Stability analysis of the reproduction operator in bacterial foraging optimization

In his seminal paper published in 2002, Passino pointed out how individual and groups of bacteria forage for nutrients and how to model it as a distributed optimization process, which he named the Bacterial Foraging Optimization Algorithm (BFOA). One of the major operators of BFOA is the reproduction phenomenon of virtual bacteria, each of which models one trial solution of the optimization problem. During reproduction, the least healthy bacteria (with a lower accumulated value of the objective function in one chemotactic lifetime) die and the other healthier bacteria each split into two, which then starts exploring the search place from the same location. The phenomenon has a direct analogy with the selection mechanism of classical evolutionary algorithms. This paper attempts to model reproduction as a dynamics and then analyses the stability of the reproductive system very near to an equilibrium point, which in this case is an isolated optimum. It also finds conditions under which a stable reproduction event can take place, to direct a worse bacterium towards a better one. Our analysis reveals that a stable reproduction event contributes to the quick convergence of the bacterial population near optima.