Periodic Property and Asymptotic Behavior for a Discrete Ratio-Dependent Food-Chain System with Delays

*e past decades have witnessed a great deal of interest in the periodic phenomena of predator-prey systems. For example, Zhang and Tian [1] investigated the multiple periodic solutions of a generalized predator-prey system with exploited terms. Zhang and Wang [2] analyzed the existence and global attractivity of a positive periodic solution for a generalized delayed prey-predator system. Li et al. [3] studied multiple positive periodic solutions of n species delay competition systems with harvesting terms. Ding et al. [4] made a detailed discussion on the periodic solution of a Gause-type predator-prey systems with impulse. Shen and Li [5] obtained a set of sufficient conditions for the existence of at least one strictly positive periodic solution and the uniqueness and global attractivity of positive periodic solution for an impulsive predator-prey model with dispersion and time delays. For more knowledge about the periodic solutions of predator-prey models, one can see [6–12]. For the papers mentioned above, it shall be pointed out that many investigations have been performed to analyze the dynamical behavior on biological species by using continuous or impulsive mathematical models [6–9]. It has been widely argued and accepted that difference equations often occur in numerous setting and forms, both in mathematics itself and in its applications to statistics, computing, electrical circuit analysis, dynamics, economics, biology, and other fields [13]. In recent years, Xu et al. [14] have studied the persistence and stability of the following ratio-dependent food-chain system with delay:


Introduction
e past decades have witnessed a great deal of interest in the periodic phenomena of predator-prey systems. For example, Zhang and Tian [1] investigated the multiple periodic solutions of a generalized predator-prey system with exploited terms. Zhang and Wang [2] analyzed the existence and global attractivity of a positive periodic solution for a generalized delayed prey-predator system. Li et al. [3] studied multiple positive periodic solutions of n species delay competition systems with harvesting terms. Ding et al. [4] made a detailed discussion on the periodic solution of a Gause-type predator-prey systems with impulse. Shen and Li [5] obtained a set of sufficient conditions for the existence of at least one strictly positive periodic solution and the uniqueness and global attractivity of positive periodic solution for an impulsive predator-prey model with dispersion and time delays. For more knowledge about the periodic solutions of predator-prey models, one can see [6][7][8][9][10][11][12]. For the papers mentioned above, it shall be pointed out that many investigations have been performed to analyze the dynamical behavior on biological species by using continuous or impulsive mathematical models [6][7][8][9]. It has been widely argued and accepted that difference equations often occur in numerous setting and forms, both in mathematics itself and in its applications to statistics, computing, electrical circuit analysis, dynamics, economics, biology, and other fields [13]. In recent years, Xu et al. [14] have studied the persistence and stability of the following ratio-dependent food-chain system with delay: , where x 1 (t), x 2 (t), and x 3 (t) denote densities of the prey, predator, and the top predator populations at time t, respectively, τ 1 ≥ 0 is constant time delay due to negative feedback of the prey, and τ 2 ≥ 0 and τ 3 ≥ 0 are constant time delays due to gestation. a i (i � 1, 2, 3), a 11 , a 12 , a 21 , a 23 , a 32 , m 12 , and m 23 are all positive constants. In detail, one can see [14].
In real life, many biological and environmental parameters do vary in time(for example, naturally subject to seasonal fluctuations). However, Xu et al. [14] did not involve the varying parameters of the food-chain model. To describe the object relationship between predator population and prey population, we modify system (1) as the following nonautonomous ratio-dependent food-chain system with varying delay: , .
Many authors [15][16][17][18] argue that discrete time models governed by difference equations are more appropriate to describe the dynamics relationship among populations than continuous ones when the populations have nonoverlapping generations. Moreover, discrete time models can also provide efficient models of continuous ones for numerical simulation.
In order to reveal the dynamic relationship of predator and prey and explain the stability law of both species by applying the computer simulations, we think that it is reasonable to study time ratio-dependent food-chain systems governed by difference equations. Following the lines of Wiener [19] and Fan and Wang [20], we obtain the discrete time analogue of system (2): , , where k � 0, 1, 2, . . . and all the variables and parameters have the same biological meanings as those in system (1). e main task of this article is to discuss the dynamics of system (3). at is, applying Mawhin's continuous theorem [21] to study the existence of positive periodic solutions of (3) and investigating the global asymptotical stability of system (3) by means of the method of Lyapunov function. e main innovation point lies in better applying computer simulation to explain the changing law of biological population. e remainder of the paper is organized as follows. In Section 2, a easily verifiable sufficient condition for the existence of positive solutions of difference equations is obtained by the continuation theorem and priori estimations. e sufficient condition for the global asymptotical stability of system (3) when all the delays are zero is presented in Section 3. In Section 4, we give some computer simulations.

