Permanence , Periodicity and Extinction of a Delayed Biological System with Stage-Structured Preference for Predator

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.


Introduction
In recent years, much attention has been paid to biological systems with stage structure [1]- [23].One important reason is that there are many species whose individual members have a life history taking them through two stages, immature and mature.Thus considering stage structure in population corresponds with the natural phenomenon.Another reason is that stage-structured ecological models are much simpler than the models governed by partial differential equations but they can exhibit phenomena similar to those of partial differential equations and many important physiological parameters can be incorporated [24].The other reason is that the biological dynamics has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [25].
In References [1]- [5], the authors have studied the stability of a class of stage-structured predator-prey systems.The authors in [6] [7] have made Hopf bifurcation analysis in delayed predator-prey systems with stage structure.As we know, environmental and biological parameters (such as the seasonal effects of weather, food supplies, and mating habits) fluctuate naturally over time; thus the effects of periodically varying environments are considered to be important selective forces in systems with fluctuating environments [19] [20].Thus, incorporating periodicity into models of stage-structured biological systems is more realistic with a changing environment.Therefore, many researchers have studied a class of periodic nonautonomous biological systems with stage structures [8]- [16] [21].Recently, Cui and Song [21] considered the following predator-prey system with stage-structured prey: ) x t , and ( ) y t denote the densities of immature prey, mature prey, and predator spe- cies, respectively.They obtained a set of sufficient and necessary conditions that guarantee the permanence of the system.
In the natural world, many predators switch to alternative prey when their favored food is in short supply [22]- [24].For example, the lynx switches to red squirrel when the snowshoe hare is scarce [25].Even if there is only one prey type, the degree of predation or the quality (including palatability) of prey is likely to vary with its stage structure, which is likely to affect the predator's preference for different stage-structured prey.This type of behavior has been reported in Asecodes hispinarum [26], who parasitizes all 5 instars of Brontispa Logissina, but prefers to parasitize the 2nd and 3rd instars when it is exposed to all the instars of larvae, and in Microplitis mediator [27], who prefers to parasitize the 2nd and 3rd instars of Mythimna separate.
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 [21], we consider a periodic predator-prey system with time delay and a predator with stage-structured preference.

Formulation of the Model
Let ( ) x t and ( ) y t represent the density of immature prey, mature prey and predator species, respec- tively.Our periodic predator-prey system with time delay and stage-structured preference of the predator can be described as following: where The coefficients in system (2.1) are all continuous positive T-periodic functions.Parameter ( ) t ω is the immature prey preference of the predator, which takes a value between 0 and 1; ( ) is the mature prey preference of the predator [28] [29].c t c t is the conversion rate of nutrients into the reproduction of the pre- dator.The parameter 2 τ is the delay due to gestation, that is to say, only the mature adult predator can contribute to the production of predator biomass.The functional response of the predator to the mature prey takes the Holling type-III form of h t x t e t x t + and ( ) ( ) h t h t denotes the conversion rate of nutrients into the reproduction of the predator.
The initial conditions for system (2.1) take the form of For the purpose of convenience, we write ( ) ( ) ( ) Obviously, ( ) B t is a T-periodic and strictly positive function.Then system (2.1) becomes 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 ( ) g t is a continuous T-periodic function.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.

Permanence and Periodicity
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 ( ) ( ) such that every positive solution of this system satisfies ( ) ( ) has a unique positive T-periodic solution which is globally asymptotically stable.Lemma 3.3.(See [31]).System has a unique positive T-periodic solution which is globally asymptotically stable with respect to for all the solution of system (3.2) with respect to be the any solution of system (3.2).By Lemma 3.3, system (3.2) has a unique globally attractive positive T-periodic solution x t x t , for any positive constant ε ( 0 1 ε < < ), there exists a 1 0 T > , such that for all By applying (3.4), we obtain ( ) ( ) We have ( ) Theorem 3.5.System (2.5) is permanent and has at least one positive T-periodic solution provided where is the unique positive periodic solution of system (3.2) given by Lemma 3.3 and x M is the upper bound of system (3.2) given by Lemma 3.4 and defined by equation (3.5).
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 , where x M is the upper bound of system (3.2) given by Lemma 3.4 and defined by equation (3.5).Furthermore, there exists a positive constant y M , such that ( ) Given any solution , , x t x t y t of system (2.5) with initial conditions (2.3) and (2.4), we have Consider the following auxiliary system By Lemma 3.3, system (3.7) has a unique globally attractive positive T-periodic solution , u t u t be the solution of system (3.7) with . By the vector comparison theorem [32], we have 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 , such that ( ) Proof.By Proposition 3.6, there exists a positive 2 0 T > such that ( ) Hence, from the first and second equations of system (2.5), we obtain t T ≥ .By Lemma 3.3, the following auxiliary system has a unique global attractive positive T-periodic solution , u t u t be the solution of system (3.13) with , by the vector comparison theorem [32], we obtain Therefore, ( ) Consider the following system with a parameter δ , Then, for the above 0 ε , there exists a sufficiently large 4 3 Using the continuity of the solution in the parameter, we have ( ) ( ) , T T T + as 0 δ → .Hence, there exists a ( ) , So, we get By applying (3.21), from the first and second equation of system (2.5), we have  ( ) , for the given 0 2 By using (3.20), we obtain Therefore, by using (3.21) and (3.22), for , , , .

c t x t h t x t y t y t d t f t f t q t t t x t t e t x t y t t
s → +∞ , ( ) m q t → +∞ as q → +∞ , and , 1 By Proposition 3.6, for a given positive integer m, there exist a ( ) for Thus, from the boundedness of ( ) ( ) By (3.18) and (3.27), there exist constants 0 P > and 0 0 By using Propositions 3.6 and 3.7, there exists a large enough 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 [33], we obtain system (2.5) has at least one positive T-periodic solution.This completes the proof of Theorem 3.5.

Extinction
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 Proof.According to (4.1), for every given positive constant ε ( ) From the first and second equations of system (2.5), we have Hence, for the above 1 ε there are exists a ( ) 3) It follows from (4.2) and (4.3) that for ( ) Firstly, we show that exists a ( ) ( ) { }  According to Theorem 3.5, system (2.5) with the above coefficients is permanent and admits at least one positive 2π-periodic solution for any nonnegative 2π-periodic function ( ) q t .Figure 1 shows the dynamic behavior of system (2.5) with the above coefficients and ( ) 0.07 q t = .Figure 2 shows the dynamic behavior of system

. 10 ) 12 ) 3 . 7 .
By(3.10)  and Lemma (3.2), we obtain that system (3.9) has a unique positive T-periodic solution which is globally asymptotically stable.Then, for the above ε given in (3.4), there exists a 2 1This completes the proof of Proposition 3.6.□Proposition There exists a positive constant x x M η <

Figure 1 .Figure 2 .
Figure 1.The periodic found by numerical integration of system (2.5) with initial condition .
This completes the proof of Proposition 3.7.□ .20)Suppose that the conclusion (3.17) is not true, then there exists C 19) has a unique positive T-periodic solution which is a contradiction.This shows that (4.5) holds.By the arbitrariness of ε , it immediately follows that ( ) 0 y t → as 0 t → .This completes the proof of Theorem 4.1.