Prof. Odo Diekmann (Utrecht University)

Abstract: In terms of numbers, the dynamics of a population is generated by mortality and reproduction. In structured population models, individuals are characterized by variables such as age, size or other (physiological) characteristics. In that case, development/maturation needs to be modelled too (a trivial task in the case of age, but certainly not in general).

Both first order PDE and delay equations are used to formulate structured population models. Here we employ 'delay equations' as a collective noun for DDE (Delay Differential Equations) and RE (Renewal Equations = Volterra Integral Equations) and conceptually define a 'delay equation' as a rule for extending a function of time towards the future, on the basis of the (assumed to be) known past. With a delay equation one associates a dynamical system by translation along the extended function.

To derive local stability and bifurcation results for these dynamical systems, one needs a variation-of-constants formula. Underlying this formula is a somewhat unusual functional analytic framework (nicknamed sun-star calculus) in which duality theory for semigroups of bounded linear operators plays a key role. The main aim of the lectures is to provide a motivated introduction to the sun-star framework for the analysis of delay equations, with special attention for the not-so-well-known RE and various population models.

Additional topics:

• model formulation: how PDE and RE relate to each other
• integration along characteristics by using the biological interpretation
• variable maturation delay (and the lack of differentiability of solution operators for size structured PDE)
• infectious disease models and waning immunity
• characteristic equations and Hopf bifurcation
• pseudo-spectral approximation to facilitate the use of numerical bifurcation tools for ODE
• twin semigroups