Existence of Positive Periodic Solutions
For convenience and simplicity in the following discussion, we always use the following notations throughout the paper: where f(k) is an ω−periodic sequence of real numbers defined for k ∈ Z. Let Z denote the integer number, R denote the real number, R + denote the nonnegative real number, and R 3 denote the three-dimensional real vector. We always assume that (H1)a i (i � 1, 2, 3), a 11 , a 12 , a 21 , a 23 , a 32 , m 12 , m 23 : , In order to explore the existence of positive periodic solutions of (3) and for the reader's convenience, we shall first introduce a few concepts and results without proof, borrowing from Gaines and Mawhin [21].
Let X and Y be normed vector spaces, L: DomL ⊂ X ⟶ Y is a linear mapping, and N: X ⟶ Y is a continuous mapping. e mapping L will be called a Fredholm mapping of index zero if dimKer L � codimIm L < + ∞ and Im L is closed in Y. If L is a Fredholm mapping of index zero and there exist continuous projectors P: X ⟶ X and Q: Y ⟶ Y such that Im P � Ker L and Im L � Ker Q � Im(I − Q), It follows that L|Dom L∩Ker P: (I − P)X ⟶ Im L is invertible. We denote the inverse of that map by K P . If Ω is an open-bounded subset of X, the mapping N will be called L−compact on Ω if QN(Ω) is bounded and K P (I − Q)N: Ω ⟶ X is compact. Since Im Q is isomorphic to Ker L, there exist isomorphisms J: Im Q ⟶ Ker L.
Lemma 1 (see [21], continuation theorem). Let L be a Fredholm mapping of index zero, and let N be Lemma 2 (see [20]). Let g: Z ⟶ R be ω periodic, i.e., g(k + ω) � g(k); then, for any fixed k 1 , k 2 ∈ I ω and any k ∈ Z, one has Lemma 3. Assume that a 2 < a 21 and a 3 < a 32 ; then, the system algebraic equations Proof. Obviously, v 1 � a 1 /a 11 > 0. Substituting v 1 � a 1 /a 11 into the second equation of system (7) and simplifying, we obtain In the following, we define the function: It is easy to see that en, it follows from the zero-point theorem and Similarly, substituting v * 2 into the third equation of system (7) and simplifying, we have We define the function Clearly, Discrete Dynamics in Nature and Society lim θ⟶+∞ g(c) � a 3 > 0. (15) en, it follows from the zero-point theorem and monotonicity of g(c) that there exists a unique v * 3 > 0 such that g(v * 3 ) � 0. e proof is complete. Define Let l ω ⊂ l 3 denote the subspace of all ω periodic sequences equipped with the usual supremum norm ‖ · ‖, i.e., ‖z‖ � |z 1 (k)| + |z 2 (k)| + |z 3 (k)| for any z � z(k): k ∈ Z { } ∈ l ω . It is easy to show that l ω is a finite-dimensional Banach space. Let en, it follows that l ω 0 and l ω c are both closed linear subspaces of l ω and In the following, we will ready to establish our result. □ Theorem 1. Let S 1 be defined by (37). Under condition (H1), suppose that the following conditions (H2) a 1 > a 12 m 12 , hold; then, system (3) has at least an ω positive periodic solution.
Proof. First, we make the change of variables x i (t) � exp(u i (t))(i � 1, 2, 3); then, (3) can be reformulated as where Let X � Y � l ω , where u ∈ X, k ∈ Z. en, it is trivial to see that L is a bounded linear operator and dim Ker L � 3 � codimIm L.
en, it follows that L is a Fredholm mapping of index zero. Define It is not difficult to show that P and Q are continuous projectors such that Im P � Ker L, 4 Discrete Dynamics in Nature and Society Furthermore, the generalized inverse (to L) k P : Im L ⟶ Ker P∩Dom L exists and is given by Obviously, QN and K P (I − Q)N are continuous. Since X is a finite-dimensional Banach space, using the Ascoli--Arzela theorem, it is not difficult to show that Suppose that u(k) � (u 1 (k), u 2 (k), u 3 (k)) T ∈ X is an arbitrary solution of system (28) for a certain λ ∈ (0, 1); summing both sides of (28) from 0 to ω − 1 with respect to k, respectively, we obtain In view of the hypothesis that u � u(k)

