A global asymptotic stability condition for a Lotka–Volterra model with indirect interactions

ABSTRACT

A two-species Lotka–Volterra model extended with an arbitrary number of indirect interactions through diffusible and renewable compounds is presented in view of its considerable interest to the microbial community modelling. After the determination of the system’s fixed points and a short discussion over their local asymptotic stability, Lyapunov’s second method is applied to derive a sufficient condition of global asymptotic stability. Biologically, this condition indicates the necessity for one microbial type to show strong self-inhibition and the compounds to be quickly replaced.

KEYWORDS:

• Global stability
• Lyapunov’s second method
• ecological modelling
• microbial interactions

AMS SUBJECT CLASSIFICATIONS:

• 34D23
• 37B25
• 37C25
• 92D40

Acknowledgements

The author would like to thank Samuel Alizon, Yannis Michalakis, Yves Dumont and Alain Rapaport for their helpful comments and the French Ministry of Higher Education and Research, CNRS and IRD for their support.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by Ministàre de l’Education Nationale, de l’Enseignement Superieur et de la Recherche.