# Math Insight

### Overview of: Spruce budworm outbreak

#### Project summary

Determine the evolution of the spruce budworm outbreak based on a model of the population dynamics in response to predation by birds and carrying capacity of the forest. Present the results with a summary diagram that show sthe evolution of the outbreak, including how it begins and ends.

#### Requirements for project report

Your project report should be no longer than three pages. It should contain the following sections.

1. The model

Introduce the model of the spruce budworm evolution. Explain the meaning of the parameters $r$ and $a$ as well as important features of the form of the function describing predation by birds.

2. The build-up toward the insect outbreak

Describe the situation of the model when $r=0.52$ and $a=3.6$, in particular, the equilibria and their stability. For any stable equilibria, describe the initial conditions for which the solution will converge to the equilibrium.

Describe what changes when you increase $a$ above $7.1$, such as to 7.5. In particular, determine the equilibria and their stability. For any stable equilibria, describe the initial conditions for which the solution will converge to the equilibrium. If there are multiple stable equilibria compare the level of bird predation $h(w)$ at the equilibria.

If we started with a small forest (i.e., a small $a$), and the forest then grew to this size where $a$ is around 7.5, describe the state of the budworm population (which equilibrium it would be near).

Describe any changes if you further increase the forest size to give a carrying capacity around $a=20$. If the forest grew to this size, describe the state of the budworm population.

3. The outbreak

Describe the situation when the carrying capacity is around $a=30$. In particular, determine the equilibria and their stability. For any stable equilibria, describe the initial conditions for which the solution will converge to the equilibrium. Assess the role the birds in controlling the outbreak. (Is their predation level much larger than if $w$ were a tenth this size?)

4. The decline of the outbreak

Describe what happens to the spruce budworm population as the forest begins to die due the outbreak, decreasing the carrying capacity back down to $a=20$. In particular, compare this situation to the case when $a=20$, above. Discuss why the budworm population is different this time.

Discuss how far the carrying capacity must be dropped before the outbreak stops. In particular, for what value of $a$ does the equilibrium corresponding to the outbreak disappear?

5. Summary figure

Your final task is to create this summary diagram with $a$ on the horizontal axis and $w$ on the vertical axis. Draw a bifurcation diagram, with solid lines representing stable equilibria and dashed lines representing unstable equilibria. On top of the bifurcation diagram, draw a curve showing the evolution of the spruce budworm population size $w$ and the carrying capacity $a$ through the whole cycle of the outbreak. This curve should have arrows, to show how the state changes with time, and it should be a loop, as this cycle will continue.

Be sure to label everything clearly on the diagram so the foresters can understand what your model is telling them. Use this diagram to tell the story of how the model predicts how the spruce budworm outbreak initiates and is terminated. Remember, the foresters may not understand mathematics very well, so include enough description so that they can follow what you did.

Total points: 30