ecological stability

When the species abundances of an ecological system are treated with a set of differential equations, it is possible to test for stability by linearizing the system at the equilibrium point.{{Cite book|title=Mathematical Models in Population Biology and Epidemiology|last=Carlos.|first=Castillo-Chávez|date=2012-01-01|publisher=Springer New York|isbn=9781461416869|oclc=779197058}} Robert May used this stability analysis in the 1970s which uses the Jacobian matrix or community matrix to investigate the relation between the diversity and stability of an ecosystem.{{Cite journal|last=May|first=Robert M.|date=1972-08-18|title=Will a Large Complex System be Stable?|journal=Nature|language=en|volume=238|issue=5364|pages=413–414|doi=10.1038/238413a0|pmid=4559589|bibcode=1972Natur.238..413M|s2cid=4262204}}

To analyze the stability of large ecosystems, May drew on ideas from statistical mechanics, including Eugene Wigner's work successfully predicting the properties of Uranium by assuming that its Hamiltonian could be approximated as a random matrix, leading to properties that were independent of the system's exact interactions.{{cite arXiv|last1=Cui |first1=Wenping |title=Les Houches Lectures on Community Ecology: From Niche Theory to Statistical Mechanics |date=2024-03-08 |eprint=2403.05497 |last2=Marsland III |first2=Robert |last3=Mehta |first3=Pankaj|class=q-bio.PE }}{{Cite book |last=Allesina |first=Stefano |url=https://stefanoallesina.github.io/Theoretical_Community_Ecology/ |title=Theoretical Community Ecology}} May considered an ecosystem with $S$ species with abundances $N\_1,\backslash ldots,N\_S$ whose dynamics are governed by the couples system of ordinary differential equations,$$\backslash frac\{\backslash mathrm\; dN\_i\}\{\backslash mathrm\; d\; t\}\; =\; g\_i(N\_1,\backslash ldots,N\_S),\backslash qquad\; i\; =\; 1,\backslash ldots,S.$$Assuming the system had a fixed point, $N\_1^\backslash star,\backslash ldots,\; N\_S^\backslash star$, May linearized dynamics as,$$\backslash frac\{\backslash mathrm\; d\; N\_i\}\{\backslash mathrm\; d\; t\}\; =\; \backslash sum\_\{j=1\}^S\; J\_\{ij\}(N\_j\; -\; N\_j^\backslash star)\; ,\backslash qquad\; i\; =1\; ,\backslash ldots,S.$$The fixed point will be linearly stable if all the eigenvalues of the Jacobian, $J\_\{ij\}$, are positive. The matrix $J$ is also known as the community matrix. May supposed that the Jacobian was a random matrix whose off-diagonal entries $J\_\{ij\}\backslash ;(i\backslash neq\; j)$ are all all drawn as random variates from a probability distribution and whose diagonal elements $J\_\{ii\}$ are all -1 so that each species inhibits its own growth and stability is guaranteed in the absence of inter-species interactions. According to Girko's circular law, when $S\backslash gg\; 1$, the eigenvalues of $J$ are distributed in the complex plane uniformly in a circle whose radius is $\backslash sqrt\{S\}\backslash sigma$ and whose center is $-1$, where $\backslash sigma$ is the standard deviation of the distribution for the off-diagonal elements of the Jacobian. Using this result, the eigenvalue with the largest real part contained in the support of the spectrum of $J$ is $-1+\backslash sqrt\{S\}\backslash sigma$. Therefore, the system will lose stability when,$$\backslash sqrt\{S\}\; >\; \backslash frac\{1\}\{\backslash sigma\}.$$This result is known as the May stability criterion. It implies that dynamical stability is limited by diversity, and the strictness of this tradeoff is related to the magnitude of fluctuations in interactions.

