Finding adequate descriptions of multi-agent systems is a central issue for modelling collective dynamics. We propose a mathematical description of multi-agent systems as discrete dynamical systems. The ground of our proposition is the influence-reaction method of Ferber and Müller. The key idea is that agents should never act directly on other components of the system (agents or environment) but release influences which are then combined to update the state of the system. We propose a method which decomposes the definitions of multi-agent system into six parts. We illustrate our method on the multi-Turmite model, also known as the multiple Langton's ants model. We exhibit two formulations of this model, which we study with three different simulation schemes. We show that for the same formulation, and the same initial conditions, the use of different simulation schemes may lead to qualitatively different evolutions of the system. As a positive spin-off of this study, we exhibit new phenomena of the multi-Turmite model such as deadlocks or gliders.