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Introduction

One proposed explanation of the high diversity of many natural communities has been that disturbance drives mechanisms of coexistence. As first proposed by Connell (1978), disturbance interrupts ecological succession and prevents domination by the most competitive species. If disturbance is expected to occur patchily in space (Connell 1978), then a natural landscape consists of a mosaic of patches in different successional states (Chesson and Huntly 1997) and thus has high diversity, at least at the landscape scale. This idea has been represented in models by various authors, beginning with Caswell (1978) and Hastings (1980), and involves a competition‐colonization trade‐off (Tilman 1994) whereby a strict competitive hierarchy drives succession and inversely ranked colonizing abilities bias colonization toward low‐ranked competitors. Although this form of the disturbance hypothesis is the most highly developed, it is only one form (Roxburgh et al. 2004). For this reason, Chesson and Huntly (1997) preferred to identify it specifically with the name “successional mosaic hypothesis.” Roxburgh et al. (2004) pointed out the existence of a variety of conceptions for the disturbance hypothesis and cogently argued that without a better appreciation of the similarities and differences between different models, the disturbance hypothesis is not scientifically useful.

Our model allows us to demonstrate a unique instance of coexistence under disturbance, driven by a trade‐off between resistant and resilient responses to disturbance. The formal mechanism that promotes coexistence is the spatial storage effect. As an example community, we have in mind shrubs differentiated from one another in their mode of regeneration after fire. It has long been argued that such life‐history differences in regeneration are likely to be common in nature (Grubb 1977), and our model could be applied to species‐rich communities such as the Fynbos (Cape Province, South Africa), the Kwongan (Western Australia), or even communities of sessile animals prone to disturbance (Paine and Levin 1981). Our intent is to keep the model general, so that it can be applied to a variety of natural systems and to show that the underlying mechanism depends less on specific biology than on the qualitative features of resistance and resilience. We obtain a general, analytical solution to this model and an explicit expression for growth from low density, which allows the unambiguous identification of the mechanism as the storage effect. The resulting formula for the storage effect in terms of covariance differences provides a route to testing the hypothesis in the field (Sears and Chesson 2007; Chesson 2008).

The Spatial Storage Effect

The spatial storage effect derives its name from the idea that persistence of populations in favorable locations in space is analogous to storage of seeds over time in a seed bank or the high survival of adult organisms in iteroparous populations (Chesson 2000a, 2000b). The idea is that these life‐history traits buffer population growth against unfavorable events in time or space. The gains made under favorable conditions persist in the face of unfavorable events and continue to fuel population growth when favorable conditions recur. Such buffered population growth alone, however, does not make a coexistence mechanism and is not the complete mechanism referred to as the storage effect (Chesson 1994, 2000a). Associated with it are two other ingredients, namely, covariance between environment and competition and species‐specific responses to the environment. Positive covariance between environment and competition means that when the physical environment is favorable to a species, that species experiences higher competition. This outcome is expected when a species is abundant and places greater demands on resources when physical environmental conditions are favorable. Thus, competition increases with the favorability of the environment. This increased competition limits a species’ ability to take advantage of favorable environmental conditions. In contrast, if a species is at low density and experiences mostly interspecific competition, this positive covariance is expected only if a species responds to the physical environment in the same way as its competitors. So in the presence of species‐specific responses to the environment, covariance between environment and competition for a species at low density is likely to be weak or negative. Thus, a species at low density is relatively unrestricted in its ability to respond to favorable conditions. Buffered population growth means that the ability to respond to favorable conditions outweighs the difficulties presented by conditions at different times and different places. The net outcome is that a species at low density is at an average advantage, compared with a species at high density, and can recover from its low‐density state, enabling species coexistence. The key active features are buffered population growth and covariance between environment and competition, the value of which increases as a species’ density increases. Together, they create the coexistence mechanism called the storage effect.