(30)
It follows from (28) and (29) that Discrete Dynamics in Nature and Society By the first equation of (29), we have which leads to By (31) and (35) and Lemma 2, we obtain us, In view of the second equation of (29) and (37), it is easy to obtain us, From the first equation of (29) and (37), we obtain en, It follows from (40) and (42) and Lemma 2 that us, By the third equation of (29), we obtain which leads to en, By the third equation of (29), we also obtain us, we obtain 6 Discrete Dynamics in Nature and Society From (48) and (50) and Lemma 2, we derive us, Obviously, S i (i � 1, 2, 3) are independent of the choice of λ ∈ (0, 1). Take S � max S 1 , S 2 , S 3 + S 0 , where S 0 is taken sufficiently large such that |lnv * T is the unique positive solution of (7). Now, we have proved that any solution u � u(k) { } � (u 1 (k), u 2 (k), u 3 (k)) T of (28) in X satisfies ‖u‖ < S, k ∈ Z.
Let Ω ≔ u � u(k) { } ∈ X: ‖u‖ < S { }; then, it is easy to see that Ω is an open-bounded set in X and verifies requirement (a) of Lemma 2. When u ∈zΩ ∩Ker L, u � (u 1 , u 2 , u 3 ) T is a constant vector in R 3 with ‖u‖ � |u 1 | + |u 2 | + |u 3 | � S. en, Now, let us consider homotopic ϕ(u 1 , u 2 , u 3 , In fact, when ϕ(u 1 , u 2 , u 3 , 0) � Gu and ϕ(u 1 , u 2 , u 3 , 1) � QNu, then ϕ is a homptopic mapping. Letting J be the identity mapping and by direct calculation, we obtain where χ 11 � −a 11 exp u * 1 < 0, By now, we have proved that Ω verifies all requirements of Lemma 2; then, it follows that Lu � Nu has at least one solution in Dom L∩Ω, namely, (20) has at least one ω Discrete Dynamics in Nature and Society 7 ) T is an ω periodic solution of system (3) with strictly positive components. e proof is complete.

Global Asymptotic Stability
In this section, we will present sufficient conditions for the globally asymptotical stability of system (3) when all the delays are zero.
Then, the positive ω-periodic solution of system (3) is globally asymptotically stable.
Proof. In view of eorem 2, there exists a positive periodic solution x * 1 (k), x * 2 (k), x * 3 (k) of system (3). We prove below that it is uniformly asymptotically stable. First, we introduce the change of variables as follows: en, it follows from (3) that From (64), we have where A * 1 � δ 1 a 11 (k) +