Wavefronts in time-delayed reaction-diffusion systems. Theory and comparison to experiment

Abstract

We review the recent theoretical progress in the formulation and solution of the front speed problem for time-delayed reaction-diffusion systems. Most of the review is focused on hyperbolic equations. They have been widely used in recent years, because they allow for analytical solutions and yield a realistic description of some relevant phenomena. The theoretical methods are applied to a range of applications, including population dynamics, forest fire models, bistable systems and combustion wavefronts. We also present a detailed account of successful predictions of the models, as assessed by comparison to experimental data for some biophysical systems, without making use of any free parameters. Areas where the work reviewed may contribute to future progress are discussed.

Vertical cultural transmission effects on demic front propagation: Theory and application to the Neolithic transition in Europe

“…After integrating Eq. (52), we assume as usual that the minimum speed is that of the front [48] and obtain c = min…”

Front Speed of Language Replacement

“…For each value of time t; in general the front pro…le p N (x; y; t) depends on both x and y (this is easily seen by considering, e.g., the case of a circular front). However, let us choose the x axis parallel to the local velocity of the front [34]. Then, it is reasonable to assume that for t !…”

“…Then, it is reasonable to assume that for t ! 1 the curvature will become negligible and the front pro…le p N will depend only on (16) and assuming that the minimum speed is the one of the front [34] we obtain the speed given by Eq. (7).…”

“…and applying linear analysis for = 1 [18,34] (as in Appendix B above), the front speed is found to be c = 2…”

Cultural Collapse and System Survival Due to Environmental Modification

“…One of the most prevalent of these is Fisher's reaction-diffusion equation [10,11], which gives rise to a "Wave of Advance" of cultural developments, the classic case being the arrival of Neolithic farming in western Europe [12][13][14][15][16]. Here, the model borrowed ideas from theoretical biology and showed that the required diffusion and logistic growth parameters required to match the archaeological data for wave speed were reasonable.…”