text:
\section*{Abstract}

The storage effect is a well-known explanation for the coexistence of competing species in temporally varying environments. Like many complex ecological theories, the storage effect is often used as an explanation for observed coexistence on the basis of heuristic understanding, rather than careful application of a detailed model.  But, a careful examination of some widely employed heuristic representations of the storage effect shows that they can be misleading. One interpretation of the storage effect states that species coexist by specializing on a small slice of all environmental states, and therefore must have a robust life-stage (e.g., long-lived adults, a seedbank) in order to "wait it out" for a good year. Another more general interpretation states that \textit{buffering} helps species "wait it out", where \textit{buffering} means populations are protected from large losses when the environment is poor and competition is high. Here, we show that both of these conventional interpretations are imperfect. Multiple models show that stage-structure, long lifespans, and overlapping generations are not required for the storage effect. Further, a species that experiences buffering necessarily grows poorly when the environment is favorable and competition is high. We review the empirical literature and conclude that species are likely to find themselves in this good-environment/high-competition scenario, and therefore, that buffering tends to hurt individual species. The buffering interpretation of the storage effect can be thought of as conflating a community-level criterion for coexistence with a population-level criterion for persistence; it is like claiming that species can persist by limiting their own growth rates, since the Lotka-Volterra model tells us that intraspecific competition must be greater than interspecific competition. To build a better understanding of the storage effect, we examine how it manifests in particular models and we demonstrate the connections between the mathematical and verbal/textual descriptions of the storage effect. While it is possible to improve one's understanding of the storage effect, the storage effect is a very general phenomenon, and so a simple ecological interpretation (in terms of a small set of life-history characteristics like stage-structure or dormancy) will not be forthcoming.

Keywords: storage effect, coexistence mechanisms, modern coexistence theory, bet-hedging, stage-structure, dormancy

\tableofcontents

\newpage

\section{Introduction}
\label{Introduction}

The temporal storage effect (often simply called the storage effect) is a general explanation for how species can coexist by specializing on different states of a temporally fluctuating environment. The storage effect has an impressive resume. First, it formalized the concept of environmental niche partitioning, which has long been thought (e.g., \cite{grinell1917niche}) to promote coexistence. Second, by showing that many species can coexist on just a single resource (\cite{Chesson1994}, Eq. 91; \cite{miller2017evolutionary}), the storage effect provided a potential resolution to \posscite{hutchinson1961paradox} paradox of the plankton. Third, the storage effect revealed the beneficent side of environmental variation, which historically was thought to undermine the ability of species to persist (\cite{Lewontin1969}; \cite{May1974}). A number of recent papers have synthesized the simultaneously stabilizing and destabilizing effects of environmental variation (\cite{adler2008environmental}; \cite{schreiber2019rarity}; \cite{pande2020mean}, \cite{Dean2020Stochasticity-inducedSynthesis}).

Unfortunately, the storage effect is not easy to understand. This is partially because the storage effect is inherently complicated, with a full explanation involving abstractions like statistical interaction effects and cross-species comparisons. One limitation to understanding is the fact that there are three ways to describe the storage effect --- a mathematical definition, a list of conditions, and the verbal/textual interpretation --- and it is not clear how these descriptions relate to each other. In Section \ref{A crash course in the storage effect}, we provide an explanation of the storage effect that explicitly relates the three ways of describing the storage effect.

It is worth belaboring the distinction between the \textit{definition} and \textit{interpretation} of the storage effect. The \textit{definition} is codified in a mathematical formula, which shows how the storage effect can be quantified. The \textit{interpretation} is a verbal or textual story about how species coexist via the storage effect, and serves several important purposes. 1) It imbues the math with biological meaning. 2) It is useful in data-poor situations; it allows one to hypothesize about whether or not the storage effect is operating in a specific system, without having to analyze a model or conduct a pilot experiment. 3) It is not always clear how the pieces of the mathematical definition of the storage effect should be defined (specifically $E_j$, $C_j$, $E_j^*$, and $C_j^*$; read on to Section \ref{The mathematical definition}). The interpretation of the storage effect can provide guidance for making these choices, which in turn can inform experimental design.

The storage effect is often given an interpretation --- which we will call \textit{the conventional interpretation v1} --- paraphrased as follows: species coexist by specializing on different states of a temporally fluctuating environment, so species must have a robust life stage in order to "wait it out"
for a favorable time period; "Storage" refers to a robust life stage can
"wait it out". The conventional interpretation v1 is incorrect, or at the very least, imprecise. One can construct models where the storage effect is positive, despite there being no stage structure (e.g., \cite{loreau1989coexistence}; \cite{loreau1992time}; \cite{klausmeier2010successional}; \cite{li2016effects}; \cite{letten2018species}; \cite{schreiber2021positively}). Despite these counterexamples, the imprecision of the conventional interpretation v1 is not widely recognized. In Section \ref{A critique of the conventional interpretation of the storage effect}, we explain why the storage effect has nothing to do with storing population vitality in a robust life stage.

Some authors present a more general interpretation, which we will call \textit{the conventional interpretation v2}: "Storage" can be more generally understood as \textit{buffering}, which is a negative interaction effect (analogous to a negative coefficient for an interaction term in multiple regression) of environment and competition on per capita growth rates; this buffering helps out rare species because it prevents extreme losses when the environment is unfavorable and competition is high. In Section \ref{A critique of the conventional interpretation of the storage effect}, we explain why the conventional interpretations v2 is also imprecise. We also speculate on the reasons for the genesis and perpetuation of these interpretations (Sections \ref{The origin of the conventional interpretation} \& \ref{The perpetuation of the conventional interpretation}), which serves as a geneological debunking argument (\cite{srinivasan2015archimedean}; \cite{kahane2011evolutionary}), i.e., undermining a proposition by showing that common belief in said proposition can be explained by historical or psychological factors.

\section{A crash course on the storage effect}
 \label{A crash course in the storage effect}

Here, we provide a comprehensive description of the storage effect. This section builds the groundwork for our critique of the conventional interpretations of the storage effect, but experts may skip to Section \ref{A critique of the conventional interpretation of the storage effect}).

The storage effect is well-demonstrated with a toy model of coral reef fish dynamics. The \textit{lottery hypothesis} (\cite{Sale1977}) states that the local diversity of coral reef fishes is generated by the random allocation of space: when an adult fish dies, the various fish species enter a lottery for the open territory with a number of tickets equal to the number of larvae that each fish species produces. The lottery hypothesis was motivated by the fact that coral reef fishes do not appear to finely partition food types, but do appear to be limited by space. Space limitation is evidenced by the observed territoriality of adults (\cite{warner1980population}), the production of larvae in massive numbers, and the weak correlation between adult population size and the subsequent number of recruits (\cite{cushing1971dependence}, \cite{szuwalski2015examining}).

While Sale's lottery hypothesis does a fine job at explaining local biodiversity, it cannot explain the maintenance of biodiversity --- coexistence. \textcite{chesson1981environmentalST} were able to attain coexistence with the addition of a single feature: temporal variability in per capita larval production. The resulting model is now known as the \textit{lottery model}, and the more general process permitting coexistence is known as \textit{the storage effect}. The exposition here follows the lottery model of \textcite{Chesson1994}, as opposed to original lottery model (\cite{chesson1981environmentalST}), which is more complex due to stochasticity in both per capita adult mortality and larval production.

Imagine a guild of fish species inhabiting discrete territories on a coral reef. Several events occur in each time-step of the lottery model, here presented in chronological order

\begin{enumerate}
    \item The fish spawn. Per capita larval-production, (i.e., per capita fecundity) fluctuates from time-step to time-step, putatively due to dependency on environmental factors that also fluctuate. Like the larvae of many marine fish, our hypothetical larvae disperse offshore (ostensibly to avoid predation) and return some time later, though still within the time-step.

    \item Adult fish die with some density-independent probability, leaving behind an empty territory. The death probability may vary across species, but unlike fecundity, does not vary across time.

    \item The larvae return to the reef and inherit the empty territories with a recruitment probability for any given larva being equal to the number of empty sites, divided by the total number of larvae. This uniform probability of per larva recruitment is the \textit{lottery} in the lottery model. The unrecruited larvae die before the next time-step begins.

\end{enumerate}

The above dynamics are expressed in the difference equations,

\begin{equation} \label{lottery}
\begin{aligned}
   n_j(t+1) =  n_j(t) \left[\overbrace{s_j}^{\text{survival prob.}} + \; \overbrace{\eta_{j}(t)}^{{\scriptstyle \text{per capita fecundity}}} \; \left[\rule{0cm}{1.25cm}\right. \frac{ \overbrace{ \sum \limits_{j \neq i}^{S} (1-s_j) n_{j}(t)}^{\text{open territories}}}{ \underbrace{\sum \limits_{j \neq i}^{S} \eta_{j}(t) n_{j}(t)}_{ {\scriptstyle \text{total larvae}}}} \left. \rule{0cm}{1.25cm}\right] \right], \quad j = (1, 2, \ldots, S),
   \end{aligned}
\end{equation}

where $n_j(t)$ is the density of species $j$ at time $t$, $s_j$ is the adult survival probability, and $\eta_j(t)$ is the time-varying per capita larval production.

\begin{figure} \label{lottery model}
  \centering
      \includegraphics[scale = 1.2]{lottery_model_fishes.png}
  \caption{Diagrammatic representation of the lottery model}
\end{figure}

In the lottery model, empty space is the only limiting resource, so the competitive exclusion principle (which states that no more than $N$ species can coexist on $N$ regulating factors; (\cite{volterra1926variationsST}, \cite{Lotka1932TheSupply}, \cite{GauzeG.F.GeorgiĭFrant︠s︡evich1934TsfeST}; \cite{levin1970community}) is transcended if even two fish species are able to coexist. Intuitively, if a species becomes rare for whatever reason (e.g., competition, catastrophes), then it must have a positive per capita growth rate if it is to recover from rarity. This is a slight simplification (see \cite[p.~293]{barabas2018chesson}), but for our current purposes, we say that coexistence is related to a \textit{rare-species advantage}, operationalized by the \textit{invasion growth rate}: the long-term average per capita growth rate of a species that has been perturbed to low density. The practice of determining coexistence based on invasion growth rates is called an \textit{invasion analysis} (\cite{turelli1978reexamination}; \cite{Grainger2019TheResearch}).

