The storage effect is obtained by allowing the environment and a single regulating factor to vary. In this scenario, the emergent growth rate includes the main effect of the environment, the spatiotemporal average effect of the regulating factor, and effects of fluctuations in pairs of regulating factors. Therefore, in order to isolate the interaction between the environment and a regulating factor, we must subtract $\overline{\mathscr{E}_i}'$. We must also subtract the appropriate terms within  ${\Delta \rho_i}^{'(e)}$ and ${\Delta N_{i}}^{'(e)}$, which amounts to $\sum_{l = 1}^{L} \E{t}{\log( \E{x}{g_r \left( E_r^*, \phi_r \left( \left\{\boldsymbol{F^{*r}}^{-\{k,l\}}, F_k, F_l \right\} \right) \right)})}$.

§.§.§ Structured population models

In a structured population model, discrete age, stage, or size classes can be treated as different regulating factors $F_k$, and thus handled via the methods above (Section <ref>). When population structure is continuous in nature, the methods above can still be applied, but instead of holding some regulating factors at their equilibrium values while allowing others to vary, one would hold a range of a single (continuous) regulating factor constant, while allowing another range to vary. In other words, sums across discrete regulating factors are replaced by an integral across the states of a single continuous regulating factor.

§.§ The maximum number of species that can coexist via fitness density covariance

When there is no temporal variation, (<ref>) in Appendix <ref> reduces to

\begin{equation}
\begin{aligned}
\E{t}{\Cov{x}{v_j}{\lambda_j}} & \approx \frac{q}{1-q} \Var{x}{\alpha_j^{(1)} (E_j(x) - E_{j}^{*}) + \beta_j^{(1)} (C_r(x) - C_{r}^{*})} \\
& \approx \frac{q}{1-q} \left[ {\alpha_j^{(1)}}^2 \Var{x}{E_j} + 2 \alpha_j^{(1)} \beta_j^{(1)} \Cov{x}{E_j}{C_j} + {\beta_j^{(1)}}^2 \Var{x}{C_j} \right].
\end{aligned}
\end{equation}

The competition parameter $C$ can be expanded as a function of $L$ regulating factors (see Appendix <ref>):$C_j = \phi_j(\boldsymbol{F})$, where $\boldsymbol{F} = (F_1, F_2, ..., F_L)$. The environment parameter can be expressed a vector of $M$ discrete states: $E_j \in \boldsymbol{E}'$, where $\boldsymbol{E}' = (E'_{1}, E'_{2}, ..., E'_{M})$. Then, there are $M \times L$ "effective regulating factors" with the form $\Cov{x}{E'_{m}}{F_k}$; and $L(L-1)/2$ "effective regulating factors" with the form $\Cov{x}{F_k}{F_l}$. This result shows that fitness density covariance can potentially support a large number of species.

§ REFERENCES

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

journaltitleTheoretical Population Biology
titleHow should resources be counted?

journaltitleProceedings of the National Academy of Sciences of the United States of America
titleClimate variability has a stabilizing effect on the coexistence of prairie grasses
Climate change,Competition,Grassland,Plant community,Population dynamics

CRC press

edition2nd ed.
titleAn introduction to stochastic processes with applications to biology

journaltitleProceedings of the National Academy of Sciences
titleFunctional tradeoffs determine species coexistence via the storage effect

journaltitlePhysical Review E
titleExtinction of metastable stochastic populations

Wiley Online Library

titleChesson's coexistence theory: reply

Wiley Online Library

journaltitleEcological Monographs
titleChesson's coexistence theory

journaltitleJournal of Mathematical Biology
titlePersistence and extinction for stochastic ecological models with internal and external variables
Coexistence,Extinction,Stochastic difference equations,Stochastic persistence

University of Chicago Press Chicago, IL

journaltitleThe American Naturalist
titleA new approach to the generation time in matrix population models

Cambridge University Press

journaltitleAdvances in Applied Probability
titleAnalysis of a stochastic approximation algorithm for computing quasi-stationary distributions

Wiley Online Library

journaltitleJournal of the Royal Statistical Society: Series A (General)
titleSampling and Bayes' inference in scientific modelling and robustness

journaltitleMathematical Biosciences
titleItô versus Stratonovich calculus in random population growth
Itô calculus,Population growth,Random environments,Stochastic differential equations,Stratonovich calculus

titleTemporal variation , dormancy , and coexistence : A field test of

Oxford University Press

edition1st ed.
titleAn Illustrated Guide to Theoretical Ecology
olar   v=2.1   it=r   linkaccess=abs   issn=00129658   p=AONE   sw=w

Sunderland, Mass.

Sinauer Associates

edition2nd ed.
titleMatrix population models: construction, analysis, and interpretation
Population biology – Computer simulation; Matrices – Data processing

booktitlePatch dynamics
titleEcological interactions in patchy environments: from patch-occupancy models to cellular automata

University of Chicago Press

titleEcological niches: linking classical and contemporary approaches

journaltitleJournal of Mathematical Biology
titleInvasibility and stochastic boundedness in monotonic competition models
Coexistence,Competition,Competitive exclusion,Invasibility,Stochastic boundedness

journaltitleTheoretical Population Biology
titleMultispecies Competition in Variable Environments
esson/Peter/Reprints/1994_Multispecies competition in variable environments.pdf

journaltitleTheoretical Population Biology
titleGeneral theory of competitive coexistence in spatially-varying environments

journaltitleJournal of Ecology
titleUpdates on mechanisms of maintenance of species diversity
Lotka–Volterra,average fitness difference,natural enemy partitioning,niche overlap,permanence,scaling factors,species coexistence,stabilizing mechanism

titleChesson's coexistence theory: Comment

numberHanski 1999
titleChapter 6: Scale Transition Theory for Understanding Mechanisms in Metacommunities

The University of Chicago Press

journaltitleThe American Naturalist
titleThe roles of harsh and fluctuating conditions in the dynamics of ecological communities

isbn1476-4687 (Electronic)\r0028-0836 (Linking)
titleThe interaction between predation and competition

journaltitleTheoretical Population Biology
titleThe storage effect due to frequency-dependent predation in multispecies plant communities

journaltitleTheoretical Population Biology
titleCoexistence of competitors in spatially and temporally varying environments: A look at the combined effects of different sorts of variability

University of Chicago Press

journaltitleThe American Naturalist
titleEnvironmental variability promotes coexistence in lottery competitive systems

journaltitleJournal of Mathematical Neuroscience
titlePath integral methods for stochastic differential equations

Wiley Online Library

journaltitleEcological Monographs
titleLarge niche differences emerge at the recruitment stage to stabilize grassland coexistence

titleThe Assembly of Species Communities: Chance or Competition?

journaltitleThe Journal of Animal Ecology
titleSpecies Interactions, Local and Regional Processes, and Limits to the Richness of Ecological Communities: A Theoretical Perspective

journaltitleTrends in Ecology and Evolution
titleSkeletons, noise and population growth: The end of an old debate?

John Wiley & Sons

titleStatistics for spatial data

journaltitleCold Spring Harbor symposia on quantitative biology
titleMaintenance of genetic heterogeneity.

