PERMANENCE OF A FOOD-CHAIN SYSTEM WITH STAGE STRUCTURE AND TIME DELAY

A non-autonomous food-chain system incorporating discrete time delay and stage-structure for each species has been presented in this paper. The sufficient conditions are derived for permanence and non-permanence of the considered system by applying the standard comparison theorem. Finally, the ecological meaning of the conclusions are discussed.


Introduction
The permanence of predation systems have been received much attention in recent years (Seeing in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]).Many ecological effects are incorporated in the predation systems when scientists investigate permanence of the considered system with stage-structure and time delay [6][7][8][9][10][11][12][13][14][15].The pioneering work on a single species growth model with stage structure and time delay, which represented a mathematically more careful and biologically meaningful formulation approach, was formulated and discussed by Aiello and Freedman [1].Motivated by their work, many authors focused on the permanence of the predation system with stage structure and time delay [Seeing in [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].Chen et al. [2] proposed the following system (1) and obtained a set of novel criteria which ensure the permanence of the considered system by introducing a new lemma and applying the standard comparison theorem.Their method and conclusions were novel and interesting.As one type of the predation system, the food-chain system are more realistic and complex than the prey-predator system.Thus, considering the food-chain system is more interesting and challenging.Liu et al. [3] proposed an impulsive reaction-diffusion periodic food-chain system with ratio-dependent functional response, they obtained the sufficient conditions for the ultimate boundedness and permanence of the foodchain system by the standard comparison theory and upper and lower solution method.Liao et al. [4] considered is a delayed discrete time Lotka-Volterra type food-chain model and obtained the sufficient conditions of the permanence.A food-chain predator-prey model with Holling IV type functional response is proposed by Shen [5] and the sufficient conditions for the permanence and the global attractivity of the system were obtained by applying the comparison theorem of the differential equation and constructing the suitable Lyapunov function.
However, the food-chain system with stage-structure and time delay are mainly confined to one or two species having stage-structure in the existing literature.Permanence of the system in which three species have stage-structure are received less attention.Motivated by these, our paper represents the following stage-structured food-chain system with time delay (2) Where I 1 (t) and M 1 (t) denote the immature and mature population densities of the prey at time t, respectively.I 2 (t) and M 2 (t) represent the immature and mature population densities of the predator at time t, respectively.I 3 (t) and M 3 (t) are the immature and mature population densities of the top predator at time t, respectively.a ii (t ) are positive and continuous functions for all t ≥ 0. The above system assumes that mature predators feed only on the mature prey population, and the mature top predators feed only on the mature predators.The birth rate of the prey population is proportional to the existing mature population with a proportionality α 1 > 0. c 1 (t) and c 2 (t) are the capturing rate of mature predators and top predators, respectively.α i (c i (t)) −1 (i = 2, 3) is the conversion rate of nutrients into the reproduction of mature predators and top predators, respectively.a ii (t) > 0 (i = 1, 2, 3) denotes the intra-species competition rate of mature prey, mature predators and top predators respectively.γ i (i = 1, 2, 3) is the death rate of immature population of prey, predator and top predator respectively.d 2 (t) and d 3 (t) denote the death rate of mature predators and top predators, respectively.τ i > 0 (i = 1, 2, 3) is the length of time from birth to maturity of ith species.
The initial conditions of system (2) are given by (3) For the continuity of initial conditions, we require further that (4) In this paper, we will mainly consider the stage-structured three species food-chain system with discrete time delay, and obtain the sufficient conditions for permanence and nonpermanence of the system (2).

Permanence and Non-Permanence
Throughout this paper, we define that (5) Definition 3.1 [1].If there exist positive constants M i and M s , such that each solution (I 1 (t), M 1 (t), I 2 (t), M 2 (t), I 3 (t), M 3 (t)) of the system (2) satisfies Then the system (2) is permanent.Otherwise, it is called non-permanent.
Lemma 3.2 [2] Considering the following equation where a, b, c and τ are positive constants, v(t) > 0 for all t ∈ [−τ, 0], we have Then the system (2) is permanent.
) is a solution of the system (2) with initial conditions (3) and ( 4).
From the second equation of the system (2), we have By Lemma 3.2 and standard comparison theorem, we get Then for any ε > 0, there exists a T 1 > 0, for any t > T 1 > 0, we have M 1 (t) < M s 1 + ε.Therefore, for any t > T 1 + τ, we obtain from the fourth equation of the system (2) Noticing that α 2 e −γ 2 τ 2 M s 1 − d i 2 > 0, by Lemma 3.2 and standard comparison theorem, we have Since ε is sufficiently small, we conclude that Then for this ε > 0, there exists a For any t > T 2 + τ, we drive from the sixth equation of the system (2) Noticing that α 3 e −γ 3 τ 2 M s 2 − d i 3 > 0, by Lemma 3.2 and standard comparison theorem, we get Since ε is sufficiently small, we obtain that lim sup Then for this ε > 0, there exists a T 3 > T 2 + τ > 0, for any t > T 3 > 0, such that M 3 (t) < M s 3 + ε.
Set T 4 = T 3 + τ > 0, for any t > T 4 > 0. Similarly we have Again, for t > T 4 + τ, we derive from the second equation of the system (2) According to the assumptions (H 1 ) and (H 2 ), it is obtained By Lemma 3.2 and standard comparison theorem, we get Since ε is sufficiently small, we have Therefore, for this ε > 0, there exists a T 5 > T 4 + τ > 0, for any t > T 5 > 0, such that M 1 (t) > M i 1 − ε.
Similarly, we have By the Definition 3.1, the system (2) is permanent.This completes the Proof.
As can be reached from the Definition 3.1, we also obtain that