Consider a two-species lottery model with a red species and a blue species  (Fig. \ref{lottery model}). The red species produces many larvae during hot years and few larvae during cold years. The blue species responds to temperature oppositely: it produces few larvae during hot years and many larvae during cold years. If the blue fish species becomes rare, will it be able to recover?

We first ask the question, what happens when the blue fish species experiences a good (i.e., cold) year (Fig. \ref{lottery model}a)? There are many blue larvae produced per capita, but there are few larvae produced in total because blue fish are rare, and although the red fish are common, their per capita fecundity is low because the environment is less favorable to them. Each blue fish larva thus experiences relatively little competition, which we may measure as larvae per empty site. In this scenario, we see that a rare species is able to capitalize on a good environment.

To uncover a potential rare-species advantage, we must now examine an analogous scenario from the perspective of the common species: what happens when the red fish species experiences a good (i.e., warm) year (Fig. \ref{lottery model}b)? Many red larvae are produced in total, since there are many red fish and the environment favors the red fish. However, there is now an excess of red larvae, which significantly decreases the probability of any one larva winning a territory. The consequence of this high competition is a small or zero-valued per capita growth rate; the last panel of Fig. \ref{lottery model}b shows no net change). Unlike the rare blue fish, the common red fish is unable to capitalize on a good environment.

We will now frame this concrete scenario (i.e., the red fish experiencing a hot year) in slightly more general terms: For a common species, a good response to the environment (e.g., high per capita larval fecundity) causes high competition (e.g., many larvae per open site), which ultimately undermines the good response to the environment. For a rare species, a good response to the environment does not lead to as much competition. It is this asymmetry between rare and common species that drives the storage effect.

\subsection{A simple interpretation of the storage effect}
\label{An okay interpretation of the storage effect}

\begin{siderules}
Good environments lead to high competition for common species, but less so for rare species. Since high competition undermines the positive effects of a good environment (via a negative interaction effect of environment and competition on per capita growth rates), rare species are better able to take advantage of a good environment than common species.
\end{siderules}

This interpretation is correct but leaves out some details. The simple interpretation corresponds to a positively-valued storage effect, but what does a negatively-valued storage effect look like? How can the asymmetry between rare species and common species be represented mathematically? For that matter, how can the negative interaction effect be represented mathematically? What, in the simple interpretation, is necessary and/or sufficient for the storage effect? Is the storage effect a species-level or community-level characteristic? These questions are best answered with the mathematical definition of the storage effect and its textual analogue: an ingredient list of conditions for the storage effect.

\subsection{The mathematical definition}
\label{The mathematical definition}

The mathematical definition of the storage effect is embedded within \textit{Modern Coexistence Theory} (\cite{Chesson1994}; \cite{chesson2000general}; \cite{barabas2018chesson}), a framework for partitioning invasion growth rates into additive terms; these terms correspond to different explanations for coexistence, and  are therefore called \textit{coexistence mechanisms}. The storage effect is one of several coexistence mechanisms. In order to arrive at the mathematical definition of the storage effect, we provide a step-wise summary of the derivation of coexistence mechanisms:

\begin{enumerate}
    \item \textbf{Choose one species to be the rare species}. This species is called the \textit{invader} and the remaining species are called \textit{residents}. Set the invader's density to zero, and let residents attain their limiting dynamics, i.e., let them equilibrate to their typical densities.

    The invader is denoted by the subscript $i$, the residents are denoted by the subscript $s$, and a generic species is denoted by the subscript $j$.

    \item \textbf{Write the per capita growth rates in terms of the environmental and competition}. Let the per capita growth rate of species $j$ be some function $g_j$ of the environmental parameter $E_j(t)$ and competition $C_j(t)$, i.e., $dn_j(t)/(n_j(t) dt) = r_j(t) = g_j(E_j(t),C_j(t))$. In discrete-time models like the lottery model, the effective per capita growth rate is the logged \textit{finite rate of increase}, i.e., $r_j(t) = \log(\lambda_j(t))$, where $\lambda_j(t) = n_j(t+1)/n_j(t)$. Extensions to structured populations can be found in \textcite{Ellner2019}. For notational simplicity, we drop the explicit dependence on time $t$.

    The parameter $E_j$ is called the environmental parameter, the environmentally-dependent parameter, or simply the environment. It is typically a demographic parameter, belonging to species $j$, that depends on the abiotic environment (e.g., germination probability depends on temperature). More generally, $E_j$ may represent the effects of density-independent factors. The parameter $C_j$ is called the competition parameter, but it may represent the effects of density-dependent factors. Of particular interest is the case where $C_j$ is a function of shared predators, potentially leading to the storage effect due to predation (\cite{kuang2010interacting}; \cite{chesson2010storage}; \cite{stump2017optimally}).

    The invasion growth rate is the long-term average per capita growth rate of the invader. More generally, the invasion growth rate is the dominant Lyapunov exponent of the dynamical system representing population dynamics (\cite{Metz1992}; \cite{dennis2003}).

    \item \textbf{Expand growth rates with respect to $E_j$ and $C_j$}. First, select equilibrium values of the environment and competition, $E_j^*$ and $C_j^*$, such that $g_j(E_j^*, C_j^*) = 0$. Next, perform a second-order Taylor series expansion of $g_j(E_j, C_j)$ about $E_j^*$ and $C_j^*$.

    The result is

\begin{equation} \label{taylor_decomp}
\begin{aligned}
 \left. {r_{j}(E_j,C_j)}%
_{\stackunder[1pt]{}{}}%
 \right|_{%
 \stackon[1pt]{$\scriptscriptstyle E_j = E_{j}^{*}$}{$\scriptscriptstyle C_j = C_{j}^{*}$}}
 \approx \; & \alpha_j^{(1)} (E_j - E_{j}^{*}) + \beta_j^{(1)} (C_j - C_{j}^{*}) \\ & +
\frac{1}{2} \alpha_j^{(2)} (E_j - E_{j}^{*})^{2} + \frac{1}{2} \beta_j^{(2)} (C_j - C_{j}^{*})^{2} +
\zeta_j  (E_j - E_{j}^{*})   (C_j - C_{j}^{*}),
\end{aligned}
\end{equation}

where the coefficients of the Taylor series,

\begin{equation}  \label{taylor_coef}
\begin{aligned}
 \alpha_j^{(1)} = \pdv{g_j\scriptstyle{(E_j^*, C_j^*)}}{E_j},  \quad
 \beta_j^{(1)} = \pdv{g_j\scriptstyle{(E_j^*, C_j^*)}}{C_j},  \quad
 \alpha_j^{(2)} = \pdv[2]{g_j\scriptstyle{(E_j^*, C_j^*)}}{E_j},  \quad
 \beta_j^{(2)} = \pdv[2]{g_j\scriptstyle{(E_j^*, C_j^*)}}{C_j,}  \quad
 \zeta_j = \pdv{g\scriptstyle{(E_j^*, C_j^*)}}{E_j}{C_j},  \quad
\end{aligned}
\end{equation}

are all evaluated at $E_j = E_j^*$ and $C_j = C_j^*$.

\item \textbf{Time-averaging}. Invasibility is determined by what happens in the long-run, so our next step is to take the temporal average of \eqref{taylor_decomp}. Temporal averages are denoted with "bars"; e.g., the average per capita growth rate of species $j$ is $\overline{r}_j$.

\begin{equation} \label{taylor_decomp_avg1}
\begin{aligned}
 \left. {\overline{r}_{j}}%
_{\stackunder[1pt]{}{}}%
 \right|_{%
 \stackon[1pt]{$\scriptscriptstyle E_j = E_{j}^{*}$}{$\scriptscriptstyle C_j = C_{j}^{*}$}}
 \approx \; & \alpha_j^{(1)} (\overline{E_j} - E_{j}^{*}) + \beta_j^{(1)} (\overline{C_j} - C_{j}^{*}) \\ & +
\frac{1}{2} \alpha_j^{(2)} \Var{}{E_j} + \frac{1}{2} \beta_j^{(2)} \Var{}{C_j} +
\zeta_j  \Cov{}{E_j}{C_j}.
\end{aligned}
\end{equation}

The above expression above rests on several assumptions about the magnitude of environmental fluctuations and the relationship between environment, population density, and competition (details can be found in \textcite{Chesson1994} and \textcite{chesson2000general}). Most crucially, we assume that environmental fluctuations, $\abs{E_j - E_j^*}$, are very small, and that average environmental fluctuations  $\abs{\overline{E_j} - E_j^*}$, are even smaller. These \textit{small-noise assumptions} ensure that the expression \eqref{taylor_decomp_avg1} is a good approximation of the true invasion growth rate, thus justifying the truncation of the Taylor series at second order. The small-noise assumptions also justify the replacement of second-order polynomial terms with central moments, e.g., $\overline{(E_j - E_j^*)^2}$ is replaced by $\Var{}{E_j}$: the equilibrium value $E_j^*$ is assumed to be very close the temporal average $\overline{E_j}$, such that little growth rate is lost by replacing the former with the latter.

\item \textbf{Invader--resident comparisons}

The long-term average growth rate of each resident must be zero (otherwise residents would go extinct or explode to infinity), so the value of the invasion growth rate is unaltered if we subtract a linear combination of the residents' long-term average growth rates.

\begin{equation} \label{inv res comparison}
    \overline{r}_i = \overline{r}_i - \sum \limits_{s \neq i}^S q_{is}  \overline{r}_s
\end{equation}

The $q_{is}$ are called \textit{scaling factors} (\cite{barabas2018chesson}) or comparison quotients (\cite{Chesson2019}). The original definition, provided by \textcite{Chesson1994}, is $q_{is} = \frac{\beta_i^{(1)}}{\beta_s^{(1)}} \frac{\partial C_i}{\partial C_s}$ (evaluated at the equilibrium values of environment and competition). Ellner et al. (\citeyear{ellner2016quantify}, \citeyear{Ellner2019}) has suggested scaling resident growth rates to create a simple average over resident species: in \eqref{inv res comparison}, replace the $q_{is}$ with $1/(S-1)$. Elsewhere, we have argued for scaling resident growth rates by a ratio of species' generation times, so as to convert the residents' population-dynamical speeds to that of the invader (\cite{johnson2022methods}). Incidentally, this approach is equivalent to the $q_{is}$ scaling factors for the simple models that we examine in this paper. Additionally, our arguments against the conventional interpretations of the storage effect do not hinge on how resident growth rates are scaled. For these reasons and for the sake of convention, we use Chesson's original scaling factors.

