Ecological stability

In

Stable ecological systems abound in nature, and the scientific literature has documented them to a great extent. Scientific studies mainly describe

). 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 of theoretical ecology in 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.^[3]

In 1997, Grimm and Wissel made an inventory of 167 definitions used in the literature and found 70 different stability concepts.^[5] 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. 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.^[6] 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.^[3]

Stability analysis

When the

May stability analysis and

random matrix theory

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.^[8][9][10] May considered an ecosystem with ${\displaystyle S}$ species with abundances ${\displaystyle N_{1},\ldots ,N_{S}}$ whose dynamics are governed by the couples system of ordinary differential equations,

Assuming the system had a fixed point, ${\displaystyle N_{1}^{\star },\ldots ,N_{S}^{\star }}$, May linearized dynamics as,The fixed point will be linearly stable if all the eigenvalues of the Jacobian, ${\displaystyle J_{ij}}$, are positive. The matrix ${\displaystyle J}$ is also known as the community matrix. May supposed that the Jacobian was a random matrix whose off-diagonal entries ${\displaystyle J_{ij}\;(i\neq j)}$ are all all drawn as random variates from a probability distribution and whose diagonal elements ${\displaystyle J_{ii}}$ 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 ${\displaystyle S\gg 1}$, the eigenvalues of ${\displaystyle J}$ are distributed in the complex plane uniformly in a circle whose radius is ${\displaystyle {\sqrt {S}}\sigma }$ and whose center is ${\displaystyle -1}$, where ${\displaystyle \sigma }$ 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 ${\displaystyle J}$ is ${\displaystyle -1+{\sqrt {S}}\sigma }$. Therefore, the system will lose stability when,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

physics.

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.[13] 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,

.

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).^[14]

Species and community stability

Stability can be studied at the species or at the community level, with links between these levels.^[14]

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

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."^[15]

Resilience, elasticity and amplitude

theory.

Lyapunov stability

Researchers applying

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.[20]

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.^[4][21] Diversity can enhance the stability of ecosystem functions at various ecological scales.

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 by

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

Notes

References

• Hall, Charles A.S. ""Ecology" (Accessed at World Book online reference centre)". Archived from the original on June 8, 2011. Retrieved 2009-06-16.
• Homepage, for publications of
  Charles A S Hall with. "Hall's publications on ecology". Archived from the original
  on 2021-03-09. Retrieved 2009-10-08. See Complete Publications List in Publications section.