The familiar textbook statement, concerning competition between two species, is that intraspecific competition must exceed interspecific competition for coexistence. What is rarely mentioned though is that this intuitive, compelling rule works exclusively in the case of two competitors. In multispecies competitive communities, such simple rules are not available. The main reason is that two-species communities have a very simple structure, while for multiple species, the exact same set of competition coefficients arranged in different ways will lead to communities with vastly different properties. Here we ask whether the two-species coexistence rule can be generalized to multispecies competitive communities, and how community structure affects coexistence.
We derive a multispecies generalization of the two-species rule in the context of symmetric Lotka-Volterra competition, and present explicit stability conditions for random competitive communities. We then explore the influence of community structure on coexistence by finding the arrangements of competition coefficients that are the most/least conducive to stability. Results show that both the most and least stabilized cases have striking global structures, with a nested pattern emerging in both cases. The distribution of intraspecific coefficients leading to the most and least stabilized communities also follows a predictable pattern that can be justified analytically. In addition, we show that the size of the parameter space allowing for feasible communities always increases with the strength of intraspecific effects in a characteristic way that is independent of the interspecific interaction structure. We conclude by presenting some extensions of our results to nonsymmetric competition.