A simple evolutionary game arising from the study of the role of IGF-II in pancreatic cancer

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 $x+\mathcal{N}_{c}$, while a cheater at $x$ gives birth at rate $\lambda>1$ times the fraction of producers in $x+\mathcal{N}_{d}$ and sends its offspring to a randomly chosen point in $x+\mathcal{N}_{c}$. We first study this game on the $d$-dimensional torus $(\mathbb{Z}\bmod L)^{d}$ with $\mathcal{N}_{d}=(\mathbb{Z}\bmod L)^{d}$ and $\mathcal{N}_{c}$ = the $2d$ nearest neighbors. If we let $L\to\infty$ then $t\to\infty$ the fraction of producers converges to $1/\lambda$. In $d\ge3$ the limiting finite dimensional distributions converge as $t\to\infty$ to the voter model equilibrium with density $1/\lambda$. We next reformulate the system as an evolutionary game with “birth-death” updating and take $\mathcal{N}_{c}=\mathcal{N}_{d}=\mathcal{N}$. Using results for voter model perturbations we show that in $d=3$ with $\mathcal{N}=$ the six nearest neighbors, the density of producers converges to $(2/\lambda)-0.5$ for $4/3<\lambda<4$. Producers take over the system when $\lambda<4/3$ and die out when $\lambda>4$. In $d=2$ with $\mathcal{N}=[-c\sqrt{\log N},c\sqrt{\log N}]^{2}$ there are similar phase transitions, with coexistence occurring when $(1+2\theta)/(1+\theta)<\lambda<(1+2\theta)/\theta$ where $\theta=(e^{3/(\pi c^{2})}-1)/2$.