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The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing. (English) Zbl 1035.34043
Summary: The Holling-Tanner model for predator-prey systems has two Hopf bifurcation points for certain parameters. The dependence of the environmental parameters on the underlying bifurcation structure is uncovered using two-timing. Emphasis is on how the bifurcation diagram changes as the Hopf bifurcation points separate. Two degenerate cases require a modification of conventional two-timing. When the two Hopf bifurcation points nearly coalesce, the two stable periodic solution branches are shown to be connected. As a ratio of linear growth rates varies, the Hopf bifurcation points separate further and one limit cycle becomes unstable. This situation can correspond to an outbreak in populations. The modified two-timing analysis analytically captures the unstable and stable limit cycles of the new branch.
MSC:
34C60Qualitative investigation and simulation of models (ODE)
34C25Periodic solutions of ODE
37G15Bifurcations of limit cycles and periodic orbits
92D25Population dynamics (general)
34C23Bifurcation (ODE)
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