Numerical Bifurcation Analysis of Physiologically Structured Population Models via Pseudospectral Approximation

Abstract

Physiologically structured population models are typically formulated as a partial differential equation of transport type for the density, with a boundary condition describing the birth of new individuals. Here we develop numerical bifurcation methods by combining pseudospectral approximate reduction to a finite dimensional system with the use of established tools for ODE. A key preparatory step is to view the density as the derivative of the cumulative distribution. To demonstrate the potential of the approach, we consider two classes of models: a size-structured model for waterfleas (Daphnia) and a maturity-structured model for cell proliferation. Using the package MatCont, we compute numerical bifurcation diagrams, like steady-state stability regions in a two-parameter plane and parametrized branches of equilibria and periodic solutions. Our rather positive conclusion is that a rather low dimension may yield a rather accurate diagram! In addition we show numerically that, for the two models considered here, equilibria of the approximating system converge to the true equilibrium as the dimension of the approximating system increases; this last result is also proved theoretically under some regularity conditions on the model ingredients.