Optimal foraging approach to patch use

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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.

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Figure 9.22 The marginal value theorem. (a) When a forager enters a patch, its rate of energy extraction is initially high (especially in a highly productive patch or where the forager has a high foraging efficiency), but this rate declines with time as the patch becomes depleted. The cumulative energy intake approaches an asymptote. (b) The options for a forager. The solid colored curve is cumulative energy extracted from an average patch, and tt is the average traveling time between patches. The rate of energy extraction (which should be maximized) is energy extracted divided by total time, i.e. the slope of a straight line from the origin to the curve. Short stays in the patch (slope = Eshort/(tt + sshort)) and long stays (slope = Elong/(tt + slong)) both have lower rates of energy extraction (shallower slopes) than a stay (sopt) which leads to a line just tangential to the curve. sopt is therefore the optimum stay-time, giving the maximum overall rate of energy extraction. All patches should be abandoned at the same rate of energy extraction (the slope of the line OP). (c) Low productivity patches should be abandoned after shorter stays than high productivity patches. (d) Patches should be abandoned more quickly when traveling time is short than when it is long. (e) Patches should be abandoned more quickly when the average overall productivity is high than when it is low.

Figure 9.23 (a) An experimental 'tree' for great tits, with three patches. (b) Predicted optimal time in a patch plotted against traveling time (-----), together with the observed mean points (± SE) for six birds, each in two environments. (c) The same data points, and the predicted time taking into account the energetic costs of traveling between patches. (After Cowie, 1977; from Krebs, 1978.)

Figure 9.23 (a) An experimental 'tree' for great tits, with three patches. (b) Predicted optimal time in a patch plotted against traveling time (-----), together with the observed mean points (± SE) for six birds, each in two environments. (c) The same data points, and the predicted time taking into account the energetic costs of traveling between patches. (After Cowie, 1977; from Krebs, 1978.)

The model therefore confirms that the optimal stay-time should be greater in more productive patches than in less productive patches (Figure 9.22c). Moreover, for the least productive patches (where the extraction rate is never as high as OP) the stay-time should be zero. The model also predicts that all patches should be depleted such that the final extraction rate from each is the same (i.e. the 'marginal value' of each is the same); and it predicts that stay-times should be longer in environments where the traveling time between patches is longer (Figure 9.22d) and that stay-times should be longer where the environment as a whole is less profitable (Figure 9.22e).

Encouragingly, there is evidence from a number of cases that lends support to the marginal value theorem. In one of the first tests of the theory, Cowie (1977) considered the prediction set out in Figure 9.22d: that a forager should spend longer in each patch when the traveling time is longer. He used captive great tits in a large indoor aviary, and got the birds to forage for small pieces of mealworm hidden in sawdust-filled plastic cups - the cups were 'patches'. All patches on all occasions contained the same number of prey, but traveling time was manipulated by covering the cups with cardboard lids that varied in their tightness and therefore varied in the time needed to prize them off. Birds foraged alone, and Cowie used six in all, subjecting each to two habitats. One of these habitats always had longer traveling times (tighter lids) than the other. For each bird in each habitat Cowie measured the average traveling time and the curve of cumulative food intake within a patch. He then used the marginal value theorem to predict the optimal stay-time in habitats with different traveling times, and compared these predictions with the stay-times he actually observed. As Figure 9.23 shows, the correspondence was quite close. It was closer still when he took account of the fact that there was a net loss of energy when the birds were traveling between patches.

Predictions of the marginal value theorem have also been examined through the behavior of the egg parasitoid, Anaphes victus, attacking the beetle Listronotus oregonensis in a laboratory setting (Boivin et al., 2004). Patches differed in quality by virtue of the varying proportions of hosts already parasitized at the start of the experiment, and in line with the theorem's predictions, parasitoids stayed longer in the more profitable patches (Figure 9.24a). However, contrary to a further prediction, the marginal rate of fitness gain (the rate of progeny production in the final 10 min before leaving a patch) was greatest in the initially most profitable patches (Figure 9.24b).

As was the case with optimal diet theory, the risk of being preyed upon can be expected to modify the predicted outcomes of optimal patch use. With this in mind, Morris and Davidson (2000) compared the giving-up food extraction rates of white-footed mice (Peromyscus leucopus) in a forest habitat (where predation risk is low) and a forest-edge habitat (where predation risk is high). They provided 'patches' (containers) with 4 g of millet grain in 11 foraging sites in the two habitat types, and in both habitat types some sites were in relatively open situations and others were beneath shrubs. They then monitored the grain remaining at the time that the patches were abandoned on two separate days. Their results (Figure 9.25) supported the predictions that mice should abandon patches at predictions of the marginal value theorem ...

... supported by some experiments optimal patch use predictions are modified when there is a risk of being preyed upon

Figure 9.25 The mass of millet grain remaining (giving-up density, g) was higher in patches in the open (riskier) than in paired patches located under shrubs (safer), and was higher in forest-edge habitat (higher predation) than in forest (lower predation). (After Morris & Davidson, 2000.)

higher harvest rates in vulnerable edge habitats than in safe forest habitats, particularly in open situations (where predation risk is highest in each habitat).

