Ecologists have long studied the effects of disturbance on species diversity. More recently, researchers have become interested in understanding how the various aspects of disturbance interact to influence community diversity. While the effects of temporal autocorrelation have also received some attention, the potential for manipulating disturbance autocorrelation to achieve management goals has not been theoretically explored.
We consider the interactions between temporal autocorrelation of disturbance occurrence and disturbance intensity at varying disturbance frequencies. Using an annual plant model, we show that when intensity and frequency are kept constant, changing the temporal autocorrelation of disturbance occurrence can also affect competitive outcomes. Additionally, we show that when species coexist, the degree of autocorrelation can affect which species reaches higher densities.
We describe several examples (including prescribed burning, grazing and mowing) that outline how manipulation of temporal autocorrelation may be used to achieve conservation and eradication goals at no additional cost.
Synthesis and applications. Our results provide important insights into, and have potential application to, land management and conservation. While changing the intensity and frequency of human-induced disturbances can be costly, adjusting the temporal autocorrelation of disturbance occurrence may be considered a ‘no-cost manipulation’. In instances where a land manager lacks the funds or resources to manipulate other aspects of disturbance, such as intensity and frequency, changing the temporal autocorrelation may provide an effective, economical alternative.
The effects of disturbance on diversity in ecological systems have been studied for decades (Grime 1977; Connell 1978; Sousa 1984; Pickett, Collins & Armesto 1987; Mackey & Currie 2001). While disturbance is not essential for species coexistence, a wide array of theoretical and empirical studies suggest that disturbance plays an important role in coexistence, especially in disturbance-prone communities (Pickett & White 1985; Petraitis, Latham & Niesenbaum 1989; Chesson & Huntly 1997; Roxburgh, Shea & Wilson 2004; Miller & Chesson 2009). The more recent of these studies utilize Chesson's (1994) general theoretical framework for species coexistence in variable environments. However, researchers have struggled to produce a unified treatment of disturbance, questioning its relative importance and the generality of observed diversity–disturbance patterns (Mackey & Currie 2001; Shea, Roxburgh & Rauschert 2004). Miller, Roxburgh & Shea (2011) point to the complex, multifaceted nature of disturbance to explain this lack of common ground. The ecological effects of disturbance are complicated, because disturbances have various distinct aspects, namely return time, intensity, extent, timing and duration (Gill 1975; Pickett & White 1985; Menges & Hawkes 1998; Shea, Roxburgh & Rauschert 2004; Haddad et al. 2008). All of these aspects are relatively independent quantities that can be measured for a particular disturbance event: the time since the last disturbance (return time, which results in the frequency for a disturbance regime), the vigour of the disturbing force (intensity), the area affected (extent), the timing of the disturbance occurrence relative to the species' growth cycle (timing or seasonality) and how long a disturbance lasts (duration). Several studies have begun to disentangle the effects of these aspects, both individually (Collins 2000; Miller, Roxburgh & Shea 2012) and in concert (Miller, Roxburgh & Shea 2011; Zhang & Shea 2011).
However, even these five aspects of disturbance events fail to fully describe disturbance processes. Many theoretical studies make the assumption that all disturbance events are independent and identically distributed (Shea, Roxburgh & Rauschert 2004; Svensson, Lindegarth & Pavia 2009). While this is a very useful assumption that has cleared the way for some invaluable advances in the theory of disturbance, we extend this work by relaxing the assumption and allowing for autocorrelation of disturbance occurrence. Indeed, empirical findings have shown that disturbance events are not necessarily independent over time. Rather, natural disturbance events are often autocorrelated, because previous disturbance occurrences can affect when future events will occur (Sousa 1984; Pickett & White 1985; Poff 1992; Gill & Allan 2008). Thus, it is also important to consider properties of the overall disturbance regime, rather than only a single disturbance event (Gill 1975; Pickett & White 1985). While all five of the aspects mentioned above may be extended to pertain to the entire disturbance regime by considering the mean value of the aspect quantifications over all the individual disturbance events (in the case of return time, it is common to consider the reciprocal, the frequency of disturbance), the temporal autocorrelation of disturbance occurrence is a characteristic unique to the disturbance regime in that it cannot be meaningfully applied to a single event. Although there has been some consideration of the effects of temporal autocorrelation of disturbance (Moloney & Levin 1996; Johst & Wissel 1997; Tuljapurkar & Haridas 2006; Snyder 2008; Schreiber & Ryan 2011) and spatial autocorrelation of disturbance events (Gill, Allan & Yates 2003; Johst & Drechsler 2003; Vuilleumier et al. 2007; Elkin & Possingham 2008), here we specifically consider how the interactions between autocorrelation and intensity at varying frequencies affect competitive outcomes and how these outcomes may be used to improve management efficacy.
