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 on the roof of the world: Himalayan langurs and the classical prey model

Optimal foraging theory has only been sporadically applied to nonhuman primates. The classical prey model, modified for patch choice, predicts a sliding “profitability threshold” for dropping patch types from the diet, preference for profitable foods, dietary niche breadth reduction as encounter rates increase, and that exploitation of a patch type is unrelated to its own abundance. We present results from a 1‐year study testing these predictions with Himalayan langurs (Semnopithecus entellus) at Langtang National Park, Nepal. Behavioral data included continuous recording of feeding bouts and between‐patch travel times. Encounter rates were estimated for 55 food types, which were analyzed for crude protein, lipid, free simple sugar, and fibers. Patch types were entered into the prey model algorithm for eight seasonal time periods and differing age‐sex classes and nutritional currencies. Although the model consistently underestimated diet breadth, the majority of nonpredicted patch types represented rare foods. Profitability was positively related to annual/seasonal dietary contribution by organic matter estimates, whereas time estimates provided weaker relationships. Patch types utilized did not decrease with increasing encounter rates involving profitable foods, although low‐ranking foods available year‐round were taken predominantly when high‐ranking foods were scarce. High‐ranking foods were taken in close relation to encounter rates, while low‐ranking foods were not. The utilization of an energetic currency generally resulted in closest conformation to model predictions, and it performed best when assumptions were most closely approximated. These results suggest that even simple models from foraging theory can provide a useful framework for the study of primate feeding behavior. Am J Phys Anthropol, 2010. © 2009 Wiley‐Liss, Inc.     

The Behavioral Ecology of Shellfish Gathering in Western Kiribati,Micronesia. 2: Patch Choice,Patch Sampling,and Risk

The prey choice model, previously applied among shellfish gatherers in Kiribati, Micronesia, has shown that they are foraging in a manner that matches the predictions of optimal foraging theory by maximizing their net energy return rates. Similar conclusions can be drawn subsequent to testing the patch choice model, including patch switching; patch sampling; and the analysis of risk. In light of these results, it is argued that natural selection probably never encouraged the persistence of conservation because individuals have nearly always benefited from short-term goals to ensure greater fitness. However, the possibility remains that as a result of changed circumstances brought about by increasing human population, more efficient extractive technologies, and expanding market opportunities, genuine, as opposed to epiphenomenal conservation, may become established in heavily impacted environments.

Foraging tactics of an aquatic insect: Partial consumption of prey

Optimal patch foraging theory has recently been used to model partial consumption of prey by predators (Cook & Cockrell 1978; Sih 1980). I tested several predictions of this model with larvae of the predaceous diving beetle Dytiscus verticalis, which are predators of anuran tapdoles. The results of the study supported the qualitative predictions of the model. The prediction of decreasing search time with increasing prey density was confirmed. Handling time and the amount ingested per prey item also decreased with decreasing search time. A quantitative test of the optimal foraging model revealed that it accurately predicted handling times when prey densities were high but failed to do so at low prey densities. Two hypotheses based on mean extraction rates are proposed in an attempt to explain these results.     

Context dependence in foraging behaviour of Achillea millefolium

Context-dependent foraging behaviour is acknowledged and well documented for a diversity of animals and conditions. The contextual determinants of plant foraging behaviour, however, are poorly understood. Plant roots encounter patchy distributions of nutrients and soil fungi. Both of these features affect root form and function, but how they interact to affect foraging behaviour is unknown. We extend the use of the marginal value theorem to make predictions about the foraging behaviour of roots, and test our predictions by manipulating soil resource distribution and inoculation by soil fungi. We measured plant movement as both distance roots travelled and time taken to grow through nutrient patches of varied quality. To do this, we grew Achillea millefolium in the centers of modified pots with a high-nutrient patch and a low-nutrient patch on either side of the plant (heterogeneous) or patch-free conditions (homogeneous). Fungal inoculation, but not resource distribution, altered the time it took roots to reach nutrient patches. When in nutrient patches, root growth decreased relative to homogeneous soils. However, this change in foraging behaviour was not contingent upon patch quality or fungal inoculation. Root system breadth was larger in homogeneous than in heterogeneous soils, until measures were influenced by pot edges. Overall, we find that root foraging behaviour is modified by resource heterogeneity but not fungal inoculation. We find support for predictions of the marginal value theorem that organisms travel faster through low-quality than through high-quality environments, with the caveat that roots respond to nutrient patches per se rather than the quality of those patches.     

