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Discrete Dynamics in Nature and Society
Volume 2009, Article ID 285934, 24 pages
http://dx.doi.org/10.1155/2009/285934
Research Article

Global Stability for a Delayed Predator-Prey System with Stage Structure for the Predator

Institute of Applied Mathematics, Shijiazhaung Mechanical Engineering College, Shijiazhuang 050003, China

Received 26 March 2009; Accepted 20 July 2009

Academic Editor: Leonid Berezansky

Copyright © 2009 Xiao Zhang 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.

Abstract

A delayed predator-prey system with stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of equilibria of the system is discussed. The existence of Hopf bifurcation at the positive equilibrium is established. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and two boundary equilibria of the system. Numerical simulations are carried out to illustrate the theoretical results.

1. Introduction

Stage-structure is a natural phenomenon and represents, for example, the division of a population into immature and mature individuals. As is common, the dynamics-eating habits, susceptibility to predators, and so forth. are often quite different in these two subpopulations. Hence, it is of ecological importance to investigate the effects of such a subdivision on the interaction of species. In [1], Chen et al. introduced the following stage-structured single-species population model:

where and denote the immature and mature population densities at time , respectively; is the birth rate of the immature population at time ; and are the death rates of the immature and mature at time ; represents the transformation rate of the immature into the mature; is the probability of the successful transformation of the immature into the mature. If the birth rate of model (1.1) obeys the Malthus rule, that is, , the death rates of the immature and mature populations are logistic, that is,

and the transformation rate of the immature into mature is proportional to the immature population, that is, , then model (1.1) becomes

Based on the idea above, many authors studied different kinds of stage-structured models, and a significant body of work has been carried out (see, for example, [28]).

In [3], Gao et al. considered the following predator-prey model with stage structure:

where represents the density of the prey at time ; and represent the densities of the mature and the immature predator at time , respectively. The parameters are positive constants in which is the intrinsic growth of the prey, is the death rate of the mature predator population, is the death rate of the immature predator population, is the intra-specific competition rate of the prey population, is the capturing rate of the predator population, is the conversion rate of nutrients into the reproduction of the predator, is the intra-specific competition rate of the mature predator, is the birth rate of the immature predator, is the transformation rate from the immature predator individuals to mature predator individuals. The predation decreases the average growth rate of prey linearly with a certain time delay , this assumption corresponds to the fact that predators cannot hunt prey when the predators are infant; predators have to mature for a duration of units of time before they are capable of decreasing the average growth rate of the prey species; is the time delay due to gestation, the delay in time for prey biomass to increase predator number. In [3], Gao et al. studied the global stability of the positive equilibrium and boundary equilibria of model (1.4) by constructing Liapunov functionals and comparison argument, respectively.

We note that most of the predator-prey models with time delays studied in the literature are all of the Kolmogorov-type. In [9], Wangersky and Cunningham proposed delayed predator-prey models that are not of the Kolmogorov-type. They considered the following delayed system:

where the delay is a constant based on the assumption that the change rate of predators depends on the number of both the prey and the predators present at some previous time.

Motivated by the work of Gao et al. [3] and Wangersky and Cunningham [9], in the present paper, we consider the following predator-prey model with stage structure and time delay:

The meanings of the positive parameters are the same as those in system (1.4). The meaning of time delay is the same as in system (1.5).

The initial conditions for system (1.6) take the form

where , the Banach space of continuous functions mapping the interval into , where .

This paper is organized as follows. In the next section, we introduce some notations and state several lemmas which will be essential to our proofs. In Section 3, we discuss the local stability of a positive equilibrium and boundary equilibria of system (1.6). The existence of Hopf bifurcation is studied. In Section 4, by means of an iterative technique and comparison argument, sufficient conditions are derived for the global stability of the positive equilibrium and boundary equilibria of system (1.6). Some numerical examples are given to illustrate the results above. A brief discussion is given in Section 6 to end this work.

2. Preliminaries

In this section, we introduce some notations and state several results which will be useful in next section. Let be the cone of nonnegative vectors in . If , we write if for . Let denote the standard basis in . Suppose and let be the Banach space of continuous functions mapping the interval into with supremum norm. If , we write when the indicated inequality holds at each point of . Let and let denote the inclusion by . Denote the space of functions of bounded variation on by . If , and , then for any , we let be defined by .

We now consider

We assume throughout this section that is continuous; is continuously differentiable in ; for all , and some . Then by [10], there exists a unique solution of (2.1) through for . This solution will be denoted by if we consider the solution in , or by if we work in the space C. Again by [10], is continuously differentiable in . In the following, the notation will be used as the condition of the initial data of (2.1), by which we mean that we consider the solution of (2.1) which satisfies .

To proceed further, we need the following results from [11, 12]. Let , and define

We write for a generic point of . Let . Due to the ecological applications, we choose as the state space of (2.1) in the following discussions.

Fix arbitrarily. Then we set , and denotes the Frechet derivation of with respect to . It is convenient to have the standard representation of as

in which satisfies

where is continuous from the left in .

We make the following assumptions for (2.1).