Wiley Online Library

titleStable coexistence in a fluctuating environment: an experimental demonstration

titleExamination of the "null" model of connor and simberloff for species co-occurrences on Islands

Wiley Online Library

journaltitleJournal of the Royal Statistical Society: Series B (Methodological)
titleAssessment and propagation of model uncertainty

Princeton University Press

titleThe aim and structure of physical theory.
Physics – Philosophy

journaltitleJournal of Mathematical Biology
titleConvergence to stationary distributions in two-species stochastic competition models
Coexistence,Competition,Continuous components,Invasibility,Markov chains,Stationary distributions,Stochastic boundedness

University of Chicago Press Chicago, IL

journaltitleThe American Naturalist
titleGeneration time in structured populations

Wiley Online Library

journaltitleEcology letters
titleHow to quantify the temporal storage effect using simulations instead of math

journaltitleEcology Letters
titleAn expanded modern coexistence theory for empirical applications
Coexistence,competition,environmental variability,model,theory

journaltitleEcology Letters
titleTechnical Comment on Pande et al. (2020): Why invasion analysis is important for understanding coexistence
Coexistence,environmental variability,extinction,invasion analysis,persistence

The Institute of Mathematical Statistics
the Bernoulli Society

journaltitleElectronic Journal of Probability
titleQuasi stationary distributions and Fleming-Viot processes in countable spaces

journaltitleInternational Institute for Applied Systems Analysis Interim Report IR-01-043, September 2001
titleInvasion fitness and adaptive dynamics in spatial population models

journaltitleEcology Letters
titleTesting for local species saturation with nonindependent regional species pools
Community assembly,Invasibility,Local versus regional species richness,Microcosms,Monte Carlo test,Protists,Species interactions,Species saturation

journaltitleSIAM Journal on Mathematical Analysis
titleRobust permanence for ecological differential equations, minimax, and discretizations

titleHandbook of stochastic methods

journaltitlearXiv preprint arXiv:2011.01808
titleBayesian workflow

journaltitleStochastic processes and their applications
titleIndependent sampling of a stochastic process

journaltitleProceedings of the National Academy of Sciences of the United States of America
titleApplying modern coexistence theory to priority effects
Competition,Fitness difference,Invasion criterion,Niche difference,Stabilizing difference

Elsevier Ltd

journaltitleTrends in Ecology and Evolution
titleThe Invasion Criterion: A Common Currency for Ecological Research
fitness difference,invasion growth rates,modern coexistence theory,mutual invasibility,stabilizing niche difference

journaltitlearXiv preprint arXiv:1206.6712
titleSimulation of quasi-stationary distributions on countable spaces

journaltitleJournal of Mathematical Biology
titleOn the impossibility of coexistence of infinitely many strategies
Ecological niche,Evolution of seed-size,Limiting similarity,Lotka-Volterra competition model,Physiologically structured populations,Regulated coexistence,Structural stability

journaltitleTheoretical Population Biology
titleCan there be more predators than prey?

journaltitleEcological Monographs
titleRegional and local species richness in an insular environment: Serpentine plants in California
Disturbance,Geologic age,Local richness,Plant diversity,Productivity-diversity,Regional richness,Serpentine,Spatial habitat structure,Species-area,Species-energy,Structural equation modeling

journaltitleJournal of mathematical biology
titleThe competitive exclusion principle in stochastic environments

journaltitleJournal of Northwest Atlantic Fishery Science
titleA Statistical Description of Recruitment in Eighteen Selected Fish Stocks
Atlantic cod & Atlantic herring & Atlantic mackere

journaltitleJournal of theoretical biology
titlePair and triplet approximation of a spatial lattice population model with multiscale dispersal using Markov chains for estimating spatial autocorrelation

journaltitleMonatshefte für Mathematik
titleA general cooperation theorem for hypercycles

Princeton University Press

titleThe unified neutral theory of biodiversity and biogeography

titleOf Miracles (section X of An Enquiry Concerning Human Understanding)

Evolutionary Ecology, Ltd.

journaltitleEvolutionary Ecology Research
titleThe role of climate in the dynamics of annual plants in a Chihuahuan Desert ecosystem

journaltitleUnpublished Manuscript
titleMethods for calculating coexistence mechanisms: Beyond scaling factors

journaltitleUnpublished Manuscript
titleThe storage effect is not about bet-hedging or population stage-structure

journaltitleUnpublished Manuscript
titleTowards a heuristic understanding of the storage effect

journaltitlePhysical Review Letters
titleHow colored environmental noise affects population extinction

Academic Press

edition1st ed.
titleA second course in stochastic processes

journaltitleJournal of Applied Probability
titleOn the extinction of the SIS stochastic logistic epidemic

journaltitleTheoretical Population Biology
titleInteracting coexistence mechanisms in annual plant communities: Frequency-dependent predation and the storage effect

Oxford University Press

titleStochastic population dynamics in ecology and conservation

Wiley Online Library

journaltitleEcology Letters
titleOpposing effects of floral visitors and soil conditions on the determinants of competitive outcomes maintain species diversity in heterogeneous landscapes

Wiley Online Library

titleThe adaptive dynamics of altruism in spatially heterogeneous populations

University of Chicago Press

journaltitleThe American Naturalist
titleCommunity equilibria and stability, and an extension of the competitive exclusion principle

Blackwell Publishing Ltd Oxford, UK

journaltitleAnnals of the New York Academy of Sciences
titleDiscussion paper: the qualitative analysis of partially specified systems

University of Chicago Press

journaltitleThe American Naturalist
titleCoexistence in a variable environment

titleCounterfactual Dependence and Time's Arrow

titleCounterfactuals and comparative possibility

journaltitleProceedings of the National Academy of Sciences of the United States of America
titleOn population growth in a randomly varying environment.

University of Chicago Press Chicago, IL

journaltitleThe American Naturalist
titleThe effects of dynamical rates on species coexistence in a variable environment: the paradox of the plankton revisited

titleExplaining the excess of rare species in natural species abundance distributions
dynamics,england,lower severn estuary,patterns,population stability

Wiley Online Library

journaltitleEcology Letters
titleDoes Mother Nature really prefer rare species or are log-left-skewed SADs a sampling artefact?

journaltitleTrends in Ecology & Evolution
titleHow should we define ‘fitness’ for general ecological scenarios?

titleA System of Logic, Ratiocinative and Inductive: 1

journaltitleTheoretical Ecology
titleEvolutionary stability of coexistence due to the storage effect in a two-season model

Oxford University Press

edition2nd ed.
titleCommunity ecology
Biotic communities; Ecology; Biotic communities

journaltitleJournal of Theoretical Biology
titleExtinction and quasi-stationarity in the Verhulst logistic model

The Royal Society London

journaltitleProceedings of the Royal Society of London. Series B: Biological Sciences
titleLifting the veil on abundance patterns

edition1st ed.
titleModelling fluctuating populations

journaltitleAdvances in Applied Probability
titleOn the distribution of the time to extinction in the stochastic logistic population model

