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Open Access
Peer-reviewed
Research Article
Game-Theoretic Methods for Functional Response and Optimal Foraging Behavior
* Ross Cressman,
Affiliation Department of Mathematics, Wilfrid Laurier University, Waterloo,
Ontario, Canada
x
* Vlastimil Krivan ,
* E-mail: [50]vlastimil.krivan@gmail.com
Affiliation Institute of Entomology, Biology Centre, Academy of Sciences of
the Czech Republic, and Faculty of Science, University of South Bohemia,
Ceské Budejovice, Czech Republic
x
* Joel S. Brown,
Affiliation Department of Biological Sciences, University of Illinois at
Chicago, Chicago, United States of America
x
* József Garay
Affiliation MTA-ELTE Theoretical Biology and Evolutionary Ecology Research
Group and Department of Plant Systematics, Ecology and Theoretical Biology,
Eötvös Loránd University, Budapest, Hungary
x
Game-Theoretic Methods for Functional Response and Optimal Foraging Behavior
* Ross Cressman,
* Vlastimil Krivan,
* Joel S. Brown,
* József Garay
PLOS
x
* Published: February 28, 2014
* [51]https://doi.org/10.1371/journal.pone.0088773
*
* [52]Article
* [53]Authors
* [54]Metrics
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* [57]Figures
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Figure 1
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Figure 7
Abstract
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.
Citation: Cressman R, Krivan V, Brown JS, Garay J (2014) Game-Theoretic Methods
for Functional Response and Optimal Foraging Behavior. PLoS ONE 9(2): e88773.
https://doi.org/10.1371/journal.pone.0088773
Editor: Robert Planque, Vrije Universiteit, Netherlands
Received: June 7, 2013; Accepted: January 16, 2014; Published: February 28, 2014
Copyright: © 2014 Cressman et al. This is an open-access article distributed
under the terms of the [58]Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.
Funding: Support was provided by NSERC of Canada, the Biology Centre
(RVO:60077344), and the Hungarian National Scientific Research Fund (OTKA
K62000,K67961). The funders had no role in study design, data collection and
analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
The functional response [59][1], [60][2] considers the number of prey (or
resource items) consumed by a single predator (or forager) as influenced by prey
abundance. By dictating the mortality rate of prey and the feeding rate of
predators, it is central to understanding consumer-resource dynamics [61][3],
[62][4]. Furthermore, the functional response can be extended to consider a
predator seeking two prey types [63][1]. Besides being more realistic for many
predators, functional responses on two food types create indirect effects between
the prey via the shared predator. For instance, if consuming a prey item takes
time or reduces motivation, then the presence of a second food type decreases the
forager's consumption of the first food type. Via the functional response, such
prey become indirect mutualists [64][5]. Conversely, short-term apparent
competition [65][6], [66][7] results if the presence of the second prey
encourages the predator to spend more time or effort searching for and capturing
prey. This happens when foragers bias their efforts towards areas rich in
resources. Regardless, the two-food functional response is central to
understanding diets, optimal foraging for multiple resources, predator mediated
indirect effects between prey, and population dynamics within food webs.
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 [67][3] is viewed as a mass action phenomenon between the predator and its
prey. The forager's overall encounter rate with prey is simply the product of
prey abundance and the predator's encounter probability on that prey. Upon
encountering a prey, the forager can elect to consume the prey at some handling
time cost, or reject the opportunity and continue the search for other prey.
Starting from Holling's [68][1] two-food functional response this approach has
generated increasingly sophisticated predictions.
In Pulliam [69][8] (see also [70][9]), a "zero-one" or "bang-bang" rule for diet
choice was derived. A forager should either always accept or always reject an
encountered food item. When encountered, the preferred food (based on a higher
reward to handling time ratio) should always be consumed. If searching for and
handling the preferred food type yields more (or less) reward than simply
handling the less preferred food, then the less preferred food should always be
rejected (or accepted) when encountered. Empirical support was encouraging but
equivocal [71][10]. Most foragers show a partial selectivity, they are neither
completely opportunistic nor completely selective. A number of mechanisms have
been proposed and modeled for why foragers sometimes only partially consume a
less preferred food; including food depletion [72][11], food bulk and digestion
limitations [73][12], complementary nutrients [74][13], local omniscience
[75][14], incorrect prey classification and sampling by predators [76][15],
[77][16], prey crypsis [78][17] etc.
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 [79][14].
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 [80][18],
[81][19]. Our goals are threefold. First, does an explicit consideration of
decision making recover the results from the classic "mass-action" models of diet
choice. Second, can these decision trees assist in uncovering the optimal
decision rules for agent-based foraging models. Third, what are the similarities
and differences between the decision tree of a forager and evolutionary games in
extensive form. To achieve these goals we imagine a forager that searches for and
handles food items of two types.
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 [82][20]), but can only handle one of
the items, the other being lost. For instance, these two prey may be together at
the same place competing over a common resource. Alternatively, the predator may
search a small area completely for any prey before deciding whether to attack. In
the third, we consider recognition time where the forager must expend additional
time if it wants to know the type of food that has been encountered prior to
handling.
Methods
Decision trees and the functional response for two prey types
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 [83]Appendix S1. We start
with a well known example that leads to the Holling type II functional response
for two prey types.
Suppose that there are two types of prey [journal.pone.0088773.e001] and
[journal.pone.0088773.e002] with fixed densities [journal.pone.0088773.e003] and
[journal.pone.0088773.e004] , respectively. We assume that these prey are
scattered randomly among [journal.pone.0088773.e005] microhabitats where
[journal.pone.0088773.e006] is much larger than the number of individuals (i.e.,
[journal.pone.0088773.e007] ). Thus, there will be at most one prey in each
microhabitat (i.e. the probability that there are two or more in some
microhabitat is negligible). Thus, the probabilities that a given microhabitat
has no prey is [journal.pone.0088773.e008] , exactly one prey
[journal.pone.0088773.e009] is [journal.pone.0088773.e010] and exactly one prey
[journal.pone.0088773.e011] is [journal.pone.0088773.e012] . These probabilities
are assumed not to change with time, which is the usual assumption when deriving
a functional response.
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 [journal.pone.0088773.e013] for it to determine whether a prey is
there or not. There are then three possible encounter events: the predator
encounters a prey of type [journal.pone.0088773.e014] , a prey of type
[journal.pone.0088773.e015] , or no prey at all. These events occur with
probabilities [journal.pone.0088773.e016] , [journal.pone.0088773.e017] and
[journal.pone.0088773.e018] respectively (see [84]Figure 1, Level 1).
[85]thumbnail
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Figure 1. The decision tree for two prey types.
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
[journal.pone.0088773.e019] . Since each entry here is simply the product of the
probabilities along the path leading to this endpoint, we do not provide them in
the decision trees from now on. With random prey distribution and
[journal.pone.0088773.e020] large, [journal.pone.0088773.e021] and
[journal.pone.0088773.e022] . If prey [journal.pone.0088773.e023] is the more
profitable type, the edge in the decision tree corresponding to not attacking
this type of prey is never followed at optimal foraging (indicated by the dotted
edge in the tree). The reduced tree is then the resulting diagram with this edge
removed.
