# Global analysis of a simplified model of anaerobic digestion and a new result for the chemostat

## TL;DR

This paper provides a comprehensive global analysis of a simplified anaerobic digestion model, proving convergence to equilibrium and exploring the impact of environmental perturbations through stochastic algorithms.

## Contribution

It introduces a global stability analysis for a simplified ADM1 model and develops algorithms to simulate environmental effects on anaerobic digestion.

## Key findings

- No periodic orbits exist even with bistability.
- Solutions always converge to an equilibrium.
- Environmental perturbations significantly affect transient dynamics.

## Abstract

A. Bornh\"oft, R. Hanke-Rauschenbach, and K. Sundmacher, [Nonlinear Dyn., 73 (2013), pp. 535-549] introduced a qualitative simplification to the ADM1 model for anaerobic digestion. We obtain global results for this model by first analyzing the limiting system, a model of single species growth in the chemostat in which the response function is non-monotone and the species decay rate is included. Using a Lyapunov function argument and the theory of asymptotically autonomous systems, we prove that even in the parameter regime where there is bistability, no periodic orbits exist and every solution converges to one of the equilibrium points. We then describe two algorithms for stochastically perturbing the parameters of the model. Simulations done with these two algorithms are compared with simulations done using the Gillespie and tau-leaping algorithms. They illustrate the severe impact environmental factors may have on anaerobic digestion in the transient phase.