The community consequences of these interactions have potential applications for land management and conservation. The premise of utilizing disturbance to achieve management goals involves the intentional introduction of a disturbance to the system, or the alteration of an existing disturbance regime. For instance, land managers often implement prescribed burning, controlled flooding, pesticide application and mowing either to promote species diversity (hence coexistence) or to control an invasive species. The level of autocorrelation of such anthropogenic disturbance occurrences may be altered at little to no cost to the manager. To illustrate, consider a land manager who can afford to mow her field 25 times over the next 50 years, a frequency of 0·5. She may choose a number of distinct disturbance regimes at this frequency (e.g. mow every other year for 50 years, mow for 25 consecutive years and then not for the final 25), all of which will carry an identical total cost (assuming no economic discounting). The variable being manipulated in these regimes is autocorrelation (see ‘'Materials and methods'’ for explanation and implementation of autocorrelation). In this paper, we demonstrate that manipulating the autocorrelation of disturbance occurrence can influence community properties, specifically species' densities and species coexistence. Using a well-known annual plant model, we show that, given a constant frequency and intensity, the level of autocorrelation of disturbance occurrence can affect whether the end result is stable coexistence or competitive exclusion. Furthermore, we show that even when manipulating the autocorrelation cannot produce the desired competitive outcome, such changes can affect the density of each species, giving the manager the ability to choose which species dominates the community. While our model is not designed to give specific recommendations for any particular real community, our results highlight that the manipulation of disturbance autocorrelation may be a potentially valuable tool for land managers. For specific recommendations in a given system, our methods can readily be applied to purpose-built community models, or even directly tested empirically.
Materials and methods
To determine the interactions between temporal autocorrelation of disturbance occurrence and intensity, we developed an annual plant model that builds upon similar models that have been used to study the effects of aspects of disturbance on coexistence (Chesson 1994; Roxburgh, Shea & Wilson 2004; Miller, Roxburgh & Shea 2011). Our model is a system of stochastic finite-difference equations and incorporates the Reciprocal-Yield Law (Shinozaki & Kira 1956; Ellner 1985) to describe the reduction in seed yield owing to competition. Here, ‘reciprocal yield’ refers to the assumed inverse relationship between the population density of mature plants in the community and the species' per capita seed yield. The strength of this relationship is controlled by a competition parameter, αjk. A high value of αjk represents the presence of species k seeds having a strong, negative impact on the ability of species j seeds to grow and survive.
Our model includes the following species' life-history parameters: germination rate (G), seed bank survival (s) and seed yield (Y) (2). We use a common notation, letting Xjt represent the number of seeds of species j at time t. We define Xj,t+1 = Xjt, where (the finite rate of increase of species j at time t) is equal to
Here, Cjt is defined as
which, ecologically, is the number of species j seeds that germinate in a given year, adjusted by the competition term αjk. Note that the intraspecific competition coefficients, αjj, are set equal to 1 without the loss of generality (Ellner 1985). Finally, Dt represents the decrease in seed yield at time t owing to disturbance and is explained further below.
The disturbance regime in our model is controlled by three parameters: autocorrelation of disturbance occurrence (ρ), intensity (I) and frequency (F). To illustrate autocorrelation, we show several examples of disturbance regimes at various autocorrelation levels (Table 1). Zeros represent years with no disturbance, and ones represent years that do have a disturbance.
Table 1. Disturbance occurrence patterns at varying levels of autocorrelation
… 010101010101 …
… 110111000100 …
(independent Bernoulli process)
… 00000001111111 …
… 1111 … OR … 0000 …
Table 2. Summary of terms in simulation model
Seed bank Survival
Competitive effect of species k on species j
Temporal autocorrelation of disturbance occurrence
Autocorrelation (ρ) is measured on a scale from −1 to 1. A disturbance occurrence process with an autocorrelation of −1 is completely deterministic. At each time step, or each year, there will be a disturbance occurrence if there was not an occurrence at the previous time step, and likewise, there will not be a disturbance occurrence if there was one at the previous time step. An autocorrelation of 0 means that the likelihood of a disturbance occurrence at any given time step is independent of the previous time step. In this case, disturbance occurrence is an independent, identically distributed Bernoulli process with the probability of success (disturbance occurrence) equal to the frequency. That is, the disturbance occurrence process is analogous to tossing a coin (weighted according to the frequency) at each time step. Finally, a process with an autocorrelation of 1 will again be deterministic, and the presence of disturbance at any given time step will be identical to that of the previous time step.
