Optimal foraging in patches: A case for stochasticity

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.     

Optimal foraging: movement patterns of bumblebees between inflorescences

Nectar-collecting bumblebees are hypothesized to employ rules of movement which result in the maximum net rate of energy gain (i.e., are optimal). The optimal movement rules are derived from a mathematical model and are used to generate predicted patterns of movement. The predicted patterns are compared with field observations. These observations support the hypothesis. An important component of the mathematical model is the memory of the foraging animal. The field data have implications concerning the memory capabilities of the bumblebees.     

Optimal foraging in hummingbirds: Rule of movement between inflorescences

The movements of hummingbirds between inflorescences of scarlet gilia (Ipomopsis aggregata) were studied. These movements exhibited the following patterns: (1) Although the hummingbirds appeared to avoid moving to the previous inflorescence, no significant correlation was found between the directions of successive inter-inflorescence movements. (2) The frequency distribution of inter-inflorescence flight distances was found to be leptokurtic. (3) The hummingbirds were more likely to move to an inflorescence the larger and/or closer it was. (4) The hummingbirds moved to inflorescences of greatest apparent size (i.e. ratio of number of flowers available to distance from present inflorescence) more often than they moved to the largest inflorescence, the closest infloresence, or the inflorescence estimated to yield the greatest rate of energy gain. (5) The frequency distribution of moves to the inflorescence having the ith greatest apparent size is well fitted by a geometric distribution. This is consistent with the hummingbrids choosing the inflorescence of greatest apparent size (excluding the previous inflorescence) from within some scanning sector. These movement patterns are consistent with the expectations of optimal foraging theory only if the hummingbirds cannot or do not determine the directions of possible inflorescences relative to the direction of arrival at the present inflorescence and if they cannot assess independently the sizes and distances of possible inflorescences.     

Optimal foraging by zooplankton within patches: the case of Daphnia

The motions of many physical particles as well as living creatures are mediated by random influences or 'noise'. One might expect that over evolutionary time scales internal random processes found in living systems display characteristics that maximize fitness. Here we focus on animal random search strategies [G.M. Viswanathan, S.V. Buldyrev, S. Havlin, M.G.E. Da Luz, E.P. Raposo, H.E. Stanley, Optimizing the success of random searches, Nature 401 (1999) 911-914; F. Bartumeus, J. Catalan, U.L. Fulco, M.L. Lyra, G.M. Viswanathan, Optimizing the encounter rate in biological interactions: Lévy versus Brownian stratagies, Phys. Rev. Lett. 88 (2002) 097901 and 89 (2002) 109902], and we describe experiments with the following Daphnia species: D. magna, D. galeata, D. lumholtzi, D. pulicaria, and D. pulex. We observe that the animals, while foraging for food, choose turning angles from distributions that can be described by exponential functions with a range of widths. This observation leads us to speculate and test the notion that this characteristic distribution of turning angles evolved in order to enhance survival. In the case of theoretical agents, some form of randomness is often introduced into search algorithms, especially when information regarding the sought object(s) is incomplete or even misleading. In the case of living animals, many studies have focused on search strategies that involve randomness [H.C. Berg, Random Walks in Biology, Princeton University, Princeton, New Jersey, 1993; A. Okubo, S.A. Levin (Eds.), Diffusion and Ecological Problems: Modern Perspectives, second ed., Springer, New York, 2001]. A simple theory based on stochastic differential equations of the motion backed up by a simulation shows that the collection of material (information, energy, food, supplies, etc.) by an agent executing Brownian-type hopping motions is optimized while foraging for a finite time in a supply patch of limited spatial size if the agent chooses turning angles taken from an exponential distribution with a specific stochastic intensity or 'noise width'. Search strategies that lead to optimization is a topic of high current interest across many disciplines [D. Wolpert, W. MacReady, No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation 1 (1997) 67].     

Optimal foraging in bumblebees: Hunting by expectation

The foraging behaviour literature contains three hypotheses concerned with hunting by expectation. These suggest possible rules animals use to decide when to leave particular feeding sites and search in other places for food. Predictions of the three hypotheses were tested experimentally by varying the quality of plants (amount and distribution of nectar) encountered by bumblebees (Bombus appositus). Results support only a rate expectation hypothesis. Bees left multiflowered plants when the amount of nectar found in the first flower was below a threshold volume. Bees stayed on plants if they received greater than the threshold volume. This threshold nectar volume is close to the amount predicted if a bee forages to maximize its rate of net energy intake.     

