Quantum chaos

From Wikipedia, the free encyclopedia		https://en.wikipedia.org/wiki/Quantum_chaos

Quantum chaos is the field of physics attempting to bridge the theories of quantum mechanics and classical mechanics. The figure shows the main ideas running in each direction.

 

Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in the limit as the ratio of Planck's constant to the action of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos (although this may not be a fruitful way of examining classical chaos). If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?

In seeking to address the basic question of quantum chaos, several approaches have been employed:

1. Development of methods for solving quantum problems where the perturbation cannot be considered small in perturbation theory and where quantum numbers are large.
2. Correlating statistical descriptions of eigenvalues (energy levels) with the classical behavior of the same Hamiltonian (system).
3. Semiclassical methods such as periodic-orbit theory connecting the classical trajectories of the dynamical system with quantum features.
4. Direct application of the correspondence principle.

History

Experimental recurrence spectra of lithium in an electric field showing birth of quantum recurrences corresponding to bifurcations of classical orbits.

During the first half of the twentieth century, chaotic behavior in mechanics was recognized (as in the three-body problem in celestial mechanics), but not well understood. The foundations of modern quantum mechanics were laid in that period, essentially leaving aside the issue of the quantum-classical correspondence in systems whose classical limit exhibit chaos.

Approaches

Comparison of experimental and theoretical recurrence spectra of lithium in an electric field at a scaled energy of ${\displaystyle \epsilon =-3.0}$.

Questions related to the correspondence principle arise in many different branches of physics, ranging from nuclear to atomic, molecular and solid-state physics, and even to acoustics, microwaves and optics. However, classical-quantum correspondence in chaos theory is not always possible. Thus, some versions of the classical butterfly effect do not have counterparts in quantum mechanics.

Important observations are often associated with classically chaotic quantum systems are spectral level repulsion, dynamical localization in time evolution (e.g. ionization rates of atoms), and enhanced stationary wave intensities in regions of space where classical dynamics exhibits only unstable trajectories (as in scattering). In the semiclassical approach of quantum chaos, phenomena are identified in spectroscopy by analyzing the statistical distribution of spectral lines and by connecting spectral periodicities with classical orbits. Other phenomena show up in the time evolution of a quantum system, or in its response to various types of external forces. In some contexts, such as acoustics or microwaves, wave patterns are directly observable and exhibit irregular amplitude distributions.

Quantum chaos typically deals with systems whose properties need to be calculated using either numerical techniques or approximation schemes (see e.g. Dyson series). Simple and exact solutions are precluded by the fact that the system's constituents either influence each other in a complex way, or depend on temporally varying external forces.

Quantum mechanics in non-perturbative regimes

Computed regular (non-chaotic) Rydberg atom energy level spectra of hydrogen in an electric field near n=15. Note that energy levels can cross due to underlying symmetries of dynamical motion.

Computed chaotic Rydberg atom energy level spectra of lithium in an electric field near n=15. Note that energy levels cannot cross due to the ionic core (and resulting quantum defect) breaking symmetries of dynamical motion.

For conservative systems, the goal of quantum mechanics in non-perturbative regimes is to find the eigenvalues and eigenvectors of a Hamiltonian of the form

${\displaystyle H=H_{s}+\varepsilon H_{ns},\,}$

where ${\displaystyle H_{s}}$ is separable in some coordinate system, ${\displaystyle H_{ns}}$ is non-separable in the coordinate system in which ${\displaystyle H_{s}}$ is separated, and ${\displaystyle \epsilon }$ is a parameter which cannot be considered small. Physicists have historically approached problems of this nature by trying to find the coordinate system in which the non-separable Hamiltonian is smallest and then treating the non-separable Hamiltonian as a perturbation.

Finding constants of motion so that this separation can be performed can be a difficult (sometimes impossible) analytical task. Solving the classical problem can give valuable insight into solving the quantum problem. If there are regular classical solutions of the same Hamiltonian, then there are (at least) approximate constants of motion, and by solving the classical problem, we gain clues how to find them.

Other approaches have been developed in recent years. One is to express the Hamiltonian in different coordinate systems in different regions of space, minimizing the non-separable part of the Hamiltonian in each region. Wavefunctions are obtained in these regions, and eigenvalues are obtained by matching boundary conditions.

Another approach is numerical matrix diagonalization. If the Hamiltonian matrix is computed in any complete basis, eigenvalues and eigenvectors are obtained by diagonalizing the matrix. However, all complete basis sets are infinite, and we need to truncate the basis and still obtain accurate results. These techniques boil down to choosing a truncated basis from which accurate wavefunctions can be constructed. The computational time required to diagonalize a matrix scales as ${\displaystyle N^{3}}$, where ${\displaystyle N}$ is the dimension of the matrix, so it is important to choose the smallest basis possible from which the relevant wavefunctions can be constructed. It is also convenient to choose a basis in which the matrix is sparse and/or the matrix elements are given by simple algebraic expressions because computing matrix elements can also be a computational burden.