[86]https://doi.org/10.1371/journal.pone.0088773.g001
For the first event when encountering prey [journal.pone.0088773.e024] , the
predator has two possible actions: Either "attack prey
[journal.pone.0088773.e025] " and "do not attack prey [journal.pone.0088773.e026]
". These actions occur with probabilities [journal.pone.0088773.e027] and
[journal.pone.0088773.e028] respectively (see [87]Figure 1, Level 2). Similarly,
in the second event when a predator encounters prey [journal.pone.0088773.e029] ,
the two possible actions of the predator are to "attack prey
[journal.pone.0088773.e030] " and "do not attack prey [journal.pone.0088773.e031]
" with probabilities [journal.pone.0088773.e032] and [journal.pone.0088773.e033]
respectively. For the third event, when no prey are found, the only predator
action is "do not attack" with probability 1. Altogether, there are five possible
predator activities, and these correspond to the five edges at Level 2 in the
decision tree of [88]Figure 1.
Let the predator's handling times of prey [journal.pone.0088773.e034] and
[journal.pone.0088773.e035] be [journal.pone.0088773.e036] and
[journal.pone.0088773.e037] respectively. The five predator activity events are:
encounter a microhabitat with prey [journal.pone.0088773.e038] and attack it;
encounter a microhabitat with prey [journal.pone.0088773.e039] and do not attack
it; encounter a microhabitat with prey [journal.pone.0088773.e040] and attack it;
encounter a microhabitat with prey [journal.pone.0088773.e041] and do not attack
it; encounter an empty habitat. The probability distribution of these activities
(i.e. the "activity distribution", [89][21]) in this order is
[journal.pone.0088773.e042] [journal.pone.0088773.e043]
[journal.pone.0088773.e044] [journal.pone.0088773.e045]
[journal.pone.0088773.e046] with duration times [journal.pone.0088773.e047]
[journal.pone.0088773.e048] [journal.pone.0088773.e049]
[journal.pone.0088773.e050] [journal.pone.0088773.e051] respectively. All this
information is included in the decision tree of [90]Figure 1. Also included in
this tree are the energy consequences ( [journal.pone.0088773.e052] ) to the
predator of each of the five activities.
Calculation of functional responses is based on renewal theory (for details, see
section Decision trees and the functional responses of [91]Appendix S1) which
proves that the long term intake rate of a given prey type can be calculated as
the mean energy intake during one renewal cycle divided by the mean duration of
the renewal cycle [92][20], [93][22]-[94][24]. A single renewal cycle is given by
a predator passing through the decision tree in [95]Figure 1. Since type
[journal.pone.0088773.e053] prey are only killed when the path denoted by
[journal.pone.0088773.e054] and then [journal.pone.0088773.e055] is followed, the
functional response to prey [journal.pone.0088773.e056] ,
[journal.pone.0088773.e057] , is given through [96]Figure 1 by
[journal.pone.0088773.e058]
Similarly, the functional response for prey [journal.pone.0088773.e059] is
[journal.pone.0088773.e060]
These are the functional responses assumed in standard two prey models (e.g.,
[97][9], [98][20], [99][25]) given in our notation. For instance, if we normalize
searching time so that [journal.pone.0088773.e061] , [journal.pone.0088773.e062]
can be rewritten in terms of prey density in the more familiar form
[journal.pone.0088773.e063] . As mentioned above, it is assumed that the
encounter rates, [journal.pone.0088773.e064] and [journal.pone.0088773.e065] ,
remain unchanged over the renewal cycle in that predation has negligible effect
on prey densities during this time. This occurs if, for example,
[journal.pone.0088773.e066] and [journal.pone.0088773.e067] are large or
[journal.pone.0088773.e068] is quite large and so predation is rare. Our decision
tree approach provides a mechanistic foundation to typical functional responses
assumed in the literature. In particular, it is obvious that the standard Holling
II functional response [100][2] given by [journal.pone.0088773.e069] is the
outcome for [101]Figure 1 when there is only one type of prey and the predator
always pursues every prey it encounters (take [journal.pone.0088773.e070] and
[journal.pone.0088773.e071] ).
The predator's rate of energy gain, [journal.pone.0088773.e072] , is given by
([102]Figure 1) [journal.pone.0088773.e073] (1)
Like others [103][9], [104][20], [105][23], [106][26], we assume that the forager
aims to maximize [journal.pone.0088773.e074] . This theory predicts that if the
two types of prey are ranked according to their "profitabilities" (i.e. their
respective nutritional values per unit of handling time
[journal.pone.0088773.e075] ), then the more profitable prey type is always
included in the diet. That is, if [journal.pone.0088773.e076] , then the optimal
foraging strategy is to attack all encountered prey [journal.pone.0088773.e077]
(i.e. [journal.pone.0088773.e078] ). Furthermore, the decision to attack the
lower ranked prey (i.e. prey B) satisfies the zero-one rule. Specifically,
[journal.pone.0088773.e079] (respectively, [journal.pone.0088773.e080] ) if its
profitability is greater than (respectively, less than) the nutritional value of
only attacking prey of type A (i.e. [journal.pone.0088773.e081] if and only if
[journal.pone.0088773.e082] ). The threshold value for including the less
profitable prey in the predator's diet depends only on the chances of
encountering the more profitable prey (i.e. only on the density of prey
[journal.pone.0088773.e083] ) since [journal.pone.0088773.e084] if and only if
[journal.pone.0088773.e085] where [journal.pone.0088773.e086] (2)[107][9],
[108][20], [109][23], [110][26].
Decision trees and extensive form games
The decision tree approach is reminiscent of games given in extensive form
[111][18], [112][19]. Because of this relationship between decision trees and
extensive form games, game theory can then be used to find the optimal foraging
strategy. First, we use the truncation method to eliminate those paths that
always yield suboptimal outcomes. When applied to [113]Figure 1, truncation
removes the dotted path of rejecting the opportunity to capture prey type A. It
is never optimal to reject the prey that offers a higher reward to handling time
ratio. But what of node B? For food B with a lower energy to handling time ratio,
we can find the optimal foraging strategy by analyzing the agent normal form
[114][19]. This method assigns a separate player (called an agent) to each
decision node. The possible decisions at this node become the agent's strategies
and its payoff is given by the total energy intake rate of the predator it
represents. When game theory is used to solve a single predator's decision tree,
all of the virtual agents have the same common payoff, and in a sense, these
agents engage in a cooperative game. The optimal foraging strategy of the single
predator is then a solution to this game.
