Animal Behavior/Modeling

Modeling Adaptive Behavior

It is a daunting (or impossible) task to study behavior by simultaneously considering all significant factors and interactions that may impact it. An alternative method attempts to reduce the associated complexities by wrapping ones expectations into a set of mathematical abstractions based on a reduced number of significant variables. Resulting quantitative predictions are necessarily incomplete but are often less ambiguous and easier to test. The goal is to model an animals rational decisions of what action to take, given some information about the world. The player then faces consequences for the decision as a function of the action, the actions of others (if applicable), and the state of the surrounding environment. The player is expected to be rational by maximizing the expected utility.

Optimality Models

Optimality models quantitatively predict the consequences for a particular animal to behave in a certain way when pitted against the environment. It assumes that animals should behave so as to maximize their fitness. Optimality models consider separate, dependent, fitness benefits (B) and fitness costs (C) over a given range of values in a decision variable (i.e., the variable of interest). The solution to an optimality model attempts to find the point where the net benefits (i.e., B-C) are maximized. Individual solutions consider the consequences for a focal animal regardless of what strategies other individuals are relying on.

The main advantages of optimality models are in that they:

• help us clarify our assumptions
• force us to think about the information the animal has
• tie many different ideas together in one place
• generate testable predictions

Consequences of behavioral decisions are described as equations where the success of decisions of one individual do not depend on what decisions other animals make.

• Decision Variable is the variable that we aim to optimize (e.g., optimize speed to obtain the best mileage when driving from here to there)
• Currency refers to the criterion used to examine the outcome for different values of the decision variable (mileage)
• Constraints are factors that limit the relationship between decision variable and currency (e.g., speed limits)

Optimality models use calculus and logic to solve for a minimum or a maximum in a specific function f(x). This can be achieved by taking the derivative of the function df[x)/dt (i.e., to obtain the slope of the function) and setting it to 0. Such an approach can also be used to explore sensitivity to changes in values of the decision variable and to determine what types of information are relevant for the relationship. Precise predictions can be formalized for empirical testing

Predator - Prey Population Cycles

In order to study predation and population oscillations, Huffaker used mite species, one being the predator and the other being the prey. By modifying the spatial structure of the habitat, he could manipulate the population dynamics and allow the overall survival rate for both species to increase. He did this by altering the distance between the prey and oranges (their food), establishing barriers to predator movement, and creating corridors for the prey to disperse. These changes increased habitat patches and in turn provided more areas for the prey to seek temporary protection. When the prey would go extinct locally at one habitat patch, they were able to reestablish by migrating to new patches before being attacked by predators. This habitat spatial structure of patches allowed for coexistence between the predator and prey species and promoted a stable population oscillation model. Predator-prey interactions that in turn influence population dynamics have been described in detail

Optimal Foraging Theory

OFT examines the choice of food items. Such prey models (even though the prey doesn’t necessarily have to be another animal) look at decisions of basic diet selection. The animal searches for food, and finds potential prey items one at a time. It has to decide whether it should stop searching and eat the food it has found, or whether to ignore the food and keep searching. We assume that the animal is trying to maximize its currency (i.e., rate of energy intake measured in calories) in a particular time period. Each type of food takes a different amount of time to handle it during consumption. For instance, a Northwestern crow searches along the water line during low tide for large molluscs called whelks. When it finds one, it flies upward and drops it on the rocks to break it open. Smaller whelks are harder to break, and need to be dropped from a greater height, so they may be ignored. Also the content of certain kinds of nuts is more difficult to access than that of others.

E₁ and E₂ refer to the caloric energy that is contained within a prey of type 1 or 2 respectively. The handling time needed to access the caloric content for such prey items is h₁ and h₂. If the net energy gained per unit of time (or the rate of energy gain) for type 1 exceeds that for item 2

E₁/h₁ > E₂/h₂

then you would do well to eat prey of type 1 whenever you find it. If you find an item of type 2 you may consider rejecting it and to keep searching if the energy lost in searching for item 1 added to its handling time is less than what is gained from eating item 2 outright.

E₂/h₂ > E₁/(S₁ + h₁)

Depending on the relationships of this equation the predator should eat just prey 1 (and be a specialist) or it should eat both prey 1 and prey 2 (and be a generalist). This decision to specialize depends on the quality and abundance of S1, regardless of S2.

Marginal Value Theorem

The MVT considers an optimally foraging creature that exploits patchy resources as it must decide when to move on to the next patch. As it attempts to optimize a cost/benefit ratio, individuals will stay longer as the distance between patches increases, or when the environment as a whole is less profitable.

Ideal Free Distribution

An ideal free distribution describes the way in which animals distribute themselves among several patches of resources - ideally (with full complement of information about the distribution of the resource) and freely (without barriers to placing oneself at the preferred location). The theory states that individual animals will aggregate in various patches proportionately to the amount of resources available in each. So for example, if patch A contains twice as much food as patch B, there will be twice as many individuals foraging in patch A as in patch B.

