## Combining behavior and population dynamics

Population dynamics interest us and behavior interests us; how much more interesting then would be the combination of behavior and population dynamics? At the same time, the study of population dynamics is hard and the study of behavior is hard. How much more difficult then

Figure 4.14. In order to couple behavior and population dynamics, we need to think about annual time scales (on which Nicholson-Bailey dynamics occur) and within-season time scales (on which behavior occurs).

nnual time scale t t + 1

Seasonal time scale will be the combination of behavior and population dynamics? Pretty difficult, but as the example in this section shows, we can make some progress. This example draws heavily on a paper by Bernie Roitberg and me (Mangel and Roitberg 1992). Citations to other literature on this subject are found in Connections.

We return to Eq. (4.1), which connects the annual dynamics of hosts and parasitoids. In order to incorporate behavior, we need to think about both the annual time scale (as in Eq. (4.1)) and about a time scale within the season, which we shall denote by s (Figure 4.14). Thus, the season runs from s = 0 to s = S (which we might set equal to 1 to obtain exactly the same form as Eq. (4.1)) and we now think of the populations as H(t, s) and P(t, s), the number of hosts and parasitoids at time s within year t. First, we observe that to obtain the between-season Nicholson-Bailey dynamics, we solve the within-season dynamics dH

in the sense that the solution of these equations is Eqs. (4.1) when we couple the appropriate within- and between-season dynamics by linking H(t, S) with H(t + 1,0) and P(t, S) with P(t + 1, 0). Our first insight thus comes virtually for free: the Nicholson-Bailey dynamics assume a constant number of parasitoids throughout the season. How might this occur? We could assume, for example, that parasitoids are emerging from last season's hosts at the same rate at which they are dying during this season. This assumption means that we can think of the number of parasitoids as a function only of the season and not worry about para-sitoid numbers within the season.

In order to incorporate behavior, we require some kind of variation in the hosts. Following the lead of the previous section, let us assume that hosts come in two phenotypes in which the first phenotype is preferred, for whatever biological reasons. We thus consider the dynamics of H1(t, s) and H2(t, s), the numbers of superior and inferior hosts at time s within year t. To capture parasitoid behavior, we assume that there is a time s* (which we will find) before which only the superior hosts are attacked and after which both hosts are attacked. The within-season dynamics in Eq. (4.32) must now be expanded to account for these two cases. We will assume that parasitoids search randomly and that when both hosts are attacked, they are attacked in proportion to their initial abundance (so that the parasitoid cannot distinguish between previously attacked hosts and unattacked hosts). The dynamics for the two host types thus become and dH 1

dH 2

"d7

and solution of these equations will tell us the within-season population dynamics of hosts, given the behavior of the parasitoids. Equations (4.33) are linear equations and are very easy to solve (so much so that I do not even make finding the solution an exercise). We conclude that at the end of season t

and Eqs. (4.34) tell us the whole story about the within-season effects of behavior.

Next, we must construct the between-season population dynamics of hosts and parasitoids. If we assume that each surviving host produces R offspring, then the total number of hosts at the start of the next season will be HT(t + 1, 0) = R{H1(t, S) + H2(t, S)}, which needs to be distributed across superior and inferior phenotypes. In general, we might imagine that these are functions of the total host population, so that Hi(t + 1, 0) = fi(HT(t + 1, 0)). The simplest function is a constant proportion and Hassell (1978) called this the ''proportional refuge model'' because the inferior hosts provide a refuge from attack by parasitoids. An alternative, which captures density-dependent effects, is

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