分叉理论的生物学应用
数学
干草叉分叉
跨临界分岔
分叉
分岔理论
图灵
边值问题
边界(拓扑)
鞍结分岔
分岔图
数学分析
统计物理学
标识
DOI:10.1007/s11538-022-01055-x
摘要
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.
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