Roxburgh et al. (2004) suggested that their disturbance models for a patchy environment could be explained as the spatial storage effect. They demonstrated buffered population growth but not covariance between environment and competition. Moreover, their models assume a strict competitive hierarchy and are not clearly distinguished from the successional‐mosaic models. Without an investigation of the covariance between environment and competition, it is not clear that observed coexistence in their models is in fact due to the storage effect. Possingham et al. (1995) have also presented a model of competition via disturbance, which demonstrates coexistence of multiple fire‐adapted eucalypt species. In their equilibrium model, coexistence is achieved when one eucalypt species creates an environment that is more favorable to the other. The mechanism for coexistence in disturbance‐prone communities offered by the Possingham et al. (1995) model is distinct from the one presented here, in that the storage effect is not present and local retention, which is crucial for covariance between environment and competition in our model, is not accounted for in their well‐mixed system. Here, we consider a quite different disturbance model in which there is no competitive hierarchy and no successional process but coexistence does result from patchy disturbance in space. We can definitively identify the underlying mechanism as the storage effect, by demonstrating buffered population growth and by showing how covariance between environment and competition changes with species abundances. In addition, we are able to measure the full magnitude of the mechanism and to show that it is the only mechanism contributing to coexistence in our model.

Model Formulation

Our model might be applied in a variety of contexts. However, for specificity, we discuss the model in terms of shrub communities in Mediterranean heathlands, where fire is the main source of disturbance. Fire is known to be an important factor such systems, in part because the combined effects of opening up space, burning biomass, and nutrient release create a variety of environmental conditions (Whelan 1995). When burns are frequent and a stable burning regime has persisted on evolutionary timescales, many species develop specific responses to fires (Wisheu et al. 2000).

Mediterranean heathlands display high levels of plant diversity as well as many fire response strategies. Here, we consider two strategies picked to represent opposing ends of the observed spectrum. Some species, known as “sprouters,” are not killed even in severe burns and are able to resprout from surviving structures, such as lignotubers or epicormic buds. Other species, known as “seeders,” die in a fire but trigger the release of a seed bank stored in the canopy, often in the form of serotinous cones (Wisheu et al. 2000).

In general, we can classify these strategies independently of the biology as “resistant” and “resilient,” respectively. The sprouter strategy is resistant, because its adaptation allows it to resist the common effect of fires killing individuals. Likewise, the seeder strategy is resilient, in a dynamical‐systems sense, because although the population densities will plunge after a fire, the resulting surge of recruitment allows the population to quickly spring back to previous levels. This terminology is consistent with that of Boucher et al. (1994), who observed resistant and resilient strategies in a field study of rain forest trees subject to hurricane disturbance. Here we develop a mathematical model of the system that demonstrates how the action of disturbance can allow a stable coexistence of two species that employ these opposing strategies, regardless of their specific biology.

We consider dynamics on two spatial scales, a regional landscape and local patches. The time evolution of the system is governed by mechanistic rules at the patch level. Regional properties then emerge from the local interactions. It is important to have this two‐scale resolution because it allows us to understand large‐scale behavior in terms of local phenomena. Since the local dynamics embody inherent nonlinearities, the regional dynamics will be qualitatively different from those of any given patch. This phenomenon is known as the scale transition (Chesson et al. 2005). To describe the effects of competition, we implicitly include a smaller scale of home sites within patches. This is the characteristic scale of competition, which is the diameter of the rooting zone of a mature individual. The effects of competition and disturbance at the smallest scale drive the dynamics of the patches, which in turn propagate up to the regional level.

In Mediterranean heathlands, competitive effects are concentrated during the postfire period of seedling establishment. Since seedlings cannot establish in a home site occupied by a mature individual and there are far more seedlings than empty sites, many seedlings will not be able to secure a home site. We assume that a species’ chances of securing a free site are proportional to its density in the current pool of competitors. This is “lottery competition,” a standard model of competition for space (Chesson and Warner 1981). Establishment conditions depend on a number of factors. In our case, the source of variation is fire. Recently burned patches are mostly open space, in contrast to the very densely populated patches that have not experienced a fire recently.

