A search theory model of patch-to-patch forager movement with application to pollinator-mediated gene flow

We present a spatially implicit analytical model of forager movement, designed to address a simple scenario common in nature. We assume minimal depression of patch resources, and discrete foraging bouts, during which foragers fill to capacity. The model is particularly suitable for foragers that search systematically, foragers that deplete resources in a patch only incrementally, and for sit-and-wait foragers, where harvesting does not affect the rate of arrival of forage. Drawing on the theory of job search from microeconomics, we estimate the expected number of patches visited as a function of just two variables: the coefficient of variation of the rate of energy gain among patches, and the ratio of the expected time exploiting a randomly chosen patch and the expected time travelling between patches. We then consider the forager as a pollinator and apply our model to estimate gene flow. Under model assumptions, an upper bound for animal-mediated gene flow between natural plant populations is approximately proportional to the probability that the animal rejects a plant population. In addition, an upper bound for animal-mediated gene flow in any animal-pollinated agricultural crop from a genetically modified (GM) to a non-GM field is approximately proportional to the proportion of fields that are GM and the probability that the animal rejects a field.     

Intelligent decisions from the hive mind: Foragers and nectar receivers of Apis mellifera collaborate to optimise active forager numbers

We present a differential equation-based mathematical model of nectar foraging by the honey bee Apis mellifera. The model focuses on two behavioural classes; nectar foragers and nectar receivers. Results generated from the model are used to demonstrate how different classes within a collective can collaborate to combine information and produce finely tuned decisions through simple interactions. In particular we show the importance of the ‘search time’ - the time a returning forager takes to find an available nectar receiver - in restricting the forager population to a level consistent with colony-wide needs.     

Using spatially explicit models to characterize foraging performance in heterogeneous landscapes

ABSTRACT The success of most foragers is constrained by limits to their sensory perception, memory, and locomotion. However, a general and quantitative understanding of how these constraints affect foraging benefits, and the trade-offs they imply for foraging strategies, is difficult to achieve. This article develops foraging performance statistics to assess constraints and define trade-offs for foragers using biased random walk behaviors, a widespread class of foraging strategies that includes area-restricted searches, kineses, and taxes. The statistics are expected payoff and expected travel time and assess two components of foraging performance: how effectively foragers distinguish between resource-poor and resourcerich parts of their environments and how quickly foragers in poor parts of the environment locate resource concentrations. These statistics provide a link between mechanistic models of individuals' movement and functional responses, population-level models of forager distributions in space and time, and foraging theory predictions of optimal forager distributions and criteria for abandoning resource patches. Application of the analysis to area-restricted search in coccinellid beetles suggests that the most essential aspect of these predators's foraging strategy is the "turning threshold," the prey density at which ladybirds switch from slow to rapid turning. This threshold effectively determines whether a forager exploits or abandons a resource concentration. Foraging is most effective when the threshold is tuned to match physiological or energetic requirements. These performance statistics also help anticipate and interpret the dynamics of complex spatially and temporally varying forager-resource systems.     

Interactions with Combined Chemical Cues Inform Harvester Ant Foragers' Decisions to Leave the Nest in Search of Food

Social insect colonies operate without central control or any global assessment of what needs to be done by workers. Colony organization arises from the responses of individuals to local cues. Red harvester ants (Pogonomyrmex barbatus) regulate foraging using interactions between returning and outgoing foragers. The rate at which foragers return with seeds, a measure of food availability, sets the rate at which outgoing foragers leave the nest on foraging trips. We used mimics to test whether outgoing foragers inside the nest respond to the odor of food, oleic acid, the odor of the forager itself, cuticular hydrocarbons, or a combination of both with increased foraging activity. We compared foraging activity, the rate at which foragers passed a line on a trail, before and after the addition of mimics. The combination of both odors, those of food and of foragers, is required to stimulate foraging. The addition of blank mimics, mimics coated with food odor alone, or mimics coated with forager odor alone did not increase foraging activity. We compared the rates at which foragers inside the nest interacted with other ants, blank mimics, and mimics coated with a combination of food and forager odor. Foragers inside the nest interacted more with mimics coated with combined forager/seed odors than with blank mimics, and these interactions had the same effect as those with other foragers. Outgoing foragers inside the nest entrance are stimulated to leave the nest in search of food by interacting with foragers returning with seeds. By using the combined odors of forager cuticular hydrocarbons and of seeds, the colony captures precise information, on the timescale of seconds, about the current availability of food.     

