# Entropy production of cyclic population dynamics

Benjamin Andrae, Jonas Cremer, Tobias Reichenbach, Erwin Frey

Systems with a large number of interacting particles are almost always too complex to analyse on the level of individual particle variables. In equilibrium thermodynamics, entropy is a global variable - applying to the system as a whole - that very efficiently characterizes the system's behaviour. However, many biological and ecological systems operate far from thermal equilibrium, and therefore the concept of thermodynamic entropy can no longer be naively applied to them. Still, it is very desirable to find global variables that provide a characterization of the system.

Noting that all non-equilibrium states require driving by external forces to avoid eventual relaxation into thermodynamic equilibrium, we consider the thermodynamic entropy constantly produced in the environment by these driving forces a good candidate for a useful global variable. This is confirmed by our investigation of a system in population dynamics where three species exhibiting cyclic dominance are described by a master equation. We relate the transition rates of this description with the external driving forces, and evaluate the entropy production in simulations and with very successful analytic approximations. The results show that the maximum of entropy production thus defined coincides almost exactly with the system's characterizing critical behaviour, proving that it is indeed a very useful global variable.