State‐dependent Bayesian foraging on spatially autocorrelated food distributions

When prey are cryptic and are distributed in discrete clumps (patches), Bayesian foragers revise their prior expectation about a patch's prey density by using their foraging success in the patch as a source of information. Prey densities are often spatially autocorrelated, meaning that rich patches are often surrounded by other rich patches, while poor patches are often in the midst of other poor patches. In that case, foraging success is informative about prey densities in the current patch and in the surrounding patches. In a spatially explicit environment where prey are cryptic and their densities autocorrelated, I modelled two types of Bayesian foragers that aim to maximize their survival rate: (1) the spatially ignorant forager which does not take account of the spatial structure in its food supply and (2) the spatially informed forager which does take this into account. Not surprisingly, the spatially informed forager has a higher survivorship than the spatially ignorant forager, simply because it is able to obtain more reliable prey density estimates than the spatially ignorant forager. Surprisingly though, the emerging policy used by the spatially informed forager is to leave patches at a lower (expected) giving‐up density (GUD) the further away from its latest prey capture. This is because this forager is willing to wait for good news: a prey capture far from the latest prey capture drastically changes the forager's expectations about prey densities in the patches that it will exploit in the near future, whereas a prey capture near its latest prey capture hardly affects these expectations. Thus, by sacrificing current intake rate for information gain, the spatially informed forager ultimately maximizes its long‐term pay‐off. Finally, as the value of food is less the more energy is stored, both types make state‐dependent giving‐up decisions: the higher their energy store levels, the higher their GUDs.     

Bayesian foraging with only two patch types

I model the optimal Bayesian foraging strategy in environments with only two patch qualities. That is, all patches either belong to one rich type, or to one poor type. This has been a situation created in several foraging experiments. In contrast, previous theories of Bayesian foraging have dealt with prey distributions where patches may belong to one out of a large range of qualities (binomial, Poisson and negative binomial distributions). This study shows that two‐patch systems have some unique properties. One qualitative difference is that in many cases it will be possible for a Bayesian forager to gain perfect information about patch quality. As soon as it has found more than the number of prey items that should be available in a poor patch, it “knows” that it is in a rich patch. The model generates at least three testable predictions. 1) The distribution of giving‐up densities, GUDs, should be bimodal in rich patches, when rich patches are rare in the environment. This is because the optimal strategy is then devoted to using the poor patches correctly, at the expense of missing a large fraction of the few rich patches available. 2) There should be a negative relation between GUD and search time in poor patches, when rich patches are much more valuable than poor. This is because the forager gets good news about potential patch quality from finding some food. It therefore accepts a lower instantaneous intake rate, making it more resistant against runs of bad luck, decreasing the risk of discarding rich patches. 3) When the energy gains required to remain in the patch are high (such as under high predation risk), the overuse of poor patches and the underuse of rich increases. This is because less information about patch quality is gained if leaving at high intake rates (after short times). The predictions given by this model may provide a much needed and effective conceptual framework for testing (both in the lab and the field) whether animals are using Bayesian assessment.     

Optimal foraging: food patch depletion by ruddy ducks

Summary I studied the foraging behavior of ruddy ducks (Oxyura jamaicensis) feeding on patchily distributed prey in a large (5-m long, 2-m wide, and up to 2-m deep) aquarium. The substrate consisted of a 4x4 array of wooden trays (1.0-m long, 0.5-m wide, and 0.1-m deep) which contained 6 cm of sand. Any tray could be removed from the aquarium and loaded with a known number of prey. One bird foraged in the aquarium at a time; thus, by removing a food tray after a trial ended and counting the remaining prey, I calculated the number of prey consumed by the bird. I designed several experiments to determine if ruddy ducks abandoned a food patch in a manner consistent with the predictions of a simple, deterministic, patch depletion model. This model is based on the premise that a predator should maximize its rate of net energy intake while foraging. To accomplish this, a predator should only remain in a food patch as long as its rate of energy intake from that patch exceeds the average rate of intake from the environment. In the majority of comparisons, the number of food items consumed by the ruddy ducks in these experiments was consistent with the predictions of the foraging model. When the birds did not forage as predicted by the model, they stayed in the patch longer and consumed more prey than predicted by the model. An examination of the relation between rate of net energy intake and time spent foraging in the food patch indicated that by staying in a patch longer than predicted, the ruddy ducks experienced only a small deviation from maximum rate of net energy intake. These results provided quantitative support for the prediction that ruddy ducks maximize their rate of net energy intake while foraging.     

