Scale transition theory with special reference to species coexistence in a variable environment

Abstract

Scale transition theory is a mathematical technique for understanding changes in population dynamics with changes in spatial or temporal scale. It explains the emergence of new properties on large scales from the interaction between nonlinearities and variation on small scales. It applies statistical theory for averaging nonlinear functions to understanding this interaction. The fundamental concepts are most easily illustrated with reference to spatial models where state variables on larger spatial scales are simply defined as averages of those on smaller scales. Scale transition theory also explains the conceptually difficult topic of how species coexistence arises from temporal fluctuations. In this case, averages of per capita growth rates over time define long-term population trends and outcomes, and these averages are critically affected by interactions between nonlinear dynamics and temporal variation in state variables and environmental variables. Two general mechanisms of species coexistence, the storage effect and relatively nonlinear competitive variance, emerge.

Keywords:

• spatial scale
• temporal scale
• variable environment
• species coexistence
• scaling up

1. Introduction

There are many different approaches to the development of theory in ecology, but there is too little interplay between these approaches. Researchers from different traditions may separately make progress on common problems with little knowledge of contributions from other traditions. Sometimes, the approach of another tradition seems so alien that its contributions are unclear. For example, those coming from a tradition in dynamical systems theory might not be so familiar with approaches coming from a tradition of statistics and probability theory, and so be uncertain as to what those approaches achieve for a problem of interest. In this article, developments in ecological dynamics from a probability and statistics tradition are reviewed for general mathematically and ecologically literate audiences. The hope is to cross some of the boundaries that impede communication between workers from different traditions who address similar problems of ecological dynamics.

Key questions in the study of population and community dynamics are how populations persist, how they fluctuate over time, and how interacting species coexist. Of concern is how the rules of population dynamics change when the spatial or temporal scale under consideration is changed. This leads to the phenomenon of emergence: rules might be formulated on one scale, say a day or year, but we seek the outcome over a year, century, or millennium 14. We might model the interactions between organisms on a small spatial scale, e.g. centimetres, where these interactions have direct and immediate effects, but we wish to understand the ultimate outcomes of these interactions on a much larger area, for example, a hectare, a whole valley, or an entire geographic province. What appears to be happening on a small scale may be very different from the outcome on a larger scale: rules for population dynamics can be scale-dependent 3, 18, 22, 47. In each of these examples, fundamental to the emergence of different rules on larger scales is variation from unit to unit on small scales. If there is no variation, there is just one rule and it applies on all scales. The question becomes, how does variation from unit to unit on one scale combine to give a different outcome on a larger scale? Nonlinearity has a critical role. If the rules are defined by linear functions, then the large scale rule is the same as the small scale rule 18. Nonlinearities lead to different emergent outcomes on different scales. Ecological theory comes from understanding this process. Theory has the property of generality: it is not the result of a particular model but applies to relationships found broadly, for example, in families of models 39. For its application in ecology, scale transition theory serves two other functions: the bridge between simulation and analytical models 37, 56, and the ability to test the theory in nature 12.

2. Scaling up

How do we formulate relationships on different scales? We answer this first for spatial scale. In ecology, population size is commonly measured as density: numbers of individual organisms of a given species per unit area. If we measure density in a small area, say 1 m², and call that N _x , where the variable x labels the particular square metre, then the density in those same units over a hectare or any other area, is just the average of N _x over x, a result 8 denoted by N¯. The dynamics of N _x in discrete time, considering only the local scale, might be given as

for some function f. The higher scale dynamic might involve movement between localities, but provided there is no mortality during that movement (or that mortality is already accounted for in $f$), then the higher scale outcome is simply the average over the lower scale outcome 8:

where the quantity on the right is the average over x (spatial location) of the quantity f(N _{x, t} ). To examine the properties of this average, suppose f is a quadratic function,

Then

where is the variance in space of N _{x, t} . The linear component of the function is unchanged by the averaging, but the quadratic component adds the term due to the general formula,

Without the quadratic term, we would simply have

and the population dynamic formula for the large scale would be identical to that on the small scale. The quadratic term, however, leads to a large-scale outcome that differs from the small-scale prediction by a term that depends on the variance from unit to unit on the small scale:

The average of a nonlinear function is generally different from the nonlinear function of the average, and when the nonlinearity is of the quadratic form, the exact deviation can be expressed in terms of variances. The fundamental mathematical concept here is the statistical inequality called Jensen's inequality 49, 50, which applies to the basic kinds of nonlinear functions: those that are concave up, or concave down. Simply stated, Jensen's inequality says that the mean, , of a concave-up function, f(N), is greater than the function of the mean, f(N¯). A concave-up function has a positive second derivative, i.e. positive acceleration. This acceleration boosts above f(N¯) as actual N values deviate from N¯. Here, a positive c gives a concave-up function, but a negative c gives a concave-down function, which simply reverses the inequality. The simplest instance of Jensen's inequality is given by Equation (5), above, viz the average of a square. This particular formula allows quadratic approximations to nonlinear functions to estimate changes in dynamics with changes of scale.

From a strictly computational perspective, expression (7) is unsatisfactory without a formula for . Viewed from a theoretical perspective, however, the equation demonstrates precisely where the phenomenon of emergence enters: through variation in space and the nonlinearity jointly. Formula (7) can also be used to determine something about the result of the interaction between variation and nonlinearity. For instance, if N¯ converges on an equilibrium with c negative, then the equilibrium in the presence of spatial variation is necessarily smaller 5, 9. Various approaches can be taken to understand the dynamics of the variance itself to determine fully the properties of Equation (7) 3 8 10 38,54–56. These will generally involve the dynamics of migration. An elementary yet instructive example is obtained when the distribution of N _x in space is determined solely by some spatially varying property of the physical environment 8 18. A spatial location x is assigned a relative attractiveness value U _x . Assuming U _x has spatial mean 1, dispersal leads to . If the statistical distribution of attractiveness is not time dependent, we write where θ is a positive constant. Thus . Substituting in Equation (7) we obtain the closed dynamical equation for the larger scale,

The full effect of variance in space is now apparent. If c is negative, indicating local-scale density-dependent regulation, this formula shows that variance in space has created stronger density regulation: the negative constant c has been increased in absolute magnitude by the proportion 1+θ². This is Lloyd's insight in the concept of mean crowding 41, revealed in a recent field study 46. Changes in the strength of density dependence due to this interaction between variance and nonlinearity can also change chaotic or cyclic dynamics to stable dynamics on the large scale 8 21.

Equations for interacting species naturally have multi-dimensional nonlinearities because they have more than one state variable. Multidimensional nonlinearities can be important also in single-species models when the effects of the physical environment on local population growth are considered. Such effects can be included by having a population parameter depend on physical location. Local dynamics then take the form

where the parameter b _x accounts for the physical environment at location x. Multi-dimensional nonlinearities are distinct from one-dimensional nonlinearities when they specify interactions between variables. The simplest of these is a product: b _x N _{x, t} , i.e. a bilinear function, obtained by replacing bN _{x, t} by b _x N _{x, t} in Equation (3). Averaging in space to obtain the higher scale dynamic,

The product nonlinearity has introduced a covariance term because the average of a product is the product of the average plus the covariance. This particular covariance can have major effects: when dispersal is limited, i.e. a fraction of a population does not disperse, or dispersal is short range, populations build up in favourable locations (fitness-density covariance 18), which can greatly enhance population persistence through a positive contribution of to population growth. It can also have a major role in species coexistence 3 18 42 55.

These are the basics of scale transition theory illustrated in a discrete-time, discrete-space setting. Generalizations include both continuous time 3 18 46, and continuous space 3 56. We now use these basic concepts to understand temporal variation and the coexistence of interacting species.

3. Scaling up temporally

How does variation on one temporal scale alter outcomes on larger temporal scales? We can address this issue in deterministic models whose state variables or parameters vary over time, and we can address this also in stochastic models. The stochastic models that we consider, however, are not the traditional stochastic population models whose state spaces are the positive integers 36, but models where the state of an individual species is a nonnegative real number, representing its abundance 13. The stochastic phenomena driving these models are stochastic variation in the parameters meant to represent stochastic temporal variation in the physical environment. In contrast, the stochastic phenomena in traditional stochastic population models represent chance events occurring independently in the lives of individual organisms, termed demographic stochasticity 45 or within-individual variability 4. Within-individual variability is believed to be a much less important component of temporal fluctuations in large populations than environmental variability 4 45, and so it is the latter that we emphasize here. We consider populations at low density, but not low absolute size.

