Periodicity in a nonlinear discrete predator–prey system with state dependent delays

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear discrete state dependent delays predator–prey systemN1(k+1)=N1(k)expb1(k)-∑i=1nai(k)(N1(k-τi(k,N1(k),N2(k))))αi-∑j=1mcj(k)(N2(k-σj(k,N1(k),N2(k))))βj,N2(k+1)=N2(k)exp-b2(k)+∑i=1ndi(k)(N1(k-ρi(k,N1(k),N2(k))))γi,where ai,cj,di:Z→R+ai,cj,di:Z→R+ are positive ωω-periodic, ωω is a fixed positive integer. b1,b2:Z→R+b1,b2:Z→R+ are ωω-periodic and ∑k=0ω-1bi(k)>0. τi,σj,ρi:Z×R×R→R(i=1,2,…,n,j=1,2,…,m)τi,σj,ρi:Z×R×R→R(i=1,2,…,n,j=1,2,…,m) are ωω-periodic with respect to their first arguments, respectively. αi,βj,γi(i=1,2,…,n,j=1,2,…,m)αi,βj,γi(i=1,2,…,n,j=1,2,…,m) are positive constants.