Coexistence by virtue of similarity versus dissimilarity: The implications of nonsmooth competition kernels

Background/Question/Methods

An active topic of debate today in ecology is whether species coexist by being sufficiently different from one another to decrease interspecific competition to tolerable levels, or because they are so similar that fitness differences between them are negligible. One way this question can be studied theoretically is to consider models of competition along a niche axis or gradient. Traditionally, the competition kernel in such models has been assumed to be a Gaussian function of niche difference, in which case limiting similarity is the expected behavior (though it is possible to have tight species packing in this model, such a coexistence will be oversensitive to perturbations). Here we consider a different class of competition kernels: those that have a sharp peak at zero niche difference, meaning that their derivative is discontinuous at zero. Several niche gradient models in use today have such kernels: apart from the classical Lotka-Volterra model, the competition-colonization tradeoff model and the tolerance-fecundity tradeoff model (Muller-Landau 2010) are examples. Using analytical and numerical studies, we explored the theoretical and biological implications of such kernels in these models. It turns out that the shape of the kernel has a large impact on model behavior.

Results/Conclusions

We find that, unlike in the case of smooth kernels, robust coexistence of similar species is indeed possible if the competition kernel has a sharp peak at zero niche difference. The reason for this discrepancy between models with peaked and smooth kernels is that a smooth kernel decreases only quadratically around its maximum so that competition between similars will be just as strong as intraspecific competition to first order, making the coexistence overly sensitive to perturbations. A peaked kernel on the other hand decreases linearly around zero even for small niche differences and therefore competition between similars can decrease fast enough to allow for their coexistence. In the context of linear resource utilization (Lotka-Volterra) models, we find that the discontinuity of the utilization function is what creates peaked kernels. Interpreted within the context of population regulation, a sharp discontinuity means that even those species that are very close on the niche axis are somewhat independently regulated‚ therefore they are in fact not similar at all. We also conjecture that sharp discontinuities in resource utilization are not biologically feasible, and so peaked competition kernels are artifacts of model idealizations. That is, in practice the competition kernel should always be smooth.