Figure 5.

Types of travelling wave solutions for the go-or-grow model under mechanism M₁. (Top) Parameter regions of (M, ρ*)-plane corresponding to different behaviours of the solutions ρ₁ and ρ₂ (for a fixed value of the sharpness parameter (α = 4)). Region 3 is defined as unstable to diffusion-driven processes (i.e. condition (12) is satisfied) and corresponds to simple travelling wave solutions. The dotted line that separates regions 1 and 2 is the result of numerical investigation. In region 2, solutions are characterized by an oscillatory profile of the moving front, where peaks of immotile cells form and remain as fixed patterns after the front went past. In region 3, the solutions degenerate behind the moving front into irregular spatio-temporal oscillations with no apparent order. (Bottom) Snapshots at time t ≃ 377 of the density profiles of the motile (ρ₁ – red line) and static (ρ₂ – green line) populations corresponding to the points P_i in the top graph. The value of M is kept fixed while ρ* is increased from left to right, which corresponds to the vertical path between P₁ and P₃ in the top graph. (Left) Simple travelling wave solutions – P₁ = (1.6, 0.57) in region 1. Note the small peak of the static population at the edge of the front. (Centre) Oscillatory moving front – P₂ = (1.6, 0.6)) in region 2. The oscillatory nature of the peak results in the formation of stationary patterns of immotile cells behind the moving front. (Right) Irregular spatio-temporal oscillations behind the moving front – P₃ = (1.6, 0.75) in region 3.