Final Activity Report Summary - CO.ST.C.A.R.E. (Continuum State Cellular Automata and Random Equations - Applications to urban growth and traffic models) The first aim of our project was to start the development of a new axiomatic theory of complex systems of interacting entities including meaningful examples (e.g. cellular automata, CA, and multi-agents systems, MAS) and with good mathematical properties. In the first year of the project we precisely defined the axioms of this new mathematical structure called Interaction Spaces (IS). Defining these axioms we included as IS: systems with behaviour depending on the past history (systems with memory, i.e. non-Markovian), CA (both deterministic and stochastic; in general with time dependant neighbourhoods), multi-agents systems, CA with non-local interactions, etc. A first example of IS developed in detail has been a decisions support mathematical model for urban growth processes. An important second example is work in progress and consists in a trans-disciplinary project (together with urban planners, environmental protection officers, architects, physicists and mathematicians) for a traffic model coupled with a urban growth model and used as a decision support system for the management of urban projects of new big traffic generators (e.g. shopping centres, free time centres, tourists attraction sites, monopolistic activities, etc.). After a sufficiently detailed study of the scientific literature we chose to focus our future attention on the following examples of IS: models of housing markets, models of Wikipedia and open-source software, pedestrian movement, patterns formation, Darwinian evolution and behavioural finance. During the first year project new validation ideas of these type of complex systems has been developed. The scientific project is underdevelopment (3 years remaining). Objectives to complete: Development of mathematical models from other fields, proof that every extensive observable in an IS verifies both an ordinary differential equation (ODE) and a random differential equation (RDE); numerical solution of the RDE; numerical study of bifurcations in IS using RDE and ODE.