Morphology of Experimental and Simulated Turing Patterns

A full derivation is given in Ref. [44]:f u + g v < 0, (2.19)f u g v − f v g u > 0, (2.20)df u + g v > 0, (2.21)(df u + g v ) 2 − 4d(f u g v − f v g u ) > 0. (2.22)Eqs. (2.19) - 2.20) give the Hopf bifurcation while Eqs. (2.21) - 2.22) give the Turingbifurcation. For the LE model the linearized reaction terms are given byf u = ∂f∂u∣ = 3α2 − 5u0 ,v 01 + α 2 , f v = ∂fα∂v ∣ = −4u0 ,v 01 + α2, (2.23)g u = ∂g∂u∣ = σ b 2α2u0 ,v 01 + α 2, g v = ∂gα∂v∣ = −σ bu0 ,v 01 + α 2 (2.24)and d = σ c. Inserting Eqs. (2.23 - 2.24) into Eqs. (2.19 - 2.22) yields the conditionsσ bα > 3α 2 − 5, (2.25)σ bα> 0,1 + α2 (2.26)σ bα < (3α 2 − 5)d, (2.27)9d 2 α 4 − 30d 2 α 2 − 26dα 3 σb + 25d 2 − 10dσbα + σ 2 b 2 α 2 > 0. (2.28)Assuming that σ,d,α > 0 and considering Eq. (2.27), Eq. (2.28) yields for the Turingbifurcationb T = c (13a 2 + 125 − 4 √ √ )10a a5a2 + 25(2.29)Additionally Eq. (2.25) yields for the Hopf bifurcationb H = 1 ( 3σ 5 a − 25 ), (2.30)aand the region of Turing instability is given byb H < b < b T , (2.31)i.e. the region of the parameter space, where the homogeneous steady state is stablewithout diffusion but unstable, when diffusion is taken into account. The curvesb H and b T are shown in section 4.1 for parameters that resemble the experimentalconditions. Additionally in regions where the homogeneous steady state is unstablewith and without diffusion the convergence to heterogeneous steady states is oftenfavorable. Only in regions, where b H > b > b T and when b T ≈ b H , oscillatory andspatio-temporal solutions can occur.24