Collocation techniques for structured populations modeled by delay equations

Collocation methods can be applied in different ways to delay models, e.g., to detect stability of equilibria, Hopf bifurcations and compute periodic solutions to name a few. On the one hand, piecewise polynomials can be used to approximate a periodic solution for some fixed values of the model parameters, possibly using an adaptive mesh. On the other hand, polynomial collocation can be used to reduce delay systems to systems of ordinary differential equations and established continuation tools are then applied to analyze stability and detect bifurcations. These techniques are particularly useful to treat realistic models describing structured populations, where delay differential equations are coupled with renewal equations and vital rates are given implicitly as solutions of external equations, which in turn change with model parameters. In this work we show how collocation can be used to improve the performance of continuation for complex models and to compute periodic solutions of coupled problems.