Memory Professor System

The advantages to a consumer of spending more time in higher profitability patches are easy to see. However, the detailed allocation of time to different patches is a subtle problem, since it depends on the precise differentials in profitability, the average profitability of the environment as a whole, the distance between the patches, and so on. The problem has been a particular focus of attention for optimal foraging theory. In particular, a great deal of interest has been directed at the very common situation in which foragers themselves deplete the resources of a patch, causing its profitability to decline with time. Amongst the many examples of this are insectivorous insects removing prey from a leaf, and bees consuming nectar from a flower.

Charnov (1976b) and Parker and Stuart (1976) produced similar models to predict the behavior of an optimal forager in such situations. They found that the optimal stay-time in a patch thresholds and giving-up times

Figure 9.21 (a) On arrival in a patch, fifth-instar Plectrocnemia conspersa larvae that encounter and eat a chironomid prey item at the beginning of the experiment ('fed') quickly cease wandering and commence net-building. Predators that fail to encounter a prey item ('unfed') exhibit much more widespread movement during the first 30 min of the experiment, and are significantly more likely to move out of the patch. (b) Directly density-dependent aggregative response of fifth-instar larvae in a natural environment expressed as mean number of predators against combined biomass of chironomid and stonefly prey per 0.0625 m2 sample of streambed (n = 40). (After Hildrew & Townsend, 1980; Townsend & Hildrew, 1980.)

should be defined in terms of the rate of energy extraction experienced by the forager at the moment it leaves a patch (the 'marginal value' of the patch). Charnov called the results the 'marginal value theorem'. The models were formulated mathematically, but their salient features are shown in graphic form in Figure 9.22.

The primary assumption of the model is that an optimal forager will maximize its overall intake of a resource (usually energy) during a bout of foraging, taken as a whole. Energy will, in fact, be extracted in bursts because the food is distributed patchily; the forager will sometimes move between patches, during which time its intake of energy will be zero. But once in a patch, the forager will extract energy in a manner described by the curves in Figure 9.22a. Its initial rate of extraction will be high, but as time progresses and the resources are depleted, the rate of extraction will steadily decline. Of course, the rate will itself depend on the initial contents of the patch and on the forager's efficiency and motivation (Figure 9.22a).

The problem under consideration is this: at what point should a forager leave a patch? If it left all patches immediately after reaching them, then it would spend most of its time traveling between patches, and its overall rate of intake would be low. If it stayed in all patches for considerable lengths of time, then it would spend little time traveling, but it would spend extended periods in depleted patches, and its overall rate of intake would again be low. Some intermediate stay-time is therefore optimal. In addition, though, the optimal stay-time must clearly be greater for profitable patches than for unprofitable ones, and it must depend on the profitability of the environment as a whole.

Consider, in particular, the forager in Figure 9.22b. It is foraging in an environment where food is distributed patchily and when should a forager leave a patch that it is depleting?

where some patches are more valuable than others. The average traveling time between patches is tt. This is therefore the length of time the forager can expect to spend on average after leaving one patch before it finds another. The forager in Figure 9.22b has arrived at an average patch for its particular environment, and it therefore follows an average extraction curve. In order to forage optimally it must maximize its rate of energy intake not merely for its period in the patch, but for the whole period since its departure from the last patch (i.e. for the period tt + s, where s is the stay-time in the patch).

If it leaves the patch rapidly then this period will be short (tt + sshort in Figure 9.22b). But by the same token, little energy will be extracted (Eshort). The rate of extraction (for the whole period tt + s) will be given by the slope of the line OS (i.e. Eshort/( tt + sshort)). On the other hand, if the forager remains for a long period (slong) then far more energy will be extracted (E[ong); but, the overall rate of extraction (the slope of OL) will be little changed. To maximize the rate of extraction over the period tt + s, it is necessary to maximize the slope of the line from O to the extraction curve. This is achieved simply by making the line a tangent to the curve (OP in Figure 9.22b). No line from O to the curve can be steeper, and the stay-time associated with it is therefore optimal (sopt).

The optimal solution for the forager in Figure 9.22b, therefore, is to how to maximize leave that patch when its extraction overall energy intake rate is equal to (tangential to) the slope of OP, i.e. it should leave at point P. In fact, Charnov, and Parker and Stuart, found that the optimal solution for the forager is to leave all patches, irrespective of their profitability, at the same extraction rate (i.e. the same 'marginal value'). This extraction rate is given by the slope of the tangent to the average extraction curve (e.g. in Figure 9.22b), and it is therefore the maximum average overall rate for that environment as a whole.

O J2

High productivity patch and/or high -— forager efficiency | |||||||||||||

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Low productivity patch and/or high forager efficiency | ||||||||||||

/Sc" |
High productivity patch and/or high forager efficiency Low productivity patch and/or high forager efficiency | ||||||||||||

Enters patch |
Hong opt short Average Low productivity patch High productivity patch Average Low productivity patch Sshort tt S!ong tt Short tt Long tt
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