Recent work has extended the approaches of May to construct phase diagrams for ecological models, like the generalized Lotka–Volterra model or consumer-resource models, with large complex communities with disordered interactions.{{Cite journal |last=Bunin |first=Guy |date=2017-04-28 |title=Ecological communities with Lotka-Volterra dynamics |url=https://link.aps.org/doi/10.1103/PhysRevE.95.042414 |journal=Physical Review E |volume=95 |issue=4 |pages=042414 |doi=10.1103/PhysRevE.95.042414|pmid=28505745 |bibcode=2017PhRvE..95d2414B }}{{Cite journal |last1=Blumenthal |first1=Emmy |last2=Rocks |first2=Jason W. |last3=Mehta |first3=Pankaj |date=2024-03-21 |title=Phase Transition to Chaos in Complex Ecosystems with Nonreciprocal Species-Resource Interactions |url=https://link.aps.org/doi/10.1103/PhysRevLett.132.127401 |journal=Physical Review Letters |volume=132 |issue=12 |pages=127401 |doi=10.1103/PhysRevLett.132.127401|pmid=38579223 |arxiv=2308.15757 |bibcode=2024PhRvL.132l7401B }} This work has relied on uses and extensions of random matrix theory, the cavity method, the replica formalism, and other methods inspired by spin-glass physics.

Although the characteristics of any ecological system are susceptible to changes, during a defined period of time, some remain constant, oscillate, reach a fixed point or present other type of behavior that can be described as stable.{{Cite journal|last=Lewontin|first=Richard C.|date=1969|title=The Meaning of Stability|journal=Brookhaven Symposia in Biology|volume=22|pages=13–23|pmid=5372787}} This multitude of trends can be labeled by different types of ecological stability.

Dynamical stability refers to stability across time.

A stable point is such that a small perturbation of the system will be diminished and the system will come back to the original point. On the other hand, if a small perturbation is magnified, the stationary point is considered unstable.

In the sense of perturbation amplitude, local stability indicates that a system is stable over small short-lived disturbances, while global stability indicates a system highly resistant to change in species composition and/or food web dynamics.

In the sense of spatial extension, local instability indicates stability in a limited region of the ecosystem, while global (or regional) stability involves the whole ecosystem (or a large part of it).{{Cite journal |last1=Jarillo |first1=Javier |last2=Cao-García |first2=Francisco J. |last3=De Laender |first3=Frederik |date=2022 |title=Spatial and Ecological Scaling of Stability in Spatial Community Networks |journal=Frontiers in Ecology and Evolution |volume=10 |doi=10.3389/fevo.2022.861537 |issn=2296-701X|doi-access=free |arxiv=2201.09683 }}

Stability can be studied at the species or at the community level, with links between these levels.

Observational studies of ecosystems use constancy to describe living systems that can remain unchanged.

Resistance and inertia deal with a system's inherent response to some perturbation.

A perturbation is any externally imposed change in conditions, usually happening in a short time period. Resistance is a measure of how little the variable of interest changes in response to external pressures. Inertia (or persistence) implies that the living system is able to resist external fluctuations. In the context of changing ecosystems in post-glacial North America, E.C. Pielou remarked at the outset of her overview,

"It obviously takes considerable time for mature vegetation to become established on newly exposed ice scoured rocks or glacial till...it also takes considerable time for whole ecosystems to change, with their numerous interdependent plant species, the habitats these create, and the animals that live in the habitats. Therefore, climatically caused fluctuations in ecological communities are a damped, smoothed-out version of the climatic fluctuations that cause them."Pielou, After the Ice Age: The Return of Life to Glaciated North America (Chicago: University of Chicago Press) 1991:13

Resilience is the tendency of a system to retain its functional and organizational structure and the ability to recover after a perturbation or disturbance.{{Cite journal|last1=Donohue|first1=Ian|last2=Hillebrand|first2=Helmut|last3=Montoya|first3=José M.|last4=Petchey|first4=Owen L.|last5=Pimm|first5=Stuart L.|last6=Fowler|first6=Mike S.|last7=Healy|first7=Kevin|last8=Jackson|first8=Andrew L.|last9=Lurgi|first9=Miguel|last10=McClean|first10=Deirdre|last11=O'Connor|first11=Nessa E.|date=2016|title=Navigating the complexity of ecological stability|url=https://onlinelibrary.wiley.com/doi/abs/10.1111/ele.12648|journal=Ecology Letters|language=en|volume=19|issue=9|pages=1172–1185|doi=10.1111/ele.12648|pmid=27432641|bibcode=2016EcolL..19.1172D |s2cid=25646033 |issn=1461-0248}} Resilience also expresses the need for persistence although from a management approach it is expressed to have a broad range of choices and events are to be looked at as uniformly distributed.{{Cite journal|last=Holling|first=C. S.|date=1973|title=Resilience and Stability of Ecological Systems|journal=Annual Review of Ecology and Systematics|volume=4|pages=1–23|issn=0066-4162|jstor=2096802|doi=10.1146/annurev.es.04.110173.000245|s2cid=53309505 |url=http://pure.iiasa.ac.at/26/1/RP-73-003.pdf}} Elasticity and amplitude are measures of resilience. Elasticity is the speed with which a system returns to its original/previous state. Amplitude measures how far a system can be moved from the previous state and still return. Ecology borrows the idea of neighborhood stability and a domain of attraction from dynamical system theory.

