A simple evolutionary game arising from the study of the role of IGF-II in pancreatic cancer
Ruibo Ma and Rick Durrett
Abstract. 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 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 with Nd = the torus 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 λ ∈ (4/3,4). Producers take over the system when λ ≤ 4/3 and die out when λ ≥ 4. There are similar results for a "long-range" model in d = 2.
Preprint
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