### Project 1.8.1. Project—The Spruce Budworm.

*Choristoneura fumiferana*or the

*eastern spruce budworm*is a species of moth native to the eastern United States and Canada. The caterpillars feed on the needles of spruce and fir trees. Populations can experience significant oscillations. The spruce budworm population remains at a relatively low, constant level most of the time. However, outbreaks have been recurring approximately every three decades, and studies suggest the spruce budworm has been breaking out in eastern North America for thousands of years.

Outbreaks of the spruce budworm have been responsible for some major deforestations in Canada and the United States. The eastern spruce budworm is considered one of the most destructive forest pests in North America.

You are acting as a consultant to the state department of forestry. Your task is to explain how outbreaks occur and how often the state can expect outbreaks.

The equation

\begin{equation}
x' = r \left( 1 - \frac{x}{K} \right) x - c \frac{x^2}{a + x^2}\tag{1.8.1}
\end{equation}

has been used to describe the dynamics of spruce budworm populations, where the variable \(x\) denotes the population or density of the insect [16]. One explanation that has been given for the occurrence of outbreaks is based on the multiple bifurcations that occur with this differential equation.

#### (a)

Explain how (1.8.1) can be used to model the spruce budworm population.

#### (b)

If \(a = 0.01\text{,}\) \(c = 1\text{,}\) and \(K = 1\text{,}\) we have a family of differential equations parameterized by \(r\text{,}\)

\begin{equation*}
x' = rx(1-x) - \frac{x^2}{0.01 + x^2}.
\end{equation*}

Solve the equation

\begin{equation*}
rx(1-x) - \frac{x^2}{0.01 + x^2} = 0
\end{equation*}

and plot the result in the \(xr\)-plane for \(0 \leq r \lt 1\text{.}\)

#### (c)

To find the bifurcation diagram for the spruce budworm equation, reflect the graph obtained in part Task 1.8.1.b about the line \(r= x\) line.

#### (d)

Estimate the two bifurcation values from your graph. Explain what happens to the population as \(r\) increases. That is, when does an outbreak occur? What happens after an outbreak?

#### (e) Your Final Report.

Your final report should contain a one-page executive summary. The executive summary should summarize your work in such a way that the reader can rapidly become acquainted with the material. It should contain a brief description of the problem, important background information, a discussion of pertinent assumptions, a short description of your methodology, concise analysis, and your main conclusions. Assume the reader is familiar with the basics of calculus and differential equations, so there is no need to walk through every step of your solution process or include equations. However, you should still describe the processes and mathematical techniques you used to reach your conclusions and explain why you used them. Refer the reader to the appendices as needed.

Appendices should be neatly formatted and present information in a logical manner.

*DO NOT*simply print out Sage code. Consolidate your results and provide a short explanation of what it is the reader is seeing while also highlighting key pieces of information in the appendix.- Appendix A—Answers and analysis
- Additional Appendices—Include additional appendices as necessary.