In traditional stochastic population models, in the absence of immigration, 0 is an absorbing state accessible from all other states. In the absence of explosion to infinity, 0 is reached in finite time 4. In contrast, in continuous state-space models driven by environmental variation, population densities cannot reach zero in finite time, but may asymptotically approach it like deterministic models. If they do not asymptotically approach 0, then the likely outcome is that their probability distributions converge on stationary distributions on the positive real numbers. These stationary distributions are the counterparts of attractors on the positive real numbers of deterministic models 13 25 27. These facts allow population growth rates to characterize population dynamics, enabling the approach to temporal scale transitions discussed here.

In studying temporal scale transitions, we focus not on state variables defined on particular temporal scales but on population per capita growth rates, how they are defined on different scales, and how the outcomes they predict depend on interactions between variation over time and nonlinearities in population growth. In continuous time, the per capita growth rate is

which has the important property

or equivalently

Here r¯ is the average of r(t) over the interval 0 to T, i.e. the integral in Equation (12) divided by T. For understanding temporal dynamics, r(t) provides the key quantity on the instantaneous timescale, while r¯ provides the appropriate measure on longer timescales. This is true by Equation (13), which shows that r¯ gives actual outcomes on longer timescales, which is what we seek. For the discrete time models, the appropriate definition of the per capita growth rate is

which is fully consistent with continuous time formulations because it is just the continuous time r(t) integrated over one time unit, preserving Equations (12) and (13).

We ask how nonlinearity of r(t), and variation over time in r(t), affect the outcome on a larger scale, i.e., affect r¯ for that scale. The focus of our questions is species coexistence, and this greatly restricts the circumstances for which we need to evaluate r¯. For these questions, we wish to know if a species will recover from low density if perturbed there or continue to decline and become extinct. We will compare r¯ for a species in two different states: an invader state, and a resident state. A resident state is where the species is long-term persistent 13. For a deterministic model, this will mean that the system has converged on an attractor, with species densities bounded away from 0 2. In a stochastic model, we assume instead that the system has converged on a stationary probability distribution with all densities positive 13 27. Equation (13) implies that the value of r¯ for a resident species necessarily converges on 0 as T→∞, because a resident species can have no long-term trend. Intuitively, an invader is thought of as a species perturbed to very low density, but mathematically, its density is set to 0. Ordinarily, r(t) is still defined, as is its average over time. The other species in the system are assumed to have achieved the resident state in the absence of the invader. The quantity r¯ for an invader thus defines the growth trend from low density.

The sign of r¯ for each species in the invader state is critically important to their coexistence. Both the permanence criterion for deterministic models 28, and the stochastic boundedness criterion for stochastic models 13 27, require r¯ to be positive for each species as the invader for species coexistence. Essentially, they require each species to be able to recover from low density in the presence of the others and so avoid extinction. Various side conditions are required to turn this necessary condition into a sufficient condition, but these side conditions will not concern us here. We focus instead on this highly intuitive necessary condition, assuming that the side conditions are satisfied. Thus, our concern is with the behaviour of r¯, and we seek its value in the limit as T→∞.

This approach ignores Allee effects, resulting from disruption of social structure, or similar effects, at low density 57, which might cause population growth rates to decline as densities become very low. Allee effects complicate the concepts of persistence and coexistence 13 52. However, with few exceptions 31 32, species coexistence studies have assumed that Allee effects can be ignored. Despite recent concerns 57, no major body of work on species coexistence accounts for Allee effects. Although the approach followed here might be modified to incorporate Allee effects, this has not yet been done.

3.1. Relatively nonlinear competitive variance (‘relative nonlinearity’)

To study the effects of nonlinearity and variation on r¯, we consider a model where each species is affected by a common factor of magnitude F(t) varying over time and regulating their densities. We think of this factor as measuring the stress faced by all species. For example, F(t) might represent resource shortage, and thus be of the form , in other words, the difference between maximum resource availability and actual resource availability at time t. Or F(t) might the density of a predator. To regulate abundance, F(t) must tend to increase as the species become more abundant, thus reducing their population growth rates.