A given Hamiltonian shares the same constants of motion for both classical and quantum dynamics. Quantum systems can also have additional quantum numbers corresponding to discrete symmetries (such as parity conservation from reflection symmetry). However, if we merely find quantum solutions of a Hamiltonian which is not approachable by perturbation theory, we may learn a great deal about quantum solutions, but we have learned little about quantum chaos. Nevertheless, learning how to solve such quantum problems is an important part of answering the question of quantum chaos.

Correlating statistical descriptions of quantum mechanics with classical behavior

Nearest neighbor distribution for Rydberg atom energy level spectra in an electric field as quantum defect is increased from 0.04 (a) to 0.32 (h). The system becomes more chaotic as dynamical symmetries are broken by increasing the quantum defect; consequently, the distribution evolves from nearly a Poisson distribution (a) to that of Wigner's surmise (h).

Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems. Random matrix theory was developed in an attempt to characterize spectra of complex nuclei. The remarkable result is that the statistical properties of many systems with unknown Hamiltonians can be predicted using random matrices of the proper symmetry class. Furthermore, random matrix theory also correctly predicts statistical properties of the eigenvalues of many chaotic systems with known Hamiltonians. This makes it useful as a tool for characterizing spectra which require large numerical efforts to compute.

A number of statistical measures are available for quantifying spectral features in a simple way. It is of great interest whether or not there are universal statistical behaviors of classically chaotic systems. The statistical tests mentioned here are universal, at least to systems with few degrees of freedom (Berry and Tabor have put forward strong arguments for a Poisson distribution in the case of regular motion and Heusler et al. present a semiclassical explanation of the so-called Bohigas–Giannoni–Schmit conjecture which asserts universality of spectral fluctuations in chaotic dynamics). The nearest-neighbor distribution (NND) of energy levels is relatively simple to interpret and it has been widely used to describe quantum chaos.

Qualitative observations of level repulsions can be quantified and related to the classical dynamics using the NND, which is believed to be an important signature of classical dynamics in quantum systems. It is thought that regular classical dynamics is manifested by a Poisson distribution of energy levels:

${\displaystyle P(s)=e^{-s}.\ }$

In addition, systems which display chaotic classical motion are expected to be characterized by the statistics of random matrix eigenvalue ensembles. For systems invariant under time reversal, the energy-level statistics of a number of chaotic systems have been shown to be in good agreement with the predictions of the Gaussian orthogonal ensemble (GOE) of random matrices, and it has been suggested that this phenomenon is generic for all chaotic systems with this symmetry. If the normalized spacing between two energy levels is ${\displaystyle s}$, the normalized distribution of spacings is well approximated by

${\displaystyle P(s)={\frac {\pi }{2}}se^{-\pi s^{2}/4}.}$

Many Hamiltonian systems which are classically integrable (non-chaotic) have been found to have quantum solutions that yield nearest neighbor distributions which follow the Poisson distributions. Similarly, many systems which exhibit classical chaos have been found with quantum solutions yielding a Wigner-Dyson distribution, thus supporting the ideas above. One notable exception is diamagnetic lithium which, though exhibiting classical chaos, demonstrates Wigner (chaotic) statistics for the even-parity energy levels and nearly Poisson (regular) statistics for the odd-parity energy level distribution.

Semiclassical methods

Periodic orbit theory

Even parity recurrence spectrum (Fourier transform of the density of states) of diamagnetic hydrogen showing peaks corresponding to periodic orbits of the classical system. Spectrum is at a scaled energy of −0.6. Peaks labeled R and V are repetitions of the closed orbit perpendicular and parallel to the field, respectively. Peaks labeled O correspond to the near circular periodic orbit that goes around the nucleus.

Relative recurrence amplitudes of even and odd recurrences of the near circular orbit. Diamonds and plus signs are for odd and even quarter periods, respectively. Solid line is A/cosh(nX/8). Dashed line is A/sinh(nX/8) where A = 14.75 and X = 1.18.

Periodic-orbit theory gives a recipe for computing spectra from the periodic orbits of a system. In contrast to the Einstein–Brillouin–Keller method of action quantization, which applies only to integrable or near-integrable systems and computes individual eigenvalues from each trajectory, periodic-orbit theory is applicable to both integrable and non-integrable systems and asserts that each periodic orbit produces a sinusoidal fluctuation in the density of states.