To illustrate the approach, we make the decision tree of [115]Figure 1 into a
two-player foraging game. Player 1 corresponds to decision node A with strategy
set [journal.pone.0088773.e087] and player 2 to node B with strategy set
[journal.pone.0088773.e088] . Their common payoff [journal.pone.0088773.e089] is
given by (1). In an extensive form game, the payoff functions are linear in the
behavioral strategy choices of all players. For our optimal foraging games, these
payoffs are nonlinear functions and so are more similar to those found in
population games [116][27], [117][28]. As a game, we seek the Nash equilibrium
(NE). This is a pair of behavioral strategies [journal.pone.0088773.e090] , one
for each player, such that neither player can gain by unilaterally changing its
strategy. That is, [journal.pone.0088773.e091] (3)for all
[journal.pone.0088773.e092] and [journal.pone.0088773.e093] . In game-theoretic
terms, [journal.pone.0088773.e094] is a NE if [journal.pone.0088773.e095] is a
best response of player 1 to [journal.pone.0088773.e096] and
[journal.pone.0088773.e097] is a best response of player 2 to
[journal.pone.0088773.e098] .
Clearly, an optimal foraging behavior [journal.pone.0088773.e099] (
[journal.pone.0088773.e100] for all [journal.pone.0088773.e101] and
[journal.pone.0088773.e102] ) corresponds to a NE since it satisfies (3). Solving
the game (i.e. finding the NE) for the classic foraging model of two types of
prey is straightforward. Since [journal.pone.0088773.e103] for all
[journal.pone.0088773.e104] and [journal.pone.0088773.e105] the behavioral
strategy of player 1 to attack (i.e. [journal.pone.0088773.e106] ) strictly
dominates all its other options (i.e. [journal.pone.0088773.e107] ) and so, at
any NE, player 1 must play [journal.pone.0088773.e108] . The NE strategy of
player 2 is then any best response to [journal.pone.0088773.e109] (i.e. any
[journal.pone.0088773.e110] that satisfies [journal.pone.0088773.e111] for all
[journal.pone.0088773.e112] ). A short calculation yields
[journal.pone.0088773.e113] (4)where [journal.pone.0088773.e114] is given by (2).
These results are shown in [118]Figure 2 where NE are indicated by solid circles
(panels (a) and (c)) and by the solid line segment on the right edge of panel
(b). In this latter case (i.e. when [journal.pone.0088773.e115] ), every point on
this vertical edge [journal.pone.0088773.e116] is a NE and the entire edge forms
a NE component (i.e. a maximal connected set of NE, cf. [119][19]). Thus, at this
critical encounter rate with the more profitable prey type, the zero-one rule of
optimal foraging which states that a given resource type in a given patch is
either always consumed when encountered or never consumed, must be modified
because the optimally foraging predator preference for the alternative prey type
can be anywhere between 0 and 1.
[120]thumbnail
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Figure 2. Qualitative outcomes of the optimal foraging strategy for the classical
foraging model (1) with two prey types as a function of the encounter probability
with the most profitable prey (i.e. of [journal.pone.0088773.e117] ).
Panel (a) assumes that [journal.pone.0088773.e118] in which case the optimal
strategy and NE is [journal.pone.0088773.e119] In panel (c),
[journal.pone.0088773.e120] and the optimal strategy (and NE) is
[journal.pone.0088773.e121] The arrows in each panel indicate the direction of
increasing energy intake per unit time at points in the unit square. For
completeness, the figure also includes the threshold case, panel (b), where
[journal.pone.0088773.e122] (i.e. the density of [journal.pone.0088773.e123] prey
is at the switching threshold). Although this case is rarely considered by
ecologists, its inclusion here is important to understand the optimal outcomes in
our more complicated models. In panel (b), the optimal strategy is
[journal.pone.0088773.e124] where [journal.pone.0088773.e125] , corresponding to
the solid right-hand edge of the unit square that forms a set of NE points.
[121]https://doi.org/10.1371/journal.pone.0088773.g002
Since [122]Figure 1 is a two-level foraging game, Theorem 3 of section Zero-one
rule and the Nash equilibrium of [123]Appendix S1 implies that the NE given by
[124]Figure 2 (i.e. by [journal.pone.0088773.e126] and
[journal.pone.0088773.e127] given by (4)) completely characterize optimal
predator foraging behavior. [125]Figure 2 also indicates the direction of
increasing energy intake per unit time at points in the unit square. This
suggests yet another connection to game theory; namely, how does the predator
learn its optimal behavior? This question is commonly studied in evolutionary
game theory [126][19], [127][29] where individual behaviors evolve in such a way
that strategies with higher payoff become used more frequently. By following the
flow of increasing payoff in the figure, it is clear from [128]Figure 2 that such
an evolutionary process will automatically lead to optimal predator behavior. We
will return to this question in section Game theory and evolutionary outcomes for
the prey recognition game where the evolutionary outcome is not so clear.
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 [129][30] based on evolutionary outcomes can then be used to discard
suboptimal NE behaviors and select only those NE corresponding to optimal
foraging behaviors as we will see in the final example that includes prey
recognition effects (see section Prey Recognition Effects).
Results
Foraging with simultaneous resource encounters
In this section, we again assume that there are two resource types (denoted as
[journal.pone.0088773.e128] and [journal.pone.0088773.e129] ) but, unlike section
Decision trees and the functional response for two prey types, some microhabitats
can contain a mixture of both types (denoted as [journal.pone.0088773.e130] ). In
this case, we assume that the consumer can forage for at most one resource type
in any encounter event. Other microhabitats can be resources free. Furthermore,
let [journal.pone.0088773.e131] , [journal.pone.0088773.e132] and
[journal.pone.0088773.e133] respectively be the proportions of these
microhabitats that contain only resource [journal.pone.0088773.e134] , only
resource [journal.pone.0088773.e135] prey and both resources
[journal.pone.0088773.e136] respectively. Finally, let
[journal.pone.0088773.e137] be the proportion of microhabitats that contain no
resources. If the consumer chooses a patch at random, the distribution of
encounter events is given by Level 1 of [130]Figure 3.
[131]thumbnail
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Figure 3. The decision tree for the simultaneous encounter game.
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.
[132]https://doi.org/10.1371/journal.pone.0088773.g003
[133]Figure 3 also contains the distribution of consumer activity events under
the assumption that the consumer is always successful when it decides to forage a
resource that it encounters. In the predator-prey interpretation, this means the
predator kills its prey whenever it attacks. As discussed in the final paragraph
of section Decision trees and the functional responses of [134]Appendix S1, our
decision tree approach to optimal foraging is also applicable when the attacking
predator is only successful with a certain probability that may depend on the
type of prey. Here [journal.pone.0088773.e138] (respectively,
[journal.pone.0088773.e139] ) is the probability the consumer forages for the
resource when it encounters only resource type [journal.pone.0088773.e140]
(respectively, type [journal.pone.0088773.e141] ). Also
[journal.pone.0088773.e142] (respectively, [journal.pone.0088773.e143] ) is the
probability the consumer forages type [journal.pone.0088773.e144] (respectively,
type [journal.pone.0088773.e145] ) resource when it chooses a microhabitat that
contains both types of resources and so [journal.pone.0088773.e146] is the
probability the consumer decides not to forage for either resource in this
encounter event.
