Detecting minimum energy states and multi-stability in nonlocal advection-diffusion models for interacting species.
Detecting minimum energy states and multi-stability in nonlocal advection-diffusion models for interacting species.
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DOI：
10.1007/s00285-022-01824-1
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发表时间：
2022-10-20
影响因子：
1.9
通讯作者：
Potts, Jonathan R.
中科院分区：
文献类型：
--
作者：
Giunta, Valeria;Hillen, Thomas;Lewis, Mark A.;Potts, Jonathan R.
关键词：
从生物过程模型中导出涌现模式是数学生物学的核心问题。在偏微分方程的背景下，这些涌现模式有时表现为相应能量泛函的局部极小值。在这里，我们给出了确定一类广泛的多物种非局部平流扩散模型的局部最小能量状态的定性结构的方法，这些模型最近被提出用于对生态系统的空间结构进行建模。我们表明，当每对物种以对称方式相互响应时（即通过相互回避或相互吸引，以相同的强度），系统允许能量泛函随着时间的推移而减少，并在下面有界。这表明系统最终将达到局部最小能量稳定状态，而不是永久波动。我们利用这种能量泛函来开发工具，包括计算代数几何的新颖应用，用于对局部最小能量解的数量和定性结构进行猜想。这些猜想为在哪里寻找数值稳态解提供了指导，我们通过数值分析对其进行了验证。我们的技术表明，即使有两种物种，也可以出现具有多达四类局部最小能量状态的多稳定性。相关的动力学包括通过物种内部和物种之间的聚集和排斥进行空间排序。新兴的空间模式包括类似领土的隔离以及狭窄的尖峰型解决方案的混合。总的来说，我们的研究揭示了移动和相互作用的物种系统中丰富的多重稳定性的概况。
Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations, these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection–diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy states can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species.
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影响因子：
3.1
作者：
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通讯作者：
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DOI：
10.1142/s0218202516400042
发表时间：
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