# The replicator equation vs the Lotka-Volterra equation

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.

• Can you please define all the parameters in both equations? – Remi.b Sep 20 '15 at 15:45
• Sorry. I've just added a description of the parameters. – falsum Sep 20 '15 at 15:59
• It'd be great if one could already prove this equaltiy between the two models. Did M. Nowak referenced his claim? Do you have references also for the replicator and the Lotka-Volterra equations? The L-V equation seems a little weird to me. It doesn't quite match with what I found on wiki. Also, it feels weird to talk about $n-1$ species, rather than replacing all the $n-1$, by $n$ in the equation and make it valid for $n$ species. +1 – Remi.b Sep 22 '15 at 0:22
• Nowak refers to this result in his "Evolutionary Dynamics" but the proof appears in Hofbauer & Sigmund's "Evolutionary Games and Population Dynamics." Regarding the L-V equation, my understanding is that Nowak is referring to the generalized version of this eqn rather than the one used to study competition between species. I could follow the proof but, like you, I still find it very weird that we are mapping a game theoretic model with $n$ strategies to a L-V equation with $n-1$ species. It's one of those cases in which you understand the math but you cannot quite understand what is going on. – falsum Sep 22 '15 at 16:18

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 $x_i\equiv y_i/\sum_j y_j$.