Bayes’ theorem and its applications in animal behavior

ArticleinOikos 112(2):243-251 · February 2006with 126 Reads
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Abstract
Bayesian decision theory can be used to model animal behaviour. In this paper we give an overview of the theoretical concepts in such models. We also review the biological contexts in which Bayesian models have been applied, and outline some directions where future studies would be useful. Bayesian decision theory, when applied to animal behaviour, is based on the assumption that the individual has some sort of “prior opinion” of the possible states of the world. This may, for example, be a previously experienced distribution of qualities of food patches, or qualities of potential mates. The animal is then assumed to be able use sampling information to arrive at a “posterior opinion”, concerning e.g. the quality of a given food patch, or the average qualities of mates in a year. A correctly formulated Bayesian model predicts how animals may combine previous experience with sampling information to make optimal decisions. We argue that the assumption that animals may have “prior opinions” is reasonable. Their priors may come from one or both of two sources: either from their own individual experience, gained while sampling the environment, or from an adaptation to the environment experienced by previous generations. This means that we should often expect to see “Bayesian-like” decision-making in nature.

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    When searching for targets whose location is not known, animals should benefit by adopting movement patterns that promote random encounters. During mate search, theory predicts that the optimal search pattern depends on the expected distance to potential partners. A key question is whether actual males and females update their mate search patterns to increase encounter probability when conditions change. Here we show that two termite species, Reticulitermes speratus and Coptotermes formosanus, adaptively alternate between sexually monomorphic and dimorphic movements during mate search. After leaving their nests in a synchronized manner, termites begin to search for a mate. The resulting pairs perform tandem runs toward potential nest sites. We found that both sexes moved faster and in straight lines before finding partners, which is known to improve encounter rates when targets have completely unpredictable positions. In stark contrast, when pairs were accidentally separated during tandem running, they showed distinct sexually dimorphic movements, where females paused for long periods while males paused only briefly and moved actively. Data-based simulations demonstrated that such sexually dimorphic movements are advantageous when a mate is located nearby but its exact location is unknown. These results emphasize the importance of biological details to evaluate the efficiency of random search in animals. By extending the concept of mutual search beyond the context of mating, the dimorphic movements between partners represent a remarkable convergence between termites and other animals including humans.
  • Article
    Full-text available
    Beck’s insight—that beliefs about one’s self, future, and environment shape behavior—transformed depression treatment. Yet environment beliefs remain relatively understudied. We introduce a set of environment beliefs— primal world beliefs or primals —that concern the world’s overall character (e.g., the world is interesting, the world is dangerous ). To create a measure, we systematically identified candidate primals (e.g., analyzing tweets, historical texts, etc.); conducted exploratory factor analysis ( N = 930) and two confirmatory factor analyses ( N = 524; N = 529); examined sequence effects ( N = 219) and concurrent validity ( N = 122); and conducted test-retests over 2 weeks ( n = 122), 9 months ( n = 134), and 19 months (n = 398). The resulting 99-item Primals Inventory (PI-99) measures 26 primals with three overarching beliefs— Safe, Enticing , and Alive (mean α = .93)—that typically explain ∼55% of the common variance. These beliefs were normally distributed; stable (2 weeks, 9 months, and 19 month test-retest results averaged .88, .75, and .77, respectively); strongly correlated with many personality and wellbeing variables (e.g., Safe and optimism, r = .61; Enticing and depression, r = −.52; Alive and meaning, r = .54); and explained more variance in life satisfaction, transcendent experience, trust, and gratitude than the BIG 5 (3%, 3%, 6%, and 12% more variance, respectively). In sum, the PI-99 showed strong psychometric characteristics, primals plausibly shape many personality and wellbeing variables, and a broad research effort examining these relationships is warranted.
  • Article
    Full-text available
    Suppose that you are looking for visual targets in a set of images, each containing an unknown number of targets. How do you perform that search, and how do you decide when to move from the current image to the next? Optimal foraging theory predicts that foragers should leave the current image when the expected value from staying falls below the expected value from leaving. Here, we describe how to apply these models to more complex tasks, like search for objects in natural scenes where people have prior beliefs about the number and locations of targets in each image, and search is guided by target features and scene context. We model these factors in a guided search task and predict the optimal time to quit search. The data come from a satellite image search task. Participants searched for small gas stations in large satellite images. We model quitting times with a Bayesian model that incorporates prior beliefs about the number of targets in each map, average search efficiency (guidance), and actual search history in the image. Clicks deploying local magnification were used as surrogates for deployments of attention and, thus, for time. Leaving times (measured in mouse clicks) were well-predicted by the model. People terminated search when their expected rate of target collection fell to the average rate for the task. Apparently, people follow a rate-optimizing strategy in this task and use both their prior knowledge and search history in the image to decide when to quit searching.
  • Article
    Status-dependent strategies represent one of the most remarkable adaptive phenotypic plasticities. A threshold value for individual status (e.g., body size) is assumed above and below which each individual should adopt alternative tactics to attain higher fitness. This implicitly assumes the existence of an "absolute" best threshold value, so each individual chooses a tactic only on the basis of its own status. However, animals may be able to assess their status on the basis of surrounding individuals. This "relative" assessment considers a threshold value to be changeable depending on individual situations, which may result in significant differences in ecological and evolutionary dynamics compared with absolute assessment. Here, we incorporated Bayesian decision-making and adaptive dynamics frameworks to explore the conditions necessary for each type of assessment to evolve. Our model demonstrates that absolute assessment is always an evolutionarily stable strategy (ESS) in a stable environment, whereas relative assessment can be an ESS in stochastic environments. The consequences of future environmental change differ considerably depending on the assessment chosen. Our results underscore the need to better understand how individuals assess their own status when choosing alternative tactics.
  • Article
    Questions: How much should animals invest in information gathering when they have no prior information about the present state of the foraging patch, and does the ability to use information have an effect on the size of the investment? Mathematical method: Optimization of the size of investment when animals use the same foraging patch repeatedly. Key assumptions: Animals determine the size of their foraging investment before each foraging bout. They make a first investment without having any prior information. They then use information gained from the experience of the first bout to estimate patch profitability. (The larger the investment during the first bout, the more accurate the estimate.) Predictions: When patch profitability can be correctly estimated irrespective of the size of investment, animals should invest as if they were at a patch of average profitability. If larger investments do produce more accurate estimates, animals should make a greater investment than that for average profitability in the environment. The more likely it is that information proves to be invalid, the smaller the optimal investment in gaining that information. Therefore, it is expected that the species that relocates its foraging patch frequently makes a relatively smaller investment in its first foraging bout, whereas the species that exhibits strong patch tenacity makes a relatively greater investment in its first bout. Animals should pay the cost of the non-maximized gain for information that will help them to estimate foraging patch profitability more accurately and to enhance their future foraging.
