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The replicator-mutator equation is a non-local reaction-diffusion system

Posted Thursday December 28 2023.

I was reading about allopatric speciation, which occurs in evolution when two groups of members of some species become geographically isolated from each other. The evolutionary paths of the two groups diverge and eventually they are considered different species.

I guess some biologists can interpret geographical location as a genomic variable, or in the other direction, had generalized their view of genome to include geography. In that case, allopatric and sympatric speciation (which occurs without geographic separation) are really the same, just with different trait variables.

Then, I was reading this other paper where they modeled the evolution of population genotypes as a non-local reaction-diffusion PDE, and I saw the connection to the replicator-mutator equation, which I first heard about in this Youtube video, “Biology as information dynamics” by John Baez.

The reaction-diffusion equation is often written as follows, which I will call equation 1: \[ \frac{du}{dt} = \Delta u + f(u) \] I actually don’t like this notation, and prefer the following, which will be equation 2: \[ \frac{du}{dt} = \Delta u + f \circ u \] The difference is subtle, but in equation 2, f is applied pointwise to the field u - this corresponds to a local reaction term. This is usually the situation being described even when equation 1 is written.

In an evolutionary model, the fitness of each individual doesn’t just depend on the population density in a neighborhood of their genotype, it depends on the whole ecology, that is, the whole distribution of genotypes. This is actually better reflected by equation 1, where the reaction term depends, at each point in genospace, on the entire distribution of genotypes across the whole genospace.

I almost forgot to mention, \(\Delta\) is the Laplace operator, and models diffusion over the genospace as a result of mutation.