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Complex dynamics in a prey predator system with multiple delays. (English) Zbl 1243.92051
Summary: The complex dynamics is explored of a prey predator system with multiple delays. Holling type-II functional response is assumed for the prey dynamics. The predator dynamics is governed by a modified P.H. Leslie and J.C. Gower scheme [Biometrika 47, 219–234 (1960; Zbl 0103.12502)]. The existence of periodic solutions via Hopf-bifurcations with respect to both delays are established. An algorithm is developed for drawing two-parametric bifurcation diagrams with respect to two delays. The domain of stability with respect to \(\tau _{1}\) and \(\tau _{2}\) is thus obtained. The complex dynamical behavior of the system outside the domain of stability is evident from the exhaustive numerical simulations. Direction and stability of periodic solutions are also determined using normal form theory and center manifold arguments.

MSC:
92D40 Ecology
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
37N25 Dynamical systems in biology
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