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Zbl 1035.34043
Braza, Peter A.
The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing. (English)
[J] SIAM J. Appl. Math. 63, No. 3, 889-904 (2003). ISSN 0036-1399; ISSN 1095-712X/e

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 2000:: *34C60 Applications of qualitative theory of ODE
34C25 Periodic solutions of ODE
37G15 Bifurcations of limit cycles and periodic orbits
92D25 Population dynamics
34C23 Bifurcation (periodic solutions)

Keywords: Holling-Tanner; predator-prey; degenerate Hopf bifurcations; limit cycles; two-timing; outbreaks

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