Cross-feeding shapes both competition and cooperation in microbial ecosystems

Recent work suggests that cross-feeding -- the secretion and consumption of metabolic biproducts by microbes -- is essential for understanding microbial ecology. Yet how cross-feeding and competition combine to give rise to ecosystem-level properties remains poorly understood. To address this question, we analytically analyze the Microbial Consumer Resource Model (MiCRM), a prominent ecological model commonly used to study microbial communities. Our mean-field solution exploits the fact that unlike replicas, the cavity method does not require the existence of a Lyapunov function. We use our solution to derive new species-packing bounds for diverse ecosystems in the presence of cross-feeding, as well as simple expressions for species richness and the abundance of secreted resources as a function of cross-feeding (metabolic leakage) and competition. Our results show how a complex interplay between competition for resources and cooperation resulting from metabolic exchange combine to shape the properties of microbial ecosystems.