Quasi-equilibrium reduction in a general class of stoichiometric producer–consumer models

Abstract

This article compares a general closed nutrient, stoichiometric producer-consumer model to a two-dimensional 'quasi-equilibrium' approximation. We demonstrate that the quasi-equilibrium system can be rigorously analysed, resulting in nullcline-based criteria for the local stability of system equilibria and for the non-existence of periodic orbits. These results are applied to a study of the dependence of the reduced system on nutrient and energy enrichment. When energy and nutrient enrichment are considered together, the associated bifurcation structures of the two models are seen to share the same essential qualitative characteristics. However, numerical simulations of the three-dimensional parent model show highly complex domains of the persistence and extinction that by Poincare-Bendixson theory are not possible for the two-dimensional reduction. This complexity demonstrates a major difference between the two models, and suggests potential challenges in the use of either model for predicting the long-term behaviour of real-world systems at specific nutrient and energy levels.

Enrichment in a general class of stoichiometric producer–consumer population growth models

“…However, there seems to be a general consistency between the two systems. A rigorous comparison of system (2.1, 2.2, 2.4) and this two--dimensional approximation is given in [44]. In summary, based on these calculations it appears that the bifurcation curves shown in Figure 4.1 partition the (b, N T ) plane into distinct regions of system dynamics.…”

“…As is the case for those models, under high energy levels the system supports a globally attracting equilibrium state. The results presented here provide useful motivation for a rigorous analysis of the quasi--equilibrium approximation (2.1, 2.2, 4.1) and an examination of its similarities and differences to the general model of this paper [44].…”

“…In contrast, the models of [36], [20], [47] and [49] assume that nutrient uptake is proportionate to P, with a proportionality constant that is a saturating function of N m , but independent of internal stores. Our assumption provides certain technical advantages, as will be seen in Section 4 and in the mathematical analysis of [43,44]. Moreover, Diehl [8 (Appendix A)] cites evidence that the details of nutrient uptake may not be a significant influence on system dynamics.…”

“…A "noncentral saddle--node homoclinic" bifurcation point [29], denoted by NCSNHC, locates the joining of the curve of homoclinics and the saddle node curve. See [44] and [45] for similar diagrams computed for specific two--dimensional, producer--consumer models derived under different producer growth and nutrient uptake assumptions. Dashed horizontal lines locate nutrient levels associated with six specific energy enrichment scenarios described in the text.…”