Fire is a stochastic environmental fluctuation, and it is the only source of variation in our model. In addition, this variation is purely spatiotemporal, meaning that time averages show no spatial variation and spatial averages show no temporal variation. Thus, each patch has the same probability of burning in a given year, and this probability does not vary through time. These assumptions are simple, to allow complete analytical development, but are complex enough to enable us to model the ecological effects of the fire that arise from species‐specific response strategies. Our assumptions for the disturbance process are common, (e.g., in Possingham et al. 1995), and they are not so limiting as they may seem at first. To wit, our key result, equation (5), holds for all probability distributions of disturbance intensity and frequency and for spatially correlated disturbance, as would be expected, especially from fire. More realistic modifications can be made, such as lag time between burns. This would introduce temporal autocorrelation in environmental responses. In this case, the coexistence criteria will not be as simple as those in equation (5), but we can still expect to have a stabilizing storage effect term, because it has been shown that the storage effect in the two‐species temporal‐lottery model does not depend on autocorrelation in the environment (Chesson and Warner 1981).

We consider the dynamics of two species, which we refer to as seeders (D) and sprouters (T). While many examples of seeding and sprouting species are known, we refer not to particular species but to two divergent fire response strategies. The sprouters are not directly affected by fire; their mortality rate (δ) and per capita seed yield (Y) are constants. They are indirectly affected, however, through competition with seeder species, which are directly affected by fire. Seeder species have yields and death rates that depend on whether a fire occurs: seeder yields are minimal when no fire has burned that year and maximal when a fire does occur. This implicitly incorporates the storage of seed in a canopy bank. Likewise, seeder death rates are maximal when a fire occurs and minimal when one does not. The maximal and minimal values of seeder yield and mortality are specified only to produce the coexistence regions in figures 1 and 2.

In our model, dispersal is treated in a simple manner. The important feature is that there are two scales involved, local and global. While most seeds are retained in their natal patch, some small percentage will travel much farther. Once seeds have escaped their natal patch, we assume that they can reach any patch in the system. This type of dispersal was considered by Comins and Noble (1985) and is consistent with the widespread presence of wind dispersal adaptations among the shrubs of Mediterranean heathlands (Hammill et al. 1998). The exact nature of the dispersal kernel is not important for our conclusions so long as the scale of dispersal is larger than the grain of environmental heterogeneity (Snyder and Chesson 2004).

We now formally define the model, incorporating the features described above. Our notation generally follows the conventions of Chesson (2000a), and a list of symbol definitions is included in table 1. Let N_jx(t) be the density of adults of species j in patch x at time t. The fitness of such an adult is defined as an individual’s expected contribution to the next generation of adults. If we denote this fitness λ_jx(t), then is the number of adults in the system at time arising from adults in patch x during time t. Because of seed dispersal, some of these individuals will be in other patches.

Table 1: List of symbols

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Three main terms contribute to λ: adult survival, establishment of seedlings in their natal patch, and establishment of seedlings elsewhere. Adult survival is simply , where δ is the disturbance‐dependent death rate. For the local establishment of species j, we consider the R portion of the yield Y_jx(t), which is retained locally in patch x at time t. In the absence of competition, the local rate of establishment would be given by . Competition acts to decrease establishment from this maximum, and the local rate of establishment under competition in a focal patch x is , where C_x is the magnitude of competition in patch x. The portion of the Y_jx(t) seeds, which disperse widely from location x, compete in some other patch, y, where their establishment rate is given by . Thus, the overall establishment rate of dispersers is the average of this quantity over patches, that is, . These three terms give us an equation for λ_jx(t):

Thus, the fitness of an individual in patch x consists of its survival rate in that patch, plus the offspring that it establishes in that patch, plus the offspring that it establishes in other patches. We define the magnitude of competition, C_x(t), as the ratio of the number of seedlings competing to the amount of space for which they are competing. This is lottery competition, meaning that the chance that a seedling of species j establishes is governed by the proportion of species j seedlings in the patch and the current amount of free home sites in the patch (Chesson and Warner 1981).

