Math Insight

Applet: Spruce budworm outbreak movie

This movie illustrates how the spruce budworm population size initially stays low as the forest grows, until finally the population explodes into a outbreak of the budworms which decimates the forest. Click the play button in the lower left corner of one of the panels to start the animation. The left panel shows a plot of $f(w)=rw(1-r/a)-w^2/(1+w^2)$, which is the right hand side of the spruce budworm model $w'(t)=f(w)$. The movie does not show the evolution of this differential equation, but just sets that the budworm size to be a stable equilibrium of the model. (The movie assumes that the evolution to the equilibrium happens faster than the time scale represented.)

The right panel shows how the population size $w$ and the carrying capacity $a$ of the forest evolve with time $T$. When the budworm population is low, the forest grows so that the carrying capacity $a$ increases steadily. Then, when the outbreak occurs, the forest dies and the carrying capacity $a$ decreases steadily. The rate of increase and decrease of $a$ is arbitrary; the time $T$ is some slow time scale over which the forest evolves.

Applet file: spruce_budworm_outbreak_movie.ggb

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