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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 $$\aligned x(t) &= x(t)\Biggl[a(t)- b(t) \int^t_{-\infty} K(t- s) x(s)\,ds\Biggr]- e(t) g\Biggl({x(t)\over y(t)}\Biggr),\\ y(t) &= y(t)\Biggl[e(t) g\Biggl({x(t)- \tau(t)\over y(t)- \tau(t)}\Biggr)- d(t)\Biggr],\endaligned\tag{$*$}$$ where $x(t)$ and $y(t)$ represent the predator and prey densities, respectively, $a(t)$, $b(t)$, $c(t)$, $d(t)$, $e(t)$ and $\tau(t)$ are positive periodic continuous functions with period $\omega> 0$, $\omega$ is a positive real constant. $K(s): \bbfR^+\to \bbfR^+$ is a measurable, $\omega$-periodic, normalized function such that $\int^{+\infty}_0 K(s)\,ds= 1$. By using the continuation theorem of the coincidence degree theory [see {\it R. E. Gaines} and {\it 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 $(*)$ when the functional response function $g$ is monotonic or nonmonotonic. As corollaries, some applications are listed.

34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
Full Text: DOI
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