y/x*of its “favorite” foodprey*x. Model*1.2is
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 constant*d*in the denominator. In fact,

*y*^{}t *r*2

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:

*x*^{}t

*r*1− *c*1*y*
*xk*1 −*b*1*x*

*x,*
*y*^{}t

*r*2− *c*2*y*
*xk*2

*y,*

1.5

where*r*1 is the percapita growth rate of the prey*x,b*1 is a measure of the strength of prey
on preyinterference competition,*c*1is the maximum value of the per capita reduction rate
of prey *x* due to predator *y,* *k*1 measures the extent to which the environment provides
protection to prey *x* k2 for predator *y,* *r*2 gives the maximal percapita growth rate of
predator*y, andc*2has a similar meaning to that of*c*1.

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 delay*τ*:

*x*^{}t

*r*1− *c*1*y*
*xk*1 −*b*1*x*

*x,*
*y*^{}t

*r*2− *c*2*yt*−*τ*
*xt*−*τ k*2

*y,*

1.6

that is, a single discrete delay*τ >* 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 of*τ* 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:

*x*^{}_{1}t *αx*2t−*γx*1t−*αe*^{−γ τ}*x*2t−*τ*,

*x*^{}_{2}t *αe*^{−γ τ}*x*2t−*τ*−*βx*^{2}_{2}t, 1.7
where*x*1t,*x*2tdenote 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 and*τ* is the “fixed” time to maturity, and the term
*αe*^{−γ τ}*x*2t−*τ*models the immature individuals who were born at time *t*−*τ* i.e.,*αx*2t−
*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 *xt*of the system, there exist
constants*M,m >* 0 such that*m* ≤ lim inf_{t→ ∞}*xt* ≤ lim sup_{t}_{→ ∞}*xt* ≤ *M*then 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 is*α >*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 constant*β >*0, and immature individuals become mature
at age*τ*. 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:

*x*^{}_{1}t *αx*2t−*γx*1t−*αe*^{−γ τ}*x*2t−*τ*,
*x*^{}_{2}t *αe*^{−γ τ}*x*2t−*τ*−*βx*_{2}^{2}t−*c*1*ytx*2t

*x*2t *k*1 *,*
*y*^{}t *yt*

*r*2− *c*2*yt*
*x*2t *k*2

*.*

1.8

The initial conditions are given by *x*2θ ≥ 0, continuous on *θ* ∈ −τ,0, and *x*10,
*x*20, y0*>*0, while*x*1t,*x*2t, and*yt*denote 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

*x*1t
_{t}

*t−τ**αe*^{−γt−s}*x*2sds, 1.9

*x*10
_{0}

−τ*αe*^{γ s}*x*2sds. 1.10

The last two equations in1.8do not contain*x*1t. Hence, if we know the properties of*x*2t
then the properties of*x*1tcan be easily obtained from1.8and1.9. Hence, we only need
to consider the following system:

*x*^{}_{2}t *αe*^{−γ τ}*x*2t−*τ*−*βx*_{2}^{2}t−*c*1*ytx*2t
*x*2t *k*1 *,*
*y*^{}t *yt*

*r*2− *c*2*yt*
*x*2t *k*2

*,*

1.11

with initial conditions*x*2θ≥0continuous on*θ*∈−τ,0and*x*20, 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:**

*x*^{}t *axt*−*τ*−*bxt*−*cx*^{2}t, 2.1

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

i*Ifa > b, then lim**t*→ ∞*xt a*−*b/c.*

ii*Ifa < b, then lim**t*→ ∞*xt 0.*

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

* Lemma 2.2. Supposex*2θ≥

*0 is continuous onθ*∈−τ,0, and

*x*20,

*y0>0, then the solution*

*of System*1.11

*satisfiesx*2t,

*yt>0 for allt >0.*

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

**Theorem 2.3. Suppose***x*2θ ≥ *0 is continuous on* *θ* ∈ −τ,0, and *x*20,*y0* *>* *0, then the*
*solutions of* 1.11*are bounded for all larget.*

*Proof. From the first equation of*1.11, we have

*x*^{}_{2}t≤*αe*^{−γ τ}*x*2t−*τ*−*βx*_{2}^{2}t. 2.2
According toLemma 2.1and the standard comparison principle28, there exists a*T*1 *>* 0
and 1*>*0 such that

*x*2t≤ *αe*^{−γ τ}

*β* 1*M*1*,* for*t > T*1*τ.* 2.3

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

*y*^{}t≤*yt*

*r*2− *c*2*yt*
*M*1*k*2

*,* for*t > T*1*τ.* 2.4

From the comparison principle, there exists a*T*2*> T*1such that, for any suﬃciently small 2,
*yt*≤ M1*k*2r2

*c*2 2*M*2*,* for*t > T*2*τ.* 2.5

The proof is complete.

