Change of a continuous character caused by gene flow. An analytical approach

Recent results [Sznajd-Weron and P ekalski, Physica A 259 (1998) 457] have shown that the competition of two forces – gene flow and natural selection (GFS) – is responsible for three basic types of population structures. Using Monte Carlo simulations a critical value above which there is no gene flow was found. In this paper we apply Fourier analysis and the properties of convolution of probability density functions in order to find this critical value analytically. The obtained critical value for an arbitrarily dimensional system is the same as found in simulations for 1-, 2-, and 3-dimensional systems. We also show that in the investigated system there exists a possibility of a dynamical phase transition in one dimension. This leads us to the conclusion that different population structures observed in nature correspond to different ground states of the GFS model [Sznajd-Weron, P ekalski, Physica A 252 (1998) 336].