In general, it possible for F(t) to be multidimensional, but the unidimensional case has special importance for then only one species would persist in the absence of temporal fluctuations 2. The key question is whether temporal fluctuations permit a diversity of species. Thus, we focus on the case of a unidimensional F, indicating later the potential extensions to multidimensional F. However, a multidimensional F can be reduced to a unidimensional F when all species are affected in the same way by each of the components of the multidimensional F 15 37, again posing difficulties for species coexistence in the absence of temporal fluctuations.

We refer to the species in question as ‘focal species’ to distinguish them from resource species or predators that may contribute to F. We assume also that the species may depend on an environmental variable, E _j (t) that fluctuates over time. Thus, the growth rate of species j in a multi-species community might then be

where E _j (t) is assumed to have long-term average Ē _j , and ϕ _j is some monotone increasing but nonlinear function of F(t). Under this model, the focal species are affected by environmental factors that are not themselves affected by the focal species, and by the factor F which is affected by the focal species. The quantity F is a density-dependent factor because it results in feed back loops from focal species densities to focal species densities. In a complete model there is an equation defining the dynamics of F(t). However, we are not concerned with what that equation is. Instead, our concern is directly how the fluctuations in F(t) over time affect species coexistence.

Evaluating r¯ means calculating

From Jensen's inequality, we know that will deviate from in a direction that depends on whether ϕ _j is concave up or down. In this sense we can think of a species with a concave down ϕ as benefiting from fluctuations in F about its mean relative to a species with a concave up ϕ. Importantly, these fluctuations, along with F¯, are affected by the resident-invader states of the species. This fact leads to a coexistence mechanism.

To see how this coexistence mechanism arises, we approximate ϕ _j by a quadratic function about a fixed value F* of F, using a second order Taylor expansion 7. The first derivative of ϕ _j at F* is denoted by β _j and can be regarded as a timescale adjustment for species j because it measures the rate at which species j responds to changes in the common limiting factor F 12. Nonlinearity is measured as τ _j , which is half the second derivative of ϕ _j divided by β _j . The Taylor expansion to the adjusted growth rate, r _j /β _j , is thus

The first term here on the RHS has a time average,

To avoid endless discussion of neutral cases that have little relevance to the question of when r¯ is positive, we assume that no two species have the same value of μ.

If we now compare these adjusted growth rates for two species labelled i and s, we obtain

The linear term (F−F*), has vanished, with important consequences. First of all, if the ϕ’s are linear, or the τ’s equal (relative linearity), then the quadratic term in Equation (19) vanishes too, and the equation is exact. This means that relative changes in these scaled growth rates are driven merely by relative fluctuations in the environmental-dependent growth rate terms E, which have no feedback to focal species density. Moreover, we have to conclude that

Because any species persisting in the long run must have an r¯ of 0, we see that it is impossible for both species i and s to persist while they have different μs. Fundamentally, the species with the largest value of μ must drive the others extinct regardless of the number of species in the system 15. Similarly, if fluctuations disappear as T→∞, Equation (19) reduces to Equation (20) by choosing F* to be the limiting value of F. Again only one species persists in the long run. In both the relatively linear case and the asymptotically constant case, the persistent species is the one with largest value of μ. Moreover, in the linear and asymptotically constant cases, that species is the one for which the solution gives the largest F*. Depending on what F actually measures, this would be the species that can tolerate the highest resource shortage, or the highest predator abundance, reproducing the classical R* and P* rules respectively 30 58.

This analysis implies that any hope of coexistence must stem from the joint action of relative nonlinearity and persistent fluctuations in F. These fluctuations might be strictly endogenous, or might involve an interaction between the nonlinear dynamics of F, and the exogenous forcing from temporal variation represented in the E _j . Assuming now that species i is in the invader state, and species s is in the resident state, r¯ _s is necessarily 0, and the long-term average of expression (19) is

For coexistence, expression (21) needs to be positive for each species as invader. Critical to this outcome is the fact that will differ depending on which species is in the invasion state, as it clearly can only reflect the dynamics of the resident species. To indicate this, we add the superscript {−i} to F to indicate that it depends on the full set of focal species except species i.

