We consider the following standard partial order relation on $\mathbb{R}^n$:
We say $X=(x_1,x_2,\ldots,x_n)\leq (y_1,y_2,\ldots,y_n)=Y$ iff $\sum_{i=1}^k x_i \leq \sum_{i=1}^k y_i,\quad \forall k: 1\leq k<n,\quad \sum_{i=1}^n x_i=\sum_{i=1}^n y_i$
Is there a vector field $Z$ on $\mathbb{R}^n$ with the monoton property $X\leq Y\implies \phi_t (X) \leq \phi_t(Y),\quad \forall t>0$ where $\phi$ is the flow of $Z$? What is a classification of all vector fields with this monoton property? Can this property be equivalent to certain algebraic or semi algebraic conditions onthe Jacobian of vector field $Z$?
Note: If the above equivalent relation would be replaced with the relation $X\leq Y \iff x_i\leq y_i$ then the problrm is fully known, the competitive system introduced by M.W. Hirsch.
