SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 1090-1103, January 2021.
To assess which ecosystems are most vulnerable it is necessary to compare the resilience of complex interaction networks in a meaningful way. A fundamental problem for the comparative analysis of ecological stability is that the organisms in ecological networks operate on different time scales. A conventional solution to this problem has been to assume the intraspecific interaction strengths in the dynamical system (and diagonal elements in the community matrix) have the same value, ignoring the time scale differences, and therefore disregarding vital ecological information. In this paper, we consider two methods that have previously been developed to deal with community matrices arising from populations with widely different time scales and which contain differing self-regulation terms (diagonal entries). One approach considers the critical self-regulation in a system by proportionally adjusting the diagonal entries until the tipping point is found. The other is a scaling procedure that translates the intraspecific information on the diagonal on to the off-diagonal entries. We show the relation between the leading eigenvalue of the latter, and the numerical diagonal parameter of the former, which in many ecologically relevant networks is exact. In addition, we show for $3 \times 3$ scaled competitive systems how the feedback determines whether the leading eigenvalue is real- or complex-valued, which is important for knowing when the scaling procedure remains ecologically sensible. While arising from an ecological setting, this work has wider implications in network theory and linear algebra.

中文翻译：

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矩阵缩放和临界点

SIAM Journal on Applied Dynamical Systems，第 20 卷，第 2 期，第 1090-1103 页，2021 年 1 月。
为了评估哪些生态系统最脆弱，有必要以有意义的方式比较复杂交互网络的弹性。生态稳定性比较分析的一个基本问题是生态网络中的生物体在不同的时间尺度上运行。这个问题的传统解决方案是假设动力系统中的种内相互作用强度（和群落矩阵中的对角元素）具有相同的值，忽略时间尺度差异，从而忽略重要的生态信息。在本文中，我们考虑了先前开发的两种方法来处理来自具有广泛不同时间尺度的人群产生的社区矩阵，这些社区矩阵包含不同的自我调节项（对角线条目）。一种方法是通过按比例调整对角线条目直到找到临界点来考虑系统中的关键自我调节。另一个是缩放过程，它将对角线上的种内信息转换为非对角线条目。我们展示了后者的主要特征值与前者的数值对角参数之间的关系，这在许多生态相关网络中是准确的。此外，我们展示了 $3 \times 3$ 缩放竞争系统的反馈如何确定领先特征值是实值还是复值，这对于了解缩放程序何时在生态上保持合理很重要。虽然源于生态环境，但这项工作在网络理论和线性代数方面具有更广泛的意义。