## Exercise 48 MH

Equation (4.39) does not include a physiological state of the parasitoid. How would it be modified to include egg complement? The equation also does not include the possibility that a host which is encountered has previously been attacked. How would it be modified to account for that?

Further exploration of these ideas requires numerical solution; the parameters that Mangel and Roitberg (1992) fix are 50 parasitoids, 500 hosts at the start of the first season, a season of length 20, host per capita reproduction R = 2, quality of inferior host 3 = 0.1, search parameter a = 0.0001, and mortality probabilities during search and oviposition of 0.001 and 0.2 respectively.

There are at least two ways that we can conceive of viewing the results. The first is through the host-parasitoid phase plane in which steady states are represented by single points and oscillatory solutions by closed orbits (limit cycles). The second is the distribution of s* across different years. These results are shown in Figure 4.15. We conclude from Figures 4.15a-c that the dynamics of the interaction between host and parasitoid can be very rich when behavior and population dynamics are coupled, including strange attractors such as Figure 4.15c (Mangel and Roitberg (1992) show some even stranger cases). Perhaps more importantly, the result ofFigure 4.15d, which shows the distribution of s* across generations and years, tells us that we should expect variation in behavior of parasitoids in the field. Nature is indeed complicated and variable, but much of that complication and variation can be captured and understood.

In both this model, Eq. (4.39), and the previous model, Eq. (4.31), a shift in host preference occurs during the season. However, the mechanisms are very different. Here, the shift in host preference occurs because there is so little time left in the season that 3 exceeds the loss in lifetime fitness that occurs when the parasitoid chooses an inferior host, i.e. 3 > (mo — ms)F(s + 1|t). On the other hand, inEq. (4.31) the shift in host preference occurs because there is too little time left in the season for the ovipositing female to find good hosts for all of her eggs. Thus, Eq. (4.31) involves a mixture of time limitation and egg limitation, whereas Eq. (4.39) is purely time limitation (via survival). It would be interesting to design experiments to separate these; in the next section we consider some additional theory.

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