Question

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Question:
In a species of deer, you find an allele that causes females to always produce an equal number of male and female offspring. Present an argument affirming that this sex ratio is an evolutionary stable strategy. Then, try to present a plausible scenario in this species in which this phenotype would not be an ESS.

Answer 1

The evolutionarily stable strategy is the one that cannot be replaced by another strategy. For example, in a population where a strategy is followed, all the individuals will get the payoff from that strategy. If a part of the population started using an alternate strategy, and if they do better than the rest, then they increase in number as time progresses and the actual original strategy is not considered as stable. But, if the new alternate strategy does not allow the small population to grow further, then the original strategy is considered as stable and called evolutionarily stable strategy (ESS).

If the population plays the role of ESS, then a mutation that makes a small part of that population implementing the alternate strategy will be eliminated. This is the example:

Example 1:

                Y                              X

2	2
1	1

X

Y

If the entire population is X type, then all of them get the full payoff. If the mutants like Y appear then they get the payoff of 1 against all the Xs they meet. If Y mutation has done bad, it cannot invade X. If the entire population started with Y all the members will get the payoff of 1. If the mutant X appears, then X will take over the Y and gets the payoff of 2. Ys only get the payoff of 0 against themselves.

In this example, the population of Y can be invaded by X while the population of X cannot be invaded by Y. Hence, X is the evolutionarily stable strategy in this gameplay.

Example 2:

                Y                              X

1	2
2	1

X

Y

Here, in the population of all Y (X=0, Y=100), if X is made as 1, the proportion of X seems to be increasing per generation. In the reverse situation, when X=100 and Y =1, then also the proportion of Y increases per generation.

In this example, Both X and Y are considered as evolutionarily stable strategies. Here, if the entire population of X is present it gets the payoff of 2, a rare mutant Y gets a payoff of 1 against all the Xs. If the entire population is Y it gets the payoff of 2 and the rare mutant X will get the payoff of 1 against all the Ys. So, X, as well as Y, are ESSs.