POSITIVE PERIODIC SOLUTIONS OF A DISCRETE MUTUALISM MODEL WITH TIME DELAYS

A discrete periodic mutualism model with time delays is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, the existence of positive periodic solutions of the model is established.


Introduction
Two species cohabit a common habitat and each species enhances the average growth rate of the other, this type of ecological interaction is known as facultative mutualism [8].In [6], the author has studied the existence of positive periodic solutions of the periodic mutualism model dN 1 ( where r i ,K i ,α i ∈ C(R,R + ), α i > K i , i = 1,2, τ i ,σ i ∈ C(R,R + ), i = 1,2, r i , K i , α i , τ i , σ i (i = 1,2) are functions of period ω > 0. Since the study on periodic solutions of a population model is of great interest in mathematical biology [5] and many authors [1,7] have argued that the discrete-time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations, then, discrete-time models can provide efficient computational types of continuous models for numerical simulations.It is reasonable to study the discrete-time mutualism model governed by difference equations.One of the ways of deriving difference equations modeling the dynamics of populations with nonoverlapping generations is based on appropriate modifications of the corresponding models with overlapping generations [2,4].In this approach, differential equations with piecewise constant arguments have been proved to be useful.Following the same idea and the same method in [2,4], one can easily derive the following discrete analog of (1.1), which takes the form of (1. 2) The exponential form of (1.2) is more biologically reasonable than that directly derived by replacing the differential by difference in (1.1).Our purpose in this paper is to use Mawhin's continuous theorem [3] to study the existence of positive periodic solutions of (1.2).
Let Z, R, and R 2 denote the sets of all integers and two-dimensional Euclidean vector space, respectively.Throughout this paper, we always assume that the following hold.

Existence of a positive periodic solution
In order to use Mawhin's continuous theorem to establish the existence of at least one positive periodic solution of (1.2), we need to make some preparations.Let X,Y be normed vector spaces, let L : DomL ⊂ X → Y be a linear mapping, and let N : X → Y be a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dimKer L = codim Im L < +∞ and ImL is closed in Y .If L is a Fredholm mapping of index zero, there exist continuous projectors P : X → X and Q : Y → Y such that ImP = Ker L, KerQ = Im L = Im(I − Q).It follows that the mapping L| DomL Ker P : (I − P)X → Im L is invertible.We denote the inverse of the mapping 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 For convenience, we introduce Mawhin's continuous theorem [3, page 40] as follows.
Lemma 2.1.Let L be a Fredholm mapping of index zero and let N be L-compact on Ω. Assume that Then the operator equation Lx = Nx has at least one solution in Ω DomL.

Yongkun Li 501
In what follows, we will use the notations where {u(k)} is an ω-periodic sequence of real numbers defined for k ∈ Z.
The following result was given by [2, Lemma 3 (2. 3) The following result was given by [6, Lemma 2.2]. and Now we state our fundamental theorem about the existence of a positive ω-periodic solution of (1.2).
Proof.Consider the following system of difference equations with delays: where r i , K i , α i , τ i , σ i (i = 1,2) are the same as those in (1.2).It is easy to see that if (2.6) has an ω-periodic solution 2).Therefore, to complete the proof, it suffices to show that system (2.6) has at least one ω-periodic solution. Define Let l ω ⊂ l 2 denote the subspace of all ω-periodic sequences equipped with the norm • , that is, (2.8) It is not difficult to show that l ω is a finite-dimensional Banach space.Let then it is easy to check that l ω 0 and l ω c are both closed linear subspaces of l ω and (2.10) Take X = Y = l ω and let then it is easy to see that L is a bounded linear operator with then 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 Furthermore, the generalized inverse (to L) K P : ImL → Ker P DomL exists, which is given by (2.16) Assume that {(y 1 (k), y 2 (k)) T } ∈ X is a solution of system (2.16) for a certain λ ∈ (0,1).Summing on both sides of (2.16) from 0 to ω − 1 with respect to k, we obtain It is easy to see that we can rewrite (2.17) and (2.18), respectively, as Thus, from (2.16) and (2.19), it follows that In a similar way, by (2.16) and (2.20), we have which implies that there exist points k 1 ,k 2 ∈ I ω such that where Similarly, by (2.20), we can obtain that there exist points k 3 ,k 4 ∈ I ω such that where Therefore, in view of (2.22)-(2.27)and Lemma 2.2, we have (2.29) Hence, Furthermore, by Lemma 2.3, we have where the degree is Brouwer degree and the isomorphism J can be chosen to be the identity mapping, since ImQ = Ker L. By now we know that Ω verifies all the requirements in Lemma 2.1 and then (2.6) has at least one ω-periodic solution.Therefore, (1.2) has at least one positive ω-periodic solution.The proof is complete.

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.