Periodic solutions of delayed ratio-dependent predator–prey models with monotonic or nonmonotonic functional responses

Abstract

In this paper, by using the continuation theorem of coincidence degree theory, we establish several existence results of positive periodic solutions for the delayed ratio-dependent predator–prey model x′(t)=x(t) a(t)−b(t) ∫ −∞ t K(t−s)x(s) ds −c(t)g x(t) y(t) y(t), y′(t)=y(t) e(t)g x(t−τ(t)) y(t−τ(t)) −d(t) , when functional response function g is a monotonic or nonmonotonic, where a( t), b( t), e( t), τ( t) and d( t) are all positive periodic continuous functions with period ω>0, K is a dense function. As corollaries, some applications are listed. In particular, our results extend some known criteria.