Ideal free and related distributions aggregation and interference

We can see, then, that consumers tend to aggregate in profitable patches where their expected rate of food consumption is highest. Yet we might also expect that consumers will compete and interfere with one another (discussed further in Chapter 10), thereby reducing their per capita consumption rate. It follows from this that patches that are initially most profitable become immediately less profitable because they attract most consumers. We might therefore expect the consumers to redistribute themselves, and it is perhaps not surprising that the observed patterns of predator distributions across prey patches vary substantially from case to case. But can we make some sense of this variation in pattern?

In an early attempt to do so, it was proposed that if a consumer forages optimally, the process of redistribution will continue until the profitabilities of all patches are equal (Fretwell & Lucas, 1970; Parker, 1970). This will happen because as long as there are dissimilar profitabilities, consumers should leave less profitable patches and be attracted to more profitable ones. Fretwell and Lucas called the consequent distribution the ideal free distribution: the consumers are 'ideal' in their judgement of profitability, and 'free' to move from patch to patch. Consumers were also assumed to be equal. Hence, with an ideal free distribution, because all patches come to have the same profitability, all consumers have the same consumption rate. There are some simple cases where consumers appear to conform to an ideal free distribution insofar as they distribute themselves in proportion to the profitabilities of different patches (e.g. Figure 9.27a), but even in such cases one of the underlying assumptions is likely to have been violated (e.g. Figure 9.27b -all consumers are not equal).

optimal foraging in plants the ideal free distribution ...

... is a balance between attractive and repellant forces

Ideal Free Distribution

Figure 9.27 (a) When 33 ducks were fed pieces of bread at two stations around a pond (with a profitability ratio of 2 : 1), the number of ducks at the poorer station, shown here, rapidly approached one-third of the total, in apparent conformity with the predictions of ideal free theory. (b) However, contrary to the assumptions and other predictions of simple theory, the ducks were not all equal. (After Harper, 1982; from Milinski & Parker, 1991.)

Figure 9.27 (a) When 33 ducks were fed pieces of bread at two stations around a pond (with a profitability ratio of 2 : 1), the number of ducks at the poorer station, shown here, rapidly approached one-third of the total, in apparent conformity with the predictions of ideal free theory. (b) However, contrary to the assumptions and other predictions of simple theory, the ducks were not all equal. (After Harper, 1982; from Milinski & Parker, 1991.)

incorporating a range of interference coefficients

The early ideas have been much modified taking account, for example, of unequal competitors (see Milinski & Parker, 1991; Tregenza, 1995, for reviews). In particular, ideal free theory was put in a more ecological context by Sutherland (1983) when he explicitly incorporated predator handling times and mutual interference amongst the predators. He found that predators should be distributed such that the proportion of predators in site i, p,, is related to the proportion of prey (or hosts) in site i, h,, by the equation:

where m is the coefficient of interference, and k is a 'normalizing constant' such that the proportions, p,, add up to 1. It is now possible to see how the patch to patch distribution of predators might be determined jointly by interference and the selection by the predators of intrinsically profitable patches.

If there is no interference amongst the predators, then m = 0. All should exploit only the patch with the highest prey density (Figure 9.28), leaving lower density patches devoid of predators.

If there is a small or moderate amount of interference (i.e. m > 0, but m < 1 - a biologically realistic range), then high-density prey patches should still attract a disproportionate number of predators (Figure 9.28). In other words, there should be an aggregative response by the predators, which is not only directly density dependent, but actually accelerates with increasing prey density in a patch. Hence, the prey's risk of predation might itself be expected to be directly density dependent: the greatest risk of predation in the highest prey density patches (like the examples in Figure 9.20c, d).

With a little more interference (m ~ 1) the proportion of the predator population in a patch should still increase with the proportion of prey, but now it should do so more or less linearly rather than accelerating, such that the ratio of predators : prey is roughly the same in all patches (Figure 9.28, and, for example,

Figure 9.20a). Here, therefore, the risk of predation might be expected to be the same in all patches, and hence independent of prey density.

Finally, with a great deal of interference (m > 1) the highest density prey patches should have the lowest ratio of predators : prey (Figure 9.28). The risk of predation might therefore be expected to be greatest in the lowest prey density patches, and hence inversely density dependent (like the data in Figure 9.20e).

It is clear, therefore, that the range of patterns amongst the data in Figure 9.20 reflects a shifting balance between the forces of attraction and of repulsion. Predators are attracted to highly profitable patches; but they are repelled by the presence of other predators that have been attracted in the same way.

This description, however, of the relationship between the distribution pseudo-interference of predators and the distribution of predation risk has been peppered with 'might be expected to's. The

m —»- 0



Proportion of prey in the /th patch (h)

Proportion of prey in the /th patch (h)

Figure 9.28 The effect of the interference coefficient, m, on the expected distribution of predators amongst patches of prey varying in the proportion of the total prey population they contain (and hence, in their 'intrinsic' profitability). (After Sutherland, 1983.)

Fgiure 9.29 (a) The aggregative response of the egg parasitoid, Trichogramma pretiosum, which aggregates on patches with high densities of its host Plodia interpunctella. (b) The resultant distribution of ill effects: hosts on high-density patches are least likely to be parasitized. (After Hassell, 1982.)

reason is that the relationship also depends on a range of factors not so far considered. For example, Figure 9.29 shows a case where the parasitoid Trichogramma pretiosum aggregates in high-density patches of its moth host, but the risk of parasitism to the moth is greatest in low-density host patches. The explanation probably lies in time wasted by parasitoids in high-density host patches, dealing with already parasitized hosts that may still attract para-sitoids because they are not physically removed from a patch (unlike preyed-upon prey) (Morrison & Strong, 1981; Hassell, 1982). Thus, earlier parasitoids in a patch may interfere indirectly with later arrivals, in that the previous presence of a parasitoid in a patch may reduce the effective rate at which later arrivals attack unpara-sitized hosts. This effect has been termed 'pseudo-interference' (Free et al., 1977); its potentially important effects on population dynamics are discussed in Chapter 10.

Expected patterns are modified further still if we incorporate learning by the predators, or the costs of migration between patches (Bernstein et al., 1988, 1991). With a realistic value of m (= 0.3), the aggregative response of predators is directly density dependent (as expected) as long as the predators' learning response is strong relative to the rate at which they can deplete patches. But if their learning response is weak, predators may be unable to track the changes in prey density that result from patch depletion. Their distribution will then drift to one that is independent of the density of prey.

Similarly, when the cost of migration is low, the predators' aggregative response remains directly density dependent (with m = 0.3) (Figure 9.30a). When the cost of migration is increased, however, it still pays predators in the poorest patches to move, but for others the costs of migration can outweigh the potential gains of moving. For these, the distribution across prey patches is random. This results in inverse density dependence in mortality rate between intermediate and good patches, and in a 'domed' relationship overall (Figure 9.30b). When the cost of migration is very high, it does not pay predators to move whatever patch they are in - mortality is inversely density dependent across all patches (Figure 9.30c).

Clearly, there is no shortage of potential causes for the wide range of types of distributions of predators, and of mortality rates, across prey patches (see Figures 9.20 and 9.29). Their consequences, in terms of population dynamics, are one of the topics dealt with in the chapter that follows. This highlights the crucial importance of forging links between behavioral and population ecology.

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