# 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

This applet is found in the pages

List of all applets

#### General information about Geogebra Web applets

This applet was created using Geogebra. In most Geogebra applets, you can move objects by dragging them with the mouse. In some, you can enter values with the keyboard. To reset the applet to its original view, click the icon in the upper right hand corner.