Hi I'm self learning in game-theory and one of the most used equations is the replicator equation, which simulates how frequencies of a certain strategy change in a population of players.
$$\frac{dp_i}{dt}= p_i[\pi(e_i,p)-\pi(p,p)] $$
where $p_i$ is the proportion of individuals in the population using strategy $i$. $\pi(e_i,p)$ is the payoff to use strategy $i$ when population is its current state $p$. Lastly, $\pi(p,p)$ is the average payoff, a simple weighted average, summing the payoffs for each strategy, times their proportion in the population.
Many sources (e.g.) say that this equation is forward invariant when $0<p_i<1 \space \forall i $. However, they put no constrain in the value of payoff. There is also no indication the authors are using some information of the function $\pi()$.
However, I can see an example where this is not true. Let $p_i$ is very small (say 0.01) and the payoff for strategy i is very small compared to the rest of the population $\pi(e_i,p)-\pi(p,p)= -100000$. $\frac{dp_i}{dt}$ and $p_i$ will become negative, no?
What am I missing? How can the replicatior equation be forward invariant?