Numerical Equilibrium Analysis for Structured Consumer Resource Models

Abstract

In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for "Daphnia consuming algae" models in C-code. The results obtained by way of this implementation are shown in the form of graphs.

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

“…The decrease of resource is due to an increasing consumption. If the mortality rate decreases further the p-birth starts decreasing due to competition for the resource E. The B-transcritical bifurcation curve in Figure 5 is the existence boundary for the positive equilibrium and coincides with the one in Figure 7 in de Roos et al (2010). Above the curve the model has a trivial, a B-trivial and a positive equilibrium, and below the curve only a trivial and a B-trivial.…”

“…For the modeling at the individual (i-) level and the step from the i-level to the population (p-) level we use an analogous formalism and reasoning as for the consumer resource models in Diekmann et al (2010) and de Roos et al (2010).…”

Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example

“…An effective way is to single out two parameters and to use ω to parametrise the curve in the corresponding parameter plane at which a Hopf bifurcation occurs (see Diekmann et al 1995;de Roos et al 2009, for details). In principle one can determine by way of additional analytic computations whether the Hopf bifurcation is sub-or supercritical.…”

“…A new edition of the former paper, employing the delay framework of the present paper, is submitted (de Roos et al 2009). …”

Integrating ecological insight derived from individual-based simulations and physiologically structured population models

“…Formulation of the partial differential equation that mathematically describes the population model does not require further assumptions (de Roos, 1997). Furthermore, specification of this partial differential equation is, not necessary for computing the equilibrium states of the PSPM, as an approach for these computations (De Roos, 2008;de Roos et al, 2010;Diekmann et al, 2003) has been developed, which only relies on providing the functions of the core life history model as input. This technique has recently been implemented in a dedicated software package PSPManalysis (de Roos, 2014) that can compute the equilibrium states of a PSPM over a particular range of values of any arbitrary model parameter.…”