A maybe wrong eco-evo idea?

I was never in theoretical ecology or really in evolutionary theory, but I've always had an interest. It's basically an open question how to deal with the interplay between the two. Ecology works on a much faster timescale than evolution, but the two both operate on a population-- so what are the true population dynamics?

I ran across https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10614967/ and it's almost like we had the same idea hit at the same time! I even wonder if maybe David Schwab went to the same APS March Meeting Session to get the idea. The basic idea is very simple: ecology is one set of equations, evolution is represented by another set of equations, and to understand the interplay between the two, you just couple the two equations. David Schwab and his co-author found the coupling-- mutations lead to new phenotypes, which then lead to a change in the ecology.

Interestingly, the way that I would approach the problem is very different. It basically boils down to a different model and a totally different analysis that leads to solutions that can approach the fully correct solution more and more accurately. The model I would use for evolution would be the Fisher-Wright model, and the model I would use for ecology would be a master equation version of the generalized Lotka-Volterra model. Then, since ecology operates on fast timescales, you could solve for the probability distribution over species number in the generalized Lotka-Volterra model first, and plug it into the evolutionary equations to get a full solution.

The generalized Lotka-Volterra model takes some explaining. Basically, in the generalized Lotka-Volterra model, there are three "chemical reactions": one birth, one death, and one death via eating by another species. If you write down the master equation for these reactions, you can use moment equations to find better and better approximations via a Maximum Entropy approach to the true probability distribution over the number of each species. The moment equations don't close, but you could artificially at some order just assume that the higher-order moments have an independence property of some sort. At the very least, you can get a Gaussian approximation to what the ecological equations might look like. Of course that's not correct, but maybe it's good enough. This avoids a thorny problem with pretending that the ecological equations solve quickly. In reality, the rate equation approximation to the master equation can yield chaos. This chaos goes away if you consider the fact that the number of species has to be a nonnegative integer. Whether or not these MaxEnt approximations are good is up to the field to decide, but you can get better and better approximations by including more and more moments.

The Fisher-Wright equations are well-known to take one generation and produce a binomial distribution over the next generation's population number with properties that depend on selection coefficients and mutation rates. So basically, the next generation is sampled, it settles down to something having to do with a MaxEnt approximation, and you plug the resultant MaxEnt approximation into the next generation (convolve it) to see the probabilistic updates. You can automatically assume that this is the right thing to do because of the separation of timescales.

Altogether, you essentially get a complete solution to the probability distribution over population number as a function of time.

This isn't my field, and my schizophrenic voices basically don't want me to do this project, so I won't, but I'm hoping somebody will. I don't think it's a terrible idea.