To obtain useful results, we need a number of technical assumptions, justified by the general development of Chesson 7. First, we assume F* is fixed independently of which species is in the invasion state, and that a small parameter σ² characterizes the magnitude of fluctuations in F. Hence . So that no terms dominate in Equation (21), we assume that . To ensure adequate approximations, we need to restrict the previously arbitrary F* to within O(σ²) of F¯, regardless of which species is resident. Such a choice is possible when fluctuations are environmentally driven 7, or arise from a Hopf bifurcation 44. It follows that , i.e., that is essentially the variance of F. Moreover, this constraint on F* makes the outcome to o(σ²) independent of the actual value of F* chosen. We can now interpret the approximation sign ≈ in expression (21) to mean ‘equal with an error o(σ²)’. Requiring Equation (21) to be positive for each of two species (labelled 1 and 2) when in the invasion state leads to the following coexistence condition,

where we have arbitrarily assumed that is positive.

Condition (22), says a number of things about coexistence. First of all, because variances are nonnegative, the ratio of the μ and τ differences must be positive. Thus, the species with the larger value of τ (more concave up or less concave down) must be the species with the larger μ. That is, it must be the species that would win in the interaction without fluctuations present. Second, that same species must experience smaller variance as invader, i.e. as the resident it must cause larger variance in F. Because the value of μ is linear in Ē, which is independent of ϕ, the value of μ₁−μ₂ can be varied essentially independently of the other quantities within the order of magnitude constraints assumed here. The other quantities in condition (22) then define the actual range of μ₁−μ₂ compatible with coexistence.

The requirement that the species with the larger τ must be associated with larger variance as resident tends to occur naturally, because a nonlinearity giving a large positive τ tends to generate fluctuations 2 17. Of course, the actual equation governing the dynamics of F is critical. For example, if F represents shortage of a biological resource, then a large positive value of τ comes from a type II functional response, which might cause a large limit cycle with large resource fluctuations 2. However, fluctuations in the physical environment, i.e., temporal fluctuations in the E _j , have a role too. Different species might be differentially sensitive to changes in the physical environment, and the resident with larger sensitivity (larger variance in E) might generate more variance in F, overriding or reinforcing any tendency coming from τ. In particular, condition (22) with the variances switched might apply, in which case founder control would occur: both species would have negative r¯s as invader, and so each would exclude the other when resident.

The accuracy of this analysis depends on the ability of quadratic functions to adequately approximate the functions ϕ over the range of variation in F applicable in a given setting, a property relatively easily checked. It is important to note that the results of the quadratic approximation are asymptotically correct for any continuously twice differentiable ϕ as the fluctuations in F become small. They, thus provide a general guide to the possibilities for species coexistence when temporal fluctuations enter a model. Although the results will not be quantitatively accurate when the ϕ’s are substantially nonquadratic over the relevant range of variation, the results of this quadratic analysis may continue to provide qualitative explanatory power beyond their range of quantitative accuracy 37. If the assumption that μ differences are O(σ²) is violated, then the μ-difference in Equation (21) dominates the other terms, meaning that simple equilibrium analysis is sufficient. So this assumption is not a genuine restriction.

Condition (22) precludes more than two species from coexisting 7. Opportunities for coexistence of more than two species, therefore, depend on conditions when quadratic analysis breaks down, i.e. when higher order nonlinearities are important. To my knowledge there are no developed examples of such cases in the literature for one-dimensional F. Broad opportunities for coexistence of more than two species as a consequence of relative nonlinearity do occur when there are more than two competitive factors. The quadratic analysis extends to multifactor situations in a relatively straightforward manner to provide a direct analogue of Equation (21) for the invader growth rate in the multifactor case 7. It has, however, received only limited study, and there are no worked examples of the application of the formula. As the number of quadratic nonlinearities increases with the square of the number of factors, and each nonlinearity has the potential to allow an additional species to persist in a community 40, this mechanism is rich with unexplored potential. There is one clear example in the literature that illustrates the potential of this mechanism. It is a study of phytoplankton species competing for several nutrient resources 34, which demonstrated up to nine species coexisting on as few as three resources 33, but this study did not make use of the multifactor generalizations of the equations present here to understand this outcome.