Optimal foraging for specific nutrients in predatory beetles

Evolutionary theory predicts that animals should forage to maximize their fitness, which in predators is traditionally assumed equivalent to maximizing energy intake rather than balancing the intake of specific nutrients. We restricted female predatory ground beetles (Anchomenus dorsalis) to one of a range of diets varying in lipid and protein content, and showed that total egg production peaked at a target intake of both nutrients. Other beetles given a choice to feed from two diets differing only in protein and lipid composition selectively ingested nutrient combinations at this target intake. When restricted to nutritionally imbalanced diets, beetles balanced the over- and under-ingestion of lipid and protein around a nutrient composition that maximized egg production under those constrained circumstances. Selective foraging for specific nutrients in this predator thus maximizes its reproductive performance. Our findings have implications for predator foraging behaviour and in the structuring of ecological communities.     

Optimal trajectories for the short-distance foraging flights of swans

Optimal flight theory relates body measurements (wing span, body cross-section, body mass) and aerodynamic variables (air density, drag, profile and induced power ratios) to the most energy-efficient velocity for long distance migration. For short-range (2-10 km) foraging flights the theory is expanded to include non-negligible costs for take-off and energy savings/losses for climbing to altitude (drag decreases with air density and therefore with altitude). The theory predicts clear differences between Tundra and Trumpeter swans. Generally speaking, for flights between 2 and 10 km Trumpeter swans can be expected to fly approximately 5-10 m lower in altitude and 1-2 ms(-1)more slowly than Tundra swans. Moreover, the total energy required for these foraging flights is approximately 150% larger for a Trumpeter than a Tundra swan (80 vs. 120 kJ of direct mechanical energy for a 5 km flight), suggesting that Trumpeter swans may be less inclined to take-off than Tundra swans. These factors indicate that even Trumpeters native to the area (as opposed to recently translocated) would be more vulnerable to hunting than native Tundra swans. The expanded theory is compared to observations made in Utah's Bear River Migratory Bird Refuge.     

Optimal foraging: food patch depletion by ruddy ducks

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

Optimal foraging: a bird in the hand released

Optimal foraging theory aims to elucidate strategies that maximize resource intake. Although traditionally used to understand animal foraging behavior, recent evolutionary experiments with viruses offer a new twist on an old idea.     

Optimal foraging and the traveling salesman

Optimal foraging models are examined that assume animals forage for discrete point resources on a plane and attempt to minimize their travel distance between resources. This problem is similar to the well-known traveling salesman problem: A salesman must choose the shortest path from his home office to all cities on his itinerary and back to his home office again. The traveling salesman problem is in a class of enigmatic problems, called NP-complete, which can be so difficult to solve that animals might be incapable of finding the best solution. Two major results of this analysis are: (1) The simple foraging strategy of always moving to the closest resource site does surprisingly well. More sophisticated strategies of “looking ahead” a small number of steps, choosing the shortest path, then taking a step, do worse if all the resource sites are visited, but do slightly better (less than 10%) if not all the resource sites are visited. (2) Short cyclical foraging routes resulted when resources were allowed to renew. This is suggested as an alternative explanation for “trap-lining” in animals that forage for discrete, widely separated resources.     

Optimal intraguild foraging and population stability

This article explores effects of adaptive intraguild predation on species coexistence and community structure in three species' food webs. Two Lotka-Volterra models that assume a trade-off between competition and predation strength are considered in detail. The first model does not explicitly model resource dynamics and is considered with both nonadaptive and adaptive intraguild predation; in the latter case predators choose their diet in order to maximize their instantaneous population growth rate. The second model includes resource population dynamics. Effects of adaptive intraguild predation on the community structure along a gradient in environment productivity are analyzed and compared with some experimental results of protist food webs. Conditions under which intraguild predation is adaptive are discussed for both models. It is proved that if intraguild predators are perfect optimizers then intraguild predation should decrease with increasing environmental productivity and adaptive intraguild predation is a stabilizing factor provided environmental productivity is high enough.     

