Mathematical modeling of dialectical emergent hybrid regimes in ecosystems

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.