Researchers applying mathematical models from system dynamics usually use Lyapunov stability.{{cite web|url=http://philsci-archive.pitt.edu/archive/00002987/01/PSA_2006_Justus_10-15-06.pdf|title=Ecological and Lyanupov Stability|last=Justus|first=James|year=2006|publisher=Paper presented at the Biennial Meeting of The Philosophy of Science Association, Vancouver, Canada}}{{cite journal|year=2008|title=Ecological and Lyanupov Stability|journal=Philosophy of Science|volume=75|issue=4|pages=421–436|doi=10.1086/595836|author=Justus, J|citeseerx=10.1.1.405.2888|s2cid=14194437}}(Published version of above paper)

Focusing on the biotic components of an ecosystem, a population, or a community possesses numerical stability if the number of individuals is constant or resilient.{{Cite book|title=The Princeton guide to ecology|last1=A.|first1=Levin, Simon|last2=R.|first2=Carpenter, Stephen|date=2012-01-01|publisher=Princeton University Press|isbn=9780691156040|pages=65|oclc=841495663}}

It is possible to determine if a system is stable just by looking at the signs in the interaction matrix. 

The relationship between diversity and stability has been widely studied.{{Cite journal|last1=Ives|first1=Anthony R.|last2=Carpenter|first2=Stephen R.|date=2007-07-06|title=Stability and Diversity of Ecosystems|journal=Science|language=en|volume=317|issue=5834|pages=58–62|doi=10.1126/science.1133258|issn=0036-8075|pmid=17615333|bibcode=2007Sci...317...58I|s2cid=11001567}}{{cite journal|last1=Furness|first1=Euan N.|last2=Garwood|first2=Russell J.|last3=Mannion|first3=Philip D.|last4=Sutton|first4=Mark D.|title=Evolutionary simulations clarify and reconcile biodiversity-disturbance models|journal=Proceedings of the Royal Society B: Biological Sciences|volume=288|issue=1949|year=2021|issn=0962-8452|doi=10.1098/rspb.2021.0240|pmid=33878917|pmc=8059584}}

