Impulsive perturbation and bifurcation of solutions for a model of chemostat with variable yield

Abstract

In this paper, we consider a variable yield model of a single-species growth in a well-stirred tank containing fresh medium, assuming the instances of time as triggering factors in which the nutrient refilling process and the removal of microorganisms by the uptake of lethal external antibiotic are initiated. It is also assumed that the periodic nutrient refilling and the periodic antibiotic injection occur with the same periodicity, but not simultaneously. The model is then formulated in terms of autonomous differential equations subject to impulsive perturbations. It is observed that either the population of microorganisms essentially washes out, or more favorably, the system is permanent. To describe this dichotomy, some biologically significant integral conditions are introduced. Further, it is shown that in a certain critical situation, a nontrivial periodic solution emerges via a bifurcation phenomenon. Finally, the dynamics of the model is illustrated with numerical experiments and computer simulations.

Global stability for a model of competition in the chemostat with microbial inputs

“…d) Other approaches: as crowding effects [7], flocculation [12], multi-substrate feeding [11], [23], impulsive input of substrate concentration [28], [29], [44], intraspecific competition [24].…”

Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties

“…However, it is commonly observed in experiments that multiple competing species can persist in chemostats with one limiting substrate. Numerous methods and theories have been developed to generate or explain coexistence in chemostats, such as crowding effects, feedback controls in which the inputs are functions of the state variables instead of being constant (e.g., in [8,27]), flocculation [11], heterogeneity properties of the medium (as noted in [10,16,36,38]), impulsive use of substrates (as explained in [31,32,47]), intra-species competition [22], multiple substrates [10,20], and deterministic or stochastic time-varying inputs (as explained, e.g., in [2,3,7,14,19,29,30,41,45]). In this work, we study another approach, based on an alternative model that we describe next.…”