Fixed point global attractors and repellors in competitive Lotka-Volterra systems

摘要

Zeeman and Zeeman [E.C. Zeeman and M.L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems, Trans. Amer. Math. Soc. 355 (2003), pp. 713-734] show that if a strongly competitive Lotka-Volterra system (ⅰ) has a unique interior fixed point p and (ⅱ) the carrying simplex Σ lies below (above) the strongly balanced tangent plane to Σ at p then the system has no periodic orbits and p is a global attractor (repellor) relative to Σ. Condition (ⅱ) is then translated into the definiteness of a certain quadratic function on the tangent plane, which is equivalent to the definiteness of an (N- 1) × (N- 1) real symmetric matrix that can be computed. Here we adapt these methods to show that the above conclusions are still true without the assumption (ⅰ). Hence, our results apply to globally attracting or repelling fixed points on the boundary, as well as in the interior, of R~N+. Moreover, the algebraic condition for global attraction also implies global asymptotic stability of the fixed point. We also show that the global attraction holds not just relative to E, but also relative to the interior of the first quadrant.