The impact of a predator on the outcome of competition in the three-trophic food web

We study the effects of predation on the competition of prey populations for two resources in a chemostat. We investigate a variety of small food web compositions: the bi-trophic food web (two resources-two competing prey) and the three-trophic food web (two resources-two prey-generalist predator) comparing different model formulations: substitutable resources and essential resources, namely Liebig's minimum law model (perfect essential resources) and complementary resources formulations. The prediction of the outcome of competition is solely based on bifurcation analysis in which the inflow of resources into the chemostat is used as the bifurcation parameter. We show that the results for different bi-trophic food webs are very similar, as only equilibria are involved in the long-term dynamics. In the three-trophic food web, the outcome of competition is manifested largely by non-equilibrium dynamics, i.e., in oscillatory behavior. The emergence of predator-prey cycles leads to strong deviations between the predictions of the outcome of competition based on Liebig's minimum law and the complementary resources. We show that the complementary resources formulation yields a stabilization of the three-trophic food web by decreasing the existence interval of oscillations. Furthermore, we find an exchange of a region of oscillatory co-existence of all three species in Liebig's formulation by a region of bistability of two limit cycles containing only one prey and the predator in the complementary formulation.