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Editor's Picks Most read. Sign up. When encounter rates are available, the prey model algorithm can be completed in linear time that scales with the number of task types. Additionally, the ratio of the number of encounters with a type to the total time will asymptoti- cally converge to the true encounter rate in the environment, and so a simple method exists for estimating the encounter rate.
Although this on-line implementation of the prey model is relatively simple to implement, Pavlic and Passino  present a much simpler decision-making heuristic that converges to prey-model-optimal behav- ior without the need for encounter rate estimation. This heuristic is the natural extension of the conventional patch model implementation to the prey model case.
Thus, on-line im- plementations of OFT-inspired decision-making are suitable for autonomous agents with strict timing requirements and simple computational abilities. Their apparatus consists of eight zones that each include a temperature sensor and a heating element. The zones are arranged so that there is significant cross coupling i. This apparatus could be a model of a large room with multiple temperature actuators or a building with multiple rooms.
Assuming that at most one heating element can be energized at a time, Quijano et al. At each instant of time, there is an error associated with each zone representing the differ- ence between the desired temperature and the temperature at its sensor. Quijano et al. The centralized controller randomly chooses which zone to monitor at each time. Hence, it encounters each error type just as a forager encounters prey types.
Similar foraging-inspired resource-allocation algorithms could be used on mobile agents deployed on factory floors that must balance queues of raw materials. If a raw material is loaded into a physical queue from one end only, the queue will frequently be overloaded on that end. A mobile robot that must move around the queue to shift resources from one location to another could prioritize its movements based on the height of each location in the queue compared to the average height.
Those areas with the greatest off-average error would be highest value and thus would attract the greatest attention from the re-allocation agent. In one example, humans are viewed as foragers that accumulate information from websites that are viewed as patches of information, and it is assumed that humans will allocate time in each web patch according to optimal foraging theory.
Hence, web developers must organize content on their web pages in order to maximize the time an optimal information forager should spend using their sites. For example, one of the key results of the patch model of optimal foraging theory is that foragers will spend less time in all patches if the average time between patch encounters decreases.
In particular, the forager leaves each patch when the patch marginal returns fall below a particular threshold, and that threshold increases as the search time between patches decreases. Consequently, web sites designed to retain visitors for as long as possible e. Here, we summarize the classical OFT optimization objective and show how it is a special case of the advantage-to-disadvantage function.
We also show how a related but different optimization objective favored by some behavioral ecologists is also an advantage-to-disadvantage function. Later, in Section 1. OFT studies behaviors that maximize Darwinian fitness, which is an unmeasurable quantity in general. Charnov  and Pyke et al. Unfortunately, for any finite lifetime, this optimization objective strongly depends on precise knowledge of how gain and time covary [23, 80].
In particular, Charnov  assumes that encounters with each type come from an independent Poisson counting process. So the process describing all encounters is the merged Poisson process, and the energetic intake is modeled by a Markov renewal— reward process corresponding to this merged process.
That is, the behavior should maximize the rate n! The expectation of ratios: Some observational evidence [e. Likewise, some applications will be ill suited for solutions inspired by OFT. Below, in Section 1. OFT inadequacies in finite-lifetime models Classical foraging theory is not well suited for modeling finite lifetimes where either success thresholds must be met or only a finite number of tasks can be processed.
For example, a small bird may perish from the heat lost during the night if it does not eat enough during the day. Likewise, an AAV dispatched for a finite periods of time e. In the infinite lifetime case, future opportunities are certain, and so waiting can be a beneficial tactic. However, in the finite-lifetime case, future opportunities are uncertain, and so successful foragers should be biased toward present returns.
For cases with survival thresholds over short times, Stephens and Charnov  describe a risk-sensitive forager that maximizes the probability that a net gain thresh- old will be achieved by some critical time. This risk-sensitive foraging model is also used by Andrews et al. Initially, the AAV specializes on targets that have a high value-to-time ratio. However, at the end of its life, if it has not accumulated enough value to reach its goal threshold, it begins to general- ize on all targets it encounters.
Hence, the risk-sensitive behavior is a perturbation of the rate-maximizing behavior that becomes most pronounced at the end of life i. However, the risk-sensitive model not only uses limiting forms of the mean and variance of the accumulated gain, but it is also based on results that follow from the central-limit theorem.
Hence, even though the formulation is meant to prescribe behaviors for short-lifetime agents, it is based on assumptions that are only true for agents with long lifetimes. As discussed by Wajnberg , OFT can be used to describe the behavior of an insect that searches for hosts to lay her eggs in. However, it is best suited to model this scenario when typical lifetimes are too short to deplete the egg supply.
However, several studies have shown that egg-limited parasitoids are not uncommon [33, 45, 64, 90, , ]. Furthermore, in AAV applications where packages e. Autonomous agent model for finite-event scenario Consider a task-processing agent similar to the one described in Section 1. For example, a forager may need to eat or store N items to survive over winter, or a female may have N eggs to lay in encountered hosts, or an AAV must deliver one of N packages to each deserving target.