  • Article
    Sensitive periods, in which experience shapes phenotypic development to a larger extent than other periods, are widespread in nature. Despite a recent focus on neural–physiological explanation, fewformal models have examined the evolutionary selection pressures that result in developmental mechanisms that produce sensitive periods. Here, we present such a model. We model development as a specialization process during which individuals incrementally adapt to local environmental conditions, while receiving a constant stream of cost-free, imperfect cues to the environmental state. We compute optimal developmental programmes across a range of ecological conditions and use these programmes to simulate developmental trajectories and obtain distributions of mature phenotypes. We highlight four main results. First, matching the empirical record, sensitive periods often result from experience or from a combination of age and experience, but rarely from age alone. Second, individual differences in sensitive periods emerge as a result of stochasticity in cues: individuals who obtain more consistent cue sets lose their plasticity at faster rates. Third, in some cases, experience shapes phenotypes only at a later life stage (lagged effects). Fourth, individuals might perseverate along developmental trajectories despite accumulating evidence suggesting the alternate trajectory is more likely to match the ecology. © 2016 The Author(s) Published by the Royal Society. All rights reserved.
  • Article
    Errors in decision-making in animals can be partially explained by adaptive evolution, and error management theory explains that cognitive biases result from the asymmetric costs of false-positive and false-negative errors. Error rates that result from the cognitive bias may differ between sexes. In addition, females are expected to have higher feeding rates than males because of the high energy requirements of gamete production. Thus, females may suffer relatively larger costs from false-negative errors (i.e. non-feeding) than males, and female decisions would be biased to reduce these costs if the costs of false-positive errors are not as high. Females would consequently overestimate their capacity in relation to the probability of predation success. We tested this hypothesis using the Japanese pygmy squid Idiosepius paradoxus. Our results show that size differences between the squid and prey shrimp affected predatory attacks, and that predatory attacks succeeded more often when the predator was relatively larger than the prey. Nevertheless, compared to male predatory attacks, female squid frequently attacked even if their size was relatively small compared to the prey, suggesting that the females overestimated their probability of success. However, if the females failed in the first attack, they subsequently adjusted their attack threshold: squid did not attack again if the prey size was relatively larger. These results suggest a sex-specific cognitive bias, that is females skewed judgment in decision-making for the first predation attack, but they also show that squid can modify their threshold to determine whether they should attack in subsequent encounters.
  • Article
    The evolutionary success of many organisms depends on their ability to make decisions based on estimates of the state of their environment (e.g., predation risk) from uncertain information. These decision problems have optimal solutions and individuals in nature are expected to evolve the behavioural mechanisms to make decisions as if using the optimal solutions. Bayesian inference is the optimal method to produce estimates from uncertain data, thus natural selection is expected to favour individuals with the behavioural mechanisms to make decisions as if they were computing Bayesian estimates in typically-experienced environments, although this does not necessarily imply that favoured decision-makers do perform Bayesian computations exactly. Each individual should evolve to behave as if updating a prior estimate of the unknown environment variable to a posterior estimate as it collects evidence. The prior estimate represents the decision-maker’s default belief regarding the environment variable, i.e., the individual’s default ‘worldview’ of the environment. This default belief has been hypothesised to be shaped by natural selection and represent the environment experienced by the individual’s ancestors.
  • Article
    The combination of spatial structure and non-linear population dynamics can promote the persistence of coupled populations, even when the average population growth rate of the patches seen in isolation would predict otherwise. This phenomenon has generally been conceptualized and investigated through the movement of individuals among patches that each holds many individuals, as in metapopulation models. However, population persistence can likewise increase as the result of individuals moving among sites (e.g. breeding territories) within in a single patch. Here I examine the latter: individuals making small-scale informed decisions with respect to where to breed can promote population persistence in poor environments. Based on a simple algebraic model, I demonstrate information thresholds, and predict that greater information use is required for population persistence under lower spatial heterogeneity in habitat quality, all else equal. Second, I implement an individual-based model to explore prior experience and prospecting on conspecific success within a more complex, and spatially heterogeneous environment. Uniquely, I jointly examine the effects of simulated habitat loss, spatial heterogeneity prior to habitat, and variation in information gathering on population persistence. I find that habitat loss accelerates population quasi-extinction risk; however, information use reduces extinction probabilities in proportion to the level of information gathering. Per capita reproductive success declines with number of breeding sites, suggesting that information-mediated Allee effects may contribute to extinction risk. In conclusion, my study suggests that populations in a changing world may be increasingly vulnerable to extinction where patch size and spatial heterogeneity constrain the effectiveness of information-use strategies. This article is protected by copyright. All rights reserved.
  • Article
    Understanding the structural complexity and the main drivers of animal search behaviour is pivotal to foraging ecology. Yet, the role of uncertainty as a generative mechanism of movement patterns is poorly understood. Novel insights from search theory suggest that organisms should collect and assess new information from the environment by producing complex exploratory strategies. Based on an extension of the first passage time theory, and using simple equations and simulations, we unveil the elementary heuristics behind search behaviour. In particular, we show that normal diffusion is not enough for determining optimal exploratory behaviour but anomalous diffusion is required. Searching organisms go through two critical sequential phases (approach and detection) and experience fundamental search tradeoffs that may limit their encounter rates. Using experimental data, we show that biological search includes elements not fully considered in contemporary physical search theory. In particular, the need to consider search movement as a non-stationary process that brings the organism from one informational state to another. For example, the transition from remaining in an area to departing from it may occur through an exploratory state where cognitive search is challenged. Therefore, a more comprehensive view of foraging ecology requires including current perspectives about movement under uncertainty.
  • Article
    Phytophagous insects choose their feeding resources according to their own requirements in addition to properties of the host plants, such as biomechanical defences. The feeding preferences of the native folivorous insects of the Andean-Patagonian forest (Argentina) have rarely been studied. These environments present a wide diversity and abundance of insects associated with trees of the Nothofagus and Lophozonia (Nothofagaceae) genera, which represent the main tree species of the forests of the southern hemisphere. In particular, Lophozonia alpina and Lophozonia obliqua are of great interest because they have a wide distribution, a high capacity for hybridization and exhibit great phenotypic plasticity. This versatility causes substantial variation in the biomechanical properties of leaves, affecting the feeding preferences of insects. The purpose of this work was to study the food selection behaviour of three leaf-chewing insects (Polydrusus nothofagii, Polydrusus roseaus (Coleoptera: Curculionidae) and Perzelia arda (Lepidoptera: Oecophoridae)) associated with L. alpina and L. obliqua as host plants. Based on their choices, our aim was to determine a preference scale for each insect species and the variables on which these preferences were based. Therefore, we selected trees of L. alpina and L. obliqua, measured several properties such as cellulose content and recorded which leaves were eaten. As a result, we determined that the three species of insects feed on both host plants but prefer the leaves of L. obliqua, with cellulose content being the main determining factor for their decisions. However, in the case of P. arda, there was a positive relationship between cellulose and host plant preference, whereas there was an opposite relationship for the weevils. We conclude that during feeding selection, there are some properties of the leaves that have a more important role than others and that the same property does not exert the same behavioural response in all folivorous insects.
