Path-integral solution of MacArthur's resource-competition model for large ecosystems with random species-resources couplings

TL;DR

This paper develops a path-integral analytical approach to solve MacArthur's resource-competition model for large ecosystems with random couplings, revealing how steady states depend on heterogeneity and stability.

Contribution

It introduces a novel dynamical path-integral method to analyze large ecosystems with random species-resource interactions, extending previous approaches.

Findings

Steady state solutions depend on heterogeneity parameters.

The model's behavior can be reduced to an effective one-species-one-resource system.

The approach is applicable to other ecosystem models.

Abstract

We solve MacArthur's resource-competition model with random species-resource couplings in the `thermodynamic' limit of infinitely many species and resources using dynamical path-integrals a la De Domincis. We analyze how the steady state picture changes upon modifying several parameters, including the degree of heterogeneity of metabolic strategies (encoding the preferences of species) and of maximal resource levels (carrying capacities), and discuss its stability. Ultimately, the scenario obtained by other approaches is recovered by analyzing an effective one-species-one-resource ecosystem that is fully equivalent to the original multi-species one. The technique used here can be applied for the analysis of other model ecosystems related to the version of MacArthur's model considered here.