Stability and Bifurcation Analysis of Discrete Dynamical Systems 2020View this Special Issue
Periodic Property and Asymptotic Behavior for a Discrete Ratio-Dependent Food-Chain System with Delays
In this paper, a discrete ratio-dependent food-chain system with delay is investigated. By using Gaines and Mawhin’s continuation theorem of coincidence degree theory and the method of Lyapunov function, a set of sufficient conditions for the existence of positive periodic solutions and global asymptotic stability of the model are established.
The past decades have witnessed a great deal of interest in the periodic phenomena of predator-prey systems. For example, Zhang and Tian  investigated the multiple periodic solutions of a generalized predator-prey system with exploited terms. Zhang and Wang  analyzed the existence and global attractivity of a positive periodic solution for a generalized delayed prey-predator system. Li et al.  studied multiple positive periodic solutions of species delay competition systems with harvesting terms. Ding et al.  made a detailed discussion on the periodic solution of a Gause-type predator-prey systems with impulse. Shen and Li  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 . In recent years, Xu et al.  have studied the persistence and stability of the following ratio-dependent food-chain system with delay:where , and denote densities of the prey, predator, and the top predator populations at time , respectively, is constant time delay due to negative feedback of the prey, and and are constant time delays due to gestation. , and are all positive constants. In detail, one can see .
In real life, many biological and environmental parameters do vary in time(for example, naturally subject to seasonal fluctuations). However, Xu et al.  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 [15–18] 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  and Fan and Wang , we obtain the discrete time analogue of system (2):where and all the variables and parameters have the same biological meanings as those in system (1).
The main task of this article is to discuss the dynamics of system (3). That is, applying Mawhin’s continuous theorem  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. The main innovation point lies in better applying computer simulation to explain the changing law of biological population.
The 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. The 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.
2. Existence of Positive Periodic Solutions
For convenience and simplicity in the following discussion, we always use the following notations throughout the paper:where is an periodic sequence of real numbers defined for . Let denote the integer number, denote the real number, denote the nonnegative real number, and denote the three-dimensional real vector.
We always assume that are periodic, i.e.,
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 .
Let be normed vector spaces, is a linear mapping, and is a continuous mapping. The mapping will be called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors and such that , It follows that is invertible. We denote the inverse of that map by . If is an open-bounded subset of , the mapping will be called compact on if is bounded and is compact. Since is isomorphic to , there exist isomorphisms .
Lemma 1. (see , continuation theorem). Let be a Fredholm mapping of index zero, and let be compact on . Suppose(a)For each , every solution of is such that (b) for each , and Then, the equation has at least one solution lying in .
Lemma 2. (see ). Let be periodic, i.e., ; then, for any fixed and any , one has
Lemma 3. Assume that and ; then, the system algebraic equationshave a unique solution with .
Proof. Obviously, . Substituting into the second equation of system (7) and simplifying, we obtainIn the following, we define the function:It is easy to see thatThen, it follows from the zero-point theorem and monotonicity of that there exists a unique such that . Similarly, substituting into the third equation of system (7) and simplifying, we haveWe define the functionClearly,Then, it follows from the zero-point theorem and monotonicity of that there exists a unique such that . The proof is complete.
DefineLet denote the subspace of all periodic sequences equipped with the usual supremum norm , i.e., for any . It is easy to show that is a finite-dimensional Banach space.
LetThen, it follows that and are both closed linear subspaces of andIn the following, we will ready to establish our result.
Proof. First, we make the change of variables ; then, (3) can be reformulated aswhereLet ,where . Then, it is trivial to see that is a bounded linear operator andThen, it follows that is a Fredholm mapping of index zero. DefineIt is not difficult to show that and are continuous projectors such thatFurthermore, the generalized inverse (to ) exists and is given byObviously, and are continuous. Since is a finite-dimensional Banach space, using the Ascoli–Arzela theorem, it is not difficult to show that is compact for any open-bounded set . Moreover, is bounded. Thus, is compact on with any open-bounded set .
Now, we are at the point to search for an appropriate open-bounded subset for the application of the continuation theorem. Corresponding to the operator equation , we haveSuppose that is an arbitrary solution of system (28) for a certain ; summing both sides of (28) from 0 to with respect to , respectively, we obtainIn view of the hypothesis that , there exist such thatIt follows from (28) and (29) thatBy the first equation of (29), we havewhich leads toBy (31) and (35) and Lemma 2, we obtainThus,In view of the second equation of (29) and (37), it is easy to obtainthenThus,From the first equation of (29) and (37), we obtainThen,It follows from (40) and (42) and Lemma 2 thatThus,By the third equation of (29), we obtainwhich leads toThen,By the third equation of (29), we also obtainThus, we obtainFrom (48) and (50) and Lemma 2, we deriveThus,Obviously, are independent of the choice of . Take , where is taken sufficiently large such that where is the unique positive solution of (7).
Now, we have proved that any solution of (28) in satisfies .
Let ; then, it is easy to see that is an open-bounded set in X and verifies requirement (a) of Lemma 2. When is a constant vector in with . Then,Now, let us consider homotopic , whereIn fact, when and , then is a homptopic mapping. Letting be the identity mapping and by direct calculation, we obtainwhereThus,By now, we have proved that verifies all requirements of Lemma 2; then, it follows that has at least one solution in , namely, (20) has at least one periodic solution in , say ; then, it follows that is an periodic solution of system (3) with strictly positive components. The proof is complete.
3. 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.
Theorem 2. Let , and are defined by (67), (68), and (69), respectively. Assume that (H1)–(H3) are satisfied. Furthermore, suppose that there exist positive constants , and such that andThen, the positive -periodic solution of system (3) is globally asymptotically stable.
Proof. In view of Theorem 2, there exists a positive periodic solution of system (3). We prove below that it is uniformly asymptotically stable. First, we introduce the change of variables as follows:Then, it follows from (3) thatwhere converges, uniformly with respect to , to zero as .
Define Lyapunov function as follows:where , and are positive constants given in (67)–(69), respectively, and satisfy . Calculating the difference of along the solution of systems (60)–(62) and using (i), (ii), and (iii), we have