Dynamics of quasi-species models with a complex spin coherent state representation

Abstract

We investigate the dynamics of the parallel mutation-selection quasi-species model of biological evolution for various fitness landscapes. By using the semiclassical propagator for spin coherent states, for the linear fitness landscape, we find the expression for the transition rate from an arbitrary initial state to an arbitrary final state and expressions for the mean fitness and the surplus. For the sharp-peak fitness landscape, we find the solutions to the dynamics equation in two time regions and the crossover time which separates these two regions. Finally, we present a semiclassical method to derive analytic expressions for the dynamics of the mean fitness and the surplus for general symmetric fitness landscapes, and in case of the quadratic fitness landscape, we obtain the dynamics of the mean fitness consistent with previous results.