## 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
\begin{aligned} x'(t) &= x(t) \Biggl[a(t)-b(t) \int_{-\infty}^t k(t-s)x(s)\,ds \Biggr]- \frac {c(t)x^2(t)y(t)} {m^2y^2(t)+ x^2(t)},\\ y'(t) &= y(t) \biggl[\frac {e(t)x^2(t-\tau)} {m^2y^2 (t-\tau)+ x^2(t-\tau)}- d(t) \biggr], \end{aligned}
is established, where $$a(t)$$, $$b(t)$$, $$c(t)$$, $$e(t)$$ and $$d(t)$$ are all positive periodic continuous functions with period $$\omega>0$$, $$m>0$$ and $$k(s)$$ is a measurable function with period $$\omega$$, and $$\tau$$ is a nonnegative constant. The permanence of the system is also considered. In particular, if $$k(s)= \delta_0(s)$$, where $$\delta_0(s)$$ 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:

 34K13 Periodic solutions to functional-differential equations 92D25 Population dynamics (general) 34K25 Asymptotic theory of functional-differential equations
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