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A delayed biological system of predator-prey interaction with stage structure and density dependent juvenile birth rate is investigated. It is assumed that the prey population has two stages: immature and mature. The growth of the immature prey is density dependent and is a function of the density of adult prey. Such phenomenon has been reported for beetles, tribolium, copepods, scorpions, several fish species and even crows. The growth of the predator is affected by the time delay due to gestation. By some Lemmas and methods of delay differential equation, the conditions for the uniform persistence and extinction of the system are obtained. Numerical simulations illustrate the feasibility of the main results and demonstrate that the density dependent coefficient has influence on the system populations’ densities though it has no effect on uniform persistence and extinction of the system.

In the natural world, there are many species whose individual members have a life history that takes them through two stages: juvenile and adult. Individuals in each stage are identical in biological characteristics, and some vital rates (rates of survival, development and recruitment) of individuals in a population almost always depend on stage structure [

All the coefficients in system (1.1) are continuous positive

Sometimes, the past state as well as current conditions can influence biological dynamics and such interactions have motivated the introduction of time delay in stage-structured predator-prey systems [

In some stage-structured populations, the intraspecific and interspecific competitions occur within each stage structure. And each stage-structured density affects not only its population but also other stage-structured populations. In two-stage single-species population, Abrams and Quince have demonstrated that adult population competition makes a low birth rate of juvenile population [

Motivated by the above facts and based on the recent work of Yang et al. [

where

pendent coefficient. The function

sponse of the predator to the immature prey and

The initial conditions for system (1.2) take the form of

where

At the same time, we adopt the following notations through this paper:

where

The remainder of this paper is organized as follows. In Section 2, we introduce some lemmas and then explore the uniformly persistence and periodicity of system (1.2). In Section 3, we investigate the extinction of the predator population in system (1.2). In Section 4, numerical simulations are presented to illustrate the feasibility of our main results. Conclusion is given in Section 5.

In this section, we analyze the uniform persistence and periodicity of system (1.2) with initial conditions (1.3). First, we introduce the following definition and lemmas, which are useful for obtaining our results.

Definition 2.1. The system

Definition 2.2. The system

constants

Lemma 2.1. (See [

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

Lemma 2.2. (See [

has a positive T periodic solution

Theorem 2.1. System (1.2) is uniformly persistent and has at least one positive T periodic solution provided that

where

We need the following propositions to prove Theorem 2.1.

Proposition 2.1. There exists a positive constant

Proof. Obviously,

Consider the following auxiliary system:

By Lemma 2.2, system (2.4) has a unique globally attractive positive T periodic solution

From the global attractivity of_{1} > 0, such that for all

By applying (2.5) and (2.6), we obtain

for all

By Lemma (2.1), we obtain that system (2.7) has a unique positive T periodic solution

Set

Proposition 2.2. There exists a positive constant

Proof. By Proposition 2.1, there exists a positive

for

has a unique global attractive positive T periodic solution

Moreover, from the global attractivity of

Combined (2.10) with (2.11), we have

Therefore,

Proposition 2.3. Suppose that (2.3) holds, then there exists a positive constant

Proof. By assumption (2.3), we can choose arbitrarily small constant

sume that

tem (2.2)), such that

where

By Lemma 2.2, system (2.14) has a unique positive T periodic solution

attractive. Let

constant

Suppose that the conclusion (2.12) is not true, then there exists

where

for

for all

By the global asymptotic stability of

So,

Therefore, by using (2.17) and (2.19), for

Integrating (2.20) from

Thus, from (2.13) we know that

Proof of Theorem 2.1. By Propositions2.2 and 2.3, system (1.2) is uniform weakly uniformly persistent. From Propositions 2.1 and Theorem 1.3.3 in [

In this section, we investigate the extinction of the predator population in system (1.2) with initial conditions (1.3) under some condition.

Theorem 3.1. In system (1.2), suppose that

where

Proof. According to (3.1), for every given positive constant

From the first and second equations of system (1.2), we have

Hence, for the above

It follows from (3.2) and (3.3), that for

First, we show that exists a

as

That is to say

where

which is a contradiction. This shows that (3.5) holds. By the arbitrariness of

From Theorem 2.1 and 3.1, we obtain that the density dependent coefficient

In this section, we provide some examples to illustrate the feasibility of our main results in Theorems 2.1 and 3.1 and then explore the effect of density dependent coefficient

Example 4.1. Let

In this case, system (2.2) given by Lemma 2.2 has a unique positive periodic solution

Let

Let

From Theorem 2.1, we know that the density dependent coefficient

Example 4.2. Let

In this case, by a simple calculation, we obtain

to Theorem 3.1, system (1.2) is impermanent and the predator population experiences extinction. The numerical simulation shown in

In this paper, we propose a stage-structured predator-prey system with time delay and density-dependent juve-

nile growth. We explore the uniformly persistent and extinction of system (1.2). By Lemma 2.2, we know that

This work was supported by the National Natural Science Foundation of China (No. 31370381), the General Project of Educational Commission in Sichuan Province (Grant No. 16ZB0357), the Major Project of Educational Commission in Sichuan Province (Grant No.16ZA0357) and the Major Project of Sichuan University of Arts and Science (Grant No.2014Z005Z).

Limin Zhang,Chaofeng Zhang, (2016) Uniform Persistence, Periodicity and Extinction in a Delayed Biological System with Stage Structure and Density-Dependent Juvenile Birth Rate. American Journal of Computational Mathematics,06,130-140. doi: 10.4236/ajcm.2016.62014