Analytic approach to the evolutionary effects of genetic exchange

We present an approximate analytic study of our previously introduced model of evolution including the effects of genetic exchange. This model is motivated by the process of bacterial transformation. We solve for the velocity, the rate of increase of fitness, as a function of the fixed population size, $N$. We find the velocity increases with ln $N$, eventually saturating at an $N$ which depends on the strength of the recombination process. The analytical treatment is seen to agree well with direct numerical simulations of our model equations.

Figure 1

(Color online) Comparison of the analytic approximation for the velocity at $\overline{x}=L∕2$ as a function of the cutoff $P_c$, Eq. (31), (“Third Order”) with the results of direct simulations of the time-dependent spatially discrete equation, Eq. (1). In addition, we show the lower order approximations, Eqs. (18) (“First Order”) and (24) (“Second Order”). The parameters employed are $L=1000,F=2.1$.

Figure 2

(Color online) Velocity vs the cutoff, $P_c$, when only birth and not death is excluded beyond the cutoff. For comparison, the velocity from the original model with both birth and death excluded beyond the cutoff, is presented as well. The results of simulation (circles) as well as the analytic WKB prediction (lines) are displayed. Parameters are $L=1000,F=2$.