Now, we show that System1.11is permanent.

**Theorem 2.4. Suppose that**

*αe*^{−γ τ}−*c*1*M*2

*k*1 *>*0, 2.6

*whereM*2*is defined by*2.5, then System1.11*is 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 system*1.11, we have

*y*^{}t≥*yt*

*r*2−*c*2*yt*
*k*2

*.* 2.7

It then follows that

*t→ ∞*liminf*yt*≥ *k*2*r*2

*c*2 *m*2*>*0. 2.8

Using the first equation of System1.11andTheorem 2.3, for suﬃciently large*T*, we have

*x*^{}_{2}t≥*αe*^{−γ τ}*x*2t−*τ*−*βx*^{2}_{2}t−*c*1*ytx*2t
*k*1

≥*αe*^{−γ τ}*x*2t−*τ*−*βx*^{2}_{2}t−*c*1*M*2*x*2t
*k*1 *.*

2.9

ByLemma 2.1and the comparison principle, we have that

*t→ ∞*liminf*x*2t≥ *αe*^{−γ τ}−*c*1*M*2*/k*1

*β* *m*1*>*0. 2.10

Therefore, the above calculations andTheorem 2.3imply that there exist*M**i*,*m**i* *>*0,*i*1,2,
such that

0*< m*1≤ lim

*t→ ∞*inf*x*2t≤ lim

*t→ ∞*sup *x*2t≤*M*1*,*
0*< m*2≤ lim

*t*→ ∞inf*yt*≤ lim

*t→ ∞*sup *yt*≤*M*2*.*

2.11

The proof is complete.

**3. Analysis of Equilibria**

System1.11has the following nonnegative equilibria:

*E*0 0,0, *E*1
*αe*^{−γ τ}

*β* *,*0

*,* *E*2

0,*k*2*r*2

*c*2

*,* *E*3

*x*^{∗}_{2}*, y*^{∗}

*,* 3.1

where

*x*^{∗}_{2}

*αe*^{−γ τ}−*βk*1−*c*1*r*2*/c*2

*αe*^{−γ τ}−*βk*1−*c*1*r*2*/c*2

2−4βc1*r*2*/c*2k2−*αe*^{−γ τ}*k*1

2β *,*

*y*^{∗} *k*2*r*2*r*2*x*^{∗}_{2}
*c*2 *.*

3.2

We see that the positive equilibrium*E*3exists if
*αe*^{−γ τ}*>* *c*1*k*2*r*2

*c*2*k*1 *.* 3.3

The characteristic equation at equilibrium*E*0is

*λ*−*αe*^{−γ τ}*e*^{−λτ}

λ−*r*2 0, 3.4
and, consequently, since it has a positive eigenvalue*λr*2,*E*0is unstable.

The characteristic equation at equilibrium*E*1is given by the transcendental equation

*λ*−

−2αe^{−γ τ}*αe*^{−γ τ}*e*^{−λτ}

λ−*r*2 0. 3.5
Again,*λr*2is a positive eigenvalue, so*E*1is also unstable.

The analysis of the stability of*E*2 requires a little more work. We have the following
results.

**Theorem 3.1. Let**

Ê0 *c*2*k*1

*c*1*k*2*r*2*αe*^{−γ τ}*,* 3.6

*then equilibriumE*2*is*
i*unstable if*^{Ê}0*>1,*

ii*linearly neutrally stable if*^{Ê}_{0} *1,*
iii*locally asymptotically stable if*^{Ê}_{0}*<1.*

*Proof.* i The characteristic equation of equilibrium*E*2is given by

*λc*1*k*2*r*2

*c*2*k*1 −*αe*^{−γ τ}*e*^{−λτ}

λ*r*2 0, 3.7

clearly, one characteristic root is*λ*−r2*<*0, others are the roots of
*Fλ λc*1*k*2*r*2

*c*2*k*1 −*αe*^{−γ τ}*e*^{−λτ} 0. 3.8

Assume that^{Ê}_{0} *>* 1, therefore*< αe*^{−γ τ} then*F*0 *<* 0 and*F∞ ∞. HenceFλ*has at
least one positive root and*E*2is unstable.

