Bayesian Foraging Research Articles

Home ranges and the value of spatial information

Animals concentrate their activities within areas we call home ranges because information about places increases fitness. Most animals, and certainly all mammals, store information about places in cognitive maps—or neurally encoded representations of the geometric relations among places—and learn to associate objects or events with places on their map. I define the value of information as a time-dependent increment it adds to any appropriate currency of fitness for an informed versus an uninformed forager, and integrate it into simple conceptual models that help explain movements of animals that learn, forget, and use information. Unlike other space-use models, these recognize that movement decisions are based on an individual's imperfect and ever-changing expectancies about the environment—rather than omniscience or ignorance. Using simple, deterministic models, I demonstrate how the use of such dynamic information explains why animals use home ranges, and can help explain diverse movement patterns, including systematic patrolling or “traplining,” shifting activity or focal areas, extra-home-range exploration, and seemingly random (although goal-directed and spatially contagious) movements. These models also provide insights about interindividual spacing patterns, from exclusive home ranges (whether defended as territories or not) to broadly overlapping or shared ranges. Incorporating this dynamic view of animal expectancies and information value into more-complex and realistic movement models, such as random-walk, Bayesian foraging, and multi-individual movement models, should facilitate a more comprehensive and empirical understanding of animal space-use phenomena. The fitness value of cognitive maps and the selective exploitation of spatial information support a general theory of animal space use, which explains why mammals have home ranges and how they use them.

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.

Bumblebees Learn to Forage like Bayesians

Bayesian foraging in patchy environments requires that foragers have information about the distribution of resources among patches (prior information), either set by natural selection or learned from past experience. We test the hypothesis that bumblebee foragers can rapidly learn prior information from past experience in two very different experimental environments. In the high-variance environment (patches of low and high quality), stochastic optimality models predicted that finding rewards should sometimes sharply increase an optimal forager's tendency to stay in a patch (an incremental response), whereas in the uniform environment, finding rewards should always decrease the tendency to stay (a decremental response). We use Cox regression models to show that, in a matter of hours, bees learned to match both predicted responses, resulting in a reward intake rate that averaged 80% of the predicted maximum. Following training in either environment, bees' adaptive behavior carried over to a common test environment, thus confirming the influence of memorized prior information. Although Bayesian foraging by learning is often presumed, this study is the first to clearly isolate the adaptive use of a learned prior expectation. More generally, it highlights the remarkable adaptive plasticity of an important generalist pollinator and agent of selection.

Bumblebees Learn to Forage like Bayesians

Bayesian foraging in patchy environments requires that foragers have information about the distribution of resources among patches (prior information), either set by natural selection or learned from past experience. We test the hypothesis that bumblebee foragers can rapidly learn prior information from past experience in two very different experimental environments. In the high‐variance environment (patches of low and high quality), stochastic optimality models predicted that finding rewards should sometimes sharply increase an optimal forager’s tendency to stay in a patch (an incremental response), whereas in the uniform environment, finding rewards should always decrease the tendency to stay (a decremental response). We use Cox regression models to show that, in a matter of hours, bees learned to match both predicted responses, resulting in a reward intake rate that averaged 80% of the predicted maximum. Following training in either environment, bees’ adaptive behavior carried over to a common test environment, thus confirming the influence of memorized prior information. Although Bayesian foraging by learning is often presumed, this study is the first to clearly isolate the adaptive use of a learned prior expectation. More generally, it highlights the remarkable adaptive plasticity of an important generalist pollinator and agent of selection.

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.

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.

