Abstract: Multiple circumstances and diffusion mechanisms in biological and economic modeling involve partial differential equations (PDEs). Functional PDEs (with discrete delays) may be even more adapted to real world problems. Some PDEs are already attached to basic concepts such as a marginal rate of substitution or an elasticity of substitution, from which we can infer the form of utility or production functions. Other PDEs are inherent to the resolution process of a problem, such as the Hamilton-Jacobi-Bellman PDE for solving continuous-time control problems (e.g. Stackelberg differential games) , and the Fokker-Planck PDE of parabolic type to obtain the probability density function of solutions in an uncertain random environment (e.g. to determine the probability that a particle will be found in a given region). In the modeling process, PDEs (with even more complications) may also formalize behaviors, such as the logistic growth of populations with migrations, and the adopters’ dynamics of new products in innovation models. In biology, these events are then related to the variations in the environment, the population densities and overcrowding, the migrations and spreading of humans, animals, plants and other cells and organisms. In economics and management science, the diffusion processes of technological innovations in the. Marketplace (e.g. the mobile phone) is a major subject. Moreover, these innovation diffusion models refer mainly to epidemic models. This contribution introduces to this powerful modeling process with PDEs and reviews the essential features of the dynamics in ecological and economic modeling. The computations are carried out by using the software Wolfram Mathematica ® 8.

Brief Biography of the Speaker:
Andre A. Keller (Prof.) is at present an associated researcher in mathematical economics at CLERSE a research unit UMR8019 of the French Centre National de la Recherche Scientifique (CNRS) by the Universite de Lille 1, Sciences et Technologies. He is also participating to the group ‘Dynamique et Complexite’ which is supported by the Federation de Physique et Interfaces. He received a PhD in Economics (Operations Research) in 1977 from the Universite de Paris Pantheon-Sorbonne. He is a WSEAS Member since 2010 and a Reviewer for the journals Ecological Modelling (Elsevier) and WSEAS Transactions on Information Science and Applications. He taught applied mathematics (optimization techniques) and econometric modeling, microeconomics, theory of games and dynamic macroeconomic analysis. His experience centers are on building and analyzing large scale macro-economic models, as well as forecasting. His research interest has concentrated on: high frequency time-series modeling with application to the foreign exchange market, on discrete mathematics (graph theory), stochastic differential games and tournaments, circuit analysis, optimal control under uncertainties. His publications consist in writing articles, books and book chapters. The book chapters are e.g. on semi-reduced forms (Martinus Nijhoff, 1984), econometrics of technical change (Springer and IISA, 1989), advanced time-series analysis (Woodhead Faulkner, 1989), stochastic differential games (Nova Science, 2009), optimal fuzzy control (InTech, 2009). One book is on time-delay systems (LAP, 2010). One another book is on nonconvex optimization techniques (WSEAS Press, forthcoming 2012).