The number of seeds of species j that remain in their natal site is given by , and the number of seeds that arrive at a site x from the dispersal pool is . Note that this term is averaged over patches, because of the assumption of random widespread dispersal. By the time independence of the environmental process, the average of the yield at each site, , is equal to the average yield times the average density, . The amount of free space in a patch is determined by how many adults have died there that year. This is given by δ_lx(t)N_lx(t). Taking sums over species l and forming the ratio gives the formal expression for magnitude of competition: The terms in the numerator add up to all seedlings attempting to establish in the patch x, and the terms in the denominator represent total free space. These features are common to lottery competition, and the general trend is for competition to increase with the number of competitors present and decrease with the amount of free space. When we consider the action of C on equation (1), we see that increased competition decreases fitness by reducing the number of offspring that establish. Also note that the main feature of C is that it is a ratio of yield to mortality, where each term is adjusted by the appropriate density. This will give us insight into interpreting the coexistence regions derived in the next section.

Model Analysis

To determine coexistence conditions for the system, we use invasion analysis (Bolker and Pacala 1999). This states that the species will coexist if each can increase from low density (as an “invader”) in the presence of the other, established as a resident. This is a definition of stable coexistence (Chesson 2000a). We use the subscripts “i” and “r” to refer to invaders and residents in this situation.

To examine invasibility, we utilize the long‐term low‐density growth rate for an invader, denoted . This is the fitness λ_i, averaged over individuals and computed for the case of limiting low density (Chesson 2000b). This quantity is useful in the following way: is the regional density of mature individuals, which is the spatial average of local densities. In general, . This equation tells us that the regional density of invaders is governed by . Thus, implies that an invader will increase from low density. If such a species in the community is brought to low regional density by disturbance, it will not become extinct but will increase because of the positive regional growth rate. This leads to persistence in the system, and hence serves as a way to study coexistence by using the method of mutual invasibility. To have mutual invasibility, we must satisfy the following pair of inequalities, one for each species: So if , we are considering the sprouter species as invader, and if , then the sprouters will increase in density while competing with established seeders. When the role of invader is changed to , has an analogous interpretation. Under these circumstances, both species will persist in the system. If one or the other is perturbed to low density, it will tend to climb back to a higher density. However, when an invader has , then perturbation to low density leads to its extinction. If we set up a system of inequalities requiring growth from low density, the conditions that satisfy the inequalities give us criteria for coexistence. These criteria involve the harmonic mean of competition taken over space, with species i as the invader. We denote this harmonic mean H(C^{−i}). It is a measure of the location of the distribution of the varying quantity C^{−i}, appropriate to the dynamics of the model we are considering. It is the reciprocal of the average of the reciprocal; that is, The coexistence criteria, which are derived in appendix A, are given by a system of inequalities: The biological interpretation of the criteria in equation (5) is straightforward. Consider the term . This is the invader species’ lifetime yield, based on mean yield and mean mortality. For this species to persist, this lifetime yield must exceed the right‐hand side, which is a measure of average competition. The left‐ and right‐hand sides of equation (5) are very similar in terms of the life‐history traits they represent. The harmonic mean of C^{−i} is also a ratio of yield to mortality (see eq. [A5]), but when yield varies in space, its harmonic mean is not the simple ratio of means . Instead, the harmonic mean can be less than this value when Y varies in space. Biologically, this represents the fact that the resident species is limited in its ability to take advantage of favorable conditions. This limitation occurs because, when a resident has high yield in a patch, it automatically suffers high competition there, especially if local retention is high (see discussion of the storage effect below). This fact makes invasion by another species easier. Indeed, it makes it possible for species that would be inferior based on average yields and mortality rates to invade and opens up a region of coexistence for species that, on the basis of such average values, are unequal.