The functional response can then be developed from the decision tree in
[135]Figure 3 that includes the searching and handling times as well as the
energy intakes of the different activity events. Proceeding as in section
Decision trees and functional response for two prey types, the functional
responses to resource type [journal.pone.0088773.e147] and
[journal.pone.0088773.e148] are given by [journal.pone.0088773.e149]
(5)respectively, where [journal.pone.0088773.e150] . Thus the total consumer
energy intake per unit time is [journal.pone.0088773.e151] (6)
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
[136]Figure 3. As shown in section Foraging with simultaneous resource encounters
of [137]Appendix S1, the behavior strategy to consume resource
[journal.pone.0088773.e152] at node A strictly dominates all other actions of
this player (i.e., [journal.pone.0088773.e153] for all
[journal.pone.0088773.e154] ), as we assume that resource
[journal.pone.0088773.e155] is more profitable to the predator than resource
[journal.pone.0088773.e156] (i.e. that [journal.pone.0088773.e157] ). It is also
shown there that any behavior strategy at node AB whereby a resource is not
always consumed (i.e. [journal.pone.0088773.e158] ) is strictly dominated. Thus
[journal.pone.0088773.e159] and [journal.pone.0088773.e160] at any NE.
From these two results, the decision tree in [138]Figure 3 can be truncated by
deleting the two edges indicated by dotted lines. With this change, the consumer
energy intake rate [journal.pone.0088773.e161] becomes
[journal.pone.0088773.e162] (7)where now [journal.pone.0088773.e163] .
Thus, the optimal strategy is a NE of the two-player game corresponding to the
reduced tree of [139]Figure 3. In this two-level foraging game, player 1
corresponds to decision node AB with strategy [journal.pone.0088773.e164] and
player 2 at node B with strategy [journal.pone.0088773.e165] . Their common
payoff is given by (7). From section Foraging with simultaneous resource
encounters of [140]Appendix S1 the best response for player 1 that encounters
both prey types simultaneously given the current strategy of player 2 is
[journal.pone.0088773.e166] (8)where [journal.pone.0088773.e167] (9)
Similarly, the best response of player 2 when encountering only resource
[journal.pone.0088773.e168] is [journal.pone.0088773.e169] (10)where
[journal.pone.0088773.e170] (11)
Then [journal.pone.0088773.e171] is a NE if and only if this strategy pair
satisfies [141]equations (8) and (10). Thus, unlike section Decision trees and
functional response for two prey types, the NE behavior at one consumer decision
node depends on the behavior at the other.
By Theorem 3 in section Zero-one rule and the Nash equilibrium of [142]Appendix
S1, NE correspond to optimal foraging behavior. Thus, the optimal foraging
behavior depends critically on the values of [journal.pone.0088773.e172] and
[journal.pone.0088773.e173] . In particular, it is important to know whether
these values are between [journal.pone.0088773.e174] and
[journal.pone.0088773.e175] , less than [journal.pone.0088773.e176] or greater
than [journal.pone.0088773.e177] . For instance, suppose that
[journal.pone.0088773.e178] and [journal.pone.0088773.e179] . Then, from (10),
[journal.pone.0088773.e180] (since [journal.pone.0088773.e181] ) and so
[journal.pone.0088773.e182] by (8). In this case, the only optimal foraging
behavior is to consume [journal.pone.0088773.e183] whenever it is encountered and
to consume [journal.pone.0088773.e184] only when it is not encountered
simultaneously with [journal.pone.0088773.e185] . In general, we observe that (i)
if [journal.pone.0088773.e186] , then [journal.pone.0088773.e187] , and (ii) if
[journal.pone.0088773.e188] , then [journal.pone.0088773.e189] . These
inequalities constrain the number of possible optimal strategies to
[journal.pone.0088773.e190] . These are the possible optimal strategies among the
vertices of the unit square in [143]Figure 4. As we will see, at certain
threshold parameter values, two of these vertices can both correspond to optimal
behavior. In this case, all points on the edge between these vertices correspond
to optimal behavior as well. In particular, the case where
[journal.pone.0088773.e191] can never occur because the two necessary conditions
[journal.pone.0088773.e192] and [journal.pone.0088773.e193] are excluded by (i)
and (ii). This intuitive result predicts that if the less profitable resource
type [journal.pone.0088773.e194] is consumed when encountering both types (i.e.,
if [journal.pone.0088773.e195] which implies that [journal.pone.0088773.e196] ),
it will always be consumed when encountered alone. Moreover, there is no interior
optimal strategy (i.e. there is no NE whereby the consumer exhibits partial diet
choice when encountering both resource types simultaneously as well as partial
diet choice when encountering resource B on its own) since this requires that
both [journal.pone.0088773.e197] and [journal.pone.0088773.e198] be strictly
between [journal.pone.0088773.e199] and [journal.pone.0088773.e200] , which does
not happen for any parameter choice.
[144]thumbnail
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Figure 4. All qualitative outcomes of the optimal foraging strategy (8) and (10)
with parameters [journal.pone.0088773.e201] .
In these plots, the energetic value [journal.pone.0088773.e202] of resource B
varies in the interval from [journal.pone.0088773.e203] to
[journal.pone.0088773.e204] (i.e. [journal.pone.0088773.e205] ). The critical
values of [journal.pone.0088773.e206] are [journal.pone.0088773.e207] ;
[journal.pone.0088773.e208] ; [journal.pone.0088773.e209] ;
[journal.pone.0088773.e210] . The arrows in each panel indicate the direction of
increasing energy intake per unit time at points in the unit square. In each case
shown, these arrows lead to a single vertex indicated by the filled in circle
which corresponds to the optimal foraging behavior (and unique NE). (a) For
[journal.pone.0088773.e211] , [journal.pone.0088773.e212] and
[journal.pone.0088773.e213] . Thus [journal.pone.0088773.e214] and
[journal.pone.0088773.e215] . (b) For [journal.pone.0088773.e216] ,
[journal.pone.0088773.e217] and [journal.pone.0088773.e218] . Thus
[journal.pone.0088773.e219] and [journal.pone.0088773.e220] . The dashed line
denotes [journal.pone.0088773.e221] As [journal.pone.0088773.e222] increases,
[journal.pone.0088773.e223] moves to the right until it coincides with the
vertical line [journal.pone.0088773.e224] when [journal.pone.0088773.e225] . At
this critical value of [journal.pone.0088773.e226] (not shown), all points
[journal.pone.0088773.e227] on this vertical line are optimal foraging strategies
(and NE). (c) For [journal.pone.0088773.e228] , [journal.pone.0088773.e229] and
[journal.pone.0088773.e230] . Thus [journal.pone.0088773.e231] and
[journal.pone.0088773.e232] . (d) For [journal.pone.0088773.e233] ,
[journal.pone.0088773.e234] and [journal.pone.0088773.e235] Thus
[journal.pone.0088773.e236] and [journal.pone.0088773.e237] . The dashed line
denotes [journal.pone.0088773.e238] (e) For [journal.pone.0088773.e239] ,
[journal.pone.0088773.e240] and [journal.pone.0088773.e241] . Thus
[journal.pone.0088773.e242] and [journal.pone.0088773.e243] .
