  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.
