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.


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


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