Bifurcation in a Model of Spruce Budworm Populations

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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 (March 2011)
Open content licensed under CC BY-NC-SA
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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.
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