Population dynamics of resource limited plants and their pollinators.

In this paper we build upon and generalize an earlier model of the interactions between a plant and its pollinator (Ingvarsson and Lundberg, 1995). In this model we assume that the performance of the pollinator population is directly linked to the size of the plant population. To avoid the problem of both populations growing exponentially we have, without loss of generality, assumed the plant population to be resource limited. Analysis of the system shows that there exists either two or no internal equilibrium points. The case with no equilibrium points corresponds to the trivial case where the system cannot persist, resulting in the extinction of both the plant and pollinator population. When the two internal equilibrium points do exist, one of them will always be unstable. This unstable equilibrium can be viewed as an equivalent of the threshold criteria derived in Ingvarsson and Lundberg (1995) in the sense that whenever the system is initiated above the unstable equilibrium point, persistence of the system is assured, while both species will go extinct whenever the system is initiated below the unstable equilibrium point. The analytical results were verified by numerical simulations of the system. We conclude that the existence of a threshold criteria, below which the system cannot persist is a general feature of plant-pollinator systems. We discuss how the existence of the threshold criteria will affect the persistence of plant-pollinator systems in light of, for instance, habitat fragmentation or stochastic reductions in the densities of either the plant or pollinator population. We further highlight some recent empirical studies that indicate the existence of a threshold in natural populations below which extinction is inevitable.