A graphical representation of the coexistence region is given in figures 1 and 2. Both are plots of the system of inequalities that determine the coexistence region, obtained from inequality (5). Because Y_T and δ_T are constant, these inequalities can be simplified. In particular, , and . Thus, the two inequalities (eq. [5]) can be fitted together, and this gives us a concise description of the coexistence region: In both plots, the vertical axis has units of lifetime yield, and the independent parameters are locally retained seeds (R) and probability of fire (f). Without stochastic variation due to disturbance, only one of the two inequalities can be satisfied. With stochastic disturbance, both inequalities can be simultaneously satisfied. Essentially, the variability brought about by fire opens up a region of parameter space where coexistence is possible. This becomes clear when we scale the long‐term low‐density growth rate, , by its sensitivity to competition (app. A), which allows us to see how different factors contribute to positive growth: The expression identified as variation independent contains only parameter averages and does not depend on variation in these quantities. This expression is an average fitness comparison between the species, in the sense of Chesson (2000b, 2008). A common fitness measure is lifetime yield, . Here it appears as the inverse, but the variation‐independent expression will still be positive whenever the traditional fitness difference is positive, that is, whenever . The variation‐dependent term is 0 unless there is environmental variation, because only then can the covariance be positive. This term measures the storage effect, as discussed below. In many general models involving the storage effect, this covariance would be positive, whichever species were resident, but here it is positive only for seeders as resident, because here sprouters are assumed to be insensitive to disturbance. This means that seeders can invade only by having a higher ratio of average yield to average mortality (as is specified in eq. [6]), but invasion of sprouters with an inferior ratio of average yield to average mortality is possible because the resident seeders have positive covariance between environment and competition, which keeps them in check and provides a relative advantage to sprouters as invaders.

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Figure 1: Coexistence criteria from inequality (6). Coexistence occurs in the hatched region, above H(C^{−T}) (dashed line) and below (solid line). Seeder minimum and maximum mortality and yield are , and , respectively. The ratio is free, and . Figure 2 shows a different cross section.

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Figure 2: Coexistence criteria from inequality (6). Coexistence occurs in the striped region, above H(C^{−T}) (dashed line) and below (solid line). Parameters are the same as in figure 1, with . Intermediate values of f give a wider coexistence region than extreme values; that is, a larger range of values for lead to coexistence.

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Equation (7) illustrates the concepts of stabilizing and equalizing mechanisms of coexistence (Chesson 2000b) and, more generally, stabilizing and equalizing components of mechanisms (Chesson 2003; Snyder et al. 2005). As we show in appendix B, the storage effect here (i.e., the variation‐dependent component of eq. [7]) acts as both an equalizing and a stabilizing mechanism. Because the storage effect is positive only for sprouters as invaders, it reduces their fitness disadvantage relative to residents. However, because the storage effect is positive when averaged over species, it means that the average over species of the invader growth rates (eq. [7]) is positive, permitting each to have a positive growth rate at low density, provided that their fitness differences are not too great. In this sense, the storage effect is also a stabilizing mechanism in this model. Thus, here the storage effect has both stabilizing and equalizing components, and we show in appendix B that each component is equal in magnitude to Reducing the fitness difference via an equalizing mechanism with no stabilizing component would broaden the coexistence region in parameter space but would not promote stable coexistence on its own. To see this, consider the model in a constant environment. The covariance in equation (7) would be 0, and changing which species play the roles of resident and invader would necessarily change the sign of . Thus, only one species could have , precluding coexistence in a constant environment. However, when we include the effects of a varying environment, the positive covariance when seeders are residents introduces a mechanism with a stabilizing component as well as an equalizing component. This stabilizing component makes it possible for both species to have positive invader growth rates and allows for stable coexistence. Thus, the coexistence demonstrated is variation dependent.

The formal mathematical definition of the spatial storage effect, according to Chesson (2000a), is given in appendix A, and we show there that the variation‐dependent term in equation (7) is the storage effect. The storage effect is normally denoted ΔI and is shown in the appendix to take the form The environmental response of a species j is measured by and the competitive response by − . Thus, ΔI shows resident‐invader difference of covariances between environment and competition, multiplied by local retention R. Note that − increases with C. This is a qualitatively similar way of expressing competition and is proportional to the standard variables developed by Chesson (1994). It has this form because of the way that C enters into equation (1). This transformation to standardized variables removes the ambiguity associated with the many qualitatively equivalent ways of expressing competition. To see intuitively that this term measures the storage effect, recall from “The Spatial Storage Effect” that, under the storage effect, the resident species is limited in its ability to respond to favorable environmental conditions because competition is greater during favorable conditions. This comes about because of the difference between resident and invader covariances between environment and competition.

