Periodic Two-Predator, One-Prey Interactions and the Time Sharing of a Resource Niche

A competition model involving two competing predator species and a single renewable resource prey species is studied under the assumption that the system parameters are periodic in time. It is shown by means of global bifurcation techniques that a continuum of positive periodic solutions exists as a function of a selected (averaged) parameter and that the stability of these solutions (at least locally near bifurcation) depends on the direction of bifurcation. In the special autonomous case of constant parameters the bifurcation is vertical and the spectrum of the continuum is a discrete point. This autonomous case supports the principle of competitive exclusion in that coexistence of the predators on the single resource prey is possible (in the sense that the system equilibrates) only on a parameter set of measure zero. In the more general case of periodic coefficients, however, it is shown that the spectrum can be an interval of positive length provided the predator parameter oscillations are out-of-phase in a certain sense and hence how such oscillations can promote the possibility of stable coexistence. The specific case, when all system parameters are constant except the predator resource consumption rates which are taken to be small amplitude cosine oscillations around a positive mean value, is studied in detail both analytically and numerically. Besides illustrating and corroborating the general results, this example clearly shows the effect on the spectral interval, and hence on the possibility of stable coexistence of the predators, which out-of-phase resource consumption rates can have.