Phloem Catastrophe: Identifying Plant Tipping Points Through Bifurcation Analysis.

This chapter presents a mathematical framework to identify conditions leading to phloem failure using bifurcation analysis and catastrophe theory. The approach assumes that at equilibrium, sucrose production through leaf photosynthesis is balanced by sucrose transport within the phloem. Leaf photosynthesis is modeled according to stomatal optimality theories, which propose that stomatal opening adjusts to maximize carbon uptake while being constrained by resource availability (usually water per unit leaf area in the rooting zone). Sucrose transport in the phloem vasculature is described by the Münch mechanism, which is the basis for the pressure flow hypothesis (PFH). The PFH argues that both water and sucrose transport are due to a pressure-driven flow based on osmotic potential differences between source and sink tissues. The proposed framework models this transport process within a leaf-xylem-phloem network, allowing for an examination of how key factors, such as xylem water potential and finite sucrose removal rate along the entire flow path, and anatomical features of the phloem (tube length and radius) influence the rate, efficiency, and failure tipping points of sucrose transport. By systematically varying these control parameters, the number and stability of equilibrium points necessary to maintain a balance between sucrose production and transport are identified. The analysis reveals that phloem failure emerges as an imperfect pitchfork bifurcation, where two stable equilibria at low and high sucrose loading concentrations are separated by an unstable equilibrium at intermediate sucrose loading. The lower stable equilibrium sucrose concentration is shown to be within the range of reported sucrose loading concentration values across many species. The unstable equilibrium branch operates at loading sucrose concentration values commensurate with those responsible for the negative feedback between elevated sucrose concentration levels and reduced leaf-level photosynthesis. This behavior characterizes a phloem catastrophe, which shares mathematical analogies with other ecological catastrophes. The framework thus provides a theoretical basis for understanding the necessary conditions under which phloem transport may collapse, offering insights into the robustness and vulnerability of plant vascular systems and how anatomical traits shape this safety-efficiency trade-off. It also foreshadows a necessary coordination between plant photosynthetic and phloem hydraulic properties, such as plant height, and suggests testable hypotheses that can be evaluated experimentally.