  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.
