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 Ecological stability 

An ecosystem is said to posses ecological stability (or equilibrium) if it does not experience abrupt large changes in its characteristics across time, or if it is capable of returning to its equilibrium state after a perturbation (a capacity known as resilience). Although the terms community stability and ecological stability are sometimes used interchangeably, community stability refers only to the characteristics of communities. It is possible for an ecosystem or a community to be stable in some of their properties and unstable in others; e.g., in response to a drought, a plant community might conserve biomass but lose biodiversity.

The concept of ecological stability emerged in the first half of the 20th century, and with the advancement of theoretical ecology in the 1970's, 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 Wisel 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 replace ecological stability with more specific terms, such as constancy, resilience and persistence. Following this strategy, an ecosystem which oscillates cyclically around a fixed point, such as the one delineated by the predator-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.

Stable ecological systems abound in nature, and the scientific literature has documented them to a great extent. Scientific studies mainly describe grassland plant communities and microbial communities. Nevertheless, it is important to mention that not every community or ecosystem in nature is stable. Also, noise plays an important role on biological systems and, in some scenarios, it can fully determine their temporal dynamics.

Stability analysis
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. Robert May developed this stability analysis in the 1970's which uses the Jacobian matrix.

Types of Ecological Stability
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 behaviour that can be described as stable. This multitude of trends can be labeled by different types of ecological stability.

Dynamical Stability
Dynamical stability referes to stability across time.

Stationary, Stable, Transient, and Cyclic points.
An 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
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.

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

Resistance and inertia (persistence)
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.'"

Resilience, elasticity and amplitude
Resilience is the tendency of a system to retain its functional and organisational structure after a perturbation. Elasticity and amplitude are measures of resilience. Elasticity is the speed with which a system returns. 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 from dynamical systems theory.

Lyapunov Stability
Researchers applying mathematical models from system dynamics usually use Lyapunov stability.

Numerical Stability
Focusing on the biotic components of an ecosystem, a population or a community posses 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.

Structural Stability
Some of the sources that will help me on the construction of the section are 1 and 2. Reference to Serguei's work. Link to wiki of structural stability.

A dynamical system is structurally stable when it has a basin of attraction that pulls towards an stable state. Is this true?

Differences between dynamical and structural stability.

Examples of ecological systems structurally stable.

Stability and diversity
The relation between diversity and stability has been widely studied.

Are more diverse systems more stable?

History of Ecological Stability Thought
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. 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. 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.