iiSince^{Ê}_{0}1, that is,*c*1*k*2*r*2*/c*2*k*1*αe*^{−γ τ},*F0 *0, so*λ*0 is a root of*Fλ *0. As
*F*^{}λ *ταe*^{−γ τ}*e*^{−λτ}1, we have*F*^{}0*>*0. The root*λ*0 is simple. If other roots are of form
*aiω, for somea*and*ω*in*R, they satisfy*

*aαe*^{−γ τ}_{2}

*ω*^{2}
*αe*^{−γ τ}_{2}

*e*^{−2aτ}*.* 3.9

Then, we must have*a* ≤ 0; that is, all other roots have nonpositive real parts. Hence*E*2 is
linearly neutrally stable.

iiiIf^{Ê}_{0}*<*1, then*c*1*k*2*r*2*/c*2*k*1*> αe*^{−γ τ}. Assume that there exists an eigenvalue*λ*with
Re*λ*≥0, then we have

Re*λ*−*c*1*k*2*r*2

*c*2*k*1 *αe*^{−γ τ}*e*^{−Re}* ^{λτ}*cosτIm

*λ*

≤*αe*^{−γ τ}*e*^{−Re}* ^{λτ}*−

*c*1

*k*2

*r*2

*c*2*k*1 *<*0.

3.10

It is a contradiction, so Re*λ <*0. This shows that all roots of*Fλ *0 must have negative real
parts, hence, the equilibrium*E*2is locally asymptotically stable.

The proof of the theorem is complete.

*Remark 3.2. Note that when the predator reaches its steady statey* *k*2*r*2*/c*2in the absence
of prey,*αe*^{−γ τ}can be interpreted as the per capita recruitment rate of prey and*c*1*k*2*r*2*/c*2*k*1
*c*1*y/k* 1approximates the per capita death rate of the prey. Therefore,^{Ê}0*αe*^{−γ τ}*c*2*k*1*/c*1*k*2*r*2is
the basic demographic number of prey when the predator’s population size reaches its steady
state*y*in the absence of prey*x. When*^{Ê}0*>*1, the population size of prey will increase, thus
*E*2is unstable. Similarly we can interpretiiandiiiinTheorem 3.1.

*Remark 3.3. The suﬃcient condition given by*2.6for the permanence of System1.11can
be rewritten in the following form

Ê0*>*1*αe*^{−γ τ}
*k*2*β*

*.* *p*0*.* 3.11

So a “large” basic demographic number ^{Ê}0 *> p*0 *>* 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 equilibrium*E*3 x_{2}^{∗}*, y*^{∗}. Recall
there exists*E*3when3.3holds, that is, when*τ*is in the interval*I* 0, τ^{∗}, where

*τ*^{∗} 1

*γ* ln *c*2*αk*1

*c*1*r*2*k*2*.* 3.12

The characteristic equation at*E*3is

*Dλ, τ λ*^{2}

*r*22βx^{∗}_{2} *c*1*k*1*y*^{∗}
*k*1*x*_{2}^{∗}_{2}

*λr*2

2βx_{2}^{∗} *c*1*k*1*y*^{∗}
*k*1*x*^{∗}_{2}_{2}

*c*1*c*2*x*^{∗}_{2}*y*^{∗}^{2}
*k*1*x*^{∗}_{2}

*k*2*x*^{∗}_{2}_{2}

*r*2−*λ*− 2c2*y*^{∗}
*k*2*x*^{∗}_{2}

*αe*^{−γ τ}*e*^{−λτ}0.

3.13

Let

*Pλ, τ λ*^{2}*P*1τλ*P*0τ,

*Qλ, τ λQ*1τ *Q*0τ, 3.14

where

*P*1τ *r*22βx^{∗}_{2} *c*1*k*1*y*^{∗}
*k*1*x*^{∗}_{2}2*,*

*P*0τ *r*2

2βx^{∗}_{2} *c*1*k*1*y*^{∗}
*k*1*x*^{∗}_{2}2

*c*1*c*2*x*^{∗}_{2}*y*^{∗}^{2}
*k*1*x*^{∗}_{2}

*k*2*x*^{∗}_{2}2*,*
*Q*1τ −αe^{−γ τ}*,* *Q*0τ −r2*αe*^{−γ τ}*.*

3.15

Then the characteristic equation at*E*3becomes

*Dλ, τ* *Pλ, τ Qλ, τe*^{−λτ} 0. 3.16

First, we will prove

*P0, τ* *Q0, τ/*0, 3.17

that is,*λ*0 cannot be a root of3.16for any*τ*∈*I.*

In fact, by the definition ofx^{∗}_{2}*, y*^{∗}, we have
*D0, τ* *P*0τ *Q*0τ *P0, τ Q0, τ*