Game Theory Models

A more comprehensive prediction of an animal's most profitable behavior may well require us to also consider what others are doing. A behavior may well be quite rewarding when it is rare in a population (e.g., deception) but may not be nearly as advantageous to its actor when it becomes common. Game theory models thus consider the value of frequency-dependent fitness benefits. Originally developed as a tool to predict rational human economic behavior, its application to many evolutionary problems has improved our understanding of situations where fitness consequences of a behavior depend on types and frequencies of behaviors exhibited by other animals within the population.

It is assumed that individuals represent players of a game in which all parties are aware of the rules, are consciously attempting to maximize their payoffs, and are attempting to predict the moves of their opponents. Players move simultaneously or at least do not observe the other player's move before making their own.

Pay-off Matrix

A Pay-off matrix for a hypothetical, normal form game that pits two individuals against each other. Player Red (rows) may choose strategy R1 or R2 when dealing with player Blue (columns), while the latter may either use strategy B1 or B2. The payoff matrix lists the color-coded consequences that each individual receives for any particular combination of strategies played.

	strategy B1	strategy B2
strategy R1	[3], [3]	[2], [-1]
strategy R2	[-1], [2]	[2], [2]

A Payoff Matrix is used to formally represent a game that includes all conceivable strategies, along with each strategy's corresponding payoff against any other strategy. The matrix of a game specified the payoffs that each player receives for the combinations of actions played. For example, if player 1 plays top and player 2 plays left, player 1 receives 4 and player 2 receives 3. In each cell, the first number represents the payoff to the row player (in this case player 1), and the second number represents the payoff to the column player (in this case player 2).

The simplified Pay-off matrix rewritten for a hypothetical, symmetrical, normal form game that pits two individuals against each other. As both player may choose strategy A or B, the payoff matrix lists the consequences that the row individual receives for all possible combinations.

	strategy 1	strategy 2
strategy 1	3	2
strategy 2	-1	2

A case of a symmetric game exists if the payoffs do not depend on which player chooses each action. In the example above, this is the case if strategy R1 (strategy 1 of player 1) is the same strategy as B1 (strategy 1 of player 2) and strategy R2 is the same as strategy B2. The game can then be represented with a single payoff per cell, i.e., the payoff representing the payoff for the row player. For example, the payoff matrices on the right represents the same game as the matrix above, if R1 = B1 and R2 = B2..

Evolutionary stable strategy

Game theory aims to identify behavioral rules that form an Evolutionary stable strategy (or ESS). Such a strategy, when adopted by a population of players, cannot be invaded by any mutant strategy. Thus once the stratgey is fixed in a population, natural selection alone is sufficient to prevent alternative strategies from invading the population as long as there are no new selective forces brought to bear. First published as a specific term in the 1972 book by John Maynard Smith, the ESS is widely used in behavioural ecology and economics, and has been used in anthropology, evolutionary psychology, philosophy, and political science.

Prisoner's dilemma

Payoff matrix for the classic example of a Prisoner's Dilemma in which two men (A and B) are accused of a crime and interrogated individually about their willingness to provide evidence for their partner's involvement|-

	B does not testify	B testifies
A does not testify	-1, -1	-12, 0
A testifies	0, 12	-3, -3

The Prisoner's Dilemma uses Game Theory to explore why individuals may cooperate in a given situation or why they may not. In this case the basic situation is adversarial but they may find common ground for working together anyway. This is the scenario ...

"Two men are arrested for committing a crime together, but there is insufficient evidence for a conviction. Following the separation of the two men, the police offer both the same deal - if he testifies against his partner and provides evidence for involvement in the crime (i.e., defects) he will derive a benefit at the other's expense. If the other remains silent (i.e., cooperates), the betrayer goes free while the cooperator receives the full sentence. If each player 'rats out' the other, both receive a somewhat reduced version of the full sentence. If both remain silent, however, both are sentenced to minimal jail time as they can only be prosecuted on a minor charge. Each prisoner must choose either to defect the partner (i.e., rat him out) or cooperate with him (i.e., remain silent). What should they do?"

As they are trying to limit their negative consequences (i.e., reducing their jail time) the logical decision leads each individual to betray the other as this decision offers the better one-off benefit regardless of what player B decides (i.e., in both columns does the second row value list a preferred payoff over the first row value). If faced with a single instance of this decision, both players can thus reasonably be expected to defect on their partner, resulting in the second worst possible outcome for both players. The greatest individual combined benefit (1 month of jail time on a lesser charge for both) would have come about, however, had they both cooperated. In scenarios where the individuals are facing this decision repeatedly, trust and cooperation will thus offer significant benefits to both players over a simple egoistic solution. The most successful strategy in repeated Prisoner's dilemma scenarios is a strategy of Tit-for-Tat where the players maximize their benefit by cooperating on the first round and subsequently play the strategy the other player used on the previous round.

Hawks and Doves Game

Game models have greatly increased our understanding of conditions under which ritualized displays may take the place of aggressive interactions. Maynard-Smith's approach involves resource-centered conflict between members of a group which will follow either a strategy of unrestrained use of weapons (hawk) or reliance on highly ritualized elements of fighting (dove):

Payoff matrix for the Hawk-Dove game. Hawk is a strategy where the individual fights until either it sustains an injury or the opponent retreats. The losing hawks suffers a cost for such injury (C). A Dove contests the interaction with displays but will retreat immediately if the opponent escalates with a use of its weapons. The winner of the encounter obtains the resource in question and gains control over its value (V). Interactions between two hawks result in the term 1/2(V-C) as hawks against hawks will win half of the time and lose half of the time. Moreover, hawks will also sustain an injury half of the time. In interactions with another dove, a dove will win half of the time and retreat without cost the other half of the time.

