A universal geometry governs the response of ecosystems to environmental perturbations*

A fundamental question in ecology is to understand how ecosystems respond to environmental perturbations. This is especially challenging due to the strong coupling between species and their environments. Here, we introduce a theoretical framework for calculating the linear response of ecosystems to environmental perturbations in generalized consumer resource models. Our construction is applicable to a wide class of systems including models with non-reciprocal interactions, cross-feeding, and non-linear growth/consumption rates. Within our framework, all ecological variables are embedded into four distinct vector spaces and ecological interactions are represented by geometric transformations between vector spaces. We show that near a steady state, such geometric transformations directly map environmental perturbations—in resource availability and mortality rates—to shifts in niche structure. We illustrate these ideas in a variety of settings including a simple model for pH-induced toxicity in bacterial denitrification. We conclude by discussing the implications of this universal niche geometry for other problems in theoretical ecology including identifying collective modes and characterizing eco-evolutionary dynamics.