Periodicity in a delayed ratio-dependent predator-prey system. (English) Zbl 0994.34058

The authors consider the following nonautonomous ratio-dependent predator-prey system \[ \begin{aligned} \frac{dx}{dt}&= x\Biggl[a(t)-b(t)\int_{-\infty}^tk(t-u)x(u) du \Biggr]-\frac{c(t)xy}{my+x},\\ \frac{dy}{dt}&= y\Biggl[-d(t)+\frac{f(t)x(t-\tau(t))}{my(t-\tau(t))+x(t-\tau(t))} \Biggr],\end{aligned} \] 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 \(a\in C(\mathbb{R},\mathbb{R})\), \(b, c, d,f,\tau\in C(\mathbb{R}, \mathbb{R}^+)\) 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.


34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
34C25 Periodic solutions to ordinary differential equations
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