  • Article
    Full-text available
    The question of when to collect new information and how to apply that information is central to much of behaviour. Theory suggests that the value of collecting information, or sampling, depends on environmental persistence and on the relative costs of making wrong decisions. However, empirical tests of how these variables interact are lacking.We tested whether bumblebee foraging decisions are indeed influenced by these two factors. We gave bees repeated choices between a resource providing a steady, mediocre reward and a resource fluctuating between a low reward and a high reward. In this paradigm, we manipulated environmental persistence by changing how long the quality of a fluctuating resource remained stable at one reward level. We manipulated the costs of decision errors by changing the relative values of the available rewards. Bees sampled the fluctuating resource more frequently when it changed quality more frequently, indicating that they measured environmental persistence and reacted to it as predicted by theory. Bees showed surprisingly suboptimal tracking, not reliably choosing the currently best resource except when the fluctuating resource was very persistent and the potential rewards high. While bees modify their choices in response to different levels of change and potential rewards, they do not always do so according to optimality predictions. © 2017 The Author(s) Published by the Royal Society. All rights reserved.
  • Article
    Probabilistic decision-making is a general phenomenon in animal behavior, and has often been interpreted to reflect the relative certainty of animals' beliefs. Extensive neurological and behavioral results increasingly suggest that animal beliefs may be represented as probability distributions, with explicit accounting of uncertainty. Accordingly, we develop a model that describes decision-making in a manner consistent with this understanding of neuronal function in learning and conditioning. This first-order Markov, recursive Bayesian algorithm is as parsimonious as its minimalist point-estimate, Rescorla-Wagner analogue. We show that the Bayesian algorithm can reproduce naturalistic patterns of probabilistic foraging, in simulations of an experiment in bumblebees. We go on to show that the Bayesian algorithm can efficiently describe the behavior of several heuristic models of decision-making, and is consistent with the ubiquitous variation in choice that we observe within and between individuals in implementing heuristic decision-making. By describing learning and decision-making in a single Bayesian framework, we believe we can realistically unify descriptions of behavior across contexts and organisms. A unified cognitive model of this kind may facilitate descriptions of behavioral evolution.
  • Article
    Increasing attention has focused on understanding how past experiences can influence and explain variation in mating preferences among individuals. We examined how previous social experiences affected courtship preferences in Drosophila melanogaster by exposing individual males to different frequency distributions of high- and low-quality (HQ and LQ) females and by allowing them to copulate with either a HQ or LQ mate. For a male, a large female represents a high-quality mate while a small female is a low-quality mate. We subsequently quantified the courtship behavior of these individuals in the presence of one HQ and one LQ female. Two aspects of male courtship behavior were significantly influenced by prior experience. Males previously exposed to a population of 75% HQ females more often initially courted the HQ than LQ female and more strongly biased overall courtship activity toward the HQ female compared to males previously exposed to a population of 25% HQ females. Furthermore, for some males, the type of female a male mated with in the experience phase influenced the type of female he first courted in the test phase: males that experienced a population containing only 25% HQ females and who mated with a LQ female in the experience phase, more often courted the LQ female first in the test phase while all other males biased courtship toward the HQ female. Our results indicate that information gained about the relative abundance of mate quality types and previous mating experience can affect future mating behavior.
  • Article
    Livestock health is economically important for agropastoral households whose wealth is held partly as livestock. Households can invest in disease prevention and treatment, but livestock disease risk is also affected by grazing practices that result in inter-herd contact and disease transmission in regions with endemic communicable diseases. This paper examines the relationships between communal grazing and antimicrobial use in Maasai, Chagga and Arusha households in northern Tanzania. We develop a theoretical model of the economic connection between communal grazing, disease transmission risk, risk perceptions, and antimicrobial use, and derive testable hypotheses about these connections. Regression results suggest that history of disease and communal grazing are associated with higher subjective disease risk and greater antimicrobial use. We discuss the implications of these results in light of the potential for relatively high inter-herd disease transmission rates among communal grazers and potential contributions to antimicrobial resistance due to antimicrobial use.
  • Article
    1.Animals are often required to make decisions about their use of current resources while minimising travel costs and risks due to uncertainty about the forthcoming resources. Passive soaring birds utilise warm rising‐air columns (thermals) to climb up and obtain potential energy for flying across large areas. However, the utilisation of such inconsistent natural resources may be challenging for soaring‐gliding birds and involve a set of decisions to maintain efficient flight. 2.To assess which temporal scales of previous experience with environmental inputs best predicted thermal‐climbing departure decisions of soaring birds, we used movement data from Eurasian griffon vultures (Gyps fulvus) tracked by GPS transmitters. We applied Cox proportional hazard regression and a model selection approach to identify thermal‐climbing departure decisions and to compare a range of temporal scales. 3.Our findings support the use of current and recent (short‐term; last 20 minutes) experiences, compared to longer term, past experiences, in predicting the time until departure from thermals. The models supported decision rules that integrated information originating from different temporal scales, implying a tendency to depart from a thermal later when the current climb rate was stronger than experienced recently and vice‐versa. Additionally, climb rates in thermals revealed significant autocorrelation over short time scales (shorter than 30 minutes). 4.The correspondence between thermals’ characteristics and the factors that best predicted thermal‐climbing departure decisions presumably reflect optimal decisions individuals make to handle their dynamic environment and to reduce movement‐related costs of such a basic activity for soaring‐gliding birds. This article is protected by copyright. All rights reserved.
  • Article
    Full-text available
    In the last decades, developmental origins of health and disease (DOHaD) has emerged as a central framework for studying early‐life effects, that is, the impact of fetal and early postnatal experience on adult functioning. Apace with empirical progress, theoreticians have built mathematical models that provide novel insights for DOHaD. This article focuses on three of these insights, which show the power of environmental noise (i.e., imperfect indicators of current and future conditions) in shaping development. Such noise can produce: (a) detrimental outcomes even in ontogenetically stable environments, (b) individual differences in sensitive periods, and (c) early‐life effects tailored to predicted future somatic states. We argue that these insights extend DOHaD and offer new research directions.