The principal result of this development is an expression for the density of states which is the trace of the semiclassical Green's function and is given by the Gutzwiller trace formula:

${\displaystyle g_{c}(E)=\sum _{k}T_{k}\sum _{n=1}^{\infty }{\frac {1}{2\sinh {(\chi _{nk}/2)}}}\,e^{i(nS_{k}-\alpha _{nk}\pi /2)}.}$

Recently there was a generalization of this formula for arbitrary matrix Hamiltonians that involves a Berry phase-like term stemming from spin or other internal degrees of freedom. The index ${\displaystyle k}$ distinguishes the primitive periodic orbits: the shortest period orbits of a given set of initial conditions. ${\displaystyle T_{k}}$ is the period of the primitive periodic orbit and ${\displaystyle S_{k}}$ is its classical action. Each primitive orbit retraces itself, leading to a new orbit with action ${\displaystyle nS_{k}}$ and a period which is an integral multiple ${\displaystyle n}$ of the primitive period. Hence, every repetition of a periodic orbit is another periodic orbit. These repetitions are separately classified by the intermediate sum over the indices ${\displaystyle n}$. ${\displaystyle \alpha _{nk}}$ is the orbit's Maslov index. The amplitude factor, ${\displaystyle 1/\sinh {(\chi _{nk}/2)}}$, represents the square root of the density of neighboring orbits. Neighboring trajectories of an unstable periodic orbit diverge exponentially in time from the periodic orbit. The quantity ${\displaystyle \chi _{nk}}$ characterizes the instability of the orbit. A stable orbit moves on a torus in phase space, and neighboring trajectories wind around it. For stable orbits, ${\displaystyle \sinh {(\chi _{nk}/2)}}$ becomes ${\displaystyle \sin {(\chi _{nk}/2)}}$, where ${\displaystyle \chi _{nk}}$ is the winding number of the periodic orbit. ${\displaystyle \chi _{nk}=2\pi m}$, where ${\displaystyle m}$ is the number of times that neighboring orbits intersect the periodic orbit in one period. This presents a difficulty because ${\displaystyle \sin {(\chi _{nk}/2)}=0}$ at a classical bifurcation. This causes that orbit's contribution to the energy density to diverge. This also occurs in the context of photo-absorption spectrum.

Using the trace formula to compute a spectrum requires summing over all of the periodic orbits of a system. This presents several difficulties for chaotic systems: 1) The number of periodic orbits proliferates exponentially as a function of action. 2) There are an infinite number of periodic orbits, and the convergence properties of periodic-orbit theory are unknown. This difficulty is also present when applying periodic-orbit theory to regular systems. 3) Long-period orbits are difficult to compute because most trajectories are unstable and sensitive to roundoff errors and details of the numerical integration.

Gutzwiller applied the trace formula to approach the anisotropic Kepler problem (a single particle in a ${\displaystyle 1/r}$ potential with an anisotropic mass tensor) semiclassically. He found agreement with quantum computations for low lying (up to ${\displaystyle n=6}$) states for small anisotropies by using only a small set of easily computed periodic orbits, but the agreement was poor for large anisotropies.

The figures above use an inverted approach to testing periodic-orbit theory. The trace formula asserts that each periodic orbit contributes a sinusoidal term to the spectrum. Rather than dealing with the computational difficulties surrounding long-period orbits to try to find the density of states (energy levels), one can use standard quantum mechanical perturbation theory to compute eigenvalues (energy levels) and use the Fourier transform to look for the periodic modulations of the spectrum which are the signature of periodic orbits. Interpreting the spectrum then amounts to finding the orbits which correspond to peaks in the Fourier transform.

Rough sketch on how to arrive at the Gutzwiller trace formula

1. Start with the semiclassical approximation of the time-dependent Green's function (the Van Vleck propagator).
2. Realize that for caustics the description diverges and use the insight by Maslov (approximately Fourier transforming to momentum space (stationary phase approximation with h a small parameter) to avoid such points and afterwards transforming back to position space can cure such a divergence, however gives a phase factor).
3. Transform the Greens function to energy space to get the energy dependent Greens function ( again approximate Fourier transform using the stationary phase approximation). New divergences might pop up that need to be cured using the same method as step 3
4. Use ${\displaystyle d(E)=-{\frac {1}{\pi }}\Im (\operatorname {Tr} (G(x,x^{\prime },E))}$ (tracing over positions) and calculate it again in stationary phase approximation to get an approximation for the density of states ${\displaystyle d(E)}$.

Note: Taking the trace tells you that only closed orbits contribute, the stationary phase approximation gives you restrictive conditions each time you make it. In step 4 it restricts you to orbits where initial and final momentum are the same i.e. periodic orbits. Often it is nice to choose a coordinate system parallel to the direction of movement, as it is done in many books.

Closed orbit theory

Experimental recurrence spectrum (circles) is compared with the results of the closed orbit theory of John Delos and Jing Gao for lithium Rydberg atoms in an electric field. The peaks labeled 1–5 are repetitions of the electron orbit parallel to the field going from the nucleus to the classical turning point in the uphill direction.

Closed-orbit theory was developed by J.B. Delos, M.L. Du, J. Gao, and J. Shaw. It is similar to periodic-orbit theory, except that closed-orbit theory is applicable only to atomic and molecular spectra and yields the oscillator strength density (observable photo-absorption spectrum) from a specified initial state whereas periodic-orbit theory yields the density of states.

Only orbits that begin and end at the nucleus are important in closed-orbit theory. Physically, these are associated with the outgoing waves that are generated when a tightly bound electron is excited to a high-lying state. For Rydberg atoms and molecules, every orbit which is closed at the nucleus is also a periodic orbit whose period is equal to either the closure time or twice the closure time.