In each case, the time to complete each mission is finite and random, but the number of tasks completed in each mission is fixed at N. The agent mission can be represented by either process, but many cycles of the former process may complete during a single cycle of the latter process. In particular, the finite-event agent can maximize: i Excess rate. Because mission durations are finite by definition, success thresholds can be added. That is, GT is the value threshold the agent must reach to be dispatched on another mission.
This threshold will often be positive, but it may be negative e. That is, when future opportunities are certain, choices should be made based on the balance between returned gain and required processing time i. However, when future opportunities are uncertain i. That is, when the agent is at risk of not meeting its success threshold, it spends relatively more time processing i. Classical OFT describes behaviors that simultane- ously maximize net gain and minimize foraging time. The relative importance of time minimization over gain maximization is varied in order to minimize the opportunity cost  of each activity.
That is, the optimal rate of gain repre- sents the maximum gain that can be returned for each unit of time. An OFT behavior accumulates gain in each activity only if there is no other activity that could return more mean gain for that amount of time. Hence, the optimal rate of gain represents the gain—time tradeoff that minimizes opportunity cost.
We include the threshold GT for completeness, but it only shifts the objective function by a constant value, and so it has no impact on the optimal solution. That is, when maximizing excess rate above, the relative value of gain and time float with the environment and the success threshold.
For high thresholds in environments where encounters return relatively low gain, high-gain opportunities have a greater value. In this case, because the relative gain—time value is fixed, the success threshold has no effect on optimal solutions. Stephens and Krebs  criticize using efficiency i. However, efficiency is a commonly used metric in engineering appli- cations. So we can define an effi- ciency metric that answers both concerns of Stephens and Krebs.
An optimally efficient behavior will maximize the advantage-to-disadvantage function n! If the task-processing agent is given a low success threshold or a large number of tasks to complete, it should not greatly perturb its behavior from the pure efficiency maximizer.
Just as the gain—time tradeoff can be fixed a priori , so can the gain—cost tradeoff. Results of the application of the algorithms described in Section 1. Here, we extend the graphical optimization approach described by Stephens and Krebs  to arbitrary advantage-to-disadvantage functions with arbitrary constraints.
We use insights from the graphical process to compare and contrast the example optimiza- tion objectives discussed in Section 1. An analytical optimization approach is given in Section 1. Because Equation 1. This process is illustrated in Figure 1. Graphical optimization of this function is shown in Figure 1. Consequently, the point of tangency between the ray and the feasible behavior frontier will move to the right.
In this region of decreasing average processing gain, the increased average processing time preempts the very costly searching i. This effect is a result of opportunity cost minimization; there is less opportunity cost for additional processing when searching is itself very costly. Processing tasks not only accumulates gain, but it prevents the loss of gain through searching.
A task-processing agent ceases processing a task when it is likely that a task with higher marginal returns will be found quickly. However, when there is a long time between encountered tasks, it is better to burn fuel processing a task longer than burning fuel searching for a new task because gain is accumulated while processing but not while searching.
These results can be extended to prey-model problems by translating increased task-processing times to increased preference for task types with higher processing times; these prey-model effects e. In this case, Equation 1. That is, in the patch case, every finite-event task-processing agent that maximizes excess rate can be transformed into an equivalent infinite-time rate maximizer by increasing search cost.
So increasing threshold GT or decreasing number of cycles N will have the same effect on the finite-event excess-rate maximizer as increasing search cost cs on the infinite-time rate maximizer, and Figure 1. This result is consistent with the idea that thresholds induce an exploration cost which is reduced when future opportunities are certain.
That is, because the agent receives no gain while searching, searching is a less desirable activity when high gain thresholds must be met. Stephens and Charnov  present a risk-sensitive model of foraging behavior that predicts the optimal combination of gain mean and variance to maximize the probability of reaching a critical energetic threshold.
Hence, present gains are increased to reduce lifetime gain variance i. Hence, lifetime gain variance is increased i. So the time-limited task-processing agent trades per-encounter gain with number of encounters to maximize the probability of reaching a success threshold. The excess-rate task-processing model modifies the classical rate-maximizing model in a similar way. However, this model has a fixed number of encoun- ters and a variable time, and the gain success threshold is essentially a forced cost.
Consequently, results are opposite the expected results from risk-sensitivity theory. In this context, a posi- tive threshold implies that the agent suffers a loss from each processed encounter, and so the opportunity cost of more processing time is reduced. The agent delays the next encounter in order to mitigate the effect of the next positive threshold. In this context, a negative threshold implies that the agent receives a gain from each processed en- counter, and so the opportunity cost of more processing is increased.
At this heightened cost, the agent cannot afford to spend more time processing when future negative thresholds are left to be encountered. In particular, Equation 1. So possible solutions come from the dark upper frontier in Figure 1. Just as in Figure 1. So optimization of smooth frontiers is depicted as finding a point of deceleration that is tangent to a line with slope w. Two such lines are shown in Figure 1. Figure 1. The shaded area used in graph- ical rate maximization is also used in a ; however, the optimal TDNG behavior corresponds to the point of tangency with a line of slope w.
