Obligate mutualism with a predator: Stability and persistence of three-species models

We present a general model for three interacting populations, where one population, called a mutualist, benefits a predator in its interaction with the prey. Biologically, there are four different ways in which the mutualist could benefit the predator: by enhancing prey growth rate, by enhancing the rate of prey capture, by providing an alternative food supply for the predator, and by enhancing the efficiency of utilization of prey, once they are ingested. We discuss examples of each type of interaction. We restrict our model to those situations in which the predator cannot survive on the prey in the absence of the mutualist. Therefore, if mutualism exists, it is obligate for the predator. Other conditions of the model include the dynamics of the prey and the mutualist alone and together in the absence of the predator. Given additional reasonable restrictions on the model, we determine the conditions for persistence, where persistence is defined as the continued existence of all three populations without any of them going extinct. There are two ways in which survival may arise in these models. Under one set of conditions, which is equivalent to the predator being able to invade a prey-mutualist system when rare, persistence will occur for any set of positive critical population sizes. Alternatively, survival will occur if there is an asymptotically stable interior equilibrium. However, the conditions for this are complex, and survival may occur only for initial populations in a limited region around the equilibrium.