Hosts that are attacked by parasitoids will come in a range of varieties. For example, hosts will vary in size and larger hosts will often provide more hemolymph for developing parasitoids than smaller hosts. When the hosts are pupae (and thus do not move around), hosts that are in the sun may provide quicker development times for the parasitoids than those that are in the shade (and if they are not hidden may also provide higher risks of mortality). For solitary parasitoids, then, an issue is in which kind of host to lay eggs.

Gregarious parasitoids have an additional problem of how many eggs to lay in a host. More eggs in a host may imply more daughters, but they could be smaller, and since size is tied to fecundity, the overall representation of genes in future generations may be reduced. For example, Rosenheim and Rosen (1992) studied clutch size in Aphytis lingnaensis, which attacks scale insects that are pests of citrus. They found that the average size of a daughter emerging from a clutch of size c laid in a host was S(c) = max{0.2673-0.0223c, 0} and that the number of eggs a female can lay depends upon her size according to E(S) = max{181.8S-26.7, 0}, where max {A, B} means take the larger of A or B. Thus, for example, there is a minimum size below which a female does not have any eggs. A simple measure of fitness for an ovipositing female is the number of grand-offspring produced from a clutch of size c and this is cE(S(c)). This computation (Figure 4.12) shows that for a gregarious parasitoid there may be an optimal number of eggs to lay in a single host.

In the early 1980s, Eric Charnov and Sam Skinner recognized that many of the ideas from foraging theory could be applied to understand the evolution of host choice in parasitoids (Charnov and Skinner 1984, 1985, 1988). I subsequently wrote on similar topics; the relevant chapters in Mangel and Clark (1988) and Clark and Mangel (2000) are entry points to the broader literature. In this section, we will consider a relatively simple version of these kinds of models, which will set up the next two sections as well as introduce new methods.

We begin with a pro-ovigenic, univoltine, solitary parasitoid in which the season length is T and the parasitoid has two different kinds of hosts to attack. Oviposition in host type i leads to an offspring in the next generation with probability fi and we assume that host type 1 is

better, in the sense thatf > f2, and that hosts are encountered at different rates. Because there is one generation per year, we choose as a measure of fitness the expected lifetime reproductive success of the parasitoid and ask for the pattern of host acceptance that maximizes expected reproductive success. The question is interesting because both time within the season and current egg complement may affect the oviposi-tion behavior. For example, early in the season, we might anticipate that females will be more choosy, for the same egg complement, than late in the season. We also might predict that females with many eggs will be less choosy than those with fewer eggs. The question is then how are we able to more formally characterize these predictions and use them in guiding our thinking about additional theory, experiments and field work.

The method we use is called stochastic dynamic programming (SDP, see Connections for more details; in a previous draft I apologized -perhaps to Strunk and White - for using a noun adjective. But Nick Wolf, upon reading this apology, wrote ''I don't see the problem: 'programming' is a noun (a gerund, actually), 'dynamic' is an adjective modifying 'programming', and 'stochastic' is an adjective acting as an adverb because it modifies another adjective. No problem!'') and is very straightforward in this particular case. To begin, we define a fitness function F(x, t) as the maximum expected reproductive success from the current time t until T (when the season ends) given that the egg complement at time t is X(t) = x. Here "maximum" refers to the choices over alternative oviposition behaviors at a particular time and egg complement, and "expected" refers to the mathematical average over possible host encounters and natural mortality. Thus, expected lifetime reproductive success at emergence is F(x, 1).

We will work in discrete time, and characterize the probability of encountering a host of type i in an unit of time by 1,, so that the sum 11 +12, which must be less than or equal to 1, is a measure of the richness of the environment. We assume that the probability of surviving a single period of time is e—m, where m is the rate of mortality.

Since the season ends at time T, there is no gain in fitness thereafter and any eggs that the parasitoid holds are wasted. This end of season condition is represented mathematically by

For previous times, we need to think about the balance between current and future fitness. At any previous time t, three mutually exclusive events may occur: no host is encountered, host type 1 is encountered or host type 2 is encountered. If no host is encountered (with probability 1 —11 —12) and the parasitoid survives, she will start the next time interval with the same number of eggs. If a host of type 1 (the better host) is encountered we assume that she oviposits in it; if she survives to the next period she begins that period with one less egg. If a host type 2 is encountered, the parasitoid may reject the host (thus beginning the next period with the same number of eggs, but having gained no fitness from the encounter) or she may oviposit in (thus beginning the next period with one fewer egg but having gained fitness from the encounter). We assume that the order of the processes is oviposition, then survival (so that she always gets credit for an oviposition, even if she does not survive to the next period). These three possibilities and their consequences lead to a relationship between fitness at time t and at time t +1:

F (x, t) = (1 — 1 — l2)e—mF (x, t + 1)+ 1 [ f 1 + e—mF (x — 1, t + 1)]

+ 12 maxfe—mF (x, t + 1); f2 + e—mF (x — 1, t + 1)} (4.31)

Equation (4.31) is called an equation of stochastic dynamic programming. We solve this equation backwards in time, since F(x, T) is known. Hence, this method is called "backwards induction"; details of doing this can be found in Mangel and Clark (1988) and Clark and Mangel (2000).

