When the exception becomes the rule: The disappearance of limiting similarity in the Lotka–Volterra model

We investigate the transition between limiting similarity and coexistence of a continuum in the competitive Lotka–Volterra model. It is known that there exist exceptional cases in which, contrary to the limiting similarity expectation, all phenotypes coexist along a trait axis. Earlier studies established that the distance between surviving phenotypes is in the magnitude of the niche width 2 σ provided that the carrying capacity curve differs from the exceptional one significantly enough. In this paper we studied the outcome of competition for small perturbations of the exceptional (Gaussian) carrying capacity. We found that the average distance between the surviving phenotypes goes to zero when the perturbation vanishes. The number of coexisting species in equilibrium is proportional to the negative logarithm of the perturbation. Nevertheless, the niche width provides a good order of magnitude for the distance between survivors if the perturbations are larger than 10%. Therefore, we conclude that limiting similarity is a good framework of biological thinking despite the lack of an absolute lower bound of similarity.