250170 SE Biomathematische Modelle (2005W)

Particular attention is being paid these days to the math.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 optimizat. problems (ACO's, i.e. Ant Colony Optimizat.). Further decentralized/parallel computing is exploiting the capabilities of discretizat. of nonlinear reaction-diffusion systems by means of stochastic interacting particle systems.
Among other interesting features, these systems lead to selforganizat. phenomena, which exhibit interesting spatial patterns. As an example, here an interacting particle system modelling the social behaviour of ants is proposed, based on a system of stochastic differential equations, driven by social aggregating/repelling "forces". Specific reference to species observed in will be made. Extensions to models of chemotaxis, such as tumor driven angiogenesis will also be presented, in which the so called organizat. process is driven by an underlying biochemical field, strongly coupled with the spatial/geometric structure of the growing tumor. Current interest concerns how do 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. We will see how by rather simple models based on stochastic differential equations we may gain remarkable insight in the behaviour of complex systems.