Stable coexistence in indefinitely large systems of competing species (2404.14031v1)
Abstract: The Lotka-Volterra system is a set of ordinary differential equations describing growth of interacting ecological species. This model has gained renewed interest in the context of random interaction networks. One of the debated questions is understanding how the number of species in the system, $n$, influences the stability of the model. Robert May demonstrated that large systems become unstable, unless species-species interactions vanish. This outcome has frequently been interpreted as a universal phenomenon and summarised as "large systems are unstable". However, May's results were performed on a specific type of graphs (Erd\H{o}s-R\'enyi), whereas we explore a different class of networks and we show that the competitive Lotka-Volterra system maintains stability even in the limit of large $n$, despite non-vanishing interaction strength. We establish a lower bound on the interspecific interaction strength, formulated in terms of the maximum and minimum degrees of the ecological network, rather than being dependent upon the network's size. For values below this threshold, coexistence of all species is attained in the asymptotic limit. In other words, the outlier nodes with large degree cause instability, rather than the large number of species in the system. Our result refines May's bound, by showing that the type of network model is relevant and can lead to completely different results.
- May, R. Will a large complex system be stable? Nature 238, 413–414 (1972).
- McCann, K. S. The diversity–stability debate. Nature 405, 228–233 (2000).
- Stability criteria for complex ecosystems. Nature 483, 205–208 (2012).
- Allesina, S. et al. Predicting the stability of large structured food webs. Nature Communications 6, 7842 (2015).
- Coexistence of many species in random ecosystems. Nature Ecology & Evolution 2, 1237–1242 (2018).
- High-order species interactions shape ecosystem diversity. Nature Communications 7, 12285 (2016).
- Higher-order interactions stabilize dynamics in competitive network models. Nature 548, 210–213 (2017).
- Eppinga, M. B. et al. Frequency-dependent feedback constrains plant community coexistence. Nature Ecology & Evolution 2, 1403–1407 (2018).
- Modularity and stability in ecological communities. Nature Communications 7, 12031 (2016).
- Self-regulation and the stability of large ecological networks. Nature Ecology & Evolution 1, 1870–1875 (2017).
- Diversity begets stability: Sublinear growth and competitive coexistence across ecosystems. Science 383, eadg8488 (2024).
- Complexity and stability of ecological networks: a review of the theory. Population Ecology 60, 319–345 (2018).
- Jordano, P. Sampling networks of ecological interactions. Functional Ecology 30, 1883–1893 (2016).
- Mutualistic networks (Princeton University Press, 2013).
- Asymmetric density dependence shapes species abundances in a tropical tree community. Science 329, 330–332 (2010).
- Sparse species interactions reproduce abundance correlation patterns in microbial communities. Proceedings of the National Academy of Sciences 121, e2309575121 (2024).
- Comes, C.-A. Banking system: Three level lotka-volterra model. Procedia Economics and Finance 3, 251–255 (2012).
- May’s instability in large economies. Phys. Rev. E 100, 032307 (2019).
- Goh, B. S. Global stability in many-species systems. The American Naturalist 111, 135–143 (1977).
- Finding large independent sets using competitive dynamics (2024). Manuscript under consideration for publication.
- Competition for time: Evidence for an overlooked, diversity-maintaining competitive mechanism. Ecology Letters 27, e14422 (2024).
- Tilman, D. Competition and biodiversity in spatially structured habitats. Ecology 75, 2–16 (1994).