As discussed by Houston and McNamara , if w is set to the maximal value of Equation 1. So TDNG optimization is equivalent to the de- coupled optimization of the n versions of this expression. So optimization of each type has an identical structure as optimization of the aggregate. As shown in Figure 1. As shown by the dark line segments in Figure 1. The graphical optimization approach shows that efficiency maximization and rate maximization can have similar optimal solutions.
In these patch problems, Equation 1. In particular, if the processing cost functions are monotonically increasing with time, changes in the environment associated with increases in optimal-rate pro- cessing time will also be associated with increases in optimal-efficiency processing time. The efficiency defined by Equation 1. In this case, those currency conversions vary among types and the environment. Under the patch assumption, the cost-discounted gain CDG function in Equation 1.
As in TDNG optimization, optimization of each type can be decoupled from the other types. In a , excess-efficiency maximization is shown to be similar to excess-rate maximization. This result qualitatively matches what is expected for rate max- imization. In b , cost-discounted gain optimization is shown for one particular type. In the special case where processing costs are linear in time, optimization is depicted by Figure 1. So not only can each of the finite-event optimization objectives be optimized using similar methods, but they all have results that are qualitatively identical to classical rate-maximization results.
Hence, these optimization objectives can be used to model behaviors that do not perfectly fit the classical foraging model. Meanwhile, to motivate a graphical opti- mization method, we observe that because Equation 1. Each of these func- tions is an advantage-to-disadvantage function nearly identical to Equation 1. Here, we apply a more rigorous analysis approach. In particular, we describe the math- ematical structure of smooth objective functions at points of optimality.
In Sec- tion 1. However, to highlight the salient features common to all of those algorithms, we first connect the characterization of a generalized task-processing optimum to the popular algorithms used in OFT-type applications Section 1. Finally, we summarize the application of algorithms from Section 1.
We give conditions that guarantee that a behavior is a strict local maximum of Equation 1. If the optimization objective is strictly convex, these conditions describe its unique global maximum. The conditions in Equa- tions 1. So Equations 1. If Equation 1. Here, we show how the conditions in Equations 1. Elements of these two cases can be found in each of the generalized algorithms. In particular, task types are ranked by some generalized profitability and then partitioned into take-most and take-few sets, and processing times are found through some generalized marginal value theorem.
Prey model as optimal task-type choice: profitability ordering When applied to Equation 1. This relation- ship is depicted in Figure 1. This ordering does not depend on the encounter rates nor the cost of search. The profitability of each type is the slope of the dotted line connecting the origin to its gain, time - coordinate.
Patch model as optimal processing-time choice: marginal value When Equation 1. Then Equation 1. The resulting graph is exactly the situation described by Equation 1. Because each algorithm has different requirements than the others, one algorithm may apply to one task-processing scenario better than another. So after ordering the types appropriately, each algorithm finds an optimal pool size and a set of optimal processing times for that pool size.
These assumptions guarantee that the objective function has a maximum. That is, types are ranked by their extended generalized profitabilities. These assumptions are nearly identical to the ones in Section 1. Generalized Patch Algorithm The algorithms in Sections 1. Here, we give a generalized patch algorithm that can be used when each generalized profitability function comes from a certain class of decreasing functions.
So the magnitude of dj is nonzero and increasing everywhere on its interior. That is, the types can be partitioned into a set with unbounded profitabilities and a set with bounded profitabilities whose bounds can be ordered. These definitions represent a generalized marginal value theorem.
In particular, we apply methods from Section 1. In each case, task types are assigned indices ordered by decreasing maximum generalized profitability, and any interior optimal processing time will satisfy the generalized MVT condition. Comparing each row reveals features distinctive to each associated objective function, and noting similarities reveals important structural features of classes of objective functions.
In all cases, an optimal behavior will have types ranked by maximum generalized profitability and will meet the generalized MVT condition. The classical MVT condition in the first row states that the optimal processing time occurs when the instantaneous rate of gain in each patch drops to the long- term rate of gain. This feature is mirrored in generalized MVT conditions for the excess rate ER case in the second row as well as the excess efficiency EE case in the fourth row.
For all three cases, the optimal behavior for one task type is coupled to the optimal behavior for another task type due to the mutual effects on the environmental average. This feature is due to the presence of decision variables in the denominator of the corresponding advantage-to-disadvantage objective functions. However, the optimal times are modulated by a common environmental parameter.
In the TDNG and CDG cases, it is the discount factor w that represents the relative importance of gain maximization and time or cost minimization. Hence, in these two cases, the encounter rates and search cost have no impact on the optimal behavior. Thus, by fixing the discount factor, the opportunity cost of searching is also fixed and thus does not vary with the environment.
