# Animal Behavior/Modeling

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[1]. 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[2]. Predator-prey interactions that in turn influence population dynamics have been described in detail[3]

#### 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 decided 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, Northwestern Crows search 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 hight, 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. 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. [4]

### 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 3, 3 2, -1 -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 3 2 -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.

#### Prisoner's dilemma

Prisoner B remains silent ("cooperates") Prisoner B confesses (defects) A serves 1 month A serves 12 months A serves no time A serves 3 months

The Prisoner's Dilemma implements Game Theory to explore why individuals may cooperate in a given situation or why they may not. Concerned with lessening their time in jail, 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. If faced with a single instance of this decision, both players can thus reasonably be expected to defect on their partner resulting in a less than optimal outcome (3 months in jail). The greatest individual benefit to each (1 month in jail) would have come about, however, had they both cooperated. In scenarios where the same individuals are facing this decision repeatedly, cooperation will offer significant benefits.

#### 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 (V−C)/2 V 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?

υ[H,D] > υ[D,D]
V > V/2

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

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 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

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).

#### Chicken, Brinkmanship, and War of Attrition Games

swerve straight 0 -1 1 -10
##### Chicken

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

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.

## References

1. Huffaker CB. 1958. Experimental Studies on Predation: Dispersion factors and predator-prey oscillations. Hilgardia 27: 83
2. Kareiva P. 1987. Habitat Fragmentation and the Stability of Predator-Prey Interactions. Nature 326: 388
3. Janssen A. et al. 1997. Metapopulation Dynamics of a Persisting Predator-Prey system
4. Fretwell SD & Lucas HL. 1970. On territorial behavior and other factors influencing habitat distribution in birds. I. Theoretical Development. Acta Biotheoretica 19: 16-36