Competition for fluctuating nutrient
A model of the competition of n species for a single essential periodically fluctuating nutrient is considered. Instead of the familiar Michaelis-Menten kinetics for nutrient uptake, we assume only that the uptake rate functions are positive, increasing and bounded above. Sufficient conditions for extinction are given. The existence of a nutrient threshold under which the Principle of Competitive Exclusion holds, is proven. For two species systems the following very general result is proven: All solutions of a τ-periodic, dissipative, competitive system are either τ-periodic or approach a τ-periodic solution. A complete description of the geometry of the Poincaré operator of the two species system is given.