Exercise 46 MH

Write your own program to solve Eq. (4.31) and create the boundary curve. Then conduct numerical experiments to investigate how the boundary curve varies as you change parameters. It is always good to make a prediction by thinking about the result that you anticipate before you run the code. (If you need help getting the coding going, I suggest that you consult Mangel and Clark (1988) and Clark and Mangel (2000).)

In nature we do not observe boundary curves or states. Rather we observe behaviors manifested in time. One way of capturing these behavioral observations is through the simulation of a large number of indivi-dualsthat followthe rules generatedbythe dynamic programming equation. Colin Clark and I (Mangel and Clark 1988, Clark and Mangel 2000) called such individual based models forward iterations (see Connections for more about these), to distinguish them from the backward iterations that generate the decision rules. To implement them, we envision simulating a large number, N, of individuals in which the egg complement of individual i at time t is denoted by X (t). We then use the random number generator to connect the state of each individual at time t to time t — 1. If we let the state — 1 correspond to death, the forward state dynamics associated with Eq. (4.31) are: X(t + 1) = —1 if the parasitoid does not survive from t to t + 1; X(t + 1) = X(t) ifno host is encountered or an inferior host is encountered and the parasitoid survives from t to t + 1; andX(t + 1) = X(t) — 1ifa superior host is encountered or an inferior host is encountered and accepted. By simulating forward, we are able to track variables that are measurable in the field or laboratory such as behaviors, mean egg complements, and survival. Sometimes these can even be done by purely analytical (Markov Chain) methods; see Mangel and Clark (1988) and Houston and McNamara (1999) for examples, but many times simulation is required because the analytical methods are simply too hard.

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