Entropy production in the cyclic lattice Lotka-Volterra model

The cyclic Lotka-Volterra model in a D-dimensional regular lattice is considered. Entropy production of its “nucleus growth” mode is investigated by analyzing the time evolution of the family of entropies $S_q=(1-\sum_i p_i^q)/(q-1)$ , with $q\in \mathbb{R}$ . This family contains as particular case (q=1) the usual entropic form $S_1=-\sum_i p_i\ln p_i$ . The rate of growth of the entropy S q , for some $q\neq 1$ , is expected to provide non-trivial information about certain complex systems. For the system here considered, it is shown, both numerically and by means of analytical considerations, that a linear increase of entropy with time, meaning finite asymptotic entropy rate, is achieved for the entropic index q c =1-1/D, as previously conjectured in the literature. However, although $q_c\neq 1$ , this relation can be explained in terms of very simple features not directly connected to the complexity of the dynamics. The relation between the characteristic entropic index and lattice dimensionality is shown to be a consequence of the fact that the system soon approaches a steady regime where the nucleus radius grows linearly with time. Copyright Springer-Verlag Berlin/Heidelberg 2004