= Cyclic imitation chain
{title2=$X^N$}
In a population of $N$ playing <rock paper scissors>, let losers copy winners. With pair-selection events of rate $(N-1)/2$, the proportion process has jumps $(e_1-e_3)/N$, $(e_2-e_1)/N$, $(e_3-e_2)/N$ at rates $Nx_1x_3$, $Nx_2x_1$, $Nx_3x_2$. Its <fluid limit> obeys $\dot x_i=x_i(x_{i-1}-x_{i+1})$, with indices interpreted cyclically. The <product of numbers> $x_1x_2x_3$ is conserved in this <ordinary differential equation>, whereas the finite chain eventually reaches a monochromatic absorbing state.
Back to article page