Scale-dependent foraging and patch choice in fractal environments

Many spatially complex environments are fractal, and consumers in these environments face scale-dependent trade-offs between encountering high densities of small resource patches versus low densities of large resource patches. I address the effects of these trade-offs on foraging by incorporating scale-dependent encounter of resources in fractal landscapes into classical optimal foraging theory. This model is then used to predict optimal scales of perception (foraging scale) and patch choice in response to spatial features of landscapes. The model predicts that, for a given density of resources, landscapes with greater extent and fractal dimension and that contain patchy (low fractal dimension) resources favour large foraging scales and specialization on a small proportion of resource patches. Fragmented (low fractal dimension) landscapes of small extent with dispersed (high fractal dimension) resources favour smaller foraging scales and generalists that use a large proportion of available resource patches. These predictions synthesize the results of other spatially explicit consumer–resource models into a simple framework and agree reasonably well with results of several empirical studies. This study thus places optimal foraging theory in a spatial context and suggests evolutionary mechanisms of consumers' responses to important spatial phenomena (e.g. habitat fragmentation, resource aggregation). This revised version was published online in July 2006 with corrections to the Cover Date.     

Flower choice in honeybees: Effects of food value and flower density

To test the effects of food value on the flower choice, individual honeybees (Apis mellifera) were offered a choice of 25 % sucrose solution (SS) and 1 of 6 different SSs, ranging from 5 % to 50 % SS, at either a low or a high flower density. Artificial flowers were filled with each SS. The honeybees showed a stronger preference for a concentrated SS to a diluted SS at a high than at a low flower density, and the degree of preference was positively correlated to the difference in the sucrose concentration between paired SSs. These foraging patterns were consistent with qualitative predictions from optimal foraging theory. Furthermore, it was found that experience in feeding on a concentrated SS lowered the foraging motivation for a diluted SS at the high flower density, but not at the low flower density. I discuss the effects of food density, food profitability and experience on the foraging behaviour of honeybees.     

Taking the rough with the smooth: foraging for particulate food in continuous time

This paper describes the development of the general dynamical model of foraging developed by Ollason (1980, Theoret. Popul. Biol. 18, 44-65) to predict foraging for particulate food in three different types of environment. In an environment containing particles of different types of food, the model predicts the selection of an approximately optimal diet; in an environment in which the particles occur in patches, the model predicts a time budget of patch occupancy that approximates to the optimal time budget; and in an environment containing patches of particles that regenerate by the addition of particles of food at constant rates, the model predicts that animals will dispose themselves among the patches approximately as predicted by the ideal free distribution. Where the predictions of the model depart from the predictions of optimal foraging theory, they are qualitatively similar to the observed departures of the behaviour of real animals from the predictions of optimal foraging theory. The model provides a general representation of the foraging decisions of animals whether they feed strictly continuously or discontinuously on particles of food, and does so without explicit reference to optimization processes.     

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.     

Optimal foraging by bacteriophages through host avoidance

Optimal foraging theory explains diet restriction as an adaptation to best utilize an array of foods differing in quality, the poorest items not worth the lost opportunity of finding better ones. Although optimal foraging has traditionally been applied to animal behavior, the model is easily applied to viral host range, which is genetically determined. The usual perspective for bacteriophages (bacterial viruses) is that expanding host range is always advantageous if fitness on former hosts is not compromised. However, foraging theory identifies conditions favoring avoidance of poor hosts even if larger host ranges have no intrinsic costs. Bacteriophage T7 rapidly evolved to discriminate among different Escherichia coli strains when one host strain was engineered to kill infecting phages but the other remained productive. After modifying bacteria to yield more subtle fitness effects on T7, we tested qualitative predictions of optimal foraging theory by competing broad and narrow host range phages against each other. Consistent with the foraging model, diet restriction was favored when good hosts were common or there was a large difference in host quality. Contrary to the model, the direction of selection was affected by the density of poor hosts because being able to discriminate was costly.     

