Figure 2. A toggle switch with diffusion can generate prolonged boundary formations.

(A)–(D): Simulations of system (1,4) with parameters D = .001mm²/min, γ = .5 min⁻¹, α = 10nM/min, A₀ = 100nM, μ = 5mm⁻¹, and σ = .5 nM. (A) Illustration of initial conditions. (B) Simulation with L = .5mm. (C) Spatial profiles of u and v at time t = 1000 min for L = .5mm. (D) Simulations showing boundary formations are transient by extending the simulation time. (E)–(F): We study the effects of the domain length L and the standard deviation σ in the random initial conditions (4) on the time t = t^* required to reach a homogeneous state. We run 50 simulations for each condition and calculate the average t^*, μ(t^*), and standard deviation σ^t* . (E) Plot of average time μ(t^*) until spatial homogeneity is achieved. We normalize by 100 min and apply a logarithmic transformation. (F) Plot of corresponding coefficient of variation σ^t*/μ(t^*).