Aim of the course is to provide basic and advanced tools needed to provide a differential-equations-based model for a selected set of applied settings from population biology, epidemiology, chemistry, cell biology and similar topics. Moreover, the course aims to provide the methods to partly solve the proposed models or to detect their qualitative behaviour. The methods proposed include energy estimates, viscous approximation methods, compactness methods. Finally, the mathematical (analytical) results are complemented with suitable comments and interpretations.
ODE models. Single species population models. ODEs with delay. Dynamical systems. Multi species models: predator-prey, competition, mutualism. Models in epidemiology. SIR and XSI models. Modelling AIDS. Modelling reaction kinetics. Michaelis-Menten approach. Singular limits. PDE models. Diffusion equations. Nonlinear diffusion, existence of solutions and asymptotic behavior. Reaction diffusion equations. Threshold phenomena and travelling waves. Reaction diffusion systems. Turing instability. Models for chemotaxis. Blow-up of solutions. Models for structured populations.