Evolutionary model with recombination and randomly changing fitness landscape
We investigate the evolutionary model with recombination and random switches in the fitness function due to change in a special gene. The dynamical behaviour of the fitness landscape induced by the specific mutations is closely related to the mutator phenomenon, which, together with recombination, plays an important role in modern evolutionary studies. It is of great interest to develop classical quasispecies models towards better compliance with the observation. However, these properties significantly increase the complexity of the mathematical models. In this paper, we consider symmetric fitness landscapes for several different environments, using the Hamilton-Jacobi equation (HJE) method to solve the system of equations at a large genome length limit. The mean fitness and surplus are calculated explicitly for the steady-state, and the relevance of the analytical results is supported by numerical simulation. We consider the most general case of two landscapes with any values of mutation and recombination rates (three independent parameters). The exact solution of evolutionary dynamics is done via a solution of a fourth-order algebraic equation. For the more straightforward case with two independent parameters, we derive the solution using a quadratic algebraic equation. For the simplest case, when there are two landscapes with the same mutation and recombination rates, we derive some effective fitness landscape, mapping the model with recombination to the Crow-Kimura model.