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Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response. (English) Zbl 1076.34085

Summary: By using the continuation theorem of the coincidence degree theory, the existence of positive periodic solutions for the delayed ratio-dependent predator-prey model with Holling type III functional response

x ' (t)=x(t)a (t) - b (t) - t k (t-s) x (s) d s-c(t)x 2 (t)y(t) m 2 y 2 (t)+x 2 (t),y ' (t)=y(t)e(t)x 2 (t-τ) m 2 y 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, and 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 a positive periodic solution.

MSC:
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
34K25Asymptotic theory of functional-differential equations