Pseudospectral methods and numerical continuation for the analysis of structured population models

In this thesis new numerical methods are presented for the analysis of models in population dynamics. The methods approximate equilibria and bifurcations in a certain class of so called structured population models. Chapter 1 consists of an introduction to structured population dynamics, where the state of the art is presented through a classical consumer-resource model [44]. The necessity of new numerical methods for analyzing structured population models is discussed and motivated by their applications to life sciences. In Chapter 2 [44] is extended to a more general class in which a structured population with a unique state at birth interacts with an environment of unstruc- tured populations and interaction variables. Equilibrium types are defined, the model is linearized and a characteristic equation is obtained. Finally, a discussion about equilibria and bifurcations under parameter variation is included. In Chapter 3 a new pseudospectral method for the computation of eigenvalues of linear VFE/DDE systems is presented. The technique consists of constructing a finite approximation of the infinitesimal generator of the solution semigroup. The spectral convergence of the method is proved, and a piecewise variation which speeds up the computations presented and validated with toy models. An exten- sion to deal with structured population models is proposed and validated with the model in [44]. Chapter 4 is devoted to the numerical continuation of equilibrium branches and bifurcation curves under parameter variation for models of the class presented in Chapter 2. A new technique for the curve continuation is presented, where a reduction of the dimension and a simplification of the equilibrium conditions result in new test functions for the detection of transcritical bifurcations, reducing the computational cost. The methods were implemented in the development of routines that were tested and validated with models from the literature