Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK.
Bull Math Biol. 2022 Aug 11;84(9):101. doi: 10.1007/s11538-022-01055-x.
Turing's theory of morphogenesis is a generic mechanism to produce spatial patterning from near homogeneity. Although widely studied, we are still able to generate new results by returning to common dogmas. One such widely reported belief is that the Turing bifurcation occurs through a pitchfork bifurcation, which is true under zero-flux boundary conditions. However, under fixed boundary conditions, the Turing bifurcation becomes generically transcritical. We derive these algebraic results through weakly nonlinear analysis and apply them to the Schnakenberg kinetics. We observe that the combination of kinetics and boundary conditions produce their own uncommon boundary complexities that we explore numerically. Overall, this work demonstrates that it is not enough to only consider parameter perturbations in a sensitivity analysis of a specific application. Variations in boundary conditions should also be considered.
图灵形态发生理论是一种从近均匀性产生空间模式的通用机制。尽管已经广泛研究，但我们仍然可以通过回归常见的教条来产生新的结果。其中一个被广泛报道的信念是，图灵分岔是通过叉形分岔产生的，这在零通量边界条件下是正确的。然而，在固定边界条件下，图灵分岔通常是跨越临界的。我们通过弱非线性分析得到这些代数结果，并将其应用于 Schnakenberg 动力学。我们观察到，动力学和边界条件的组合产生了它们自己的不常见的边界复杂性，我们通过数值方法进行了探索。总的来说，这项工作表明，在对特定应用的敏感性分析中，仅仅考虑参数扰动是不够的。边界条件的变化也应该被考虑。