Though the average growth rate of each resident is zero, the components of the average growth rate (i.e., the additive terms in \eqref{taylor_decomp_avg1}) are not necessarily zero. Therefore, we can draw meaningful comparisons between the invader and the residents by substituting the right-hand side of the Taylor series expansion (\eqref{taylor_decomp_avg1}) into the invader--resident comparison (\eqref{inv res comparison}) and grouping like-terms:

\begin{equation} \label{MCT full}
\begin{aligned}
    \overline{r}_i \approx & \underbrace{\alpha_i^{(1)} (\overline{E_i} - E_{i}^{*}) + \frac{1}{2} \alpha_i^{(2)} \Var{}{E_i} +  \beta_i^{(1)} C_{i}^{*}  - \sum \limits_{s \neq i}^S q_{is} \left((\overline{E_s} - E_{s}^{*}) + \frac{1}{2} \alpha_s^{(2)} \Var{}{E_s} +  \beta_s^{(1)} C_{s}^{*} \right) }_{r'_{i}, \text{Density-independent effects}}  \\
    + & \underbrace{\beta_i^{(1)} \overline{C_i} - \sum \limits_{s \neq i}^S q_{is}  \beta_s^{(1)} \overline{C_s} }_{\Delta \rho_i, \text{Linear density-dependent effects}} \\
    + & \underbrace{\frac{1}{2}\left[ \beta_i^{(2)} \Var{}{C_i}  - \sum \limits_{s \neq i}^S q_{is}   \beta_s^{(2)} \Var{}{C_s} \right]}_{\Delta N_i, \text{Relative nonlinearity}} \\
    + & \underbrace{\zeta_i  \Cov{}{E_i}{C_i}  - \sum \limits_{s \neq i}^S q_{is}  \zeta_s  \Cov{}{E_s}{C_s}.}_{\Delta I_i,  \text{The storage effect}}
\end{aligned}
\end{equation}

The new symbols ($r'_i$, $\Delta \rho_i$, $\Delta N_i$, and $\Delta I_i$) denote coexistence mechanisms.

Because the coexistence mechanisms constitute the invasion growth rate, they measure a \textit{rare-species advantage}. Because the coexistence mechanisms make cross-species comparisons, they capture (respectively) different types of \textit{specialization}. Because explanations for coexistence have historically featured types of specialization that result in a rare-species advantage, it is fair to say that coexistence mechanisms measure the importance of different classes of explanations for coexistence.

\end{enumerate}

The mathematical definition of the storage effect, denoted $\Delta I_i$, is

\begin{equation} \label{SE math def}
    \Delta I_i = \zeta_i \;  \Cov{}{E_i}{C_i} - \sum \limits_{s \neq i}^S q_{is} \; \zeta_j \; \Cov{}{E_s}{C_s}.
\end{equation}

When ecologists talk colloquially about a storage effect, they are typically talking about a positive $\Delta I_i$ that is mediated through competition. However, the storage effect can also be negative, and/or mediated through apparent competition. In the case of a negative storage effect, there is a tendency for rarity to cause lower per capita growth rates. Therefore, negative storage effects can mediate a stochastic priority effect (\cite{chesson1988interactions}; \cite{schreiber2021positively}).

Coexistence mechanisms are often divided by the invader's sensitivity to competition, which we may operationalize here as $\abs{\beta_i^{(1)}}$. The rationale here is that $\abs{\beta_i^{(1)}}$ can be interpreted as the speed of population dynamics (at least in the lottery model and annual plant model; sensu \cite{Chesson1994}), so dividing by it enables a better comparison of species with slow and fast life-cycles (\cite{Chesson2018}). Note that this type of scaling is distinct from the aforementioned $q_{is}$ scaling factors (though they are related; \cite{johnson2022methods}).

Scaled coexistence mechanisms are sometimes averaged over species (see \cite{chesson2003quantifying}, \cite{barabas2018chesson}), either to make comparisons between communities or to quantify how a mechanism affects species \textit{in general}. The \textit{community-average storage effect} is defined as

\begin{equation} \label{community average SE}
   \overline{ \left(\frac{\Delta I}{\abs{\beta^{(1)}}} \right)} = \frac{1}{S} \sum_{i = 1}^{S} \frac{\Delta I_i}{\abs{\beta_i^{(1)}}}.
\end{equation}

\subsection{The ingredient-list definition}
\label{The ingredient list definition}

The storage effect depends on three ingredients:

\begin{enumerate}
    \item species-specific responses to the environment,

    \item a non-zero interaction effect with respect to fluctuations in the environment and competition (also known as \textit{nonadditivity}), and

    \item covariance between environment and competition (\textit{$EC$ covariance} for short).

\end{enumerate}

This ingredient-list definition (\cite{chesson1989short}; \cite{Chesson1994}) describes what is \textit{generally} required for the storage effect to be positive (or negative) for many species. To see that the ingredients are associated with a systematically positive (or negative) storage effect, consider a model with \textit{symmetric species}: each species responds the environment in accordance with a symmetric covariance matrix with diagonal elements $\sigma^2$ and off-diagonal elements $\rho \sigma^2$, where $\rho$ is the between-species correlation in $E_j$); species otherwise have identical demographic parameters. Just below (in Section \ref{Species-specific responses to the environment}), we show that the storage effect is then
\begin{equation} \label{symmetry}
    \Delta I = - \zeta  (1 - \rho) \theta
\end{equation}%
for every species. The three ingredients are captured  in the above formula: species-specific responses to the environment is $(1- \rho)$; the interaction effect is $\zeta$; and covariance is $\sigma^2 \theta$, where $\theta$ is a constant that converts the environmental responses of residents into competition (see Section \ref{Species-specific responses to the environment} below). Mathematical expressions for the storage effect are generally more complicated in the non-symmetric case (see Eq. 29 in \cite{Chesson1994}).

Contrary to the claims of previous research, the three ingredients are neither necessary nor sufficient for a single species' storage effect to be non-zero. To see why the ingredients are not necessary, consider the following non-symmetric scenario in which all species respond to the environment identically, but only species $k$ has a non-zero interaction effect (i.e., $\zeta_k \neq 0$). In this scenario, the condition \# 1 (species-specific responses to the environment) is not satisfied, yet species $k$ may nonetheless have a non-zero storage effect. In Appendix \ref{Quantifying the storage effect in the lottery model}, we show that the ingredients are not sufficient for the storage effect: when all ingredients are present in the lottery model, the storage effect can be zero if different species have different adult survival probabilities.

While it may seem pedantic to emphasize that the ingredients are neither necessary nor sufficient, we will see later (Section \ref{Discussion: Rethinking storage}) that this has implications for the interpretation of the storage effect. We do not want to give the impression that the ingredients are unimportant. Very few high-level features of the world have necessary and sufficient properties; in everyday life and in ecology, concepts are fuzzy, being held together by an open-ended set of correlated properties (Wittgenstein famously called these \textit{family resemblance concepts}; \cite{wittgenstein1968philosophical}). Because the storage effect is generally related to the three ingredients, we can better understand the the storage effect by studying the three ingredients in-depth.

\subsubsection{Ingredient \#1: Species-specific responses to the environment}
\label{Species-specific responses to the environment}

The function of ingredient \#1 is rather obvious: to establish the presence of niche differences, which are necessary for coexistence via any mechanism (\cite{GauzeG.F.GeorgiĭFrant︠s︡evich1934TsfeST}; \cite{chesson1991need}). In the absence of \textit{species-specific responses to the environment}, there would be no rare-species advantage. In terms of the lottery model,  a good (bad) year for the blue species would automatically be a good (bad) year for the common red species, such that both species would always experience the same level of competition. In our story about coral reef fishes, species had negatively correlated responses to the environment. However, \textit{species-specific responses to the environment} only entails that species' responses not be perfectly correlated. As we will see (in Section \ref{A critique of the conventional interpretation of the storage effect}), real-world species tend to have positively correlated responses to the environment.

To see how \textit{species-specific responses to the environment} are implied by the mathematical definition, we must re-express the competition parameter in terms of the environment. For illustrative purposes, suppose that there are only two species, and that both share a competition parameter $C$ that can be written as a smooth function $h$ of the resident's density and environmental response: $C = h(E_s, n_s)$. The the competition parameter can then be approximated as a first-order Taylor series expansion: $C = C^* + \frac{\partial h(E_s^*, n_s^*)}{\partial E_s}(E_s - E_s^*) + \frac{\partial h(E_s^*, n_s^*)}{\partial n_s}(n_s - n_s^*)$. With the additional assumption that environmental states are uncorrelated across time (which precludes covariance between environment and population density), this approximation of $C$ can be substituted into the $EC$ covariance. For the invader, $\Cov{}{E_i}{C} \approx \frac{\partial h(E_s^*, n_s^*)}{\partial E_s} \Cov{}{E_i}{E_s} = \frac{\partial h(E_s^*, n_s^*)}{\partial E_s} \rho \sigma_i \sigma_s$, where $\sigma_i$ is the scale of fluctuations in $E_i$, $\sigma_s$ is the scale of fluctuations in $E_s$, and $\rho$ is the cross-species correlation between $E_i$ and $E_s$. By contrast, the residents' covariance term is $\Cov{}{E_s}{C} \approx \frac{\partial h(E_s^*, n_s^*)}{\partial E_s} \Var{}{E_s} = \frac{\partial h(E_s^*, n_s^*)}{\partial E_s} \sigma_{2}^2$.