3.2. The storage effect

In what ways can temporal fluctuations in the physical environment promote species coexistence? In the models reviewed here, the physical environment is represented implicitly through the population parameter E, expressing the response of the species to environmental conditions. Examples are the direct growth responses of species to weather or climate, season within a year, or the cycle of day and night. The growth rates of all organisms fluctuate in response to varying environmental conditions, which can be extreme when events like fire, hurricanes, and floods are considered. Although temporal variation in the parameter E has the potential to contribute to coexistence by the relative nonlinearity mechanism, as discussed above, a realistic modification of the model (15) introduces a more powerful mechanism, the storage effect 7.

The way environmental fluctuations enter a model has a major role in the effects that they have. The storage effect occurs when environmental fluctuations are closely linked to fluctuations in resource consumption or exposure to predators. Such linkage is likely because many species vary their activity patterns depending on the physical environmental conditions. Thus, the extent to which they actively seek resources, or are exposed to predators, varies greatly over time 16 17. To model these effects, Equation (15) is replaced with the equation

where a _j (t) and E _j (t) may both vary with the environment, but the variation of most significance to the storage effect is the variation in E _j (t) because E _j (t) modifies the effect of F(t), representing changes in resource consumption or exposure to predators. To make the point stronger, Equation (23) assumes that the ϕ’s are the same function for all the species, eliminating relative nonlinearity as a possible mechanism. In Equation (23) species differ from one another at any one time due to the slope of r _j as a function of the limiting factor F. These slope differences are consistent across all values of F, but change with time. In contrast, in the model (15) the critical issue is the fact that the rate of change in slope, as F changes, is different for different species; in other words, the graphs of r against F for different species have different shapes.

If the density-dependent limiting factor F reflects resource shortage, we can interpret the quantity ϕ(F) as the effect of competition on the growth rates of the species. If, instead, F reflects predators, the corresponding concept for density-dependent predation, termed apparent competition 29, applies. For simplicity, therefore, we use the natural notation C for ϕ(F) (‘competition’) so that Equation (23) can be written in the shorthand

We assume ergodicity: the environmental responses E _j and a _j have well-defined averages in the limit as time tends to infinity. The response E _j will also be assumed to be positive. The important time average of the growth rate is

where the product nonlinearity in E _j C has introduced the covariance, and therefore an effect of temporal fluctuations on long-term growth. Like the study of relative nonlinearity, it is helpful to make a timescale adjustment to the average growth rate. We use here average slope, Ē, defining the average rate of response to the common competition. The quantity μ is then defined as ā/Ē, and then subtracting the zero adjusted growth rate, , of a resident species from that of an invader, with , yields

Mean competition, C¯, vanishes from the equation, and we are left with a long-term invader growth rate that consists of a term reflecting relative performance in the mean environment, plus a resident-invader comparison of covariance over time between environment and competition. This covariance comparison is the only part of the equation dependent on fluctuations. Without fluctuations, these covariances are zero, and only one species would persist, the one with the largest μ value, as we saw above in the discussion of relative nonlinearity. The most important point is that in the presence of fluctuations, the covariance comparison would often be positive, and if large enough, would make the invader growth rate positive overall, counteracting a negative μ _i −μ _s term. Indeed, n species coexistence is possible 7.

The way the covariance difference becomes positive is intuitive: For the resident, higher activity in consuming resources (higher E), increases resource shortage. Similarly, with apparent competition higher activity allows predator build-up, and increases mortality from predation. Thus, we expect the resident covariance to be positive. However, since the invader is at zero density, it has no direct influence on C, and so is only nonzero to the extent that E _i is correlated with E _s over time, making a positive covariance difference likely. This fact is simplest to appreciate in a simple discrete-time model.

