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This study considers a delayed biological system of predator-prey interactions where the predator has stage-structured preference. It is assumed that the prey population has two stages: immature and mature. The predator population has different preference for the stage-structured prey. This type of behavior has been reported in Asecodes hispinarum and Microplitis mediator. By some lemmas and methods of delay differential equation, the conditions for the permanence, existence of positive periodic solution and extinction of the system are obtained. Numerical simulations are presented that illustrate the analytical results as well as demonstrate certain biological phenomena. In particular, overcrowding of the predator does not affect the persistence of the system, but our numerical simulations suggest that overcrowding reduces the density of the predator. Under the assumption that immature prey is easier to capture, our simulations suggest that the predator’s preference for immature prey increases the predator density.

In recent years, much attention has been paid to biological systems with stage structure [

In References [

where

In the natural world, many predators switch to alternative prey when their favored food is in short supply [

However, previous studies on prey age preference only have been done in laboratory tests. Few researchers have investigated the phenomenon with mathematical models and carried out theoretical analysis together with numerical simulation. To extend research in this area, and based on the recent study by Cui and Song [

Let

where

The coefficients in system (2.1) are all continuous positive T-periodic functions. Parameter

The term

survive, and those progressing from the immature stage to the mature stage at time t. The death rate of the mature prey population is logistic in nature and it is proportional to the square of the population with proportionality

The initial conditions for system (2.1) take the form of

where

mapping the interval

For continuity of initial conditions, we require

For the purpose of convenience, we write

Obviously,

In this paper, we consider system (2.5) with initial conditions (2.3) and (2.4). At the same time, we adopt the following notation through this paper:

where

The rest of the paper is arranged as follows. In the following section, we introduce some lemmas and then explore the permanence and periodicity of system (2.5). In Section 4, we investigate the extinction of the predator population in system (2.5). In Section 5, numerical simulations are presented to illustrate the feasibility of our main results. Furthermore, the simulated results are explained according to the biological perspective. In section 6, a brief discussion is given to conclude this work.

In this section, we analyze the permanence and periodicity of system (2.5) with initial conditions (2.3) and (2.4). Firstly, we introduce the following definition and Lemmas which are useful to obtain our result.

Definition 3.1. The system

Lemma 3.2. (See [

has a unique positive T-periodic solution which is globally asymptotically stable.

Lemma 3.3. (See [

has a unique positive T-periodic solution

Lemma 3.4. There exists a positive constant

for all the solution of system (3.2) with respect to

Proof. Let

By applying (3.4), we obtain

Let

We have

Theorem 3.5. System (2.5) is permanent and has at least one positive T-periodic solution provided

where

where

We need the following propositions to prove Theorem 3.5.

Proposition 3.6. For all the solutions of system (2.5) with initial conditions (2.3) and (2.4), we have

Proof. Obviously,

Consider the following auxiliary system

By Lemma 3.3, system (3.7) has a unique globally attractive positive T-periodic solution

By applying (3.8) and Lemma 3.4, we obtain

In addition, from the third equation of (2.5) we have

Consider the following auxiliary equation:

According to the condition (3.6), we have

By (3.10) and Lemma (3.2), we obtain that system (3.9) has a unique positive T-periodic solution

By applying (3.11), we obtain

Set

This completes the proof of Proposition 3.6. □

Proposition 3.7. There exists a positive constant

Proof. By Proposition 3.6, there exists a positive

for

has a unique global attractive positive T-periodic solution

Moreover, from the global attractivity of

Combined (3.14) with (3.15), we have

, (3.16)

Therefore,

Proposition 3.8. Suppose that (3.6) holds, then there exists a positive constant

Proof. By assumption (3.6), we can choose arbitrarily small constant

assume that

system (3.2)), such that

where

Consider the following system with a parameter

By Lemma 3.3, system (3.19) has a unique positive T-periodic solution

attractive. Let

Then, for the above

Using the continuity of the solution in the parameter, we have

So, we get

Suppose that the conclusion (3.17) is not true, then there exists

where

By applying (3.21), from the first and second equation of system (2.5), we have

for all

asymptotic stability of

So,

By using (3.20), we obtain

Therefore, by using (3.21) and (3.22), for

Integrating (3.23) from

Thus, from (3.18) we know that

Proposition 3.9. Suppose that (3.6) holds, then there exists a positive constant

Proof. Suppose that (3.24) is not true, then there exists a sequence

On the other hand, by Proposition 3.8, we have

Hence, there exist time sequences

and

By Proposition 3.6, for a given positive integer m, there exist a

Because of

for

or

Thus, from the boundedness of

By (3.18) and (3.27), there exist constants

and

for

Let

comparison theorem, we have

By using Propositions 3.6 and 3.7, there exists a large enough

for

Therefore, by using (3.20),

According to (3.27), there exists a positive integer

as

Integrating (3.31) from

that is,

This is a contradiction. This completes the proof of Proposition 3.9. □

Proof of Theorem 3.5. By using Propositions 3.6-3.9, system (2.5) is permanent. Using result given by Teng and Chen in [

In this section, we investigate the extinction of the predator population in system (2.5) with initial conditions (2.3) and (2.4) under some condition.

Theorem 4.1. Suppose that

where

where

Proof. According to (4.1), for every given positive constant

where

From the first and second equations of system (2.5), we have

Hence, for the above

It follows from (4.2) and (4.3) that for

Firstly, we show that exists a

That is to say

Secondly, we show that

where

is bounded for

By the continuity of

(4.3), we have

which is a contradiction. This shows that (4.5) holds. By the arbitrariness of

In this section, we give some examples to illustrate the feasibility of our main results in Theorems 3.5 and 4.1.

Example 5.1. Let

In this case, system (3.2) given by Lemma 3.3 has a unique positive periodic solution

calculation, we have

According to Theorem 3.5, system (2.5) with the above coefficients is permanent and admits at least one positive 2p-periodic solution for any nonnegative 2p-periodic function

(2.5) with the above coefficients and

Example 5.2. In system (2.5), let

Example 5.3. In system (2.5), let

According to Theorem 4.1, system (2.5) is impermanent and the predator population is extinction. Numerical simulation given in

In this paper, we propose and analyze a periodic predator-prey system with time delay and prey stage-structured preference by the predator. The permanence and existence of positive periodic solutions of system (2.5) are explored. The conditions for the impermanence of the system and the extinction of the predator population are obtained. By Lemma 3.3, we know that

more palatable and more easily captured by the predator than the mature. Example 5.3 illustrates the correctness of Theorem 4.1.

We would like to mention here that we are unable to solve the following questions:

1) How many positive periodic solutions exist in system (2.5)?

2) Is the solution global attractivity if system (2.5) has only one positive periodic solution?

We leave these for future work.

The authors express sincere gratitude to the anonymous referees for their helpful comments and suggestions that led to an improvement of our original manuscript.

This work was supported by the Major Project of Sichuan University of Arts and Science (Grant No. 2014Z005Z), by the General Project of Educational Commission in Sichuan Province (Grant No. 16ZB0357).

Limin Zhang,Chaofeng Zhang, (2016) Permanence, Periodicity and Extinction of a Delayed Biological System with Stage-Structured Preference for Predator. Journal of Applied Mathematics and Physics,04,546-560. doi: 10.4236/jamp.2016.43060