Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator–prey model with Holling type functional response

By using the continuation theorem of coincidence degree theory, the existence of positive periodic solutions for a delayed ratio-dependent predator–prey model with Holling type III functional response x′(t)=x(t)[a(t)−b(t)∫−∞tk(t−s)x(s) ds]−c(t)x2(t)y(t)m2y2(t)+x2(t),y′(t)=y(t)e(t)x2(t−τ)m2y2(t−τ)+x2(t−τ)−d(t),is established, where a(t),b(t),c(t),e(t) and d(t) are all positive periodic continuous functions with period ω>0, m>0 and k(s) is a measurable function with period ω, τ is a nonnegative constant. The permanence of the system is also considered. In particular, if k(s)=δ0(s), where δ0(s) is the Dirac delta function at s=0, our results show that the permanence of the above system is equivalent to the existence of positive periodic solution.