The foraging benefits of information and the penalty of ignorance

Patch use theory and the marginal value theorem predict that a foraging patch should be abandoned when the costs and benefits of foraging in the patch are equal. This has generally been interpreted as all patches being abandoned when their instantaneous intake rate equals the foraging costs. Bayesian foraging – patch departure is based on a prior estimate of patch qualities and sampling information from the current patch – predicts that instantaneous quitting harvest rates sometimes are not constant across patches but increase with search time in the patch. That is, correct Bayesian foraging theory has appeared incompatible with the widely accepted cost–benefit theories of foraging. In this paper we reconcile Bayesian foraging with cost–benefit theories. The general solution is that a patch should be left not when instantaneous quitting harvest rate reaches a constant level, but when potential quitting harvest rate does. That is, the forager should base its decision on the value now and in the future until the patch is left. We define the difference between potential and instantaneous quitting harvest rates as the foraging benefit of information, FBI. For clumped prey the FBI is positive, and by including this additional benefit of patch harvest the forager is able to reduce its penalty of ignorance.     

Optimal Bayesian foraging policies and prey population dynamics-some comments on Rodriguez-Girones and Vasquez

In this paper we show the density-dependent harvest rates of optimal Bayesian foragers exploiting prey occurring with clumped spatial distribution. Rodríguez-Gironés and Vásquez (1997) recently treated the issue, but they used a patch-leaving rule (current value assessment rule) that is not optimal for the case described here. An optimal Bayesian forager exploiting prey whose distribution follows the negative binomial distribution should leave a patch when the potential (and not instantaneous) gain rate in that patch equals the best long-term gain rate in the environment (potential value assessment rule). It follows that the instantaneous gain rate at which the patches are abandoned is an increasing function of the time spent searching in the patch. It also follows that the proportion of prey harvested in a patch is an increasing sigmoidal function of the number of prey initially present. In this paper we vary several parameters of the model to evaluate the effects on the forager's intake rate, the proportion of prey harvested per patch, and the prey's average mortality rate in the environment. In each case, we study an intake rate maximizing forager's optimal response to the parameter changes. For the potential value assessment rule we find that at a higher average prey density in the environment, a lower proportion of the prey is taken in a patch with a given initial prey density. The proportion of prey taken in a patch of a given prey density also decreases when the variance of the prey density distribution is increased and if the travel time between patches is reduced. We also evaluate the effect of using predation minimization, rather than rate maximization, as the currency. Then a higher proportion of the prey is taken for each given initial prey density. This is related to the assumption that traveling between patches is the most risky activity. Compared to the optimal potential value assessment rule, the current value assessment rule performs worse, in terms of long-term intake rate achieved. The difference in performance is amplified when prey density is high or highly aggregated. These results pertain to the foraging patch spatial scale and may have consequences for the spatial distribution of prey in the environment.     

Does group foraging promote efficient exploitation of resources?

Increased avoidance of food patches previously exploited by other companions has been proposed as one adaptive benefit of group foraging. However, does group foraging really represent the most efficient way to exploit non- or slowly-renewing resources? Here, I used simulations to explore the costs and benefits of exploiting non-renewing resources by foragers searching for food patches independently or in groups in habitats with different types of resource distribution. Group foragers exploited resources in a patch more quickly and therefore spent proportionately more time locating new patches. Reduced avoidance of areas already exploited by others failed to overcome the increased time cost of searching for new food patches and group foragers thus obtained food at a lower rate than solitary foragers. Group foraging provided one advantage in terms of a reduction in the variance of food intake rate. On its own, reduced avoidance of exploitation competition through group foraging appears unlikely to increase mean food intake rate when exploiting non-renewing patches but may provide a way to reduce the risk of an energy shortfall.     