Next, we plug these covariance approximations into the definition of the storage effect, define $\theta = \frac{\partial h(E_2^*, n_2^*)}{\partial E_2}$, and assume that species have identical $\zeta_j$, $\sigma_j$, $\beta_j^{(1)}$ (which implies that $q_{is} = 1$). Then, we arrive at \eqref{symmetry} above: $\Delta I_i = - \zeta  (1 - \rho) \theta$.
Generalizing to the multi-species case involves redefining the $h$ so that competition is a function of all residents' environmental responses $\myvect{E}$, and population densities $\myvect{n}$; redefining $\theta = \frac{1}{S-1} \sum_{s \neq i}^S \frac{\partial h(\myvect{E^*}, \myvect{n^*})}{\partial E_s}$. If $\zeta$ is negative (which indicates \textit{buffering}), then the storage effect is proportional to $(1-\rho)$, demonstrating that (all else being equal) the storage effect increases as environmental niche differences increase.
asdf
\subsubsection{Ingredient \#2: An interaction effect between environment and competition}
\label{An interaction effect between environment and competition}

When a red fish experiences a hot year, it is not merely the case that the negative effects of high competition offset the positive effects of a good environment. Rather, the environment and competition act synergistically to reduce per capita growth rates further. This synergy is the \textit{interaction effect}, akin to an interaction effect in multiple regression. In fact, the coefficient $\zeta_j$ in the mathematical definition of the storage effect (\eqref{SE math def}) \textit{is} the interaction effect of a multiple regression where $E_j - E_j^*$ and $C_j - C_j^*$ are predictors, in the limit of small environmental noise. The causal interpretation of an interaction effect in multiple regression is that the level of one predictor modulates another predictor's effect on the response variable (\cite{gelman2007data}), which is why the simple interpretation of the storage effect (Section \ref{An okay interpretation of the storage effect}) states that "high competition undermines the positive effects of a good environment". Put yet another way, high competition means that the population is less sensitive to changes in the environment.

In our exposition thus far, we have described a negative interaction effect. However, both negative or positive interaction effect can lead to either a positive or negative storage effect. In the jargon of Modern Coexistence Theory, a negative interaction effect (i.e., $\zeta < 0$) is called \textit{subadditivity} or \textit{buffering} (\cite{Chesson1994}). The term \textit{subadditive} comes from the fact that the joint effects of environment and competition are less than than the sum of their parts. The term \textit{buffering} comes from the fact that the doubly deleterious effect of a poor environment and high competition is somewhat abated: the term $\zeta_j (E_j - E_j^*)(C_j-C_j^*)$ is positive when $\zeta_j < 0$, $(E_j - E_j^*) < 0$, and $(C_j-C_j^*) >0$. A positive interaction effect (i.e., $\zeta_j >0$) is synonymous with \textit{superadditivity} or \textit{amplifying} (\cite{Chesson1989}). More generally, both positive and negative interaction effects are referred to as \textit{nonadditivity}. The storage effect is generally positive in systems with subadditivity and positive $EC$ covariances, \textit{or} systems with superadditivity and negative $EC$ covariances. Conversely, the storage effect is generally negative in systems with subadditivity and negative $EC$ covariances, \textit{or} systems with superadditivity and positive $EC$ covariances.

At a high level of abstraction, the interaction effect can be thought of combining the environment and competition into a large number of density-dependent factors. Coexistence requires negative feedback loops where species demonstrate some degree of specialization on density-dependent factors (\cite{meszena2006competitive}), but some communities do not have enough density-dependent factors to be specialized upon (\cite{Hutchinson1959}; \cite{hutchinson1961paradox};  but see \cite{levin1970community}; \cite{haigh1972can}; \cite{abrams1988should}). On the other hand, species may readily specialize on different environmental states, but environmental variation alone cannot promote coexistence (\cite{chesson1997roles}). The interaction effect combines the competition parameter with the environmental parameter to get the best of both worlds: the density-dependent factors (implicit in the competition parameter) provide the negative feedback while the species-specific environmental parameter provides the specialization.

But what is an interaction effect in more concrete terms? What ecological concepts lead to an interaction effect in particular models? In coexistence theory, a negative interaction effect has primarily been associated with differential sensitivities of different life-stages. \textcite{chesson1988community} write, "... iteroparous plant and sessile marine organisms, can buffer by participating in reproduction over a number of years... Semelparous species can experience these buffering effects if the offspring of an individual mature over a range of years..." More generally, a negative interaction effect may arise from other forms of population structure: dormancy (\cite{caceres1997temporal}; \cite{ellner1987alternate}), phenotypic variation (\cite{chesson2000mechanisms}), or spatial variation (\cite{chesson2000general}). In studies of the population genetic storage effect (which promotes allelic diversity), negative interaction effects can be produced by heterozygotes (\cite{Dempster1955}; \cite{Haldane1963}), sex-linked alleles (\cite{reinhold2000maintenance}), epistasis (\cite{gulisija2016phenotypic}), and maternal effects (\cite{yamamichi2017roles}).

Though there are plenty of models in which population structure produces an interaction effect, population structure is neither necessary nor sufficient for an interaction effect. To see why it is not sufficient, consider a modified version of the lottery model in which adult mortality occurs before the spawning event. In this model, the logged finite rate of increase (i.e., the discrete-time analogue of per capita growth rate) is $r_j = \log(s_j(1 - \exp{E_j - C}))$ (compare with the original lottery model, \eqref{lottery}), and therefore, $\zeta_j = 0$. Here, a storage effect is impossible, even though there is stage structure, and even though the larval stage is more sensitive than the adult stage to both environment and competition (as in the original lottery model).

To see why population structure is not necessary for an interaction effect, consider a phytoplankton species that \textcite{monod1949growth} dynamics,

\begin{equation}
    \frac{d n_j(t)}{d t} = n_j(t) \left[ b_j(t) \frac{R(t)}{K_j + R(t)} - d_j \right],
\end{equation}

where $n_j(t)$ is the density of phytoplankton species $j$, $b_j(t)$ is the temporally-fluctuating maximum rate of increase, $R(t)$ is nitrogen concentration, $K_j$ is the half-saturation constant, and $d_j$ is the death rate. Defining $E_j = \log(b_j)$ and $C_j = \log \left( \frac{K_j + R}{R} \right)$, we find that $\zeta_j = d_j$. Here, we see an interaction effect (buffering increases as the death rate increases), despite the lack of stage structure. This phytoplankton model also demonstrates that "buffering" (i.e., a negative interaction effect) is not necessarily a result of life-history traits (e.g., dormancy, iteroparity) that have a clear adaptive purpose, but rather a by-product of banal population dynamics: the maximum uptake rate varies with the environment (which follows from the fact that the rate of a chemical reaction tends to increase with temperature; \cite{eyring1935activated}; \cite{kingsolver2009well}), growth rates are a saturating function of resource concentration (which follows from ion channels having a nutrient handling time; \cite{aksnes1991theoretical}). The consequence of these non-adaptive and putatively inescapable features is a negative interaction effect.

We conclude that at the current moment, there is no ecological interpretation of the interaction effect that is satisfactorily general. That is, an interaction effect cannot be intuited purely from knowledge about species' life-history characteristics, such as dormancy, iteroparity/non-overlapping generations, stage-structure, or more generally, a verbal description of a new model. Instead, the ecological interpretation of an interaction effect (or lack thereof) must be determined on a model-by-model basis, either with mathematical analysis or analogy with previously-studied models.

\subsubsection{Ingredient \#3: Covariance between environment and competition}
\label{Covariance between environment and competition}

The final ingredient, covariance between environment and competition, is immediately evident in the mathematical definition of the storage effect (\eqref{SE math def}). In more biological terms, the covariance captures the causal relationship between environment and competition. The most obvious way in which a good environment causes high competition (and vice-versa) is through intergenerational population growth: a good environment produces a larger population, and a larger population usually corresponds to higher competition. However, temporal autocorrelation in the environment is required for $EC$ covariance via intergenerational population growth  (\cite{li2016effects}; \cite{letten2018species}; \cite{Ellner2019}; \cite{schreiber2021positively}); the past environment determines the present competition, but the $EC$ covariation involves the current environment and current competition, so the current environment must resemble the past environment. The storage effect also arises when species have phenology differences in periodic environments (\cite{loreau1989coexistence}; \cite{loreau1992time}; \cite{klausmeier2010successional}), since a periodic environment is just a special case of a temporally autocorrelated environment.  In a stage-structured model, a good environment can lead to high competition within a single time-step (\cite{chesson1988community}). Consider the lottery model: a good environment (i.e., high per capita fecundity) at the spawning stage leads to high competition (i.e., many larvae per territory) at the recruitment stage. Note here that there is still temporal autocorrelation in the sense that the larvae carry the effects of the environment through time.

Although the archetypical storage effect is mediated through resource competition, the storage effect may also be mediated through apparent competition; the parameter $C_j$ may be generally understood as the effects of all density-dependent factors. When the storage effect is mediated through resource competition, the $EC$ covariance is generically positive, though it may be negative when the environment is negatively autocorrelated (\cite{schreiber2021positively}).

When the storage effect is mediated through apparent competition, a negative, positive or zero-valued covariance is possible. For the sake of the current discussion, assume that the competition parameter is the density of the shared predator $P$, times the predator's functional response $f_j(N_j)$, all divided by prey density i.e., $C_j = P * (f_j(N_j) / N_j)$. \textcite{stump2017optimally} analyzed a variant of the annual plant model and found that a type 2 functional response leads to a negative $EC$ covariance: good environments lead to a large number of seeds, which satiate predators, thus lowering the per-seed predation pressure. Kuang and Chesson (\cite*{kuang2010interacting}, \cite*{chesson2010storage}) found that the a type 3 functional response (i.e., frequency-dependent predation) leads to a positive $EC$ covariance: good environments lead to a large number of seeds, which are then preferentially consumed. In the previous two examples, the predator demonstrates a fast behavioral response to changes in prey density. If the predator demonstrates only a numerical response to prey density (corresponding to a type 1 functional response) and the environment is temporally autocorrelated, then the covariance will be positive. If instead the environment is uncorrelated through time, the storage effect due to the predation is not possible (\cite{Kuang2009CoexistenceEffect}): the environment changes before it appreciably affects predation pressure, and therefore does not produce the necessary covariance.

\section{A critique of the conventional interpretations of the storage effect}
\label{A critique of the conventional interpretation of the storage effect}

The \textit{conventional interpretation v1} of the storage effect is built around the idea that species have special adaptations or life-history traits that help them persist through temporarily hostile environments. This theme is evidenced by various quotes:

\begin{itemize}

\item "\ldots adults must be able to survive over periods of poor recruitment, such that the population declines only slowly during these periods. Under these conditions, a species tends to recover from low densities, and competitive exclusion is opposed. .... We refer to this phenomenon as the storage effect because strong recruitments are essentially stored in the adult population, and are capable of contributing to reproduction when favorable conditions return (\cite{Chesson1985}).