Wiley Online Library

journaltitleEcology letters
titleMean growth rate when rare is not a reliable metric for persistence of species

journaltitlePhysical Review E
titleTaming the diffusion approximation through a controlling-factor WKB method

titleThe book of why: the new science of cause and effect
l=en   lr=   id=9H0dDQAAQBAJ   oi=fnd   pg=PT7   dq=pearl+book+of+why   ots=ZgBDhNtui_   sig=5O-yqZMGOaJf8poQ1S_D9qTk84U

New York

abstractPopulation dynamics; Spatial patterns in one-species populations; Spatial relations of two or more species; Many-species populations.
titleAn introduction to mathematical ecology
Ecology – Mathematical models; ecology; mathematical models; Écologie – Modèles mathématiques; Ecology; Räumliche Verteilung; Ökologie; Biometry – Statistical methods; Mathematics

Wiley Online Library

titleDiversity: between neutrality and structure

journaltitleEcological Modelling
titleRecruitment variability and stochastic population growth of the soft-shell clam, Mya arenaria
Bivalves,Elasticity,Larval ecology,Lognormal distribution,Matrix population models,Quasi-extinction,Recruitment,Sensitivity,Stochastic models,Stochastic population growth rate

journaltitleJournal of Mathematical Biology
titlePersistence in fluctuating environments for interacting structured populations

New York

titleTheory of population genetics and evolutionary ecology : an introduction
Population genetics; Population biology; Evolution

University of Chicago Press

journaltitleThe American Naturalist
titleMaintenance of High Diversity in Coral Reef Fish Communities

journaltitleJournal of Differential Equations
titleCriteria for Cr robust permanence

The University of Chicago Press Chicago, IL

journaltitleThe American Naturalist
titlePositively and negatively autocorrelated environmental fluctuations have opposing effects on species coexistence

journaltitleJournal of Mathematical Biology
titlePersistence in fluctuating environments

titleNew Methods for Quantifying the Spatial Storage Effect: an Illustration With Desert Annuals
coexistence mechanisms,competition along gradients,environmental variation,facilitation,neighborhood competition,spatial heterogeneity,spatial storage effect

journaltitleTheoretical Ecology
titleWhen does environmental variation most influence species coexistence?
Coexistence,Colored noise,Disturbance,Environmental variation,Life history traits

The University of Chicago Press

journaltitleThe American Naturalist
titleExamining the relative importance of spatial and nonspatial coexistence mechanisms

Wiley Online Library

journaltitleEcology letters
titleLocal dispersal can facilitate coexistence in the presence of permanent spatial heterogeneity

Elsevier Ltd

journaltitleTrends in Ecology and Evolution
titleTowards a Probabilistic Understanding About the Context-Dependency of Species Interactions
antagonism,competition,context-dependency,environmental variability,interaction switches,mutualism,probability diagrams,structural stability

booktitleJ. Biosci. |
titleDaniel Bernoulli (1738): evolution and economics under risk

journaltitleTheoretical Population Biology
titleDistance-responsive predation is not necessary for the Janzen–Connell hypothesis

journaltitleTheoretical Population Biology
titleHow optimally foraging predators promote prey coexistence in a variable environment

journaltitleJournal of theoretical biology
titleHow spatial structure and neighbor uncertainty promote mutualists and weaken black queen effects

Princeton University Press

titleResource competition and community structure

Wiley Online Library

titleRequirements for the spatial storage effect are weakly evident for common species in natural annual plant assemblages

journaltitleTheoretical Population Biology
titleA reexamination of stability in randomly varying versus deterministic environments with comments on the stochastic theory of limiting similarity

Nature Publishing Group

titleTemporal coexistence mechanisms contribute to the latitudinal gradient in forest diversity

Wiley Online Library

titleCoexistence in tropical forests through asynchronous variation in annual seed production

Oxford University Press

titleExplanation in causal inference: methods for mediation and interaction

journaltitleMathematical Biosciences
titleOn the asymptotic behavior of the stochastic and deterministic models of an epidemic

Elsevier Inc.

journaltitleTheoretical Population Biology
titleThe relative importance of relative nonlinearity and the storage effect in the lottery model
variable environment

University of Minnesota

titleMathematical analysis of a population model with several predators on a single prey

§ RECYCLING

he determination of mechanisms that permits coexistence of species, both theoretically and em-5pirically, has been a central question in ecology. Since these explanations for coexistence (being6reductive by definition) are certainlypartial explanationsmore than one is operating in any situ-7ation and the real challenge is to determine the relative importance of any particular explanation8under particular circumstances. Here, we present a methodology for quantifying the relative im-9portance of different explanations for coexistence, based on an extension ofModern Coexistence10Theory. Current versions of Modern Coexistence Theory only allow for the analysis of communi-11ties that are affected by spatialortemporal environmental variation, but not both. We show how12to analyze communities with spatiotemporal fluctuations, how to parse the importance of spatial13variation and temporal variation in promoting coexistence, and how to measure everything with14either mathematical expressions or simulation experiments. This approach shows how to determine15the relative importance of spatial versus temporal explanations for coexistence rather than having16simply to attribute coexistence to one or the other

§.§ Relative nonlinearity in models with one resource, two consumers, and an opportunist-gleaner trade-off (NOTE: incomplete)

To better understand the processes by which relative nonlinearity can promote coexistence, we will start simple. Consider a community with two consumers and one resource (fig??). Both consumer species' share a density-independent death rate (solid lines), but have different birth rates (solid lines) which are saturating functions of resource concentration. In the absence of fluctuations, species 1 will exclude species 2 - Resource concentrations will decrease as both consumers' populations grow, until the resource concentration dips below the minimum resource requirement of species 2. This is nothing more than a verbal explanation of Tilman's (1982) $R^*$ rule. What happens when we allow resource levels to fluctuate? Species 1 (the winner in the no-fluctuations-case) will be harmed more than species 2 by resource variation, due to jensen's inequality and the relatively large concavity of its birth rate function. Is this enough to allow species 2 to coexist?

To find out, we will use a heuristic that originates from Tilman's graphical analysis of resource consumer models: For coexistence to occur, species must consume proportionately more of that which most limits their own growth. In our example (fig <ref>) Species 1 is hurt more by resource variation, so it is most limited by mean resource levels. Thus, for species 2 to coexist (i.e., invade), species 1 must increase variation in resource levels when it is abundant.

Species 1 can increase variation (and thus promote coexistence) by inducing cyclical resource-consumer dynamics (Armstrong and Mcgehee 1976 1980); this outcome is contingent upon model parameters, but it is not a quirk of meticulously selected parameters - species 1 has a faster resource consumption rate than species 2, and is therefore inherently more destabilizing (cite). That being said, cyclical and chaotic population dynamics appear to be rare in the real world (cite).

If resource dynamics are subject to environmental stochasticity or demographic stochasticity, then resource variation should scale monotonically with mean resource levels (lande and engen; gardiner). Here, since species 1 has a lower $R^*$ than species 2, species 1 tends to decrease resource variation, thus undermining coexistence. There is some reason to believe that this outcome is common in the real-world - Taylor's law (cite) suggests that the aforementioned relationship between the mean and variance is common, at least for biotic resources.

