Radial propagation in population dynamics with density-dependent diffusion

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.

Figure 1

Demonstration of evolution of the radially symmetric population density profile $u(r,t)$ (10) in 2D ($\gamma =1$) in comparison with the numerical solution. The solid lines represent the exact solutions and the circle markers represent the numerical solutions. The parameters are as follows: $p=2$, $\rho =0.05$, and $r_0=5$. The density profiles are initiated at $t=0$ and evolve until $t=102$.

Figure 2

Corresponding front position $R\left(t\right)$ extracted from the density profiles in Fig. 1. The solid lines represent the exact solutions and the circle markers represent numerical solutions.