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 twocomponent 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 phaseplane orbit, nullclines, and equilibrium points.
The classic singleequation model of a spruce budworm population is , where , , , and are positive parameters [1]. 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 [2], 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 . [1] 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. [2] R. M. May, "Thresholds and Breakpoints in Ecosystems with a Multiplicity of Stable States," Nature, 269, 1977 pp. 471–477.