[145]https://doi.org/10.1371/journal.pone.0088773.g004
It is interesting to analyze dependence of the optimal strategy
[journal.pone.0088773.e244] on the energetic value ( [journal.pone.0088773.e245]
) of the less profitability prey. We consider only those energetic values for
which B is the less profitable prey type (i.e., [journal.pone.0088773.e246] ). To
this end, we need to know the critical values of [journal.pone.0088773.e247] when
either [journal.pone.0088773.e248] or [journal.pone.0088773.e249] are equal to 0
or 1. Let [journal.pone.0088773.e250]
Then [journal.pone.0088773.e251] and [journal.pone.0088773.e252] . We divide the
analysis into two cases. For the first, assume that the handling time of resource
A is longer than or equal to that of resource B ( [journal.pone.0088773.e253] ).
As we assume that resource A is more profitable than B, it follows that the
energy content in food items must be larger in resource A (
[journal.pone.0088773.e254] ) and so [journal.pone.0088773.e255] by (9). Thus,
the optimal foraging strategy is to consume the [journal.pone.0088773.e256]
resource when both are encountered (i.e. [journal.pone.0088773.e257] ).
Furthermore, [journal.pone.0088773.e258] if [journal.pone.0088773.e259] and
[journal.pone.0088773.e260] if [journal.pone.0088773.e261] (i.e. the
[journal.pone.0088773.e262] resource is consumed when encountered on its own only
if its energy value is sufficiently high). The dependence of the optimal strategy
as a function of prey B energetic value is shown in [146]Figure 5A.
[147]thumbnail
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Figure 5. Dependence of the optimal foraging strategy [journal.pone.0088773.e263]
((8), solid line) and [journal.pone.0088773.e264] ((10), dashed line) on the
energy content of the less profitable prey type B.
Panel A assumes a larger handling time of prey type A (
[journal.pone.0088773.e265] , [journal.pone.0088773.e266] ), while panel B
assumes the opposite case ( [journal.pone.0088773.e267] ,
[journal.pone.0088773.e268] ). Other parameters [journal.pone.0088773.e269]
[journal.pone.0088773.e270] [journal.pone.0088773.e271]
[journal.pone.0088773.e272] [journal.pone.0088773.e273]
[148]https://doi.org/10.1371/journal.pone.0088773.g005
The more interesting case where prey A handling time is shorter than prey B
handling time ( [journal.pone.0088773.e274] ; [149]Figure 5B) is analyzed in
section Foraging with simultaneous resource encounters of [150]Appendix S1. When
energy content of prey B is smaller than [journal.pone.0088773.e275] the optimal
strategy is [journal.pone.0088773.e276] . For intermediate energy content
satisfying [journal.pone.0088773.e277] the optimal strategy is
[journal.pone.0088773.e278] For relatively large energy content
[journal.pone.0088773.e279] the optimal strategy is [journal.pone.0088773.e280] .
These results are also included in [151]Figure 4 that in addition provides the
direction of increasing energy intake per unit time at all points in the unit
square. In all cases analyzed in the previous two paragraphs, the outcome
satisfies the zero-one rule (i.e. either always consume a given resource type in
a given patch or never consume it) as suggested by [152][31].
It is particularly interesting to see what happens at the critical values
[journal.pone.0088773.e281] where [journal.pone.0088773.e282] , and
[journal.pone.0088773.e283] where [journal.pone.0088773.e284] These values
correspond to transitions (b)-(c) and (d)-(e) respectively in [153]Figure 4
because the dashed vertical (panel (b)) and horizontal (panel (d)) lines
respectively are then on the boundary of the unit square. Straightforward
calculations show that [journal.pone.0088773.e285] when
[journal.pone.0088773.e286] . Thus [journal.pone.0088773.e287] is independent of
[journal.pone.0088773.e288] and the optimal foraging behavior is any strategy
pair of the form [journal.pone.0088773.e289] for [journal.pone.0088773.e290] .
Similarly, when [journal.pone.0088773.e291] , [journal.pone.0088773.e292] and the
optimal foraging behavior is any strategy pair of the form
[journal.pone.0088773.e293] for [journal.pone.0088773.e294] . Once again, the
zero-one rule must be modified at these critical values. For instance, when
[journal.pone.0088773.e295] , resource B is always consumed under optimal
foraging when encountered on its own. However, if both resources are encountered
simultaneously, optimal foraging occurs for any preference for the less
profitable prey type. In section Zero-one rule and the Nash equilibrium of
[154]Appendix S1, the modified zero-one rule states that there is at least one
optimal foraging behavior where the corresponding NE is a pure strategy, i.e.,
where the predator preference for a prey is either 0 or 1. After such
modification the zero-one rule holds even at [journal.pone.0088773.e296] because
the (pure) strategy [journal.pone.0088773.e297] is optimal. This extension of the
zero-one rule applies to situations where optimal preferences for prey types as a
function of a parameter switch suddenly at some critical values from 1 to 0 or
vice versa.
These results can be partially explained through the patch choice model of
[155][31]. Specifically, since patches A and AB have the same maximum
profitabilities [journal.pone.0088773.e298] (which is higher than in patch B),
both are included in the consumer's diet. However, as shown by [156][31], this
does not mean that the most profitable resource is chosen in patch AB. From (9),
we see whether [journal.pone.0088773.e299] or [journal.pone.0088773.e300] depends
both on the ranking of [journal.pone.0088773.e301] and
[journal.pone.0088773.e302] profitabilities (the denominator in (9)) as well as
on the difference in energy gain [journal.pone.0088773.e303] per unit consumed.
When resource type A is both more profitable and also has a higher energetic
value ( [journal.pone.0088773.e304] ), or search time is short, then
[journal.pone.0088773.e305] and, consequently, [journal.pone.0088773.e306] i.e,
only resource A will be consumed in patches containing both resource types. Only
when type B is energetically more valuable than type A and either search time is
long enough, or the probability of encountering patch A and patch AB is low
enough, can resource type B be selected when both resources are encountered
simultaneously. In this case, resource B will also be consumed when encountered
on its own.
Prey recognition effects
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
[journal.pone.0088773.e307] beyond the time required to search the microhabitat.
That is, the predator has an option of paying this extra cost to gain information
on the prey type encountered before it decides whether to attack. This
information is said to be gathered in the facultative sense [157][32]. Kotler and
Mitchell [158][32] point out instances of facultative information that occur in
host-parasite and in mate selection models. They also discuss optimal foraging
when information is gathered in the obligate sense (i.e. the predator gathers
this information on prey type in the process of handling the prey and has the
option of rejecting it at that point). Although we do not consider the model for
obligate information in this paper, its decision tree (and analysis of optimal
foraging) is simpler than [159]Figure 6 for facultative information.
[160]thumbnail
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Figure 6. Decision tree for prey recognition game.
