Positive almost periodic solutions for a predator-prey Lotka-Volterra system with delays ∗

In this paper, by using Mawhin’s continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive almost periodic solutions are obtained for the predator-prey Lotka-Volterra competition system with delays 8 > > dui(t) dt = ui(t) � ai(t) − n P l=1 ail(t)ul(t − �il(t)) − m P j=1 bij(t)vj(t − �ij(t)) � ,i = 1,...,n, dvj(t) dt = vj(t) � − rj(t) + n P l=1 djl(t)ul(t − �jl(t)) − m P h=1 ejh(t)vh(t − �jh(t))

Since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a periodically or almost periodically varying environment are considered as important selective forces on systems in a fluctuating environment.Therefore, on the one hand, models should take into account both the seasonality of the periodically changing environment and the effects of time delays [6-11, 13, 14, 17-27].However, on the other hand, in fact, it is more realistic to consider almost periodic system than periodic system.
There are many works on the study of the Lotka-Volterra type periodic systems that have been developed in [6-9, 11, 17, 19, 21, 24].But, relatively few papers have been published on the existence of almost periodic solutions for the Lotka-Volterra type almost periodic systems.Recently, by using the definition of almost periodic function, the contraction mapping, fixed point theory, appropriate Lyapunov functionals and almost periodic functional hull theory some authors have done many good works in theory on almost periodic systems [10,26,[28][29][30].Motivated by above, in this paper, we are concerned with the following predator-prey Lotka-Volterra system with delays where Our main purpose of this paper is by using the coincidence degree theory [30] to study the existence of positive almost periodic solutions of (1.1).Our result obtained in this paper is completely new and our methods used in this paper can be used to study the existence of positive almost periodic solutions to other types of Lotka-Volterra systems with delays.

Preliminaries
Let X, Y be normed vector spaces, L : Dom L ⊂ X → Y be a linear mapping and N : X → Y be a continuous mapping.The 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 exists continuous projectors P : X → X and Q : Y → Y such that Im L = Ker L, Ker Q = Im L = Im (I −Q), it follows that the mapping L Dom L∩KerP : (I − P )X → Im L is invertible.We denote the inverse of that mapping by K P .If Ω is an open bounded subset of X, then the mapping N will be called L-compact on Ω if QN( Ω) is bounded and We introduce the Mawhin's continuation theorem [30] as follows.(2) QNy = 0 for every y ∈ ∂Ω ∩ Ker L; (3) deg{JQN, Ω ∩ Ker L, 0} = 0.
Then Ly = Ny has at least one solution in Dom L ∩ Ω.
For convenience, we denote AP (R, R n ) is the set of all vector valued, almost periodic functions on R and for f ∈ AP (R, R n ) we denote by the set of Fourier exponents and the module of f , respectively.Suppose that f (t, φ) is almost periodic in t, uniformly with respect to φ ∈ S. E{f, ε, S} denotes the set of ε-almost periods for f with respect to S ⊂ C([−σ, 0], R n ), l(ε, S) denotes the length of the inclusion interval and M(f ) = lim T →∞ 1 T T 0 f (s) ds denotes the mean value of f .The following lemma will paly an important role in the proof of our main result.
Then for all t ∈ R, the following hold and Proof.For any t ∈ R, there exists τ ∈ E{f, ε} such that t Hence, (2.1) holds.
Similarly, we also have Thus, (2.2) holds.The proof is complete. where

Main results
By making the substitution Eq.(1.1) is reformulated as e jh (t)e y h (t−θ jh (t)) , j = 1, . . ., m. Proof.If {z n } ⊂ V 1 and z n converges to z 0 , then it is easy to show that z 0 ∈ AP (R, R n ) with mod(z 0 ) ⊂ mod(Π).Indeed, for all |λ| < α we have z 0 e −iλs ds = 0, which implies that z 0 ∈ V 1 .One can easily see that V 1 is a Banach space endowed with the norm • .The same can be concluded for the spaces X and Y.The proof is complete.
From the definitions of φ(t) and φ V 1 (t), one can deduce that t φ(s) ds and t φ V 1 (s) ds are almost periodic functions and thus φ V 2 (t) ≡ (0, 0, . . ., 0) T := 0, which implies that φ(t) Thus Furthermore, one can easily show that Im L is closed in Y and dimKer L = n = codimIm L. Therefore, L is a Fredholm mapping of index zero.The proof is complete. where

It is clear that
In view of Im P = Ker L and Im we can conclude that the generalized inverse (of L) K P : Im L → Ker P ∩ Dom L exists and is given by where G[z] is defined by QN and (I − Q)N are obviously continuous.Now we claim that K P is also continuous.By our hypothesis, for any ε < 1 and any compact set S ⊂ C([−σ, 0], R n ), where σ = max 1≤i,l≤n 1≤j,h≤m sup t∈R {σ il (t), τ ij (t), δ jl (t), θ jh (t)}, let l(ε, S) be the inclusion interval of E{F, ε, S}.Suppose that {z k (t)} ⊂ ImL = V 1 and z k (t) uniformly converges to z 0 (t).Since t 0 z k (s) ds ∈ Y (n = 0, 1, 2, . ..), there exists ρ (0 < ρ < ε) such that E{F, ρ, S} ⊂ E{ t 0 z n (s) ds, ε}.Let l(ρ, S) be the inclusion interval of E{F, ρ, S} and l = max{l(ρ, S), l(ε, S)}.It is easy to see that l is the inclusion interval of both E{Π, ε, S} and E{Π, ρ, S}.Hence, for all t ∈ [0, l], there exists τ t ∈ E{F, ρ, S} ⊂ E{ t 0 z k (s) ds, ε} such that t + τ t ∈ [0, l].Therefore, by the definition of almost periodic functions we observe that By applying (3.2), we conclude that t 0 z(s) ds (z ∈ Im L) is continuous and consequently K P and K P (I − Q)Nz are also continuous.
From (3.2), we also have that t 0 z(s) ds and K P (I − Q)Nz are uniformly bounded in Ω.In addition, we can easily conclude that QN( Ω) is bounded and K P (I − Q)Nz is equicontinuous in Ω. Hence by the Arzelà-Ascoli theorem, we can immediately conclude that If the following condition is satisfied: (H) The system of linear algebraic equations M(e jh )y h , j = 1, . . ., m Then Eq.( 1.1) has at least one positive almost periodic solution.
Proof.In order to apply Lemma 2.1, we set the Banach spaces X and Y the same as those in Lemma 3.1 and the mappings L, N, P, Q the same as those defined in Lemmas 3.2 and 3.3, respectively.Thus, we can obtain that L is a Fredholm mapping of index zero and N is a continuous operator which is L-compact on Ω.It remains to search for an appropriate open and bounded subset Ω.
where Ω is any open bounded subset of X. EJQTDE, 2012 No. 65, p. 5 Proof.The projections P and Q are continuous such that Im P = Ker L and Im L = Ker Q.