Equilibrium and surviving species in a large Lotka-Volterra system of differential equations.
Large random matrices
Linear complementarity problems
Lotka–Volterra equations
Stability of food webs
Journal
Journal of mathematical biology
ISSN: 1432-1416
Titre abrégé: J Math Biol
Pays: Germany
ID NLM: 7502105
Informations de publication
Date de publication:
19 06 2023
19 06 2023
Historique:
received:
27
10
2022
accepted:
19
05
2023
revised:
25
04
2023
medline:
21
6
2023
pubmed:
19
6
2023
entrez:
19
6
2023
Statut:
epublish
Résumé
Lotka-Volterra (LV) equations play a key role in the mathematical modeling of various ecological, biological and chemical systems. When the number of species (or, depending on the viewpoint, chemical components) becomes large, basic but fundamental questions such as computing the number of surviving species still lack theoretical answers. In this paper, we consider a large system of LV equations where the interactions between the various species are a realization of a random matrix. We provide conditions to have a unique equilibrium and present a heuristics to compute the number of surviving species. This heuristics combines arguments from Random Matrix Theory, mathematical optimization (LCP), and standard extreme value theory. Numerical simulations, together with an empirical study where the strength of interactions evolves with time, illustrate the accuracy and scope of the results.
Identifiants
pubmed: 37335417
doi: 10.1007/s00285-023-01939-z
pii: 10.1007/s00285-023-01939-z
doi:
Types de publication
Journal Article
Research Support, Non-U.S. Gov't
Langues
eng
Sous-ensembles de citation
IM
Pagination
13Informations de copyright
© 2023. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
Références
Akjouj I, Najim J (2022) Feasibility of sparse large Lotka–Volterra ecosystems. J Math Biol 85(6–7):66
doi: 10.1007/s00285-022-01830-3
Akjouj I, Hachem W, Maïda M, Najim J (2023) Equilibria of large random Lotka–Volterra systems with vanishing species: a mathematical approach. arXiv preprint arXiv:2302.07820
Allesina S, Tang S (2012) Stability criteria for complex ecosystems. Nature 483(7388):205
doi: 10.1038/nature10832
Arnoldi J-F, Bideault A, Loreau M, Haegeman B (2018) How ecosystems recover from pulse perturbations: a theory of short- to long-term responses. J Theor Biol 436:79–92
doi: 10.1016/j.jtbi.2017.10.003
Bai ZD, Silverstein JW (2010) Spectral analysis of large dimensional random matrices, 2nd edn. Springer series in statistics. Springer, New York
doi: 10.1007/978-1-4419-0661-8
Balkema A, De Haan L (1978) Limit distributions for order statistics. I. Theory Prob Appl 23(1):77–92
doi: 10.1137/1123006
Baskerville EB, Dobson AP, Bedford T, Allesina S, Anderson TM, Pascual M (2011) Spatial guilds in the Serengeti food web revealed by a Bayesian group model. PLoS Comput Biol 7(12):e1002321
doi: 10.1371/journal.pcbi.1002321
Bizeul P, Najim J (2021) Positive solutions for large random linear systems. Proc Am Math Soc 149(6):2333–2348
doi: 10.1090/proc/15383
Bordenave C, Chafaï D (2012) Around the circular law. Probab Surv 9:1–89
doi: 10.1214/11-PS183
Brose U, Archambault P, Barnes A, Bersier L-F, Boy T, Canning-Clode J, Conti E, Dias M, Digel C, Dissanayake A, Flores A, Fussmann K, Gauzens B, Gray C, Häussler J, Hirt M, Jacob U, Jochum M, Kéfi S, McLaughlin O, MacPherson M, Latz E, Layer-Dobra K, Legagneux P, Li Y, Madeira C, Martinez N, Mendonça V, Mulder C, Navarrete S, O’Gorman E, Ott D, Paula J, Perkins D, Piechnik De, Pokrovsky I, Raffaelli D, Rall B, Rosenbaum B, Ryser R, Silva A, Sohlström Es, N. Sokolova, M. Thompson, R. Thompson, F. Vermandele, C. Vinagre, S. Wang, J. Wefer, R. Williams, E. Wieters, G. Woodward, and A. Iles. (2019) Predator traits determine food-web architecture across ecosystems. Nat Ecol Evol 3(6):919–927
Bunin G (2017) Ecological communities with Lotka–Volterra dynamics. Phys Rev E 95(4):042414
doi: 10.1103/PhysRevE.95.042414
Busiello DM, Suweis S, Hidalgo J, Maritan A (2017) Explorability and the origin of network sparsity in living systems. Sci Rep 7(1):12323
doi: 10.1038/s41598-017-12521-1
Capitaine M, Donati-Martin C, Féral D (2009) The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations. Ann Probab 37(1)
Clenet M (2022) Equilibrium and surviving species in a large Lotka–Volterra system of differential equations. https://github.com/maxime-clenet/Equilibrium-and-surviving-species-in-a-large-Lotka-Volterra-system