  • Article
    1.Foraging is a three‐stage process during which animals visit patches, consume food and quit. Foraging theory exploring relative patch quality has mostly focused on patch use and quitting decisions, ignoring the first crucial step for any forager: finding food. Yet, the decision to visit a patch is just as important as the decision to quit, as quitting theories can only be used if animals visit patches in the first place. Therefore, to better understand the foraging process and predict its outcomes, it is necessary to explore its three stages together. 2.We used the common brushtail possum (Trichosurus vulpecula) as a model to investigate foraging decisions in response to food varying in quality. Specifically, we tested whether patch nutritional quality affected: (1) patch visits; (2) behaviours at the patch during a foraging visit and (3) patch quitting decisions (quantified using Giving‐Up‐Density – GUD). Free‐ranging possums were presented with diets varying in nitrogen content and concomitantly volatile organic compound (VOC) composition at feeding stations in the wild. 3.We found that possums were able to distinguish between different quality foods from afar, despite the location of the diets changed daily. Possums used VOC (i.e. odour cues) emitted by the diets to find and select patches from a distance. High quality diets with higher protein and lower fiber, were visited more often and for longer. Possums spent more time foraging on diets high in nutritional content, resulting in lower GUDs. 4.Our study provides important quantitative evidence that foraging efficiency plays out during all the three stages of the foraging process (i.e. visit, consume and quit), and demonstrates the significance of considering all these stages together in future studies and foraging models. Sensory cues such as food odours play a critical role in helping foragers, including mammalian herbivores, find high quality food. This allows foragers to make quick, accurate and important decisions about food patches well before patch quitting decisions come into play. This article is protected by copyright. All rights reserved.
  • Preprint
    Many biological, psychological and economic experiments have been designed where an organism or individual must choose between two options that have the same expected reward but differ in the variance of reward received. In this way, designed empirical approaches have been developed for evaluating risk preferences. Here, however, we show that if the experimental subject is inferring the reward distribution (to optimize some process), they will never agree in finite time that the expected rewards are equal. In turn, we argue that this makes discussions of risk preferences, and indeed the motivations of behaviour, not so simple or straightforward to interpret. We use this particular experiment to highlight the serious need to consider the frame of reference of the experimental subject in studies of behaviour.
  • Preprint
    Full-text available
    Superorganisms such as social insect colonies are very successful relative to their non-social counterparts. Powerful emergent information processing capabilities would seem to contribute to their abundance, as they explore and exploit their environment collectively. In this series of three papers, we develop a Bayesian model of collective information processing, starting here with nest-finding, then examining foraging (part II) and externalised memories (pheromone territory markers) in part III. House-hunting Temnothorax ants are adept at discovering and choosing the best available nest site for their colony. Essentially, we propose that they estimate the probability each choice is best, and then choose the highest probability. Viewed this way, we propose that their behavioural algorithm can be understood as a sophisticated statistical method that predates recent mathematical advances by some tens of millions of years. Here, we develop a model of their nest finding that incorporates insights from approximate Bayesian computation as a model of collective estimation of alternative choices; and Thompson sampling, as an effective regret-minimising decision-making rule by viewing nest choice in terms of a multi-armed bandit problem (Robbins, 1952). Our Bayesian framework points to the potential for further bio-inspired statistical techniques. It also facilitates the generation of hypotheses regarding individual and collective movement behaviours when collective decisions must be made.
  • Article
    Science requires replication. The development of many cloned or isogenic model organisms is a testament to this. But researchers are reluctant to use these traditional animal model systems for certain questions in evolution or ecology research, because of concerns over relevance or inbreeding. It has largely been overlooked that there are a substantial number of vertebrate species that reproduce clonally in nature. Here we highlight how use of these naturally evolved, phenotypically complex animals can push the boundaries of traditional experimental design and contribute to answering fundamental questions in the fields of ecology and evolution.
  • Article
    Full-text available
    There is enduring debate over the question of which early-life effects are adaptive and which ones are not. Mathematical modelling shows that early-life effects can be adaptive in environments that have particular statistical properties, such as reliable cues to current conditions and high autocorrelation of environmental states. However, few empirical studies have measured these properties, leading to an impasse. Progress, therefore, depends on research that quantifies cue reliability and autocorrelation of environmental parameters in real environments. These statistics may be different for social and non-social aspects of the environment. In this paper, we summarize evolutionary models of early-life effects. Then, we discuss empirical data on environmental statistics from a range of disciplines. We highlight cases where data on environmental statistics have been used to test competing explanations of early-life effects. We conclude by providing guidelines for new data collection and reflections on future directions. This article is part of the theme issue ‘Developing differences: early-life effects and evolutionary medicine'.
  • Article
    We investigated patch quality assessment by the parasitoid Aphidius ervi Haliday (Hymenoptera: Braconidae) when exploiting colonies of the lettuce aphid Nasonovia ribisnigri Mosley (Hemiptera: Aphididae). Three host patches of different qualities were sequentially offered to a naïve female parasitoid. High quality patches (HQ) consisted of 20 N. ribisnigri reared on susceptible lettuce; low quality patches (LQ) consisted of 20 N. ribisnigri reared on partially resistant lettuce and mixed quality patches (MQ) consisted of equal numbers (10) of aphids reared on both lettuce types. Parasitized aphids were reared until parasitoid emergence; number of mummies and sex ratio were noted. On the first host patch encountered, the mean number of A. ervi mummies produced was significantly higher for HQ host patches (X ± SD; 7.1 ± 5.0) than for LQ host patches (3.8 ± 4.5). This suggests that female A. ervi do not need prior experience to assess patch quality; females probably using an innate estimate of patch quality when encountering a first host patch. This initial patch quality assessment can be modified with experience. When females encountered a MQ patch, they kept their exploitation level constant on the following patch, whatever its quality. Females increased their level of host acceptance on the third patch when three LQ patches were offered successively; accepting low-quality hosts could be preferable when better hosts are not currently available in the habitat. These results suggest that A. ervi females behave in a manner consistent with a Bayesian updating process when foraging for hosts.
  • Article
    Full-text available
    A technique for using patch giving up densities to investigate habitat preferences, predation risk, and interspecific competitive relationships is theoretically analyzed and empirically investigated. Giving up densities, the density of resources within a patch at which an individual ceases foraging, provide considerably more information than simply the amount of resources harvested. The giving up density of a forager, which is behaving optimally, should correspond to a harvest rate that just balances the metabolic costs of foraging, the predation cost of foraging, and the missed opportunity cost of not engaging in alternative activities. In addition, changes in giving up densities in response to climatic factors, predation risk, and missed opportunities can be used to test the model and to examine the consistency of the foragers' behavior. The technique was applied to a community of four Arizonan granivorous rodents (Perognathus amplus, Dipodomys merriami, Ammospermophilus harrisii, and Spermophilus tereticaudus). Aluminum trays filled with 3 grams of millet seeds mixed into 3 liters of sifted soil provided resource patches. The seeds remaining following a night or day of foraging were used to determine the giving up density, and footprints in the sifted sand indicated the identity of the forager. Giving up densities consistently differed in response to forager species, microhabitat (bush versus open), data, and station. The data also provide useful information regarding the relative foraging efficiencies and microhabitat preferences of the coexisting rodent species.