\item "\ldots the storage-effect coexistence mechanism relies on such buffering effects of persistent stages, because these prevent catastrophic population decline when poor recruitment occurs." (\cite{chesson2003quantifying})

\item "Persistence of adults limits the damage from unfavourable conditions, but does not prevent strong growth at other times. \ldots Similarly, the dormant seeds of annual plants are relatively insensitive to environmental factors and competition in comparison with the actively growing plants." (\cite{Chesson2004})

\item "Seed banks or long-lived adults ''store'' the effects of favorable years, which buffer the effects of bad years when population sizes may decline." (\cite{sears2007new})

\item "First, organisms must have some mechanism for persisting during unfavourable periods, such as a seedbank, quiescence or diapause. This condition, which gives the storage effect its name, buffers negative population growth; without it, populations would go extinct after a brief unfavourable period and environmental variation could never promote coexistence." (\cite{adler2014testing})

\end{itemize}

To be clear, we are not making any judgements about what the quoted authors do or do not know about the storage effect; it is often pedagogically useful to present definitions that are evocative but not 100\% precise. Our purpose in quoting the above authors is to provide evidence that there exists an interpretation of the storage effect, paraphrased as \ldots

\begin{quote}
\textbf{The conventional interpretation of the storage effect v1}: Species coexist by specializing on different parts of a fluctuating environment, so species must have a robust life stage in order to "wait it out" for a favorable time period. Thus, "storage" refers to a robust life stage can "wait it out".
\end{quote}

By showing that age or stage structure is not necessary for the storage effect, a number of papers have shown that the conventional interpretation v1 is imprecise. The temporal storage effect can be generated in models without any age or stage structure if the environment is temporally autocorrelated (\cite{loreau1989coexistence}; \cite{loreau1992time}; \cite{klausmeier2010successional}; \cite{li2016effects}; \cite{letten2018species}; \cite{Ellner2019}; \cite{schreiber2021positively}). All of these unstructured population models can be interpreted as having non-overlapping generations; thus, the notion of longevity or an individual "waiting it out" for a good year is not necessary for the storage effect.

A simple argument shows why the conventional interpretation v1 is fundamentally flawed: An invader population has functionally has no individuals in which population vitality or years of good recruitment can be stored. The fact that we can approximate the dynamics of a low density population with a zero density population means that the presence of conspecific individuals has a negligible effect on per capita growth rates. To be clear, in structured population models, invasion growth rates depend on the quasi-stable age/stage distribution (\cite{CaswellHal2001Mpm:}), even if the invader is at infinitesimal density. However, in the lottery model, all larvae die within a single time-step, which means the stable stage distribution in the lottery model is entirely adult fish, regardless of the adult survival parameter. Again, the conventional interpretation v1 cannot be correct because the whole premise of invasion analysis is that individuals of a rare species have negligible effects on per capita growth rates.

It is true that in the lottery model and the annual plant model, coexistence is not possible when generations are non-overlapping (i.e., adult or seed survival equals zero). However, this is because the particular structure of these models do not allow for an interaction effect when the survival probability equals zero, not because anything is stored in a particular life stage. To see this more clearly, we will consider the classic annual plant model (also called the seedbank model; \cite{Chesson1994}),
\begin{equation} \label{annual plants}
    X_j(t+1)= X_j(t) \left[ s_j (1 - G_j(t)) + \frac{G_j(t) Y_j}{\sum_{k =1}^{S} c_k G_k(t) X_k(t)} \right],
\end{equation}
where $X_j$ is the density of seeds of species $j$, $s_j$ is the probability that a seed survives the growing season if it does not germinate, $G_j(t)$ is the time-varying germination probability, $c_k$ are competition coefficients, and $Y_j$ is proportional to yield: the number of seeds per germinant. Defining  $C_j = C = \log\left(\sum_{k =1}^{S} c_k G_k(t) X_k(t)\right)$ and $E_j = \log(G_j(t))$, we find that $\zeta_j = -s_j$: As previously stated, the interaction effect is zero when seed survival is zero. In general, the interaction effect will vanish for any growth rate function that takes the form $r = \log(u(E) * v(C)) = \log(u(E)) + \log(v(C))$, where $u$ and $v$ are arbitrary but smooth functions.

Now, consider the slightly-modified annual plant model, where competition is a piece-wise function:

\begin{equation}
    X_j(t+1)= X_j(t) \left[ s_j (1 - G_j(t)) + G_j(t) Y_j - C_j(t)\right]
\end{equation}

\begin{equation}
C_j(t) =
\begin{cases}
        \sum_{k =1}^{S} c_k G_k(t) X_k(t) & \text{if} \; \epsilon + \sum_{k =1}^{S} c_k G_k(t) X_k(t) \leq  s_j (1 - G_j(t)) \\
        \epsilon & \text{if} \; \epsilon + \sum_{k =1}^{S} c_k G_k(t) X_k(t)  >  s_j (1 - G_j(t)). \\
\end{cases}
\end{equation}

Like the original annual plant model, the per capita yield is $G_j Y_j$ when $C_j = 0$. Unlike the original annual plant model, the finite rate of increase, $\lambda_j$, decreases linearly with competition. As competition increases further, the finite rate of increase levels-out at some small number $\epsilon$, which prevents $\log(\lambda_j) = -\infty$ (a mathematical nuisance), but may have no influence on actual dynamics if competition never increases to that point. In this modified annual plant model, the interaction effect is  $\zeta_j = Y_j$ when when there is no seedbank (i.e., $s_j = 0$). This demonstrates that stage structure is by no means necessary for the storage effect, it is just one of many ways in which an interaction effect may arise. Although the meaning of an interaction effect can be stated in abstract terms, it is difficult to explain in terms of ecological or population dynamical concepts (see Section \ref{An interaction effect between environment and competition}).

Some authors employ a more general interpretation of the storage effect based on the abstract notion of \textit{buffering}. \textcite{Chesson2004} explains: "Buffered population growth means that less is lost during unfavorable conditions (high competition and an unfavorable environment) than is gained during favorable conditions (low competition and a favorable environment)...". Buffering can be measured as minus the interaction effect between environment and competition, which is, in mathematical terms, the cross partial derivative of per capita growth rates with respect to environment and competition, evaluated around equilibrium values, i.e., $\text{buffering} =  -\zeta = - \partial^2 r(E^*,C^*) / (\partial E \partial C)$ (\cite{Chesson1985}; \cite{chesson2000mechanisms}; \cite{snyder2012storage}). Again, quotes provide evidence that this more general interpretation exists in the ecology milieu:

\begin{itemize}

    \item "\ldots there is some way to "store" the effects of good times, to get organisms through bad ones." (\cite{barabas2018chesson})

    \item "More generally, mechanisms leading a positive $\Delta I$ value involve storage of the benefits of favorable periods in the population, whether this storage can be traced to a seed bank of something else. The term storage is a metaphor for the potential for periods of strong positive growth rate that cannot be canceled by negative growth at other times." (\cite{Chesson1994})

    \item "Storage effects happen when the invader experiences low competition in favorable environments and has the ability to store that double benefit." (\cite{ete:Snyder})

    \item "However, these gains by the rare come to nothing if they are wiped out in bad years. The storage effect can therefore maintain coexistence only if species are buffered against sudden rapid declines. One natural way for this to occur is if generations overlap and established individuals are immune to the causes of temporal variation (e.g., viability selection on offspring, no selection on adults)." (\cite{messer2016can})

\end{itemize}

\begin{quote}
\textbf{The conventional interpretation storage effect v2}: "Storage" can be more generally understood as 'buffering', which is a negative interaction effect of environment and competition on per capita growth rates. This buffering helps out rare species because it prevents extreme losses when the environment is unfavorable and competition is high.
\end{quote}

The convention interpretation v2 is imprecise because the storage effect tends to decrease as invader "storage" (operationalized as buffering) increases. To see this, we derive a mathematical expression for the storage effect in the two-species lottery model (Appendix \ref{Quantifying the storage effect in the lottery model}). When species-specific responses to the environment are partially correlated (i.e., $0 < \rho < 1$), the storage effect for the species 1 is proportional to $\left[ s_2 - \rho s_1 \right]$, where $s_j$ is the adult survival probability of species $j$. As the invader's adult survival probability $s_1$ increases, the storage effect decreases. Since the invader's adult survival probability is measure of the invader's "storage" or "buffering", we see that storage can decrease the storage effect.

If species have positively correlated environmental responses, then species $i$'s storage effect will tend to increase as species $i$ experiences more buffering (i.e., as $\zeta_i$ becomes more negative). This is precisely what we expect to see in nature. After all, it is well-known that plants tend to have strong and positive responses to increases in temperature and precipitation (\cite{rosenzweig1968net}; \cite{lieth1973primary}; \cite{sala1988primary}. The probability of germination --- which is often identified as the environmental response in models of annual plants --- can display a complex interdependency on temperature and precipitation (\cite{facelli2005differences}), but nevertheless tends to increases as either abiotic variable increases (\cite{baskin1998seeds}).

Indeed, there is good empirical evidence that species do have positively correlated environmental responses. We reviewed empirical studies that explicitly attempted to quantify or provide evidence for/against the storage effect (\cite{caceres1997temporal}; \cite{venable1993diversity}; \cite{pake1995coexistence}; \cite{pake1996seed}; \cite{adler2006climate}; \cite{sears2007new}; \cite{descamps2005stable}; \cite{angert2009functional}; \cite{usinowicz2012coexistence}; \cite{facelli2005differences}; \cite{chesson2012storage}; \cite{kelly2002coexistence}; \cite{kelly2005new}; \cite{usinowicz2017temporal}; \cite{ignace2018role}; \cite{hallett2019rainfall}; \cite{armitage2019negative}; \cite{armitage2020coexistence}; \cite{zepeda2019fluctuation}; \cite{zepeda2019fluctuation}; \cite{towers2020requirements}; \cite{jiang2007temperature}; \cite{holt2014variation}; \cite{ellner2016quantify}). In the 24 studies we were able to find, there were 15 distinct communities. Of these 15 communities, $7/15\; (47\%)$ showed evidence of positive correlations in species' environmental responses, $3/15\; (20\%)$ showed zero correlation on average, $2/15\; (13\%)$ showed negative correlations, and $3/15\; (20\%)$ did not provide sufficient information to make a determination about the average sign of pairwise correlations. Two of the communities were only studied in the context of microcosm experiments. When we only consider natural communities for which sufficient information is available, $7/10\; (70\%)$ communities showed positive correlations and $2/10\; (20\%)$ show uncorrelated responses. Only $1$ community showed evidence of negative correlations, which is what is needed for buffering to help rare species. For more details on our analysis of the literature, see \textit{empirical\_E\_correlations.pdf} at \url{https://github.com/ejohnson6767/storage_effect_critique}.

