Stability of the Positive Steady-state Solutions of Systems of Nonlinear Volterra Difference Equations of Population Models with Diffusion and Infinite Delay

An open problem given by Kocic and Ladas in 1993 is generalized and considered. A sufficient condition is obtained for each solution to tend to the positive steady-state solution of the systems of nonlinear Volterra difference equations of population models with diffusion and infinite delays by using the method of lower and upper solutions and monotone iterative techniques.


Introduction.
We consider the r -dimensional Euclidean space R r .For x = (x 1 ,..., x r ) T ∈ R r , we define its norm x = max i∈I |x i |, where I = {1,...,r }.In R r , we introduce a cone P = {x : x i ≥ 0, i ∈ I}.Then it is a solid cone in R r .It is easy to show that P is normal, regular, minimal, strong minimal and regenerated (see Amann [3]).For two elements x and y = (y 1 ,...,y r ) T in P , we introduce a partial ordering ≤ such that x < (or =)y if and only if x i < (or =)y i for i ∈ I and x ≤ y means that x i ≤ y i for i ∈ I. So, (R r , ≤) becomes a partial ordered Banach space.In R r , we also define an operation of multiplication ⊗ by x⊗y = (x 1 y 1 ,...,x r y r ) T .In this way, (R r , +, ⊗) is a partially ordered commutative ring by installing both this operation ⊗ and the ordinary addition + with the zero element 0 = (0,...,0) T and the unit element u = (1,...,1) T .Define an In the r ×r -dimensional matrix space R r ×r , we also introduce a partial ordering ≤.If X = (x ij ) r ×r and Y = (y ij ) r ×r are two elements in R r ×r , then define that X < (or =)Y if and only if x ij < (or =)y ij for i, j ∈ I and X ≤ Y means that x ij ≤ y ij for i, j ∈ I. Therefore, R r ×r also becomes a partially ordered Banach space.
Consider the following systems of nonlinear Volterra difference equations of population model with diffusion and infinite delays: ..,M s } × {0, 1,...}, where ∆ 1 and ∆ 2 are forward partial difference operators, ∆ 2 1 is a discrete Laplacian operator (see [7,14,15]), A, C > (0) r ×r are diagonal matrices, b ∈ R r and b > 0, u •,• ∈ R r is a double vector sequence (only in form), D 0 = (0) r ×r and D i ∈ R r ×r for i ∈ Z + (0).Together with (1.1), we consider the homogeneous Neumann boundary condition and the initial condition where ∆ N is the normal difference, ∂Ω is the boundary of Ω (see [15]) and φ m,j ∈ P for (m, n) ∈ Ω × Z − (0).By a solution, we mean a double vector sequence (in form) For any given initial and boundary condition (1.2) and (1.3), we can show that the initial and boundary value problem (1.1), (1.2), and (1.3) have a unique solution (see [16]).
We suppose that ( We write throughout this paper that Then D n , D ± n are all nonnegative, nondecreasing and bounded above by D. Since P is regular, we can let D ± = lim n→∞ D ± n .It is easy to see that From Berman and Plemmons [4] or Siljak [18], we know that C −D − is a nonsingular and inverse-positive Metzlerian matrix, i.e., C −D − is invertible and det(C we know, from Metzlerian matrix theory, that C + δ is invertible and det(C + δ) −1 > 0.
In addition, we let It is obvious that the nonlinear Volterra difference equation of population model is a special case when r = 1 and without diffusion, where ∆ is the forward difference operator (cf.[2,7]).
In Kocic and Ladas [10], the following open problem was given.
Open problem.Obtain stability and oscillation results for (1.10).
It is easy to show that (1.1) has only two steady-state solutions u m,n ≡ 0 and u m,n ≡ (C +δ) −1 b.The purpose of this paper is to give a sufficient condition for each solution of (1.1) to tend to the positive steady-state solution u m,n ≡ (C + δ) −1 b of (1.1) by using the method of lower and upper solutions and monotone iterative techniques (cf.[1,13]).
By (2.6), we have that ∆p n } is nonincreasing.The proof is thus complete.
Similarly to the proof of Lemma 2.2, we can easily show that p (2) n is nonincreasing, which completes the proof.
Remark 2.5.As a matter of fact, we can directly use the maximum principle (see Cheng [5]) to obtain the contradiction.

.42)
Now, we consider the Cauchy problems and

.44)
Similarly to the above argument, we can obtain (2.45) Letting → 0, we see that (2.40) holds for k = 1.Again, by repeating the above process, we have that (2.40) holds.

Main results and remarks.
Using the seven lemmas in Section 2, together with the property that P is normal, we get the following main result.Theorem 3.1.Let (1.4), (1.7), (1.8), and (1.9) hold.Assume that {u m,n } is the unique solution of (1.1), (1.2), and (1.3).Then (3.3) Remark 3.3.It is well known that (1.1) describes the growth of r -species alive in Ω, that the densities of the r -populations at place m and time n is u m,n , and that the summation represents the effects of the past history on the present growth rate in mathematical ecology.Therefore, we can only consider the case φ > 0. If this is not the case, these species do not exist.The condition φ < ∞ means that the densities of these species should be finite in practice.Relation (3.1) means that the growth of these species goes to an equilibrium state under ordinary conditions.Equation (1.10) is the case that we do not consider the places and diffusion.