From local to global behavior in competitive Lotka-Volterra systems

• Published in: Transactions of the American Mathematical Society Vol. 355; no. 2; pp. 713 - 734
• Main Authors: Zeeman, E. C., Zeeman, M. L.
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-02-2003
• Subjects: Algebra; Animal and plant ecology; Animal, plant and microbial ecology; Biological and medical sciences; Coordinate systems; Demecology; Eigenvalues; Eigenvectors; Exact sciences and technology; Fundamental and applied biological sciences. Psychology; General aspects; Global behavior; Hyperplanes; Liapunov functions; Mathematical analysis; Mathematical theorems; Mathematics; Ordinary differential equations; Periodic orbits; Sciences and techniques of general use; Carrying simplex; split Liapunov function; ruling out recurrence; Volterra-Liapunov; Fix point; Stability; Existence theorem; Differential equation; Dynamical system; Volterra Lyapunov stability; Hopf bifurcation; Lotka-Volterra equations; Hyperplane; Lyapunov function; Equation system
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-02-03103-3

In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out non-trivial recurrence. We thus deduce the global dynamics of a system from its local dynamics. The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point p\in\operatorname{int}{{\mathbf R}^n_+} and the carrying simplex of the system lies to one side of its tangent hyperplane at p, then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.