The spruce budworm is a serious pest in forests dominated by fir trees. In eastern Canada, this pest attacks the balsam fir tree in cycles of about 4 years, where the budworms defoliate and kill most of the trees in the forest. Ludwig, Jones, and Holling [1] developed a model to simulate the population of spruce budworms, and their model is written:

where r represents the growth rate for the spruce budworms and k represents the amount of foliage available for the worms to eat. The last term in the expression represents the predation of the worms (primarily by birds). In this problem, we will use the qualitative methods of differential equations developed in class to understand some of the dynamics of an outbreak. (This problem will only begin to explore this problem, so for students who are more interested in the subject, they may want to go to the original source.) The growth of trees is slow relative to the population dynamics of the spruce budworm, so the parameter k changes slowly.

a. For a young forest, k is relatively small. Assume that r = 0.55 and k = 6. Use Maple's DEplot function to graph the slope field for this model for t [0, 20] and P [0, 5]. Show the solution trajectories that have the initial conditions P(0) = 0.1, 1, 2, and 5.

b. With the parameters in Part a., use Excel to graph the function f(P) for P [-1, 4]. Find all equilibria, giving their stability, and create the phase-line diagram (on your Excel graph). Use open circles to represent unstable equilibria and closed circles for stable equilibria. Explain how this analysis represents an endemic infestation of spruce budworms in the young forest.

c. As the forest grows, k becomes larger. Let k = 10 and use Maple's DEplot function to graph the slope field for this model for t [0, 20] and P [0, 10]. Show the solution trajectories that have the initial conditions P(0) = 0.1, 1, 2, 5, and 10.

d. With the parameters in Part c., use Excel to graph the function f(P) for P [-1, 10]. Find all equilibria, giving their stability, and create the phase-line diagram (on your Excel graph). Use open circles to represent unstable equilibria and closed circles for stable equilibria. Starting with the initial level of spruce budworms at the endemic level above, what level of spruce budworms will occur in this more mature forest? If there are a large number of spruce budworms in this forest, then what level of spruce budworms will be in this forest?

e. As the forest continues to grow, k becomes larger. Let k = 20 and use Maple's DEplot function to graph the slope field for this model for t [0, 20] and P [0, 20]. Show the solution trajectories that have the initial conditions P(0) = 1, 5, 10, 15, and 20.

f. With the parameters in Part e., use Excel to graph the function f(P) for P [-1, 20]. Find all equilibria, giving their stability, and create the phase-line diagram (on your Excel graph). Use open circles to represent unstable equilibria and closed circles for stable equilibria. Starting with the initial level of spruce budworms at the endemic level above, what level of spruce budworms will occur in this more mature forest?

g. Bonus: Find what a hysteresis effect is and explain how this model exhibits hysteresis resulting in periodic outbreaks of spruce budworms in growing balsam fir forests.

[1] Ludwig, Jones, and Holling (1978), Qualitative analysis of insect outbreak systems: The spruce budworm and forest, J. Anim. Ecol. 47, 315.