*r*2

2βx^{∗}_{2} *c*1*k*1*y*^{∗}
*k*1*x*^{∗}_{2}_{2}

*c*1*c*2*x*_{2}^{∗}*y*^{∗}^{2}
*k*1*x*^{∗}_{2}

*k*2*x*^{∗}_{2}_{2} −*r*2*αe*^{−γ τ}
*r*2

2βx^{∗}_{2} *c*1*k*1*y*^{∗}

*k*1*x*^{∗}_{2}_{2} −*αe*^{−γ τ}

*c*1*c*2*x*^{∗}_{2}*y*^{∗}^{2}
*k*1*x*^{∗}_{2}

*k*2*x*^{∗}_{2}_{2}
*r*2

⎛

⎝−βk1−*c*1*r*2

*c*2

*αe*^{−γ τ}−*βk*1−*c*1*r*2

*c*2

_{2}

−4β
*c*1*r*2

*c*2 *k*2−*αe*^{−γ τ}*k*1

⎞⎠

*>*0.

3.18

Therefore,*λ*0 is not a root of3.16.

The characteristic equation3.16at*τ* 0 is

*P*λ,0 *Qλ,*0 0, 3.19

that is,

*λ*^{2} P10 *Q*10λ*P*00 *Q*00 0. 3.20

Then,

*λ*^{2}

*r*22βx^{∗}_{2} *c*1*k*1*y*^{∗}
*k*1*x*^{∗}_{2}_{2} −*α*

*λr*2

2βx^{∗}_{2} *c*1*k*1*y*^{∗}

*k*1*x*^{∗}_{2}_{2} *c*1*c*2*x*_{2}^{∗}*y*^{∗}^{2}
*k*1*x*_{2}^{∗}

*k*2*x*^{∗}_{2}_{2} −*α*

0.

3.21

Since*P*0τ *Q*0τ*>*0 for all*τ*∈0, τ^{∗}, then*P*00 *Q*00*>*0. Notice that

*P*10 *Q*10 *r*22βx^{∗}_{2} *c*1*k*1*y*^{∗}

*k*1*x*_{2}^{∗}_{2} −*α.* 3.22

If*P*10Q10*>*0, then3.20has two solutions with negative real parts. Hence,*E*3is locally
asymptotically stable at*τ*0. If*P*10 *Q*10*<*0, then*E*3is unstable at*τ*0.

To determine the local stability of the interior equilibrium*E*3 x^{∗}_{2}*, y*^{∗}, we proceed as
follows29.

Assume that*λ*±iωτ,*ωτ>*0 satisfy3.16, we have
*Piω, τ* −ω^{2}*iωP*1τ *P*0τ,
*P**R*iω, τ *P*0τ−*ω*^{2}*,* *P**I*iω, τ *ωP*1τ,

*Qiω, τ* *iωQ*1τ *Q*0τ,

*Q**R*iω, τ *Q*0τ, *Q**I*iω, τ *ωQ*1τ.

3.23

The first step is to look for the positive roots*ωτ>*0 of

*Fω, τ* |Piω, τ|^{2}− |Qiω, τ|^{2}0 3.24

in*I* 0, τ^{∗}. Since

*Fω, τ* *ω*^{4}*ω*^{2}

−2P0τ *P*_{1}^{2}τ−*Q*_{1}^{2}τ

*P*_{0}^{2}τ−*Q*^{2}_{0}τ, 3.25

we have

*Fω, τ ω*^{4}*bτω*^{2}*cτ* 0,
*bτ* −2P0τ *P*_{1}^{2}τ−*Q*^{2}_{1}τ,

*cτ* *P*_{0}^{2}τ−*Q*^{2}_{0}τ.

