Strictly Cooperative Systems with a First Integral

Source abstract

We consider systems of differential equations ${{dx_i } / {dt}} = F_i (x_1 , \cdots ,x_n )$ in the nonnegative orthant in the n-space satisfying the following hypotheses: i) $F(0) = 0$; ii) if $x_i < y_i$ and $x_j = y_j$ for $j \ne i$ then $F_k (x) < F_k (y)$ for $k \ne i$; iii) F possesses a first integral with positive gradient. We prove that every solution to such a system either converges to an equilibrium or eventually leaves any compact set.