#### Abstract

We propose and study the permanence of the following periodic Holling III predator-prey system with stage structure for prey and both two predators which consume immature prey. Sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.

#### 1. Introduction

The aim of this paper is to investigate the permanence of the following periodic stage-structure predator-prey system with Holling III functional response:

where , , , , , , , , and , , , are all continuous positive -periodic functions. Here and denote the density of immature and mature prey species, respectively, and is the density of the predators.

The periodic functions in (1.1) have the following biological meanings. The birth rate into the immature population is given by ; that is, it is assumed to be proportional to the existing mature population, with a proportionality coefficient . The death rate of the immature population is proportional to the existing immature population and to its square with coefficients and , respectively. The death rate of the mature population is of a logistic nature, with proportionality coefficient . The transition rate from the immature individuals to the mature individuals is assumed to be proportional to the existing immature population, with proportionality coefficient . Similarly, gives the density-dependent death rate of the predators. and give the coefficients that relate to the conversion rate of the immature prey biomass into predator biomass. More details about the biological background for (1.1) can be found in [1–10].

The function represents the functional response of predator to immature prey. Let , then we have

The functional response of predator species to immature prey species takes the Holling type III, that is, . Holling type III is the third function in which Holling proposed three kinds of functional response of the predator to prey based on numerous experiments for different species. The Holling type form of functional response is intituled prey-dependent model form. It is applied to almost invertebrate that is one of the most extensive applied functional responses.

In [2], Cui and Takeuchi considered the following periodic predator-prey system with a stage structure:

where

Different predators usually consume prey in different stage structures. Some predators only prey on immature prey, and some predators only prey on mature prey [5]. Based on system (1.3), we also consider another predator species which also consumes immature prey. Assuming that the predator consumes immature prey according to Holling III functional response while the other predator consumes mature prey also according to the Holling III functional response, we get model (1.1).

To the best of the authors' knowledge, for the nonautonomous case of predator-prey systems with two predators which consume immature prey and stage structure for prey, whether one could obtain the sufficient and necessary conditions which insure the permanence of the system or not is still an open problem.

The aim of this paper is, by further developing the analysis technique of Cui and Takeuchi [2], to derive a set of sufficient and necessary conditions which ensure the permanence of the system (1.1). The rest of the paper is arranged as follows. In Section 2, we introduce some lemmas and then state the main result of this paper. The result is proved in Section 3.

Throughout this paper, for a continuous -periodic function we set

#### 2. Main Results

*Definition 2.1. *The system
is said to be permanent if there exists a compact set in the interior of , such that all solutions starting in the interior of ultimately enter and remain in .

The following lemma can be found in [4].

Lemma 2.2. *If , , , , and are all -periodic, then system
**
has a positive -periodic solution which is globally asymptotically stable with respect to .*

Lemma 2.3 (see [11]). *If and are all -periodic, and if and for all , then the system
**
has a positive -periodic solution which is globally asymptotically stable.*

Theorem 2.4. *Suppose that
**
holds then system (1.1) is permanent, where is the unique positive periodic solution of system (2.2) given by Lemma 2.2.*

Theorem 2.5. *System (1.1) is permanent if and only if (2.4) holds.*

#### 3. Proof of the Main Results

We need the following propositions to prove Theorems 2.4 and 2.5. The hypothesis of the lemmas and theorems of the preceding section are assumed to hold in what follows.

Proposition 3.1. *There exist positive constants and such that
*

*Proof. *Obviously, is a positively invariant set of system (1.1). Given any positive solution of (1.1), we have

By Lemma 2.2, the following auxiliary equation
has a globally asymptotically stable positive -periodic solution . Let be the solution of (3.3) with . By comparison, we then have
for . By (2.4), we can choose a positive small enough such that
Thus, from the global attractivity of , for the above given , there exists a such that
Equation (3.4) combined with (3.6) leads to
In addition, for , from the third and fourth equations of (1.1) and (3.7) we get
Consider the following auxiliary equation:
It follows from (3.5) and Lemma 2.3 that (3.9) has a unique positive -periodic solution which is globally asymptotically stable. Similarly to the above analysis, there exists a such that for the above , one has
Let , , then we have
This completes the proof of Proposition 3.1.

Proposition 3.2. *There exist positive constants , such that
*

*Proof. *By Proposition 3.1, there exists such that
Hence, from the first and second equations of system (1.1), we have
for . By Lemma 2.2, the following auxiliary equation
has a globally asymptotically stable positive -periodic solution . Let be the solution of (3.15) with ; by comparison, we have
Thus, from the global attractivity of , there exists a , such that
Equation (3.17) combined with (3.16) leads to
That is, we have
This completes the proof of Proposition 3.2.

Proposition 3.3. *There exists a positive constant such that
*

*Proof. *By assumption (2.4), we can choose a constant and the same constant as in Proposition 3.1 such that
where
Consider the following equation with a parameter :
By Lemma 2.2, (3.23) has a unique positive -periodic solution , which is globally asymptotically stable. Let be the solution of (3.23) with initial condition , ; then, for the above , there exists a , such that
By continuity of the solution in the parameter, we have uniformly in as . Hence, for there exists such that
So we have
Since and are all -periodic, we have
Choose a constant and
Suppose that the conclusion (3.20) is not true. Then there exists a such that, for the positive solution of (1.1) with an initial condition , we have
So there exists a such that
By applying (3.30), from the first and second equations of system (1.1) it follows that for all ,
Let be the solution of (3.23) with and , ; we know that , ,

By the global asymptotic stability of , for the given , there exists such that
So we have
and hence
From (3.7) and (3.34), we have

By (3.35) and (1.2), from the third and fourth equations of system (1.1) we have
Integrating (3.36) from to yields
By (3.21), we know that as , , which is a contradiction. This completes the proof.

*Proof of Theorem 2.4. *By Propositions 3.2 and 3.3, system (1.1) is uniform weak persistent [2]. From [12, Propositions and Theorem ], system (1.1) is permanent. This completes the proof of Theorem 2.4.

*Proof of Theorem 2.5. *The sufficiency of Theorem 2.5 now follows from Theorem 2.4. We thus only need to prove the necessity of Theorem 2.5. Suppose that
We will show that
In fact, by (3.38), we know that, for any given , there exist and such that
Note that for . Since
we know that, for the given , there exists such that
By (3.40), (1.2), and (3.42), we have
We now show that there must exist such that . Otherwise, by (3.43), we have
This implies , which is a contradiction.

Let
We know that is bounded (Proposition 3.1 given). We now show that
Otherwise, there exists such that
By the continuity of , there must exist such that and for . Let be the nonnegative integer such that ; by (3.43) we have
which is a contradiction. This implies that (3.46) holds. We then conclude, by the arbitrariness of , that as , . This completes the proof of Theorem 2.5.