Dynamics of infinite populations evolving in a landscape of uni- and bimodal fitness functions

It is known that the distribution of a population evolving in a landscape of a single adaptive hill is unimodal and centered at the optimum. Evolution of a population in a landscape of two adaptive hills is a more interesting example with consequences to theoretical interpretations and applications in optimization. Its formal analysis is not a trivial task. This paper considers theoretical aspects of evolution. A study of a very simple model of asexual phenotypic evolution is presented under assumptions of infinite populations and a one-dimensional search space. For an infinite population, the evolution of the population is equivalent to the evolution of a density function describing distribution of trials with a given fitness. The evolution of the density distributions is analyzed as the evolution of density parameters, means and variances, in the landscapes of unimodal and bimodal fitness functions. As a result, discrete-time recursive equations on parameters of density distribution in the next generation are obtained based on the parameters of the current generation. Of particular interest is the location of the mean of the stationary distribution and the dynamics of crossing a saddle between optima of the bimodal fitness function. Theoretical considerations are supported by simulations. The evolutionary process is able to localize the global optimum and to pass through saddles between optima. In particular, it is demonstrated that under certain conditions, the equilibrium distribution of traits can be multimodal.