Bifurcation analysis for a three-species predator-prey system with two delays.(English)Zbl 1243.92055

Summary: A three-species predator-prey system with two delays is investigated. By choosing the sum $$\tau$$ of two delays as a bifurcation parameter, we first show that Hopf bifurcations at the positive equilibrium of the system can occur as $$\tau$$ crosses some critical values. Secondly, we obtain formulas determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.

MSC:

 92D40 Ecology 34K18 Bifurcation theory of functional-differential equations 34K13 Periodic solutions to functional-differential equations 65C20 Probabilistic models, generic numerical methods in probability and statistics
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