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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

x(t)=x(t)a (t) - b (t) - t K (t-s) x (s) d s-e(t)gx(t) y(t),y(t)=y(t)e (t) g x(t)-τ(t) y(t)-τ(t) - d (t),(*)

where x(t) and y(t) represent the predator and prey densities, respectively, a(t), b(t), c(t), d(t), e(t) and τ(t) are positive periodic continuous functions with period ω>0, ω is a positive real constant. K(s): + + is a measurable, ω-periodic, normalized function such that 0 + K(s)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 ω-periodic solution of system (*) when the functional response function g is monotonic or nonmonotonic. As corollaries, some applications are listed.

MSC:
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)