A practical approach to computing Lyapunov exponents of renewal and delay equations.

We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists of the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided.

A pseudospectral method for investigating the stability of linear population models with two physiological structures.

The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. In this paper, we propose a general numerical method to approximate this spectrum. In particular, we first reformulate the problem in the space of absolutely continuous functions in the sense of Carathéodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.

Host coexistence in a model for two host-one parasitoid interactions.

Building from a continuous-time host-parasitoid model introduced by Murdoch et al. (Am Nat 129:263-282, 1987), we study the dynamics of a 2 host-parasitoid model assuming, for the sake of simplicity, that larval stages have a fixed duration. If each host is subjected to density-dependent mortality in its larval stage, we obtain explicit conditions for the existence of an equilibrium where the two host species coexist with the parasitoid. However, if host demography is density-independent, equilibrium coexistence is impossible. If at least one of the 1 host-parasitoid systems has an oscillatory dynamics (which happens under some parameter values), we found, through numerical bifurcation, that coexistence is favoured. Coexistence between the two hosts may occur along a periodic solution even without density-dependence. Models of this type may be relevant for the use of parasitoids as biocontrol agents of insect pests.