Quantifying the likelihood of co-existence for communities with asymmetric competition

Trade-offs in performance of different ecological functions within a species are commonly offered as an explanation for co-existence in natural communities. Single trade-offs between competitive ability and other life history traits have been shown to support a large number of species, as a result of strong competitive asymmetry. We consider a single competition-fecundity trade-off in a homogeneous environment, and examine the effect of the form of asymmetry on the likelihood of species co-existing. We find conditions that allow co-existence of two species for a general competition function, and show that (1) two species can only co-exist if the competition function is sufficiently steep when the species are similar; (2) when competition is determined by a linear function, no more than two species can co-exist; (3) when the competition between two individuals is determined by a discontinuous step function, this single trade-off can support an arbitrarily large number of species. Further, we show analytically that as the degree of asymmetry in competition increases, the probability of a given number of species co-existing also increases, but note that even in the most favourable conditions, large numbers of species co-existing along a single trade-off is highly unlikely. On this basis, we suggest it is unlikely that single trade-offs are able to support high levels of bio-diversity without interacting other processes.