"From individual dynamics at the microscopic scale to continuum
dynamics at the macroscopic scale: The ant colony paradigm"
Prof. Dr. Vincenzo Capasso (Mailand)
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 suffciently 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.