  • Some learning rules for acquiring information
    • A I Houston
    • A Kacelnik
    • J M Mcnamara
    Houston, A. I., Kacelnik, A. and McNamara, J. M. 1982. Some learning rules for acquiring information. Á/ In: McFarland, D. J. (ed.), Functional ontogeny. Pitman, pp. 140 Á/191.
  • Conference Paper
    Numerous behavioral models assume individuals combine knowledge in the form of a prior distribution with current sample information using Bayesian updating to estimate the quality of environmental parameters. I examine this assumption by reviewing 11 empirical studies. Six studies compared observed behavior to predictions of Bayesian and non-Bayesian models, while five studies manipulated prior distributions directly and observed how such manipulations altered behavior. Eight species of birds, three mammals, one fish and one insect exhibited behavior consistent with Bayesian updating models; one studied bird species failed to show evidence of Bayesian updating. Most studies examined how individuals estimated food patch quality but two investigated mating decisions. These studies suggest a variety of animals in different ecological contexts behave in manners consistent with predictions of Bayesian updating models. Future work on decision-making should focus on understanding how animals learn prior distributions and on decision-making in additional ecological contexts.
  • Chapter
    For many years, one of the central concerns of natural history was the food habits of animals; that is, what food animals eat, and how they go about obtaining it. The question was, How do animals forage? More recently, optimal foraging theory has posed the question, How should animals forage? This question may be asked in many forms, and the answers have cast light on the old subject of animals’ food habits. Optimal foraging theory has helped make the study of foraging more interesting, which may account for the theory’s remarkable popular success noted by Krebs, Stephens and Sutherland (1983).
  • Article
    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 ratio σ²/ μ of 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.
  • Article
    We develop two criteria for measuring patch assessment ability. First, we examine the ability of foragers to equalize benefits and costs at manipulated resource patches. Second, we compare patch utilization patterns of four possible foraging strategies (prescient, fixed time, Bayesian, and rate assessor) with actual foraging patterns. Experiments with several desert rodent and avian species suggest that Merriam's kangaroo rat may obtain the best estimate of patch quality, followed by the round-tail ground squirrel and Arizona pocket mouse. Kangaroo rats exhibited both a prescient and Bayesian strategy. Pocket mice and ground squirrels exhibited both a fixed-time and Bayesian strategy. Gambel's Quail appeared to be the least sophisticated forager and exhibited only a fixed-time strategy. The fixed-time strategy was observed most frequently in the low variance environment where patch differences were relatively minor. In general, increased patch variation led to poorer patch estimates but allowed employment of sophisticated foraging strategies. Avian group foragers did not obtain better estimates of patch quality than solitary foragers.
  • Article
    This paper discusses a renewal process whose time development between renewals is described by a Markov process. The process may be controlled by choosing the times at which renewal occurs, the objective of the control being to maximize the long-term average rate of reward. The time at which the (n plus 1)th renewal is made is determined by solving a stopping problem for the Markov process. In general, implementation of this policy requires a knowledge of the transition probabilities of the Markov process. An example is presented in which one needs to know essentially nothing about the details of this process or the fine details of the reward structure in order to implement the policy. The example is based on a problem in biology.
  • Article
    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.
  • Article
    The information that social foragers use to estimate the quality of food patches might include (1) patch-sample information alone, or it might be used in combination with (2) pre-harvest and/or (3) public information. Four possible patch-assessment strategies resulting from combinations of these sources of information make different predictions concerning how individual foraging success should influence patch persistence and the order of patch departure of individuals in the group. Laboratory data on pairs of budgerigars, Melopsittacus undulatus, are consistent with the hypothesis that these birds combine patch-sample and pre-harvest information to estimate the quality of artificial resource patches.
  • Article
    Full-text available
    Models of mate choice tactics have assumed that females randomly encounter males when collecting information and the information is perfect. Empirical observations of four bird species show that females selectively visit males and repeat visits to males before mating. This suggests that the assumptions of previous models have been too restrictive. An alternative model of information gathering and mate choice, which relaxes the assumptions of random encounters and perfect information, is presented. In this Comparative Bayes model, the decision of when and from whom to collect information is made using Bayesian estimates of each male's quality. Predictions from the model are that: (1) the occurrence of mate assessment will increase as initial uncertainty about the quality of males increases, as the cost of gathering information decreases, and as the signal perceived by the female becomes a better representation of males' actual qualities; (2) the occurrence of repeat visits to males will be highest when signals from males are of medium reliability; and (3) the decision of which male to assess will depend on the estimated qualities of males, prior certainty about each male's quality, the reliability of each male's signal, and the costs of assessment. Simulations compare the fitness outcomes of the Comparative Bayes tactic to other mate choice tactics. The fitness from the Comparative Bayes tactic is significantly higher than from the fixed threshold tactic and than from the best-of-n tactic when the cost of assessment is low.
  • Article
    Full-text available
    Classical treatments of problems of sequential mate choice assume that the distribution of the quality of potential mates is known a priori. This assumption, made for analytical purposes, may seem unrealistic, opposing empirical data as well as evolutionary arguments. Using stochastic dynamic programming, we develop a model that includes the possibility for searching individuals to learn about the distribution and in particular to update mean and variance during the search. In a constant environment, a priori knowledge of the parameter values brings strong benefits in both time needed to make a decision and average value of mate obtained. Knowing the variance yields more benefits than knowing the mean, and benefits increase with variance. However, the costs of learning become progressively lower as more time is available for choice. When parameter values differ between demes and/or searching periods, a strategy relying on fixed a priori information might lead to erroneous decisions, which confers advantages on the learning strategy. However, time for choice plays an important role as well: if a decision must be made rapidly, a fixed strategy may do better even when the fixed image does not coincide with the local parameter values. These results help in delineating the ecological-behavior context in which learning strategies may spread.
  • Article
    A model which relates optimal food preference relationships and caloric yield per unit time of potential food sources is derived. It is suggested, on the basis of this model, that: 1) Food preferences can be adequately described only if a number of factors other than relative frequencies in the diet and relative abundances of the food types are known. 2) Animals should be more selective in their choice of foods when satiated or when food is common, more indiscriminate when starved or when food is scarce. 3) Animals may eat one food type with greater frequency, relative to its abundance, than another even if the other food is richer and more efficiently exploited. This occurs in situations of high relative abundance of the first food type. 4) The extent to which predators tend to pass by potential food items may be used to evaluate the role of food in the population limitation of a predator species. 5) Food preferences appear to change readily and appropriately to changes in the environment. preferences a ...
