The following case study is based on Bogich & Shea (2008). We will use it again in the monitoring and learning section of the course. Gypsy moths are a devastating invasive forest pest in North America. As the manager of a large forest complex thats just ahead of the invasion front, you are interested in minimizing the damage to your forest stand while slowing the advance of the invasion. Your forest is divided into 50 patches of about equal size. After speaking with gypsy moth population experts, you decide to categorize the state of each of your patches into one of three states (or categories), based on the degree of gypsy moth infestation: empty patches (empty of gypsy moths, full of trees); patches of medium infestation; and patches with large infestations. A patch with a medium infestation costs $1000 per year in lost revenue (penalty); a patch with a large infestation costs $3000 per year in lost revenue. Empty patches (call the number of empty patches E) can become medium patches (call this probability c), and medium patches (call the number of medium patches M) can grow to become large patches (call this probability g). Large patches (call the number of large patches L) can decline to become medium patches (call this probability d), or can become extinct (call this probability e(L)). Medium patches can also go extinct (call this probability e(M)).

Build a conceptual model (i.e., a diagram showing the possible states and arrows indicating the possible transitions among states, each arrow labeled with the probability of the transition) for the annual dynamics of gypsy moths in these patches. (2pts) Assuming that the transition rates are deterministic, write a quantitative model for these dynamics, in the absence of management (like the BIDE model in the lecture), write equations that describe how Lt becomes Lt+1. Mt becomes Mt+1, and Et becomes Et+1. (Hint: Mt+1 = Lt*d + Mt*(1 g e(M)) + Et*c). (2pts) You have three management options at your disposal: (1) doing nothing; (2) reducing colonization; and (3) eradicating two large patches.

Given your available resources, you can apply one of these actions each year. You expect action 2 to reduce the colonization rate by 75%. When you apply action 3, the eradication occurs before the other transitions, and the eradicated patches cannot be colonized in that year. Incorporate these dynamics and modify the equations of the model in part 2 appropriately. (Assume all of these options cost the same to implement.) (2pts) Now build your model in Excel, using the Excel shell provided. (Several notes: you can have fractional patches; also, you may find that you need to trap 0s, especially when using action 3that is, make sure you dont eradicate more large patches than exist.)

Assume the five rate parameters are: colonization rate c = 0.05, growth rate g = 0.1, extinction rate of large patches e(L) = 0.01, extinction rate of medium patches e(L) = 0.05, and rate of decline of large patches to medium d = 0.03. Show that in the absence of action, the equilibrium (when the parameters are constant over time then it should reach an equilibrium state) distribution of patches is 25 large, 10 medium, and 15 empty. (3pts) Suppose you begin with 3 large, 5 medium, and 42 empty patches in the initial (t=0) year. What is the total cost incurred over ten years (through t=10) if (a) no action is taken, (b) action 2 (reducing colonization) is taken each year, or (c) action 3 (eradicating 2 large patches) is taken each year. Why do you observe the trends you do? Whats the best action to take, and why? (2pts) There is uncertainty about several of the rate parameters. An alternative model has both the parameters associated with extinction of medium patches and colonization set at 0.3 (rather than 0.05). What is the expected 10-yr cost for the 3 possible actions? (1pt) (a) If the initial weights on the two models are 50%, what action would be optimal in the face of uncertainty? (b)

What is the expected value of perfect information (EVPI)? That is, how much would costs be reduced by completely resolving uncertainty about which model is, in fact, true. (c) Given this information, how much would you be willing to pay for a monitoring program that could resolve this uncertainty? (a) What if the initial weight on the first model were 0.3, what would the EVPI be? So is EVPI the same no matter when the current weights on models are? What if the initial state of the system was 25 large, 10 medium, and 15 empty patches? What action would be optimal under each of the models, and under the averaged model? What is the expected value of perfect information in this case? Explain.