Fluctuation spectra of large random dynamical systems reveal hidden structure in ecological networks

Understanding the relationship between complexity and stability in large dynamical systems—such as ecosystems—remains a key open question in complexity theory which has inspired a rich body of work developed over more than fifty years. The vast majority of this theory addresses asymptotic linear stability around equilibrium points, but the idea of ‘stability’ in fact has other uses in the empirical ecological literature. The important notion of ‘temporal stability’ describes the character of fluctuations in population dynamics, driven by intrinsic or extrinsic noise. Here we apply tools from random matrix theory to the problem of temporal stability, deriving analytical predictions for the fluctuation spectra of complex ecological networks. We show that different network structures leave distinct signatures in the spectrum of fluctuations, and demonstrate the application of our theory to the analysis of ecological time-series data of plankton abundances.

[1]  R. L. Stratonovich.  On a Method of Calculating Quantum Distribution Functions , 1957 .

[2]  R. Kubo.  The fluctuation-dissipation theorem , 1966 .

[3]  L. Stone.  The feasibility and stability of large complex biological networks: a random matrix approach , 2018, Scientific Reports.

[4]  Mike S. Fowler,et al.  Navigating the complexity of ecological stability. , 2016, Ecology letters.

[5]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[6]  Yang-Yu Liu,et al.  Mapping the ecological networks of microbial communities , 2017, Nature Communications.

[7]  A J McKane,et al.  Predator-prey cycles from resonant amplification of demographic stochasticity. , 2005, Physical review letters.

[8]  Tobias Galla,et al.  Dynamically evolved community size and stability of random Lotka-Volterra ecosystems , 2018, EPL (Europhysics Letters).

[9]  Diego Garlaschelli,et al.  Uncovering hidden functional brain organization by random matrix theory , 2017, 1708.07046.

[10]  Feng Luo,et al.  Constructing gene co-expression networks and predicting functions of unknown genes by random matrix theory , 2007, BMC Bioinformatics.

[11]  Si Tang,et al.  Stability criteria for complex ecosystems , 2011, Nature.

[12]  Stefano Allesina,et al.  The stability–complexity relationship at age 40: a random matrix perspective , 2015, Population Ecology.

[13]  K. Foster,et al.  The ecology of the microbiome: Networks, competition, and stability , 2015, Science.

[14]  Lin Jiang,et al.  Different Effects of Species Diversity on Temporal Stability in Single‐Trophic and Multitrophic Communities , 2009, The American Naturalist.

[15]  Stefano Allesina,et al.  Effect of population abundances on the stability of large random ecosystems. , 2017, Physical review. E.

[16]  Richard Levins,et al.  Coexistence in a Variable Environment , 1979, The American Naturalist.

[17]  T. Rogers,et al.  Cavity approach to the spectral density of non-Hermitian sparse matrices. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  MARK R. GARDNER,et al.  Connectance of Large Dynamic (Cybernetic) Systems: Critical Values for Stability , 1970, Nature.

[19]  Koujin Takeda,et al.  Cavity approach to the spectral density of sparse symmetric random matrices. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  ROBERT M. MAY,et al.  Will a Large Complex System be Stable? , 1972, Nature.

[21]  David Tilman,et al.  Biodiversity, Stability, and Productivity in Competitive Communities , 2000, The American Naturalist.

[22]  J. Ginibre.  Statistical Ensembles of Complex, Quaternion, and Real Matrices , 1965 .

[23]  C. Gardiner.  Stochastic Methods: A Handbook for the Natural and Social Sciences , 2009 .

[24]  Christian Wissel,et al.  Babel, or the ecological stability discussions: an inventory and analysis of terminology and a guide for avoiding confusion , 1997, Oecologia.

[25]  Giorgio Parisi,et al.  SK Model: The Replica Solution without Replicas , 1986 .

[26]  M. Pascual,et al.  Stochastic amplification in epidemics , 2007, Journal of The Royal Society Interface.

[27]  Gunnar Rätsch,et al.  Ecological Modeling from Time-Series Inference: Insight into Dynamics and Stability of Intestinal Microbiota , 2013, PLoS Comput. Biol..

[28]  P. Reich,et al.  Biodiversity and ecosystem stability in a decade-long grassland experiment , 2006, Nature.

[29]  A. Ives,et al.  Stability and variability in competitive communities. , 1999, Science.

[30]  Tim Rogers,et al.  Stochastic Pattern Formation and Spontaneous Polarisation: The Linear Noise Approximation and Beyond , 2012, Bulletin of mathematical biology.

[31]  P. Archambault,et al.  No complexity–stability relationship in empirical ecosystems , 2016, Nature Communications.

[32]  Eric J Alm,et al.  High resolution time series reveals cohesive but short-lived communities in coastal plankton , 2018, Nature Communications.

[33]  James Rosindell,et al.  Unified neutral theory of biodiversity and biogeography , 2010, Scholarpedia.

[34]  A. Maritan,et al.  Effect of localization on the stability of mutualistic ecological networks , 2015, Nature Communications.

[35]  M. Loreau,et al.  Resilience, reactivity and variability: A mathematical comparison of ecological stability measures. , 2015, Journal of theoretical biology.

[36]  Jean-Philippe Bouchaud,et al.  Will a Large Economy Be Stable ? , 2019 .

[37]  Stefano Allesina,et al.  Predicting the stability of large structured food webs , 2015, Nature Communications.

[38]  J. Hubbard.  Calculation of Partition Functions , 1959 .

[39]  Jeffrey Pennington,et al.  Nonlinear random matrix theory for deep learning , 2019, NIPS.

[40]  T. Rogers,et al.  Spectral theory of sparse non-Hermitian random matrices , 2018, Journal of Physics A: Mathematical and Theoretical.

[41]  Andrew L Jackson,et al.  On the dimensionality of ecological stability. , 2013, Ecology letters.

[42]  F. Barraquand,et al.  Strong self‐regulation and widespread facilitative interactions in phytoplankton communities , 2020, Journal of Ecology.

[43]  Tamara N. Romanuk,et al.  Experimental design and the outcome and interpretation of diversity-stability relations , 2011 .

[44]  Alex James,et al.  Constructing Random Matrices to Represent Real Ecosystems , 2015, The American Naturalist.

[45]  Stefan B. Williams,et al.  BioTIME: A database of biodiversity time series for the Anthropocene , 2018, Global Ecology and Biogeography.

[46]  Diego Garlaschelli,et al.  Uncovering functional signature in neural systems via random matrix theory , 2017, PLoS Comput. Biol..

[47]  Kurt Wiesenfeld,et al.  Noisy precursors of nonlinear instabilities , 1985 .

[48]  M. Loreau,et al.  Species Synchrony and Its Drivers: Neutral and Nonneutral Community Dynamics in Fluctuating Environments , 2008, The American Naturalist.

[49]  F. Barraquand,et al.  Strong self-regulation and widespread facilitative interactions in phytoplankton communities , 2019, bioRxiv.

[50]  Tobias Galla,et al.  Intrinsic noise in game dynamical learning. , 2009, Physical review letters.

[51]  Stefano Allesina,et al.  Modularity and stability in ecological communities , 2016, Nature Communications.

[52]  E. Wigner.  On the Distribution of the Roots of Certain Symmetric Matrices , 1958 .

[53]  T. Tao,et al.  RANDOM MATRICES: THE CIRCULAR LAW , 2007, 0708.2895.

[54]  Terence Tao,et al.  Random matrices: Universality of ESDs and the circular law , 2008, 0807.4898.