Continuation of Bifurcations as an Approach to Parametric Dependency in a Model of Snowshoe Hare Demography.

Figure 2. Application of continuation to the problem of determining parametric dependencies in complex ecological models. The parameters chosen determine the rate at which hares consume winter browse - k_B being the maximum rate of browse consumption and C_B, the half-saturation constant in the hares' functional response.

Left. Resonance horns corresponding to periodic, quasiperiodic and chaotic dynamics of different base periods are projected onto two dimensions. Each horn is delimited by curves of saddle-node bifurcations which give rise to periodic orbits of increasing period as one moves up and to the left along the the dashed curve. The latter is a locus of Neimark-Sacher bifurcations whereon the resonance horns all terminate. Within the horns additional bifurcations induce periodic, quasiperiodic and chaotic modulations of the base cycle. The box (dotted line) indicates the plausible range of parameter values as reported in the literature. Regions A, B, C and D correspond respectively to A)fixed point dynamics, B) cycles of period less than 8 years, C) cycles of period 8-12 years and D) cycles of period greater than 12 years (D).

Right. Time series corresponding to points 1, 2 and 3 in the period- ten resonance horn.

(Reproduced from King, A. A. and W. M. Schaffer. 2001. The geometry of a population cycle: A mechanistic model of snowshoe hare demography. Ecology. 82: 814-830.)

(Reproduced from King, A. A. and W. M. Schaffer. 2001. The geometry of a population cycle: A mechanistic model of snowshoe hare demography. Ecology 82: 814-830.)