Numerical bifurcation analysis of renewal equations via pseudospectral approximation

TL;DR

This paper introduces a pseudospectral method to approximate nonlinear renewal equations with ordinary differential equations, enabling efficient bifurcation analysis in ecological and epidemiological models.

Contribution

It presents a novel pseudospectral approximation approach for renewal equations, with convergence proofs and improved computational efficiency over previous methods.

Findings

The new method is ten times faster than previous approaches.

It accurately captures equilibria and periodic solutions in models.

The approach simplifies numerical bifurcation analysis of complex systems.

Abstract

We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system is ten times more efficient than the one originally proposed in [Breda et al, SIAM Journal on Applied Dynamical Systems, 2016], as it avoids the numerical inversion of an algebraic equation.