Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere

作者：Morris W. Hirsch · 发表于：SIAM Journal on Mathematical Analysis · 年份：1985 · DOI：10.1137/0516030 · 被引用次数：546 · 研究领域：Advanced Differential Equations and Dynamical Systems、Mathematical and Theoretical Epidemiology and Ecology Models、Nonlinear Dynamics and Pattern Formation

A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all of the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The main results in this article are the following. A cooperative system cannot have nonconstant attracting periodic solutions. In a cooperative system whose Jacobian matrices are irreducible the forward orbit converges for almost every point having compact forward orbit closure. In a cooperative system in 2 dimensions, every solution is eventually monotone. Applications are made to generalizations of positive feedback loops.