Combining Bifurcation Analysis and Population Heterogeneity to Ask Meaningful Questions

  • Irina KarevaEmail author
Part of the STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health book series (STEAM)


Classical approaches to analyzing dynamical systems, such as 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 the 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 classical bifurcation analysis. A combination of these methods allows visualizing evolutionary trajectories and making meaningful predictions about system dynamics 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 application of the HKV method to the large existing literature of fully analyzed models can reveal new and meaningful dynamical behaviors, if the right questions are asked.



The author would like to thank anonymous reviewers for helpful and insightful comments. This research received no external funding.

Disclosure of Potential Conflicts of Interest

IK is an employee of EMD Serono, U.S. subsidiary of Merck KGaA. Opinions expressed in this paper do not necessarily reflect the opinions of Merck KGaA.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Mathematical and Computational Sciences Center, School of Human Evolution and Social ChangeArizona State UniversityTempeUSA

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