Co-existence of a period annulus and a limit cycle in a class of predator-prey models with group defense. (English) Zbl 1478.34059
Summary: For a family of two-dimensional predator-prey models of Gause type, we investigate the simultaneous occurrence of a center singularity and a limit cycle. The family is characterized by the fact that the functional response is nonanalytical and exhibits group defense. We prove the existence and uniqueness of the limit cycle using a new theorem for Liénard systems. The new theorem gives conditions for the uniqueness of a limit cycle which surrounds a period annulus. The results of this paper provide a mechanism for studying the global behavior of solutions to Gause systems through bifurcation of an integrable system which contains a center and a limit cycle.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 92D25 | Population dynamics (general) |
| 34C25 | Periodic solutions to ordinary differential equations |
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