  • Article
    SYNOPSIS. This paper deals with risk sensitivity in amount of food, and is concerned with modelling the risk-sensitive behaviour exhibited by an animal which is attempting to survive a period such as winter. I argue that, in maximising survival probability, risk-averse behaviour is much more important than risk-prone behaviour. I also argue that animals in the laboratory will continue to use rules which are adapted to food sources which change over time. A model of environmental change is investi- gated. This model predicts that a consequence of change is that less risk- prone behaviour than that predicted by standard models is to be expected in laboratory studies. The need to learn about a stochastic food source in a changing environment is predicted to further reduce the incidence of risk-prone behaviour. It is difficult to learn about highly variable food sources. This is shown to lead to risk-aversion even when the optimisation criterion is maximisation of mean long-term rate of food gain. Implications of the general modelling philosophy are discussed.
  • Article
    A mathematically tractable stochastic model is presented in which animals forage systematically for prey distributed in patches, and 3 possible strategies (stopping rules) are considered. The best rule is found and compared with the giving-up time (GUT) rule and a fixed-time rule in which the forager remains until it has exhausted each patch. The GUT rule is better than the fixed-time rule for variable patches, and it is relatively insensitive to environmental changes, but it is not as good as the best rule (the assessment rule). The assessment rule is robust in the sense that the rate of finding prey achieved by a rule of the 'assessment' type changes very little as the rule changes. This also means that a rule that is best for one environment will be quite good for environments that are substantially different. -from Author
  • Article
    It is well documented that animals take risk of predation into account when making decisions about how to behave in particular situations, often trading-off risk against opportunities for mating or acquiring energy. Such an ability implies that animals have reliable information about the risk of predation at a given place and time. Chemosensory cues are an important source of such information. They reliably reveal the presence of predators (or their presence in the immediate past) and may also provide information on predator activity level and diet. In certain circumstances (eg, in the dark, for animals in hiding) they may be the only cues available. Although a vast literature exists on the responses of prey to predator chemosensory cues (or odours), these studies are widely scattered, from marine biology to biological control, and not well known or appreciated by behavioural ecologists. In this paper, we provide an exhaustive review of this literature, primarily in tabular form. We highlight some of the more representative examples in the text, and discuss some ecological and evolutionary aspects of the use of chemoscensory information for prey decision making. Curiously, only one example illustrates the ability of birds to detect predator odours and we have found no examples for terrestrial insects, suggesting a fruitful area for future study.
  • Article
    A forager that is optimising its behaviour oil 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 foragers 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 important inter-territorial ecological factor in the lesser spotted woodpecker, and how much this factor contributes to the reproductive success in this species.
  • Article
    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.
  • Article
    Full-text available
    We present a model of the survival-maximizing foraging behavior of an animal searching in patches for hidden prey with a clumped 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 clumped, 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 value 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 will 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.
  • Article
    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.
  • Prey distribu-tion as a factor determining the choice of optimal foraging strategy 710 Á/723. and mating preferences: a review of causes and conse-quences
    • Y Iwasa
    • M Higashi
    • N Yamamura
    Iwasa, Y., Higashi, M. and Yamamura, N. 1981. Prey distribu-tion as a factor determining the choice of optimal foraging strategy. Á/ Am. Nat. 117: 710 Á/723. and mating preferences: a review of causes and conse-quences. Á/ Biol. Rev. 72: 283 Á/327.
  • Article
    A genotype is said to show phenotypic plasticity if it can produce a range of environmentally dependent phenotypes. Plasticity may or may not be adaptive. We consider plasticity as a genetically determined trait and thus find the optimal response of an animal to its environment. Various aspects of this optimal response are illustrated with examples based on reproductive effort. We investigate the selection pressure for plastic as opposed to fixed strategies. An example with spatial heterogeneity is used to compare our approach with that of Stearns and Koella (1986).
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    Experiments in which animals work for food can be classified as involving either an open economy or a closed economy. In a closed economy, an animal's interaction with its environment determines how much food it obtains, whereas in an open economy the intake is determined in part by the experimenter. In this paper these economies are modelled by allowing the animal to allocate its time to various activities including foraging. Maximization of reproductive success involves a trade-off between starvation and predation, and this is used to predict the amount of time devoted to the various activities. The optimal proportion of time spent foraging, u∗, depends on the rate, g, at which food can be obtained. This dependence is investigated and it is shown that in the model of the open economy u∗ increases with g, whereas in the model of the closed economy u∗ decreases with g. The decrease in u∗ during an open economy experiment (‘satiation’) is also explored. The comparison of behaviour in open and closed economies leads to a discussion of the advantages and disadvantages of the economies as experimental procedures. A general conclusion is that open economies are more suitable for studies of foraging behaviour, and closed economies are more suitable for time allocation studies when activities other than foraging are also important.
  • Article
    This paper considers an animal foraging on prey which are distributed in well-defined patches. It is assumed that the environment may be stochastic and that the animal can gain information on patch type as it forages. The foraging policy which maximises mean reward rate for the environment is characterised in terms of a function of state called the potential function. This policy is shown to be given by the rule: continue foraging on the present patch while the potential is positive, when the potential falls to zero move on to the next patch. Let r denote the current reward rate on a patch and let γ denote the maximum mean reward rate for the environment. It is shown that r ⩽ γ if it is optimal to leave. Conditions which ensure r < γ are also given. For a large class of environments the optimal policy is stated in terms of a revised reward rate r̃, and is given by the rule: continue on the present patch while , when r̃ falls to γ move on to the next patch. Finally, it is shown that the stay time on a patch is a decreasing function of γ.
  • Article
    In this paper we consider a simple model of an environment in which prey are distributed in patches. It is assumed that each patch contains at most one item, but items may vary in the ease with which they can be found. The time spent in unsuccessful search on a patch gives information of whether a patch contains an item, and if it does, how difficult that item is to find. We show how this information can be used to find the policy which maximizes the mean rate of reward for the environment. The analysis is illustrated by two examples.
  • Article
    Full-text available
    Several alternative decision rules have been proposed for how individuals assess and choose options, such as mates and territories. Three of these rules are the threshold rule, where individuals choose the first option that exceeds a preset level of quality, the best-of- n rule, where individuals assess a fixed number of options and then choose the best of those options, and the comparative Bayes rule, where individuals use estimates of options to selectively assess and choose options. It has been previously concluded that the threshold rule produces higher average fitness than the best-of- n rule when assessment costs are not trivial. However, previous comparisons assumed that time and options are infinite, individuals can estimate the distribution of option quality without uncertainty or mistakes, and individuals receive perfect information about the quality of assessed options. I found that the best-of- n rule produces higher average fitness than the threshold rule despite significant assessment costs, when time for choosing an option is limited, when individuals are choosing from a small pool of options, when estimates of the distribution of option quality are error-prone, and when there is uncertainty about the distribution of option quality. I also found that the comparative Bayes rule produces higher average fitness than the threshold and best-of-n rules when time or options are limited and when individuals receive imperfect information about the quality of assessed options. Therefore, the optimality of alternative decision rules depends on more than the size of assessment costs and the previous conclusions of empirical studies that have assumed such need to be re-examined. Copyright 2002 The Association for the Study of Animal Behaviour. Published by Elsevier Science Ltd. All rights reserved.
