Radial propagation in population dynamics with density-dependent diffusion

Abstract

Population dynamics that evolve in a radial symmetric geometry are investigated. The nonlinear reaction-diffusion model, which depends on population density, is employed as the governing equation for this system. The approximate analytical solution to this equation is found. It shows that the population density evolves from the initial state and propagates in a traveling-wave-like manner for a long-time scale. If the distance is insufficiently long, the curvature has an ineluctable influence on the density profile and front speed. In comparison, the analytical solution is in agreement with the numerical solution.

Traveling wavefronts for density‐dependent diffusion reaction convection equation with time delay

“…from which we can directly get (18). For c * < c < 2𝑓 ′ (0) 𝛼 , we see that v c (u) ∼ 𝑓 ′ (0) c u 2 or v c (u) ∼ cu from Equation (13). By Lemma 2.3, v c (u) is decreasing with respect to the wave speed c > 0.…”

“…In recent years, reaction diffusion models with density-dependent diffusion mechanism are widely found from porous media and population dynamics. [8][9][10][11][12][13][14][15] If the movement of an individual is modeled as a random walk, the diffusion coefficient is constant. However, the motion of a biological population is not purely random but moves with sense.…”

“…In fact, convection phenomenon widely exists in nature, such as in chemotactic cells, turbulent flow, porous media, viscous liquid and plankton distributions in the sea, 9,10,13,[21][22][23] so the effect of convection terms on the propagation of traveling wave solutions of reaction diffusion equations is of much interest to researchers. However, not much has been undertaken on the delayed reaction diffusion equations with constant convection.…”

“…Here, the density-dependent diffusion coefficient linearly depends on population density u and g(u(x, t), u(x, t − 𝜏)) is the reaction term with time delay. Equation (4) arises in several scientific fields such as biological population genetics and cellular ecology; see previous works 9,12,13,16,22,24,26 and the references therein.…”

“…Colour figure can be viewed at wileyonlinelibrary.com] For any c ≥ c * > 0, the initial-value problem(13) has a positive solution satisfying v c (0) = 0.…”