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Updated: Aug 10, 2026

The Use of Chemostats in Microbial Systems Biology
Published on: October 15, 2013
Numerical proof for chemostat chaos of Shilnikov's type
Bo Deng1, Maoan Han1, Sze-Bi Hsu2
1Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China.
Abstract:
A classical chemostat model is considered that models the cycling of one essential abiotic element or nutrient through a food chain of three trophic levels. The long-time behavior of the model was known to exhibit complex dynamics more than 20 years ago. It is still an open problem to prove the existence of chaos analytically. In this paper, we aim to solve the problem numerically. In our approach, we introduce an artificial singular parameter to the model and construct singular homoclinic orbits of the saddle-focus type which is known for chaos generation. From the configuration of the nullclines of the equations that generates the singular homoclinic orbits, a shooting algorithm is devised to find such Shilnikov saddle-focus homoclinic orbits numerically which in turn imply the existence of chaotic dynamics for the original chemostat model.
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