However, in the EoR case, even though optimal processing times can be determined independently, they all simultaneously respond to changes in search cost or encounter rates in a qualitatively similar way as the optimal processing times in the classical case. In fact, for the single-type patch case, the EoR and classical cases match. However, as shown in the second row of the table, in prey or general cases, the profitability ordering for the ER and classical cases will not match. In the patch case, higher success thresholds imply longer optimal processing times because of a greater premium on accumulating gain to reach the threshold.
Similarly, for the general ER case, higher success thresholds lead to a shift in profitability orderings toward task types with higher gain. The invariance of profitability ordering is a key result of classical OFT. Although the risk-sensitive foraging model of Stephens and Charnov  that is applied to an autonomous vehicle problem by Andrews et al. However, the ER maximization analysis above suggests that task types that a task-processing agent would specialize on at low thresholds may be excluded entirely from very high threshold cases.
A similar pref- erence reversal is also predicted by a stochastic dynamic programming analysis of foragers facing mortality i. As discussed in Section 1. Hence, the invariance of task-type ordering may be a result of the many-task long-run-time assumptions present in popular foraging models. In engineering applications where there are relatively few tasks or high success thresholds, bio-inspired task-type order- ing should be evaluated carefully. These simulations are similar to those by Andrews et al.
The three rows that correspond to each gain threshold GT show the mean and standard error of the mean SEM for total net gain and total time accumulated in each run as well as the percentage of runs where the total gain met or exceeded the success threshold. The particular numerical details of the simulation e. Because the simulation represents a task-choice problem i. Along with the five strategies used to generate Table 1. This strategy meets each of the four GT thresholds given, but each mission is much longer.
The four scenarios shown differ in their success threshold GT. Particularly notable GT rows have been emphasized in bold. In this case, the average total gain is then , and the average total time is time units. This simple strategy achieves all four success thresholds in less than a third of the time required for the strategy in item a that also is uniformly successful.
Both of the single-type strategies in items a and c are successful, but they depend upon a low search cost rate cs and a high encounter rate for their preferred task type. If the environment is relatively sparse in tasks of the desired type, the agent will engage primarily in costly searching as it ignores encounters with other types that may be more frequent.
A better strategy is to balance the benefits of waiting for more profitable types with the benefits of reducing costly search time. Additionally, reducing the time of missions allows mobile agents to be re-deployed more quickly thus increasing the value returned overall. An agent fol- lowing the take-all strategy does not discriminate; the agent processes every task encounter and the mission ends after exactly N encounters.
As this strat- egy does not depend upon the success threshold GT , its performance does not vary across different GT selections. The encounter rates could also be esti- mated as in Andrews et al. Because the CR strategy is based on a rate-maximization assumption, the CR strategy has a very high total-gain—total-time ratio. Hence, for missions limited by time as opposed to number of tasks, it would likely return relatively high gain.
However, when task-processing opportunities are limited, the strategy gives too much priority to task types with low processing times. Consequently, its task- choice priorities vary with GT. As with the CR strategy, the ER simulation here is performed with a priori knowledge of encounter rates to compare its performance with the estimated ecess rate eER strategy described below.
The heuristic makes no use of encounter rates. Instead, it compares the recognized profitability to the present total-gain—total-time ratio in order to determine whether an encountered task should be processed. The eCR strategy has similar performance as the CR strategy.
Consequently, its performance follows behind the performance of the ER strategy. Thus, the ER and eER strategies show that simple strategies exist that adapt to different mission success thresholds by waiting longer for high-gain tasks without depending on maximally long mission times.
Moreover, the intuitive nature and simple implementation of these strategies is ported from classical OFT through the generalized framework described in this chapter. We then introduce a single advantage-to-disadvantage optimization objective that generalizes several of the existing objectives used in OFT, and we also give four new models of finite-run-time optimality and show how each of them are special cases of advantage-to-disadvantage optimization. Each finite-run- time objective function includes a success threshold that mixes elements of classical rate maximization with shortfall minimization i.
Additionally, these four models provide optimization frameworks for the design of task-processing agents that can only engage in a finite number of tasks e. As we show in simulation, the general framework allows for the design decision-making algorithms with similar attractive structures as OFT-inspired algorithms but better performance in engineering applications.
We also show how a generic optimization framework provides a substrate on which different optimal task-processing behaviors can be compared. For example, our anal- ysis shows a relationship between rate and efficiency maximization, two approaches that are usually viewed in opposition to each other. These com- parisons reveal which key features of different optimization metrics can lead to vastly different behaviors in application.
Most applications of foraging theory to engineering focus on problems amenable to optimal prey i. However, Pavlic and Passino  apply foraging behaviors described by Gendron and Staddon  to a fixed-wing AAV that may also choose its search speed. In this case, the speed of the vehicle affects its detection accuracy. Increased speed increases the encounter rate with task types that are easy to detect but decreases the encounter rate with task types that are difficult to detect, and so predicting the optimal combination of task-type choice and search speed is nontrivial.