3.26

Depending on the signs of*bτ*and*cτ, System*3.26may have no positive real roots, or
the root

*ω*τ
1

2

−bτ *bτ*^{2}−4cτ
_{1/2}

*,* *τ* ∈*I*⊆*I,* 3.27

or otherwise the root
*ω*−τ

1 2

−bτ− *bτ*^{2}−4cτ
_{1/2}

*,* *τ* ∈*I*−⊆*I,* 3.28

or, as the last case, both*ω*τand*ω*−τ. Note that if System3.26has no positive roots*ωτ*
in*I, then no stability switches can occur.*

From the structure of*P*10 *Q*10, a suﬃcient condition for*E*3at*τ* 0 to be locally
asymptotically stable is given by

*α*−2βk1−2c1*r*2

*c*2 *>*0, 3.29

which implies*P*10 *Q*10*>* 0. Stability switches for increasing*τ* in*I* 0, τ^{∗}may occur
only with a pair of roots*λ*±iωτ ωτreal positivethat cross the imaginary axis.

Next, we state the following theorem on the local asymptotic stability of equilibrium
*E*3.

* Theorem 3.4. The positive equilibriumE*3

*of System*1.11

*is locally asymptotically stable if*

*α*−2βk1−2c1

*r*2

*c*2 *>*0, *c*2−2c1*>*0. 3.30
*Remark 3.5. From*3.30, we know that if the birth rate of immature preyαis suﬃciently
large and the maximum value of the per capita reduction rate of*x*due to*y*is smaller than
the maximum value of the per capita reduction rate of*y*due to*x*then the positive equilibrium
*E*3is locally asymptotically stable.

*Proof. We only need to prove thatE*3 has no stability switches as*τ* increases and that*E*3 is
stable at*τ* 0. Consider the roots of3.20, by the above discussion, we know if3.30holds
then

*P*10 *Q*10 *r*22βx^{∗}_{2} *c*1*k*1*y*^{∗}

*k*1*x*^{∗}_{2}_{2} −*α >*0. 3.31

So the roots of3.20must have negative real parts, hence*E*3is stable at*τ* 0. Next, we prove
that*E*3has no stability switches as*τ*increases in0, τ^{∗}. We only need to prove that System
3.26has no positive roots*ωτ*in*I*.

From3.26, we have

*cτ* *P*_{0}^{2}τ−*Q*^{2}_{0}τ P0τ *Q*0τP0τ−*Q*0τ. 3.32

We know that*P*0τ *Q*0τ*>*0 and

*P*0τ−*Q*0τ *r*2

2βx^{∗}_{2} *c*1*k*1*y*^{∗}

*k*1*x*^{∗}_{2}2 *αe*^{−γ τ}

*>*0. 3.33

So*cτ>*0.

By3.26, we also have
*bτ *−2P0τ *P*_{1}^{2}τ−*Q*^{2}_{1}τ

−2

2c2*y*^{∗}
*k*2*x*^{∗}_{2} −*r*2

2βx^{∗}_{2} *c*1*k*1*y*^{∗}
*k*1*x*^{∗}_{2}_{2}

−2 *c*1*c*2*x*_{2}^{∗}*y*^{∗}^{2}
*k*1*x*^{∗}_{2}

*k*2*x*^{∗}_{2}_{2}

−r2 2c2*y*^{∗}

*k*2*x*^{∗}_{2} 2βx^{∗}_{2} *c*1*k*1*y*^{∗}
*k*1*x*^{∗}_{2}2

_{2}

−*α*^{2}*e*^{−2γ τ}

2c2*y*^{∗}
*k*2*x*_{2}^{∗}−*r*2

2

2βx^{∗}_{2} *c*1*k*1*y*^{∗}
*k*1*x*_{2}^{∗}2

_{2}

−2 *c*1*c*2*x*_{2}^{∗}*y*^{∗}^{2}
*k*1*x*^{∗}_{2}

*k*2*x*^{∗}_{2}2 −*α*^{2}*e*^{−2γ τ}

*c*2*y*^{∗}
*k*2*x*_{2}^{∗}

_{2}

−2 *c*1*c*2*x*^{∗}_{2}*y*^{∗}^{2}
*k*1*x*^{∗}_{2}

*k*2*x*^{∗}_{2}_{2}

2βx^{∗}_{2} *c*1*k*1*y*^{∗}

*k*1*x*^{∗}_{2}_{2} *αe*^{−γ τ}

2βx_{2}^{∗} *c*1*k*1*y*^{∗}

*k*1*x*^{∗}_{2}_{2} −*αe*^{−γ τ}

*>*

2βx^{∗}_{2} *c*1*k*1*y*^{∗}

*k*1*x*^{∗}_{2}_{2} *αe*^{−γ τ}

2βx^{∗}_{2} *c*1*k*1*y*^{∗}

*k*1*x*_{2}^{∗}_{2} −*αe*^{−γ τ}

*c*^{2}_{2}*y*^{∗2}

*k*2*x*^{∗}_{2}_{2} − 2c1*c*2*y*^{∗2}
*k*2*x*^{∗}_{2}_{2}

*>*0,

3.34
the last inequality holds because3.30and therefore we have that*bτ* *>* 0 and*cτ* *>* 0.