Decisions under uncertainty: Bayesian foraging – a meeting held at Lund University in August 2003

Decisions under uncertainty: Bayesian foraging – a meeting held at Lund University in August 2003

A simpler, more general method of finding the optimal foraging strategy for Bayesian birds

Oaten's (1977)stochastic model for optimal foraging in patches has been solved for a number of particular cases. A few cases, such as Poisson prey distribution and either systematic or random search, are easy to solve. In other cases, such as binomial prey distribution and random search, the form of the optimal strategy may be found using a theorem of McNamara, although more work is required to find which particular rule of the proper form is actually best. More generally (but not completely generally), optimal strategies may be found using dynamic programming. This requires that the number of prey found up to a particular time is a sufficient statistic for the number of prey remaining in a patch. This requirement cannot be dispensed with, but other simplifying assumptions that were used in the past are not necessary. In particular, it is not necessary, even for the sake of convenience, to assume that prey distribution has a form convenient for Bayesian analysis, such as a beta mixture of binomials or a gamma mixture of Poissons. Any prey distribution may be used if whatever prey are in a patch are located at random, and if search either is systematic for discrete time or for continuous time, or is random for continuous time. In earlier work, some pains had to be taken to find the rate of finding prey achieved by a given candidate strategy, but this is not necessary if expected gains and expected times are calculated routinely for each potential stopping point during dynamic programming. A new, simple method of finding optimal strategies is illustrated for discrete time and systematic search. This paper is based on a talk given at the Fifth Hans Kristiansson Symposium held in Lund, Sweden in August, 2003. The subject of the symposium was Bayesian foraging.

Optimal Bayesian Foraging Policies and Prey Population Dynamics—Some Comments on Rodrı́guez-Gironés and Vásquez

In this paper we show the density-dependent harvest rates of optimal Bayesian foragers exploiting prey occurring with clumped spatial distribution. Rodrı́guez-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.

Gaining Ecological Information about Bayesian Foragers through Their Behaviour. II. A Field Test with Woodpeckers

The lesser spotted woodpecker Dendrocopos minor is a territorial species with its food resources distributed in patches, i.e. insect larvae in dead branches of trees. The species is threatened in Northern Europe, and therefore it is of interest to identify factors responsible for its decline. We hypothesise that (a) foraging behaviour is maximising fitness, (b) the woodpeckers, through their foraging behaviour, reveal their own opinion about their territory or their individual quality, and (c) that an inter-territorial comparison of behaviour and reproduction shows which factor is responsible for the breeding success of the woodpeckers. We collected data on giving-up density of prey (GUD) and patch residence time (PRT) during branch visits about one month before egg laying. The data was collected from a natural breeding population in southern Sweden, 1993-1996. A positive correlation between GUD and PRT within individuals is diagnostic for a Bayesian forager exploiting a clumped distribution of prey. This correlation was found in 22 of 28 investigated individuals. Average GUD of the breeding individuals was positively related to breeding success, i.e. there was a significant negative correlation between average GUD and commencement of egg laying, and a significant positive correlation between average GUD and number of fledglings. For the number of eggs laid there was a weak (not significant) positive relation with average GUD. Average GUD was not influenced by the age or sex of the individuals. Neither did these individual quality measures influence the relations between average GUD and reproductive success. Average GUD and average PRT of individuals was positively correlated. In concert, these results suggest that differences in average prey density across territories prior to breeding is the factor explaining most of the variation in breeding success of the lesser spotted woodpeckers in the studied population.

Gaining Ecological Information about Bayesian Foragers through Their Behaviour. I. Models with Predictions

A forager that is optimising its behaviour on the basis of some information is likely to reveal that particular information through its behaviour. We investigated the cases of completely informed and Bayesian foragers searching for food with a clumped distribution (the negative binomial distribution) between patches. Two different currencies were analysed: maximised intake rate and minimised predation risks. Predictions were made concerning two foraging variables: the average giving-up density of food items in the patches and the average patch residence time. We present how these variables correlate with each other and with a fitness variable (depending on the currency) when each of the model parameters is varied. The model is based on nine ecological parameters: (1) the forager's searching efficiency, (2) its metabolic rate during foraging and predator scanning. (3) its metabolic rate during the travel between patches. (4) the mean density of food items in patches. (5) the contagiousness (skew) of the prey density distribution. (6) the energy content of a food item, (7) predation rate during foraging, (8) predation rate during predator scanning, and (9) predation rate during travelling between patches. The values of the model parameters are selected to be representative for a lesser spotted woodpecker, Dendrocopos minor, foraging in its territory. Four qualitatively unique foraging and fitness variable relationships were found. Hence, empirically found relations between these variables can be used to infer the ecological factors causing the variation in foraging behaviour and fitness. The model is applied in a companion paper to pinpoint the most mportant inter-territorial ecological factor in the lesser spotted woodpecker, and how much this factor contributes to the reproductive success in this species.

