The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing

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 modi. cation 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.