In equation (7), the covariance difference reduces to a single term for the residents, because the invader’s covariance between environment and competition is 0 as a result of our assumptions of time independence in the disturbance process and insensitivity of one species to disturbance. Moreover, these same assumptions mean that only the covariance for the seeder as resident is not 0. In spite of this high degree of degeneracy, however, this single positive covariance is responsible for coexistence here. The environmental response of the resident is measured by and the competitive response by − . Thus, ΔI shows resident‐invader difference in covariances between environment and competition multiplied by the local retention R. It is worth considering what happens when the invader’s covariance between environment and competition is not 0, for instance, when the insensitive species had some sensitivity to disturbance or disturbance had some autocorrelation. A seeder invader would still not suffer the limitations of covariance between environment and competition, because it is at low density (and so has a small total competitive effect) and would still have a more limited environmental response than the resident. As a consequence, invader covariance between environment and competition would still be low, maintaining invader advantage because the invader could still respond to favorable conditions without much limitation from competition. The storage effect would appear in the invasion rate of the sprouter in this case, but it would be small, because both covariances contributing to the storage effect would be small, and so not greatly perturb the analysis above, as confirmed by simulations.

As we can see from equation (9), the strength of the storage effect depends on the retained fraction R, because only the retained fraction is involved with covariance between environment and competition. Here, the life‐history trait responsible for creating covariance between environment and competition is the local retention during dispersal, combined with the distribution of the population over a landscape on a scale larger than an individual fire. Thus, the “storage” of the storage effect refers here to the spatial storage of population in unburned areas. This is analogous to the temporal “storing” of population in a seed bank or as long‐lived adults. Interestingly, the magnitude of the storage effect increases with the magnitude of local retention R, because of its role in covariance between environment and competition. This can be seen from the form of equation (9) and is also evident from figure 1. This increasing relationship remains until , at which point the lack of dispersal lessens the buffering and weakens the storage effect to its minimum value (app. C). Equation (9) is a numerical measure of the storage effect, equal to the degree of buffering times the difference between resident and invader covariances between environment and competition. This expression is derived in appendix A from the formal definition of the storage effect in Chesson (2000a). While the storage effect may be operating as a stabilizing mechanism in many contexts, it is important to verify this through the method of invader‐resident comparisons, as outlined in appendix A. This has the advantage of providing a quantitative strength of the mechanism and also prevents confounding with other potential mechanisms.

Discussion

Since the influential work of Connell (1978), there has been considerable effort to understand the role of disturbance in diversity maintenance. Connell's intermediate disturbance hypothesis (IDH) is multifaceted, and different mechanisms can provide for coexistence under this conceptual rubric (Roxburgh et al. 2004). It seems that Connell had in mind a situation similar to the successional‐mosaic hypothesis discussed by Chesson and Huntly (1997). These views describe coexistence under disturbance through the effects of a successional process, where species are adapted to do best at different stages of succession. One way a successional mosaic can be created is by the well‐known competition‐colonization trade‐off (Tilman 1994; Roxburgh et al. 2004). In light of these works, we can ask whether all models of coexistence with between‐patch disturbance must rely on competition‐colonization trade‐offs or competitive hierarchies (Comins and Noble 1985; Chesson and Huntly 1997; Roxburgh et al. 2004).

Our model shows behavior not described by the above‐mentioned instances of the IDH, in that it neither relies on competitive hierarchies nor requires any distinctions in dispersal ability. Instead, we have a nonsuccessional, spatiotemporal disturbance system that also predicts coexistence at intermediate disturbance levels. The intermediacy can be seen in figure 2, which shows that the coexistence region is largest at intermediate fire frequencies. The coexistence occurs because of a trade‐off between disturbance response strategies. Following the more general terminology used by Boucher et al. (1994), we say that coexistence in our model is achieved through a resistance‐resilience trade‐off in life‐history traits. The key features of these strategies are that resistant species minimize death by disturbance, while resilient species respond to disturbance by a pulse of recruitment.

We may ask how many distinct disturbance response strategies can coexist through the resistance‐resilience trade‐off. Here, we have shown coexistence of two species, with one type of fire. Simulations show that three species can coexist, if there are two different levels of fire intensity. Essentially, a new, “facultative‐seeder” strategy becomes available for species that die only in severe fires. In general, we hypothesize that n such strategies can coexist in this manner, as long as there are different types of disturbance, and this will be the subject of future work.