A further complication is that increased speed can have increased costs e. Pavlic and Passino are able to extend the methods of Gendron and Staddon from two task types to an arbitrary number of task types, but it comes at the cost of a simplistic model of detection accuracy. However, the optimization objective is an advantage-to- disadvantage function with an additional decision variable representing search speed.
Extending the methods described here to handle this case is a valuable future direction that should provide more insights into complex task-processing behavior e. To attain maximum gain in minimum time, optimal foraging theory OFT predicts that natural selection will favor behaviors that maximize the long-term rate of gain .
However, in operant binary-choice experiments in the laboratory [e. Conversely, as reviewed by Giraldeau and Caraco [38, pp. That is, natural selection has gone so far as to bestow on animals the ability to calculate rate-maximizing behaviors in real time. In this chapter, we propose a simple behavioral strategy consistent with real-time rate maximization in nature and show how it appears to be impulsive in laboratory binary-choice experiments.
Additionally, we show how rate maximization in the laboratory can be restored with appropriate pre-experiment treatment. Animals are assumed to discount future rewards by some decreasing function of the time until the reward. As long as the discounting function is sufficiently concave, a smaller-sooner reward can have greater value than a larger-later reward.
However, Stephens  argues that realistic discount rates will be too shallow to impact laboratory experiments. In particular, animals with relatively long lifetimes may value rewards today more than rewards tomorrow, but a difference in delay of less than a minute should not be enough to cause a prefer- ence reversal.
Additionally, Henly et al. To explain impulsiveness without discounting, recent attention focuses on how lab- oratory methods may artificially bias subjects toward impulsive behaviors. Arguing that impulsiveness is the result of an informational constraint, Stephens and Ander- son  and Stephens et al. To investigate state-dependent effects, Houston and McNamara  use a dynamic programming model to show that an im- pulsive strategy minimizes probability of starvation when an animal is in a low-energy state.
This result agrees with studies [e. Because food deprivation accompanies conventional operant methodology, this result may explain all observed laboratory impulsiveness; however, the model of Houston and McNamara is based on a sequential-choice assumption that is rarely met in the mutually exclusive binary-choice experiments used in most tests of impulsive- ness.
Here, we describe a simple behavioral mechanism that leads to impulsiveness under typical operant conditioning and rate maximization otherwise. Here, we summarize a central OFT result and recreate a useful graphical interpretation first presented by Charnov . Assuming that natural selection favors foraging behaviors that maximize lifetime gain, the optimal behavior should trade large reward per encounter for increased number of encounters i.
Each behavior is represented by a particular collection p1 , p2 ,. The profitability of each type is the slope of the dotted line connecting the origin to its gain, time -coordinate. As encounter rates of acceptable types increase or decrease, the dashed line rotates to exclude or include more types. In the example in Figure 2. If an environmental disaster causes encounter rates to fall sharply, then more- inclusive heritable preferences present in the background population will successfully invade and dominate future generations of the population.
So optimal foraging theory posits that rate-maximizing behaviors will be the adaptive outcome of the gradual process of natural selection. However, Giraldeau and Livoreil  show that birds in the laboratory respond to a sharp change in the environment with a new behavior that matches the long-term rate-maximizing behavior predicted by OFT. This response is consistent with real-time calculation of the optimal rate-maximizing behavior.
More- over, in a survey by Sih and Christensen  of 74 recent foraging studies, 22 studies are given that show animals flexibly adapting to their experimental scenarios in a way that is at least qualitatively consistent with optimal foraging theory. Thus, OFT may also be viewed as describing the proximate outcome of dynamic behaviors that adapt to changing environments. In the recent survey by Sih and Christensen  of 60 foraging studies originally surveyed by Stephens and Krebs  as well as 74 more recent studies, optimal foraging theory is shown to have mixed success in describing animal behavior.
However, even in herbivores [48, ] and molluscivores [94, ] that have essentially immobile prey, optimal foraging theory fails to predict the observed diet preferences. In these cases, the digestive rate model [DRM; 47, 49, ] is shown to better predict animal diet preference, but the DRM itself is the result of generalizing the prey model i. It maximizes the same rate expression from Equation 2. Depending on whether the material is a nutrient or a toxin, the intake rate X will have a lower or upper bound, respectively.
In the molluscivore shorebird examples described by van Gils et al. When the gizzard of one of these cropping birds is full, the bird must pause its foraging behavior until its gizzard has emptied enough for it to continue. Thus, these shorebirds have a maximum intake rate of ballast material that is not negligible. A key result of the DRM is that prey types should not be ordered by profitability as in Equation 2.
The DRM gives an algorithm to find the prey type whose quality partitions the group into take-always and take-never groups. Hence, even though the DRM is the result of rate maximization under a constraint, its result both re-orders prey preferences and appears to violate the zero—one rule. Although our main focus is to present a simple model of foraging that explains impulsiveness in the laboratory, we shall also show how our simplified model has similar qualitative features as the DRM.
