STABLE STEADY-STATE SOLUTIONS OF SOME BIOLOGICAL AGGREGATION MODELS
STABLE STEADY-STATE SOLUTIONS OF SOME BIOLOGICAL AGGREGATION MODELS
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DOI：
10.1137/20m1348066
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发表时间：
2021-01-01
影响因子：
1.9
通讯作者：
Painter, Kevin J.
中科院分区：
文献类型：
--
作者：
Potts, Jonathan R.;Painter, Kevin J.
聚集现象发生在整个生物科学领域，从细胞粘附到昆虫群，从动物家园到人类城市。因此，了解它们自发出现的机制引起了应用数学家的极大兴趣。具有非局部平流的偏微分方程 (PDE) 为研究聚合提供了一种流行的形式。然而，固有的非定域性通常是确保连续统模型适定所必需的，这使得他们的研究在技术上具有挑战性。在这里，我们通过研究离散空间系统来采用不同的方法，该系统可以通过限制过程与经典非局部偏微分方程方法正式相关。我们展示了如何通过能量泛函方法找到该离散空间系统渐近稳定稳态的表达式。这使我们能够根据个体生物体的基本运动机制来预测聚集的大小。我们将其应用于最近的细胞粘附模型，揭示了一种滞后特性，即即使粘附趋势降低到分叉点之后，现有的聚集也可能持续存在。我们将其与相关非局部偏微分方程系统的数值解进行比较，表明离散空间表达式预测的滞后特性也存在于连续统系统中。
Aggregation phenomena occur across the biological sciences, from cell adhesion to insect swarms, animal home ranges to human cities. Understanding the mechanisms by which they may spontaneously emerge has therefore generated much interest from applied mathematicians. Partial differential equations (PDEs) with nonlocal advection offer a popular formalism for studying aggregations. However, the inherent nonlocality, often necessary for ensuring continuum models are well-posed, makes their study technically challenging. Here, we take a different approach by studying a discrete-space system that can be formally related to classical nonlocal PDE approaches via a limiting procedure. We show how to find expressions for the asymptotically stable steady-states of this discrete-space system via an energy functional approach. This allows us to predict the size of aggregations as a function of the underlying movement mechanisms of individual organisms. We apply this to a recent model of cell adhesion, revealing a hysteresis property whereby the existing aggregations may persist even as the adhesion tendency decreases past the bifurcation point. We compare this to numerical solutions of the associated nonlocal PDE system, showing that the hysteresis property predicted by the discrete-space expressions is also present in the continuum system.