A thermodynamical model for population growth with relaxation phenomena

Reaction-diffusion models were used in dynamic fluid, population growth, pulse propagation in nerves and other biological phenomena. Some of these models have been expanded to describe memory effects in diffusion and therefore with the use of hyperbolic equations deriving from the generalization of the Fourier and Fick laws. These generalizations come from the theory of extended irreversible thermodynamics (EIT) which is based on kinetic theory arguments. Recently it has been shown that, using the procedures of the classical irreversible thermodynamics with internal variables (CIT-IV), we can obtain equations for the dissipative flows that generalize the laws of Fourier-Fick and Cattaneo-Vernotte. In this paper, using the methodology of CIT-IV, we propose a new model that includes the effect of memory in the diffusion highlighting the presence of two relaxation times. The diffusion flow obtained is characterized by the sum of a parabolic and a hyperbolic contribution which allows the formulation of a dynamic system. As example the traveling waves solutions in the case of the logistic growth are characterized.