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 subcommunities 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 Löffler (Theor Popul Biol 115:24–34, 2017), does not hold without qualification.

对于许多管理类相互作用的Lotka经典- Volterra系统，我们建立一个排除原则，即排除了线性渐进稳定的稳定状态的在承认一个稳定的状态社区的子社区的存在，这是内部d-稳定的。已知这种类型的稳定性可以通过对角优势或 Volterra-Lyapunov 稳定性条件来确保。因此，这种类型的稳定稳态的数量受偏序组中反链上的 Sperner 引理限制。然而，如果有许多物种强烈抑制外来者但它们之间的相互作用很弱，那么稳定稳态的数量可能会非常大。通过例子，我们还表明，通常一个稳定的群落可能包含一个由单一物种组成的稳定子群落。因此，在 Lischke 和 Löffler (Theor Popul Biol 115:24–34, 2017) 对随机竞争系统的研究中，最近的一项相反的实证发现并非没有资格证明。