Population and community consequences of spatial subsidies derived from central-place foraging

Central-place foragers, such as ants, beavers, and colonial seabirds, can act as biological conduits, subsidizing local communities with allochthonous resources. To explore the consequences of such biologically vectored resource redistribution, we draw on an example from cave ecology and develop a population-level model of central-place foraging based on the dispersal kernel framework. We explore how the size of the patch in which central-place foraging occurs and the spatial distribution of foragers within that patch feed back to influence the population dynamics of the central-place forager and the species richness of the associated recipient community. We demonstrate that the particular way in which a population of central-place foragers uses space has two important effects. First, space use determines the stability of the forager population and establishes patch size thresholds for persistence, stable equilibria, and limit cycles. Second, alternative foraging kernels lead to qualitatively different scaling relationships between the size of the foraging patch and species richness back at the central place. These analyses provide a new link among elements of ecology related to animal behavior, population dynamics, and species diversity while also providing a novel perspective on the utility of integrodifference equations for problems in spatial ecology.     

Population consequences of movement decisions in a patchy landscape

Complex, human‐dominated landscapes provide unique challenges to animals. In landscapes fragmented by human activity, species whose home ranges ordinarily consist of continuous habitat in pristine environments may be forced to forage among multiple smaller habitat patches embedded in an inhospitable environment. Furthermore, foragers often must decide whether to traverse a heterogeneous suite of landscape elements that differ in risk of predation or energetic costs. We modeled population consequences of foraging decisions for animals occupying patches embedded in a heterogeneous landscape. In our simulations, animals were allowed to use three different rules for moving between patches: a) optimal selection resulting from always choosing the least‐cost path; b) random selection of a movement path; and c) probabilistic selection in which path choice was proportional to an animal's probability of survival while traversing the path. The resulting distribution of the population throughout the landscape was dependent on the movement rule used. Least‐cost movement rules (a) produced landscapes that contained the highest average density of consumers per patch. However, optimal movement resulted in an all‐or‐none pattern of occupancy and a coupling of occupied patches into pairs that effectively reduced the population to a set of sub‐populations. Random and probabilistic rules, (b and c), in relatively safe landscapes produced similar average densities and 100% occupancy of patches. However, as the level of risk associated with travel between patches increased, random movement resulted in an all‐or‐none occupancy pattern while occupied patches in probabilistic populations went extinct independently of the other patches. Our results demonstrate strong effects of inter‐patch heterogeneity and movement decisions on population dynamics, and suggest that models investigating the persistence of species in complex landscapes should take into account the effects of the intervening landscape on behavioral decisions affecting animal movements between patches.     

Optimal movement between patches under incomplete information about the spatial distribution of food items

If the food distribution contains spatial pattern, the food density in a particular patch provides a forager with information about nearby patches. Foragers might use this information to exploit patchily distributed resources profitably. We model the decision on how far to move to the next patch in linear environments with different spatial patterns in the food distribution (clumped, random, and regular) for foragers that differ in their degree of information. An ignorant forager is uninformed and therefore always moves to the nearest patch (be it empty or filled). In contrast, a prescient forager is fully informed and only exploits filled patches, skipping all empty patches. A Bayesian assessor has prior knowledge about the content of patches (i.e. it knows the characteristics of the spatial pattern) and may skip neighbouring patches accordingly by moving to the patch where the highest gain rate is expected. In most clumped and regular distributions there is a benefit of assessment, i.e. Bayesian assessors achieve substantially higher long-term gain rates than ignorant foragers. However, this is not the case in distributions with less strong spatial pattern, despite the fact that there is a large potential benefit from a sophisticated movement rule (i.e. a large penalty of ignorance). Bayesian assessors do also not achieve substantially higher gain rates in environments that are relatively rich or poor in food. These results underline that an incompletely informed forager that is sensitive to spatial pattern should not always respond to existing pattern. Furthermore, we show that an assessing forager can enhance its long-term gain rate in highly clumped and some specific near-regular food distributions, by sampling the environment in slightly larger spatial units.     

