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by Rick O'Gorman (Submitted: 07/22/2014)

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This is a model based on evolutionary game theory. This theory was first applied to evolutionary processes by John Maynard Smith. Game theory is based on sub-groups of interacting agents, drawn from a meta-population, with certain payoffs occurring between the agents. These payoffs depend on the behavioral strategies of each of the interacting agents. In biology, the classic example is "doves and hawks" where two behavioral strategies exist in a population of organisms, the "dove" strategy, which is cooperative, and the "hawk" strategy, which is exploitative. When the agents encounter a resource, they can access it by working together. If two doves cooperate, then they share the resource equally, 0.5v (where v = value of resource). However, if a hawk and dove come together on a resource, the hawk grabs everything, so the dove gets zero and the hawk gets v. The catch comes if two hawks interact--they both try to grab the resource, fight at a cost (cost = c) and so, on average, get 0.5v-c.

Depending on the value of the resource, v, and the cost of fighting, c, hawks can go to fixation (eliminate the doves) or a stable polymorphism can exist, where the level of doves and hawks balances, though not necessarily at 50% each in the population.

In this model, I have also added a third strategy, "retaliator." Retaliators act like doves with actual doves and with other retaliators, but act like hawks with hawks. Retaliators thus have an advantage over hawks because they will only pay the cost, c, of fighting if they interact with a hawk but won't pay that cost if interacting with a dove or another retaliator. Hawks, however, will pay the cost c if they interact with another hawk or retaliator. A population with all three strategies can have a number of different outcomes, depending on whether doves go extinct.


Turtles wander from patch to patch in a somewhat random fashion (they change their heading plus or minus 45 degrees). Each move costs energy, but they can get energy from patches. If they arrive alone, then they get all the energy but if there is another turtle, then they get a payoff depending on thee type of turtle they are there with. Up to two turtles can occupy the same patch. If the energy of a turtle reaches a certain level, it reproduces asexually, and if its energy reaches zero, it dies.

The reason that a turtle gets energy if alone is that otherwise, in situations where there is a net cost for hawks interacting with other hawks and there are two strategies, hawk and dove, then the hawks would become fixed, and then all die!!!

Patches require a certain amount of time before they recover their resource value. This controls the population of turtles. Patches with resources avalable are green; they are a lighter color if their resources are not available.


Run the model with only doves and hawks. Start with equal numbers of each type, and with the value of the resource greater than twice the cost of fighting. The doves will go extinct.

Now try the same again, but have the value of resource greater than the cost, but less than twice as great. A stable polymorphism should result.

Next, set the cost of fighting greater than the value of the resource. The polymorphism is still stable, but the level has changed.

Finally, put some retaliators in the population, set v > 2c and run again. Now, there is an unstable polymorphism. Try different runs with the same settings. Vary the intial numbers of each type of organism.


Rick O'Gorman
Binghamton NY 13902-6000

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