From local to global behavior in competitive Lotka-Volterra systems

Zeeman, E.; Zeeman, M.

doi:10.1090/S0002-9947-02-03103-3

From local to global behavior in competitive Lotka-Volterra systems
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by E. C. Zeeman and M. L. Zeeman PDF

Trans. Amer. Math. Soc. 355 (2003), 713-734 Request permission

Abstract:

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.

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