Intensity (I) is defined as the proportion of the seed yield that is destroyed by the disturbance. It should be noted that this is a measure of the response of species and not a physical quantity and that this response is the same for each species. Frequency (F) is defined as the proportion of years during which a disturbance occurs. Therefore, both I and F are values between 0 and 1. In disturbance years, we set Dt = (1–I). In non-disturbance years, we set Dt = 1. Thus, Dt reduces the seed yield by 1 per cent for disturbed years and has no effect for non-disturbed years.
We use invasion analysis methods to establish stable coexistence. In brief, stable coexistence is indicated when both species have a positive long-term, low-density growth rate, which describes how species grow when introduced at a low density in the presence of an established resident (Chesson 1994). In order to calculate the value of of both species for a range of autocorrelations, intensities and frequencies, we use three simulation parameters: number of autocorrelation steps, number of intensity steps and number of time steps. The simulation can be run at any fixed frequency and calculates the value of for both species at each combination of autocorrelation and intensity. Using the methods of Lunn & Davies (1998) to generate correlated binary data, we ran the simulation at frequency 0·5 using 100 autocorrelation steps ranging from −0·99 to 0·99, 100 intensity steps running from 0·01 to 0·99 and 10,000 time steps, representing 10,000 years. In this paper, we use frequency 0·5 as our primary example, although our results are qualitatively robust to changes in frequency. We created streams of autocorrelated binary data over the full range of autocorrelation (from −1 to 1) at this frequency by using a simple Markov process. It is difficult to create streams of autocorrelated binary data over the full range of autocorrelation for arbitrary frequencies, and for values near −1, it may be unknown (Emrich & Piedmonte 1991). Therefore, we only consider positive autocorrelations when we expand the simulation to other frequencies. However, our results show that interactions between autocorrelation and intensity are much stronger at positive autocorrelations, making our focus on this range appropriate. If is positive for both species, then stable coexistence is established for that particular combination of autocorrelation and intensity. We illustrate coexistence regions by shading the portion of the parameter space for which stable coexistence has been established in an autocorrelation vs. intensity plot. To ensure that our results do not depend strongly on species' life-history parameters, we conducted sensitivity analyses by generating coexistence regions analogous to those in Fig. 1b for small relative and absolute changes to each of the eight species parameters; these changes make no substantial difference to the shape of the coexistence regions (results not shown). Finally, we interpret our model in terms of three commonly used management strategies, to illustrate the management potential of our work.
Our results show that the level of autocorrelation of disturbance occurrence can have a significant impact on species coexistence (Fig. 1). Figure 1b shows that at a fixed intensity, species 1 may competitively exclude species 2, the species may coexist, or species 2 may competitively exclude species 1, depending on the level of autocorrelation. The shaded area represents combinations of autocorrelation and intensity where both species coexist. The area to the left of this coexistence region represents the combinations for which species 1 will competitively exclude species 2. The area to the right of the coexistence region shows combinations of autocorrelation and intensity that will lead to species 2 competitively excluding species 1. Manipulating autocorrelation with respect to this coexistence region has the potential to alter competitive outcomes, or relative species densities, and these effects have potential benefits when applied to management. To illustrate, we provide examples for three management strategies: prescribed burning, grazing and mowing.
Prescribed burning is a common disturbance used by land managers to control community composition, manage the spread of invasive species, conserve endangered species and reduce potential damage to human settlements (Gill 1977; Richards, Possingham & Tizard 1999; Van Dyke et al. 2004). We consider a land manager who desires to manage two species of annual plants using this method. Assume that he is interested in conserving diversity, and the objective is for the species to coexist. If species and disturbance parameters are as indicated by point X in Fig. 1b, we see that species 1 will competitively exclude species 2. A slight increase in intensity will produce coexistence, but suppose the land manager lacks the necessary funds to control hotter fires. Figure 1b shows that coexistence may also be achieved with a slight decrease in autocorrelation (i.e. a downward movement from point X into the coexistence region). Because the average number of disturbances per unit time remains fixed, this change requires no additional man-hours. Thus, manipulating autocorrelation can provide an effective means to meet management goals without incurring additional costs.
Another common management strategy is the use of grazing (Vujnovic, Wein & Dale 2002; Hickman et al. 2004). Instead of conserving diversity, suppose the land manager desires to eradicate an invasive annual plant species (species 2). If species and disturbance parameters are as point Y in Fig. 1b, we see that the species will coexist. The manager desires species 1 (the native species) to competitively exclude species 2 (the invasive species); this outcome occurs in the area to the left of the coexistence region. To move the system to this area, she may either decrease intensity or increase autocorrelation. Depending on specific circumstances, decreasing the intensity (reducing stocking rates) may result in financial losses, but increasing autocorrelation is likely to be a no-cost decision.
