For conservative linear systems (finite-state Markov processes in discrete or continuous time), the relative entropy of two distinct trajectories is a monotonically decreasing function of time. These results naturally raise the question whether distinct trajectories of nonlinear conservative systems also display monotonically decreasing relative entropy. For binary interacting Lotka-Volterra systems with anti-symmetry, the relative entropy oscillates under the motion. The main new result of this paper is that, for ternary interacting Lotka-Volterra systems with anti-symmetry, the relative entropy of two distinct trajectories is a monotonically decreasing function of time near equilibrium. Far from equilibrium, distinct trajectories of ternary Lotka-Volterra systems with anti-symmetry need not have monotonically decreasing relative entropy.