Stability and coexistence in large ecological systems

Abstract

Understanding the conditions that govern the stability and coexistence of species in ecological systems remains a central challenge in theoretical ecology. In this thesis, we explore this problem through three distinct but complementary approaches grounded in random matrix theory and consumer-resource dynamics. We begin by revisiting May’s stability criterion for large random ecosystems and extend it using the Circular Law. By incorporating sparsity and variance scaling, we derive analytical stability conditions and validate them through numerical simulations. Next, we revisit the competitive exclusion principle within the framework of consumerresource models, demonstrating how resource availability and half-saturation constants (which quantify the resource level at which a species achieves half its maximum growth rate) govern species persistence and extinction. Finally, we investigate how asymmetric migration between habitats can promote coexistence beyond classical resource-based constraints. We derive conditions for stable stationary states and analyze how e↵ective competition coefficients shape biodiversity outcomes. Together, these investigations o↵er insights into the spectral and ecological mechanisms that underpin the stability and diversity of complex ecosystems.