Review -- What is a Mixed ESS?

We previously considered examples of mixed ESSs in the Hawks and Doves game. There, we found that as long as I (the cost of injury) was greater than V (the value of the resource) then the solution was a mixed ESS.

We showed that in a mixed ESS, either

• an individual plays different behaviors with a certain probability (in which case we term the individual a "mixed strategist") or
• members of a population play various pure strategies at equilibrial frequencies.

So if we use "Hawks and Doves" as a familiar example:

• mixed strategist: an individual who plays hawk at some frequency (h) and dove at the frequency f (1 - h)
• or the population could consist of h individuals who played hawk and (1-h) who played dove.

As in any ESS, an individual who attempted to enter this population playing different hawk and dove frequencies from the evolutionarily stable mix would be selected against. Likewise if the evolutionarily stable population consisted of h hawks and (1-h) doves, any dove or hawk which entered by itself would tend to lower the fitness of itself and all other like strategists.

 Let's Compare A Mixed ESS to the War of Attrition with
the Hawk and Dove Mixed ESS We Considered Earlier

The only significant difference between a mixed ESS in the war of attrition and the Hawks and Doves game is one of complexity. In Hawks and Doves, there were only two strategies in the mix, each at a frequency that resulted in equal fitness for practitioners of both strategies. Any deviation from these frequencies resulted in a disequilibrium accompanied by a relative decrease in the fitness of the strategy whose frequency had increased above the equilibrium value; this difference in fitness drives the frequencies back to the equilibrium.

If there is an ESS to the war of attrition, then there must also be a point where the fitnesses of each strategy included in the mix are the same. The only real difference is that there are potentially:

• an infinite number of different supporting strategies (maximum costs)
• each (possibly) with its own unique frequency

This incredible difference in complexity means that we cannot find a mixed ESS in the war of attrition by setting two simple linear equations to being equal and then solving for frequency as we did with Hawks and Doves. Instead we will need to use a bit of calculus. But there is not reason to be scared -- all the relevant points will be fully explained. And following the steps should enhance your appreciation for the use of higher mathematics in biology and show how the tools of mathematics are invaluable to evolutionary biologists.