Numerical continuation and delay equations: A novel approach for complex models of structured populations

Recently, many realistic models of structured populations are described through delay equations which involve quantities defined by the solutions of {it external} problems. For instance, the size or survival probability of individuals may be described by ordinary differential equations, and their maturation age may be determined by a nonlinear condition. When treating these complex models with existing continuation approaches in view of analyzing stability and bifurcations, the external quantities are computed from scratch at every continuation step. As a result, the requirements from the computational point of view are often demanding. In this work we propose to improve the overall performance by investigating a suitable numerical treatment of the external problems in order to include the relevant variables into the continuation framework, thus exploiting their values computed at each previous step. We explore and test this {it internal} continuation with prototype problems first. Then we apply it to a representative class of realistic models, demonstrating the superiority of the new approach in terms of computational time for a given accuracy threshold.