I'm attempting to use a forward Kolmogorov differential equation to model a birth-death process. This is fairly trivial when there's only one population, but I'm working with an unspecified and time-varying number of populations, so simply developing a system of equations doesn't seem viable.
I'm trying to solve for the dynamics equivalent to this paper by Desai and Fisher (genetics.org/content/genetics/176/3/1759.full.pdf), but using a slightly different approach. More importantly, I'm trying to extend their work, but can't seem to get the basic formatting down for solving their original system.
Supposing a set of populations each undergoing a separate birth-death process (with mutations feeding in from less fit populations to more fit ones) with fitness denoted as f, I have some set of population couts n(f,t) and probabilities of the a population being at that number at time p(n(f,t)).
My first and second attempts at solving this in Mathematica are below:
DSolve[Prob[n[f,t]]'==\[Mu]*n[f-1,t]*Prob[n[f-1,t]] + (f-Mean[f] - \[Mu])*n[f,t]*Prob[n[f,t]], Prob[n[f,t]],t]
DSolve[Prob[n,f,t]'==\[Mu]*n*Prob[n,f-1,t] + (f-Mean[f] - \[Mu])*n*Prob[n,f,t], Prob[n,f,t],t]
In words, the equations depict what happens as individuals from a population with fitness f-1 feed into the population of fitness f by acquiring mutations at rate Mu. Meanwhile, individuals from population of fitness f are growing or dying off at rate proportional to their fitness relative to the mean fitness, and are also being drained into the population of fitness f+1 at rate Mu.
I'm not sure how to get DSolve to solve for the general dynamics of all populations. Any assistance or advice would be greatly appreciated!