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. Basically, what the replicator equation is saying is that the rate of increase of the strategy $i$ (i.e., $\dot x_i/ x_i$) is the difference between the fitness of $i$ (i.e., $\sum_{j=1}^{n} a_{ij}x_{j}$) and the average fitness $\phi$ of the whole population.
The Lotka-Volterra equation with $n-1$ species is: \begin{equation} \dot y_{i} = y_{i} \left(r_{i} + \sum_{j=1}^{n-1} b_{ij}y_{j}\right) \qquad i=1, \ldots , n-1 \end{equation} where $y_i$ is the abundance of species $i$, $r_i$ the intrinsic growth rate, and $b_{ij}$ describes how the species $i$ and $j$ interact (wiki entry on the Lotka-Volterra eqn).
According to Nowak's book Evolutionary Dynamics, these two equations are equivalent to each other (the same result appears in Hofbauer and Sigmund's Evolutionary Games and Population Dynamics). This is a neat result because it shows that results in ecology based on the Lotka-Volterra equation has a game theoretic interpretation and vice-versa.
Question
This would mean that the replicator equation for two strategies (e.g., the Hawk-Dove game) is equivalent to a single species interacting with itself under the Lotka-Volterra equation. How could that be? The replicator equation would describe the frequency of two strategies in the population, but the equivalent Lotka-Volterra equation would describe the evolution of a single and undifferentiated population. My problem is not so much with the proof of the equivalence between these two equations (Hofbauer and Sigmund give a proof of this equivalence in their book), but with the interpretation of the equivalence between the two eqns.