Herbivore-induced plant volatiles as cues for habitat assessment by a foraging parasitoid

1. Animals usually require information about the current state of their habitat to optimize their behaviour. For this, they can use a learning process through which their estimate is continually updated according to the cues they perceive. Identifying these cues is a long-standing but still inveterate challenge for ecologists. 2. The use of plant cues by aphid parasitoids for the assessment of habitat profitability and the adaptation of patch exploitation was studied. Grounding on predictions from optimal foraging theory, we tested whether parasitoids exploited host patches less intensively after visiting heavily infested plants than after visiting plants bearing fewer aphids. 3. As predicted, after visiting heavily infested plants parasitoids reduced their residence time and attacked fewer hosts in the next patch. This was the case regardless of whether the aphids were actually present on the first plant, indicating that the cue came from the plant. Moreover, the level of infestation of a plant at some distance from the first plant visited affected parasitoid patch exploitation on the second plant in a similar manner, indicating that the cue was volatile. 4. These results highlight a novel role of herbivore-induced volatiles in parasitoid foraging behaviour, different from the widely studied attraction at a distance.     

Heritability of selectively advantageous foraging behaviour in a small passerine

Summary I measured the heritability of foraging patch choice in a laboratory population of zebra finches (Taeniopygia guttata). Mothers and offspring were tested for their ability to discriminate between four foraging patches which provided four different rates of energy gain. Use of a foraging patch with a high rate of energy gain has been shown to confer a selective advantage on zebra finches in a similar experimental system. In this population of zebra finches there was a large amount of variation in foraging patch choice behaviour both within and among individuals. I determined that foraging patch choice was a phenotypically labile trait with a degree of stereotypy or repeatability, much lower than those typically recorded for morphological traits. The mating behaviour of zebra finches required that heritability be determined from a mother—offspring regression, which showed that narrow sense heritability of foraging patch choice was approximately 0.346. This heritability was significantly different than zero, as was heritability when it was limited by repeatability to 0.246. Foraging patch choice, a behaviour that has a demonstrated fitness consequence, had a heritable component in this laboratory population of zebra finches.     

Foraging and community consequences of seed size for coexisting Negev Desert granivores

We examined the effects of seed size on patch use and diet selection for three co-existing Negev Desert granivores: Allenby's gerbil ( Gerbillus allenbyi ), greater Egyptian sand gerbil ( Gerbillus pyramidum ), and crested lark ( Galerida cristata ). We manipulated size and spatial distribution of seeds in experimental food patches and quantified foraging behavior by measuring giving-up densities (GUDs: the amount of food remaining in a resource patch following exploitation by a forager). In one experiment, we presented small (<1.4 mm in diameter cracked wheat), medium (2.0–3.3 mm), and large (>3.4 mm) seeds in separate trays; in a second, we presented small and medium seeds separately and mixed together. Gerbils had a higher handling time efficiency on smaller seeds, but a much higher encounter probability on larger seeds (20 times higher on large than medium seeds, and 2–5 times higher on medium than small seeds). This led gerbils to have significantly lower GUDs on larger seeds than smaller seeds and to harvest a higher proportion of the larger seeds. When presented with rich and poor patches, G. allenbyi tended to equalize GUDs in both patches, indicating a quitting harvest rate rule for patch exploitation. In contrast, larks appeared to use a fixed time rule for patch exploitation. For larks, seed size did not influence encounter probabilities, and they showed no seed-size selectivity. Still, larks had higher handling efficiencies on smaller than larger seeds, and consequently had a significantly lower GUD on small than medium seeds. Despite large differences between the gerbils and larks in their foraging, our results do not support species coexistence via seed-size partitioning: the larks had much higher GUDs than the gerbils on all seed sizes. Nonetheless, seed size, seed abundance, seed distribution and the animal's patch use behavior all played major roles in determining gerbils' and larks' diet selectivities and GUDs.     

