Pattern formation in various biological systems has been attributed to Turing instabilities in systems of reaction-diffusion equations. In this paper, a rigorous mathematical description for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis is presented. We identify a generalized nonlinear degenerate chemotaxis model where a destabilization mechanism may lead to spatially non homogeneous solutions. Given any general perturbation of the solution nearby an homogenous steady state, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along the finite number of fastest growing modes. The theoretical results are tested against two different numerical results in two dimensions showing an excellent qualitative agreement.

中文翻译：

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非线性退化趋化模型解的渐近行为

各种生物系统中模式的形成已归因于反应扩散方程系统中的图灵不稳定性。在本文中，对具有趋化性的生物个体的聚集区域的模式动力学进行了严格的数学描述。我们确定了广义的非线性简并化趋化模型，其中失稳机制可能导致空间上非均匀解。给定溶液在均匀稳态附近的任何一般扰动，我们证明其非线性演化由沿着有限数量的最快增长模态的相应线性动力学控制。理论结果针对二维的两个不同数值结果进行了测试，显示出极好的定性一致性。