The previous example of small environmental noise explicates Chesson's statement that "the limited ability for relative nonlinearity to promote coexistence when acting alone means that it is best viewed as modifying other mechanisms ... by decreasing the degree of dominance of a superior competitor with a relatively concave growth rate ... " (<cit.>) - When resource variation originates from environmental or demographic stochasticity, the dominant species does not tend to increase resource variation, so relative nonlinearity by itself does not promote coexistence. Relative nonlinearity is nevertheless important because it can change competitive outcomes (e.g., if resource variation is severe enough, then species 2 will exclude species 1).

Resource variation may also originate from variation in resource inputs. In this class of scenarios, the source of resource variation is exogenous, but (unlike the case of environmental or demographic stochasticity) there is no hard-wired relationship between mean resource levels and resource variation. When resource inputs fluctuate through time, we find that the dominant competitor (species 1) tends to increase resource variation, thus promoting coexistence. The reason for the increase is related to the different slopes the two consumer's birth rate curves around equilibrium resource levels (i.e., $R^{*}_1$ and $R^{*}_2$; see fig <ref>). If the slope is steep, a resource surplus causes a dramatic increase in consumer birth rates; The subsequently large consumer population then reduces resource levels. Conversely, when resources are scare, birth rates will dramatically decline, leading to a small consumer population which is ineffective at depleting resources further. Therefore, resource levels are regulated via a negative feedback loop with consumers (fig <ref>), and the strength of this negative feedback is proportional to the slope of the birth rate function. When the dominant competitor is a 'gleaner' species (which is necessary if there is to be any trade-off between the two consumers), it necessarily has a shallower slope.

Rapid fluctuations in resource inputs will not lead to large variation in resource levels. An analogy may illustrate this point: During a warm summer, rapidly turning an air conditioner on and off will not cause large variations in the temperature of a room, compared to leaving the air conditioner on for hours, and then turning it off for hours.  Fluctuations on long time scales will similarly not lead to large resource variation - when there is plenty of time for the consumer species to grow or decay to a compensatory population size, the aforementioned negative feedback between resources and consumers is maximized. Resource variation, and thus the propensity for coexistence, is maximized when fluctuations in resource inputs occur on intermediate timescales.

Can spatial resource variation also promote coexistence via relative nonlinearity? Population cycles are necessarily a temporal phenomenon, so there is no purely spatial analogue of the endogenously generated resource-consumer cycles (armstrong 1977, 1980). If there is spatial variation in the per capita (or 'per-concentration') parameters of resource dynamics, then we can apply the same argument that we used in the case of temporal environmental (or demographic) stochasticity: resource variation is proportional to mean resource levels, so the dominant competitor (species 1) tends to decrease resource variation, thus undermining coexistence.

What happens when there is spatial variation in the resource inputs? When there is no dispersal, all patches converge to the same equilibrium resource level - relative nonlinearity cannot possibly be in effect. However, when a small proportion of the consumer species are allowed to disperse, we do find spatial variation in resource levels. Whether or not this generates coexistence is a matter of how model parameters are selected. Unlike the purely temporal case, there is nothing inherent about the opportunist-gleaner trade-off that causes the dominant competitor to induce more resource variation than the subordinate competitor.

((<ref>) in the main text)
want to do the Chesson and huntly thing. But you can't re-scale the invader, so you compensate by dividing everything by the invader's sensitivity to competition.

when L > S, no solution, whereas with Chesson, L > S-1 means no solution. So you can have more species and still have an exact answer.

when di equals zero, the whole thing falls apart. Example on page

In a resource-consumer model with an opportunist-gleaner trade-off, when does relative nonlinearity promote coexistence?

Source of resource variation     Promotes or hinders coexistence?

temporal variation

endogenous population cycles     Promotes

environmental/demographic stochasticity     Hinders

fluctuating resource supply rate     Promotes

spatial variation

endogenous population cycles     N/A

environmental/demographic stochasticity     Hinders

fluctuating resource supply rate     Depends on parameters

One can easily simulate a model forward in time and then check for coexistence (operationalized as co-occurrence for a sufficiently long time); But this only tells us whether species are coexisting, not how they are coexisting, or failing to coexist. Instead, we would like a coexistence criterion which, through its simplicity, transparently relates coexistence to patterns in demographic parameters or patterns in the structure of population-dynamical equations. MCT successfully isolates and quantifies these patterns via a partition of the invasion growth rate into coexistence mechanisms. However, MCT struggles to clearly and universally relate invasion growth rates to coexistence.

Stochastic Persistence Theory (<cit.>; <cit.>; <cit.>; <cit.>; <cit.>) shows how invasion growth rates can still be used to formulate a sufficient condition for coexistence in complex communities, which also becomes a necessary condition in the face of large, infrequent perturbations (<cit.>). This criterion is satisfied by picking weights $p_j$ (i.e., positive constants that sum to one), one for each species $j$, such that the weighted sum of the $r_j$, the realized per capita growth rates, is positive for each and every sub-community $\mu$ in which one or more species is missing. In mathematical form, the criterion is

Though spatiotemporal MCT has produced some theoretical insights, its primarily value is as a methodology for inferring the mechanisms of coexistence in real communities. Most attempts to quantify coexistence mechanisms in real communities have utilized models that do not represent spatial variation (see <cit.>, adler, usinowitz for counterexamples), despite a consensus that spatial heterogeneity can easily promote coexistence (cite). This can likely be attributed to a mixture of several challenges: a general perception that temporal mechanisms are more interesting than spatial mechanisms, the arduousness of collecting data over space and time, the debilitating number of plausible model-based representations of ecological communities, perverse incentives for publishing quickly (<cit.>), and the unavailability of a framework capable of analyzing models with spatiotemporal fluctuations. In this paper, we address the last barrier.

The irrelevancy of the competitive exclusion principle in spatiotemporal models also strikes against the usage of  scaling factors (<cit.>; <cit.>), a seminal part of MCT that were designed to cancel the linear density-dependent effects $\Delta \rho_i$. There are good reasons to abandon the scaling factors (<cit.>), but even if

, so there are effectively $K \times L$ resources. What naturally follows is that different groups of species coexist in each patch (<cit.>  <cit.> Fig??). It has also been observed that fitness density covariance behaves much like the spatial storage effect, in that it naturally engendered by environmental heterogeneity and local retention (i.e., not all individual disperse after every time-step). Through analogy

The small-noise assumptions imply that $\lambda_j(x,t) = 1 + \log(\lambda_j(x,t)) + \mathcal{O}(\sigma^2)$, which implies that $\sigma \lambda_1 = $

statements: $\lambda_j(x,t) = 1 + \log(\lambda_j(x,t)) + \mathcal{O}(\sigma^2)$ (see (<ref>) specifically) and $\widetilde{\lambda}_j(x,t) = 1 + \mathcal{O}(\sigma^2)$.

\begin{equation}
\nu_j(x,t+1) = p_j \nu_j(x,t)\left( 1 + \log(\lambda_j(x,t)) \right) + 1 - p_j + \mathcal{O}(\sigma^2)
\end{equation}