Optimal movement between patches under incomplete information about the spatial distribution of food items

If the food distribution contains spatial pattern, the food density in a particular patch provides a forager with information about nearby patches. Foragers might use this information to exploit patchily distributed resources profitably. We model the decision on how far to move to the next patch in linear environments with different spatial patterns in the food distribution (clumped, random, and regular) for foragers that differ in their degree of information. An ignorant forager is uninformed and therefore always moves to the nearest patch (be it empty or filled). In contrast, a prescient forager is fully informed and only exploits filled patches, skipping all empty patches. A Bayesian assessor has prior knowledge about the content of patches (i.e. it knows the characteristics of the spatial pattern) and may skip neighbouring patches accordingly by moving to the patch where the highest gain rate is expected. In most clumped and regular distributions there is a benefit of assessment, i.e. Bayesian assessors achieve substantially higher long-term gain rates than ignorant foragers. However, this is not the case in distributions with less strong spatial pattern, despite the fact that there is a large potential benefit from a sophisticated movement rule (i.e. a large penalty of ignorance). Bayesian assessors do also not achieve substantially higher gain rates in environments that are relatively rich or poor in food. These results underline that an incompletely informed forager that is sensitive to spatial pattern should not always respond to existing pattern. Furthermore, we show that an assessing forager can enhance its long-term gain rate in highly clumped and some specific near-regular food distributions, by sampling the environment in slightly larger spatial units.     

An Adaptive Difference in the Relationship between Foraging Mode and Responses to Prey Chemicals in two Congeneric Scincid Lizards

Sensory abilities must allow efficient detection of prey, but the senses used and their relative importance may vary with hunting methods. In lizards, ambush foragers locate prey visually and active foragers use a combination of vision and vomerolfaction, the chemical sense associated with the vomeronasal system. Active foragers, but not ambush foragers, discriminate between prey chemicals and other chemical stimuli sampled by tongue-flicking. In active foragers, features of the tongue that may improve chemical sampling, such as elongation and forking are more pronounced and density of vomeronasal chemoreceptors is greater, than in ambush foragers. Foraging mode is fixed in most lizard families, and correlated evolution has been demonstrated among foraging mode, discrimination of prey chemicals, and lingual-vomeronasal morphology by interfamilial comparisons. Here I present information on a rare case of an intrageneric difference in foraging mode in the genus Mabuya . Laboratory experiments on the discrimination of prey chemicals showed that the active forager M . striata sparsa exhibits prey chemical discrimination, but the ambush forager M . acutilabris does not. The active forager also has a slightly more elongated tongue with deeper notching at the tip than the ambush forager, which might be a response to a change in foraging behavior or a reflection of unrelated differences in head shape. These findings confirm predictions based on correlated evolution between the hunting method and use of the chemical sense to locate food. They further show that chemosensory behavior is adjusted to change in foraging mode more rapidly than was previously known and suggest that behavioral changes may occur more rapidly than associated modifications of chemosensory morphology.     

Intake rate at differently scaled heterogeneous food distributions explained by the ability of tactile-foraging mallard to concentrate foraging effort within profitable areas

The ability to respond to spatial heterogeneity in food abundance depends on the scale of the food distribution and the foraging scale of the forager. The aim of this study is to illustrate that a foraging scale exists, and that at larger scaled food distributions foragers benefit from the ability to subdivide a continuous (non-discrete) heterogeneous environment into profitable and non-profitable areas. We recorded search patterns of mallards Anas plathyrhynchos foraging in shallow water on cryptic prey items (millet seeds), distributed at different scales. A small magnet attached to the lower mandible allowed us to record in great detail the position and movements of the bill tip within a feeding tray underlain by magnet sensors. Instantaneous intake rate was determined in a subsequent experiment. We successfully determined the foraging scale (about 2×2 cm), defined as the scale above which foragers do respond (coarse scaled distribution) and below which foragers do not respond (fine scaled distribution) to spatial heterogeneity, by concentrating foraging effort within areas of high food density. A response resulted in a significantly higher intake rate, compared to a homogeneous distribution with an equal overall density. Unlike systematic search cell revisitation was common in trials, and at coarse scaled food distributions even slightly (but significantly) more frequently observed than predicted for random search. Mallards respond to food capture by restricting displacement (area restricted search) at food distributions that are considered to be clumped for the forager (large scaled coarse distributions). We argue that partitioning the environment at the foraging scale in itself could be a mechanism to concentrate foraging efforts within profitable areas, because mallard were able to respond to heterogeneity at coarse scaled food distributions even when non-clumped (i.e. without conducting area restricted search).     

The efficiency of adaptive search tactics for different prey distribution patterns: a simulation model based on the behaviour of juvenile plaice

Juvenile plaice Pleuronectes platessa are particularly useful for studying forager search behaviour because their search paths are essentially two dimensional, and punctuated by natural stops. Their prey occur in a range of natural distributions from highly aggregated to over‐dispersed. Juvenile plaice use area‐restricted search near aggregated prey and extensive search, consisting of longer moves and fewer turns, between aggregations and when searching for dispersed prey. They search for less conspicuous prey items mainly in the pauses between movements. This saltatory search behaviour contrasts with the continuous search that is usually assumed in search models. A simulation model of saltatory search behaviour showed that a strategy combining extensive and intensive search allows the efficient exploitation of a range of natural prey distribution patterns, and that it is particularly effective when the search behaviour is controlled by perceived prey density. This allows the predator to respond to the localized aggregations which often occur in nature. The selective use of intensive search was more efficient than the continuous use of extensive search even in prey distribution patterns that were statistically over‐dispersed.     

