Partial Differential Equations to Diffusion-Based Population and Innovation Models

Multiple instances and diffusion mechanisms in biological and economic modeling involve partial differential equations (PDEs). Functional PDEs (with time delays) may be even more adequate to real world problems. In the modeling process, PDEs can 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 modeling process with PDEs and reviews the essential features of the dynamics in biological, ecological and economic modeling. The computations are carried out by using the software WolframMATHEMATICA® 7. Key-Words: Diffusion process, partial differential equation, reaction-diffusion equation, traveling wave solution, critical patch size, population dynamics, innovation diffusion.