Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: Application to the theory of Neolithic transition

Abstract

We introduce a general method for the systematic derivation of nonlinear reaction-diffusion equations with distributed delays. We study the interactions among different types of moving individuals (atoms, molecules, quasiparticles, biological organisms, etc). The motion of each species is described by the continuous time random walk theory, analyzed in the literature for transport problems, whereas the interactions among the species are described by a set of transformation rates, which are nonlinear functions of the local concentrations of the different types of individuals. We use the time interval between two jumps (the transition time) as an additional state variable and obtain a set of evolution equations, which are local in time. In order to make a connection with the transport models used in the literature, we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so-called generalized master equations. The method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations, hyperbolic reaction-diffusion equations, and delay-differential reaction-diffusion equations. Thus in the analysis of a given problem we can fit to the data the type of reaction-diffusion equation and the corresponding physical and kinetic parameters. The method is illustrated, as a test case, by the study of the neolithic transition. We introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction-diffusion equation for the population density. We derive a delay evolution equation for the rate of advance of agriculture, which illustrates an application of our analysis.

Mathematical Modelling of Subdiffusion-reaction Systems

“…The first term on the right-hand side of (3.12) corresponds to the decrease of the density of particles that belong to a definite generation due to subdiffusion, and the second term describes its change due to the chemical reaction, which can be either negative or positive, unlike the models with "rejuvenation" of particles [23], [24], where it can be only negative. Equation (3.12) is supplemented by the boundary condition…”

“…In the case of "rejuvenation" of particles after the chemical reaction [23], it is necessary to distinguish between the "birth" rate, r + (u), and the "death" rate, r − (u), which leads to the following integrodifferential equation,…”

“…Another approach is adopted in [23], [24] where it is assumed that a molecule born in the course of reaction has zero age, as if it had arrived at the reaction point from somewhere else: the argument of the waiting time destribution is set to zero after the chemical transformation of the molecule. In that case, equation (3.12) includes only reactions leading to the "death" of molecules, which are assumed to be independent of the particle age, thus…”

“…In the case of a linear reaction a closed system of fractional PDEs governing concentrations of species can be constructed [21], [22]. In the case of a nonlinear reaction, the use of two temporal variables, the actual time and the time elapsed since the last jump (the "age" of the particle) is convenient [23] - [26]. A somewhat controversial point is the age prescribed to a particle after the reaction event.…”

“…A somewhat controversial point is the age prescribed to a particle after the reaction event. In [23], [24] the particle is "rejuvenated" after it reacts, i.e., its waiting time distribution is set as if the particle has performed a jump. In [21], [27], [28], [25] it is assumed that the reaction does not change the age of the particle.…”