Earth as a Closed Life-Support System: Finite Buffers, Dynamical Thresholds, and Planetary Sustainability

Abstract

Earth is nearly closed with respect to matter but open with respect to energy. We formulate planetary sustainability as a minimal nonequilibrium dynamical problem for coupled finite stocks and fluxes, using the operational language of closed ecological and engineered life-support systems: inventories, flux balances, recycling efficiencies, processing capacities, reserve times, and failure thresholds. The framework describes a competition between two finite buffers: resource stocks that can be depleted and waste-processing capacities that can be saturated. The physiological contrast between fasting and renal failure motivates a general reserve-time principle: the dominant constraint is the buffer that becomes critical first. Our nonlinear model couples a regenerative resource stock, an accumulated waste stock, and an aggregate activity variable. It exhibits a soft waste threshold, where accumulated waste suppresses growth while processing remains formally capable, and a hard threshold, where waste production exceeds maximal processing capacity. We extend the model to recycling and multiple resource and waste classes, give an explicit numerical example, and relate it to Spaceship Earth, planetary boundaries, Limits to Growth, and stock-pollution growth models. A multiscale analogy--cell, organism, planet--emphasizes that life-support functions combine degradation, recycling, transport, regulation, and removal. Carrying capacity thus emerges as a time-dependent property of coupled resource, waste, recycling, technological, and institutional dynamics: either resource depletion or elimination failure may become the binding constraint, depending on their characteristic reserve times.