Irreversibility of biological systems

In a context of Markov processes, an ecological model of three interacting species is studied, which dynamics is described by the master equation. It is presented a theoretical method through which to estimate the entropy production of the system along its evolution and its link with the concept of irreversibility of the system state. At each step only two species interact each others through a birth-death or mutation mechanism, bringing an increasing/decreasing by one unit of the number of the specific individuals. All the information about the interactions are encoded in the transition rates of the model which depend by a competition matrix and a mutation matrix. It is also imposed a cyclical competition in these coefficients. Using a van Kampen system’s size expansion the deterministic equations to which the model approaches when the total number of individuals tend to infinity are obtained. These assumes the form of antisymmetric Lotka-Volterra differential equations.The Fokker-Planck equations for this model are derived analytically. Numerical simulations of the system evolution at different initial conditions are performed. It is shown that the system present two different type of dynamics. In particular using the Fokker-Planck equations is shown that if the mutation coefficients are above the critical value of 1, any species can cyclically dominates over the others. The entropy production in this case is significantly higher than a condition in which a species prevails. We study the analytic expression of the production of entropy on the plane in which the sum of the three species concentrations is constantly 1. Furthermore it peaks at the point in which all the concentration are equal. In the final plot are shown the two entropy productions regimes for higher/lower values of critical mutation rate.