Prior knowledge about spatial pattern affects patch assessment rather than movement between patches in tactile-feeding mallard

1. Heterogeneity in food abundance allows a forager to concentrate foraging effort in patches that are rich in food. This might be problematic when food is cryptic, as the content of patches is unknown prior to foraging. In such case knowledge about the spatial pattern in the distribution of food might be beneficial as this enables a forager to estimate the content of surrounding patches. A forager can benefit from this pre-harvest information about the food distribution by regulating time in patches and/or movement between patches. 2. We conducted an experiment with mallard Anas platyrhynchos foraging in environments with random, regular, and clumped spatial configurations of full and empty patches. An assessment model was used to predict the time in patches for different spatial distributions, in which a mallard is predicted to remain in a patch until its potential intake rate drops to the average intake rate that can be achieved in the environment. A movement model was used to predict lengths of interpatch movements for different spatial distributions, in which a mallard is predicted to travel to the patch where it expects the highest intake rate. 3. Consistent with predictions, in the clumped distribution mallard spent less time in an empty patch when the previously visited neighbouring patch had been empty than when it had been full. This effect was not observed for the random distribution. This shows that mallard use pre-harvest information on spatial pattern to improve patch assessment. Patch assessment could not be evaluated for the regular distribution. 4. Movements that started in an empty patch were longer than movements that started in a full patch. Contrary to model predictions this effect was observed for all spatial distributions, rather than for the clumped distribution only. In this experiment mallard did not regulate movement in relation to pattern. 5. An explanation for the result that pre-harvest information on spatial pattern affected patch assessment rather than movement is that mallard move to the nearest patch where the expected intake rate is higher than the critical value, rather than to the patch where the highest intake rate is expected.     

On the role of patch density and patch variability in central-place foraging

Existing models of the foraging behavior of single-prey loaders in patchy environments differ on whether the optimal forager is predicted to stay in a patch until a prey is found, or to leave a patch for a next one if a prey is not found by a certain “deadline.” This article examines conditions on the probability distribution of prey density across patches that are necessary or sufficient for the existence of a finite, optimal deadline. It is shown that, for environments in which prey density is variable but never falls below some strictly positive level, a finite, optimal deadline exists when and only when the spatial density of patches is “high.” Also, a characterization is given of a large class of distributions (including the gamma distribution) for which a finite, optimal deadline exists for all levels of spatial density of patches.     

Patch use behaviour in benthic fish depends on their long-term growth prospects

Animals foraging in a heterogeneous environment may combine prior information on patch qualities and patch sample information to maximize intake rate. Prior information dictates the long-term expectations, whereas prior information in combination with patch sample information determines when to leave an individual food patch. We examined patch use behaviour of benthic feeding fish in their natural environment at different spatial scales to test if they could determine patch quality and if patch use behaviour was correlated with environmental quality. In seven lakes along a gradient of environmental quality (measured as maximum benthivore size), we made repeated measurements of giving-up density (GUD) in artificial food patches of different qualities. At the largest spatial scale, between lakes, we tested if giving-up densities revealed the long-term growth expectation of benthic fish. At the local scale of patches and micro patches we tested for the ability of benthic fish to assess patch quality, and how this ability depended on the patch exploitation levels between the different lakes. We found that GUD was positively related to maximum size of bream, suggesting that short-term behavioural decisions reflected long-term growth expectations. Benthic fish discriminated between nearby rich and poor patches, but not between rich and poor micropatches within a food patch. This suggests that the foraging scale of benthic fish lies between the patch and micro patch scale in our experiments. We conclude that patch use behaviour of benthic fish can provide a powerful measure of habitat quality that reveals how benthic fish perceive their environment.     

