# Math Insight

### Applet: Spruce budworm model

Illustration of the dynamical system modeling an outbreak of the spruce budworm population. The evolution of the budworm population $w(t)$ is modeled by the autonomous differential equation $\diff{w}{t}=f(w)$, where \begin{align*} f(w)= r w \left(1 - \frac{w}{a}\right) - \frac{w^2}{1+w^2}. \end{align*} The left panel shows a plot of $f(w)$ (black curve), which changes depending on the value of the parameters $r$ and $b$ (changeable via sliders). If you click the play button in the lower left corner of one of the panels or increase $t$ manually via the red slider in the right panel, the evolution of $w(t)$ from the initial condition $w(t_0)=w_0$ is shown by the blue curves. In the left panel, $w(t)$ is a line on the horizontal axis. In the right panel, $w(t)$ is plotted versus time $t$. The direction of $w(t)$ can be determined by the sign of $f(w)$ or by turning on the vector field (with check box). Equilibria are displayed by circles in the left panel and horizontal lines in the right panel if you check the box.

Applet file: spruce_budworm_nondimensionalized.ggb