According to closed-orbit theory, the average oscillator strength density at constant ${\displaystyle \epsilon }$ is given by a smooth background plus an oscillatory sum of the form

${\displaystyle f(w)=\sum _{k}\sum _{n=1}^{\infty }D_{\it {nk}}^{i}\sin(2\pi nw{\tilde {S_{k}}}-\phi _{\it {nk}}).}$

${\displaystyle \phi _{\it {nk}}}$ is a phase that depends on the Maslov index and other details of the orbits. ${\displaystyle D_{\it {nk}}^{i}}$ is the recurrence amplitude of a closed orbit for a given initial state (labeled ${\displaystyle i}$). It contains information about the stability of the orbit, its initial and final directions, and the matrix element of the dipole operator between the initial state and a zero-energy Coulomb wave. For scaling systems such as Rydberg atoms in strong fields, the Fourier transform of an oscillator strength spectrum computed at fixed ${\displaystyle \epsilon }$ as a function of ${\displaystyle w}$ is called a recurrence spectrum, because it gives peaks which correspond to the scaled action of closed orbits and whose heights correspond to ${\displaystyle D_{\it {nk}}^{i}}$.

Closed-orbit theory has found broad agreement with a number of chaotic systems, including diamagnetic hydrogen, hydrogen in parallel electric and magnetic fields, diamagnetic lithium, lithium in an electric field, the ${\displaystyle H^{-}}$ ion in crossed and parallel electric and magnetic fields, barium in an electric field, and helium in an electric field.

One-dimensional systems and potential

For the case of one-dimensional system with the boundary condition ${\displaystyle y(0)=0}$ the density of states obtained from the Gutzwiller formula is related to the inverse of the potential of the classical system by ${\displaystyle {\frac {d^{1/2}}{dx^{1/2}}}V^{-1}(x)=2{\sqrt {\pi }}{\frac {dN(x)}{dx}}}$ here ${\displaystyle {\frac {dN(x)}{dx}}}$ is the density of states and V(x) is the classical potential of the particle, the half derivative of the inverse of the potential is related to the density of states as in the Wu-Sprung potential.

Recent directions

One open question remains understanding quantum chaos in systems that have finite-dimensional local Hilbert spaces for which standard semiclassical limits do not apply. Recent works allowed for studying analytically such quantum many-body systems.

The traditional topics in quantum chaos concerns spectral statistics (universal and non-universal features), and the study of eigenfunctions (Quantum ergodicity, scars) of various chaotic Hamiltonian ${\displaystyle H(x,p;R)}$.

Further studies concern the parametric (${\displaystyle R}$) dependence of the Hamiltonian, as reflected in e.g. the statistics of avoided crossings, and the associated mixing as reflected in the (parametric) local density of states (LDOS). There is vast literature on wavepacket dynamics, including the study of fluctuations, recurrences, quantum irreversibility issues etc. Special place is reserved to the study of the dynamics of quantized maps: the standard map and the kicked rotator are considered to be prototype problems.

Works are also focused in the study of driven chaotic systems, where the Hamiltonian ${\displaystyle H(x,p;R(t))}$ is time dependent, in particular in the adiabatic and in the linear response regimes. There is also significant effort focused on formulating ideas of quantum chaos for strongly-interacting many-body quantum systems far from semiclassical regimes.

Berry–Tabor conjecture

In 1977, Berry and Tabor made a still open "generic" mathematical conjecture which, stated roughly, is: In the "generic" case for the quantum dynamics of a geodesic flow on a compact Riemann surface, the quantum energy eigenvalues behave like a sequence of independent random variables provided that the underlying classical dynamics is completely integrable.

Species–area relationship

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Species%E2%80%93area_relationship 

 

The species-area relationship for a contiguous habitat

The species-area relationship or species-area curve describes the relationship between the area of a habitat, or of part of a habitat, and the number of species found within that area. Larger areas tend to contain larger numbers of species, and empirically, the relative numbers seem to follow systematic mathematical relationships. The species-area relationship is usually constructed for a single type of organism, such as all vascular plants or all species of a specific trophic level within a particular site. It is rarely if ever, constructed for all types of organisms if simply because of the prodigious data requirements. It is related but not identical to the species discovery curve.

Ecologists have proposed a wide range of factors determining the slope and elevation of the species-area relationship. These factors include the relative balance between immigration and extinction, rate and magnitude of disturbance on small vs. large areas, predator-prey dynamics, and clustering of individuals of the same species as a result of dispersal limitation or habitat heterogeneity. The species-area relationship has been reputed to follow from the 2nd law of thermodynamics. In contrast to these "mechanistic" explanations, others assert the need to test whether the pattern is simply the result of a random sampling process.