Hence*Fω, τ/*0 for all*τ* ∈ *I* 0, τ^{∗}, that is, there are no stability switches for*τ* ∈ *I*
0, τ^{∗}. The proof is complete.

**4. Global Stability and Attractiveness**

In this section, we establish conditions for the global stability of equilibria*E*2 0, k2*r*2*/c*2
and*E*3 x^{∗}_{2}*, y*^{∗}of System1.11. The following theorems hold.

**Theorem 4.1. Suppose that**

*M*1*k*1

*c*1*m*2 *αe*^{−γ τ}*<*1, 4.1

*wherem*2*k*2*r*2*/c*2*,M*1*αe*^{−γ τ}*/β* 1*, then the equilibriumE*2 0, k2*r*2*/c*2*of System*1.11*is*
*globally asymptotically stable.*

*Remark 4.2. From*4.1, we also find that*γτ*has a positive eﬀect on the extinction of prey in
that a proper increase of*γτ* 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. Inequality*4.1is equivalent to

Ê0*<* 1
1*αe*^{−γ τ}*/βk*1

*.* *p*1*.* 4.2

That is, a small basic demographic number^{Ê}0 *< p*1 *<*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. From*Theorem 3.1, we know that*E*2 is locally asymptotically stable. Now, we only
need to prove global attractiveness of*E*2. By the first equation of System1.11, the proof of
Theorems2.3and2.4, and*x*2tis nonegative, we have that

*x*^{}_{2}t *αe*^{−γ τ}*x*2t−*τ*−*βx*^{2}_{2}t−*c*1*ytx*2t
*x*2t *k*1

≤*αe*^{−γ τ}*x*2t−*τ*−*βx*^{2}_{2}t−*c*1*m*2*x*2t
*M*1*k*1 *.*

4.3

FromLemma 2.1and4.1, we obtain that

*t*lim→ ∞*x*2t 0. 4.4

Then, there is a*T*0such that, for*t > T*0, we have− < x2t*< , where* is suﬃciently small.

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

*y*^{}t≤*yt*

*r*2−*c*2*yt*
*k*2

*,* 4.5

and, by the comparison principle, we conclude that
*yt*≤ k2* r*2

*c*2 *,* 4.6

and consequently lim_{t}_{→ ∞}inf*yt*≥*k*2*r*2*/c*2. Hence, we have that

*t→ ∞*lim*yt * *k*2*r*2

*c*2 *.* 4.7

The proof is complete.

Next, we study the global attractivity of the interior equilibrium*E*3of System1.11.

Consider the following system:

*v*^{}t *a*4*vt*−*τ*−*a*3*v*^{2}t− *a*1*vt*
*vt a*2*,*
*vt ϕt*≥0, for*t*∈−τ,0,

*v0>*0,

4.8

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

* Lemma 4.4. System*4.8

*has a unique positive equilibrium*

*v*^{∗}

*a*4−*a*2*a*3 a4−*a*2*a*3^{2}4a3a2*a*4−*a*1

2a3 4.9

*which is globally asymptotically stable ifa*2*a*4−*a*1*>0.*

Finally, we have the following result.

**Theorem 4.5. Suppose that**

*αe*^{−γ τ}*k*1−*r*2

*c*1

*k*2*βαe*^{−γ τ}
*c*2*β*

*>*0, *β >*1,
*αe*^{−γ τ}−*βk*1−*c*1*r*2

*c*2 *>*0,

4.10

*then the positive equilibriumE*3*in System*1.11*is globally attractive.*

*Remark 4.6. From*4.10, we know that *γτ* has a negative eﬀect on the global attractivity of
positive equilibrium; that is, an increase in the value of*γτ*can destroy Condition4.10.

