Modeling the rate of change of choosing probability for a group

Question

Suppose $x\in[0,1]$ and $1-x$ are weight factors (or say choosing probabilities) of two groups $G_1$ and $G_2$.The rate of change of $x$ is proportional to the payoff difference of the two groups, say $P=\pi(G_1)-\pi(G_2)\in[-1,1]$. I tried with $\frac{dx}{dt}=kP$, but this does not restrict $x$ in $[0,1]$. Any idea regarding this would be appreciated. Thanks.