A simple NSFD inspired method for Monod kinetics with small half saturation constants in the chemostat setting

Monod kinetics is an important nonlinearity that appears in mathematical modelling of microbial systems, but (under different names) also in many other applications in Mathematical Biology and Process Engineering. Although seemingly innocuous, for some extreme parameter values (notably very small half saturation concentrations and large decay rates), sophisticated high order solvers for ordinary differential equations have been known to fail. We explore this breakdown situation and suggest a simple, low order, easy to implement method that is inspired by so-called Nonstandard Finite Difference or Mickens schemes. We find that these can be a viable alternative to modern initial value problem solvers, in the problematic cases of extreme parameter values.