The replicator equation vs the Lotka-Volterra equation
The key point is that the first equation is describing frequencies, i.e., $\sum_{i=1}^n x_i = 1$, so there are only $n-1$ degrees of freedom. For instance, if $n=2$ (as in Hawk-Dove), you can totally describe the state of the system with just $x_1$, because $x_2$ is just $x_2=1-x_1$. This constraint is enforced by adjusting $\phi$.

To convert the Lotka-Volterra model to the first equation, define $x_i\equiv y_i/\sum_j y_j$.