Let's say there are two populations $P_1$ and $P_2$, with population profiles $\boldsymbol{x}_1$ and $\boldsymbol{x}_2$. Given an arbitrary $\boldsymbol{x}_1$, $P_2$ converges to an ESS equal $\boldsymbol{x}^*_2$. Given an arbitrary $\boldsymbol{x}_2$, $P_1$ converges to an ESS equal $\boldsymbol{x}^*_1$.
Is is reasonable to let $f,g$ be "reaction functions" and simultaneously solve for $\boldsymbol{x}_2 = f(\boldsymbol{x}_1)$ and $\boldsymbol{x}_1 = g(\boldsymbol{x}_2)$ to derive a "Nash equlibrium" $(\boldsymbol{x}_1,\boldsymbol{x}_2)$ like in the Cournot game? I am not sure about a solid game theoretical framework to do that.