From individual dynamics at the microscopic scale to continuum dynamics at the macroscopic scale: The ant colony paradigm

Particular attention is being paid these days to the mathematical modelling of the social behaviour of individuals in a biological population, for different reasons; on one hand there is an intrinsic interest in population dynamics of herds, on the other hand agent based models are being used in complex optimization problems (ACO's, i.e. Ant Colony Optimization). Further decentralized/parallel computing is exploiting the capabilities of discretization of nonlinear reaction-diffusion systems by means of stochastic interacting particle systems. Among other interesting features, these systems lead to self organization phenomena, which exhibit interesting spatial patterns. As a working example, an interacting particle system modelling the social behaviour of ants is proposed here, based on a system of stochastic differential equations, driven by social aggregating/repelling "forces". Specific reference to observed species in nature will be made. Current interest concerns how properties on the macroscopic level depend on interactions at the microscopic level. Among the scopes of the seminar, a relevant one is to show how to bridge different scales at which biological processes evolve; in particular suitable "laws of large numbers" are shown to imply convergence of the evolution equations for empirical spatial distributions of interacting individuals to nonlinear reaction-diffusion equations for a so called mean field, as the total number of individuals becomes sufficiently large. In order to support a rigorous derivation of the asymptotic nonlinear integrodifferential equation, problems of existence of a weak/entropic solution will be analyzed. Further the existence of a nontrivial invariant probability measure is analyzed for the stochastic system of interacting particles. As a further application of the same paradigm, a multiscale model for tumour-driven angiogenesis will be presented. REFERENCES [1] Boi S., Capasso V., and Morale D., Modelling the aggregating behaviour of ants of the species Polyergus Rufescens. Spatial heterogeneity in ecological models. Nonlinear Analysis. Real World Appl. 1:163-176, 2000. [2] Burger M., Capasso V., and Morale D., On an aggregating model with long and short range interactions. Nonlinear Analysis. Real World Appl. 2006. [3] Morale D., Capasso V. and Oelschlaeger K., An interacting particle system modelling aggregation behaviour: from individuals to populations. J. Mathematical Biology. 50:49-66, 2005. [4] Capasso V., and Morale D., Stochastic modelling of tumour-induced angiogenesis