On Stability of the Chemotactic Dynamics in Bacterial-Foraging Optimization Algorithm

Bacterial-foraging optimization algorithm (BFOA) attempts to model the individual and group behavior of E.Coli bacteria as a distributed optimization process. Since its inception, BFOA has been finding many important applications in real-world optimization problems from diverse domains of science and engineering. One key step in BFOA is the computational chemotaxis, where a bacterium (which models a candidate solution of the optimization problem) takes steps over the foraging landscape in order to reach regions with high-nutrient content (corresponding to higher fitness). The simulated chemotactic movement of a bacterium may be viewed as a guided random walk or a kind of stochastic hill climbing from the viewpoint of optimization theory. In this paper, we first derive a mathematical model for the chemotactic movements of an artificial bacterium living in continuous time. The stability and convergence-behavior of the said dynamics is then analyzed in the light of Lyapunov stability theorems. The analysis indicates the necessary bounds on the chemotactic step-height parameter that avoids limit cycles and guarantees convergence of the bacterial dynamics into an isolated optimum. Illustrative examples as well as simulation results have been provided in order to support the analytical treatments.