3.3. An example

A discrete-time version of multispecies competition is obtained from the model of Schoener 51. A resource, R, on which the species depend, is supplied to the system continuously at the rate K, and is lost at a rate proportional to its abundance. The proportionality constant can be set equal to 1 by the choice of time units. An individual of species j consumes the resource at the rate . Thus, the total resource loss rate is . Assuming that resource dynamics are fast relative to consumer dynamics means the resource density equilibrates at the value

The per capita growth rate must reflect individual resource consumption. The simplest assumption is that the per capita growth rate is simply the individual resource consumption rate minus a resource maintenance requirement, d _j . Thus,

As this is a discrete-time model, the equation for population dynamics is

The density-dependent factor F can be defined as K−R, i.e. as the reduction in the resource abundance due to consumption. Since ϕ is then the identity function, C=F. Thus,

Finally defining , we obtain the required form (24) for r _j (t).

To analyze the model, we restrict attention to the two-species case. With species i as invader and s as resident, competition takes the form

Because C is an increasing function of resident E, positive covariance between resident E′ and C is to be expected. However, the complication is that C depends on N, which might be correlated with E′. This possibility is removed if the sequence of environmental responses is assumed to vary independently over time. Although N _s (t) is a function of past values of , it is independent of the current one. In general, this assumption is not a serious restriction because unless autocorrelation in the environment is very strong, the dominant influence on the covariance between C and E′ comes from the direct functional relationship between C and E′, not that through any correlation between the current value of E and the current value of N. However, in the absence of autocorrelation, there is no doubt that C ^{−i}(t) has positive covariance with . The issue now is to see that invader covariance between environment and competition is smaller than the resident value.

Clearly, there would be no difference in covariance if the two species had the same activity patterns, regardless of any fluctuations in them that might be present. On the other hand if they had independent activity patterns, would be independent of C ^{−i}(t), and so would have zero covariance. However, because the species are responding to their common environment, the E′s for different species would be expected to be correlated, but that correlation is normally less than 1. If we assume that they have the same variance, correlation ρ, and a suitable distribution on the positive real numbers, such as bivariate gamma 35, then we can assume that the conditional expectation of given has slope ρ. It follows that

which gives the invasion rate

Thus, the invasion rate of each species is boosted by differing in its activity patterns from the other species (ρ<1). There may be some difference between species in how they benefit because resident covariance between environment and competition, although always positive in magnitude, may differ between species. The requirement that each invasion rate be positive, then yields the coexistence condition,

In the absence of fluctuations, this region is null. It is null also with ρ=1, when the species have the same response to the environment. Otherwise, it is a non-empty region. Note that the lower boundary in Equation (34) may depend on μ₂ but is independent of μ₁, while the upper boundary is independent of μ₂.

In contrast to relative nonlinearity, any number of species can coexist by this storage-effect mechanism with just a single density-dependent factor provided the differences in the μ values are small enough. For example, if species have independent responses to the environment, then invader is always 0 in discrete-time models like the one discussed here, for then ρ=0, while resident covariance is always positive. Regardless of the number of species, there is always some nonempty range of μ differences compatible with coexistence. However, this range tends to decrease with the number of species 7, making the addition of species to the system progressively more difficult as the number of species increases.

The storage effect depends critically on the covariance difference between the invader and resident states. The presence of the covariance in the long-term growth rate depends on the fact that E and C have an interactive effect on the growth of the population. This interactive effect has the property that when environmental conditions are poor, i.e. when E is low, the growth rate is less sensitive to competition. This can be thought of as a buffering effect that protects gains made under favourable conditions from being lost due to competition at other times. This protection of gains is the reason for the name ‘storage effect’: gains in favourable times are in effect stored to be used at later favourable times. However, this interaction alone is not sufficient for a coexistence promoting effect: the covariance itself has to be nonzero, and higher for residents than invaders. Positive covariance for residents tends to occur because taking up resources under favourable environmental conditions can create resource shortages, as seen in the example above. With apparent competition, the positive covariance arises because of predator build up during favourable conditions. These outcomes are not guaranteed, however, because the environment might change before the resource shortage or predator build up is felt. When residents do have positive covariance, asynchrony between species in their activity patterns leads to the required covariance difference.

