Reaction transport systems in biological modelling

Memory effects in a turbulent dynamo: Generation and propagation of a large-scale magnetic field

“…Due to this nonphysical property of the diffusion solution, the FKPP equation yields an overestimation of the propagation speed of travelling fronts [14]- [16]. We expect that a similar situation might take place in magnetohydrodynamics.…”

“…In fact, it can be reduced to a famous Fisher-Kolmogorov-PetrovskiiPiskunov (FKPP) equation [5,9], which has become a basic mathematical tool in the theory of propagating fronts travelling into the unstable state of the reaction-diffusion systems. There has been an increased interest in this topic, because of the large number of physical, chemical and biological problems that can be treated in terms of the FKPP equation (see, for example, [12]- [14]). Recently, there has been a tremendous activity to extend this analysis by introducing more realistic description of the transport processes.…”

Supersymmetry in random two-velocity processes

“…(2) implies that probability propagates with an infinite velocity. This feature may lead to difficulties for the physical interpretation see for instance [2,3].…”

“…(3) goes back to M. Kac [7] (see, [8] for a recent comprehensive overview on the history of random evolutions). As emphasised in [2,3], the persistent random walk provides a better description of spatial spread in population dynamics than Brownian motion. Defining the total density P (x, t) and the current Q(x, t) as:…”

Class of self-limiting growth models in the presence of nonlinear diffusion

“…Since in many living organisms concentration dependent diffusivity [4,5,[11][12][13][14][15] has been found to be essential to the modeling of reaction-diffusion systems we investigate the interplay of this nonlinear diffusion and selflimiting growth process in the dynamics. We show that the model and its variant with a finite memory transport [16][17][18][19][20][21][22][23][24][25] admit of exact solutions. The dependence of the rate of spread of the wave front on various parameters is explored.…”