To be clear, it \textit{is} true that in the lottery model (and the annual plant model), no species can have a positive storage effect if all species have zero adult (or seed) survival probability. However, \textit{this} is a community-level condition for coexistence, reflected properly in the ingredient list definition of the storage effect (Section \ref{The ingredient list definition}). The conventional interpretations v2 may be thought of as conflating the community-level condition for coexistence (i.e., some species must have some "storage" for some species to coexist via the storage effect) with a species-level condition for persistence (i.e., one species must have "storage" in order for said species' storage effect to be positive). The conflation is analogous to falsely claiming that a species can persist by strongly competing with itself, since the competitive Lotka-Volterra model shows us that coexistence occurs when intraspecific competition is greater than interspecific competition.

\subsection{The origin of the conventional interpretation}
\label{The origin of the conventional interpretation}

The storage effect was discovered by \textcite{chesson1981environmentalST} and coined by \textcite{chesson1983coexistence}, though the phenomenon had been discovered much earlier in the context of population genetics (\cite{Dempster1955};\cite{Haldane1963}; \cite{Lloyd1966}; \cite{Harper1971}; \cite{Gillespie1977}). In the 1980's, a sequence of papers (\cite{chesson1981environmentalST}; \cite{chesson1982stabilizing}; \cite{chesson1983coexistence}; \cite{chesson1984storage}; \cite{warner1985coexistence}) analyzed the two-species lottery model and repeatedly highlighted an interesting result: coexistence is not possible if both species have non-overlapping generations (i.e., the  adult survival probability equals zero). Additionally, the species with the lower average per capita fecundity cannot persist if it has non-overlapping generations. Analogous results were found for a model of annual plants (\cite{ellner1985essa}; \cite{ellner1985essb}; \cite{ellner1987alternate}; \cite{Chesson1994}): coexistence is not possible if neither species has a seed bank (i.e., the seed survival probability is zero). At first glance, these results seems to support the conventional interpretation v1 and v2.

This superficial paradox can be resolved by first recognizing that coexistence mechanisms are not independent (\cite{Song2020}; \cite{kuang2010interacting}; \cite{Yuan2015}). For example, every coexistence mechanism in the lottery model is modulated by the survival parameter. When invader survival increases, the invasion growth rate unsurprisingly increases, despite the fact that the storage effect may decrease. Now the question becomes, why did early expositions of the storage effect conflate the contemporary storage effect with the invasion growth rate?

When the storage effect was first discovered, Modern Coexistence Theory (the framework for partitioning invasion growth rates into coexistence mechanisms) had not yet been invented. The storage effect could not be separated out from other coexistence mechanisms. Consequentially, early formulae defined the storage effect as the sum of multiple coexistence mechanisms. For example, Eq.10 of \textcite{warner1985coexistence} suggests measuring the storage effect as the combined effects of relative nonlinearity (a different coexistence mechanism; \cite{armstrong1980competitive}; \cite{Chesson1994}) and the contemporary storage effect, which would indeed imply that increasing invader survival increases the storage effect (Appendix \ref{Quantifying the storage effect in the lottery model}).

\subsection{The perpetuation of the conventional interpretation}
\label{The perpetuation of the conventional interpretation}

Why is the conventional interpretation perpetuated? We have two speculative answers: 1) the conventional interpretation is "mentally sticky": the ability to persist is intuitively tied to the ability to weather bad years. 2) Coexistence is often studied in models where species behave symmetrically. In such models, the formula for the species-level storage effect (e.g., \eqref{symmetry}) increases with a shared (across species) survival parameter, thus giving the misleading impression that in general, the storage effect increases with invader survival.

It is entirely reasonable to think that if species specialize on a fluctuating environment, they must have some way to slow the exponential loss of individuals over a sequence of bad years, and that this must be particularly important for rare species, which are inherently extinction-prone (\cite{MacArthurRobertH1967Ttoi}; \cite{lande1998demographic}). However important this phenomenon might be, it is not what the storage effect is measuring. Most models used to demonstrate the storage effect feature infinite populations (an assumption made for mathematical/computational convenience), which obviates the possibility of stochastic extinction. Unless $\lambda = 0$ or $r = -\infty$ (which in most models is only possible as $n \rightarrow \infty$, a biological impossibility), extinction for infinite populations will take an infinite amount of time even without any "storage".

The notion of "waiting it out" implies that the finite rate of increase (i.e., $\lambda$, the discrete-time analogue of the per capita growth rate) is drawn from a distribution with right skew. Intuition dictates that right skew in $\lambda$ --- many slightly bad years and few extremely good years --- should increase the risk of stochastic extinction. However, using a Taylor series of $\lambda$ about its mean, one can show that right skew increases the invasion growth rate when fluctuations in $\lambda$ are small. This apparent contradiction suggests that a theory of large environmental fluctuations in finite populations is needed to understand the phenomenon which the conventional interpretation v1 points to.

Because coexistence mechanisms are defined as invader--resident comparisons (\ref{inv res comparison}), coexistence mechanisms capture both a rare-species advantage \textit{and} intrinsic differences between species (\cite{johnson2022methods}). To isolate the rare-species advantage, and to make the math easier, theoretical ecologists often study the storage effect in models where all species are identical except for their responses to the environment. Such \textit{symmetrical species} models produce formulas for the storage effect where the storage effect increases as a shared survival parameter increases (see \eqref{symmetry}). This may give the misleading impression that in general, an individual species' storage effect increases with said species' survival probability.

\section{Discussion}
\label{Discussion: Rethinking storage}

The storage effect is an important concept in community ecology, but it remains widely misunderstood. In Section \ref{An okay interpretation of the storage effect}, we gave a simple interpretation of the storage effect:  Good environments lead to high competition for common species, but less so for rare species; since high competition undermines the positive effects of a good environment, rare species are better able to take advantage of a good environment. This interpretation is a good starting point, but is lacking crucial details. It can be supplemented with the mathematical and ingredient-list definitions of the storage effect, but the math can be confusingly abstract, the ingredient list recapitulates the math, and toy models (though more concrete) can lead to confusion if one overgeneralizes from the results of particular models. In this paper, we have taken a pluralistic approach to explanation, providing many details with the hope that some will be cause the storage effect to "click" for any given reader. At the very least, we hope to give the impression that the storage effect is an intrinsically tricky concept, and therefore, that one ought to have a baseline of skepticism towards heuristic uses of the storage effect.

The conventional interpretation of the storage effect v1 first operationalizes "storage" as a robust or long-lived life stage, and then claims that such "storage" helps rare species. A number of models act as counterexamples to this claim. Further, because we measure the strength of coexistence by perturbing species to zero density, there are effectively no individuals in which years of good recruitment can be stored. While the invasion growth rate often increases with an invader's survival probability, this merely reflects lower per capita death rates, not any sort of bet-hedging strategy as implied by the conventional interpretation v1.

The conventional interpretation v2 first operationalizes "storage" as \textit{buffering} --- protection against the double-whammy of a bad environment and high competition (i.e., $\zeta_j < 0$; Section \ref{An interaction effect between environment and competition}) --- and then claims that such storage helps rare species. If a species experiences buffering, then by definition, it grows comparatively poorly in a good environment with high competition. According to our review of papers that have attempted to quantify the storage effect, species are more likely to find themselves in the good environment/high competition scenario than the good environment/low competition scenario, and thus, buffering tends to hurt rare species.

An imprecise definition or interpretation can often serve a pedagogical purpose. One example comes from \textcite{godfrey2009darwinian}: Many evolution textbooks state that there are three conditions for evolution by natural selection: phenotypic variation, heritability, and fitness differences. However, these conditions are not sufficient for evolution, since all conditions can be met, and yet allele frequencies remain stagnant in the face of stabilizing selection. Does the conventional interpretation v2 of the storage effect similarly fall into the category of imperfect-but-useful? By questioning the generality of the conventional interpretation v2, are we undermining its (implicitly pedagogical) purpose? Perhaps; buffering promotes persistence when species' responses to the environment are negatively correlated, and perhaps it is easier to think of niche differentiation (a necessary ingredient for coexistence via any mechanism) as a negative correlation between environmental responses, rather than positively but not perfectly correlated responses.

Our critique of the conventional interpretation v2 (but not v1) implicitly assumes that the contemporary mathematical definition of the storage effect is the right way to define the storage effect. Elsewhere (\cite{johnson2022methods}), we have argued that the definition is essentially correct (though the scaling factors can be problematic in certain contexts), since the invader--resident comparison captures the notion of \textit{specialization} / \textit{ecological differentiation}, a key element in historical explanations for coexistence. It is possible to devise a definition such that a species' storage effect will always increase with buffering, but such a definition would not automatically validate the conventional interpretation v2 (Appendix \ref{An alternative definition of the storage effect}).

There is a tendency for the storage effect to increase as the buffering of all species increases. This can be seen clearly in formulas for the community-average storage effect (see \eqref{community average SE lottery}, Appendix \ref{Quantifying the storage effect in the lottery model}), and the storage effect in the symmetrical case (see \eqref{symmetry}). The specious similarity between this tendency and the conventional interpretation v2 suggests that 1) the conventional interpretation v2 can be thought of as taking a community-level criterion for coexistence and mistaking it for a species-level explanation for persistence, and 2) that the storage effect is best understood in a community context. The ingredient-list definition of the storage effect (Section \ref{The ingredient list definition}) gives the community-level "conditions", in the sense that the ingredients \textit{tend} to lead to a positive (or negative) storage effect for many species.