3.4. More general models

The features considered in the models for the two coexistence mechanisms above occur jointly in more general models that take the form

where g _j is an arbitrary continuously twice differentiable function. A nonzero cross partial derivative, , leads to an interaction that generalizes the product term EC in the models above. Thus, given covariance also, the storage effect can occur, but it might be negative promoting founder control if γ is positive. Relative nonlinearity might occur if the C _j take the form , for some nonlinear functions h _j , but relative nonlinearity now needs to be assessed by the behaviour of the as functions of F. Alternatively, the C _j (t) might be functions of different competitive factors, representing resource partitioning, that is, different species depending to different degrees on a range of resources 43. Resource partition would allow species coexistence in the absence of temporal fluctuations. Equation (35) is compatible with simultaneous inclusion of three different classes of coexistence mechanisms: classical equilibrium coexistence (such as resource partitioning), the storage effect and relative nonlinearity 7. This becomes apparent through quadratic approximations along the lines of those presented above for relative nonlinearity. They lead to the following general expression for invader growth rates 7:

Here r¯′ _i represents the equilibrium mechanisms, Δ N represents relative nonlinearity (conventionally preceded by a negative sign) and Δ I represents the storage effect. These three different sorts of mechanisms are exhaustive for this general model, and are each represented by formulae that generalize those given in the examples above 7 11. Moreover, Equation (36) has a spatial analogue that includes an additional mechanism due to fitness-density covariance, as discussed above under scaling up 10 56.

4. Discussion

There have been numerous models predicting species coexistence as a result of temporal variation 1 2 19 20 23 26 34 48, including some based on detailed understanding of particular systems 24 34. The distinguishing feature of the developments reviewed here is the attempt to provide a general scheme uniting a variety of specific models with key variables and attributes interpretable in terms of relatively concrete properties of real systems. For instance, the quantities E in specific cases can be measured by life-history variables such as germination rates and resource uptake rates that are known to depend directly on environmental conditions. They lead to an understanding of covariance between environment and competition through natural submodels of the competitive process. Derived quantities such as carrying capacities and competition coefficients, are not used directly as environmental responses because they are composites of the direct responses of the organisms to the conditions they experience. Thus, it is not clear a priori how such derived quantities should vary over time or space.

One way of understanding the scale transition, both spatially and temporally, is by use of quadratic approximations. These allow estimation of the first order effects of the interaction between nonlinearity and variation, under the assumption that variation is small. However, the concepts are much more general. For example, in the discussion of coexistence by means of temporal environmental variation, the storage effect is understood as the mechanism arising from the interaction between the response to the physical environment and the response to competition, when they both fluctuate over time. A geometric approach leads to the understanding of this mechanism free of quadratic approximations 6. Moreover, relative nonlinearity can also be understood directly from Jensen's inequality without recourse to approximations 10. The role of the quadratic approximations is to estimate the magnitudes of these mechanisms, not to define the mechanisms as such.

The form of the results presented here also deserves special mention. The theory above for the storage effect, for example, expresses the analytical results as a measure of the strength of the mechanism in terms of covariance differences. In more general models, this covariance difference remains, but the interaction measure γ multiplies it. The interaction measure is easy to obtain analytically, but the covariance difference is more difficult to get, and may have to be calculated numerically or by simulation. This outcome lends itself to a hybrid approach where analytical results lead to an interpretable measure that might then be quantified by simulation. This hybrid approach leads to more understanding about the underlying mechanism than approaches that use numerics or simulation to go directly to the coexistence region in multidimensional parameter space 56. This same feature makes these ideas particularly amenable to field tests. For example, the real action in the storage effect is the covariance difference. These covariances are directly measurable in the field through standard experimental designs originally intended for measuring changes in the intensity of competition along environmental gradients 12 53. These covariances are far simpler to obtain than a fully parameterized model for the system. Because covariances characterize the mechanism, they provide particularly powerful tests of it 12.

Finally, these results are developed directly within a framework for understanding scaling up local phenomena to those scales where predictions are needed. In all cases, this scaling is achieved by focusing explicitly on the interaction between nonlinearity and variation. This process is interpretable in terms of the biology involved and leads to quantities that aid understanding and testing of the fundamental mechanisms. This process produces not just the final answer so often sought in modelling exercises, but valuable intermediate steps that themselves lead to general theory. I believe this process could be taken advantage of more generally.

Acknowledgements

This work was supported by NSF grant DEB-0542991. Comments of two anonymous reviewers have greatly improved the manuscript.