Mathematical modeling of dialectical emergent hybrid regimes in ecosystems

David G. Angeler, Jeffrey H. Allen, Craig R. Allen

Traditional resilience theory often models complex systems as toggling between discrete alternative regimes, such as clear-water and turbid states in shallow lakes, each stabilized by internal feedback. While analytically powerful, this binary paradigm overlooks more nuanced dynamics observed in many real-world systems: the emergence of hybrid regimes that blend structural
and functional elements of opposing regimes. These
configurations are not transient midpoints, but stable, self
organized outcomes shaped by legacy effects, feedback
recombination, and historical memory—a process that is
fundamentally dialectical in nature. This paper proposes a
conceptual scaffold for formalizing such dialectical dynamics
using mathematical tools. Using shallow lakes as model systems,
we show how established methods, including bifurcation and
catastrophe theory, stochastic differential equations, agent-based
models, network theory, and machine learning, can be
reinterpreted to analyze the ontological distinctiveness, spatial
organization, feedback structure and management implications of
hybrid regimes. Rather than advancing a single unifying model,
we provide a roadmap for adapting existing techniques to better
capture the complexity of ecological transitions. In doing so, we
open space for a richer, more process-relational understanding of
resilience.