Numerical Bifurcation Analysis of Physiologically Structured Populations: Consumer–Resource, Cannibalistic and Trophic Models

Abstract

With the aim of applying numerical methods, we develop a formalism for physiologically structured population models in a new generality that includes consumer resource, cannibalism and trophic models. The dynamics at the population level are formulated as a system of Volterra functional equations coupled to ODE. For this general class we develop numerical methods to continue equilibria with respect to a parameter, detect transcritical and saddle-node bifurcations and compute curves in parameter planes along which these bifurcations occur. The methods combine curve continuation, ODE solvers and test functions. Finally we apply the method to the above models using existing data for Daphnia magna consuming Algae, and for Perca fluviatilis feeding on Daphnia magna. In particular we validate the methods by deriving expressions for equilibria and bifurcations with respect to which we compute errors, and by comparing the obtained curves with curves that were computed earlier with other methods. We also present new curves to show how the methods can easily be applied to derive new biological insight. Schemes of algorithms are included.

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

“…The dashed curve with diamonds is, instead, the result of a natural continuation with secant prediction, recall Figure 1 (bottom-right), always implementd in Matlab. Following [32], the correction step is made by using the Broyden's update, integrals and external IVPs are solved simultaneously via the embedded Runge-Kutta pair DOPRI54 (see, e.g., [22]: basically, it is also behind ode45) and the maturation age is directly obtained as an output of the latter, which is indeed capable of event detection, i.e., it automatically detects when (10) is satisfied during the integration of (8). As for the previous approach, integrals, IVPs and maturation age are computed from scratch at every continuation step, independently of the same quantities computed at the previous step.…”

“…In the following we refer to the two approaches described above by simply citing the works of reference, i.e., [6] and [32], respectively. Let us note that other works exist on the numerical equilibrium analysis of the Daphnia model, see, e.g., [12] and the references therein.…”

“…Let us note that other works exist on the numerical equilibrium analysis of the Daphnia model, see, e.g., [12] and the references therein. Yet, from the numerical point of view, [32] is largely inspired from [12], so that the applied numerical techniques are basically the same. A similar comment holds also with regards to the software PSPManalysis [10].…”

“…We describe Daphnia in Section 3, highlighting the external problems and the technicalities involved with it. In this work we concentrate on the continuation of its equilibria as a starting point, thus the rest of Section 3 focuses on how to approach this problem, either with the general method proposed in [6] or with more specific tools as proposed in [32]. In Section 4 we explain our internal approach in full detail.…”

“…At the end we report on some computational results on the numerical continuation of its equilibria, obtained by using the technique recently proposed for general REs/DDEs [6], as well as by emulating algorithms already available for Daphnia-like models [32] (see also [10,12]).…”