Note that each term on the right hand side involves current accumulation to fitness (which may be 0 if no host is encountered) and future accumulations of fitness, discounted by the chance of mortality. The most interesting term, of course, is the third one in which the balance is complicated by the loss of an egg that can be used in the future accumulation of fitness.

The solution of this equation generally must be done by numerical methods, which means that specific parameter values must be chosen. For Eq. (4.31), these parameters are the encounter and mortality rates, the fitnesses associated with oviposition in the two kinds of hosts, the time horizon and the maximum egg complement that the parasitoid may have. Once these are specified, the solution of Eq. (4.31) comes rapidly (Figure 4.13a), especially these days: in Mangel and Clark (1988), we had to introduce a variety of means for getting around the limited computer power of then extant machines and software.

Although Figure 4.13a is interesting, we are often more interested in the behavior of the parasitoids than in their lifetime reproductive success. Happily, predictions about behavior come freely as we solve Eq. (4.31). That is, when we consider the maximization step in this equation, we also determine the predicted optimal behavior b*(x, t), which is to either accept the inferior host or reject it for oviposition at time t when X(t) = x. We thus are able to construct a boundary curve that separates the x — t plane into regions in which the parasitoid is predicted to reject the inferior host and regions in which the parasitoid is predicted to accept the inferior host (Figure 4.13b). Studying this figure as we move horizontally (forward in time with egg complement fixed), we see a formalization of the intuition that individuals are predicted to become less choosy as time increases. Holding time constant and moving vertically upwards, we see a formalization of the intuition that individuals are predicted to become less choosy as they have higher egg complements.

How might such an idea be tested? One method is to use a photoperiod manipulation to signal to the parasitoids that it is either earlier or later in the season than it is. For example, by rearing parasitoids in a late summer photoperiod, we send the signal that t is closer to T than it actually is and the consequence would be that if the real point in the x — t plane were at A in Figure 4.13b, we predict that the parasitoids will behave as if they were located at point B in the plane. That is, the photoperiod manipulation is predicted to cause parasitoids that would otherwise reject an inferior host to accept it. Roitberg et al. (1992) did exactly that manipulation, using a theory somewhat more complicated than the one here. The more elaborate theory lead to a wide range of predictions and the experimental results were in concordance with the predictions.

Our predictions will change as parameters vary. For example, in Figure 4.13c, I show the boundary curves for the previous case in which the mortality rate was 0.05 and for the case in which the mortality rate is 0.10. In the latter case, we predict that the balance between current and

Figure 4.13. The total lifetime expected reproductive success F(x, 1) (panel a) and the boundary for oviposition in the inferior host (panel b) obtained by the solution of Eq. (4.31) for the parameters = 0.3,12 = 0.3, m = 0.05, T = 40, f1 = 1, f2 = 0.3, and maximum egg complement 20. If egg complement exceeds the boundary value at a particular time, we predict that the parasitoid will oviposit in the inferior host; otherwise we predict that she will reject the inferior host. (c) The boundary curve changes as parameters change. When mortality increases, the balance between current and future fitness shifts towards current fitness and thus the boundary curve lowers.

Figure 4.13. The total lifetime expected reproductive success F(x, 1) (panel a) and the boundary for oviposition in the inferior host (panel b) obtained by the solution of Eq. (4.31) for the parameters = 0.3,12 = 0.3, m = 0.05, T = 40, f1 = 1, f2 = 0.3, and maximum egg complement 20. If egg complement exceeds the boundary value at a particular time, we predict that the parasitoid will oviposit in the inferior host; otherwise we predict that she will reject the inferior host. (c) The boundary curve changes as parameters change. When mortality increases, the balance between current and future fitness shifts towards current fitness and thus the boundary curve lowers.

future reproduction should favor current reproduction (because the chance of surviving to achieve future reproduction is lower) and that the parasitoids will accept inferior hosts both earlier and at lower egg complements. Roitberg et al. (1993) tested these ideas in an experiment that simulated an impending thunder storm (via a dropping barometric pressure), which has the potential of high mortality rates for small insects, and found that the predictions and experimental results were once again consistent.

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