(h0)If , and for some , then .(h1)For all with for .(h2)The matrix defined by is irreducible for each . (h3)For each , for which , there exist such that for all and for positive constant sufficiently small, .(h4)If , then for all .

The following result was established by Wang et al. [12].

Lemma 2.1. Let (h1)–(h4) hold. Then the hypothesis (h0) is valid and the following.(i)If and are distinct elements of with and with is the intersection of the maximal intervals of existence of and , then (ii)If and is defined on with , then

Lemma 2.1 shows that if (h1)–(h4) hold, then the positivity of solutions of (2.1) follows.

The following definitions and results are useful in proving our lemma.

Definition 2.2. System (2.1) is cooperative if whenever .

Definition 2.3. A square matrix is said to be a reducible matrix if and only if for some permutation matrix the matrix is block upper triangular. If a square matrix is not reducible, it is said to be an irreducible matrix. System (2.1) is called irreducible if the Jacobian matrix is irreducible.

Lemma 2.4 (Smith [11]). If (2.1) is cooperative and irreducible in , where is an open subset of , and the solution with positive initial data is bounded, then the trajectory of (2.1) tends to some single equilibrium.

We now consider the following delay differential system:

with initial conditions

System (2.8) always has a trivial equilibrium . If , then system (2.8) has a unique positive equilibrium , where

The characteristic equation of system (1.6) at the equilibrium is of the form

Let

If , then it is easy to see that, for real,

Hence, has a positive real root. Therefore, the equilibrium is unstable. If , when , it is easy to see that the equilibrium is stable. Therefore, if , by Kuang and So [13, Lemma ?B], we see that the equilibrium is locally stable for all .

If , the characteristic equation of the positive equilibrium takes the form

When , it is easy to see that the equilibrium is stable. Therefore, if , by Kuang and So [13, Lemma ?B], we see that the equilibrium is locally stable for all .

Lemma 2.5. For system (2.8), one has the following. (i)If , then the positive equilibrium of system (2.8) is globally stable.(ii)If , the equilibrium of system (2.8) is globally stable.

Proof. We represent the right-hand side of (2.8) by and set By direct calculation we have We now claim that the hypotheses (h1)–(h4) hold for system (2.8). It is easily seen that (h1) and (h4) hold for system (2.8). We need only to verify that (h2) and (h3) hold.
The matrix takes the form
Clearly, the matrix is irreducible for each .
From the definition of and , it is readily seen that for ; and for ; and , where is a positive Borel measure on . Therefore, . Thus, for each , there is such that for all and sufficiently small, . Hence, (h3) holds.
Thus, the conditions of Lemma 2.1 are satisfied. Therefore, the positivity of solutions to system (2.8) follows. It is easy to see that system (2.8) is cooperative. By Lemma 2.4 we see that any solution starting from converges to some single equilibrium. However, system (2.8) has only two equilibria: and . Note that if , the equilibrium is locally stable. Hence, any solution starting from converges to . Using a similar argument one can show the global stability of the equilibrium when . This completes the proof.

By a similar argument one can show that all solutions of system (1.6) with initial conditions (1.7) are defined on and remain positive for all .

3. Local Stability

In this section, we discuss the local stability of each equilibria and the existence of Hopf bifurcation of system (1.6).

It is easy to show that system (1.6) always has two equilibria and If the following holds:

(H1)

then system (1.6) has another boundary equilibrium , where Further, if the following holds:

(H2)

then system (1.6) has a unique positive equilibrium , where

We now study the local stability of each of the nonnegative equilibrium of system (1.6).

Let be any arbitrary equilibrium. Then the characteristic equation of system (1.6) at the equilibrium is given by

The characteristic equation of system (1.6) at the equilibrium reduces

Clearly, is a positive real root. Hence, is always unstable.

The characteristic equation of system (1.6) at the equilibrium reduces

Clearly, is a negative real root of (3.5). All other roots are give by the roots of equation

Let

If , then it is easy to see that for real,

Hence, has a positive real root. Therefore, the equilibrium is unstable. If , when , it is easy to see that the equilibrium is stable. Therefore, if , by Kuang and So [13, Lemma ?B], we see that the equilibrium is locally stable for all .

The characteristic equation of system (1.6) at the equilibria reduces to

Clearly, if , is a positive real root. Hence, if , then is unstable. Noting that

if , then (3.9) only has negative real root, and is stable.

The characteristic equation of system (1.6) at the positive equilibria is

where

It is easy to show that

It is easy to see that . Hence, by the Routh-Hurwitz theorem, when the positive equilibrium of system (1.6) is locally asymptotically stable.

If is a root of (3.11), separating the real and imaginary parts, we obtain

Squaring and adding the two equations of (3.14), it follows that

where

It is easy to show that

Noting that if , , hence . The positive equilibrium of system (1.6) is locally asymptotically stable for all . If , we know that , (3.15) has a unique positive root . Define

Then solves (3.11). This means that when , (3.11) has a pair of purely imaginary roots . Noting that the positive equilibrium is locally stable for , by the general theory on characteristic equations of delay differential equations from [14, Theorem ?4.1], remains stable for .

