Background The replicator equation with $n$ strategies is given by the differential equation: \begin{equation} \dot x_{i} = x_{i} \left( \sum_{j=1}^{n} a_{ij}x_{j} - \phi \right) \qquad i = 1, \ldots, n \end{equation} where $x_i$ is the frequency of the strategy $i$, $A=[a_{ij}]$ is the payoff matrix, and $\phi$ is the average fitness of the population.
Question: According to books on game theory (e.g., Nowak's "Evolutionary Dynamics"), if $x_i\neq 0$ at some time, $x_i$ might converge to zero but it will never reach zero. Why is that?