Cottle RW, Pang J-S, Stone RE (2009) The linear complementarity problem. SIAM
Dougoud M, Vinckenbosch L, Rohr RP, Bersier L-F, Mazza C (2018) The feasibility of equilibria in large ecosystems: a primary but neglected concept in the complexity-stability debate. PLoS Comput Biol 14(2):e1005988
doi: 10.1371/journal.pcbi.1005988
Fukami T, Nakajima M (2011) Community assembly: Alternative stable states or alternative transient states? Ecol Lett 14(10):973–984
doi: 10.1111/j.1461-0248.2011.01663.x
Galla T (2018) Dynamically evolved community size and stability of random Lotka–Volterra ecosystems (a). Europhys Lett 123(4):48004
doi: 10.1209/0295-5075/123/48004
Geman S, Hwang C-R (1982) A chaos hypothesis for some large systems of random equations. Z Wahrsch Verw Gebiete 60(3):291–314
doi: 10.1007/BF00535717
Gibbs T, Grilli J, Rogers T, Allesina S (2018) Effect of population abundances on the stability of large random ecosystems. Phys Rev E 98(2):022410
doi: 10.1103/PhysRevE.98.022410
Grilli J, Adorisio M, Suweis S, Barabás G, Banavar JR, Allesina S, Maritan A (2017) Feasibility and coexistence of large ecological communities. Nat Commun 8(1):14389
doi: 10.1038/ncomms14389
Hastings A (2001) Transient dynamics and persistence of ecological systems. Ecol Lett 4(3):215–220
doi: 10.1046/j.1461-0248.2001.00220.x
Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press
Jost L (2006) Entropy and diversity. Oikos 113(2):363–375
doi: 10.1111/j.2006.0030-1299.14714.x
Jost L (2007) Partitioning diversity into independent alpha and beta components. Ecology 88(10):2427–2439
doi: 10.1890/06-1736.1
Kéfi S, Dakos V, Scheffer M, Van Nes EH, Rietkerk M (2013) Early warning signals also precede non-catastrophic transitions. Oikos 122(5):641–648
Lamperski A (2019) Lemke’s algorithm for linear complementarity problems. https://github.com/AndyLamperski/lemkelcp
Law R, Morton RD (1996) Permanence and the assembly of ecological communities. Ecology 77(3):762–775
doi: 10.2307/2265500
Logofet DO (2018) Matrices and graphs: stability problems in mathematical ecology, 1st edn. CRC Press
Lotka AJ (1925) Elements of physical biology. Williams & Wilkins
May RM (1972) Will a large complex system be stable? Nature 238(5364):413
doi: 10.1038/238413a0
Murty KG (1988) Linear complementarity, linear and nonlinear programming. Sigma series in applied mathematics, vol 3. Heldermann
Murty KG (1972) On the number of solutions to the complementarity problem and spanning properties of complementary cones. Linear Algebra Appl 5(1):65–108
doi: 10.1016/0024-3795(72)90019-5
Neubert MG, Caswell H (1997) Alternatives to resilience for measuring the responses of ecological systems to perturbations. Ecology 78(3):653–665
doi: 10.1890/0012-9658(1997)078[0653:ATRFMT]2.0.CO;2
Nolting BC, Abbott KC (2016) Balls, cups, and quasi-potentials: quantifying stability in stochastic systems. Ecology 97(4):850–864
doi: 10.1890/15-1047.1
Opper M, Diederich S (1992) Phase transition and 1/f noise in a game dynamical model. Phys Rev Lett 69(10):1616–1619
doi: 10.1103/PhysRevLett.69.1616
Pettersson S, Savage VM, Nilsson Jacobi M (2020) Predicting collapse of complex ecological systems: quantifying the stability-complexity continuum. J R Soc Interface 17(166):20190391
doi: 10.1098/rsif.2019.0391
Serván CA, Capitán JA, Grilli J, Morrison KE, Allesina S (2018) Coexistence of many species in random ecosystems. Nat Ecol Evol 2(8):1237–1242
doi: 10.1038/s41559-018-0603-6
Smirnov NV (1949) Limit distributions for the terms of a variational series. Trudy Matematicheskogo Instituta imeni VA Steklova 25:3–60
Stone L (2018) The feasibility and stability of large complex biological networks: a random matrix approach. Sci Rep 8(1):8246
doi: 10.1038/s41598-018-26486-2
Takeuchi Y (1996) Global dynamical properties of Lotka–Volterra systems. World Scientific
Takeuchi Y, Adachi N (1980) The existence of globally stable equilibria of ecosystems of the generalized Volterra type. J Math Biol 10(4):401–415
doi: 10.1007/BF00276098
Tao T (2013) Outliers in the spectrum of IID matrices with bounded rank perturbations. Probab Theory Relat Fields 155(1):231–263
doi: 10.1007/s00440-011-0397-9
Volterra V (1931) Théorie mathématique de la lutte pour la vie. Gauthiers-Villars
Volterra V (1926) Fluctuations in the abundance of a species considered mathematically. Nature 118(2972):558–560
doi: 10.1038/118558a0