Authors have classified the species-area relationship according to the type of habitats being sampled and the census design used. Frank W. Preston, an early investigator of the theory of the species-area relationship, divided it into two types: samples (a census of a contiguous habitat that grows in the census area, also called "mainland" species-area relationships), and isolates (a census of discontiguous habitats, such as islands, also called "island" species-area relationships). Michael Rosenzweig also notes that species-area relationships for very large areas—those collecting different biogeographic provinces or continents—behave differently from species-area relationships from islands or smaller contiguous areas. It has been presumed that "island"-like species-area relationships have higher slopes (in log–log space) than "mainland" relationships, but a 2006 metaanalysis of almost 700 species-area relationships found the former had lower slopes than the latter.

Regardless of census design and habitat type, species-area relationships are often fitted with a simple function. Frank Preston advocated the power function based on his investigation of the lognormal species-abundance distribution. If ${\displaystyle S}$ is the number of species, ${\displaystyle A}$ is the habitat area, and ${\displaystyle z}$ is the slope of the species area relationship in log-log space, then the power function species-area relationship goes as:

${\displaystyle S=cA^{z}}$

Here ${\displaystyle c}$ is a constant which depends on the unit used for area measurement, and equals the number of species that would exist if the habitat area was confined to one square unit. The graph looks like a straight line on log–log axes, and can be linearized as:

${\displaystyle \log(S)=\log(cA^{z})=\log(c)+z\log(A)}$

In contrast, Henry Gleason championed the semilog model:

${\displaystyle S=\log(cA^{z})=\log(c)+z\log(A)}$

which looks like a straight line on semilog axes, where the area is logged and the number of species is arithmetic. In either case, the species-area relationship is almost always decelerating (has a negative second derivative) when plotted arithmetically.

species-area relationships are often graphed for islands (or habitats that are otherwise isolated from one another, such as woodlots in an agricultural landscape) of different sizes. Although larger islands tend to have more species, a smaller island may have more than a larger one. In contrast, species-area relationships for contiguous habitats will always rise as areas increases, provided that the sample plots are nested within one another.

The species-area relationship for mainland areas (contiguous habitats) will differ according to the census design used to construct it. A common method is to use quadrats of successively larger size so that the area enclosed by each one includes the area enclosed by the smaller one (i.e. areas are nested).

In the first part of the 20th century, plant ecologists often used the species-area curve to estimate the minimum size of a quadrat necessary to adequately characterize a community. This is done by plotting the curve (usually on arithmetic axes, not log-log or semilog axes), and estimating the area after which using larger quadrats results in the addition of only a few more species. This is called the minimal area. A quadrat that encloses the minimal area is called a relevé, and using species-area curves in this way is called the relevé method. It was largely developed by the Swiss ecologist Josias Braun-Blanquet.

Estimation of the minimal area from the curve is necessarily subjective, so some authors prefer to define the minimal area as the area enclosing at least 95 percent (or some other large proportion) of the total species found. The problem with this is that the species area curve does not usually approach an asymptote, so it is not obvious what should be taken as the total. the number of species always increases with area up to the point where the area of the entire world has been accumulated.

 

Storage effect

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Storage_effect 

The storage effect is a coexistence mechanism proposed in the ecological theory of species coexistence, which tries to explain how such a wide variety of similar species are able to coexist within the same ecological community or guild. The storage effect was originally proposed in the 1980s to explain coexistence in diverse communities of coral reef fish, however it has since been generalized to cover a variety of ecological communities. The theory proposes one way for multiple species to coexist: in a changing environment, no species can be the best under all conditions. Instead, each species must have a unique response to varying environmental conditions, and a way of buffering against the effects of bad years. The storage effect gets its name because each population "stores" the gains in good years or microhabitats (patches) to help it survive population losses in bad years or patches. One strength of this theory is that, unlike most coexistence mechanisms, the storage effect can be measured and quantified, with units of per-capita growth rate (offspring per adult per generation).

The storage effect can be caused by both temporal and spatial variation. The temporal storage effect (often referred to as simply "the storage effect") occurs when species benefit from changes in year-to-year environmental patterns, while the spatial storage effect occurs when species benefit from variation in microhabitats across a landscape.

The concept

For the storage effect to operate, it requires variation (i.e. fluctuations) in the environment and thus can be termed a "fluctuation-dependent mechanism". This variation can come from a large degree of factors, including resource availability, temperature, and predation levels. However, for the storage effect to function, this variation must change the birth, survival, or recruitment rate of species from year to year (or patch to patch).

For competing species within the same community to coexist, they have to meet one fundamental requirement: the impact of competition from a species on itself must exceed its competitive impact on other species. In other words, intraspecific competition must exceed interspecific competition. For example, jackrabbits living in the same area compete for food and nesting grounds. Such competition within the same species is called intraspecific competition, which limits the growth of the species itself. Members from different species can also compete. For instance, jackrabbits and cottontail rabbits also compete for food and nesting grounds. Competition between different species is called interspecific competition, which limits the growth of other species. Stable coexistence occurs when any one species in the community limits its own growth more strongly than the growth of others.

The storage effect mixes three essential ingredients to assemble a community of competing species that fulfill the requirement. They are 1) correlation between the quality of an environment and the amount of competition experienced by a population in that environment (i.e. covariance between environment and competition), 2) differences in species response to the same environment (i.e. species-specific environmental responses), and 3) the ability of a population to diminish the impact of competition under worsening environment (i.e. buffered population growth). Each ingredient is described in detail below with an explanation why the combination of the three leads to species coexistence.

