Lecture XII

Concepts to Develop

1) Stable polymorphism vs. oscillatory dynamics

or Hawk-Dove vs. Rock-Paper-Scissors

2) Diminishing returns and frequency dependent selection leads to stabilizing selection

The sex-ratio game

Local Mate competition

3) Questions for groups:

a) Why is the frequency of males:females so often very near to 50:50?

b) What must be true at the equilibrium sex ratio?

R - P - S

Consider a situation where the value of the resource is 1.

In a Rock - Paper game, where Rock and Paper are pure strategies, the payoff matrix looks as follows:

Clearly, Paper is an evolutionary stable strategy and can invade a population of Rock strategists when rare.

Now, let us consider the Rock - Paper - Scissors game. The payoff matrix looks as follows:

Clearly there is no ESS (look at diagonal elements and compare with column elements). Each strategy has a rare advantage when one of the others is common.

This is another example of frequency-dependent selection. The expected fitness of a strategy depends on the relative frequency of other strategies in the population.

The evolutionary stable state in this situation is 1/3 Rock, 1/3 Paper, and 1/3 Scissors. Unlike the Hawk - Dove game, where the population converged to this stable state, the Rock - Paper - Scissors game results in oscillatory dynamics around the attractor at equilibrium. Therefore, in the R - P - S game, evolution doesn't stop (frequencies of the strategies overshoot the attractor.

Are there data supporting R-P-S in nature?

(see lizard paper on reserve in the Life Sciences Library for figures and discussion)

Basically:

There are three different types of male lizards in the population

ORANGE: Aggressively defend territory against intruders. Can recognize other orange males and blue males.

YELLOW: Female mimic. Invades ORANGE territory (is not recognized by the orange lizard) and mates with females while the orange male is patrolling.

BLUE: Not as aggressive as ORANGE (i.e, lose battles to orange), but can recognize YELLOW males.

Females come to the territories to mate. Yellow males have an advantage over orange males (sneak in), blue males have an advantage over yellow males (beat up), and orange males have an advantge over blue males (beat up).

Data show that the frequency of male morphs in the population oscillate in the predicted direction over time (O --> Y --> B --> O ...)

Does this prove R - P - S? No, but it doesn't eliminate it as a possiblity.

Frequency-dependent selection usually means rare-advantage and is a form of stablizing selection.

The sex-ratio game

Frequency-dependent selection usually means rare-advantage and is a form of stablizing selection.

Think about sea urchins

a) assume 1 male can fertilize all females in the population

b) assume all eggs produced by females are fertilized

c) hence: male fitness is limited by access to eggs and female fitness is limited by access to resources (Bateman's principle)

Consider now the fitness of a rare mutant parent having a mutation that changes the number of sons and daughters it makes.

The fitness of the parent is equal to the number of daughters + the number of sons scaled by the number of females in the population relative to the number of males in the population.

parental fitness = # daughters + # sons (#females / #males)

At the equilibrium point for indivdual investment in daughters and sons (which manifests itself as the equilibrium sex ratio) what must be true?

Fitness gains through daughter and son production must be equal

For randomly mating, infinite populations, this occurs at 50:50 male:female.