State dependent behavior and the Marginal Value Theorem

The Marginal Value Theorem (MVT) is the dominant paradigm inpredicting patch use and numerous tests support its qualitativepredictions. Quantitative tests under complex foraging situationscould be expected to be more variable in their support becausethe MVT assumes behavior maximizes only net energy-intake rate.However across a survey of 26 studies, foragers rather consistently"erred" in staying too long in patches. Such a consistent directionto the errors suggests that the simplifying assumptions ofthe MVT introduce a systematic bias rather than just imprecision. Therefore, I simulated patch use as a state-dependent responseto physiological state, travel cost, predation risk, prey densities,and fitness currencies other than net-rate maximization (e.g.,maximizing survival, reproductive investment, or mating opportunities).State-dependent behavior consistently results in longer patchresidence times than predicted by the MVT or another foragingmodel, the minimize µ/g rule, and these rules fail to closely approximate the best behavioral strategy over a widerange of conditions. Because patch residence times increasewith state-dependent behavior, this also predicts mass regulationbelow maximum energy capacities without direct mass-specificcosts. Finally, qualitative behavioral predictions from theMVT about giving-up densities in patches and the effects oftravel costs are often inconsistent with state-dependent behavior.Thus in order to accurately predict patch exploitation patterns,the model highlights the need to: (1) consider predator behavior(sit-and-wait versus actively foraging); (2) identify activitiesthat can occur simultaneously to foraging (i.e., mate searchor parental care); and (3) specify the range of nutritional states likely in foraging animals. Future predictive modelsof patch use should explicitly consider these parameters.     

Sharing, consumption, and patch choice on Ifaluk atoll

Anthropological tests of patch choice models from optimal foraging theory have primarily employed acquisition rates as the currency of the model. Where foragers share their returns, acquisition rates may not be similar to consumption rates and thus may not be an appropriate currency to use when modeling foraging decisions. Indeed, on Ifaluk Atoll the distribution patterns of fish vary by fishing method and location. Previous analyses of Ifaluk patch choice decisions suggested that if Ifaluk fishers are trying to maximize their production rates they should rarely torch fish for dogtoothed tuna. However, some men do spend considerable time and energy exploiting the dogtoothed tuna patch. To improve our understanding of the constraints and motivations influencing men’s decisions to exploit this patch, here I use per capita consumption rates as a currency, rather than production rates, to evaluate predictions generated from a patch choice model. Results indicate that although fish caught in other patches are more widely distributed than fish caught in the dogtoothed tuna patch, the consumption rates of torch fishers and their kin are still considerably lower than the consumption rates of men pursuing fish in other patches. Although these results are unable to explain why Ifaluk men exploit the dogtoothed tuna patch, an important explanatory hypothesis is eliminated.     

Two gerbils of the Negev: A long-term investigation of optimal habitat selection and its consequences

Optimal foraging theory has entered a new phase. It is not so much tested as used. It helps behavioural ecologists discover the nature of the information in an animals brain. It helps population ecologists reveal coefficients of interaction and their patterns of density-dependent variation. And it helps community ecologists examine niche relationships. In our studies on two species of Negev desert gerbil, we have taken advantage of the second and third of these functions. Both these gerbils prefer semi-stabilized dune habitat, and both altered their selective use of this habitat and stabilized sand according to experimental changes we made in their populations. Their changes in selectivity agree with a type of optimal foraging theory called isoleg theory. Isoleg theories provide examples of dipswitch theories – bundles of articulated qualitative predictions – that are easier to falsify than single qualitative predictions. By linking behaviour to population dynamics through isoleg theory, we were able to use the behaviour of the gerbils to reveal the shapes of their competitive isoclines. These have the peculiar non-linear shapes predicted by optimal foraging theory. Finally, when owl predation threatens, the behaviour of Gerbillus allenbyi reveals the shape of their victim isocline. As has long been predicted by predation theory and laboratory experiments, it is unimodal.     

