Figure 13

The parameter \(\pmb{a-b}\) diagram of stability and bifurcation. (A) Parameter region of stability and instability for \(E_{0}\) under spatially homogeneous perturbations. The curves \(C_{1}\), \(C_{2}\) and \(C_{3}\) divide the parameter \(a-b\) plane into four regions (respectively denoted by I, II, III and IV), where the curves \(C_{1}\) and \(C_{3}\) come from the equation \({\frac{ 4 ( a+b) ^{4}}{ ( ( a+b) ^{3}+ a-b) ^{2}}} = \tan^{2} ( \frac{2 \pi }{5} ) +1\) and \(C_{2}\) from \((a+b)^{3}=b-a\). (B) Parameter \(a-b\) diagram showing three parameter regions, homogeneous steady state region (consisting of regions 3 and 4), pure Turing instability (consisting of regions 2, 5 and 6) and Hopf-Turing instability (only one region 1). The curves \(C_{4}\) and \(C_{5}\) come from the equation \({ { ( ( a+b) ^{3}d _{{1}}+ ( a-b) d_{{2}} ) ^{2}}}=4d_{{1}}d_{{2}} ( a+b) ^{4} \), the curve \(C_{6}\) from \(d_{1}(a+b)^{3}=d _{2}(b-a)\). Set \(\alpha =0.8\), \(d_{1}=0.01\) and \(d_{2}=0.25\).