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In this paper we consider the following system of partial linear homogeneous difference equations: xs(i+2,j)+asxs+1(i+1,j+1)+bsxs(i,j+2)=0,s=1,2,...,n−1,xn(i+2,j)+anx1(i+1,j+1)+bnxn(i,j+2)=0and the system of partial linear nonhomogeneous difference equations: ys(i+2,j)+asys+1(i+1,j+1)+bsys(i,j+2)=fs(i,j),s=1,2,...,n−1,yn(i+2,j)+any1(i+1,j+1)+bnyn(i,j+2)=fn(i,j)where n=2,3,..., xs(0,j)=ϕs(j), j=2,3,..., xs(1,j)=ψs(j),j=1,2,... (resp. ys(0,j)=ϕs(j), j=2,3,..., ys(1,j)=ψs(j),j=1,2,...) for the first system (resp. for the second system); as, bs, are real constants; fs:N2→R are known functions; ϕs(j),ψs(j) are given sequences; and s=1,2,...,n and the domain of the solutions of the above systems are the sets Nm={(i,j),i+j=m}, m=2,3,.... More precisely, we find conditions so that every solution of the first system converges to 0 as i→∞ uniformly with respect to j. Moreover, we study the asymptotic stability of the trivial solution of the first system. In addition, under some conditions on fs, we prove that every solution of the second system is bounded, and finally, we find conditions on fs so that every solution of the second system converges to 0 as i→∞ uniformly with respect to j. [ABSTRACT FROM AUTHOR]

In this paper we prove the stability of the zero equilibria of two systems of difference equations of exponential type, which are some extensions of an one-dimensional biological model. The stability of these systems is investigated in the special case when one of the eigenvalues is equal to -1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. In addition, we study the existence and uniqueness of positive equilibria, the attractivity and the global asymptotic stability of these equilibria of some related systems of difference equations. [ABSTRACT FROM AUTHOR]