Periodic trajectories of distributed parameter biochemical systems with time delay

This paper deals with a model of a biochemical reactor system with distributed parameters and with a time delay in the growth response. Time delay has been introduced in microbial growth systems to explain the time lapse between the consumption of (liquid) substrate and its conversion to (solid) biomass. We study here the properties of the resulting system of partial functional differential equations. We first prove the existence, positivity, and a compactness property of the system trajectories. We then prove the existence of periodic solutions of the system for large values of the delay. Numerical simulations illustrate the existence of such solutions. The aim of this work is to study the existence of periodic solutions for a model of a biochemical reactor system with distributed parameters and a time delay in the growth response. Periodic solutions (or oscillations) appear in many chemostat experiments (see [17,18,22] and references therein), where both transient evolutionary and stable autonomous oscillations have been observed. These experimental studies reveal a deficiency in classical growth models (such as Monod or Haldane models), since these are unable to account for oscillations. Considering a time delay in the growth response as a potential cause of oscillations has been emphasized by many authors. The pioneers in this direction are Fin and Wilson [14] who observed sustained oscillations of a yeast population in a chemo-stat and discussed a linear model with discrete delay. Caperon [8] used a modified Monod model and found sustained oscillations for large enough discrete delays in the growth response of Isochrysis galbana population. By considering a distributed delay and using numerical methods, he observed damped oscillations. Droop [12] introduced the notion of internal nutrient and added an additional variable to the system of differential equations. However, in [9], the authors observed that the Droop's model cannot produce the oscillations in cell numbers that they observed experimentally in the chemostat. In [17,28], the effect of delay in simple chemostat models was discussed. Thingstad and Langland found that for a large enough delay the equilibrium becomes unstable and by computation they exhibit a limit cycle. The delay influence on nonmonotone growth response or on models with multi-species pure and simple competition was also explored (see [7,27,29,31,32]). For additional review and discussion on time delay in chemostat models and their qualitative effect on transient dynamics, see [18].