Doelman A, Sewalt L, Zagaris A
Leiden University, Niels Bohrweg 1, 2333CA Leiden, The Netherlands.
University of Twente, P.O. Box 217, 7500AE Enschede, The Netherlands.
Chaos. 2015 Mar;25(3):036408. doi: 10.1063/1.4913484.
We consider a two-component system of evolutionary partial differential equations posed on a bounded domain. Our system is pattern forming, with a small stationary pattern bifurcating from the background state. It is also equipped with a multiscale structure, manifesting itself through the presence of spectrum close to the origin. Spatial processes are associated with long time scales and affect the nonlinear pattern dynamics strongly. To track these dynamics past the bifurcation, we develop an asymptotics-based method complementing and extending rigorous center manifold reduction. Using it, we obtain a complete analytic description of the pattern stability problem in terms of the linear stability of the background state. Through this procedure, we portray with precision how slow spatial processes can destabilize small patterns close to onset. We further illustrate our results on a model describing phytoplankton whose growth is co-limited by nutrient and light. Localized colonies forming at intermediate depths are found to be subject to oscillatory destabilization shortly after emergence, whereas boundary-layer type colonies at the bottom persist. These analytic results are in agreement with numerical simulations for the full model, which we also present.
我们考虑一个在有界域上提出的由两个分量组成的演化偏微分方程组。我们的系统是模式形成的，有一个从背景状态分岔出来的小驻定模式。它还具有多尺度结构，通过靠近原点的频谱表现出来。空间过程与长时间尺度相关，并强烈影响非线性模式动力学。为了跟踪分岔后的这些动力学，我们开发了一种基于渐近分析的方法，对严格的中心流形约化进行补充和扩展。利用该方法，我们根据背景状态的线性稳定性，获得了模式稳定性问题的完整解析描述。通过这个过程，我们精确地描绘了缓慢的空间过程如何使接近起始点的小模式失稳。我们进一步在一个描述浮游植物生长受营养和光照共同限制的模型上说明了我们的结果。发现在中间深度形成的局部菌落出现后不久会受到振荡失稳，而底部的边界层型菌落则持续存在。这些解析结果与我们也给出的完整模型的数值模拟结果一致。
