Viability Space Decomposition: A geometric partition of survival outcomes in single- and multi-agent systems

TL;DR

This paper introduces viability space decomposition, a geometric framework for analyzing survival outcomes in dynamical systems with viability constraints, applicable to biological and ecological models.

Contribution

The paper develops a novel geometric decomposition method that characterizes global survival outcomes in constrained dynamical systems, extending traditional analysis tools.

Findings

Complete analysis of three biological models using viability portrait

Introduction of new classes of manifolds: mortality, ordering, and collapse

Framework scales to complex systems and guides future research

Abstract

What determines whether an organism or collective will survive under particular conditions? This question is asked across the life sciences when determining adaptive fit, developing efficacious treatments for diseases, and assessing the risks posed by ecological shifts. To aid their investigations, researchers employ models of agents which must respect particular constraints to remain alive. By constraining the dynamics of these agents to bounded viability regions, these models form a class of extended dynamical systems where transient dynamics can lead to death, making traditional attractors and separatrices insufficient for characterizing the global space of possible behaviors. To remedy this, we develop viability space decomposition, an analysis framework for ordinary differential equation models of agents with viability constraints. We first introduce the general theory, revealing how several new classes of manifolds (mortality, ordering, and collapse) permit a complete decomposition of state space into regions of qualitatively similar survival outcomes: a viability portrait. We then demonstrate the method by completely analyzing the global behavior of three models: a subcellular network, a behaving cell with the same physiology, and two coupled cell networks. Finally, we finish by discussing how the framework scales and future directions for its development and application.