Ecological stability

Inecology,anecosystemis said to possessecological stability(orequilibrium) if it is capable of returning to its equilibrium state after a perturbation (a capacity known asresilience) or does not experience unexpected large changes in its characteristics across time.Although the termscommunity stabilityand ecological stability are sometimes used interchangeably,community stability refers only to the characteristics ofcommunities.It is possible for an ecosystem or a community to be stable in some of their properties and unstable in others. For example, avegetation communityin response to adroughtmight conservebiomassbut losebiodiversity.

Stable ecological systems abound in nature, and the scientific literature has documented them to a great extent. Scientific studies mainly describegrasslandplant communities andmicrobialcommunities.Nevertheless, it is important to mention that not every community or ecosystem in nature is stable (for example,wolves and moose on Isle Royale). Also, noise plays an important role on biological systems and, in some scenarios, it can fully determine their temporal dynamics.

The concept of ecological stability emerged in the first half of the 20th century. With the advancement oftheoretical ecologyin the 1970s, the usage of the term has expanded to a wide variety of scenarios. This overuse of the term has led to controversy over its definition and implementation.

In 1997, Grimm and Wissel made an inventory of 167 definitions used in the literature and found 70 different stability concepts.One of the strategies that these two authors proposed to clarify the subject is to replaceecological stabilitywith more specific terms, such asconstancy,resilienceandpersistence.In order to fully describe and put meaning to a specific kind of stability, it must be looked at more carefully. Otherwise the statements made about stability will have little to no reliability because they would not have information to back up the claim.Following this strategy, an ecosystem whichoscillatescyclically around a fixed point, such as the one delineated by thepredator-prey equations,would be described as persistent and resilient, but not as constant. Some authors, however, see good reason for the abundance of definitions, because they reflect the extensive variety of real and mathematical systems.

Stability analysis

When thespecies abundancesof an ecological system are treated with a set of differential equations, it is possible to test for stability bylinearizingthe system at the equilibrium point.Robert Mayused this stability analysis in the 1970s which uses theJacobian matrixorcommunity matrixto investigate the relation between the diversity and stability of an ecosystem.

May stability analysis andrandom matrix theory

To analyze the stability of large ecosystems, May drew on ideas fromstatistical mechanics,includingEugene Wigner's work successfully predicting the properties ofUraniumby assuming that itsHamiltoniancould be approximated as arandom matrix,leading to properties that were independent of the system's exact interactions.May considered an ecosystem with$S$species with abundances$N_{1},\ldots,N_{S}$whose dynamics are governed by the couples system of ordinary differential equations,$${\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}=g_{i}(N_{1},\ldots,N_{S}),\qquad i=1,\ldots,S.$$Assuming the system had a fixed point,$N_{1}^{\star },\ldots,N_{S}^{\star }$,May linearized dynamics as,$${\frac {\mathrm {d} N_{i}}{\mathrm {d} t}}=\sum _{j=1}^{S}J_{ij}(N_{j}-N_{j}^{\star }),\qquad i=1,\ldots,S.$$The fixed point will belinearly stableif all theeigenvaluesof theJacobian,$J_{ij}$,are positive. The matrix$J$is also known as thecommunity matrix.May supposed that the Jacobian was a random matrix whose off-diagonal entries$J_{ij}\;(i\neq j)$are all all drawn as random variates from aprobability distributionand 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 toGirko's circular law,when$S\gg 1$,the eigenvalues of$J$are distributed in the complex plane uniformly in a circle whose radius is${\sqrt {S}}\sigma$and whose center is$-1$,where$\sigma$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$-1+{\sqrt {S}}\sigma$.Therefore, the system will lose stability when,$${\sqrt {S}}>{\frac {1}{\sigma }}.$$This result is known as the May stability criterion. It implies that dynamical stability is limited bydiversity,and the strictness of this tradeoff is related to the magnitude of fluctuations in interactions.

Recent work has extended the approaches of May to constructphase diagramsfor ecological models, like thegeneralized Lotka–Volterra modelorconsumer-resource models,with large complex communities withdisorderedinteractions.This work has relied on uses and extensions ofrandom matrix theory,thecavity method,thereplica formalism,and other methods inspired byspin-glassphysics.

Types

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.This multitude of trends can be labeled by different types of ecological stability.

Dynamical stability

Dynamical stability refers to stability across time.

Stationary, stable, transient, and cyclic points

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.

Local and global stability

In the sense of perturbation amplitude,local stabilityindicates that a system is stable over small short-lived disturbances, while global stability indicates a system highly resistant to change inspecies compositionand/orfood 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).

Species and community stability

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

Constancy

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

Resistance and inertia (persistence)

Resistanceandinertiadeal 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.Resistanceis 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 changingecosystemsin post-glacial North America,E.C. Pielouremarked 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."

Resilience, elasticity and amplitude

Resilienceis the tendency of a system to retain its functional and organizational structure and the ability to recover after a perturbation or disturbance.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.Elasticityandamplitudeare measures of resilience. Elasticity is the speed with which a system returns to its original / previous state. Amplitude is a measure of 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 fromdynamical systemstheory.

Lyapunov stability

Researchers applyingmathematical modelsfrom systemdynamicsusually useLyapunov stability.

Numerical stability

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.

Sign stability

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

Stability and diversity

The relationship between diversity and stability has been widely studied. Diversity can enhance the stability of ecosystem functions at various ecological scales.For example, genetic diversity can enhance resistance to environmental perturbations.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 thetrophic coherenceof the network.At the level of landscapes,environmental heterogeneityacross locations has been shown to increase the stability of ecosystem functions.A stability diversity tradeoff has also been recently observed in microbial communities from human and sponge host environments.In the context of large and heterogeneous ecological networks, stability can be modeled using dynamic Jacobian ensembles.These show that scale and heterogeneity can stabilize specific states of the system in the face of environmental perturbations.

History of the concept

The term 'oekology' was coined byErnst Haeckelin 1866.Ecologyas 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.Frederic ClementsandHenry Gleasoncontributed knowledge of community structure; among other things, these two scientists introduced the opposing ideas that a community can either reach astable climaxor that it is largelycoincidental and variable.Charles Eltonargued in 1958 that complex, diverse communities tended to be more stable.Robert MacArthurproposed a mathematical description of stability in the number of individuals in afood webin 1955.After much progress made with experimental studies in the 60's,Robert Mayadvanced 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.

See also

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

References

• Hall, Charles A.S.""Ecology" (Accessed at World Book online reference centre) ".Archived fromthe originalon June 8, 2011.Retrieved2009-06-16.
• Homepage, for publications ofCharles A S Hallwith."Hall's publications on ecology".Archived fromthe originalon 2021-03-09.Retrieved2009-10-08.SeeComplete Publications ListinPublicationssection.