It is possible that much of the confusion around the storage effect can be explained by a psychological tendency towards explaining coexistence in terms of how a single species is able to recover from rarity. We speculate that this tendency exists for the same reason that we use an invasion analysis: it breaks up an abstract property --- coexistence --- into a set of simpler, concrete scenarios (i.e., can a species invade in the context determined by the limiting dynamics of resident species?). When formulating heuristic explanations for coexistence, one naturally considers a scenario in which a single species is rare, and then asks what allows this species to recover from rarity. In the case of coexistence via classical mechanisms, the heuristic explanation is, "If a species were ever to become rare, the resource that it specializes on will become more abundant, thus increasing per capita growth rates". Here, the density-dependent feedback loop contains intuitive state variables (i.e., species densities and resource concentrations, specifically their mean levels) which interact in an obvious way (i.e., more resources per capita $\rightarrow$ higher per capita growth rates). The analogous heuristic explanation for coexistence via the storage effect is "If a species were ever to become rare, then the environmental states it specializes on will not lead to as much competition (as they did previously), the covariance between environment and competition will decrease, thus increasing per capita growth rates". Here, the density-dependent feedback loop contains an abstract state variable (i.e., the covariance between environment and competition) that interacts with a species' density in a non-obvious way (i.e., a negative interaction effect, deduced via a Taylor series). This explanation is unintuitive and requires an understanding of the math behind the storage effect. It is simply easier to say that rare species "store good years of recruitment" or "are buffered against unfavorable environmental conditions".

The generality of the storage effect is both a strength and weakness. It is a strength because it gives us a way to analyze, talk about, and quantify a phenomenon that occurs in disparate systems. It is a weakness because a fully-general description of the storage effect requires abstractions like "interaction effects" and "$EC$ covariance", which makes it difficult to formulate a general ecological interpretation of the storage effect --- to connect the storage effect to an ecological concept like bet-hedging, or a small set of life-history traits like age-structure or phenotypic variation. Sure, the storage effect can be associated with stage structure, or non-overlapping generations, or dormancy. But it can also be associated by none of those things, or temporal autocorrelation, or maternal effects, or whatever mechanism theorists will dream up next.

\section{Acknowledgements}
We would like to thank Karen Abbott for helpful suggestions.

\section{Appendixes}
\label{Appendixes}

\subsection{The storage effect in the lottery model}
\label{Quantifying the storage effect in the lottery model}

The lottery model with pure temporal variation (\textit{sensu} \cite{Chesson1994}) is written as
\begin{equation} \label{lottery}
   n_{j}(t+1) =  n_{j}(t) \left[ s_j + \eta_{j}(t)   \left(\rule{0cm}{1.25cm}\right. \frac{ \sum \limits_{k = 1}^{S} (1 - s_k) n_{k}(t)}{\sum \limits_{k = 1}^{S} \eta_{k}(t) n_{k}(t)} \left.\rule{0cm}{1.25cm}\right) \right], \quad j = (1, 2, \ldots, S).
\end{equation}
Note that we do not need extra equations to track larvae because they die if they are not recruited. In order to fit the lottery model into the mold of Modern Coexistence Theory, we must define the environmental parameter and the competition parameter. Selecting $E_j = \log(\eta_j)$ and $C = \log( \frac{\sum \limits_{k = 1}^{S} \eta_{k} n_{k} }{\sum \limits_{k = 1}^{S} (1-s_k) n_{k}})$, the effective per capita growth rate takes the form

\begin{equation}
   r_j(E_k,C_j) = \log(s_j + \exp{E_j - C})
\end{equation}

With the log-scale specifications of $E_j$ and $C$, all Taylor series coefficients can be expressed purely as a function of $s_j$, which leads to tidy expressions of the coexistence mechanisms. If one uses the non-log-scale specification, the results are qualitatively identical.

Next, we choose the equilibrium parameters $E_j^*$ and $C_j^*$. The shared equilibrium level of competition is defined as the temporal average of competition experienced by the invader, averaged over all species acting as the invader: $C^* = \frac{1}{S} \sum_{i=1}^{S} \overline{C_{i}}$. With this choice made, the equilibrium environmental parameters are fixed at $E_j^* = \log(1 - s_j) + C^*$.

With the equilibrium parameters defined, the partial derivatives of $g_j$ may be computed:

\begin{equation}  \label{taylor_coef}
\begin{aligned}
& \alpha_j^{(1)}  = \pdv{g_j(E_j^*, C_j^*)}{E_j} = 1-s_j\\
 & \beta_j^{(1)} = \pdv{g_j(E_j^*, C_j^*)}{C_j} = -(1-s_j)\\
 & \alpha_j^{(2)} = \pdv[2]{g_j(E_j^*, C_j^*)}{E_j} = s_j (1-s_j) \\
 & \beta_j^{(2)} = \pdv[2]{g_j(E_j^*, C_j^*)}{C_j} = s_j (1-s_j)\\
 & \zeta_j = \pdv{g(E_j^*, C_j^*)}{E_j}{C_j} = -s_j(1-s_j). \\
\end{aligned}
\end{equation}

The general definition of the scaling factors is $q_{is} = \frac{\beta_i^{(1)}}{\beta_s^{(1)}} \frac{\partial C_i}{\partial C_s}$ (see Section \ref{The mathematical definition}). In the case where species share a single competition parameter, this reduces to $q_{is} = \frac{\beta_i^{(1)}}{\beta_s^{(1)}}$.

Under the standard small-noise assumptions of Modern Coexistence Theory, the competition parameter can be approximated by a function of residents' environmental responses: $(C - C^*) \approx \sum_{s \neq i}^S \frac{\partial h(\myvect{E^*}, \myvect{n^*})}{\partial E_s} (E_s - E_s^*)$. In the lottery model with only one invader and one resident, there is a one-to-one conversion between environment and competition: $\frac{\partial  h(E_s^*, n_s^*)}{\partial E_s} = 1$.

In a two-species system with species $1$ as the invader and species $2$ as the resident, the covariance between species $1$'s environment response and competition is $\Cov{}{E_1}{C} = \Cov{}{E_1}{E_2} = \rho \sigma^2$, where $\rho$ is the correlation between the two species' responses to the environment, and $\sigma^2$ is the variance of environmental responses (shared by both species for maximum simplicity).

Utilizing the general mathematical expression of the storage effect (\eqref{SE math def}), we find that the storage effect of species $1$ is
\begin{equation} \label{SE simple}
    \Delta I_1 \approx  \sigma^2 \left( s_2(1-s_1) - \rho \; s_1(1-s_1) \right).
\end{equation}
Setting $\Delta I_1$ equal to zero and solving for $s_2$, we find that $s_2 = \rho s_1$, which has a solution when $\rho$ is positive. This shows that the storage effect can be zero even when there are species-specific responses to the environment (i.e., $\rho < 1$), subadditivity (i.e., $s_j$ > 0), and covariance between environment and competition (i.e., $\sigma^2 \neq 0$), as claimed in Section \ref{The ingredient list definition}.

Coexistence mechanisms are often divided by species' sensitivity to competition, here operationalized as $\abs{\beta_i^{(1)}}$. This scaled version of species 1's storage effect is

\begin{equation} \label{SE simple}
    \frac{\Delta I_1}{(1-s_1)} \approx  \sigma^2 \left( s_2 - \rho \; s_1 \right).
\end{equation}

Following the definitions of coexistence mechanisms (\eqref{MCT full}), the sum of the scaled storage effect and the scaled relative nonlinearity for species $1$ is
\begin{equation} \label{SE simple}
    \frac{\Delta I_1}{(1-s_1)} + \frac{\Delta N_1}{(1-s_1)} \approx \sigma^2 \, s_1(1-\rho),
\end{equation}
which clearly increases with invader survival, as claimed in the main text (Section \ref{The origin of the conventional interpretation}).

In the two-species lottery model, the scaled community-average storage effect (\eqref{community average SE}) is
\begin{equation} \label{community average SE lottery}
   \overline{ \left(\frac{\Delta I}{1-s} \right)} = \frac{\sigma^2}{2}(s_1 + s_2)(1-\rho),
\end{equation}
which demonstrates that at the community-level, survival increases the (community-average) storage effect, as claimed in Section \ref{Discussion: Rethinking storage}.

\subsection{An alternative definition of the storage effect}
\label{An alternative definition of the storage effect}

An alternative definition of the storage effect could use an \textit{invader--invader comparison} (\cite{johnson2022methods}). Instead of comparing an invader to residents (as in \eqref{inv res comparison}), we could compare a single focal species to itself at high versus low density. Using this alternative definition of the storage effect in the two species lottery model, we would find that the storage effect increases with adult survival: when going from the high-density state to the low-density state, the $EC$ covariance decreases (if there are species-specific responses to the environment) but the survival parameter is unchanged.

There are problems with the invader--invader comparison that make it a poor method for defining coexistence mechanisms generally: it is not defined when the invasion growth rate is negative (a community with the focal species at high density cannot be prepared for the purpose of the comparison), and it does not necessarily capture the notion of specialization/differentiation. However, even if these problems did not exist, the invader--invader comparison would not validate the conventional interpretation v2. The invader--invader comparison does not say that buffering helps rare species. Rather, the invader--invader comparison shows that buffering helps a rare species relative to said species at high density: it is as much about self-limitation when abundant as it is about recovery when rare.

\section{References}
\printbibliography

\end{document}

\subsection{Coexistence mechanisms as invader--invader comparisons}
\label{Coexistence mechanisms as invader--invader comparisons}

TBD

does not describe what is necessary or sufficient for the storage effect, as prvious papers have claimed. Rather, the ingredient-list definition

Fin

If asked to explain coexistence in resource-consumer systems, one might reasonably say "If a species were ever to become rare, the resource that it specializes on will become more abundant, thus increasing per capita growth rates". If asked to explain coexistence via the storage effect, one considers a scenario in which a species is rare, and then asks what allows this species to recover from rarity. In the case of the storage effect, it is slightly mind-bending to say "If a species were ever to become rare, then if the species specializes on environmental states, EC covariance will go decrease, thus increasing per capita growth rates (assuming subadditivity)".

