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Periodic solutions of delayed ratio-dependent predator–prey models with monotonic or nonmonotonic functional responses. (English) Zbl 1069.34098

The authors consider the following delayed ratio-dependent predator-prey system

$\begin{array}{cc}\hfill x\left(t\right)& =x\left(t\right)\left[a\left(t\right)-b\left(t\right){\int }_{-\infty }^{t}K\left(t-s\right)x\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds\right]-e\left(t\right)g\left(\frac{x\left(t\right)}{y\left(t\right)}\right),\hfill \\ \hfill y\left(t\right)& =y\left(t\right)\left[e\left(t\right)g\left(\frac{x\left(t\right)-\tau \left(t\right)}{y\left(t\right)-\tau \left(t\right)}\right)-d\left(t\right)\right],\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $x\left(t\right)$ and $y\left(t\right)$ represent the predator and prey densities, respectively, $a\left(t\right)$, $b\left(t\right)$, $c\left(t\right)$, $d\left(t\right)$, $e\left(t\right)$ and $\tau \left(t\right)$ are positive periodic continuous functions with period $\omega >0$, $\omega$ is a positive real constant. $K\left(s\right):{ℝ}^{+}\to {ℝ}^{+}$ is a measurable, $\omega$-periodic, normalized function such that ${\int }_{0}^{+\infty }K\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds=1$. By using the continuation theorem of the coincidence degree theory [see R. E. Gaines and J. L. Mawhin, Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics. 568. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0339.47031)], the authors establish two main theorems on the existence of at least one positive $\omega$-periodic solution of system $\left(*\right)$ when the functional response function $g$ is monotonic or nonmonotonic. As corollaries, some applications are listed.

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