Covariance between environment and competition

The growth of a population can be strongly influenced by the environment it experiences. An environment consists of not only physical elements such as resource abundance, temperature, and level of physical disturbance, but also biological elements such as the abundance of natural enemies and mutualists. Usually organisms reproduce more in a favorable environment (i.e. either during a good year, or within a good patch), build up their population densities, and lead themselves to a high level of competition due to this increasing crowding. Such a trend means that higher quality environments usually correlate with a higher strength of competition experienced by the organisms in those environments. In short, a better environment results in stronger competition. In statistics, such correlation means that there will be a non-zero covariance between the change of population density in response to the environment and that to the competition. That is why the first ingredient is called "covariance between environment and competition".

Species-specific environmental responses

Covariance between environment and competition suggests that organisms experience the strongest competition under their optimal environmental conditions because their populations grow most rapidly in those conditions. In nature, we often find that different species from the same community respond to the same conditions in distinctive manners. For example, plant species have different preferred levels of light and water availability, which affect their germination and physical growth rates. Such differences in their response to the environment, which is called "species-specific environmental response," means no two species from a community will have the same best environment in a given year or a given patch. As a result, when a species is under its optimal environmental conditions and thus experiencing the strongest intraspecific competition, other species from the same community only experience the strongest interspecific competition coming from that species, but not the strongest intraspecific competition coming from themselves.

Buffered population growth

A population can decline when environmental conditions worsen and when competition intensifies. If a species cannot limit the impact of competition in a hostile environment, its population will crash, and it will become locally extinct. Marvelously, in nature organisms are often able to slow down the rate of population decline in a hostile environment by alleviating the impact of competition. In so doing, they are able to set up a lower limit on the rate of their population decline. This phenomenon is called "buffered population growth", which occurs under a variety of situations. Under the temporal storage effect, it can be accomplished by the adults of a species having long life spans, which are relatively unaffected by environmental stressors. For example, an adult tree is unlikely to be killed by a few weeks of drought or a single night of freezing temperatures, whereas a seedling may not survive these conditions. Even if all seedlings are killed by bad environmental conditions, the long-lived adults are able to keep the overall population from crashing. Moreover, the adults usually adopt strategies such as dormancy or hibernation under a hostile environment, which make them less sensitive to competition, and allows them to buffer against the double blades of the hostile environment and competition from their rivals. For a different example, buffered population growth is attained by annual plants with a persistent seed bank. Thanks to these long-lived seeds, the entire population cannot be destroyed by a single bad year. Moreover, the seeds stay dormant under unfavorable environmental conditions, avoiding direct competition with rivals who are favored by the same environment, and thus diminish the impact of competition in bad years. There are some temporal situations in which buffered population growth is not expected to occur. Namely, when multiple generations do not overlap (such as Labord's chameleon) or when adults have a high mortality rate (such as many aquatic insects, or some populations of the Eastern Fence Lizard), buffered growth does not occur. Under the spatial storage effect, buffered population growth is generally automatic, because the effects of a detrimental microhabitat will only be experienced by individuals in that area, rather than the population as a whole.

Outcome

The combined effect of (1) covariance between environment and competition, and (2) species-specific response to the environment decouple the strongest intraspecific and interspecific competition experienced by a species. Intraspecific competition is strongest when a species is favored by the environment, whereas interspecific competition is strongest when its rivals are favored. After this decoupling, buffered population growth limits the impact of interspecific competition when a species is not favored by the environment. As a consequence, the impact of intraspecific competition on the species favored by a particular environment exceeds that of the interspecific competition on species less favored by that environment. We see that the fundamental requirement for species coexistence is fulfilled and thus storage effect is able to maintain stable coexistence in a community of competing species.

For species to coexist in a community, all species must be able to recover from low density. Not surprisingly, being a coexistence mechanism, the storage effect helps species when they become rare. It does so by making the abundant species’ effect on itself greater than its effect on the rare species. The difference between species’ response to environmental conditions means that a rare species’ optimal environment is not the same as its competitors. Under these conditions, the rare species will experience low levels of interspecific competition. Because the rare species itself is rare, it will experience little impact from intraspecific competition as well, even at its highest possible levels of intraspecific competition. Free from the impact of competition, the rare species is able to make gains in these good years or patches. Moreover, thanks to the buffered population growth, the rare species is able to survive the bad years or patches by "storing" the gains from the good years/patches. As a result, the population of any rare species is able to grow due to the storage effect.

One natural outcome from the covariance between environment and competition is that the species with very low densities will have more fluctuation in its recruitment rates than species with normal densities. This occurs because in good environments, species with high densities will often experience large amount of crowding by members of the same species, thus limiting the benefits of good years/patches, and making good years/patches more similar to bad years/patches. Low-density species are rarely able to cause crowding, thus allowing significantly increased fitness in good years/patches. Since the fluctuation in recruitment rate is an indicator of covariance between environment and competition, and since species-specific environmental response and buffered population growth can normally be assumed in nature, finding much stronger fluctuation in recruitment rates in rare and low-density species provides a strong indication that the storage effect is operating within a community.

