Statistical Mechanics Approximation of Biogeography-Based Optimization

Biogeography-based optimization (BBO) is an evolutionary algorithm inspired by biogeography, which is the study of the migration of species between habitats. This paper derives a mathematical description of the dynamics of BBO based on ideas from statistical mechanics. Rather than trying to exactly predict the evolution of the population, statistical mechanics methods describe the evolution of statistical properties of the population fitness. This paper uses the one-max problem, which has only one optimum and whose fitness function is the number of 1s in a binary string, to derive equations that predict the statistical properties of BBO each generation in terms of those of the previous generation. These equations reveal the effect of migration and mutation on the population fitness dynamics of BBO. The results obtained in this paper are similar to those for the simple genetic algorithm with selection and mutation. The paper also derives equations for the population fitness dynamics of general separable functions, and we find that the results obtained for separable functions are the same as those for the one-max problem. The statistical mechanics theory of BBO is shown to be in good agreement with simulation.