Stochastic dynamics of the delayed chemostat with Lévy noises

Abstract

This paper formulates and studies a delayed chemostat with Lévy noises. Existence of the globally positive solution is proved first by establishing suitable Lyapunov functions, and a further result on exact Lyapunov exponent shows the growth of the total concentration in the chemostat. Then, we prove existence of the uniquely ergodic stationary distribution for a subsystem of the nutrient, based on this, a unique threshold is identified, which completely determines persistence or not of the microorganism in the chemostat. Besides, recurrence is studied under special conditions in case that the microorganism persists. Results indicate that all the noises have negative effects on persistence of the microorganism, and the time delay has almost no effects on the sample Lyapunov exponent and the threshold value of the chemostat.