The Importance of Predation Risk and Missed Opportunity Costs for Context-Dependent Foraging Patterns

Correct assessment of risks and costs of foraging is vital for the fitness of foragers. Foragers should avoid predation risk and balance missed opportunities. In risk-heterogeneous landscapes animals prefer safer locations over riskier, constituting a landscape of fear. Risk-uniform landscapes do not offer this choice, all locations are equally risky. Here we investigate the effects of predation risk in patches, travelling risk between patches, and missed social opportunities on foraging decisions in risk-uniform and risk-heterogeous landscapes. We investigated patch leaving decisions of 20 common voles (M. arvalis) in three experimental landscapes: safe risk-uniform, risky risk-uniform and risk-heterogeneous. We varied both the predation risk level and the predation risk distribution between two patches experimentally and in steps, assuming that our manipulation consequently yield different distributions and levels of risk while foraging, risk while travelling, and costs of missed, social opportunities (MSOCs). We measured mean GUDs (giving-up density of food left in the patch) for both patches as a measure of foraging gain, and delta GUD, the differences among patches, as a measure of the spatial distribution of foraging effort over a period of six hours. Distribution of foraging effort was most even in the safe risk-uniform landscapes and least even in the risk-heterogeneous landscape, with risky risk-uniform landscapes in between. Foraging gain was higher in the safe than in the two riskier landscapes (both uniform and heterogeneous). Results supported predictions for the effects of risk in foraging patches and while travelling between patches, however predictions for the effects of missed social opportunities were not met in this short term experiment. Thus, both travelling and foraging risk contribute to distinct patterns observable high risk, risk-uniform landscapes.     

Foraging for intermittently refuged prey: theory and field observations of a parasitoid

Many insect herbivores feed in concealed locations but become accessible intermittently, creating windows of greater vulnerability to attack, and generating a proportion of the prey population that is readily accessible to foraging natural enemies. We incorporated accessible prey into an extant optimal foraging model, and found that this addition allowed opportunistic exploitation of prey that have already emerged from refugia (the leaving strategy) as a viable strategy, in addition to waiting at refugia for prey to emerge (the waiting strategy). We parameterized the model empirically for the parasitoid Macrocentrus grandii and its host, Ostrinia nubilalis, under field conditions. The model predicted that M. grandii should adopt a leaving strategy when host patch density is high (travel time between patches is short), but a waiting strategy when host patch density is low (travel time between patches is long). Field observations of M. grandii patch tenure were consistent with model predictions, indicating that M. grandii exhibited flexible behaviour based on experience within a foraging bout, and that these behavioural shifts improved foraging efficiency. Behaviour of M. grandii was responsive to heterogeneity in host emergence rates, and appeared to be driven by the relatively small proportion of the host population that became accessible at a fast rate. Therefore understanding forager responses to intermittently refuged prey may require characterization of the behaviour of a subset of the prey population, rather than the average prey individual. The model can potentially be used as a framework for comparative studies across forager taxa, to understand when foragers on intermittently accessible prey should adopt fixed waiting or leaving strategies vs. a flexible strategy that is responsive to the current environment.     

Coexistence and community structure in a consumer resource model with implicit stoichiometry

We combine stoichiometry theory and optimal foraging theory into the MacArthur consumer-resource model. This generates predictions for diet choice, coexistence, and community structure of heterotroph communities. Tradeoffs in consumer resource-garnering traits influence community outcomes. With scarce resources, consumers forage opportunistically for complementary resources and may coexist via tradeoffs in resource encounter rates. In contrast to single currency models, stoichiometry permits multiple equilibria. These alternative stable states occur when tradeoffs in resource encounter rates are stronger than tradeoffs in elemental conversion efficiencies. With abundant resources consumers exhibit partially selective diets for essential resources and may coexist via tradeoffs in elemental conversion efficiencies. These results differ from single currency models, where adaptive diet selection is either opportunistic or selective. Interestingly, communities composed of efficient consumers share many of the same properties as communities based on substitutable resources. However, communities composed of relatively inefficient consumers behave similarly to plant communities as characterized by Tilman’s consumer resource theory. The results of our model indicate that the effects of stoichiometry theory on community ecology are dependent upon both consumer foraging behavior and the nature of resource garnering tradeoffs.     

Assessing models of optimal diving

Many birds and mammals forage under water and have to return to the surface to breathe. Models of optimal diving attempt to explain the behaviour of such animals in terms of selection for successful foraging given the constraints imposed by physiology. Several recent papers have questioned the accuracy of both the assumptions and the predictions of these models. Here, I provide a critical review of these papers, arguing that they misrepresent both the models and the data. As a result, they focus on inappropriate tests. I use the debate to suggest various new models and to explore the general relationship between theory and data in behavioural ecology. In particular, I consider the merits of qualitative and quantitative predictions.