Diversity can enhance the stability of ecosystem functions at various ecological scales.{{Cite journal|last1=Oliver|first1=Tom H.|last2=Heard|first2=Matthew S.|last3=Isaac|first3=Nick J.B.|last4=Roy|first4=David B.|last5=Procter|first5=Deborah|last6=Eigenbrod|first6=Felix|last7=Freckleton|first7=Rob|last8=Hector|first8=Andy|last9=Orme|first9=C. David L.|title=Biodiversity and Resilience of Ecosystem Functions|journal=Trends in Ecology & Evolution|language=en|volume=30|issue=11|pages=673–684|doi=10.1016/j.tree.2015.08.009|pmid=26437633|year=2015|url=http://nora.nerc.ac.uk/id/eprint/512028/1/N512028PP.pdf}} For example, genetic diversity can enhance resistance to environmental perturbations.{{Cite journal|last1=Forsman|first1=Anders|last2=Wennersten|first2=Lena|date=2016-07-01|title=Inter-individual variation promotes ecological success of populations and species: evidence from experimental and comparative studies|journal=Ecography|language=en|volume=39|issue=7|pages=630–648|doi=10.1111/ecog.01357|issn=1600-0587|doi-access=free|bibcode=2016Ecogr..39..630F }} At the community level, the structure of food webs can affect stability. The effect of diversity on stability in food-web models can be either positive or negative, depending on the trophic coherence of the network.{{cite journal |author=Johnson S, Domı́nguez-Garcı́a V, Donetti L, Muñoz MA |year=2014 |title=Trophic coherence determines food-web stability |journal=Proc Natl Acad Sci USA |volume=111 |issue=50 |pages=17923–17928 |doi=10.1073/pnas.1409077111|pmid=25468963 |pmc=4273378 |arxiv=1404.7728 |bibcode=2014PNAS..11117923J |doi-access=free }} At the level of landscapes, environmental heterogeneity across locations has been shown to increase the stability of ecosystem functions.{{Cite journal|last1=Wang|first1=Shaopeng|last2=Loreau|first2=Michel|date=2014-08-01|title=Ecosystem stability in space: α, β and γ variability|journal=Ecology Letters|language=en|volume=17|issue=8|pages=891–901|doi=10.1111/ele.12292|pmid=24811401|bibcode=2014EcolL..17..891W |issn=1461-0248}} A stability diversity tradeoff has also been recently observed in microbial communities from human and sponge host environments.{{Cite journal|last1=Yonatan|first1=Yogev|last2=Amit|first2=Guy|last3=Friedman|first3=Jonathan|last4=Bashan|first4=Amir|date=2022-04-28|title=Complexity–stability trade-off in empirical microbial ecosystems|journal=Nature Ecology & Evolution|volume=6 |issue=5 |pages=693–700|language=en|doi=10.1038/s41559-022-01745-8 |pmid=35484221 |bibcode=2022NatEE...6..693Y |s2cid=248432081 }} In the context of large and heterogeneous ecological networks, stability can be modeled using dynamic Jacobian ensembles.{{Cite journal|last=C. Meena, C. Hens, S. Acharyya, S. Haber, S. Boccaletti and B. Barzel|date=2023|title=Emergent stability in complex network dynamics|journal=Nature Physics|volume= 19|issue= 7|pages= 1033–1042|doi=10.1038/s41567-023-02020-8|arxiv=2007.04890|bibcode=2023NatPh..19.1033M |s2cid=234358850 }} These show that scale and heterogeneity can stabilize specific states of the system in the face of environmental perturbations.

The term 'oekology' was coined by Ernst Haeckel in 1866. Ecology as a science was developed further during the late 19th and the early 20th century, and increasing attention was directed toward the connection between diversity and stability.{{Cite book|url=https://books.google.com/books?id=lZvgTuB9Gh4C&pg=PA1|title=Animal Ecology|last=Elton|first=Charles S.|date=1927-01-01|publisher=University of Chicago Press|isbn=9780226206394|language=en}} Frederic Clements and Henry Gleason contributed knowledge of community structure; among other things, these two scientists introduced the opposing ideas that a community can either reach a stable climax or that it is largely coincidental and variable. Charles Elton argued in 1958 that complex, diverse communities tended to be more stable. Robert MacArthur proposed a mathematical description of stability in the number of individuals in a food web in 1955.{{Cite journal|last=MacArthur|first=Robert|date=1955-01-01|title=Fluctuations of Animal Populations and a Measure of Community Stability|jstor=1929601|journal=Ecology|volume=36|issue=3|pages=533–536|doi=10.2307/1929601|bibcode=1955Ecol...36..533M }} After much progress made with experimental studies in the 60's, Robert May advanced the field of theoretical ecology and refuted the idea that diversity begets stability. Many definitions of ecological stability have emerged in the last decades while the concept continues to gain attention.

• Dynamic equilibrium
• Ecological resilience
• Keystone species
• Principle of faunal succession
• Systems analysis
• Trophic coherence
• Random generalized Lotka–Volterra model

{{reflist}}

• {{cite web|url=http://www.worldbookonline.com/wb/Article?id=ar173240 |title="Ecology" (Accessed at World Book online reference centre) |last=Hall |first=Charles A.S. |date= |access-date=2009-06-16 |url-status=dead |archive-url=https://web.archive.org/web/20110608192045/http://www.worldbookonline.com/wb/Article?id=ar173240 |archive-date=June 8, 2011 }}
• {{cite web| url= http://www.esf.edu/EFB/hall/| title= Hall's publications on ecology| last= Homepage| first= for publications of Charles A S Hall with| access-date= 2009-10-08| archive-date= 2021-03-09| archive-url= https://web.archive.org/web/20210309203749/http://www.esf.edu/EFB/hall/| url-status= dead}} See Complete Publications List in Publications section.

{{modelling ecosystems|expanded=other}}

Category:Ecological processes