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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

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I can't find the replicator equation that you stated in the Wikipedia article. Your equation is linear. In the Wikipedia article on the replicator equation I see $\dot x_i = x_i((Ax)_i-x^T A x)$, which is a nonlinear system. – user53153 Feb 26 '13 at 3:42

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