	Hawk	Dove
Hawk	(V−C)/2	V
Dove	0	V/2

The different cells of the payoff matrix represent combinations of strategies for the two players, where each would prefer to win, prefer to tie rather than lose, and prefer to lose over injury. Game theory can be used to address the conditions under which such strategies are evolutionarily stable (i.e., an ESS). First we ask if a population playing one strategy can be invaded by a few animals playing the other strategy. To examine whether dove is an ESS we examine whether a population of doves can be invaded by a hawk? Specifically we cask whether the utility (u) of a hawk in a population of doves exceeds that of a dove in this scenario.

υ[H,D] > υ[D,D] V > V/2

As long as the benefit of winning the resource (V) is greater than zero, a hawk will always trump a dove when pitted against other doves (V/2). Dove thus cannot be an ESS. Hawks will be able to invade a population of doves until instances in which a hawk encounters another hawk becomes a common event. To examine whether a dove can invade a population of hawks, we test whether its payoff is greater than that of a hawk when faced with a population of hawks. Formally this evaluates whether ...

υ[D,H] > υ[H,H] 0 > (V-C)/2 0 > V/2-C/2 0 > V-C

The conclusion on whether a dove will be able to invade a population of hawks will thus depend on whether V > C. If the resource value exceeds the cost of injury, the benefit to hawks will be positive (i.e., greater than 0), and a population of hawks cannot be invaded. Hawk thus represents an ESS under conditions where a valuable resource is at stake. However if V<C then payoff to hawks will be negative and doves can invade. Doves become increasingly good at invading a population of hawks as opportunities for damage increase. In fact a population of hawks would go extinct in situations where the cost of injury exceeds the resource's value. To obtain the proportion of hawk and dove strategies that stably coexist when C>V we examine at what proportion of Hawks (p) and Doves (1-p) the fitness of individuals playing either of the two strategies is equal

W[H] = p*((V-C)/2) + (1-p)*(V) equals W[D] = p*0 + (1-p)*(V/2) p*(V/2-C/2) + (1-p)*(V) = (1-p)*(V/2) p(V/2) - p(C/2) + V - pV = (V/2) - p(V/2) p(V/2) + p(V/2) - pV - p(C/2) = (V/2) - V - p(C/2) = -(V/2) pC = V p = V/C

Game theory thus provides a theoretical underpinning for why animals tend to only fight with great ferocity when a resource of great value is at stake. Fighting, for instance, is particularly intense in elephant seals where victorious males are able to monopolize a large area of beach which contains many females. In the great majority of instances, however, resources are rarely worth being injured over and competing individuals will resolve most conflict with ritualized displays.

Sequential Assessment Games

Fighting in juvenile American lobsters (Homarus americanus) begins with a series of threat displays. If opponents are evenly matched, the encounter progressively escalates through ritualized components of fighting, restrained forms of physical combat, and finally brief periods of unrestrained fighting where opponents may even inflict injuries on each other (Huber & Kravitz, 1995).

War of Attrition

In this model aggression two contestants compete for a resource of value V by persisting while constantly accumulating costs over the time t that the contest lasts. In War of attrition scenarios fighters attempt to grind down the opponent's defenses. There is no fixed cost associated with losing or contesting, but as the encounter wears on, each player accumulates incremental costs. A decision to give up arises when one individual backs down, relinquishing access to the contested resource, rather than continuing to sustain further insults.

The Game of "Chicken", Brinkmanship, and Nuclear War

Payoff matrix for a game of chicken.

	swerve	straight
swerve	0	-1
straight	1	-50

The game Chicken is a variant of the Hawks and Doves game. In it each player prefers not to yield to the other, but the outcome where neither player yields is by far the worst possible one for both players. It has its origins in the situation where two drivers head towards each other on a collision course: one must swerve, or both may die in the crash. The driver who does swerve is called a "chicken". The game's most important feature is an assured, and significant cost to the situation where both individuals chose conflict over conciliation. "Losing" by deciding to swerve is so trivial compared to the crash that occurs if nobody swerves, the reasonable strategy would be to swerve. Yet, knowing this one may well decide not to swerve, gambling that the opponent will be reasonable.

Insights gained on the predictability of behavior based on the presence of overwhelming punitive consequences have formed a central component of the cold war military doctrine of Mutual assured destruction, where the use of nuclear weapons by either side would assure the destruction of the aggressor along with its target. This theory of deterrence states that the deployment of strong weapons as a threat to the enemy is essential in order to prevent the use of the very same weapons. Success in discouraging either side from resorting to these weapons is only effective as it is credible to both sides that a chosing conflict would result in the worst possible outcome for all - nuclear annihilation of the aggressor resulting from a credible second strike capability.

References

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