*Remark 4.7. Comparing Theorems*4.1and 4.2 with Theorems 4.1 and 4.2 in Song et al.25, we
also see the inclusion of an extra term*e*^{−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 System*1.11, we have

*x*^{}_{2}t≤*αe*^{−γ τ}*x*2t−*τ*−*βx*_{2}^{2}t, 4.11
then byLemma 2.1and the comparison principle, for suﬃciently small* >*0, there is a*T*1*>*0
such that

*x*2t*<* *αe*^{−γ τ}

*β* *u*1 4.12

for*t*≥*T*1*τ. Replacing this inequality into the second equation of*1.11, we have

*y*^{}t≤*yt*

*r*2− *c*2*yt*
*u*1*k*2

*,* *t*≥*T*1*.* 4.13

Again by the comparison principle, there is a*T*2 *> T*1*τ >*0 such that

*yt<* u1*k*2r2

*c*2 *ν*1*,* *t*≥*T*2*.* 4.14

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

*x*_{2}^{}t≥*αe*^{−γ τ}*x*2t−*τ*−*βx*^{2}_{2}t− *c*1*ν*1*x*2t

*x*2t *k*1*.* 4.15

Consider the following equation:

*z*^{}t *αe*^{−γ τ}*zt*−*τ*−*βz*^{2}t− *c*1*ν*1*zt*

*zt k*1*.* 4.16

From the first inequality of4.14andLemma 4.4, we see that4.16has a unique positive
equilibrium*z*^{∗} αe^{−γ τ} −*βk*1 αe^{−γ τ}−*βk*1^{2}4βαe^{−γ τ}*k*1−*c*1*ν*1/2βwhich is globally
asymptotically stable. Using the comparison principle, for suﬃciently small * >* 0, we see
that there is a*T*3*> T*2*τ*such that

*x*2t*> z*^{∗}− *u*_{1}*>*0. 4.17

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

*y*^{}t≥*yt*

*r*2− *c*2*yt*
*u*_{1}*k*2

*,* *t*≥*T*3*.* 4.18

By the comparison principle, there is*T*4*> T*3such that

*yt>*

*u*_{1}*k*2

*r*2

*c*2 − *ν*1*,* *t*≥*T*4*.* 4.19

Hence, we have

*u*_{1}*< xt< u*1*,* *ν*1*< yt< ν*1*,* *t*≥*T*4*.* 4.20

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

*x*_{2}^{}t≤*αe*^{−γ τ}*x*2t−*τ*−*βx*^{2}_{2}t− *c*1*ν*_{1}*x*2t

*x*2t *k*1*.* 4.21

From a similar use of the comparison principle, we conclude that there is*T*5*> T*4*τ*such that
*x*2t*< z*^{∗}_{1} *u*2*>*0, *t*≥*T*5*,* 4.22

where *z*^{∗}_{1} αe^{−γ τ} −*βk*1 αe^{−γ τ}−*βk*1^{2}4αe^{−γ τ}*k*1−*c*1*ν*_{1}β/2β > 0 is the positive
equilibrium for the equation

*z*^{}t *αe*^{−γ τ}*zt*−*βz*^{2}t− *c*1*ν*_{1}*zt*

*zt k*1*.* 4.23

From4.10, we have

*u*2 *< u*1*.* 4.24

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

*y*^{}t≤*yt*

*r*2− *c*2*yt*
*u*2*k*2

*,* *t*≥*T*5*.* 4.25

A similar discussionas aboveimplies that for suﬃciently small* >*0, there is a*T*6*> T*5such
that

*yt<* u2*k*2r2

*c*2 *ν*2*.* 4.26

Since*u*2*< u*1, we get

*ν*2 *< ν*1*.* 4.27

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

*x*^{}_{2}t*> αe*^{−γ τ}*x*2t−*τ*−*βx*^{2}_{2}t− *c*1*ν*2*x*2t

*x*2t *k*1*,* *t*≥*T*6*.* 4.28
From4.10,Lemma 4.4and the comparison principle, we see that for suﬃciently small* >*0,
there is a*T*7*> T*6*τ*such that

*x*2t*> z*^{∗}_{2}− *u*_{2}*>*0, *t*≥*T*7*,* 4.29

where *z*^{∗}_{2} αe^{−γ τ} −*βk*1 αe^{−γ τ}−*βk*1^{2}4αe^{−γ τ}*k*1−*c*1*ν*2β/2β > 0 is the positive
equilibrium for the equation

*z*^{}t *αe*^{−γ τ}*zt*−*βz*^{2}t− *c*1*ν*2*zt*

*zt k*1*.* 4.30

Moreover, since*ν*2*< ν*1we have that*u*_{2}*> u*_{1}.

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

*y*^{}t≥*yt*

*r*2− *c*2*yt*
*u*_{2}*k*2

*,* *t*≥*T*7*.* 4.31

Arguments similar to those used above guarantee the existence of a*T*8*> T*7such that

*yt>*

*u*_{2}*k*2

*r*2

*c*2 − *ν*2*,* *t*≥*T*8*,* 4.32

from which we get that*ν*2*> ν*1.