A much fuller review of tests of the marginal value theorem is provided, for example, by Krebs and Kacelnik (1991). The picture this conveys is one of encouraging but not perfect correspondence - much like the balance of the results presented here. The main reason for the imperfection is that the animals, unlike the modelers, are not omniscient. As was clear in the case of the white-footed mice, they may need to spend time doing things other than foraging (e.g. avoiding predators). Foragers may also need to spend time learning about and sampling their environment, and are none the less likely to proceed in their foraging with imperfect information about the

Figure 9.24 (a) When the parasitoid Anaphes victus attacked the beetle Listronotus oregonensis in patches of 16 hosts, a varying percentage of which had already been parasitized, parasitoids remained longer in the more profitable patches: those with the smaller percentage of parasitized hosts. (b) However, the marginal gain rate in fitness - the number of progeny produced per minute in the final 10 min before leaving a patch - was greatest in the initially most profitable patches. Bars represent standard errors. (After Boivin et al., 2004.)

distribution of their hosts. For the parasitoids in Figure 9.24, for example, Boivin et al. (2004) suggest that they seem to base their assessment of overall habitat quality on the quality of the first patch they encounter; that is, they 'learn' but their learned assessment may still be wrong. Such a strategy would be adaptive, though, if there was considerable variation in quality between generations (so that each generation had to learn anew), but little variation in quality between patches within a generation (so that the first patch encountered was a fair indication of quality overall).

Nevertheless, in spite of their limited information, animals seem often to come remarkably close to the predicted strategy. Ollason (1980) developed a mechanistic model to account for this in the great tits studied by Cowie. Ollason's is a memory model. It assumes that an animal has a 'remembrance of past food', which Ollason likens to a bath of water without a plug. Fresh remembrance flows in every time the animal feeds. But remembrance is also draining away continuously. The rate of input depends on the animal's feeding efficiency and the productivity of the current feeding area. The rate of outflow depends on the animal's ability to memorize and the amount of remembrance. Remembrance drains away quickly, for example, when the amount is large (high water level) or the memorizing ability is poor (tall, narrow bath). Ollason's model simply proposes that an animal should stay in a patch until remembrance ceases to rise; an animal should leave a patch when its rate of input from feeding is slower than its rate of declining remembrance.

An animal foraging consistently with Ollason's model behaves in a way very similar to that predicted by the marginal value theorem. This is shown for the case of Cowie's great tits in Figure 9.26. As Ollason himself remarks, this shows that to forage in a patchy environment in a way that approximates closely to optimality, an animal need not be omniscient, it does not need to sample and it does not need to perform numerical analyses to find the maxima of functions of many variables: all it predicted and observed behaviors do not correspond perfectly mechanistic models of optimal foraging

Figure 9.26 Cowie's (1977) great tit data (see Figure 9.23) compared to the predictions of Ollason's (1980) mechanistic memory model.

needs to do is to remember, and to leave each patch if it is not feeding as fast as it remembers doing. As Krebs and Davies (1993) point out, this is no more surprising than the observation that the same birds can fly without any formal qualification in aerodynamics.

Mechanistic models have also been developed and tested for a range of patterns of parasitoid attack (like that in Figure 9.24) (see Vos et al., 1998; Boivin et al., 2004). These highlight the important distinction between 'rule of thumb' behavior, where animals follow innate and unvarying rules, and learned behavior, where rules are subject to modification in the light of the forager's immediate experience. The weight of evidence suggests that learning plays at least some role in most foragers' decisions. There is an important distinction, too, between 'incremental' and 'decremental' behavior. With incremental behavior each successful attack in a patch increases the forager's chance of staying there. This is likely to be adaptive when there is considerable variation in quality between patches, because it encourages longer stay-times in better quality patches. With 'decremental' behavior each successful attack in a patch decreases the forager's chance of staying there. This is likely to be adaptive when all patches are of approximately the same quality, because it encourages foragers to leave depleted patches.

Thus, Ollason's model for great tits incorporated rule of thumb, incremental behavior. Boivin et al., on the other hand, found their parasitoids to be exhibiting learned, decremental behavior: a parasitoid attacking a healthy host, for example, was 1.43 times more likely to leave a patch thereafter, and one rejecting a host that had already been attacked was 1.11 times more likely to leave. Vos et al. (1998), by contrast, found incremental behavior when the parasitoid Cotesia glomerata attacked its butterfly larva host, Pieris brassicae: each successful encounter increased its tendency to remain in a patch. For both the great tit and parasitoids, therefore, optimal foraging and mechanistic models are seen to be compatible and complementary in explaining how a predator has achieved its observed foraging pattern, and why that pattern has been favored by natural selection.

Finally, the principles of optimal foraging are also being applied to investigations of the foraging strategies of plants for nutrients (reviewed by Hutchings & de Kroon, 1994). When does it pay to produce long stolons moving rapidly from patch to patch? When does it pay to concentrate root growth within a limited volume, foraging from a patch until it is close to depletion? Certainly, it is good to see such intellectual cross-fertilization across the taxonomic divide.

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