Bifurcation and chaos in a ratio-dependent predator-prey system with time delay. (English) Zbl 1197.37028
Summary: A ratio-dependent predator-prey model with time delay is investigated. We first consider the local stability of a positive equilibrium and the existence of Hopf bifurcations. By using the normal form theory and center manifold reduction, we derive explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions. Finally, we consider the effect of impulses on the dynamics of the above time-delayed population model. Numerical simulations show that the system with constant periodic impulsive perturbations admits rich complex dynamic, such as periodic doubling cascade and chaos.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.
MSC:
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 92D25 | Population dynamics (general) |
| 34K18 | Bifurcation theory of functional-differential equations |
| 37N25 | Dynamical systems in biology |
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