  • Article
    Allan Oaten (1977, Theor. Pop. Biol.12, 263–285) has argued that stochastic models of optimal foraging may produce results qualitatively different from those of the analogous deterministic models. Oaten's model is very general and difficult to understand intuitively. In this paper a simple, tractable model is considered in which the predator searches each patch systematically (without going over the same area twice) until he exhausts the patch or decides the patch is not very good. It is assumed that each patch contains a fixed number of bits, each of which may contain a prey. The number of prey per patch is assumed to have a binomial distribution with n equal to the number of bits and p being a random variable having a beta distribution. After searching each bit the predator decides whether to leave the patch or not according to how many prey it has found. In this paper the best strategy is determined and the long-term rate of feeding is compared with that of the naive animal that searches each patch completely. The advantage of being a Bayesian is determined for a variety of environmental conditions.
  • Article
    When a predator exploits an environment in which food is patchily distributed, it has to continually make the decision of how long to stay in a patch. In this paper we examine this question using captive black-capped chickadees foraging in a large aviary for small pieces of mealworm hidden in artificial pine cones. The results of our experiments show the following: (a) when the birds encounter a long sequence of patches (groups of pine cones) each containing the same number of prey, they do not learn to expect a fixed number of prey per patch (Gibb's hypothesis of hunting by expectation); (b) similarly, the birds do not learn to spend constant time on each patch (although the data on this are less clear cut); (c) the birds have a constant giving-up time (interval between last catch and leaving the patch) for all types of patches within an environment, and the giving-up time is inversely related to the average capture rate for the environment. These two findings are consistent with an optimal foraging model of patch use developed by E. L. Charnov. Thus our results are more consistent with the predictions of the optimal foraging than with the hypothesis of hunting by expectation.
  • Article
    The optimal-patch-use problem in predation theory is investigated by use of a stochastic discrete model to match experimental situations when deterministic continuous models are inappropriate. We first consider three elementary strategies, differing in when to leave the patch in which the predator has been foraging; namely, (1) a fixed time has passed, (2) a fixed number of prey has been captured, and (3) the interval between two successive catches has exceeded a fixed time. Each of these fixed quantities has same value that optimizes the strategy concerned, given certain conditions. The optimized strategies are compared to determine the most efficient. It is shown that the ranking of the strategies depends critically on the type of prey distribution between patches, other things being equal; e.g., strategy 3, the best when the variance of prey distribution is sufficiently high, tends to be the worst when the variance is minimal. The above strategies do not use full information for estimating the number o...
  • Article
    A graphical method is discussed which allows a specification of the optimal diet of a predator in terms of the net amount of energy gained from a capture of prey as compared to the energy expended in searching for the prey. The method allows several predictions about changes in the degree of specialization of the diet as the numbers of different prey organisms change. For example, a more productive environment should lead to more restricted diet in numbers of different species eaten. In a patchy environment, however, this will not apply to predators that spend most of their time searching. Moreover, larger patches are used in a more specialized way than smaller patches.
  • Article
    Optimal foraging thoery usually assumes that certain key environmental parameters are known to a foraging animal, and predicts the animal's behaviour under this assumption. However, an animal entering a new environment has incomplete knowledge of these parameters. If the predictions of optimal foraging theory are to hold the animal must use a behavioural rule which both learns the parameters and optimally exploits what it has learnt. In most circumstances it is not obvious that there exists any simple rule which has both these properties. We consider an environment composed of well-defined patches of food, with each patch giving a smooth decelerating flow of food (). We present a simple rule which (asymptotically) learns about and optimally exploits this environment. We also show the rule can be modified to cope with a changing environment. We discuss what is meant by optimal behaviour in an unknown and possibly changing environment, using the simple rule we have presented for illustrative purposes.
  • Article
    Early models of central-place foraging treated animals that search for prey in identical, homogenous patches. If patches vary in quality, then optimal foraging requires strategies based on time spent in a patch, and not simply on the type or number of prey found. In particular, a forager that takes no more than one prey from a patch should leave a patch after searching unsuccessfully for a certain fixed time. When patches are more variable, the forager should stay a shorter time in each patch, and the resulting rate of delivering prey to the central place will be lower. This implies that aggregation should be favored by prey faced with a single-prey-loading predator.
  • Article
    Numerous behavioral models assume individuals combine knowledge in the form of a prior distribution with current sample information using Bayesian updating to estimate the quality of environmental parameters. I examine this assumption by reviewing 11 empirical studies. Six studies compared observed behavior to predictions of Bayesian and non-Bayesian models, while five studies manipulated prior distributions directly and observed how such manipulations altered behavior. Eight species of birds, three mammals, one fish and one insect exhibited behavior consistent with Bayesian updating models; one studied bird species failed to show evidence of Bayesian updating. Most studies examined how individuals estimated food patch quality but two investigated mating decisions. These studies suggest a variety of animals in different ecological contexts behave in manners consistent with predictions of Bayesian updating models. Future work on decision-making should focus on understanding how animals learn prior distributions and on decision-making in additional ecological contexts.
  • Chapter
    Optimal foraging theory is one of the most popular areas in modern evolutionary biology. In this paper, I outline some of the main ideas of optimal foraging theory and describe two basic foraging models in detail. I argue that one of these models is less interesting than it should be because it does not include realistic assumptions about the environment or about the behavior of the forager. Then I describe a model that does make explicit behavioral and ecological assumptions. I argue for the use of particular, quantitative models and describe an experimental test of one such model. I discuss some criticisms of optimal foraging theory and try to predict what the future holds. In particular, I show how optimal foraging theory might help to bridge the gap between individual behavior and population dynamics.
  • Article
    Full-text available
    Some simple stochastic models of optimal foraging are considered. Firstly, mathematical renewal theory is used to make a general model of the combined processes of search, encounter, capture and handling. In the case where patches or prey items are encountered according to a Poisson process, the limiting probability distribution of energy gain is found. This distribution is found to be normal and its mean and variance are specified. This result supports the use of Holling’s disc equation to specify the rate of energy intake in foraging models. Secondly, a model based on minimization of the probability of death due to an energetic shortfall is presented. The model gives a graphical solution to the problem of optimal choices when mean and variance are related. Thirdly, a worked example using these results is presented. This example suggests that there may be natural relationships between mean and variance which makes solutions to the problems of ‘energy maximization’ and ‘minimization of the probability of starvation’ similar. Finally, current trends in stochastic modeling of foraging behavior are critically discussed.