Noting that $\log(\lambda_j(x,t)) = \mathcal{O}(\sigma)$

\begin{equation}
\nu_{j,1}(x,t+1) = p_j \nu_{j,1}(x,t) + p  \lambda_{j,1}(x,t)
\end{equation}

We now substitute $\alpha_j^{(1)} (E_j(x,t) - E_{j}^{*}) + \beta_j^{(1)} (C_j(x,t) - C_{j}^{*})$ for $\lambda_{j,1}(x,t)$, and approximate the invader's competition parameter fluctuation in terms of the resident's environmental response $C_i(x,t) - C_{i}^{*} \approx \frac{\partial C_i}{\partial E_r}( E_r(x,t) - E_{r}^{*}) =  E_r(x,t) - E_{r}^{*}$.

\begin{equation}
\begin{aligned}
\Cov{x}{v_i(t)}{\lambda_i(t)}  \approx  \sum \limits_{s=1}^{\infty} p^s \mathrm{Cov}_x \bigl( & \alpha_i^{(1)} (E_i(x,t-i) - E_{i}^{*}) + \beta_i^{(1)} (E_r(x,t-s) - E_{r}^{*}) ,  \\
& \alpha_i^{(1)} (E_i(x,t) - E_{i}^{*}) + \beta_i^{(1)} (E_r(x,t) - E_{r}^{*}) \bigl)
\end{aligned}
\end{equation}

We must add a third assumption. The relative density of species $j$, defined as $\nu_j(x,t)  = \frac{n_j(x,t)}{\E{x}{n_j(x,t)}}$, must not be too large:

\begin{equation} \label{ass:relative_density}
\nu_j - 1 = \mathcal{O}(\sigma).
\end{equation}

This assumption helps to bound fluctuations in $C_j$, thus ensuring the commensurability of terms within the Taylor series expansion of the local finite rate of increase ((<ref>)). If there is widespread dispersal, where in each time-step all individuals disperse evenly across patches, then $\nu_j(x,t) = 1$ for every $x$ and $t$, and (<ref>) is automatically true. It is also true via perturbation theory whenever there are small deviations from the widespread dispersal case. In general, however, the truth value of (<ref>) will depend on the details of dispersal and local population growth.

For readers without a background in applied math, the bounds on the size of fluctuations in competition and relative density ((<ref>)) may be confusing. By contrast, it makes sense that the magnitude of environmental fluctuations are controlled by the parameter $\sigma$, because the variance of a random variable is conventionally denoted by $\sigma^2$. But what does it mean for the magnitude of fluctuations in relative density to be controlled by the very same parameter? The key is that $\sigma$ is not a parameter in the usual sense (i.e., a constant in equations of population dynamics), but rather a technical place-holder for the word "small". Therefore, (<ref>) is not saying that fluctuations in the environment causally bound fluctuations in relative-density, though they certainly do in some models. Instead, (<ref>) is an assertion that leads to a desirable outcome: no terms in our final partition of the invader growth rate will be negligibly small.

The determination of mechanisms that permit coexistence, both theoretically and empirically, has been a central question in ecology.

Our notation in (<ref>) implies that $\lambda_j$ is a realization of the random variable $\lambda_j$. In reality, $\lambda_j(x,t)$ itself is usually a random variable with a cumulative distribution function that depends on space and time. Therefore, it would be more correct to use the notation $\lambda_j(x,t, \omega)$, where $\omega$ may be thought of as the seed number of a random number generator (<cit.>). We prefer to use the simpler notation $\lambda_j(x,t)$, all the while keeping in mind that we will be computing spatiotemporal averages (and other statistics) using a single realization $\omega$, but that this should not affect the invasion growth rate (since the time-average is the same as the ensemble average by assumption; see Section <ref>).

§.§ The maximum number of species that can coexist via fitness density covariance

Here, we consider an arbitrary model with scalar populations inhabiting discrete patches (indexed by $x$), with discrete-time dynamics (indexed by $t$), and with no temporal variation. In each time-step, there are two events. First is a bout of local population growth, determined by the local finite rate of increase, $\lambda_j(x,t)$. Second is a dispersal event, where in each patch a proportion of individuals, $p_j$, disperse and are distributed uniformly over all patches (including the patch of origin). It follows that a proportion of individuals, $q_j = (1-p_j)$, are retained locally; We call $q$ the retention proportion. To simplify the expressions for coexistence mechanisms, we assume that there is no temporal variation, and that population densities $N_j$ and relative densities $\nu_j$ settle to an equilibrium in each patch.

The time-evolution of population density $N_j$ at patch $x$ is given by

\begin{equation}
N_j(x,t+1) =  q_j N_j(x,t) \lambda_j(x,t) + \frac{1 - q_j}{K} \sum \limits_{s=1}^{K} N_j(s,t) \lambda_j(s,t).
\end{equation}

Often in MCT, the competition parameter is a function of both species densities and the environmental parameter; However, in such models, there is an implicit time-lag between the effects of environment and competition on population dynamics, such that the environment has enough time to affect competition within a time-step. For instance, in the lottery model (Section <ref>), per capita fecundity (the environmental parameter) affects the per-larva recruitment probability (the competition parameter), but recruitment occurs weeks or months after reproduction, a fact which is hidden by the simple structure of the lottery model equations. Here, we only which can be approximated by expanding the competition parameter. In this appendix, we only consider models in which the competition parameter is a function of species densities: $C_j = h_j(N_r)$. This simplifies things because it prevents the environment from affecting competition on two separate time-scales - within a time-step (as in the lottery model) and between time-steps (via inter-generational population growth).

To obtain the spatial storage effect, we must obtain the quantity $\Cov{x}{E_j}{C_j}$, which in the two-species/single-resident case can be approximated as $\Cov{x}{E_j}{\theta_{jr} N_r}$, where $r$ is the index of the resident species, and $\theta_{jr}$ helps convert the resident's density to the competition parameter, and is defined as $\theta_{jr} = \frac{d h_j(N_r^*)}{d N_r}$ (recall that $C_j = h_j(N_r)$).

In appendix <ref>, we derived an approximation of fitness density covariance, (<ref>),
which - assuming that the competition parameter is a function of only the resident's density, and that there is no temporal variation - can be approximated as

\begin{equation}
\begin{aligned}
\E{t}{\Cov{x}{v_j}{\lambda_j}} \approx \frac{q_j}{1-q_j} \left[ \left(\alpha_j^{(1)}\right)^2 \Var{x}{E_j} + 2 \alpha_j^{(1)} \beta_j^{(1)} \Cov{x}{E_j}{\theta_{jr} N_r} + \left(\beta_j^{(1)}\right)^2 \Var{x}{\theta_{jr} N_r} \right].
\end{aligned}
\end{equation}

Now, it is clear that simplifying the expressions for the coexistence mechanisms will require us to find the residents' density, $N_r$. Specifically, re-expressing $N_r$ in terms of the environmental parameter, $E_r$, will allow us to express the fitness-density covariance in terms of spatial variation and between-species correlation in the environment.

