Thermodynamic and variational aspects in modelling and simulation of biological evolution

Entropy-based models, quantifying the critical phenomena with an increase and reduction of organs in multiorgan organisms, are extended in this paper by inclusion of non-classical statistical entropies, e.g. q-entropies of Tsallis or Renyi, that may modify magnitudes of unstable regions in the space of process probabilities. We report computer-aided modeling and simulation of evolution in biological systems with living organisms as an effect of extremum properties of classical statistical entropy of Gibbs-Boltzmann type or its associates, e.g. Tsallis q-entropy. Evolution for animals with multiple organs is considered and rationale for an initial increase of organ number is substantiated. A variational problem searches for the maximum entropy subject to geometric constraint of the constant thermodynamic distance in the non-Euclidean space of independent probabilities pi plus possibly other constraints. Tensor form of dynamics is obtained.Some developmental processes are shown to progress in a relatively undisturbed way, whereas others may terminate rapidly due to inherent instabilities. For processes with variable number of states the extremum principle provides quantitative investigation of biological development. The results show that a discrete gradient dynamics (governed by the entropy) can be predicted from variational principles for shortest paths and suitable transversality conditions.