Aim of this project is to study some Partial Differential Equations (PDE) arising as models of biological processes and other systems
(such as e.g., crowds, animal groups, cell colonies)
involving individuals who possess some decisional abilities.
Indeed complex systems have attracted interest in various fields, from Sociology to Economyy and Biology since they pose new and
stimulating scientific challenges with respect to more traditional systems:
"Today, most of science is biology" (Reed, "Mathematical biology
is good for mathematics", Notices of AMS, 62, 2015) and also in Mathematics biological applications are becoming the main driving force of innovation.
Many biological processes are controlled by complex interactions which are not well described by a classical setting even in the case of Euclidean geometry and therefore require differential models with a more flexible structure. In addition often such processes evolve on irregular spatial domains and the Euclidean setting is only a first approximation to the complexity of the problem. Hence the increasing interest in the study of nonlinear differential models on networks, on ramified spaces and involving nonlocal terms.
In this project we concentrate on differential problems as outlined
above with the following objectives:
a) A correct mathematical formulation of interacting models with special attention to
- PDE defined on networks and other irregular geometric structures
- PDE with nonlocal terms
b) The extension to the new setting of classical functional analysis techniques in order to study the well posed-ness of the previous problems.
c) The development of numerical methods and algorithms for the
validation of the theoretical results.