In the reduced tree, the dotted edges are deleted.
[161]https://doi.org/10.1371/journal.pone.0088773.g006
As in section Decision trees and functional response for two prey types, we
assume that the two prey types are distributed among [journal.pone.0088773.e308]
micohabitats with at most one prey in each. To ease notational difficulties, we
now label these prey types as species 1 and 2 with densities
[journal.pone.0088773.e309] and [journal.pone.0088773.e310] respectively and
nutritional values [journal.pone.0088773.e311] and [journal.pone.0088773.e312]
respectively to the predator. If the predator chooses a microhabitat at random,
the encounter event distribution (see [162]Figure 6) is the same as in
[163]Figure 1 (with our change of notation). In particular,
[journal.pone.0088773.e313] [journal.pone.0088773.e314] and
[journal.pone.0088773.e315] .
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 [journal.pone.0088773.e316] [journal.pone.0088773.e317]
[journal.pone.0088773.e318] respectively (where [journal.pone.0088773.e319] ).
The horizontal dashed line in [164]Figure 6 joining these two encounter events
indicates that this decision must be made without knowing the type of prey. Thus,
in the terminology of extensive form games [165][19], the set of these two nodes
forms an "information set" of the predator and is represented by a single player
in the three-player game corresponding to [166]Figure 6. We remark that the two
nodes that form this single information set require only one player because, at
both nodes, the information available is the same (the information is that the
searching predator encountered a prey).
If the predator decides to spend recognition time to determine the encountered
prey is of type [journal.pone.0088773.e320] , then it must subsequently decide
whether to attack this prey or not with probabilities [journal.pone.0088773.e321]
and [journal.pone.0088773.e322] respectively (see the third level of [167]Figure
6). It is not necessary that [journal.pone.0088773.e323] . If we assume that the
predator is always successful when attacking a prey, the tree diagram is given in
[168]Figure 6 where [journal.pone.0088773.e324] and [journal.pone.0088773.e325]
are the handling times for prey of type 1 and 2 respectively. We also assume that
the time needed to recognize either type of prey is the same (i.e.
[journal.pone.0088773.e326] ). Proceeding as in section Decision trees and
functional response for two prey types, the functional response to prey type
[journal.pone.0088773.e327] is given by [journal.pone.0088773.e328] where
[journal.pone.0088773.e329] Thus the total predator nutritional value per unit
time is [journal.pone.0088773.e330] for fixed prey distribution
[journal.pone.0088773.e331] and [journal.pone.0088773.e332] .
The optimal predator foraging behavior corresponds to the maximum of
[journal.pone.0088773.e333] as a function of [journal.pone.0088773.e334]
[journal.pone.0088773.e335] [journal.pone.0088773.e336]
[journal.pone.0088773.e337] [journal.pone.0088773.e338] . This maximum is
considerably harder to determine than in section Decision trees and functional
response for two prey types. However, game-theoretic methods to solve for NE are
effective at simplifying the analysis. [169]Figure 6 is a three-player foraging
game with player 1 representing the predator decision at the two-node information
set at level 2 and players 2 and 3 assigned to the respective decision nodes at
level 3. From section The Nash equilibria of the prey recognition game of
[170]Appendix S1, any strategy [journal.pone.0088773.e339] of player 1 with
[journal.pone.0088773.e340] is strictly dominated and so any NE behavior of this
player must satisfy [journal.pone.0088773.e341] . Thus, at the optimal strategy,
the predator should never move to another microhabitat when it first finds a prey
since, by abandoning this prey, the predator wastes the time spent searching for
it. In contrast to section Decision trees and functional response for two prey
types where the predator could reject the prey type with low profitability on
first encounter, this is not possible here without rejecting the better prey type
as well (because upon an initial encounter the predator does not know the prey
type).
Since player 1 has strategy of the form [journal.pone.0088773.e342] for some
[journal.pone.0088773.e343] , we will denote the NE behavior of player 1 by
[journal.pone.0088773.e344] and assume that [journal.pone.0088773.e345] from now
on. Section The Nash equilibria of the prey recognition game of [171]Appendix S1
also shows that, if prey type 1 is more profitable than type 2 (i.e. if
[journal.pone.0088773.e346] as in section Decision trees and functional response
for two prey types, then the predator must attack any prey 1 that it recognizes.
We will assume this throughout this section. Thus, we will also assume that
[journal.pone.0088773.e347] in the decision tree of [172]Figure 6 and analyze the
truncated foraging game that eliminates the three edges indicated by dotted lines
there.
The reduced tree corresponds to a two-player game with strategy set
[journal.pone.0088773.e348] for player 1 at level 2 and
[journal.pone.0088773.e349] for player 2 representing the predator decision
whether to attack a recognized prey 2 at level 3. The energy intake rate is then
[journal.pone.0088773.e350] (12)where [journal.pone.0088773.e351] .
The NE of this truncated game is easy to determine when
[journal.pone.0088773.e352] . In this situation, the profitability of prey type 2
is at least as high as the nutritional value of only attacking prey type 1 when
there is no recognition time (i.e. [journal.pone.0088773.e353] ). In section
Decision trees and the functional response for two prey types (cf. [173]equation
(1)), the NE behavior of player 1 is then to attack any prey encountered and this
continues to be the NE strategy when recognition time is non-zero. Thus
[journal.pone.0088773.e354] at any NE and, in fact, all NE are of the form
[journal.pone.0088773.e355] for some [journal.pone.0088773.e356] (see section The
Nash equilibria of the prey recognition game of [174]Appendix S1 for the formal
derivation). This corresponds to the predator being opportunistic (sensu
[175][32]). Note that, if the predator immediately attacks an observed prey, the
decision whether to attack after recognizing the type of prey is no longer
relevant since this choice is never needed.
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., [journal.pone.0088773.e357] ). In this case, the
predator should consider whether to determine the prey type it encountered,
because including the less profitable prey type in its diet may decrease the mean
energy intake rate.
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 [176]Appendix S1, the best response of player 1 to a given strategy
[journal.pone.0088773.e358] of player 2 is [journal.pone.0088773.e359] (13)where
[journal.pone.0088773.e360]
Conversely, the best response of player 2 to a given strategy
[journal.pone.0088773.e361] of player 1 is [journal.pone.0088773.e362] (14)where
[journal.pone.0088773.e363]
That is [journal.pone.0088773.e364] is a NE foraging behavior if and only if it
satisfies (13) and (14). We remark that [journal.pone.0088773.e365] and
[journal.pone.0088773.e366] are both less than [journal.pone.0088773.e367] , and
[journal.pone.0088773.e368] when [journal.pone.0088773.e369] .
