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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.

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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