A delay equation is a rule for extending a function of time towards the future, on the basis of the known past. Renewal Equations prescribe the current value, while Delay Differential Equations prescribe the derivative of the current value. With a delay equation one can associate a dynamical system by translation along the extended function.
I will illustrate how such equations arise in the description of age- or size-structured populations. These population models are traditionally formulated as first order PDE with non-local boundary conditions. The delay formulation amounts to restriction to a forward invariant attracting set and so the information that is lost concerns transient behaviour only.
Next I will explain how one can use sun-star calculus for adjoint semigroups to derive conclusions concerning stability and bifurcation from an analysis of a characteristic equation.
Throughout, the interaction of a size-structured consumer (Daphnia) and its resource (algae) will serve as the motivating example. If time permits, I will also briefly discuss size-structured cell cycle models.
The lecture is based on joint work with Mats Gyllenberg and Hans Metz.