When recognition time, [journal.pone.0088773.e370] , is small (
[journal.pone.0088773.e371] ), [journal.pone.0088773.e372] is negative and
[journal.pone.0088773.e373] is positive ([177]Figure 7(a)). There are then two
possible NE outcomes; namely, attack any encountered prey immediately
corresponding to the NE [journal.pone.0088773.e374] where
[journal.pone.0088773.e375] (shown as the gray segment of the line in [178]Figure
7(a)) or never attack immediately and then only attack prey type 1 when
recognized (with NE [journal.pone.0088773.e376] , the solid dot in [179]Figure
7(a)). From section Zero-one rule and the Nash equilibrium of [180]Appendix S1,
the optimal foraging behavior must be a NE outcome but because our decision tree
has three levels, every NE may not be an optimal strategy. However, even in this
case finding all NE substantially simplifies the problem of finding the optimal
strategy, because it is now enough to evaluate function
[journal.pone.0088773.e377] given by (12) only at these NE points. Moreover, if
there is a NE component (such as the gray segment in [181]Figure 7(a)) the value
of [journal.pone.0088773.e378] at any point in this segment must be the same. By
evaluating [journal.pone.0088773.e379] and [journal.pone.0088773.e380] , we find
[journal.pone.0088773.e381] and so the optimal behavior is to never attack
immediately and then only attack prey type 1 when recognized. As recognition time
increases, [journal.pone.0088773.e382] increases and [journal.pone.0088773.e383]
decreases. However, the NE outcomes and optimal behavior remain the same as long
as [journal.pone.0088773.e384] . Optimal predator behavior is then either
described as being selective [182][32] or as being intentional [183][33].
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Figure 7. Qualitative outcomes of the optimal foraging strategy (13) and (14) for
increasing recognition time [journal.pone.0088773.e385] .
Panel (a) assumes [journal.pone.0088773.e386] for which
[journal.pone.0088773.e387] and [journal.pone.0088773.e388] . The optimal
foraging strategy is at [journal.pone.0088773.e389] (i.e. always pay the cost of
recognition and then never attack the less profitable prey type) and the NE
component (shown as the gray line segment) [journal.pone.0088773.e390]
(corresponding to the NE outcome of attacking immediately) is suboptimal. In each
of the other three panels, the (union of the) thick edges forms a strict
equilibrium set (SES, for definition see section Zero-one rule and the Nash
equilibrium of [185]Appendix S1) that is the globally stable evolutionary
outcome. Panel (b) assumes [journal.pone.0088773.e391] ,
[journal.pone.0088773.e392] and [journal.pone.0088773.e393] . The union of the
two edges [journal.pone.0088773.e394] and [journal.pone.0088773.e395] forms one
NE component corresponding to optimal foraging behavior. Panel (c) assumes
[journal.pone.0088773.e396] , [journal.pone.0088773.e397] and
[journal.pone.0088773.e398] . The edge [journal.pone.0088773.e399] forms a NE
component corresponding to optimal foraging behavior. Panel (d) assumes
[journal.pone.0088773.e400] for which [journal.pone.0088773.e401] and
[journal.pone.0088773.e402] The edge [journal.pone.0088773.e403] forms a NE
component corresponding to optimal foraging behavior. The arrows in each panel
indicate the direction of increasing energy intake per unit time at points in the
unit square. Other parameters [journal.pone.0088773.e404]
[journal.pone.0088773.e405] , [journal.pone.0088773.e406] ,
[journal.pone.0088773.e407] , [journal.pone.0088773.e408] and
[journal.pone.0088773.e409] .
[186]https://doi.org/10.1371/journal.pone.0088773.g007
For recognition time satisfying [journal.pone.0088773.e410] ,
[journal.pone.0088773.e411] while [journal.pone.0088773.e412] remains negative
and so all strategy pairs of the form [journal.pone.0088773.e413] and
[journal.pone.0088773.e414] are NE ([187]Figure 7(b)). Moreover, each corresponds
to an optimal foraging strategy since [journal.pone.0088773.e415] in this case.
For still larger recognition times, [journal.pone.0088773.e416] , thus
[journal.pone.0088773.e417] at any NE. When [journal.pone.0088773.e418] ,
[journal.pone.0088773.e419] and the corresponding optimal foraging behavior is
shown in [188]Figure 7(c), while for recognition time larger than
[journal.pone.0088773.e420] , [journal.pone.0088773.e421] ([189]Figure 7(d)). In
both cases, the NE strategy pairs are of the form [journal.pone.0088773.e422] and
these all yield optimal foraging behavior.
Game theory and evolutionary outcomes for the prey recognition game
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 [190][30]) is commonly studied in
evolutionary game theory where individual behaviors evolve in such a way that
strategies with higher payoff become used more frequently. There are several
standard models that examine the evolutionary outcome of these behaviors changing
over time [191][29], [192][34].
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
[193]Figures 2 and [194]5, respectively). These arrows indicate the direction of
increasing energy intake rate (e.g. in [195]Figure 4, this rate increases as
[journal.pone.0088773.e423] is used more frequently if and only if the vertical
arrow is pointing upward). In all cases, the predator learns to use the NE
strategy that corresponds to the optimal behavior for the foraging games of
[196]Figures 1 and [197]3 respectively. (This is true for [198]Figure 2b as well
since the arrows lead to some point on the vertical side of the unit square with
[journal.pone.0088773.e424] , all of which correspond to optimal behavior in this
threshold case when [journal.pone.0088773.e425] .)
The evolutionary outcome is also clear for the prey recognition game of this
section when recognition time is large from [199]Figure 7. Specifically, for
[journal.pone.0088773.e426] (panels (b), (c) and (d)), the predator will evolve
to a strategy on an edge consisting of NE points which correspond to optimal
foraging. In the language of evolutionary game theory, this set of NE forms a
globally stable set that attracts any initial predator behavioral choice as long
as behaviors evolve in the direction of increasing energy intake rate.
However, for short prey recognition time (i.e. [journal.pone.0088773.e427] with
[journal.pone.0088773.e428] in [200]Figure 7(a)), the NE
[journal.pone.0088773.e429] (corresponding to the optimal foraging behavior of
always spending the time to recognize the type of prey encountered and then only
attacking prey of type 1) may not be globally stable. If the predator initially
attacks recognized prey 2 with probability greater than
[journal.pone.0088773.e430] , behavior may evolve to a point in the suboptimal NE
component where [journal.pone.0088773.e431] with [journal.pone.0088773.e432]
(i.e. to a point on the gray line segment in [201]Figure 7(a)). That is, the
predator may become trapped at this suboptimal behavior, especially if evolution
increases the strategy of attacking immediately faster than it decreases the
strategy of attacking recognized prey 2 (i.e. if the arrow to the right in the
top half of [202]Figure 7(a) is much bigger than the one pointing down).
