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+.c, while a cheater at x gives birth at rate .>1times the fraction of producers in x+.d and sends its offspring to a randomly chosen point in x+.c.We first study this game on the d-dimensional torus (.modL)d with .d=(.modL)d mod and .c= the 2d nearest neighbors.If we let L.. then t..the fraction of producers converges to 1/..In d.3 the limiting finite dimensional distributions converge as t..to the voter model equilibrium with density 1/..We next reformulate the system as an evolutionary game with .birth-death. updating and take .c=.d=..Using results for voter model perturbations we show that in d=3 with .= the six nearest neighbors, the density of producers converges to (2/.).0.5 for 4/3<.<4.Producers take over the system when .<4/3 and die out when .>4 with .=[.clogN......,clogN......]2 there are similar phase transitions, with coexistence occurring when (1+2.)/(1+.)<.<(1+2.)/. where .=(e3/(.c2).1)/2.