Mathematical formulation

It is important to note that the storage effect is not a model for population growth (such as the Lotka–Volterra equation) itself, but is an effect that appears in non-additive models of population growth. Thus, the equations shown below will work for any arbitrary model of population growth, but will only be as accurate as the original model. The derivation below is taken from Chesson 1994. It is a derivation of the temporal storage effect, but is very similar to the spatial storage effect.

The fitness of an individual, as well as expected growth rate, can be measured in terms of the average number of offspring it will leave during its lifetime. This parameter, r(t), is a function of both environmental factors, e(t), and how much the organism must compete with other individuals (both of its own species, and different species), c(t). Thus,

${\displaystyle r(t)=g(e(t),c(t))\,}$

where g is an arbitrary function for growth rate. Throughout the article, subscripts are occasionally used to represent functions of a particular species (e.g. r _j(t) is the fitness of species j). It is assumed that there must be some values e* and c*, such that g(e*, c*) = 0, representing a zero-population growth equilibrium. These values need not be unique, but for every e*, there is a unique c*. For ease of calculation, standard parameters E(t) and C(t) are defined, such that

${\displaystyle E(t)=g(e(t),c^{*})\,}$

${\displaystyle C(t)=-g(e^{*},c(t))\,}$

Both E and C represent the effect of deviations in environmental response from equilibrium. E represents the effect that varying environmental conditions (e.g. rainfall patterns, temperature, food availability, etc.) have on fitness, in the absence of abnormal competitive effects. For the storage effect to occur, the environmental response for each species must be unique (i.e. E _j(t) ≠ E _i(t) when j ≠ i). C(t) represents how much average fitness is lowered as a result of competition. For example, if there is more rain during a given year, E(t) will likely increase. If more plants begin to bloom, and thus compete for that rain, then C(t) will increase as well. Because e* and c* are not unique, E(t) and C(t) are not unique, and thus one should choose them as conveniently as possible. Under most conditions (see Chesson 1994), r(t) can be approximated as

${\displaystyle r(t)=E(t)-C(t)+\gamma E(t)C(t)\,}$

where

${\displaystyle \gamma ={\frac {\partial ^{2}r}{\partial E\,\partial C}}}$

γ represents the nonadditivity of growth rates. If γ = 0 (known as additivity) it means that the impact of competition on fitness does not change with the environment. If γ > 0 (superadditivity), it means that the adverse effects of competition during a bad year are relatively worse than during a good year. In other words, a population suffers more from competition in bad years than in good years. If γ < 0 (subadditivity, or buffered population growth), it means that the harm done by competition during a bad year is relatively minor when compared to a good year. In other words, the population is able to diminish the impact of competition as the environment worsens. As stated above, for the storage effect to contribute to species coexistence, we must have buffered population growth (i.e. it must be the case that γ < 0).

The long-term average of the above equation is

${\displaystyle {\bar {r}}={\bar {E}}-{\bar {C}}+\gamma ({\bar {E}}{\bar {C}}+{\text{Cov}}(E(t),C(t)))\,}$

which, under environments with sufficient variation relative to mean effects, can be approximated as

${\displaystyle {\bar {r}}\approx {\bar {E}}-{\bar {C}}+\gamma {\text{Cov}}(E(t),C(t))}$

For any effect to act as a coexistence mechanism, it must boost the average fitness of an individual when they are at below-normal population density. Otherwise, a species at low density (known as an `invader') will continue to dwindle, and this negative feedback will cause its extinction. When a species is at equilibrium (known as a `resident'), its average long-term fitness must be 0. For a species to recover from low density, its average fitness must be greater than 0. For the remainder of the text, we refer to functions of the invader with the subscript i, and to the resident with the subscript r.

A long-term average growth rate of an invader is often written as

${\displaystyle {\bar {r_{i}}}=\Delta E-\Delta C+\Delta I\,}$

where,

${\displaystyle \Delta E={\bar {E_{i}}}-\sum _{i\neq r}q_{ir}{\bar {E_{r}}}\,}$

${\displaystyle \Delta C={\bar {C_{i}}}-\sum _{i\neq r}q_{ir}{\bar {C_{r}}}\,}$

and, ΔI, the storage effect,

${\displaystyle \Delta I=\gamma _{i}{\text{Cov}}(E_{i},C_{i})-\sum _{i\neq r}q_{ir}\gamma _{r}{\text{Cov}}(E_{r},C_{r})\,}$

where

${\displaystyle q_{ir}={\frac {\partial C_{i}}{\partial C_{r}}}}$

In this equation, q_ir tells us how much the competition experienced by r affects the competition experienced by i.

