Geometric method for global stability and repulsion in Kolmogorov systems

ABSTRACT

A class of autonomous Kolmogorov systems that are dissipative and competitive with the origin as a repellor are considered when each nullcline surface is either concave or convex. Geometric method is developed by using the relative positions of the upper and lower planes of the nullcline surfaces for global asymptotic stability of an interior or a boundary equilibrium point. Criteria are also established for global repulsion of an interior or a boundary equilibrium point on the carrying simplex. This method and the theorems can be viewed as a natural extension of those results for Lotka-Volterra systems in the literature.

KEYWORDS:

• Equilibrium point
• global attraction
• global repulsion
• global asymptotic stability
• geometric method

AMS 2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary: 37B25
• Secondary: 37C70
• 34D23
• 34D05
• 92D25

Acknowledgements

The author is grateful to the anonymous referees for their kind comments and helpful suggestions in revising the previous draft of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.