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The authors have declared that no competing interests exist.

Wrote the paper: RC VK JB JG.

We develop a decision tree based game-theoretical approach for constructing functional responses in multi-prey/multi-patch environments and for finding the corresponding optimal foraging strategies. Decision trees provide a way to describe details of predator foraging behavior, based on the predator's sequence of choices at different decision points, that facilitates writing down the corresponding functional response. It is shown that the optimal foraging behavior that maximizes predator energy intake per unit time is a Nash equilibrium of the underlying optimal foraging game. We apply these game-theoretical methods to three scenarios: the classical diet choice model with two types of prey and sequential prey encounters, the diet choice model with simultaneous prey encounters, and a model in which the predator requires a positive recognition time to identify the type of prey encountered. For both diet choice models, it is shown that every Nash equilibrium yields optimal foraging behavior. Although suboptimal Nash equilibrium outcomes may exist when prey recognition time is included, only optimal foraging behavior is stable under evolutionary learning processes.

The functional response

Two modeling approaches have addressed the question of diet choice for a forager that searches for and then handles encountered prey items. The first is found in classic optimal foraging models. The forager's encounter probability or attack rate

In Pulliam

A second approach to diet choice is emerging from spatially-explicit models such as agent based models. A forager may move through a lattice or some form of continuous space. Prey items may occur at fixed locations or may also move through the defined space. The forager possesses some detection radius. Upon detecting a prey, the forager can choose to ignore the prey or attempt a capture. Such approaches lead to greater realism by considering the roles of space and individual contingencies. While they move through the same landscape, each individual forager becomes more or less unique based on its own personal history of movement, food encounters, and foraging decisions. Some individuals may experience unusually high or low harvest rates as a consequence of runs of good or bad luck, respectively. Like the classical models of diet choice, the foragers can still make optimal foraging decisions by deciding which encountered foods to handle or reject. The simulations can be run with a myriad of decision rules, and the performance of these rules can be compared. While a best diet choice rule may emerge from a particular scenario, the explicit nature of the agent based models may obscure the elegance or simplicity of the decision rule. Such agent based models may approximate more or less the optimal decision rules from the first approach to diet choice

Here we develop a decision theory approach to diet choice. We use an explicit decision tree to evaluate the costs and benefits of different choices. Such a decision tree has similarities to extensive form games from game theory

We consider three different scenarios based on the nature of searching for food and the ability to recognize a food's type upon encounter. In the first, search is undirected in terms of food type, but upon encountering a food item the forager instantly recognizes its type. This accords with the assumptions that generate Holling's two-food functional response and an “all or nothing” decision rule of food type acceptability. In the second the forager may encounter one prey of each type (called simultaneous encounter

In this section, we develop a decision tree method to derive the predator's functional response. The tree details the predator-prey interactions under consideration. We envision several prey types spatially distributed among many patches (that we will call microhabitats). The encounter events are then partially determined by the prey through their spatial distribution before the predator arrives. For instance, if prey are territorial, then the predator can encounter at most one solitary prey in a given microhabitat. At another extreme, if the different types of prey aggregate, then the predator can encounter different prey types at the same time. Thus, encounter events depend on the spatial behavior of the prey.

We break the predation process into different stages. A typical predation process has at least three stages that answer the following questions: 1. What prey (or types of prey) does the predator encounter? 2. What does the predator do in a given encounter situation (e.g. does the predator attack a prey, what type does it attack, etc.)? 3. Is the predator successful or not if it attacks? Here, we construct functional responses from the underlying decision trees based on three scenarios. This construction is, however, quite general and described fully in section Decision trees and the functional responses of

Suppose that there are two types of prey

Suppose the predator chooses a microhabitat to search at random, that it always finds the prey in this microhabitat if there is one, and that it takes a searching time

The first level gives the prey encounter distribution. The second level gives the predator activity distribution. The final row of the diagram gives the probability of each predator activity event and so sum to

For the first event when encountering prey

Let the predator's handling times of prey

Calculation of functional responses is based on renewal theory (for details, see section Decision trees and the functional responses of

Similarly, the functional response for prey

These are the functional responses assumed in standard two prey models (e.g.,

The predator's rate of energy gain,

Like others

The decision tree approach is reminiscent of games given in extensive form

To illustrate the approach, we make the decision tree of

Clearly, an optimal foraging behavior

Panel (a) assumes that

Since

In these more general games where the decision tree has more than 2 levels, there may be NE that do not correspond to optimal foraging behavior. However, so long as the number of encounter events at level 1 and predator activities remain finite, these decision trees generate the predator's energy intake rate and its functional responses on each type of prey. Game-theoretic equilibrium selection techniques

In this section, we again assume that there are two resource types (denoted as

At optimal foraging, two edges of this tree diagram are never followed. These are indicated by dotted lines in the tree. The reduced tree is then the resulting diagram with these edges removed.