In particular, our DRM-consistent behavioral model reconciles the differences between rate maximization and DRM using an internal handling time approach, which is similar to the ecological—physiological hybrid method described by Whelan and Brown . By accepting prey with super-R profitability and rejecting those with sub-R profitability, processed prey only cause the estimated R to rise. Between processed encounters, the R estimate falls due to the effect of increased time without increased gain.
We caution that this is a loaded interpretation. Among other things, it neglects the effects of non-foraging activities e. For example, it may be maladaptive for momentary hunger to drive foraging decisions, but it persistent hunger is an important signal that the forager should generalize more. Furthermore, we show later that a small adjustment to this model accounts for optimal foraging under digestive rate constraints. In that case, a stronger connection exists between energy state and R e.
Our algorithm is depicted in Figure 2. The solid line represents the trajectory of the gain of the forager over time. Figure 2. Encounters shown with a circle are chosen for processing and encounters shown with a diamond are ignored. Hence, jumps in the gain trajectory occur at circled encounters. The estimated rate R at each encounter is the slope of an imaginary line drawn from the origin to the gain trajectory at that time.
If the profitability of an encountered prey is steeper than the present R estimate, then the prey is processed, and the current R estimate increases; otherwise, the prey is ignored. Encounters are shown in this depiction as points on the gain trajectory. Those encounters that are processed are shown as circles, and encounters that are ignored are shown as diamonds.
So an encounter is processed if its graph location, which corresponds to the estimated R at the time of the encounter, is below the dotted profitability line of its task type. For this example trajectory, the first four encounters are chosen for processing because they fall below their corresponding profitability lines.
The fifth encounter is with a prey of type 3, and it is ignored because it falls above the third profitability line. Even though the sixth encounter is also with a prey of type 3, it is chosen for processing because enough time has passed to depress the estimated rate R.
The prey type corresponding to the switching line will also represent the prey that is only partially preferred. The other prey types will be preferred via the zero—one rule. This modified process is depicted in Figure 2. Coopting their ex- ample, we notice that in our model, as R increases e. Although this algorithm may not accurately described the proximate mechanisms for decision making in foragers with digestive constraints, it may provide helpful in- tuition for visualizing DRM results.
Additionally, if the ballast-time proportionality constant i. Hence, the previous model is a limiting case of this model; as ballast times become vanishingly small, the acceptance proportion of the partially accepted prey type approaches unity or nullity. Thus, this simple behavioral model reconciles differences between classical optimal foraging theory models and di- gestive constraint models.
Moreover, this model justifies the otherwise questionable use of energy availability e. In the case of pure rate maximization, an animal in a should only use its energy state as an estimator of a low R if its energy stores are persistently low. This assumption does not lack realism. A forager that searches for sparse stationary morsels of food may rarely encounter items simultaneously, and even when it does, its choices need not be mutually exclusive.
Here, we show how our real-time algorithm is impulsive under traditional laboratory conditions and maximizes long-term rate of gain under natural conditions. We also show how our DRM-inspired real-time algorithm results in foraging preferences consistent with results from the DRM. So type 1 has the highest profitability and type 2 has the lowest average handling time.
In this simulation, encounters with each type come from separate and independent Poisson processes, and so simultaneous encounters occur with zero probability. This assumption of biased attention is motivated by the training experiment of Siegel and Rachlin  and the suggestion by Monterosso and Ainslie  that impulsiveness may be the result of attention span and not deliberate choice.
Because this behavior eventually achieves the maximum long-term rate of gain, it is eligible for proliferation by natural selection. In Figure 2. Under these experimental conditions, the estimated rate is initially very low, and so the low-profitability prey are initially acceptable for processing. Because the forager attends to these prey first and the experimenter immediately removes the other prey, the new estimated rate R remains too low to exclude the next encountered low- profitability type.
In particular, the performance in b is optimal in an environment of only type 2 prey. If these operant laboratory conditions i. The trajectory in Figure 2. As a consequence, the trajectory rises very quickly and then plateaus off until the estimated rate of gain falls below the profitability of type 1. After that time, the forager follows a typical rate-maximizing trajectory. Similarly, the trajectory in Figure 2. As with the previous case, the gain rises steeply and then plateaus until the point where the estimated gain falls below the profitability of type 1.
After that time, the previously impulsive behavior follows a true rate-maximizing trajectory. That is, because the estimated rate is so high, the forager never attends to the low-profitability type; it ceases to appear impulsive. This restoration of rationality will rarely occur under con- ventional operant methods because those subjects are typically intentionally deprived of food outside of the experimental apparatus [e.
In both cases, the behavior is optimal. However, in these examples, the forager has unbiased atten- tiveness. That is, in the case of a simultaneous encounter with two prey, the forager attends first to one of them at random following a uniform distribution. However, when encounters are artificially made by an experimenter to be simultaneous and mutually exclusive, the foraging behavior is not guaranteed to maximize the long-term rate.
In particular, runs can be put into two groups characterized by Figures 2. In the former case of Figure 2. In the latter case of Figure 2. However, as with the impulsive behavior, a rate-maximizing result like the one shown in Fig- ure 2. Again, treating animals with pre-experiment feeding is uncommon in conventional operant laboratory methods because animals are starved to enforce compliance with the experimental apparatus.
