I have been reading in Wikipedia about the Replicator equation, and I have a doubt about it.
The replicator equation can be stated as:
$$ \frac{d{x}_{i}}{dt}=\sum_{j=1}^{n}{x}_{j}{f}_{j}{q}_{ji}-{x}_{i}\bar{f} $$
where xi is the abundance of sequence i, q is the probability that i gives j, f is the fitness of the j sequence, and n is the number of generated sequences
The above formula is not so difficult to understand. In the part that I have the problem is when is mentioned that this equation is equivalent to the Lotka Volterra equation in n-1 dimensions. For what I know the Lotka-Volterra equation is:
$$ \frac{d{x}_{i}}{dt}={x}_{i}[{f}_{i}(\mathbf{x}-\bar{f}] $$
where x is a vector of sequences, and fi is the fitness of the distribution of the population.
Now in Wikipedia says that the replicator equation can be used to describe the generalized Lokta-Volterra equation which is:
$$ \frac{d{x}_{i}}{dt}={x}_{i}{f}_{i}(\mathbf{x}) $$
$$ \mathbf{f}=\mathbf{r}+A\mathbf{x} $$
where the bold parts are vectors and A is the linear transformation of the Lotka-Volterra Equation at the equilibrium point.
Now they use the following transformation in the replication equations to change to the Lotka-Volterra in n-1 dimensions:
$$ {x}_{i}=\frac{{y}_{i}}{1+y} $$
$$ {x}_{n}=\frac{{1}}{1+y} $$
$$ y=\sum_{i=1}^{n-1}{y}_{i} $$
where yi is the Lotka-Volterra variable
I don't understand how they make to use to transformations so that the replicator equation is similar to the Lotka-Volterra equation. Any help?
Thanks everybody