Our notion of the resistance‐resilience trade‐off further supports the ideas developed by Roxburgh et al. (2004), who viewed the IDH as an umbrella concept that comprises distinct theoretical mechanisms. As discussed in “Introduction,” the storage effect is defined through a resident‐invader difference in covariances between environment and competition. To be sure that a mechanism is the storage effect, it is necessary to measure this covariance. The within‐ and between‐patch models of Roxburgh et al. (2004) depict two species competing in a disturbance‐prone environment. However, their models are based on an assumption of strict competitive hierarchy, and so their mechanism does not appear to be distinct from the competition‐colonization trade‐off (Tilman 1994). Our model presents a distinct mechanism because it does not depend on strict competitive hierarchies. An interesting relationship between the competition‐colonization trade‐off and resistance was pointed out by Hastings (1980). Although poor competitors are often assumed to be better dispersers, similar coexistence through multiple competitive factors will arise if competitors low in the hierarchy have higher resistance to disturbance instead of better dispersal. This finding suggests possible co‐occurrence of resistance‐resilience and competition‐colonization trade‐offs, both of which contribute to diversity maintenance in high‐diversity disturbance systems.

We have demonstrated how resistant and resilient fire response strategies can form a mechanistic basis for a stable, competitive coexistence via the storage effect. The presence of disturbance provides species with a way to differentiate themselves with respect to their responses. Local retention measured by R causes covariance between environment and competition to be present in equation (7). When combined with species‐specific responses to the environment, this covariance will be different for residents and invaders, causing coexistence to be promoted via the storage effect, as defined by Chesson (2000a). The storage effect provides complete understanding of the functioning of disturbance in this model. Moreover, we find an exact expression for the long‐term growth rate. This growth rate is then partitioned into a fitness inequality and stabilizing components, which leave no doubt that the sole stabilizing mechanism in this model is the storage effect. We identify the storage effect through the covariance between environment and competition, where the environmental variation comes from disturbance. This quantitative partitioning of the growth rate allows mechanism strength to be assessed, facilitates comparison of different mechanisms, and provides a route to testing mechanisms in nature. The technique we illustrate in the analysis of our disturbance model is a powerful tool for model analysis that we propose should become standard in the field.

Here, we have shown a concrete example of how these mathematical features can naturally arise in a disturbance system. This constitutes the first complete demonstration of the storage effect as the primary mechanism enabling coexistence in a disturbance‐driven system and may offer insight as to how the storage effect can operate via resistance‐resilience trade‐offs in other systems. A critical feature in this instance is the presence of both stabilizing and equalizing components due to the asymmetric action of the storage effect on different species, a situation not previously explored to any significant degree and likely characteristic of the sorts of trade‐offs studied here. Our work is most similar in spirit to the simulation model of Lavorel and Chesson (1995), focusing on annual species with local disturbances and differing germination strategies. Like our model, that of Lavorel and Chesson presented evidence that differences in regeneration strategies favor coexistence. The analytical approach that we use here, however, sharply defines the mechanism.

Coexistence of such high numbers of species as are found in fire‐prone heathlands most likely depends on many distinct mechanisms. Niche differentiation can depend on leaf morphology, dispersal ability, and other life‐history traits (Cody 1986; Yeaton and Bond 1991). We believe that there is a need to consider multiple mechanisms when attempting to explain such diverse systems. As an example, including density dependence for the fire process would not alter the presence of the storage effect in the model and could allow for coexistence of additional species through the flammability niches of Possingham et al. (1995). Our method of partitioning long‐term growth rates into fitness inequalities and stabilizing terms can serve as a general tool for comparing mechanisms in different systems and can also serve to quantify the strengths of mechanisms within models. We hope that further use of this technique will allow researchers to better understand the complex phenomenon of coexistence under disturbance. Likewise, we believe that a better understanding of disturbance response strategies and the associated resistance‐resilience trade‐off will prove useful in explaining high levels of diversity found in disturbance‐prone systems.