The biological meaning of the storage effect is expressed in the mathematical form of ΔI. The first term of the expression is covariance between environment and competition (Cov(E C)), scaled by a factor representing buffered population growth (γ). The difference between the first term and the second term represents the difference in species responses to the environment between the invader and the sum of the residents, scaled by the effect each resident has on the invader (q_ir).

Predation

Recent work has extended what is known about the storage effect to include apparent competition (i.e., competition mediated through a shared predator).

These models showed that generalist predators can undermine the benefits of the storage effect that from competition. This occurs because generalist predators depress population levels by eating individuals. When this happens, there are fewer individuals competing for resources. As a result, relatively abundant species are less constrained by competition for resource in favorable years (i.e., the covariance between environment and competition is weakened), and therefore the storage effect from competition is weakened. This conclusion follows the general trend that the introduction of a generalist predator will often weaken other competition-based coexistence mechanisms, and which result in competitive exclusion.

Additionally, certain types of predators can produce a storage effect from predation. This effect has been shown for frequency-dependent predators, who are more likely to attack prey that are abundant, and for generalist pathogens, who cause outbreaks when prey are abundant. When prey species are especially numerous and active, frequency-dependent predators become more active, and pathogens outbreaks become more severe (i.e., there was a positive covariance between the environment and predation, analogous to the covariance between the environment and competition). As a result, abundant species are limited during their best years by high predation – an effect that is analogous to the storage effect from competition.

Empirical studies

The first empirical study that tested the requirements of the storage effect was done by Pake and Venable, who looked at three desert annual plants. They experimentally manipulated density and water availability over a two-year period, and found that fitness and germination rates varied greatly from year to year, and over different environmental conditions. This shows that each species has a unique environmental response, and implied that likely there is a covariance between environment and competition. This, combined with the buffered population growth that is a product of a long-lived seed bank, showed that a temporal storage effect was probably an important factor in mediating coexistence. This study was also important, because it showed that variation in germination conditions could be a major factor promoting species coexistence.

The first attempt made at quantifying the temporal storage effect was by Cáceres in 1997. Using 30 years of water-column data from Oneida Lake, New York, she studied the effect the storage effect had on two species of plankton (Daphnia galeata mendotae and D. pulicaria). These species of plankton lay diapausing eggs which, much like the seeds of annual plants, lay dormant in the sediment for many years before hatching. Cáceres found that the size of reproductive bouts were fairly uncorrelated between the two species. She also found, in the absence of the storage effect, D. galeata mendotae would have gone extinct. She was unable to measure certain important parameters (such as the rate of egg predation), but found that her results were robust to a wide range of estimates.

The first test of the spatial storage effect was done by Sears and Chesson in the desert area east of Portal, Arizona. Using a common neighbor-removal experiment, they examined whether coexistence between two annual plants, Erodium cicutarium and Phacelia popeii, was due to the spatial storage effect or resource partitioning. The storage effect was quantified in terms of number of inflorescences (a proxy for fitness) instead of actual population growth rate. They found that E. cicutarium was able to outcompete P. popeii in many situations, and in the absence of the storage effect, would likely competitively exclude P. popeii. However, they found a very strong difference in the covariance between environment and competition, which showed that some of the most favorable areas for P. popeii (the rare species), were unfavorable to E. cicutarium (the common species). This suggests that P. popeii is able to avoid strong interspecific competition in some good patches, and that this may be enough to compensate for losses in areas favorable to E. cicutarium.

Colleen Kelly and colleagues have used congeneric species pairs to examine storage dynamics where species similarity is a natural outcome of relatedness and not dependent on researcher-based estimates. Initial studies were of 12 species of trees coexisting in a tropical deciduous forest at the Chamela Biological Station in Jalisco, Mexico. For each of the 12 species they examined age structure (calculated from size and species-specific growth rate), and found that recruitment of young trees varies from year to year. Grouping the species into 6 congeneric pairs, the locally rarer species of each pair unanimously had a more irregular age distributions than the more common species. This finding strongly suggests that between closely competing tree species, the rarer species experiences stronger recruitment fluctuation than the commoner species. Such difference in recruitment fluctuation, combined with evidence of greater competitive ability in the rarer species of each pair, indicates a difference in covariance between the environment and competition between rare and common species. Since species-specific environmental response and buffered population growth can be naturally assumed, their finding strongly suggests that the storage effect operates in this tropical deciduous forest so as to maintain the coexistence between different tree species. Further work with these species has shown that the storage dynamic is a pairwise, competitive relationship, between congeneric species pairs, and possibly extending as successively nested pairs within a genus.

Angert and colleagues demonstrated the temporal storage effect occurring in the desert annual plant community on Tumamoc Hill, Arizona. Previous studies had shown the annual plants in that community exhibited a trade-off between growth rate (a proxy for competitive ability) and water use efficiency (a proxy for drought tolerance). As a result, some plants grew better during wet years, while others grew better during dry years. This, combined with variation in germination rates, produced an overall community average storage effect of 0.103. In other words, the storage effect is expected to help the population of any species at low density to increase, on average, by 10.3% each generation, until it recovers from low density.