Approximate analytic solutions to a nonlinear digester problem

Biological reactors are employed in industrial applications to break down organic waste from a range of sources into components that may be used in other applications. Such reactors may involve complex processes and many components linked by complicated interrelations. These reactions are represented mathematically as nonlinear initial value problems that must be solved numerically. Even smaller systems, more amenable to analytical analysis, require numerical solution methods due to their nonlinearity. We study a simple reactor with only two interacting components—a bacteria consuming a substrate (waste), represented by a 2×2 autonomous nonlinear initial value problem not solvable analytically. We describe a process to convert this problem to an approximating linear one that can be solved exactly to provide a closed form approximate representation of the evolving system. We assess the results of this approach and show they often agree favourably with numerical computations of the original nonlinear problem, although not always.