A simple evolutionary game arising from the study of the role of igf-II in pancreatic cancer
We study an evolutionary game in which a producer at x gives birth at rate 1 to an offspring sent to a randomly chosen point in x + Nc, while a cheater at x gives birth at rate λ > 1 times the fraction of producers in x + Nd and sends its offspring to a randomly chosen point in x + Nc. We first study this game on the d-dimensional torus (Z mod L)d with Nd = (Z mod L)d and Nc = the 2d nearest neighbors. If we let L → ∞ then t → ∞ the fraction of producers converges to 1/λ. In d ≥ 3 the limiting finite dimensional distributions converge as t → ∞ to the voter model equilibrium with density 1/λ. We next reformulate the system as an evolutionary game with “birth-death” updating and take Nc = Nd = N. Using results for voter model perturbations we show that in d = 3 with N = the six nearest neighbors, the density of producers converges to (2/λ) − 0.5 for 4/3 < λ < 4. Producers take over the system when λ < 4/3 and die out when λ > 4. In d = 2 with N = [−clog N, clog N]2 there are similar phase transitions, with coexistence occurring when (1 + 2θ)/(1 + θ) < λ < (1 + 2θ)/θ where θ = (e3/(πc2) − 1)/2.
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