The situation depicted in [203]Figure 7(a) is remarkably similar to that of the
two-player extensive form Chain store game [204][19], [205][35] (also known as
the Ultimatum mini-game [206][36] or the Entry deterrence game [207][37]). In the
large literature on this game, it is often argued that the evolutionary outcome
will be the point [journal.pone.0088773.e433] ) since neutral drift near the
suboptimal NE component will inevitably lead at some time to the strategy choice
shifting to [journal.pone.0088773.e434] after which selection will quickly lead
to [journal.pone.0088773.e435] . To see this, consider a point on this gray line
segment. If the predator decides once in a while to spend some time to recognize
the type of prey it encounters, its strategy will move to the left of the
segment. As strategies with higher payoff are then to the right and down in the
vicinity of the line segment, it is likely that [journal.pone.0088773.e436] will
regularly decrease until it reaches the lower end of the segment. Any further
strategy experimentation on the part of the predator will lead to
[journal.pone.0088773.e437] , after which the only evolutionary outcome can be
[journal.pone.0088773.e438] . In terms of evolutionary game theory again, the
suboptimal NE component is not stable whereas the optimal NE is.
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
[journal.pone.0088773.e439] ). The analysis of optimal foraging theory for this
example illustrates anew the potential of game-theoretic methods to gain a better
understanding of issues that arise in behavioral ecology.
Discussion
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 [208][9], [209][20]. The approach here is
based on methods from extensive form games [210][18], [211][19]. The importance
of these game-theoretic approaches for functional response is two-fold. First,
decision trees similar to those used in extensive form games are a natural way to
describe details of predator behavior based on the sequence of choices the
predator makes at different decision points. This facilitates writing down the
corresponding functional response. Second, we show that optimal foraging behavior
that maximizes energy intake per unit time can be determined by solving the
underlying foraging game for its Nash equilibrium. We documented these game
theory methods through three examples: the classical diet choice model,
simultaneous encounter with prey, and a model in which recognition time is
considered. We remark that, although the calculation of the optimal foraging
behavior in the first example is straightforward, it is not as easy in the last
two cases where our game theory methods lead readily to the solution.
Decision trees are often used in evolutionary ecology to describe possible
decision sequences of individuals in biological systems [212][28], including
models of kleptoparasitism [213][38] and of producers and scroungers [214][39].
They have been used less often in connection with functional responses, even
though the predation process can be conveniently described by such trees (e.g.,
[215][26], [216][40], [217][41]). Optimal foraging behavior that maximizes animal
fitness is then often described as a sequence of single choices at each decision
node faced by the predator. Such outcomes are reminiscent of those found by
applying the backward induction technique to extensive form games that also
chooses one strategy at each decision node [218][18], [219][19]. However, there
are essential differences. Specifically, under backward induction, the optimal
choice at such a node depends only on the comparisons of payoffs along paths
following this node. Unfortunately, the time constraint in our foraging game
means that decisions at one node have payoff consequences as to what is optimal
at another node, a connection between these decision nodes of the tree that has
no counterpart in extensive form games. That is, the payoff concept for "foraging
games" such as [220]Figure 1 combines both the nutritional values and the
duration of each activity given at all the end nodes of the decision tree. On the
other hand, as shown in all three examples, the extensive form technique
connected to backward induction of forming the reduced decision tree by
truncating those paths corresponding to dominated strategies remains an effective
means of considerably simplifying the NE analysis.
Dynamic programming (a form of backward induction) has also been used to find
optimal foraging behavior [221][23], [222][42]. Specifically, the approach
developed by Houston and McNamara [223][23] shows that the optimal foraging
strategy must maximize the difference between the expected energy intake during a
single renewal cycle and the product of the mean optimal energy intake rate and
the duration of the cycle. This approach specifies the optimal choice at each
decision node provided the energy intake rate under the optimal strategy is
known. Since the optimal choice in one part of the decision tree then requires
knowing the overall optimal strategy, the solution is typically obtained by
numerical iteration.
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 [224][31], [225][43]. These authors
showed that under simultaneous encounter the predictions based on the prey
profitabilities (i.e., energy content over handling time) are not sufficient to
predict the optimal foraging strategy. In fact, the optimal foraging strategies
can be quite complicated as they depend now also on the relation between the
energy content in different food types. In particular, [226]Figure 5A shows that
when the less profitable prey type 2 contains also less energy than the more
profitable prey type 1, then the more profitable prey type 1 will be selected
when both prey types are encountered. However, when the energy content of the
less profitable prey type is large (but still small enough that prey type 2
continues to be less profitable), it will be preferred when both prey types are
encountered ([227]Figure 5B). The solid line in [228]Figure 5B shows the
preference for prey type A when encountered with prey type B. When this
preference switches from 1 to 0 above [journal.pone.0088773.e440] , predator
preference for prey type B when encountered with prey type A switches from 0 to
1. All possible optimal foraging strategies as a function of the alternative prey
type energy content are shown in [229]Figure 4. In particular, it cannot happen
that the less profitable prey type is included in the predator's diet when
encountered simultaneously with the more profitable prey type but not taken when
encountered alone.
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 [230][32]. This example is more complex for several
reasons, including the fact that the corresponding decision tree now has three
different levels (whereas the previous two examples are described by two-level
trees). While the NE corresponds exactly to the optimal strategy in two-level
decision trees, this is not the case here. When recognition time is small, we
show that there are NE of the optimal foraging game that lead to suboptimal
foraging ([231]Figure 7(a)). However, these suboptimal NE are easy to exclude by
equilibrium selection techniques borrowed from evolutionary game theory
[232][30]. Specifically, optimal foraging is always given by the unique NE
outcome that corresponds to the stable equilibrium point (or set of equilibrium
points) as the predator learns its optimal strategy (i.e. as its strategy evolves
in the direction of the arrows in [233]Figures 2, [234]4, [235]7).
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 [236][32] show that the zero-one rule yields just two
possible optimal outcomes: either complete opportunism or completely selective.
Instead of analyzing for the effects of increasing recognition time as we have
done, they concentrate on what happens when the abundance of the less profitable
prey increases (which, in our notation, means [journal.pone.0088773.e441]
increases). They emphasize the somewhat counterintuitive result that, with low
abundance, the less profitable prey is excluded from the diet. At intermediate
abundances it is included, and then with high abundance it is excluded again.
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 [237]Figure 7(a)) and the
energy intake rate at the other NE point [journal.pone.0088773.e442] . Our
analysis shows that, when recognition time is small, the optimal foraging
strategy is to always pay the extra time to recognize the encountered prey type
(i.e., never attack the encountered prey item immediately
[journal.pone.0088773.e443] , [238]Figure 7(a)) and to include it in the diet if
it is the more profitable prey type (i.e., not to include the alternative prey
type 2, [journal.pone.0088773.e444] ). As the recognition time increases, the
optimal foraging strategy is not to waste time recognizing the encountered prey
type ([239]Figure 7c, d). In this case, all encountered prey types are included
in predator's diet and so [journal.pone.0088773.e445] is not uniquely defined.
That is, since all encountered prey are immediately included in predator's diet,
the question whether to include the recognized prey type in the diet becomes
irrelevant and so the preference for the alternative prey type is any number
between 0 and 1.
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.
Supporting Information
[240]Appendix S1.
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).
[241]https://doi.org/10.1371/journal.pone.0088773.s001
(PDF)
Acknowledgments
Appreciated are the suggestions from the anonymous reviewer for improvements in
the original version of the article.
Author Contributions
Wrote the paper: RC VK JB JG.
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