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

We present a model of the survival-maximizing foraging behavior of an animal searching in patches for hidden prey with a dumped distribution. We assume the forager to be Bayesian: it updates its statistical estimate of prey number in the current patch while foraging. When it arrives at the patch, it has an expectation of the patch's quality, which equals the average patch quality in the environment While foraging, the forager uses its information about the time spent searching in the patch and how many prey has been caught during this time. It can estimate both the instantaneous intake rate and the potential intake rate during the rest of the patch visit When prey distribution is dumped, potential intake rate may increase with time spent in the patch if prey is caught in the near future. Being optimal, a Bayesian forager should therefore base its patch-leaving decision on the estimated potential patch value, not on the instantaneous patch value. When patch vahie is measured in survival rate and mortality may occur either as starvation or predation, the patch should be abandoned when the forager estimates that its potential survival rate during the rest of the patch visit equals the long term survival rate in the environment This means that the instantaneous intake rate, when the patch is left, is not constant but is an increasing function of searching time in the patch. Therefore, the giving-up densities of prey in the patches wiD also be higher the longer the search times. The giving-up densities are therefore expected to be an increasing, but humped, function of initial prey densities. These are properties of Bayesian foraging behavior not included in previous empirical studies and model tests. Key words: Bayesian foraging, dynamic optimization, giving-up density, patch model, patch residence time. [Behav Ecol 9:345-353 (1998)]

Density-Dependent Patch Exploitation and Acquisition of Environmental Information

We study density-dependent resource harvest patterns due to Bayesian foraging for different distributions of resources. We first consider a forager with information about the stochastic properties of its environment. In this case we show that when the number of food items per patch follows a distribution from the exponential family, the density dependence is given by the ratioσ2/μof the distribution of number of food items per patch. Bayesian foraging can therefore lead to positive (negative binomial distribution) or negative (binomial distribution) density dependent resource harvest and even to density independent (Poisson distribution) resource harvest, depending on the distribution of resources in the environment. In a second stage we incorporate learning about the distribution of resources in the whole environment. The mean of the distribution of number of food items per patch of a given environment is learnt faster than the variance of the distribution. Learning occurs faster in poorer than richer environments.

Patch use in cranes: a field test of optimal foraging predictions

The marginal value theorem states that foragers should leave patches when the instantaneous capture rate has fallen to the average capture rate for the habitat. This predicts that patch residence time should increase with increasing patch quality and decrease with increasing habitat quality. These and other predictions from prescient, Bayesian and fixed-time models were tested using observations on 14 radio-tagged free-living common cranes,Grus grus, foraging in cereal farmland. Cranes behaved as Bayesian foragers. Their intake rates on leaving patches changed with patch and habitat quality. The behaviour of cranes was consistent with marginal value theorem predictions only in patches where energy return was lower than required to meet daily food requirements. In contrast, birds left richer patches earlier than expected and at higher intake rates than poor patches. In addition, cranes stayed longer in larger flocks. These results suggest that cranes changed their foraging rules according to their expected energy balance.

Bayesian and prescient assessment: foraging with pre-harvest information

Pre-harvest information is information that foragers can obtain about patch quality before exploitation. It may be used by Bayesian foragers, which have knowledge of the distribution of resources in the environment, or prescient foragers, which have sensory information or memory of patches that are spatially and temporally predictable in quality. Prescience leads to higher energy intake rates and should be used if the appropriate information is available. Patch assessment was investigated in Inca doves, Columbina inca, feeding in artificial resource patches to determine the kinds of information doves are capable of using to assess the quality of resource patches. As expected, doves exploited patches in a manner consistent with prescient foraging when patch quality was temporally predictable. The same individuals exploited patches in a manner consistent with Bayesian foraging when prescient foraging would not be likely because patch quality was temporally unpredictable.