Ecosystems are systems made up of large collections microscopic entities. Therefore, a statistical mechanical approach to these systems may yield reliable predictions of their macroscopic properties. Here we examine the predictions of an ecosystem model which assigns to each possible macroscopic trajectory of the system, during a given time interval, a probability which is proportional to its microscopic multiplicity, that is the number of ways it can be realized microscopically. We show that, for sufficiently large systems, this model predicts quasi-deterministic macroscopic dynamics, even though all microscopic trajectories are equiprobable. We give an explicit formalism for these most probable dynamics and point out some of its strikingly realistic properties. We argue that the interplay of microscopic multiplicities reflected in our model could provide a sufficient and basic explanation for macroscopic ecosystem dynamics.