Smart,smarter, smartest: foraging information states and coexistence

Animals possess different abilities to gain and use information about the foraging patches they exploit. When ignorant of the qualities of encountered patches, a smart forager should leave all patches after the same amount of fixed search time. A smarter forager can be Bayesian by using information on cumulative harvest and time spent searching a patch to better inform its patch‐departure decision. The smartest forager has immediate and continuous knowledge about patch quality, and can make a perfect decision about when to leave each patch. Here we let each of these three strategies harvest resources from a slowly regenerating environment. Eventually a steady‐state distribution of prey among patches arises where the environment‐wide resource renewal just balances the environment‐wide harvest of the foragers. The fixed time forager creates a distribution with the highest mean and highest variance of patch qualities, followed by the Bayesian and the prescient in that order. The less informed strategies promote distributions with both more resources and more exploitable information than the more informed strategies. While it is true that a better‐informed strategy will always out‐perform a less well‐informed, its increase in performance may not compensate it for any costs associated with being better informed. We imagine that the fixed time strategy may be least expensive and the prescient strategy most expensive in terms of sensory organs and associated assess and respond capabilities. To consider competition between such strategies with varying costs, we introduced a single individual of each of the strategies into the environments created by populations of the other strategies. There are threshold costs associated with the better‐informed strategy such that it can or cannot outcompete a less‐informed strategy. However, over a relatively narrow range of foraging costs, less‐informed and better‐informed strategies will coexist. Furthermore, for the prescient and the Bayesian strategies, some combinations of foraging costs produce alternate stable states – whichever strategy establishes first remains safe from invasion by the other.     

Agent-based model approach to optimal foraging in heterogeneous landscapes: effects of patch clumpiness

Optimal foraging theory concerns animal behavior in landscapes where food is concentrated in patches. The efficiency of foraging is an effect of both the animal behavior and the geometry of the landscape; furthermore, the landscape is itself affected by the foraging of animals. We investigated the effect of landscape heterogeneity on the efficiency of an optimal forager. The particular aspect of heterogeneity we considered was "clumpiness"– the degree to which food resource patches are clustered together. The starting point for our study was the framework of the Mean Value Theorem (MVT) by Charnov. Since MVT is not spatially explicit, and thus not apt to investigate effects of clumpiness, we built an agent-based (or individual-based) model for animal movement in discrete landscapes extending the MVT. We also constructed a model for generating landscapes where the clumpiness of patches can be easily controlled, or "tuned", by an input parameter. We evaluated the agent based model by comparing the results with what the MTV would give, i.e. if the spatial effects were removed. The MVT matched the simulations best on landscapes with random patch configuration and high food recovery rates. As for our main question about the effects of clumpiness, we found that, when landscapes were highly productive (rapid food replenishment), foraging efficiency was greatest in clumped landscapes. In less productive landscapes, however, foraging efficiency was lowest in landscapes with a clumped patch distribution.     

Foraging behaviour of Blue Tits Parus caeruleus in a patchy environment under contrasting levels of natural food supply

We analysed the foraging behaviour of free-ranging Blue Tits Parus caeruleus in open holm oak Quercus ilex woodlands of western Spain during winter. Such woodlands are patchy for foraging tits because of the scattered distribution of trees and the patterns of abundance of canopy arthropods within and among trees. Results were compared with those obtained in spring of the same year, when we found that the foraging behaviour and spatial distribution of Blue Tits were largely unaffected by food availability (Pulido and Díaz 1997). Patch (tree) residence time was highly variable both within and among individual birds, and it was uncorrelated with either previous travel time or patch quality. Contrary to a priori expectations, the behaviour of tits did not conform to short-term energy maximizing rules in winter, in spite of a 2.5-fold decrease in food supply from spring to winter and a likely 2-fold increase in bird requirements. Instead, birds tended to fly towards patches that were further away than locally available. Overall, we conclude that energy intake rate was not the fitness-related currency that birds were trying to maximize while foraging.     

Feeding rate as valuable information in primate feeding ecology

In this review I outline studies on wild non-human primates using information on feeding rate, which is defined as the food intake per minute on a dry-weight basis; further, I summarize the significance of feeding rate in primate feeding ecology. The optimal foraging theory has addressed three aspects of animal feeding: (1) optimal food patch choice, (2) optimal time allocation to different patches, and (3) optimal food choice. In order to gain a better understanding of these three aspects, the feeding rate itself or its relevance indices (e.g., rates of calorie and protein intake) could be appropriate measures to assess the quality of food and food patches. Moreover, the feeding rate plays an essential role in estimation of total food intake, because it varies greatly for different food items and the feeding time is not a precise measure. The feeding rate could also vary across individuals who simultaneously feed on the same food items in the same food patch. Body size-dependent and rank-dependent differences in the feeding rate sometimes cause individuals to take strategic behavioral options. In the closing remarks, I discuss the usefulness of even limited data on feeding rate obtained under adverse observational conditions in understanding primate feeding ecology.     

