Computing the eigenvalues of realistic Daphnia models by pseudospectral methods

This work deals with physiologically structured populations of the Daphnia type. Their biological modeling poses several computational challenges. In such models, indeed, the evolution of a size structured consumer described by a Volterra functional equation (VFE) is coupled tothe evolution of an unstructured resource described by a delay differential equation (DDE), resulting in dynamics over an infinite dimensional state space. As additional complexities, the right-hand sides are both of integral type (continuous age distribution) and given implicitly through external ordinary differential equations (ODEs). Moreover, discontinuities in the vital rates occur at a maturation age, also given implicitly through one of the above ODEs. With the aim at studying the local asymptotic stability of equilibria and relevant bifurcations, we revisit a pseudospectral approach recently proposed to compute the eigenvalues of the infinitesimal generator of linearized systems of coupled VFEs/DDEs. First, we modify it in view of extension to nonlinear problems for future developments. Then, we consider a suitable implementation to tackle all the computational difficulties mentioned above: a piecewise approach to handle discontinuities, numerical quadrature of integrals, and numerical solution of ODEs. Moreover, we rigorously prove the spectral accuracy of the method in approximating the eigenvalues and how this outstanding feature is influenced by the other unavoidable error sources. Implementation details and experimental computations on existing available data conclude the work.