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

摘要

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)∫ from x=-∞ to x=t k(t-s)x(s)ds] - (c(t)x~2(t)y(t))/(m~2y~2(t)+x~2(t)), y'(t) = y(t) [(e(t)x~2(t-τ)/(m~2y~2(t-τ) + x~2(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) = δ_0(s), where δ_0(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.