无序动力系统中的广义关联：来自多种群Lotka-Volterra模型的洞见
Generalised correlations in disordered dynamical systems: Insights from the many-species Lotka-Volterra model
AI总结：
本研究通过动态平均场分析，以广义Lotka-Volterra方程为模型，揭示了单索引关联（行内及转置行列关联）对无序系统动力学、噪声水平、反馈效应、物种存活率及生态平衡稳定性的影响。
摘要：
在研究无序系统时，人们通常选择一个由独立同分布相互作用系数构成的矩阵来表示组分之间的淬火随机耦合，或许带有某种对称约束或对角相对元素对之间的关联。然而，一组更一般的耦合（仍保持组分的统计可交换性）可能涉及仅共享单个行或列索引的相互作用系数之间的关联。这些关联已被证明在广义Lotka-Volterra方程（gLVEs）等系统中自然产生。在本工作中，我们进行了动态平均场分析，以理解单索引关联如何影响无序系统的动力学和稳定性，并以gLVEs为例。我们表明，即使相互作用系数的总体方差保持不变，行内关联也会提高平均场过程中的噪声水平。我们还看到，行和列的转置对之间的关联可以根据相关系数的符号增强或抑制反馈效应。在gLVEs的背景下，行内和转置行列关联因此既影响物种存活率，也影响生态平衡的稳定性。
英文摘要：
In the study of disordered systems, one often chooses a matrix of independent identically distributed interaction coefficients to represent the quenched random couplings between components, perhaps with some symmetry constraint or correlations between diagonally opposite pairs of elements. However, a more general set of couplings, which still preserves the statistical interchangeability of the components, could involve correlations between interaction coefficients sharing only a single row or column index. These correlations have been shown to arise naturally in systems such as the generalised Lotka-Volterra equations (gLVEs). In this work, we perform a dynamic mean-field analysis to understand how single-index correlations affect the dynamics and stability of disordered systems, taking the gLVEs as our example. We show that in-row correlations raise the level of noise in the mean field process, even when the overall variance of the interaction coefficients is held constant. We also see that correlations between transpose pairs of rows and columns can either enhance or suppress feedback effects, depending on the sign of the correlation coefficient. In the context of the gLVEs, in-row and transpose row/column correlations thus affect both the species survival rate and the stability of ecological equilibria.
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