Bifurcation in a Model of Spruce Budworm Populations
Spruce budworm populations have in recent decades become a classic case study in mathematical biology. This Demonstration illustrates the bifurcation that occurs as a certain parameter is varied in a two-component model of budworm and foliage densities. The parameter is directly related to the size of a predatory population of birds. The plot on the left shows the graphs of these densities versus time, and the plot on the right shows the corresponding phase-plane orbit, nullclines, and equilibrium points.
The classic single-equation model of a spruce budworm population is , where , , , and are positive parameters . The first term on the right side is the usual logistic growth term. The second term models predation by a constant population of birds. The parameter is a (scaled) measure of the average foliage density. The form of this term takes the following into account: (1) When foliage density is large, budworms are better able to hide from the birds, and (2) when the budworm population is small, birds will seek other sources of food.
Following , we let be a measure of the average foliage density in the forest. We also assume that (3) satisfies a logistic equation when and is scaled so that its carrying capacity is 1, (4) budworms consume foliage at a rate proportional to , and (5) the carrying capacity in the budworm equation is proportional to . The system, then, is , where , , , , , and are positive parameters. For the purposes of this Demonstration we have chosen specific values for all parameters except . These are , , and .
 D. Ludwig, D. D. Jones, and C. S. Holling, "Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest," J. Animal Ecology, 47, 1978 pp. 315–332.
 R. M. May, "Thresholds and Breakpoints in Ecosystems with a Multiplicity of Stable States," Nature, 269, 1977 pp. 471–477.