Curvature dependences of wave propagation in reaction–diffusion models

Abstract

Reaction–diffusion waves in multiple spatial dimensions advance at a rate that strongly depends on the curvature of the wavefronts. These waves have important applications in many physical, ecological and biological systems. In this work, we analyse curvature dependences of travelling fronts in a single reaction–diffusion equation with general reaction term. We derive an exact, non-perturbative curvature dependence of the speed of travelling fronts that arises from transverse diffusion occurring parallel to the wavefront. Inward-propagating waves are characterized by three phases: an establishment phase dominated by initial and boundary conditions, a travelling-wave-like phase in which normal velocity matches standard results from singular perturbation theory and a dip-filling phase where the collision and interaction of fronts create additional curvature dependences to their progression rate. We analyse these behaviours and additional curvature dependences using a combination of asymptotic analyses and numerical simulations.

Materials Design Innovations in Optimizing Cellular Behavior on Melt Electrowritten (MEW) Scaffolds

“…[149] Similar to other mathematical biology approaches, emerging cell growth models are often categorized as continuum, discrete, or hybrid. [150] Continuum models may resemble systems of ordinary or partial differential equations (ODE/PDEs) and have been described as population balance models as they describe the average behavior of communities of cells in time and space [151][152][153] Tissue growth is often nonlinear, with feedback loops and complex regulatory mechanisms. Population dynamics can account for this nonlinearity as they operate on individual interactions and behaviors of entities within a population.…”

Discrete and continuous mathematical models of sharp-fronted collective cell migration and invasion

“…Continuum partial differential equation (PDE) models have been used for over 40 years to model and interpret the spatial spreading, growth and invasion of populations of cells [1][2][3]. PDE models have been used to improve our understanding of various biological processes including wound healing [4][5][6][7][8][9], embryonic development [10][11][12][13], tissue growth [14][15][16] as well as disease progression, such as cancer [17][18][19][20][21][22][23][24]. For a homogeneous population of cells with density u ≥ 0, a typical PDE model can be written as…”

“…A standard choice for the source term is to specify a logistic term to represent carrying capacity-limited proliferation, true0 MJX-tex-caligraphicscriptS = λ u ( 1 − u / K ) , where true0 λ > 0 is the proliferation rate and true0 K > 0 is the carrying capacity density [3,7,8]. These choices of true0 J and true0 MJX-tex-caligraphicscriptS mean that equation (1.1) is a multi-dimensional generalization of the well-known Fisher–Kolmogorov model [26–29], which has been successfully used to interpret a number of applications including in vivo tumour progression [18], in vivo embryonic development [10], in vitro wound healing [7,8] and tissue growth [15,16].…”