A numerical approach to investigate the stability of equilibria for structured population models

We are interested in the asymptotic stability of equilibria of structured populations modeled in terms of systems of Volterra functional equations coupled with delay differential equations. The standard approach based on the study of the characteristic equation of the linearized system is often involved or even unattainable. Therefore we propose and investigate a numer- ical method to compute the eigenvalues of the associated infinitesimal generator. The latter is discretized by using a pseudospectral approach, and the eigenvalues of the resulting matrix are the sought approximations. An algorithm is presented to explicitly construct the matrix from the model coefficients and parameters. The method is tested first on academic exam- ples, showing its suitability also for a class of mathematical models much larger than that above mentioned, including neutral- and mixed-type equations. Applications to cannibalism and consumer-resource models are then provided to illustrate the efficacy of the proposed technique, especially for studying bifurcations.