Understanding of the main mathematical tools, local and global, analytical and geometrical, necessary for the study of mechanical and biological models based upon equations and systems of ordinary differential equations.
Understanding of the main evolutionary models of one or more interacting populations, both in a discrete and continuous setting. Understanding of physical, medical and neurological models.
General features of discrete and continuous dynamical systems. Linear and nonlinear systems, integrability, flow, first integrals. Equilibria and stability, eigenvalue analysis, Lyapunov function. Euler-Lagrange equations, Legendre transforms, Hamilton equations and Hamiltonian systems. Application to biological model of populations growth of logistic or Malthusian type, the Lotka-Volterra predator-prey system. Modelization and analysis of some physical phenomena.
Equations and systems of PDE of parabolic type which emerge in mathematical biology, in particular reaction-diffusion systems of Lodka-Volterra type. Trapping regions. Qualitative behaviour of dynamics.
The course includes various numerical simulations of the models we consider, which will be included into 12 extra laboratory lectures.
Some extra one-hour lectures complementing the topics of the course could be available, held by external teachers.
The course is mainly based upon:
For the first part:
Introduzione all'Analisi Qualitativa delle Equazioni Differenziali Ordinarie
Marco Squassina, Simone Zuccher
Apogeo Editore 2008, ISBN 9788850310845
For the second part:
Shock waves and reaction-diffusion equations / Joel Smoller . - 2. ed. - New York [etc.] : Springer, c1994. - XXII, 632 p. ; 25 cm.
|M. Squassina, S. Zuccher||Introduzione all'Analisi Qualitativa delle Equazioni Differenziali Ordinarie. 332 pagine, 365 figure.||Apogeo Editore||2008||9788850310845|
|J. Murray||Mathematical Biology||Springer||2002||0-387-95223-3|
|G. Gaeta||Modelli Matematici in Biologia||Springer||2007||978-0-7923|
Final oral examination.