Stability analysis of a self‐cycling fermentation model with state‐dependent impulse times

Abstract

Artículo de publicación ISIA self-cycling fermenter is a batch fermenter subject to recurrent emptyings of liquid volume followed by the refillingwith new fresh substrate. This article constructs amodified model of self-cycling fermenter, which is described by an impulsive differential equation at impulse-dependent times, which have been recently introduced. The main result is a set of conditions depending of the fraction of removed volume, the concentration of new substrate introduced, and the maximal length between two impulses, which ensure the existence and attractiveness of a periodic cycle. A second result provides alternative conditions for the biomass extinction.This work was supported by FONDECYT 1120218 and FONDECYT 1120709

Growth on two limiting essential resources in a self-cycling fermentor

“…Let u ± k = (s 1 (t ± k ), s 2 (t ± k )). By (6), the trajectory of (s 1 (t), s 2 (t)), t ∈ (t k , t k+1 ), is a line segment with slope R 21 and endpoints u + k and u − k+1 . The conditions for impulses to occur are given in (2).…”

“…Fan and Wolkowicz [7] extended this model to include the possibility that the resource is limiting at large concentrations. Córdova-Lepe, Del Valle, and Robledo [6] also modeled single species growth in the SCF process, but used impulse dependent impulse times instead of the state dependent impulses used by the other models. For references on the theory of impulsive differential equations, see e.g.…”

“…Proof. Solving the separable ODE, ds1 ds2 = R 12 , yields (5), and solving ds2 ds1 = R 21 , yields (6). Dividing the s 1 (t) equation in (3) by the x (t) equation, substituting for s 2 (t) using (6), and then integrating both sides, yields (7).…”

Application of the method of averaging to boundary value problems for differential equations with non-fixed moments of impulse

“…In our paper, we will use the notation and some facts from this monograph. The study of real-life problems with statedependent impulse effects can be found, for example, in [2,6,9].…”