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Periodicity in a delayed ratio-dependent predator-prey system. (English) Zbl 0994.34058
The authors consider the following nonautonomous ratio-dependent predator-prey system \align \frac{dx}{dt}&= x\Biggl[a(t)-b(t)\int_{-\infty}^tk(t-u)x(u) du \Biggr]-\frac{c(t)xy}{my+x},\\ \frac{dy}{dt}&= y\Biggl[-d(t)+\frac{f(t)x(t-\tau(t))}{my(t-\tau(t))+x(t-\tau(t))} \Biggr],\endalign where $x(t)$ and $y(t)$ stand for the prey’s and the predator’s density at time $t$, respectively; $m$ is a positive constant and $a\in C(\bbfR,\bbfR)$, $b, c, d,f,\tau\in C(\bbfR, \bbfR^+)$ are $\omega$-periodic functions. By using Gaines and Mawhin’s coincidence degree theory, some sufficient conditions which guarantee the existence of positive periodic solutions are obtained.

##### MSC:
 34K13 Periodic solutions of functional differential equations 92D25 Population dynamics (general) 34C25 Periodic solutions of ODE
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##### References:
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