Proteins searching for their target on DNA by one-dimensional diffusion: overcoming the “speed-stability” paradox

The sequence dependence of DNA-protein interactions that allows proteins to find the correct reaction site also slows down the 1D diffusion of the protein along the DNA molecule, leading to the so-called “speed-stability paradox,” wherein fast diffusion along the DNA molecule is seemingly incompatible with stable targeting of the reaction site. Here, we develop diffusion-reaction models that use discrete and continuous Gaussian random 1D diffusion landscapes with or without a high-energy cut-off, and two-state models with a transition to and from a “searching” mode in which the protein diffuses rapidly without recognizing the target. We show the conditions under which such considerations lead to a predicted speed-up of the targeting process, and under which the presence of a “searching” mode in a two-state model is nearly equivalent to the existence of a high-energy cut-off in a one-state model. We also determine the conditions under which the search is either diffusion-limited or reaction-limited, and develop quantitative expressions for the rate of successful targeting as a function of the site-specific reaction rate, the roughness of the DNA-protein interaction potential, and the presence of a “searching” mode. In general, we find that a rough landscape is compatible with a fast search if the highest energy barriers can be avoided by “hopping” or by the protein transitioning to a lower-energy “searching” mode. We validate these predictions with the results of Brownian dynamics, kinetic Metropolis, and kinetic Monte Carlo simulations of the diffusion and targeting process, and apply these concepts to the case of T7 RNA polymerase searching for its target site on T7 DNA.     

The Regulation of Ant Colony Foraging Activity without Spatial Information

Many dynamical networks, such as the ones that produce the collective behavior of social insects, operate without any central control, instead arising from local interactions among individuals. A well-studied example is the formation of recruitment trails in ant colonies, but many ant species do not use pheromone trails. We present a model of the regulation of foraging by harvester ant (Pogonomyrmex barbatus) colonies. This species forages for scattered seeds that one ant can retrieve on its own, so there is no need for spatial information such as pheromone trails that lead ants to specific locations. Previous work shows that colony foraging activity, the rate at which ants go out to search individually for seeds, is regulated in response to current food availability throughout the colony's foraging area. Ants use the rate of brief antennal contacts inside the nest between foragers returning with food and outgoing foragers available to leave the nest on the next foraging trip. Here we present a feedback-based algorithm that captures the main features of data from field experiments in which the rate of returning foragers was manipulated. The algorithm draws on our finding that the distribution of intervals between successive ants returning to the nest is a Poisson process. We fitted the parameter that estimates the effect of each returning forager on the rate at which outgoing foragers leave the nest. We found that correlations between observed rates of returning foragers and simulated rates of outgoing foragers, using our model, were similar to those in the data. Our simple stochastic model shows how the regulation of ant colony foraging can operate without spatial information, describing a process at the level of individual ants that predicts the overall foraging activity of the colony.     

Optimal feeding under stoichiometric constraints: a model of compensatory feeding with functional response

When the nutrient content of food is limited, herbivores often increase their feeding rates. Such an increase in the feeding rate is called ‘compensatory feeding’. Although it has a number of implications for herbivore population and plant–forager dynamics, the compensatory feeding is not yet functionally formulated especially in relation with ecological stoichiometry. Therefore, we constructed a simple mathematical model by incorporating the optimal feeding rate into the type II functional response to maximize a forager's growth rate under constraints of carbon or nutritionally important element like phosphorus (P). We used the planktonic herbivore Daphnia as a model herbivore. The model revealed that the optimal feeding rate increased by using excess carbon when relative P content of food was less than a certain level, which is known as the threshold elemental ratio. This level changed with the change of food abundance. It also showed that whether or not foragers should exhibit compensatory feeding depends on their stoichiometric characteristics and digestive traits, and also on the assimilability of a given food. These findings are helpful to test the feeding conditions under which compensatory feeding is advantageous for a given animal. Our model can be easily incorporated into forager population dynamics and prey‐consumer interaction models because the optimal feeding rate can be analytically given.     

