Existence of Positive Periodic Solutions of a Lotka-Volterra System with Multiple Time Delays

The research is supported by National Natural Science Foundation of China (No. 10771215) Abstract In this paper, a class of Lotka-Volterra system with multiple time delays is considered. By using the continuation theorem of coincidence degree theory, we derive a set of easily verifiable sufficient conditions that guarantees the existence of at least a positive periodic solution.


Introduction
Lotka-Volterra system is an important population system and has been studied by many authors, see (Chen Shihua, et.al, 2004, Fan Meng, et.al,1999, Li Yongkun, et.al, 2001) and the reference therein.but most of the previous results focused on the stability, attractiveness, persistence and periodicity of solution to the ordinary differential systems or time delay systems with constant delays.Rare work has been done for the systems with varying delays and varying coefficients.In 1991, Weng di Wang et.al (Wang Wending,et. al, 1991) had considered a two-dimensional predator-prey system with a finite constant number discrete delays with initial conditions x(s) = ϕ(s) ≥ 0, s ∈ [−τ, 0]; ϕ(0) > 0, where r 1 , r 2 are real constants with r 1 > 0; a i j , b i j , τ i j , ρ i j (i = 1, 2; j = 1, 2, ..., m) are non-negative constants.Not all of a 1 j and not all b 1 j ( j = 1, 2, ..., m) are zero; Both ϕ(s) and ψ(s) are continuous on the interval [−τ, 0] in which τ = max{τ i j , ρ i j : i = 1, 2; j = 1, 2, ..., m}.And obtained the conclusion that the time delays are harmless for uniform persistence of the solutions to the system.

February, 2010
We note that any biological or environmental parameters are naturally subject to fluctuation in time.It is necessary and important to consider models with periodic ecological parameters or perturbations which might be naturally exposed ( for example, those due to seasonal effects of weather, food supply, mating habits, hunting or harvesting seasons, etc.).Thus, the assumption of periodicity of the parameters is a way of incorporating the periodicity of the environment.
In this paper, we are concerned with the effects of periodicity of ecological and environmental parameters and time delays.
Throughout the paper, we always assume that (H 1 ) r i (t), a i j (t), b i j (t), τ i j (t), ρ i j (t)(i = 1, 2; j = 1, 2, ..., m) are ω periodic, i.e., The principle object of this article is to find a set of sufficient conditions that guarantees the existence of at least a positive periodic solution for system (2) (3).

Basic lemma
In order to explore the existence of positive periodic solutions of (2) (3) and for the reader , s convenience, we shall first summarize below a few concepts and results without proof, borrowing from (Yang Zhihui, et. al, 2007).
Let X, Y be normed vector spaces, L : DomL ⊂ X → Y is a linear mapping, N : X → Y is a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dimKerL = codimImL < +∞ and ImL is closed in Y.
If L is a Fredholm mapping of index zero and there exist continuous projectors P : X → X and 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 Lemma 2.2.

Existence of positive periodic solutions
For convenience and simplicity in the following discussion ,we always use the notations below throughout the paper: Then σ i j (t) and θ i j (t) have inverse functions denoted by µ i j (t), ε i j (t), (i = 1, 2; j = 1, 2, ..., m), respectively.Obviously, In the following, we will ready to state and prove our result.
Proof.Since solutions of (2) (3) remained positive for all t ≥ 0, we let (5) It is easy to see that if system (5) has one ω periodic solution (u * 1 (t), u * 2 (t)) T , then (x * (t), y * (t) ) T is a positive solution of system (2).Therefore, to complete the proof, it suffices to show that system (5) has at least one ω periodic solution.
Then X and Z are Banach spaces when they are endowed with the norm . .Let L : DomL ⊂ X → Z and N : X → Z be the following: Define continuous projective operators P and Q: We can see that (ImL), then it follows that L is a fredholm mapping of index zero.Moreover, it is easy to check that

Journal of Mathematics Research
February, 2010 By easily computation, we have where 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 Now we are at the point to search for an appropriate open, bounded subset Ω for the application of the continuation theorem.Corresponding to the operator equation Ly = λNy, λ ∈ (0, 1), we have Suppose that u(t) = (u 1 (t), u 2 (t)) T ∈ X is an arbitrary solution of system (8) for a certain λ ∈ (0, 1), integrating both sides of ( 8) over the interval [0, ω] with respect to t , we obtain In view of the following: exp{u 2 (s)}ds, (i = 1, 2; j = 1, 2, ..., m).( 11 By the mean value theorem for improper integral, there exist ξ ik j ∈ [0, ω] (i = 1, 2; k = 1, 2; j = 1, 2, ..., m) such that where Then, So we have where , , , .
By the condition of theorem 3.1, there exist t i ∈ [0, ω], i = 1, 2 such that February, 2010 By the condition (H 4 ) of theorem 3.1, there exist B 1 , B 2 > 0 such that In view of the following: then it follows from (22) (23) (24) that Obviously, B 1 , B 2 , B 3 , B 4 are independent of λ ∈ (0, 1).By the condition (H 5 ) of Theorem 3.1, it is easy to show that the algebraic equations has a unique positive solution ( ũ1 , ũ2 ) T ∈ R 2 .Take M = max{B 5 , B 6 } + B 0 , where B 0 is taken sufficiently large such that the unique positive solution (ũ 1 , ũ2 Let Ω := {u = {u(t)} ∈ X : u < M}, then it is easy to see that Ω is an open, bounded set in X and verifies requirement (a) of Lemma 2.1.When (u 1 (t), u 2 (t) Letting J be the identity mapping and by direct calculation, we get This proves that condition (b) in Lemma 2.1 is satisfied.By now, we have proved that Ω verifies all requirements of Lemma 2.1, then it follows that Lu = Nu has at least one solution (u 1 (t), u 2 (t)) T in DomL ∩ Ω, that is to say, (5) has at least one ω periodic solution in DomL ∩ Ω.Then we know that ((x(t), y(t)) T = (exp{u 1 (t)}, exp{u 2 (t)}) T is an ω periodic solution of system (2) (3) with strictly positive components.This completes the proof.
Remark 3.1.Theorem 3.1 remains valid if some or all terms are replaced by corresponding terms with discrete time delays, distribute delays (finite or infinite), or deviating arguments respectively.At this point, we would like to point out that, when one applies the continuation theorem from the coincidence degree theory to explore the existence of periodic solutions to the system of differential equations or difference equations, time delays of any type or the deviating arguments have no effect on the existence of positive solutions.
Lemma 2.1.(Robert E.Gaines et.al, 1991)(Continuation Theorem ) Let L be a fredholm mapping of index zero and let N be L−compact on Ω. Suppose (a) for each λ ∈ (0, 1), every solution x of Lx = λN x is such that x ∂Ω; (b) QN x 0 for each x ∈ KerL ∂Ω, and deg{JQN, Ω ∂KerL, 0} 0; Then the equation Lx = N x has at least one solution lying in DomL Ω.