The functional response can then be developed from the decision tree in

To find the optimal foraging strategy, we solve for the NE of the three-player game that assigns one player to each of the consumer decision nodes in

From these two results, the decision tree in

Thus, the optimal strategy is a NE of the two-player game corresponding to the reduced tree of

Similarly, the best response of player 2 when encountering only resource

Then

By Theorem 3 in section Zero-one rule and the Nash equilibrium of

In these plots, the energetic value

It is interesting to analyze dependence of the optimal strategy

Then

Panel A assumes a larger handling time of prey type A (

The more interesting case where prey A handling time is shorter than prey B handling time (

These results are also included in

It is particularly interesting to see what happens at the critical values

Similarly, when

These results can be partially explained through the patch choice model of

The functional response developed in section Decision trees and functional response for two prey types assumes the predator immediately recognizes the type of prey found during its search and then decides whether or not to attack it. In this section, we model the situation where the predator cannot distinguish the type of prey it encounters unless it is willing to spend extra “recognition” time

In the reduced tree, the dotted edges are deleted.

As in section Decision trees and functional response for two prey types, we assume that the two prey types are distributed among

On finding a prey in a microhabitat, the predator decides immediately whether to attack, move to another microhabitat to begin a new search, or spend recognition time to determine the type of prey encountered. Suppose these choices are taken with probabilities

If the predator decides to spend recognition time to determine the encountered prey is of type

The optimal predator foraging behavior corresponds to the maximum of

Since player 1 has strategy of the form

The reduced tree corresponds to a two-player game with strategy set

The NE of this truncated game is easy to determine when

For the remainder of this section, assume that the profitability of resource 2 is lower than is the mean energy intake rate obtained when feeding on the more profitable prey type only (i.e.,

To calculate the NE behavior, we proceed as in section Foraging with simultaneous resource encounters. From section The Nash equilibria of the prey recognition game of

Conversely, the best response of player 2 to a given strategy

That is

When recognition time,

Panel (a) assumes

For recognition time satisfying

For still larger recognition times,

The existence of suboptimal NE in the prey recognition game makes the interesting question considered briefly in section Decision trees and extensive form games even more important here; namely, how does the predator manage to learn its optimal behavior and avoid suboptimal equilibrium behavior. This type of question (on the so-called equilibrium selection problem

The evolutionary outcome is clear for all choices of parameters in the two diet choice models of sections Decision trees and the functional response for two prey types and Foraging with simultaneous resource encounters (see arrows in

The evolutionary outcome is also clear for the prey recognition game of this section when recognition time is large from

However, for short prey recognition time (i.e.

The situation depicted in

In summary, the optimal foraging behavior is selected in the prey recognition game as the NE component that is the stable outcome of the evolutionary learning process whether or not prey recognition time is short (i.e. for arbitrary

In this article, we develop a game-theoretic approach for constructing functional responses in multi-prey environments and for finding optimal foraging strategies based on these functional responses

Decision trees are often used in evolutionary ecology to describe possible decision sequences of individuals in biological systems

Dynamic programming (a form of backward induction) has also been used to find optimal foraging behavior

Instead, the approach we take in this article avoids such numerical methods by solving the game analytically. In this game, virtual players (also called agents) are associated with each decision point. These players are virtual because their payoff is derived from the functional response of a single individual only. Nevertheless, these players play a game because their decisions are linked, one player's optimal strategy depends on the other players' decision. We showed that solving this game by finding all the Nash equilibria will lead to the optimal foraging strategy. In those cases where some NE are not optimal foraging strategies, we showed it is easy to select the optimal ones among them by calculating their mean energy intake rate. Even when the game has infinitely many Nash equilibria that form a segment of a line (such Nash equilibrium components often arise in extensive form games), we showed that the energy intake rate at all these Nash equilibria will be the same. This means that once there are a finite number of isolated Nash equilibria points or Nash equilibrium components, finding the optimal strategy corresponds to comparing a finite number of values, which is trivial.

We documented these game-theoretic methods by applying them to three examples. The classic diet choice model with two prey types where predators encounter prey sequentially was considered first since it has been historically analyzed without game theory and yet provides an informative introduction to our new approach. Then we moved to a more complicated situation where a searching predator can simultaneously encounter both prey types

The last model discussed in this article examines whether a predator should spend time to recognize which type of prey it encountered before deciding whether to attack the prey or not

This method taken from evolutionary game theory to determine optimal foraging behavior differs from the more traditional approach based on the (modified) zero-one rule. This latter approach can be applied to the prey recognition game. Kotler and Mitchell

Game-theoretic methods play an important role in the traditional approach as well. Specifically, because the energy intake rate is the same at all points of the NE component, we need to compare only two numbers; the energy intake rate at any point of the NE component (the gray line segment in

For the three optimal foraging games modeled in this paper, the predator's encounter probabilities with different prey types do not change over the system's renewal cycle. In particular, there are no interactions among predators, such as competition for the same prey, that may alter the length of this cycle as the predator's behavior in these interactions changes. On the other hand, interactions among predators can be added to their decision trees. Our analysis of optimal foraging behavior through extensive form game-theoretic methods can then be generalized to the resultant multi-level trees, an important area of future research.

The first section of the Appendix, Decision trees and the functional responses, describes a general approach to construct functional responses from decision trees. The second section, Zero-one rule and the Nash equilibrium, generalizes the classical zero-one rule of the optimal foraging theory derived for the multi-prey Holling type II functional response to a more general functional responses. This section also shows how the zero-one rule relates to the Nash equilibrium of the underlying optimal foraging game. Appendix Foraging with simultaneous resource encounters derives the Nash equilibrium strategy (8), (10) and Appendix The Nash equilibria of the prey recognition game derives the Nash equilibrium (13), (14).

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Appreciated are the suggestions from the anonymous reviewer for improvements in the original version of the article.