 Cyclic imitation chain

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 $N{x}_1{x}_3$, $N{x}_2{x}_1$, $N{x}_3{x}_2$. Its fluid limit obeys ${\dot{x}}_i=x_i(x_{i-1}-x_{i+1})$, with indices interpreted cyclically. The product of numbers $x_1{x}_2{x}_3$ is conserved in this ordinary differential equation, whereas the finite chain eventually reaches a monochromatic absorbing state.