Optimal diving behaviour for foraging in relation to body size

Overall, large animals dive longer and deeper than small animals; however, after the difference in body size is taken into account, smaller divers often tend to make relatively longer dives. Neither physiological nor theoretical explanations have been provided for this paradox. This paper develops an optimal foraging diving model to demonstrate the effect of body size on diving behaviour, and discusses optimal diving behaviour in relation to body size. The general features of the results are: (1) smaller divers should rely more heavily on anaerobic respiration, (2) larger divers should not always make longer dives than smaller divers, and (3) an optimal body size exists for each diving depth. These results explain the relatively greater diving ability observed in smaller divers, and suggest that if the vertical distribution of prey in the water column is patchy, there is opportunity for a population of diving animals to occupy habitat niches related to body size.     

Optimal foraging area: Size and allocation of search effort

A model is derived for the optimal spatial allocation of foraging effort for an animal returning with food to a central place in a uniform habitat. The forager is assumed to maximize its yield of food during a given period. Foraging effort is expended on search for food, and on transportation to the central place. It is shown that the allocation of search has been optimal if and only if the “marginal cost” of additional food is equal throughout the foraging area when the period has elapsed. The model is used to predict the optimal area radius and allocation of search time. With realistic parameter values, the optimal time per unit area roughly decreases linearly with the distance from the central place. The influence of food density and forager characteristics is examined.     

Optimal foraging by bacteriophages through host avoidance

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

Optimal foraging of sticklebacks on swarming prey

Hungry sticklebacks, Gasterosteus aculeatus, preferentially attacked the densest region of a swarm of water fleas, but with decreasing attack readiness they increasingly preferred less dense regions. Such a hunger dependent change in feeding preference has not yet been dealt with by optimal foraging theory. A model, which assumes that high swarm densities provide high feeding rates (because of small inter-prey distances) but also high costs of confusion, predicts that a predator should always choose the lowest prey density in which it can achieve a feeding rate sufficient to satisfy its hunger. Some predictions of the model were experimentally verified. Hungry fish have a higher feeding rate in a high prey density than in a lower density and less hungry fish have a higher rate in a low density than in a high density.     

Optimal foraging and community structure: implications for a guild of generalist grassland herbivores

Summary A particular linear programming model is constructed to predict the diets of each of 14 species of generalist herbivores at the National Bison Range, Montana. The herbivores have body masses ranging over seven orders of magnitude and belonging to two major taxa: insects and mammals. The linear programming model has three feeding constraints: digestive capacity, feeding time and energy requirements. A foraging strategy that maximizes daily energy intake agrees very well with the observed diets. Body size appears to be an underlying determinant of the foraging parameters leading to diet selection. Species that possess digestive capacity and feeding time constraints which approach each other in magnitude have the most generalized diets. The degree that the linear programming models change their diet predictions with a given percent change in parameter values (sensitivity) may reflect the observed ability of the species to vary their diets. In particular, the species which show the most diet variability are those whose diets tend to be balanced between monocots and dicots. The community-ecological parameters of herbivore body-size ranges and species number can possibly be related to foraging behavior.     

Optimal compensation for temporal uncertainty in movement planning

Motor control requires the generation of a precise temporal sequence of control signals sent to the skeletal musculature. We describe an experiment that, for good performance, requires human subjects to plan movements taking into account uncertainty in their movement duration and the increase in that uncertainty with increasing movement duration. We do this by rewarding movements performed within a specified time window, and penalizing slower movements in some conditions and faster movements in others. Our results indicate that subjects compensated for their natural duration-dependent temporal uncertainty as well as an overall increase in temporal uncertainty that was imposed experimentally. Their compensation for temporal uncertainty, both the natural duration-dependent and imposed overall components, was nearly optimal in the sense of maximizing expected gain in the task. The motor system is able to model its temporal uncertainty and compensate for that uncertainty so as to optimize the consequences of movement.     

Optimal foraging: the difficulty of exploiting different feeding strategies simultaneously

Summary The foraging efficiency of a visually feeding fish, perch (Perca fluviatilis) was studied on two prey species (Daphnia magna and Chaoborus obscuripus) presented either separately or combined. It is shown that when both prey species are present, the foraging efficiency of the predator is reduced. This is due to the predator's inability to simultaneously cope with prey species with different anti-predatory behaviour. In the mixed-meal experiment the predator captured both prey species in equal proportions in disagreement with optimal foraging models assuming that handling time and encounter rate for a prey species are independent of other prey species. The results are, however, in agreement with optimal foraging models assuming that handling time and encounter rate are influenced by short time learning.