All encounters are simultaneous and are generated by a Poission process with a 1. On simultaneous encounters, the forager randomly attends to one of the encountered prey. In the typical run shown in a , the gain trajectory converges to the rate-maximizing optimal result. In another typical run shown in b , the gain trajectory converges to a result congruent with rate maximization if encounter rates were halved.
The gain trajectory switches across the profitability line for type 2. Moreover, as shown in Figure 2. We have shown that animals that are impulsive in binary-choice experiments can still be rate maximizers in nature. In this example scenario, there is no search cost i. On simultaneous encounters which occur with probability 0 due to the merged Poisson process , the forager attends to the prey with lowest handling time first i.
The gain trajectory is shown in a , and the proportion of encounters that are accepted is shown in b. In the rare case when multiple prey are encountered at the same time, the animal arbitrarily attends to the one with shortest handling time first. However, because laboratory encounters are always simultaneous and mutually exclusive, the food-deprived animal fails to make good decisions.
To test this hypothesis, animals can be given a short ad libitum feeding period directly before testing. This feeding period should raise their gain—time ratios so that the purely impulsive choice is not a viable alternative. If this hypothesis is robust, then impulsiveness may not be a curiously irrational behavior so much as a behavioral polymorphism that is normally masked to the effects of natural selection. Addition- ally, this simple behavioral algorithm may provide intuition for understanding com- plex state-based animal behavior.
An important future direction is the testing of this theory in the laboratory; however, it is consistent with previous experiments [e. Additionally, the R process is is a discrete-time denumerable Markov chain, and it is likely that the convergence we have demonstrated in simulation can be shown analytically via mathematical proof.
To establish confidence that this algorithm is a robust rate maximizer, the stochastic convergence of R should be analytically char- acterized. A notable weakness of the central hypothesis of this chapter is that it takes for granted that the animal attends first to prey with short handling times when facing simultaneous encounters. If the forager only sees simultaneous encounters with a low-profitability short-time prey and a high-profitability long-time prey and always attends to the short-time prey first, it is as if the high-profitability prey has a null encounter rate.
Moreover, an equal-opportunity forager facing all simultaneous en- counters may also appear to be overly inclusive to short-time prey. During the period of time where its initial rate estimate is low enough to consider the low-profitability short-time prey, the forager forced to make mutually exclusive choices on each si- multaneous encounter will behave as if it were facing encounter rates that are halved and thus will continue to be overly inclusive.
It is true that a forager that attends first to high-profitability prey on simultaneous encounters will outperform others in binary-choice experiments. However, if simultaneous encounters are rare in nature, it is unlikely that natural selection will have shaped these attention mechanisms, and a wide variation of attention schemes may be present in a background population.
Fur- thermore, the assumption that animals exhibit their natural preferences when starved seems flawed regardless of the underlying behavioral mechanism. This chapter serves as an example of how starved preferences can be unique.
To construct digestive profitability, we defined an internal ballast time that is likely proportional to the material quantity used in DRM constraints. Hence, di- gestive profitability for each type is the ratio of its average gain to the sum of its handling time and ballast time. Like the approach of Whelan and Brown , this model achieves qualitatively similar results as DRM without a hard digestive rate constraint.
Like the short-time attentive rate-maximizing mechanism we introduced, this DRM-consistent mechanism may also be prone to maladaptive impulsiveness in the case of simultaneous encounters. However, our objective in presenting this simple DRM-consistent behavioral model is to suggest intuition for visualizing DRM results and reconciling differences between ecological and physiological schools of foraging thought. For example, our model shows clearly how a forager with high energy stores may prefer the less bulky of two items with equal profitability while a forager with low energy stores will accept both.
An important future direction is to precisely define what a ballast time is in nature. Techniques developed by Pyke et al. These techniques have been popularized by Stephens and Krebs  as the prey and patch models of classical optimal foraging theory OFT.
They respectively describe which prey foragers should include in their diet and how long foragers should exploit a patch of prey, which are the two central questions of solitary foraging theory. The rate-maximizing prey and patch models have mixed success in explaining behavioral observations in the field. The prey model accurately describes patterns of preference, and the patch model makes accurate predictions about how foraging durations should change when background parameters change, but the patch model has poor success in many cases when predicting actual foraging durations [69, , ].
In a review by Nonacs , several examples are picked that show foraging durations tend to be longer than expected by the classical patch model. However, recent studies show how observed behaviors that are inconsistent with classical OFT are indeed rate maximizing under an adjusted foraging model.
For example, shorebirds that appear to violate classical OFT are shown to be rate maximizers when explicitly modeling digestive constraints  and the value of information discovery . Additionally, we show how the same modifications lead to foraging theoretic explanations of the sunk-cost effect [10, 11, 56, ], which describes behaviors that invest more time in the more costly of two otherwise equivalent resources.
