Exclusion and multiplicity for stable communities in Lotka-Volterra systems

For classic Lotka-Volterra systems governing many interacting species, we establish an exclusion principle that rules out the existence of linearly asymptotically stable steady states in subcommunitites of communities that admit a stable state which is internally D-stable. This type of stability is known to be ensured, e.g., by diagonal dominance or Volterra-Lyapunov stability conditions. By consequence, the number of stable steady states of this type is bounded by Sperner's lemma on anti-chains in a poset. The number of stable steady states can nevertheless be very large if there are many groups of species that strongly inhibit outsiders but have weak interactions among themselves. By examples we also show that in general it is possible for a stable community to contain a stable subcommunity consisting of a single species. Thus a recent empirical finding to the contrary, in a study of random competitive systems by Lischke and L\"offler (Theo. Pop. Biol. 115 (2017) 24--34), does not hold without qualification.