Let and be defined in (3.18). Denoting

the root of (3.11) is such that

In the following we claim that

This will signify that there exists at least one eigenvalue with positive real part for . Moreover, the conditions for the existence of a Hopf bifurcation [10] are then satisfied yielding a periodic solution. To this end, differentiating equation (3.11) with respect , we obtain that

which leads to

If , . Therefore, system (1.6) undergoes a Hopf bifurcation.

We therefore obtain the following results.

Theorem 3.1. For system (1.6), let be defined as in (3.18), one has the following.(i)The positive equilibrium of system (1.6) is always unstable.(ii)If , the equilibrium of system (1.6) is locally stable for all ; and if , is unstable for all .(iii)Let hold. If , the equilibrium of system (1.6) is unstable; if , is stable for all .(iv)Let (H2) hold. If , then the positive equilibrium of system (1.6) is locally asymptotically stable for all . If , then is locally stable for ; and is unstable for ; if system (1.6) undergoes a Hopf Bifurcation at the positive equilibrium when .

4. Global Stability

In this section, we are concerned with the global stability of the equilibria of system (1.6). The strategy of proofs is to use an iteration technique and comparison arguments, respectively.

Theorem 4.1. Let (H2) hold. Then the positive equilibrium of system (1.6) is globally asymptotically stable provided that (H3)

Proof. Let be any positive solution of system (1.6) with initial conditions (1.7). Let We now claim that . The strategy of the proof is to use an iteration technique.
We derive from the first equation of system (1.6) that
A standard comparison argument shows that Hence, for sufficiently small there exists a such that if , .
We derive from the second and the third equations of system (1.6) that for ,
Consider the following auxiliary equations: Since (H2) holds, by Lemma 2.5 it follows from (4.5) that By comparison, we obtain that Since these inequalities are true for arbitrary sufficiently small, it follows that , where Hence, for sufficiently small, there is a such that if , , .
For sufficiently small, we derive from the first equation of system (1.6) that, for ,
By comparison it follows that Since this is true for arbitrary sufficiently small, we conclude that , where Therefore, for sufficiently small, there is a such that if , .
For sufficiently small, we derive from the second and the third equations of system (1.6) that, for ,
Consider the following auxiliary equations: Since (H2) and (H3) hold, by Lemma 2.5 it follows from (4.13) that By comparison, we obtain that Since these two inequalities hold for arbitrary sufficiently small, we conclude that , where Therefore, for sufficiently small, there exists a such that if , ,
For sufficiently small, it follows from the first equation of system (1.6) that, for ,
A comparison argument yields Since this is true for arbitrary , we conclude that , where Hence, for sufficiently small, there exists a such that if , .
Again, we derive from the second and the third equations of system (1.6) that for ,
Since (H2) and (H3) hold, by Lemma 2.5, a comparison argument shows that Since these inequalities are true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that if , , .
For sufficiently small, it follows from the first equation of system (1.6) that, for ,
By comparison we obtain that Since this is true for arbitrary sufficiently small, we conclude that , where Therefore, for sufficiently small, there is a such that if , .
For sufficiently small, we derive from the second and the third equations of system (1.6) that for ,
Since (H2) and (H3) hold, by Lemma 2.5 and by comparison, it follows from (4.26) that Since these two inequalities hold for arbitrary sufficiently small, we conclude that , where Continuing this process, we derive six sequences such that, for , It is readily seen that We derive (4.29) that Noting that and , it follows from (4.31) that Thus, the sequence is nonincreasing. Hence, exists. Taking , it follows from (4.31) that We therefore obtain from (4.30) and (4.33) that It follows from (4.30), (4.33) and (4.34) that We therefore have Noting that if , then . By Theorem 3.1, the positive equilibrium is locally stable. We therefore conclude that is globally stable. The proof is complete.

Theorem 4.2. If , the equilibrium of system (1.6) is globally asymptotically stable.

Proof. Let be any positive solution of system (1.6) with initial conditions (1.7). It follows from the first equation of system (1.6) that A standard comparison argument shows that Choose sufficiently small such that Hence, for sufficiently small satisfying (4.39), there is a such that if , then . We derive from the second and the third equations of system (1.6) that for , Consider the following auxiliary equations: Since (4.39) holds, by Lemma 2.5 it follows from (4.41) that By comparison, we obtain that Hence, for sufficiently small satisfying (4.39), there is a such that if .
For sufficiently small satisfying (4.39) and , we derive from the first equation of system (1.6) that for
By comparison, we obtain that which, together with (4.38), yields
By Theorem 3.1, if , the boundary equilibrium is locally stable. We therefore conclude that is globally stable in this case. This completes the proof.

Theorem 4.3. Let hold. If , then the equilibrium of system (1.6) is globally asymptotically stable.

Proof. Let be any positive solution of system (1.6) with initial conditions (1.7). We derive from the second and the third equations of system (1.6) that, for , Consider the following auxiliary equations: Since (H1) holds, by Lemma 2.5 it follows from (4.48) that By comparison we derive that