Animal Behavior/Modeling
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[edit] 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 exptectations 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.
[edit] 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.
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
Optimal Foraging Theory examines the choice of food items: marginal value theorem, central place foragers, risk-sensitivity
[edit] 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.
| strategy ! | strategy 2 | |
|---|---|---|
| strategy A | 3, 3 | 2, -1 |
| strategy B | -1, 2 | 2, 2 |
A Payoff Matrix is used to formally represent a game that includes all conceivable strategies, along with each strategie'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).
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 A (player 1) is the same as strategy 1 (player 2) and strategy B of player 1 is the same as strategy 2 of player 2. The game can then be represented with a single payoff per cell, i.e., the payoff for the row player. For example, the payoff matrices on the right represents the same game as the matrix above, if 1 = A and 2 = B..
| strategy A | strategy B | |
|---|---|---|
| strategy A | 3 | 2 |
| strategy B | -1 | 2 |
Game theory aims to identify behavioral rules that form an Evolutionary stable strategy (or ESS). This strategy, if adopted by a population of players, cannot be invaded by any mutant strategy. An ESS is "evolutionarily" stable meaning that once it is fixed in a population, natural selection alone is sufficient to prevent alternative strategies from invading the population.
[edit] Prisoner's dilemma
[edit] 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):
| 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?
A hawk can invade a population of doves as its payoff (V) is greater than that of a dove pitted against other doves (V/2). Dove thus cannot be an ESS. Hawks will be able to invade a population of doves until encountering 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
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 costof 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 such conditions. 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
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.
[edit] Chicken, Brinkmanship, and War of Attrition Games
| swerve | straight | |
|---|---|---|
| swerve | 0 | -1 |
| straight | 1 | -10 |
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 both sides.
War of attrition scenarios refer to the situation where 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 indidivual backs down, relinquishing access to the contested resource, rather than continuing to sustain further insults.
