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Numerical methods for computing the eigenvalues of the Daphnia model

Julia Sanchez Sanz

Dottoranda del Basque Center for Applied Mathematics di Bilbao

Abstract
The Daphnia model is a physiologically structured population model that consists of a size structured consumer population that feed on an unstructured resource. The dynamics at the population level, given as a Volterra functional equation coupled to a delay differential equation, are determined by the processes that occur at the individual level: reproduction, mortality and food ingestion. Those processes depend on the size of the individual, then it is necessary to consider an individual state space, which dynamics (growth) are given by nonlinear ODE. For this type of models the principle of linearised stability and the Hopf bifurcation theorem hold, but due to the complexity of the model the computation of the eigenvalues was considered a challenge. A numerical method for computing a discrete number of eigenvalues that approximate the eigenvalues of the Daphnia model is presented in this talk, the main idea of the method is to approximate the infinitesimal generator operator by a finite dimensional operator. Finally we present numerical results of the computed eigenvalues, that converge spectrally to the exact ones.

The work is in collaboration with D. Breda (Udine), Ph. Getto (Dresden) and R. Vermiglio (Udine).