Competitive and Cooperative Tridiagonal Systems of Differential Equations

Abstract

A vector field in $\mathbb{R}^n $ determines a competitive (or cooperative) system of differential equations provided all the off-diagonal terms of the Jacobian matrix are nonpositive (or nonnegative). We show that when in addition ${{\partial F_i } / {\partial X_j }}$ is zero for $|i - j| > 1$, all solutions must converge to equilibria or diverge in a strong sense.