Stable coexistence in indefinitely large systems of competing species.

The Lotka-Volterra system is a set of ordinary differential equations describing growth of interacting ecological species. One of the debated questions is understanding how the number of species in the system, influences the stability of the model. Robert May studied how large systems may become unstable when species-species interactions do not vanish. This outcome has frequently been interpreted as a universal phenomenon and summarised as "large systems are unstable". By exploring general interaction networks, we show that the competitive Lotka-Volterra system may maintain stability even for large networks, despite non-vanishing interaction strength. We establish sufficient conditions for stability as a threshold on the interspecific interaction strength, formulated in terms of the maximum and minimum degrees (or weights) rather than the network's size. For values below this threshold, coexistence of all species is attained, regardless of the system size.Our finding generalises May's result by showing that it is the outlier nodes with large degree that cause instability, rather than the large number of species in the system.洛特卡-沃尔泰拉系统（Lotka-Volterra system）是一组描述相互作用生态物种种群增长的常微分方程。学界长期争论的核心议题之一，便是探究系统内物种数量如何影响模型的稳定性。罗伯特·梅（Robert May）曾开展研究：当物种间相互作用强度不为零时，大型系统为何会趋于不稳定。这一结论常被解读为普适现象，并被概括为"大型系统不稳定"。通过对通用相互作用网络的探索，我们的研究表明，即便相互作用强度不为零，竞争性洛特卡-沃尔泰拉系统仍可在大型网络中维持稳定性。我们推导得到了种间相互作用强度阈值下的稳定性充分条件，该条件以节点的最大、最小度（或权重）而非网络规模进行表述。当相互作用强度低于该阈值时，无论系统规模大小，所有物种均可实现共存。我们的发现推广了梅的研究结论，证明引发不稳定的是具有大度的离群节点，而非系统内庞大的物种数量。