Using mathematical modeling to ask meaningful biological questions through combination of bifurcation analysis and population heterogeneity

Abstract

Classical approaches to analyzing dynamical systems, including bifurcation analysis, can provide invaluable insights into underlying structure of a mathematical model, and the spectrum of all possible dynamical behaviors. However, these models frequently fail to take into account population heterogeneity, which, while critically important to understanding and predicting the behavior of any evolving system, is a common simplification that is made in analysis of many mathematical models of ecological systems. Attempts to include population heterogeneity frequently result in expanding system dimensionality, effectively preventing qualitative analysis. Reduction Theorem, or Hidden keystone variable (HKV) method, allows incorporating population heterogeneity while still permitting the use of previously existing classical bifurcation analysis. A combination of these methods allows visualization of evolutionary trajectories and making meaningful predictions about dynamics over time of evolving populations. Here, we discuss three examples of combination of these methods to augment understanding of evolving ecological systems. We demonstrate what new meaningful questions can be asked through this approach, and propose that the large existing literature of fully analyzed models can reveal new and meaningful dynamical behaviors with the application of the HKV-method, if the right questions are asked.