To find $N_r$, we take a perturbative approach, where both $N_j$ and $\lambda_j$ are expanded in powers of the small parameter $\sigma$: $N_{j}(x,t) = N_{j,0}(x,t) + \sigma N_{j,1}(x,t) + ...$; and $\lambda_{j}(x,t) = \lambda_{j,0}(x,t) + \sigma \lambda_{j,1}(x,t) + ...$. With this, we re-write the population dynamics ((<ref>)) as

\begin{equation}
\begin{aligned}
N_{j,0}(x,t+1) + \sigma N_{j,1}(x,t+1) + \ldots = &  q_j (N_{j,0}(x,t) + \sigma N_{j,1}(x,t) + \ldots )(\lambda_{j,0}(x,t) + \sigma \lambda_{j,1}(x,t) + \ldots )  \\
& + \frac{1 - q_j}{K} \sum \limits_{s=1}^{K}  (N_{j,0}(s,t) + \sigma N_{j,1}(s,t) + \ldots )(\lambda_{j,0}(s,t) + \sigma \lambda_{j,1}(s,t) + \ldots ).
\end{aligned}
\end{equation}

The zeroth-order dynamics are the same in every patch (because environmental fluctuations are $\mathcal{O}(\sigma)$). Assuming that there are no complex dynamics, the resident density reaches a stable equilibrium, denoted $N_{r,0}^*$, that is the same in each patch and which can be obtained by solving

\begin{equation}
N_r = N_r \, g_{r}(E_r^*, h_r(N_r))
\end{equation}

for $N_r$. For example, if the population model is $\lambda_r(x,t) = s + E(x)/(1+c N_r(x,t))$, then $N_{r,0}^* = \frac{E_r^*}{c(1-s)} - \frac{1}{c}$.

Noting that $\lambda_{j,0}^* = g_{j}(E_j^*, h_j(N_{r,0}^*)) = 1$, the first-order dynamics can be written as

\begin{equation}
\begin{aligned}
\sigma N_{j,1}(x,t+1) = & \sigma \left[ q_j \left(N_{j,0}^* \lambda_{j,1}(x,t) + N_{j,1}(x,t) \right) + (1-q_j)\left(N_{j,0}^* \E{x}{\lambda_{j,1}(t)} + \E{x}{N_{j,1}(t)} \right) \right].
\end{aligned}
\end{equation}

The small-noise assumptions can be used to match terms from the perturbative expansion above and the Taylor series expansion of $\lambda_j$ (see (<ref>)), resulting in $\sigma \lambda_{j,1}(x,t) = \alpha_j^{(1)} (E_j(x,t) - E_{j}^{*}) + \beta_j^{(1)} (C_j(x,t) - C_{j}^{*})$. The small-noise assumptions also mean that $\E{x}{\lambda_{j,1}(t)}$ and $\sigma \E{x}{N_{j,1}}$ are $\mathcal{O}(\sigma^2)$, which simplifies (<ref>) to

\begin{equation}
\begin{aligned}
\sigma N_{j,1}(x,t+1) = \sigma \left[ q_j \left(N_{j,0}^* \lambda_{j,1}(x,t) + N_{j,1}(x,t) \right) \right].
\end{aligned}
\end{equation}

Using the relationship $\sigma \lambda_{j,1}(x,t) = \alpha_j^{(1)} (E_j(x,t) - E_{j}^{*}) + \beta_j^{(1)} (C_j(x,t) - C_{j}^{*}) \approx \alpha_j^{(1)} (E_j(x,t) - E_{j}^{*}) + \beta_j^{(1)} \theta_{jr} N_{r,1})$ to substitute for $\lambda_{j,1}(x,t)$, we can solve for the equilibrium density of the resident

\begin{equation}
N_{r}^* \approx N_{r,0}^* + \sigma N_{r,1}^*, \; \text{where}
\end{equation}

\begin{equation}
\sigma N_{r,1}^* = \frac{q_r N_{r,0}^* \alpha_r^{(1)}(E_r - E_r^*)}{1-q_r(\theta_{r,r} N_{r,0}^* \beta_r^{(1)} +1)}
\end{equation}

Plugging the above expression into the formulas for coexistence mechanisms ((<ref>) - (<ref>)), and writing the covariance between the two species' environmental responses as $\phi \sigma_1 \sigma_2$ (where $\phi$ is the correlation coefficient), we find that the spatial storage effect is

\begin{equation}
\Delta I_i = \frac{q_r N_{r,0}^* \alpha_r^{(1)}}{1-q_r(\theta_{r,r} N_{r,0}^* \beta_r^{(1)} +1)} \left[ \zeta_i \phi \sigma_i \sigma_r - \zeta_r \sigma_r^2 \right],
\end{equation}

and fitness density covariance is

\begin{equation}
\begin{aligned}
\Delta \kappa_i = & \frac{q_i}{1-q_i} \left[\left(\alpha_i^{(1)}\right)^2 \sigma_i^2 +
\frac{2 \alpha_i^{(1)} \alpha_r^{(1)} \theta_{i,r} \beta_i^{(1)} \phi \sigma_i \sigma_r q_r N_{r,0}^*}{1-q_r(\theta_{r,r} N_{r,0}^* \beta_r^{(1)} +1)} +
\left( \frac{\alpha_r^{(1)} \theta_{i,r} \beta_i^{(1)} \sigma_r q_r N_{r,0}^*}{1-q_r(\theta_{r,r} N_{r,0}^* \beta_r^{(1)} +1)} \right)^2 \right] \\
- &\frac{q_r}{1-q_r} \left[\left(\alpha_r^{(1)}\right)^2 \sigma_r^2 +
\frac{2 \left(\alpha_r^{(1)}\right)^2 \theta_{r,r} \beta_r^{(1)} \sigma_r^2 q_r N_{r,0}^*}{1-q_r(\theta_{r,r} N_{r,0}^* \beta_r^{(1)} +1)} +
\left( \frac{\alpha_r^{(1)} \theta_{r,r} \beta_r^{(1)} \sigma_r q_r N_{r,0}^*}{1-q_r(\theta_{r,r} N_{r,0}^* \beta_r^{(1)} +1)} \right)^2 \right].
\end{aligned}
\end{equation}

Here, we can see that the invader's dispersal dynamics (i.e., the value of $q_i$) does not play a role in the spatial storage effect, but does play a role in fitness-density covariance. The resident's dispersal dynamics, on the other hand, play a role in both mechanisms.

