Mathematical modeling of bioprocesses with the use of fractional order derivatives

Abstract

This work brings together two recently discussed topics: mathematical modeling of a bioreactor and working with derivatives of non-integer order. Generally, it turns out that it is reasonable to replace the integer order derivatives in some of the already well known mathematical models describing bioprocesses with fractional order ones. However, the specific structure of such type of derivatives makes the study of the properties of the models a real challenge. This work contains primary results for modeling of a bioreactor with appropriately selected numerical approximations. Different scenarios are taken into consideration: starting from the simplest one - without mortality and then complicating by adding nonzero mortality term. In the classical case the solution of the system of differential equations describing the process has a specific behaviour in terms of monotonicity. Therefore, the focus of the further examinations is to find out whether it is possible to generalize the model into a fractional order one such that the key properties considering monotonicity still hold. The results show that the latter requires certain dependencies between the orders of the derivatives in the mathematical model. The hypothesis is based on two types of experiments which are described in detail. Lotka-Volterra and Monod specific growth rate are used in the mathematical model. The paper contains figures which illustrate the results from different numerical computations performed via Wolfram Mathematica software.