A co-evolutionary model of mutualism from a commensal association on Lotka-Volterra dynamics

This paper considers a Lotka-Volterra type of model of competition between a commensal pair of species and a mutualistic pair, presumed to have descended from a single ancestral pair of species. The species are behaviourally isolated but compete for resources. We have studied in detail the effects of the benefits of mutualism over commensalism. Specifically, it is shown that co-evolution of the mutalistic pair from the commensal association is not possible if (i) some of the interacting parameters coincide for the two pairs, or (ii) mutualists suffer more from competition, or (iii) mutualists produce a smaller competitive effect. If the costs of mutualism are density-independent, co-evolution of the mutualism is possible provided (i) the specific growth rate(s) of one or both of the mutualistic species is positive at the steady state of the commensal pair and (ii) the specific growth rate(s) of one or both of the commensal species is positive at the steady state of the mutualistic pair. Lastly, whenever the mutualists co-exist with the commensalists and the feasible interior equilibrium of the model is unstable, we have shown that the co-evolutionary model of mutualism collapses to the stable two-species mutualistic relationship, causing the ultimate extinction of both the species of the original commensal association.