Wolfram Demonstrations Project

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.

Contributed by: Selwyn Hollis

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The classic single-equation 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

References:

[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.