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: Random movement by pollen collecting bumblebees

Summary Two bumblebee species, Bombus bifarius and B. flavifrons, forage randomly with respect to direction when gathering pollen on Potentilla gracilis. Bees avoid revisiting flowers by being able to differentiate recently visited from unvisited flowers. This recognition occurs while bees are flying over open flowers and appears to be a response to the amount of available pollen within flowers. Random foraging with respect to direction is the optimal strategy when the probability of flower revisitation is low. Bumblebees appear to be moving preferentially between nearest neighbors, again as predicted by foraging theory. This behavior causes the establishment of pollen patches in the P. gracilis population. Unlike other pollinators studied in similar situations, bumblebees on P. gracilis do not forage utilizing an area-restricted searching behavior. Because floral reward quality can be assessed at low cost by bees foraging on P. gracilis, their tendency to move to nearby flowers even after encountering a poor quality blossom apparently yields a higher rate of net energy intake than does area-restricted searching. The data indicate that bumblebees exhibit great plasticity in foraging behavior and that they are able to forage efficiently under a wide range of environmental conditions.     

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 in bumblebees: Rule of movement between flowers within inflorescences

It is hypothesized that nectar-collecting bumblebees will be found to forage in ways that maximize their net rate of energy intake. Attention is focused, in this paper, on the manner in which these bumblebees move from one flower to another within inflorescences. Observations were made on workers of Bombus appositus, which were collecting nectar from Aconitum columbianum (monkshood). The rule of movement of the bumblebees was determined and compared, in terms of net rate of energy intake, with several possible alternative rules. Two of these alternatives gave equally high net rates of energy intake. The observed rule was very similar in nature to one of these and indistinguishable from both in terms of net rate of energy intake.     

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].     

Search and foraging behaviors from movement data: A comparison of methods

Search behavior is often used as a proxy for foraging effort within studies of animal movement, despite it being only one part of the foraging process, which also includes prey capture. While methods for validating prey capture exist, many studies rely solely on behavioral annotation of animal movement data to identify search and infer prey capture attempts. However, the degree to which search correlates with prey capture is largely untested. This study applied seven behavioral annotation methods to identify search behavior from GPS tracks of northern gannets (Morus bassanus), and compared outputs to the occurrence of dives recorded by simultaneously deployed time–depth recorders. We tested how behavioral annotation methods vary in their ability to identify search behavior leading to dive events. There was considerable variation in the number of dives occurring within search areas across methods. Hidden Markov models proved to be the most successful, with 81% of all dives occurring within areas identified as search. k‐Means clustering and first passage time had the highest rates of dives occurring outside identified search behavior. First passage time and hidden Markov models had the lowest rates of false positives, identifying fewer search areas with no dives. All behavioral annotation methods had advantages and drawbacks in terms of the complexity of analysis and ability to reflect prey capture events while minimizing the number of false positives and false negatives. We used these results, with consideration of analytical difficulty, to provide advice on the most appropriate methods for use where prey capture behavior is not available. This study highlights a need to critically assess and carefully choose a behavioral annotation method suitable for the research question being addressed, or resulting species management frameworks established.     

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: 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: 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 for multiple nutrients in an unpredictable environment

Foraging theory has typically been concerned with the acquisition of a single resource even though organisms from mammals to protozoa are capable of balancing their requirements for multiple resources. Existing theory concerning multiple nutrients from multiple foods does not predict the sequence of food selection. We constructed an optimisation model of the simplest case of two foods containing differing amounts of two nutrients. We begin with the well-supported assumption that reproductive value declines with the distance from target nutrient intake. We show that nutrient space divides into two distinct areas where the animal should exclusively consume one food or the other. The organism thus initially concentrates on one food type until the border between the areas is reached and then moves as closely as possible along the border to approach the target. This strategy is commonly observed in a range of organisms, suggesting that the assumed fitness function is common.     

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.     

Prey selection by plovers: Optimal foraging in mixed-species groups

Prey (earthworm) size selection was investigated in lapwings (Vanellus vanellus) and golden plovers (Pluvialis apricaria) feeding in mixed species flocks and compared with that predicted by an optimal foraging model based on energy intake. As well as the usual constraints of searching and handling time, our model incorporated the difficulty of capturing concealed prey, the orientation time needed to locate prey and the risk of theft by gulls (Larus ridibundus). When costs were taken into account, small worms turned out to be the most profitable. The relative profitability of size classes changed when gulls were present and birds shifted their intake accordingly so that they always took mainly the most profitable worms. Birds were expected to do best by taking the three most profitable size classes and the size range taken was consistent with this. In addition there was an inverse relationship between the probabilities of taking profitable and unprofitable worm sizes. Observations of birds were supported by field enclosure experiments which prevented birds feeding in certain areas. Departures from predictions of the model are interpreted as sampling errors due to birds using depth as an approximate indicator of worm 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.