Consider Tilman's heuristic explanation for coexistence in resource consumer systems, which states tat each species must consume proportionately more the of the resource that most limits its own growth. The phrase "proportionately more" indicates that coexistence can only be understood by comparing species, and indeed, this is born out in the math, and in Tilman's graphical theory - what matters is the angle formed by multiple species' impact vectors. Yet, if asked for a more mechanistic explanation of coexistence in resource-consumer systems, one might reasonably say "If a species were ever to become rare, the resource that it specializes on will become more abundant, thus increasing per capita growth rates". Here, the notion of specialization

but there we are also considering a situation where

Now we are forced to consider a concrete situation (one species is rare) rather than an abstract property (coexistence), and the imagination fills the gaps. Reality, which is simply that the invader suffers less from EC covariance than the resident, is not nearly as exciting as "storage" or "buffering"

On the flip side, we explain coexistence with an \textit{invader--invader comparison}, wherein a focal species is compared to itself at high and low density. Consider the following explanation for coexistence via resource partitioning: "If a species were ever to become rare, the resource that it specializes on will become more abundant, thus increasing per capita growth rates". The phrase "to become rare" betrays an implicit comparison between the high density and low density states of a focal species (not a comparison between the high density and low density states of different species, an invader--resident comparison). The discoverer of the storage effect, Peter Chesson, has alluded to this psychological tendency: "Often the mechanism is most easily understood in terms of how the conditions encountered by an individual species change between its resident and invader states." (\cite{Chesson2008}), and "...within-species comparison is more reliable if appropriate within-species resident and invaders states can be prepared" (\cite{chesson2013species}).

Paragraph about how if we can get comfortable with the idea of specializing on resource variation, we can get comfortable with the idea of specializing on EC covariance.

This bias exists for the same reason that invasion analysis exists - it breaks

Much of the confusion around the storage effect - in particular, the prevalence of the conventional interpretations (v1 and v2) -  can be traced to the fact that we typically measure coexistence differently than we explain coexistence. We measure coexistence with an invasion analysis, which transforms a complex problem (can they coexist?) into a set of simpler problems (can this species invade, in this specific context determined by resident densities?). In Modern Coexistence Theory, invasion growth rates are expressed as a series of invader--resident comparisons. This makes intuitive sense, seeing that the invader's environment is determined by the resident species. However, upon deeper reflection, the invader--resident appears flawed: the essence of coexistence is a rare-species advantage, but the invader--resident comparison mixes both genuine rare-species advantages and intrinsic (i.e., density-independent) differences between species. Although the partition of the invasion growth rate contains a term called \textit{density-independent effects} (see \eqref{MCT full}), this term does not perfectly capture \textit{all} intrinsic differences between species.

On the flip side, we explain coexistence with an \textit{invader--invader comparison}, wherein a focal species is compared to itself at high and low density. Consider the following explanation for coexistence via resource partitioning: "If a species were ever to become rare, the resource that it specializes on will become more abundant, thus increasing per capita growth rates". The phrase "to become rare" betrays an implicit comparison between the high density and low density states of a focal species (not a comparison between the high density and low density states of different species, an invader--resident comparison). The discoverer of the storage effect, Peter Chesson, has alluded to this psychological tendency: "Often the mechanism is most easily understood in terms of how the conditions encountered by an individual species change between its resident and invader states." (\cite{Chesson2008}), and "...within-species comparison is more reliable if appropriate within-species resident and invaders states can be prepared" (\cite{Chesson2013}).

By focusing entirely on the invader species, both conventional interpretations implicitly endorse the invader--invader comparison. It is no surprise then, that there is a disconnect between the conventional interpretations and the mathematical definition of the storage effect (which is based on an invader--resident comparison). If one computes the storage effect using an invader--invader comparison, one finds that a species' storage effect always increases as the focal species' "storage" increases (we perform such a computation in Appendix \ref{Coexistence mechansisms as invader--invader comparisons}). While this "invader--invader storage effect" may be a desirable and interpretable quantity, it does not usually admit analytical expressions (see Appendix \ref{Coexistence mechansisms as invader--invader comparisons}). Even further, it is not possible to compute when there is no "stable" high-density invader state, as is the case when the focal species cannot persist, or only becomes temporarily abundant throughout a heteroclynic cycle. The invader--invader comparison is underdetermined when there are multiple stable states with the focal species at high densities. In light of these problems, we recommend using the invader--resident comparison. Elsewhere (Johnson, 2021 \textit{in prep}) we have argued that "speed conversions factors" can be used to make the resident species behave more similarly to the invader species, thus approximating an invader--invader comparison with an invader--resident comparison. The aforementioned "speed conversion factors" replace the "scaling factors" (\cite{barabas2018chesson}) of previous iterations of coexistence theory.

The generality of the storage effect is undoubtedly a strength. It gives us a way to analyze, talk about, and quantify a phenomenon that occurs is disparate systems. However, the generality of storage effect can only be achieved through abstractions; through concepts like "interaction effects", "environmental niche differences", "causation", "density-dependent factors", and "density independent factors". Therefore, the generality of the storage effect is also a weakness because the necessary abstractness makes it difficult to formulate a robust ecological interpretation. Sure, the storage effect can be caused by stage structure, or non-overlapping generations, or dormancy. But it can also be caused by none of those things, or temporal autocorrelation, or transgenerational plasticity, or whatever mechanism theoretical ecologists dream up next.

Like the original annual plant model, the per capita yield is $G_j Y_j$ when $C_j = 0$. Unlike the original annual plant model (where per capita yield decreases to 0 as $C$ increases), the per capita yield asymptotes at $G_j Y_j - \gamma_j$ as $C$ goes to infinity. This limit on density dependence is perhaps more realistic, reflecting the fact that competition is partly \textit{contest competition} (cite mittlebach) and that some individuals, by virtue of good genes or pure luck, are functionally insensitive to competition (cite snyder?). In this modified annual plant model,  the interaction effect is  $\zeta_j = $. Note that when there is no seedbank (i.e., $s_j = 0$), the interaction effect is negative. This demonstrates that stage structure is by no means necessary for the storage effect, it is just one of many ways in which an interaction effect may arise. Although the meaning of an interaction effect can be stated in abstract terms, it is difficult to explain in terms of ecological or population dynamical concepts (see Section \ref{An interaction effect between environment and competition}).

In the simple models have suggested alternative schema for re-scaling resident growth rates. The results in this paper do not qualitatively  scaling factors, so we will use Chesson's original scaling factors for the sake of convention.
A full discussion scaling factors is beyond the scope of this paper (but we refer interested readers to \cite{barabas2018chesson}; \cite{Chesson2019}; Johnson 2021 \textit{in prep}).

The invader--resident comparison helps us measure a \textit{rare-species advantage that results from specialization}. The \textit{rare-species advantage} is simply the invasion growth rate; the residents' long-term average growth rates are necessarily zero, so an positive invasion growth rate corresponds to a rare-species advantage. The notion of \textit{specialization} comes from the invader--resident comparison, since specialization or differentiation will tend to help species when they become rare.

But what is the biological meaning of the an interaction effect?
The second ingredient, \textit{an interaction effect between environment and competition}, can be explained via analogy with interaction effects in regression models. However, even when we have a particular model where $E_j$ and $C_j$ have concrete meanings, it is difficult to interpret the interaction effect in terms life history characteristics.

Similarly, the spatial storage effect (\cite{chesson2000general}) can be generated in models without any age or stage structure if the environment is spatially heterogeneous (\cite{Chesson2000}; \cite{chesson2005scale}; \cite{Chesson2012}).

Elsewhere (\cite{JohnsonScalingFactors}), we have argued that all coexistence mechanisms (including the storage effect) should measure the rare-species advantage that results from specialization/differentiation, and that the invader--resident comparison (unlike the invader--invader comparison) captures this notion of specialization/differentiation. We have also argued that resident growth rates should be scaled by scaled by a ratio of generation times, since this corrects for between-species differences in population dynamical speed, a difference that can greatly modulate coexistence mechanisms but does not typically constitute a trade-off that would lead to coexistence. In the lottery model with two species or symmetric species (two models we have analyzed here, see \eqref{??}, \eqref{??}) the conventional scaling factors of Modern Coexistence Theory are precisely the ratio of generation times. For these reasons, we treat the mathematical definition of the storage effect as fundamental.

In our view, the storage effect is best understood in a holistic manner (Section \ref{A crash course in the storage effect}) -- by first understanding the connection between the mathematical definition and ingredient-list definition of the storage effect, and then understanding how the ingredients manifest is particular models.

each year (represented by a single time-step) starts out with a spawning event. Per capita larval production is denoted by $\beta_j(t)$, where $j$ is the species index. Fecundity fluctuates from year-to-year, hence the dependence on year $t$. After spawning, adult fish survive with the density-independent probability $s_j$; conversely, adult fish die with probability $1-s_j$. Finally, the larvae inherit the empty territories, with a recruitment probability for any given larva being equal to the number of empty sites, divided by the total number of larvae. If there are $S$ species, the dynamics of population density, $n_j$, can be encoded in a $S$-dimensional difference equation:

\begin{equation}
   \lambda_{j}(t) = \frac{n_j(t+1)}{n_j(t)} = s_j + \eta_{j}(t) \left( \frac{\sum \limits_{k = 1}^{S} (1-s_k) n_{k}(t)}{\sum \limits_{k = 1}^{S} \eta_{k}(t) n_{k}(t)} \right).
\end{equation}

This is likely due to the fact that ecologists study the coexistence of species with similar life histories, since theory putative coexistence is more surprising (and thus, worth explaining) in light of the competitive exclusion principle.

Generalizing to the multi-species case involves redefining the $h$ so that competition is a function of all residents' environmental responses, $\overrightarrow{E}$, and population densities, $\overrightarrow{E}$. In the symmetric species case, we define $\theta = \frac{1}{S-1} \sum_{s \neq i}^S \frac{\partial h(\overrightarrow{E^*}, \overrightarrow{n^*})}{\partial E_s}$.

Plugging these covariance approximations into the definition of the storage effect and assuming that species have identical $\zeta_j$, $\sigma_j$, $\beta_j^{(1)}$ (which implies that $q_{is} = 1$) and $\frac{\partial h(E_2^*, n_2^*)}{\partial E_j}$, we arrive at \eqref{symmetry} above. If $\zeta_i$ is negative (which indicates \textit{buffering}), then the storage effect is proportional to $(1-\rho)$, demonstrating that (all else being equal) the storage effect increases as niche differences increase.