The role of prey taxis in biological control: a spatial theoretical model

We study a reaction-diffusion-advection model for the dynamics of populations under biological control. A control agent is assumed to be a predator species that has the ability to perceive the heterogeneity of pest distribution. The advection term represents the predator density movement according to a basic prey taxis assumption: acceleration of predators is proportional to the prey density gradient. The prey population reproduces logistically, and the local population interactions follow the Holling Type II trophic function. On the scale of the population, our spatially explicit approach subdivides the predation process into random movement represented by diffusion, directed movement described by prey taxis, local prey encounters, and consumption modeled by the trophic function. Thus, our model allows studying the effects of large-scale predator spatial activity on population dynamics. We show under which conditions spatial patterns are generated by prey taxis and how this affects the predator ability to maintain the pest population below some economic threshold. In particular, intermediate taxis activity can stabilize predator-pest populations at a very low level of pest density, ensuring successful biological control. However, very intensive prey taxis destroys the stability, leading to chaotic dynamics with pronounced outbreaks of pest density.     

From random walks to informed movement

The analysis of animal movement is a large and continuously growing field of research. Detailed knowledge about movement strategies is of crucial importance for understanding eco‐evolutionary dynamics at all scales – from individuals to (meta‐)populations. This and the availability of detailed movement and dispersal data motivated Nathan and colleagues to published their much appreciated call to base movement ecology on a more thorough mechanistic basis. So far, most movement models are based on random walks. However, even if a random walk might describe real movement patterns acceptably well, there is no reason to assume that animals move randomly. Therefore, mechanistic models of foraging strategies should be based on information use and memory in order to increase our understanding of the processes that lead to animal movement decisions. We present a mechanistic movement model of an animal with a limited perceptual range and basic information storage capacities. This ‘spatially informed forager’ constructs an internal map of its environment by using perception, memory and learned or evolutionarily acquired assumptions about landscape attributes. We analyse resulting movement patterns and search efficiencies and compare them to area restricted search strategies (ARS) and biased correlated random walks (BCRW) of omniscient individuals. We show that, in spite of their limited perceptual range, spatially informed individuals boost their foraging success and may perform much better than the best ARS. The construction of an internal map and the use of spatial information results in the emergence of a highly correlated walk between patches and a rather systematic search within resource clusters. Furthermore, the resulting movement patterns may include foray search behaviour. Our work highlights the strength of mechanistic modelling approaches and sets the stage for the development of more sophisticated models of memory use for movement decisions and dispersal.     

Interactions Increase Forager Availability and Activity in Harvester Ants

Social insect colonies use interactions among workers to regulate collective behavior. Harvester ant foragers interact in a chamber just inside the nest entrance, here called the ''entrance chamber''. Previous studies of the activation of foragers in red harvester ants show that an outgoing forager inside the nest experiences an increase in brief antennal contacts before it leaves the nest to forage. Here we compare the interaction rate experienced by foragers that left the nest and ants that did not. We found that ants in the entrance chamber that leave the nest to forage experienced more interactions than ants that descend to the deeper nest without foraging. Additionally, we found that the availability of foragers in the entrance chamber is associated with the rate of forager return. An increase in the rate of forager return leads to an increase in the rate at which ants descend to the deeper nest, which then stimulates more ants to ascend into the entrance chamber. Thus a higher rate of forager return leads to more available foragers in the entrance chamber. The highest density of interactions occurs near the nest entrance and the entrances of the tunnels from the entrance chamber to the deeper nest. Local interactions with returning foragers regulate both the activation of waiting foragers and the number of foragers available to be activated.     

Classifying area-restricted search (ARS) using a partial sum approach

Many animals perform two distinct alternating movement strategies when foraging: intensive searches with low speed and high turning to cover a small area in high detail and extensive searches with high speed and low turning to cover a large area in low detail. Observed movement paths will tend to exhibit differences in speed and correlation between these different search strategies. Identifying transitions between strategies can enable one to acquire information regarding both the distribution of resources and the underlying behavioural mechanisms performed by a foraging animal. Methods such as the moving average, first-passage time, residence time and fractal landscape methods have been used to identify behavioural states of various real and simulated foragers. We provide a review of these current methods and identify a set of common limitations associated with each procedure. We develop a new mathematical approach: the partial sum method, which is designed to avoid these limitations. A comprehensive test is undertaken to evaluate and compare the performance of the partial sum and the existing methods using a carefully constructed set of computer-generated movement paths. Each simulated track was designed to replicate the possible paths performed by an animal under different foraging conditions. Our results provide strong evidence that the partial sum method is better than existing analytical methods for identifying transitions between two different search strategies.