Pattern Formation in Populations with Density-Dependent Movement and Two Interaction Scales
Fig 8
Formation of ring-like structures in the 1D model.
Evolution of the 1D version of the model starting from an initial condition for ρ(x, t = 0) consisting on a pulse of height unity and length 2Rl (displayed in the upper panel). The legend indicates the quantity represented by the different lines. The two vertical lines indicate the minima of the function g(x) (i.e. the diffusion coefficient in units of D0) at t = 0 where particles will initially tend to accumulate. The bottom panel represent the same quantities (although ρ(x, t) has been divided by a factor 5 to fit into the same scale as the other curves) after a very long integration time (t = 6 × 105). A double-peak structure has developed. The inset displays this long-time configuration in logarithmic scale, showing that g(x) ≈ 4 × 10−6 in the central region. Parameters: a = 1, b = 3.33, c = 2.67, D0 = 10−4, Rl = 0.75, Rs = 0.4, N = ∫ dxρ(x, t = 0) = 1.5