Given monotone operators on the positive orthant in n-dimensional Euclidean space, we explore the relation between inequalities involving those operators, and induced monotone dynamical systems. Attractivity of the origin implies stability for these systems, as well as a certain inequality. Under the right perspective the converse is also true. In addition we construct an unbounded path in the set where trajectories of the dynamical system decay monotonically, i.e., we solve a positive continuous selection problem.