If the invader and resident have identical demographic parameters but partially uncorrelated environmental responses, the coexistence mechanisms simplify to

\begin{equation}
\Delta I = \frac{q N_{r,0}^* \alpha^{(1)}}{1-q(\theta_{r,r} N_{r,0}^* \beta^{(1)} +1)} \left[ \zeta \sigma^2 (\phi - 1) \right], \text{and}
\end{equation}

\begin{equation}
\begin{aligned}
\Delta \kappa_i = & \frac{q}{1-q}
\frac{2 \left(\alpha^{(1)}\right)^2 \theta_{i,r} \beta^{(1)} (\phi - 1) \sigma^2 q N_{r,0}^*}{1-q(\theta_{r,r} N_{r,0}^* \beta^{(1)} +1)}
\end{aligned}
\end{equation}

\begin{equation}
\begin{aligned}
\Delta \kappa_i = & \frac{q N_{r,0}^* \alpha^{(1)}}{1-q(\theta_{r,r} N_{r,0}^* \beta^{(1)} +1)} \left[\frac{q}{1-q}  2 \alpha^{(1)} \theta_{i,r} \beta^{(1)} \sigma^2 (\phi - 1)\right]
\end{aligned}
\end{equation}

In order for both terms in (<ref>) to be of equal magnitude, we must assume that fluctuations in relative density are $\mathcal{O}(\sigma)$ ((<ref>)).

In models where per capita growth rates are given by a formula, invader abundance can usually be fixed at zero. In more complicated models (i.e., stochastic, spatially explicit simulation models), the per capita growth rates must be determined via simulation

In simple models with finite populations, we can often approximate the infinite-population case by fixing the invader's abundance at zero. In more complicated models, the per capita growth rates must be measured using simulation data, and therefore, the invader must be introduced at some low but non-zero density; we discuss strategies

However, the invasion growth rate can still be calculated in models with finite populations by measuring per capita growth rates when the invader is at low density. In models where per capita growth rates are given by a formula, invader abundance can usually be fixed at zero. In more complicated models (e.g., individual-based models), the invader must be introduced at some low but non-zero density.

These log-scale specifications of $E_j$ and $C$ are convenient. By construction, $g_{j}(E_j^*, C^*)$ = 1  (<ref>), so $\exp{E_j^* - C_j^*} = \delta_j$. This fact, combined with the fact that $e^x$ is its own derivative, leads to very simple expressions for the Taylor series coefficients ((<ref>)).

The competition parameter can be expressed as function of residents' environmental responses, via the relation $(C_j - C_j^*) = \sum_{k \neq i}^S \frac{\partial C_j}{\partial E_k} (E_k - E_k^*)$. In the focal case of one invader and one resident, there is a one-to-one conversion between the environment and competition: $\frac{\partial C}{\partial E_r} = 1$.

See Appendix <ref> for strategies for selecting the equilibrium parameters. The resulting second-order polynomial will lead to an accurate approximation of the invasion growth rate, but only if some assumptions are met. Specifically, it is assumed that 1) environmental fluctuations, $E_j - E_j^*$, are small; 2) that the average deviation, $\E{x,t}{E_j} - E_j^*$, is even smaller (because positive and negative fluctuations each other out) and 3) that dispersal can only lead to small increases in local population density and local competition. See Appendix <ref> for an extended discussion of these small-noise assumptions.

These assumptions are explained further in Appendix <ref>. Conditioned on the former assumption, the latter assumption is true when there is very little dispersal (such that variation in population density reflects variation in the environment, not the interplay between population growth and dispersal), or when there is widespread global dispersal (such that population density is distributed more-or-less uniformly across space).

Consider a community whose dynamics are represented by a system of difference equations

\begin{equation} \label{local_lambda_app}
n_j(x,t+1) = n_{j}(x,t) \lambda_{j}(x,t) + c_j(x,t) - e_j(x,t) \qquad j = (1, 2, ..., S),
\end{equation}

Although  $E_j^* = \E{x,t}{E_j}$ may be a reasonable choice in many models, one should not automatically make this selection, as it may fix $C_j^*$ at a value that is difficult to interpret. For example, choosing $C_j^* = C^* = \frac{1}{S} \sum_{i=1}^{S} \E{x,t}{C_{i}}$ (as in <cit.>) allows a positive value of $\left(\E{x,t}{C_i} - C^*\right)$ (which is part of the linear density-dependent effects, (<ref>)) to be interpreted as the focal invader species experiencing less competition than the other species when they are invaders.

To understand the effects of environmental noise, theoretical ecologists often start with a deterministic skeleton (i.e., a model without any noise) and then add a little noise. The difference in the behavior of the two models reveals the effect of noise. In these kinds of studies, there is a canonical way to select $E_j^*$ and $C_j^*$: select $E_j^*$ such that the noisy model collapses to the deterministic skeleton, and then select $C_j^*$ based on the constraint $g_j(E_j^*,C_j^*) = 1$. For example,Chesson1989 studied the noisy Lotka-Volterra model,

\begin{equation}
\frac{1}{n_j(t)} \frac{d n_j(t)}{d t} = b_j - \sum_{k = 1}^S \alpha_{jk} n_{k}(t) + E_{j}(t),
\end{equation}

which collapses to the usual Lotka-Volterra model when $E_j(t) = 0$. This suggests the choice $E_j^* = 0$. When competition is defined as $C_j(t) = \sum_{k = 1}^S \alpha_{jk} n_{k}(t)$ (a natural choice) this choice of $E_j^*$ fixes equilibrium competition at $C_j^* = b_j$.

, some coexistence mechanisms could negligibly small. This possibility is not problematic for empirical work, since the numerical estimation of coexistence mechanisms will reveal large disparities in magnitude between coexistence mechanisms. Mathematical expressions do not have fixed parameter values in this way, hence the small-noise assumptions and regularity conditions are more relevant in analytical work.

We will now attempt to justify our space-time decomposition by showing that the terms of the decomposition measure the strength of causal relationships: $S_j$ represents the causal effect of spatial variation on total variation, $T_j$ represents the causal effect of temporal variation on total variation, and $R_j$ represents the causal effect of the interaction between spatial variation and temporal variation, on total variation. To show how causation is involved, we will draw on the philosophy literature. This section is does not aim to use philisophical accounts of causation as a post-hoc argument from authority. Our contention is that a good space-time decomposition ought to measure the causal effects of spatial and temporal variation, but since causation is a vexed notion, we resort to the difference-making account of causation to make the goalposts clearer.

There is one caveat: if population dynamics are described in continuous-time and the fluctuations in regulating factors are temporally autocorrelated (i.e., $C_j(t)$ is continuous with respect to time), then it is still possible to meet condition 1b) (see Appendix <ref>).

, but the principle can still be applied in fluctuating environments, as long as there are no fluctuation-dependent coexistence mechanisms (<cit.>, p. <ref>).

which means that in order for the theory to apply, we must grant that there is no variation in the limiting dynamics of the environment or regulating factor

The discovery of fluctuation-dependent coexistence mechanisms circumvented the competitive exclusion principle and thus helped to clarify the circumstances under which the competitive exclusion principle does hold.