Repeating the above process leads to the construction of the sequencesu*n*^{∞}* _{n1}*,u

_{n}^{∞}

*, ν*

_{n1}*n*

^{∞}

*,ν*

_{n1}

_{n}^{∞}

*, and*

_{n1}*T*4n

*>*0. For

*t*≥

*T*4n, we have that

0*< u*_{1}*< u*_{2}*<*· · ·*< u*_{n}*< x*2t*< u**n**<*· · ·*< u*2*< u*1*,*

0*< ν*_{1} *< ν*_{2}*<*· · ·*< ν*_{n}*< yt< ν**n* *<*· · ·*< ν*2*< ν*1*.* 4.33

Hence, the limits ofu*n*^{∞}* _{n1}*,u

_{n}^{∞}

*,ν*

_{n1}*n*

^{∞}

*,ν*

_{n1}

_{n}^{∞}

*exist. Denote that*

_{n1}*u* lim

*t→ ∞**u**n**,* *ν* lim

*t→ ∞**ν**n**,* *u* lim

*t*→ ∞*u*_{n}*,* *ν* lim

*t*→ ∞*ν*_{n}*,* 4.34

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

By the definition of*ν**n* and*ν** _{n}*, we have

*ν*_{n}

*u*_{n}*k*2

*r*2

*c*2 −* ,* *ν**n* u*n**k*2r2

*c*2 * ,* 4.35

thus

*ν**n*−*ν*_{n}*r*2

*c*2

*u**n*−*u*_{n}

2 . 4.36

According to the definitions of*u**n*,*u** _{n}*and4.36, we have

*u*

*n*−

*u*

_{n} *αe*^{−γ τ}−*βk*1 *αe*^{−γ τ}−*βk*1

_{2}
4β

*αe*^{−γ τ}*k*1−*c*1*ν** _{n}*
2β

−*αe*^{−γ τ}−*βk*1 *αe*^{−γ τ}−*βk*1

_{2}

4βαe^{−γ τ}*k*1−*c*1*ν**n*

2β 2

− 4c1

*ν** _{n}*−

*ν*

*n*

*β*
2β *αe*^{−γ τ}−*βk*1

_{2}
4β

*αe*^{−γ τ}*k*1−*c*1*ν*_{n}

*αe*^{−γ τ}−*βk*1

_{2}

4βαe^{−γ τ}*k*1−*c*1*ν**n*

2

*<*− *c*1

*ν** _{n}*−

*ν*

*n*

*αe*^{−γ τ}−*βk*1

2 .

4.37

Let*n* → ∞, we have

*u*−*u*

≤ *c*1r2*/c*2
*u*−*u*

2

*αe*^{−γ τ}−*βk*1 2 , 4.38

hence

*αe*^{−γ τ}−*βk*1−*c*1*r*2

*c*2

*u*−*u*

≤

1*αe*^{−γ τ}−*βk*1

2 . 4.39

By4.10, we know that*αe*^{−γ τ}−*βk*1−c1*r*2/c2*>*0 and1*αe*^{−γ τ}−*βk*1*>*0. Note that can
be arbitrarily small, that is, letting → 0 leads to the conclusion that*uu. From*4.36and
letting*n* → ∞, we also conclude that*νν. 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 ^{Ê}0 for the
prey, that is the value of ^{Ê}_{0} 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 ^{Ê}0 *> p*0*>*1. Second, we derive suﬃcient
conditions for the local stability of nonnegative equilibria of Model 1.11. We show that
*E*0 0,0and*E*1 αe^{−γ τ}*/β,*0are unstable,*E*2 0, k2*r*2*/c*2is unstable if^{Ê}0*>*1stable if

Ê0 *<*1, and the positive equilibrium*E*3exists when^{Ê}0*>*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

*x*2

*y*

**Figure 1: The boundary equilibrium***E*2 0, k2*r*2*/c*2of System1.11is globally asymptotically stable.

0 100 200 300 400 500

0 1 2 3 4 5 6 7 8

Time*t*

*x,**y*

*x*2

*y*

**Figure 2: The positive equilibrium***E*3of System1.11is globally attractive.

of the comparison principle, suﬃcient conditions for the global attractivity of nonnegative
equilibria are obtained. We prove that*E*2is globally asymptotically stable when^{Ê}0*< p*1*<*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.