Considering Subpopulations in Modelling Facultative Mutualism Reveals a New Approach to Model Interspecific Interactions
Mathematical modelling of mutualism usually uses generalized Lotka-Volterra equations in which the mutualistic benefit is represented by a positive bilinear interaction term. We propose a minimal ODE model for facultative mutualism between two species that is based on the differentiation of two subpopulations per species.
In facultative mutualism between two species, each species benefits from interacting with the other species but does not rely on this interaction to exist and grow. At the species level there is a subdivision into the mutualistic subpopulation that actively interacts with the other species, and the non-mutualistic subpopulation that does not interact with the other species. The non-mutualistic subpopulation behaves the same way the total species’ population would in absence of the other species, but the mutualistic subpopulation additionally receives a benefit due to the mutualistic interaction with the other species’ mutualistic subpopulation. Therefore, in our model the interaction term is not dependent on the density of the two species, as it is the case in the generalized Lotka-Volterra equations, but only on the density of the two mutualistic subpopulations. We also took the intraspecific switch of individuals between the two subpopulations into account.
Every mutualistic benefit automatically includes a cost that must be spent in order to provide the benefit for the other species. We investigated to what extent the cost influences the proportion of mutualistic subpopulation in the total species population. If the cost exceeds the benefit for one species, the net-effect of the interaction becomes negative for that species. In that case, the situation no longer refers to mutualism, but to parasitism. We claim that due to this mechanism our model is able to represent several interspecific interactions.