The following conditions are needed (<cit.>; <cit.>):  1a) There are no environmental fluctuations, or 1b) if the environment does fluctuate, it does not generate a covariance between environment and competition. Condition 1 precludes any storage effects. Condition 2a) limiting resources must not fluctuate, or 2b) limiting resources can fluctuate, but species must respond to resource availability in a linear fashion. Condition 2 precludes coexistence via relative nonlinearity.

armstrong1976coexistence showed that the competitive exclusion principle can be violated if species have nonlinear responses to regulating factors. Spatiotemporal MCT shows that species' responses to limiting resources cannot simultaneously be linear with respect to fluctuations on the natural scale (i.e.$\pdv[2]{g_j}{C_j} = \beta_j^{(2)} = 0$), which is necessary for spatial averaging, and linear with respect to fluctuations on the log-scale (i.e., $\pdv[2]{\log(g_j)}{C_j} = \beta_j^{(2)} - \beta_j^{(1)^2}  = 0$), which is necessary for temporal averaging. Thus, Condition 1b) above cannot possibly be met. Since Condition 1a) and 1c) are unlikely, prima facie, and since real communities experience spatiotemporal fluctuations, it is unlikely that the competitive exclusion principle is applicable in the real world. There is one caveat – if population dynamics are described in continuous-time and the fluctuations in regulating factors are temporally autocorrelated (i.e., $C_j(t)$ is continuous with respect to time), then it is still possible to meet condition 1b) (see Appendix <ref>).

We might hope that the difference between the mean and the median is very small, so that a space-time decomposition which utilizes the mean is functionally equivalent to a space-time decomposition that utilizes the median. In general, however, the absolute difference between the mean and median is bounded by one standard deviation (<cit.>), $\sqrt{\Var{x}{E_j(t)}} = \mathcal{O}(\sigma)$, such that the difference may remain noticeably large under the small-noise assumptions of MCT. While we cannot prove it, the mean and median may be very close in practice, particularly since the $E_j$ is often on a log-scale (as in the spatiotemporal lottery model, see Section <ref>). There is no space-time decomposition that is singularly appropriate, but we select the sequence of spatial means to feature in our definition so that the law of total variance may be used. We ought to be aware that this arbitrariness exists, but it should not produce skepticism with regard to the efficacy of MCT. In MCT, there already exists some degree of arbitrariness in how $E_j$ and $C_j$ are defined, how $E_j^*$ and $C_j^*$ are defined, and how the speed conversion factors,$a_i /a_r$, are defined, just as there is vagueness imbued in the most beloved ecological concepts, like stability, diversity, and competition.

, and 2) because they transparently relate to biological interpretations. For instance, because we know that the $C$ cannot affect $E$, and because there are no "third variables" to cause spurious correlations, a positive covariance between $E$ and $C$ implies a causal relationship between $E$ and $C$. Therefore, $\Cov{x,t}{E}{C}$ transparently captures the idea of a good environment leading to high competition, which is an integral part of the interpretation of the storage effect (<cit.>).

Unfortunately, this clean interpretation disappears when we consider abiotic factors that vary over both space and time. This scenario carries with it different small-noise assumptions — averages of fluctuations in abiotic factors over only space or time are now $\mathcal{O}(\sigma)$ — resulting in expressions for $S_j$, $T_j$, and $R_j$ that are no more insightful than the definitions ((<ref>)–(<ref>)) themselves. This does not mean that the definitions are wrong, but it does mean that another justification of the space-time decomposition must carry the torch.

Before moving onto the second justification, a discussion of the remainder term $R$ is in order. First, a careful reader may have noticed that in this edge case (i.e., abiotic factors varying over only space or time),  both $S$ and $T$ are $\mathcal{O}(\sigma^2)$, but the remainder term $R$ is $\mathcal{O}(\sigma^4)$, and therefore $\Var{x,t}{E_j} \approx S + T$. This disparity between the size of space-time decomposition components disappears when abiotic factors vary over both space and time.

For example, when devising a hypothetical world without temporal variation, squashing temporal variation by selecting one constraint is that at every single point in time, the environmental parameter must be the same at every point in space. Under this constraint, the sequence of spatial averages $A(x) = \E{t}{E_j(x)}$ minimizes the squared error $\sum_{x,t} \left( E_j(x,t) -A(x) \right)^2$. Another reasonable choice is the sequence of temporal medians, which minimizes the sum of absolute errors. In our conception of Spatiotemporal MCT, we use the mean.

Furthermore, this spatially constant term must be a function of only the environmental parameters at the current point in time - otherwise, we could give our hypothetical worlds the exact same total variance, and nothing would be learned from the decomposition. This constraint also corresponds to Lewis1979 guidelines for making sure that hypothetical worlds are similar:  "...maximize the spatiotemporal region thorough-out which perfect match of particular fact prevails". Here, the "particular fact" is taken to be $\E{x}{E_j(t)}$. Under these constraints, the sequence of spatial averages $A(x) = \E{t}{E_j(x)}$ minimizes the squared error $\sum_{x,t} \left( E_j(x,t) -A(x) \right)^2$. Another reasonable choice is the sequence of temporal medians, which minimizes the sum of absolute errors. In our conception of Spatiotemporal MCT, we use the mean.

The most straightforward way to define $S_j$ is the difference in total variance between the actual world (i.e., $\Var{x,t}{E_j}$), and a world in which there is no spatial variation (obtained by setting set $\Var{x,t}{E_j}$ to $\Var{x,t}{\E{x}{E_j}}$). This will indeed capture the total effect of spatial variation, which includes both the main effect of spatial variation and the interaction effect between spatial variation and temporal variation. However, we want $S_j$ to capture only the main effect of spatial variation. The solution is to define a reference category where both causal factors (here, spatial variation and temporal variation) are turned off (<cit.>), and then define $S_j$ as as the difference in total variance between the possible world in which there is spatial variation, but no temporal variation, and the possible world in which there is no spatial variation or temporal variation. From this perspective, we recover the definition in (<ref>): $\Var{x,t}{\E{t}{E_j}} - \Var{x,t}{\E{x,t}{E_j}} = \Var{x}{\E{t}{E_j}} = S_j$. The time term $T_j$ is defined analogously.

The interaction effect is the degree to which the joint effect of spatial and temporal variation exceeds the sum of their individual effects. More simply, the interaction term is the difference between the whole and the sum of the parts, the parts being $S_j$ and $T_j$. Interaction effects can have multiple interpretations (<cit.>), but because spatial and temporal variance are abstract objects, we find it difficult to interpret the interaction term in any deeper sense without a more detailed underlying model.

Now we must show that the possible worlds being compared are similar. We consider similarity with respect to $E_j$'s, the ultimate inputs of the total variance, given some constraints. When devising the time term, $T_j$, one constraint is that at a single point in time, the environmental parameter must be the same at every point in space. Furthermore, this spatially constant term must be a function of only the environmental parameters at the current point in time - otherwise, we could give our hypothetical worlds the exact same total variance, and nothing would be learned from the decomposition. This constraint also corresponds to Lewis1979 guidelines for making sure that hypothetical worlds are similar:  "...maximize the spatiotemporal region thorough-out which perfect match of particular fact prevails". Here, the "particular fact" is taken to be $\E{x}{E_j(t)}$. Under these constraints, the sequence of spatial averages $A(x) = \E{t}{E_j(x)}$ minimizes the squared error $\sum_{x,t} \left( E_j(x,t) -A(x) \right)^2$. Another reasonable choice is the sequence of temporal medians, which minimizes the sum of absolute errors. In our conception of Spatiotemporal MCT, we use the mean.