Convergence and rate of convergence of simple ant models

We present an original ant model to solve the foraging problem. We describe simulations and provide a convergence analysis. We prove the convergence of the model in the discrete and in the continuous cases. We show that the ant population computes the solution of an optimal control problem and converges in a well defined sense. We discuss the rate of convergence with respect to the number of ants for the discrete case: we give experimental and theoretical arguments that suggest that this convergence rate is superlinear with respect to the number of agents. Furthermore, we explain how this model can be extended in order to solve optimal control problems and more generally any problem that involves the computation of the fixed point of a contraction mapping. This allows us to design a large class of formally well understood ant-like algorithms for problem solving.