Maintaining social cohesion is a more important determinant of patch residence time than maximizing food intake rate in a group-living primate,Japanese macaque (Macaca fuscata)

Animals have been assumed to employ an optimal foraging strategy (e.g., rate-maximizing strategy). In patchy food environments, intake rate within patches is positively correlated with patch quality, and declines as patches are depleted through consumption. This causes patch-leaving and determines patch residence time. In group-foraging situations, patch residence times are also affected by patch sharing. Optimal patch models for groups predict that patch residence times decrease as the number of co-feeding animals increases because of accelerated patch depletion. However, group members often depart patches without patch depletion, and their patch residence time deviates from patch models. It has been pointed out that patch residence time is also influenced by maintaining social proximity with others among group-living animals. In this study, the effects of maintaining social cohesion and that of rate-maximizing strategy on patch residence time were examined in Japanese macaques (Macaca fuscata). I hypothesized that foragers give up patches to remain in the proximity of their troop members. On the other hand, foragers may stay for a relatively long period when they do not have to abandon patches to follow the troop. In this study, intake rate and foraging effort (i.e., movement) did not change during patch residency. Macaques maintained their intake rate with only a little foraging effort. Therefore, the patches were assumed to be undepleted during patch residency. Further, patch residence time was affected by patch-leaving to maintain social proximity, but not by the intake rate. Macaques tended to stay in patches for short periods when they needed to give up patches for social proximity, and remained for long periods when they did not need to leave to keep social proximity. Patch-leaving and patch residence time that prioritize the maintenance of social cohesion may be a behavioral pattern in group-living primates.     

Gibbon foraging decisions and the marginal value model

We use data from an observational field study of frugivory in two sympatric gibbons, lar (Hylobates lar) and siamang (H. syndactylus), to test assumptions and predictions of the marginal value model (MVM). A key prediction of the MVM is that marginal gain rates at the time of leaving the patch are equal across patch types. We found that this is not the case for gibbons: rates of energy intake at the end of feeding sessions were significantly different for different types of fruit, and we could not attribute this to temporal variation in fruit availability. Initial and final caloric intake rates were highly correlated. This suggests that gibbons do not adjust the time spent in patches in order to maximize the average rate of energy intake. Similar results were obtained for all other currencies considered. Gibbon foraging appears to satisfy several, but not all, assumptions of the MVM. As required by the model, fruit patches occur as discrete units, patches are encountered sequentially, travel time between patches exceeds search time between items within a patch, search for and search within patches are incompatible activities, and intake rates decline over time spent in a patch. However, the declining rates we detected may be an effect of satiation instead of patch depletion, patches probably are not encountered at random, and group members may not forage independently. Thus, our results suggest that the MVM is not an adequate model of gibbon foraging behavior, but they do not invalidate the MVM per se.     

Effects of food quality, diet preference and water on patch use by Nubian ibex

Measuring patch use of a forager can reveal not only its cost and benefits from foraging, but also the importance of environmental factors and the significance of energy, nutrients and predation risk to its fitness. In order to assess the effects of various variables that may affect the foraging behavior of free-ranging Nubian ibex in the Negev Desert, Israel, giving-up densities (GUD) in artificial food patches were measured following Kotler et al. In particular, we tested the effects of food quality and water availability on Nubian ibex foraging behavior. To do so, we (1) tested whether the tannic acid content of food affected diet preferences, (2) assayed their diet selection strategy, (3) tested if the foraging decisions of the Nubian ibex were affected by the availability of water and (4) determined the nutritional relationship between food resources and water. Nubian ibex had lower GUDs and used resources patches more intensively where water was available, the food quality was higher and the location was closer to the escape terrain. Nubian ibex showed an expanding specialist diet selection when exploiting resource patches with a mix of items that differ in quality. Overall, food and water were complementary resources for Nubian ibex, and tannins reduced food quality. These factors help to determine patch foraging behavior decisions in Nubian ibex and contribute to habitat quality.     

