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nature of factors that limit parasitoid reproductive success (a set of citations to these questions can be found in Driessen and Hemerik (1992), Heimpel and Collier (1996), Rosenheim (1996), Heimpel and Rosenheim (1998), Heimpel et al. (1998), Sevenster et al. (1998), Rosenheim (1999), Rosenheim and Heimpel (2000), and van Baalen (2000)). Factors limiting parasitoid reproductive success include mortality risk while searching and foraging, egg complement, and time available for searching for oviposition sites and ovipositing. Natural selection could act on the latter two through development, either by increasing the capacity for eggs or the rate of egg maturation or by selecting for individuals who are more efficient at handling hosts.

We then come to the question: is reproductive success of parasitoids limited by time or by eggs (that is, are they likely to die with some eggs still in their bodies, unlaid and thus ''wasted,'' or are they likely to spend some of the end of their lives without eggs, but with opportunities for laying them, thus being reproductively senescent).

One example of how this problem might be attacked is provided by van Baalen (2000). If at the start of the season there are H(t) hosts and P(t) parasitoids, then the number of hosts per parasitoid is on average h(t) = H(t)/P(t). When the parasitoid searches randomly - as assumed in the Nicholson-Bailey model - the number of hosts attacked by an individual parasitoid will follow a Poisson distribution with parameter a. However, the number of attacks will be limited by either the number of hosts per parasitoid or the egg complement, e, of the parasitoid. Thus, the expected number of hosts attacked per parasitoid

Efnumber of hosts attacked per parasitoid} =

and using this gives a sense of the limitation due either to eggs or to hosts by comparison with the average egg complement of individual parasitoids (and provides another way of stabilizing the Nicholson-Bailey dynamics).

Here, I take a slightly different approach than either van Baalen or my own work in collaboration with George Heimpel and Jay Rosenheim (Heimpel et al. 1996, 1998; Rosenheim and Heimpel 2000). We will use the method of stochastic dynamic programming and begin by considering a pro-ovigenic, univoltine parasitoid. The expected lifetime reproductive success of such a parasitoid will be characterized by her maximum egg complement, xmax, and the handling time per host, h. We can formally write this as F(xmax, 1|h, T) to emphasize egg complement, lifetime expected reproductive success at the beginning of a season of length T when handling time is h.

Now we need to define what is meant by egg limitation and time limitation. I propose the following: a parasitoid is egg-limited rather than time-limited if an increase in maximum egg complement of one egg will increase her lifetime expected reproductive success more than a decrease in handling time of one unit (appropriately defined) does. This suggests that there will be a boundary in egg-complement/ handling-time space separating regions in which parasitoids are egg-limited from those in which they are time-limited, and we are going to find the boundary, defined by the condition

F (xmax, 1|h — 1, T)= F (xmax + 1, 1|h, T) (4.41)

because the left hand side of Eq. (4.41) is expected lifetime fitness with current maximum egg complement but a decreased handling time and the right hand side is expected lifetime fitness if handling time were the same but maximum egg complement were increased by one egg. The solution of Eq. (4.41) is the sought-after boundary curve h(xmax).

To begin, consider a solitary parasitoid searching for hosts that, as before, are of two kinds (clearly many generalizations are possible). We fully characterize the problem by describing the encounter rate and fitness accrued from oviposition in host type i, 1 and f respectively, and the mortality rate while searching or ovipositing, m. The dynamic programming equation is then similar to Eq. (4.31):

F (x, t|h, T)= ( 1 — 11 — 12)e—mF (x, t + 1|h, T) + 11[f 1 + e—mF (x — 1, t + h|h, T)] + max 12{e—mF (x, t + 1|h, T);f2 + e—mF (x — 1, t + h|h, T)}

Also as before, we solve this equation backwards in time, starting with the condition that F(x, T|h, T) = 0. One additional decision needs to be made, however, and that concerns how we treat oviposition for situations in which t + h > T (so that the oviposition being considered is the last). The decision that I made for the results presented here is to credit the parasitoid with this last oviposition, even if she dies immediately thereafter.

The numerical results that I show correspond to T = 100, f1 = 1, f2 = 0.1, 11 = 0.4, 12 = 0.4, and m = 0.01 or 0.001. (I have also limited handling time to a maximum of 15 time units.) The boundary curve (Figure 4.16) separates regions in which another egg would increase fitness more than a decrease in handling time. The general shape of the boundary should make sense; for example, parasitoids with lots of eggs are more likely to be time-limited than egg-limited. In addition, the dependence of the boundary on mortality rate should also make sense in that when mortality rates are higher the tendency will be towards a greater likelihood of time limitation than egg limitation.

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