This effect is also known as the Concorde fallacy  because it describes an apparent logical fallacy analogous to the continued development of a supersonic jet that never returns a profit. How- ever, by giving a foraging theoretic explanation, we show that the fallacy is actually an optimal behavior. This explanation is also consistent with observations of ani- mals that commit longer feeding times after moving into areas where prey requires greater energy to acquire .
This chapter is organized as follows. First, in Section 3. Next, in Section 3. In Section 3. This analysis shows how explicitly including costs leads to longer patch residence times that are more consistent with observed behavior. We use similar graphical methods in Section 3. Finally, in Section 3. The forager pays a cost e. Assuming that natural selection favors foragers that maximize their lifetime gain, behaviors should trade increased reward per encounter for increased number of encounters in a lifetime.
That is, behav- iors that equate per-prey rate of gain with per-lifetime rate of gain will appropriately balance present reward with future opportunity cost so as to be optimal over a life- time. That is, if behaviors in nature could be described by the MVT, then every per-type rate of average gain i.
In the studies that Nonacs reviews, foragers stop exploiting different prey types at different speeds, which leads him to the conclusion that the MVT does not hold in reality. As discussed by Stephens and Krebs , the function gi is not meant to be a gross observable reward to the forager. Instead, it models the energetic reward to the forager minus the internal handling cost, which is usually not externally observable. From Equation 3. Differences in terminal handling densities across types reflect differences in the handling burden of those types.
For example, for types with the same reward—time curve and linear handling cost functions i. Equation 3. Each point in the shaded area corresponds to a particular choice of preferences, p1 , p2 ,. The upper boundary of the shaded area is the optimal frontier on which all optimal behaviors are found. An increase in the average handling time may be due to an increase in the optimal exploitation time for each patch i.
Specific op- timal diet content and patch exploitation times can be found using algorithms from classical OFT [23, 24, 80, ]. The graphical example in Figure 3. In both cases, to maximize lifetime reward, the forager must increase present gains to compensate for future search losses—the opportunity cost of more exploitation decreases when search time or cost increase. This prediction suggests an explanation for the ostensibly anomalous observation by Nolet et al.
The increased costs necessarily decrease the maximal long-term rate of gain, which is the opportunity cost of increased exploitation, and so patches are exploited longer. This effect is investigated in more detail in Section 3. In economics, the propensity to continue a costly task after paying some initial cost is sometimes called an escalation commitment, escala- tion behavior, or escalation error, but it is more commonly known as the sunk-cost effect [56, ].
This nomenclature is also used by psychologists [e. In fact, psychologists Arkes and Ayton  note that the sunk-cost effect is equivalent to the Concorde fallacy first described by biologists Dawkins and Carlisle . Al- though none of these terms are used, the same phenomena is also observed by Nolet et al. In every context, the observation of the sunk-cost effect is an enigma because intuition suggests that this behavior is suboptimal. Here, we show how optimization of Equation 3.
We also revert to interpreting g1 as the average net handling gain i. Similar arguments can be made about the general multivariate form in Equation 3. Consider the case when gain function g1 is initially: i negative i. Functions meeting items ii and iii are treated by Stephens and Krebs . How- ever, these functions are all initially zero.
For example, rather than treating recognition cost as a separate quantity [e. Alternatively, as in the case of the tundra swans observed by Nolet et al. The graphical optimization procedure for such a initially negative gain function is shown in Figure 3. The same gain function is shown in Figures 3. Figure 3. For two equally shaped gain functions, the one with the higher initial cost will also have the higher optimal residence time. This propensity to exploit items longer when the initial cost is increased is exactly the sunk-cost effect [10, 11].
However, it also maximizes the long-term net rate of gain, and so it is the rational behavior. This result can be explained using an opportunity cost interpretation [51, ]. When the initial cost of handling increases on all encounters, the total gain decreases, and so the opportunity cost for additional exploitation decreases e.
Thus, the marginal handling gain must decrease further in order for marginal costs and marginal benefits to equalize. This effect manifests itself through the increase in optimal exploitation time. By increasing exploitation time when initial cost increases, the forager reduces the amount of time it spends paying high costs. Leaving patches at an earlier time is equivalent to volunteering for paying recognition costs more frequently. It is not the increased initial cost on a single encounter that is important; it is the increased initial cost on all encounters that causes the decrease in opportunity cost for additional exploitation.
The model that the authors use to explain the tundra swan behavior predicts that there will be a decrease in feeding time in areas where feeding has larger power requirements. However, in deep water where the power requirements to up-end are apparently larger than the requirements to head-dip, the observed tundra swans spent longer times feeding on tuber patches.
Although this result is a contradiction of the model used by Nolet et al. For each patch type i, the average net gain function gi is assumed to be initially negative in order to model the energetic burden of acquiring the belowground tubers.
Hence, we use the classical marginal value theorem result in Equation 3. The curve in Figure 3. As the average entry cost of a patch type increases, the maximum rate decreases, and so average exploitation time increases.
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