The advantages of social foraging in downy woodpeckers

I investigated the advantages gained by downy woodpeckers (Picoides pubescens) which join mixed-species winter flocks. Woodpeckers foraging alone showed high levels of vigilance as measured by head-cocking rates, and low feeding rates. Woodpeckers foraging with one or two flock members showed intermediate rates of head-cocking and feeding, while woodpeckers foraging with flocks of three or more birds showed low head-cocking rates and high feeding rates. Although local enhancement and copying may contribute to the woodpeckers' increased foraging efficiency in a flock, these do not appear to be the main factors. As downy woodpeckers spend less time on vigilance, they devote more time to foraging, thereby increasing their foraging efficiency     

The short-term regulation of foraging in harvester ants

In the seed-eating ant Pogonomyrmex barbatus, the return ofsuccessful foragers stimulates inactive foragers to leave thenest. The rate at which successful foragers return to the nestdepends on food availability; the more food available, the morequickly foragers will find it and bring it back. Field experimentsexamined how quickly a colony can adjust to a decline in therate of forager return, and thus to a decline in food availability,by slowing down foraging activity. In response to a brief, 3-to 5-min reduction in the forager return rate, foraging activityusually decreased within 2–3 min and then recovered within5 min. This indicates that whether an inactive forager leavesthe nest on its next trip depends on its very recent experienceof the rate of forager return. On some days, colonies respondedmore to a change in forager return rate. The rapid colony responseto fluctuations in forager return rate, enabling colonies toact as risk-averse foragers, may arise from the limited intervalover which an ant can track its encounters with returning foragers.     

Foraging and predator avoidance: a test of a patch choice model with juvenile bluegill sunfish

Summary In situations where foraging sites vary both in food reward and predation risk, conventional optimal foraging models based on the criterion of maximizing net rate of energy intake commonly fail to predict patch choice by foragers. Recently, an alternative model based on the simple rule when foraging, minimize the ratio of mortality rate (u) to foraging rate (f) was successful in predicting patch preference under such conditions (Gilliam and Fraser 1987). In the present study, I compare the predictive ability of these two models under conditions where available patches vary both in predation hazard and foraging returns. Juvenile bluegill sunfish (Lepomis macrochirus) were presented with a choice between two patches of artificial vegetation differing in stem density (i.e. 100, 250, and 500 stems/m²) in which to forage. Each combination (100:250, 250:500, or 100:500) was presented in the absence, presence, and after exposure to a bass predator (Micropterus salmoides). Which patch of vegetation bluegills chose to forage in, and foraging rate within each patch were recorded. Independent measurements of bluegill foraging rate and risk of mortality in the three stem densities provided the data for predicting patch choice by the two models. With no predator, preference between plots was consistent with the maximize energy intake per unit time rule of conventional optimality models. However, with a predator present, patch preference switched to match a minimize u/f criterion. Finally, when tested shortly after exposure to a predator (i.e. 15 min), bluegill preference appeared to be in a transitional phase between these two rules. Results are discussed with respect to factors determining the distribution of organisms within beds of aquatic vegetation.     

The survival-rate-maximizing policy for Bayesian foragers: wait for good news

We present a model of the survival-maximizing foraging behaviorof an animal searching in patches for hidden prey with a clumpeddistribution. We assume the forager to be Bayesian: it updatesits statistical estimate of prey number in the current patchwhile foraging. When it arrives at the parch, it has an expectationof the patch's quality, which equals the average patch qualityin the environment While foraging, the forager uses its informationabout the time spent searching in the patch and how many preyhas been caught during this time. It can estimate both the instantaneousintake rate and the potential intake rate during the rest ofthe parch visit. When prey distribution is clumped, potentialintake rate may increase with time spent in the parch if preyis caught in the near future. Being optimal, a Bayesian foragershould therefore base its patch-leaving decision on the estimatedpotential patch value, not on the instantaneous parch value.When patch value is measured in survival rate and mortalitymay occur either as starvation or predation, the patch shouldbe abandoned when the forager estimates that its potential survivalrate dining the rest of the patch visit equals the long termsurvival rate in the environment This means that the instantaneousintake rate, when the patch is left, is nor constant but isan increasing function of searching time in the patch. Therefore,the giving-up densities of prey in the patches will also behigher the longer the search times. The giving-up densitiesare therefore expected to be an increasing, but humped, functionof initial prey densities. These are properties of Bayesianforaging behavior not included in previous empirical studiesand model tests.