  • Article
    Like much mathematical modeling in biology, most optimal foraging theory is developed from deterministic analogs of basically stochastic processes. Unlike other models, however, it cannot depend on laws of large numbers to justify this simplification; ignoring stochasticity can lead to wrong answers. This is demonstrated for a predator searching spatially separated patches of prey; it is shown that the choice of an optimal procedure for deciding when to leave a patch must be based on a stochastic model—a predator whose procedure is based on a deterministic model can do arbitrarily badly by comparison with the stochastic optimizer. A general solution is given, and its complexity suggests some objections to standard optimality arguments, and some possible alternatives.
  • Article
    Statistical decision theory is discussed as a general framework for analysing how animals should learn. Attention is focused on optimal foraging behaviour in stochastic environments. We emphasise the distinction between the mathematical procedure that can be used to find optimal solutions and the mechanism an animal might use to implement such solutions. The mechanisms might be specific to a restricted class of problems and produce suboptimal behaviour when faced with problems outside this class. We illustrate this point by an example based on what is known in the literature on animal learning as the partial reinforcement effect.
  • Article
    The aim of this review is to consider variation in mating preferences among females. We define mating preferences as the sensory and behavioural properties that influence the propensity of individuals to mate with certain phenotypes. Two properties of mating preferences can be distinguished: (1) "preference functions'-the order with which an individual ranks prospective mates and (2) "choosiness'-the effort an individual is prepared to invest in mate assessment. Patterns of mate choices can be altered by changing the costs of choosiness without altering the preference function. We discuss why it is important to study variation in female mating behaviour and identify five main areas of interest: Variation in mating preferences and costs of choosiness could (1) influence the rate and direction of evolution by sexual selection, (2) provide information about the evolutionary history of female preferences, (3) help explain inter-specific differences in the evolution of secondary sexual characteristics, (4) provide information about the level of benefits gained from mate choice, (5) provide information about the underlying mechanisms of mate choice. Variation in mate choice could be due to variability in preference functions, degree of choosiness, or both, and may arise due to genetic differences, developmental trajectories or proximate environmental factors. We review the evidence for genetic variation from genetic studies of heritability and also from data on the repeatability of mate-choice decisions (which can provide information about the upper limits to heritability). There can be problems in interpreting patterns of mate choice in terms of variation in mating preferences and we illustrate two main points. First, some factors can lead to mate choice patterns that mimic heritable variation in preferences and secondly other factors may obscure heritable preferences. These factors are divided into three overlapping classes, environmental, social and the effect of the female phenotype. The environmental factors discussed include predation risk and the costs of sampling; the social factors discussed include the effect of male-male interactions as well as female competition. We review the literature which presents data on how females sample males and discuss the number of cues females use. We conclude that sexual-selection studies have paid far less attention to variation among females than to variation among males, and that there is still much to learn about how females choose males and why different females make different choices. We suggest a number of possible lines for future research.
  • Article
    How can anyone be rational in a world where knowledge is limited, time is pressing, and deep thought is often an unattainable luxury? Traditional models of unbounded rationality and optimization in cognitive science, economics, and animal behavior have tended to view decision-makers as possessing supernatural powers of reason, limitless knowledge, and endless time. But understanding decisions in the real world requires a more psychologically plausible notion of bounded rationality. In Simple heuristics that make us smart (Gigerenzer et al. 1999), we explore fast and frugal heuristics--simple rules in the mind's adaptive toolbox for making decisions with realistic mental resources. These heuristics can enable both living organisms and artificial systems to make smart choices quickly and with a minimum of information by exploiting the way that information is structured in particular environments. In this précis, we show how simple building blocks that control information search, stop search, and make decisions can be put together to form classes of heuristics, including: ignorance-based and one-reason decision making for choice, elimination models for categorization, and satisficing heuristics for sequential search. These simple heuristics perform comparably to more complex algorithms, particularly when generalizing to new data--that is, simplicity leads to robustness. We present evidence regarding when people use simple heuristics and describe the challenges to be addressed by this research program.
  • Article
    Full-text available
    Individuals may join groups for several reasons, one of which is the possibility of sharing information about the quality of a foraging area. Sharing information in a patch-foraging scenario gives each group member an opportunity to make a more accurate estimate of the quality of the patch. In this paper we present a mathematical model in which we study the effect of group size on patch-leaving policy and per capita intake rate. In the model, group members share information equally in a random search for food. Food is distributed in patches according to a negative binomial distribution. A prediction from our model is that, the larger the group, the earlier each group member should leave the current patch. We also find that the benefit from enhanced exchange of information does not exceed the cost of sharing food with group members. The per capita intake rate decreases as the group size increases. Therefore, animals should only form groups when other factors outweigh the costs, which is easiest to achieve when the travelling time is short.
  • Article
    Full-text available
    Foragers that feed on hidden prey are uncertain about the intake rate they can achieve as they enter a patch. However, foraging success can inform them, especially if they have prior knowledge about the patch quality distribution in their environment. We experimentally tested whether and how red knots (Calidris canutus) use such information and whether their patch-leaving decisions maximized their long-term net energy intake rate. The results suggest that the birds combined patch sample information with prior knowledge by making use of the potential value assessment rule. We reject five alternative leaving rules. The potential encounter rate that the birds choose as their critical departure threshold maximized their foraging gain ratio (a modified form of efficiency) while foraging. The high experimental intake rates were constrained by rate of digestion. Under such conditions, maximization of the foraging gain ratio during foraging maximizes net intake rate during total time (foraging time plus digestive breaks). We conclude that molluscivore red knots, in the face of a digestive constraint, are able to combine prior environmental knowledge about patch quality with patch sample information to obtain the highest possible net intake over total time.
  • Article
    We look at a simple model in which an animal makes behavioural decisions over time in an environment in which all parameters are known to the animal except predation risk. In the model there is a trade-off between gaining information about predation risk and anti-predator behaviour. All predator attacks lead to death for the prey, so that the prey learns about predation risk by virtue of the fact that it is still alive. We show that it is not usually optimal to behave as if the current unbiased estimate of the predation risk is its true value. We consider two different ways to model reproduction; in the first scenario the animal reproduces throughout its life until it dies, and in the second scenario expected reproductive success depends on the level of energy reserves the animal has gained by some point in time. For both of these scenarios we find results on the form of the optimal strategy and give numerical examples which compare optimal behaviour with behaviour under simple rules of thumb. The numerical examples suggest that the value of the optimal strategy over the rules of thumb is greatest when there is little current information about predation risk, learning is not too costly in terms of predation, and it is energetically advantageous to learn about predation. We find that for the model and parameters investigated, a very simple rule of thumb such as 'use the best constant control' performs well.