Beyond the Critical Depth: The Metabolic and Physical Drivers of Phytoplankton Persistence in a Changing Ocean

Abstract

While the classical Critical Depth Hypothesis (CDH) effectively explains the onset of blooms as transient instabilities, it does not fully capture the seasonal decoupling of biological rates and the long-term persistence of phytoplankton communities in fluctuating thermal environments. To address these limitations, we introduce a parsimonious framework that leverages the theory of non-autonomous dynamical systems to diagnose the stability of phytoplankton communities throughout the entire annual cycle. By linearizing the dynamics around the extinction equilibrium, we identify the invasion growth rate -formally the Floquet exponent-and derive the critical nutrient requirement ($\gamma$crit) as a bifurcation point for uniform persistence. Using end-of-the-century projections from the GFDL-ESM4 model under a high-emission scenario (SSP5-8.5), we identify a global regime shift characterized by a widespread expansion of metabolic-driven regimes, which increasingly displace regions where stability was historically governed by physical mixing. Relevance to Life Sciences. Quantitative analysis of system stability challenges CDH by demonstrating that metabolic constraints increasingly modulates phytoplankton persistence in a changing ocean. Our results, based on high-emission projections, reveal a profound physical-biological decoupling at the poles: while warming reduces the critical nutrient requirement ($\gamma$crit) facilitating persistence in previously marginal waters, this metabolic expansion is offset at poles. A 1:4 ratio between newly viable niches and ice-free deserts suggests that cryospheric retreat does not guarantee a proportional expansion of life. In addition, we identify the North Atlantic Subpolar Gyre as a ''metabolic refuge'' where mixing dynamics still anchor the ecosystem against global thermalization. By providing a ''radiography'' of the future ocean's complexity, this methodology offers a mechanistic basis to deconstruct how the dynamic balance between environmental energy and metabolic demands may determine the functional integrity of the marine biosphere under extreme anthropogenic forcing. Mathematical Content. The temperature dependence of biological rates is modeled using a thermodynamic equation, coupling population dynamics with seasonal variations in mixed layer depth and temperature. Given the non-autonomous nature of the system under annual forcing, we characterize the stability of the extinction equilibrium through its associated invasion growth rate. This rate is analytically derived as the Floquet exponent $\lambda$P , which provides a rigorous condition for uniform persistence (Theorem 3.2). The numerical analysis of this exponent, projected onto a global scale, quantifies the relative influence of environmental drivers on the stability threshold $\gamma$crit. This allows for the definition of the thermal dominance index (DT ), a metric that identifies the geographic transition from mixing-driven to metabolic-driven ecological control.