Unlike grazing and burning, mowing allows the land manager a higher degree of precision (DiTomaso 2000; Van Dyke et al. 2004). Suppose we reverse the roles of the species so that species 1 is now invasive. The manager desires species 2 to competitively exclude species 1. The current disturbance regime is identical to the previous example, so the species currently coexist. To achieve the stated management goal, he must select a combination of intensity and autocorrelation to the right of the coexistence region. This situation is similar to the first example in that the manager must either increase intensity or decrease autocorrelation. There may be situations where the height or the density of field vegetation, risk of soil erosion or the need for new equipment may render such high-intensity mowing impractical. In these cases, changing the autocorrelation may present a viable alternative.
Figure 2 illustrates the changes in the autocorrelation vs. intensity coexistence region that take place when we fix the frequency of disturbance at different values. Three cross-sections show that the shape of the coexistence region is robust to changes in frequency. However, we note that the base of these coexistence regions corresponds with the coexistence region in the intensity–frequency plane (Miller, Roxburgh & Shea 2011). Figure 2 as a whole illustrates how the frequency vs. intensity coexistence region fits together with the autocorrelation vs. intensity coexistence regions in three-dimensional space.
The effects of autocorrelation are not confined to influencing competitive outcomes. Even within the coexistence region, different patterns of species population dynamics emerge. Figure 3a, b shows that for the highest autocorrelation levels that result in coexistence, species 1 consistently exhibits a higher population density than species 2. As autocorrelation is decreased, the density of species 1 decreases and the density of species 2 increases. With the decrease in autocorrelation, the species switch roles. At the lowest levels of autocorrelation, species 2 exhibits the higher population density (Fig. 3e, f).
We illustrate this control of species density further in Fig. 4. For all results presented so far, we have used a stochastic disturbance occurrence process. Recognizing that this stochasticity may be difficult to implement from a logistical perspective, here we use a completely deterministic disturbance occurrence process. From year zero, the disturbance occurs in four-year blocks, that is, 00001111. At year 50, we change the autocorrelation of disturbance occurrence from 0·5 to −1, so that disturbance occurs every other year, 0101, etc. In just 30 years following this change, the density of species 1 decreases substantially; manipulating autocorrelation can generate significant impacts in reasonable time frames.
Since the influential work of Connell (1978), great strides have been made towards understanding the role of disturbance in promoting species coexistence (Petraitis, Latham & Niesenbaum 1989; Haddad et al. 2008; Svensson, Lindegarth & Pavia 2009). However, despite the clear potential for real-world applications, much of the previous work has been largely focused on developing a deeper basic understanding of disturbance (Moloney & Levin 1996; Roxburgh, Shea & Wilson 2004; Snyder 2008; Miller, Roxburgh & Shea 2011). In this paper, we are particularly interested in using and extending insights provided by such theoretical work to explore potential applications to management (including pest control, conservation and restoration). While manipulation of disturbance aspects may be logistically or financially prohibitive, alteration of autocorrelation in these aspects will often have no (or relatively low) cost. We acknowledge that high levels of autocorrelation may require a large initial expenditure followed by a large saving or a large initial saving followed by a large expenditure. While these may carry some important financial implications, the general notion that the same amount of money is being spent is still valid. Moreover, a change in how the expenditure is allocated across time is equally likely to result in a slight savings as it is to result in a slight additional cost. Thus, we here focus on the potential management benefits of manipulating temporal autocorrelation.
From the three examples in the results section, we see that manipulating autocorrelation can alter species coexistence results. However, even if a manager cannot achieve his coexistence or eradication goals, changing the autocorrelation level may still yield benefits at the population dynamics level. We recall that in example three, the goal of eradicating species 1 can be attained by increasing intensity. If the manager's financial situation is such that he cannot afford to increase the intensity higher than that of point Z in Fig. 1b, we see that no change in autocorrelation (vertical movement on Fig. 1b) will move the system parameters to the right of the coexistence region as desired. However, by changing the autocorrelation level, the manager may still be able to mitigate the effects of the invasive species by lowering its population density. From the species density plots in Fig. 3, we see that even though all of the plots show species coexistence, the population dynamics change as the level of autocorrelation is changed. The invasive species (species 1) has a high density at the highest levels of autocorrelation that result in coexistence. As autocorrelation is decreased, the density of the invasive species decreases, while the density of the native species increases. At autocorrelation −0·99, we see that the native species (species 2) consistently exhibits a higher density than the invasive species. So while manipulating the autocorrelation fails, in this case, to produce the desired competitive exclusion outcome, a manager may still depress the density of the invasive species by lowering the autocorrelation, thus minimizing its undesirable effects, especially if such a decrease reduces the invader below an economic damage threshold. Of course, it is important for the manager to identify his specific management goal, as the optimal strategies for complete eradication and density reduction may be different (Shea et al. 2010).
Taking a step back from specific examples, we consider the general shape of the coexistence region in Fig. 1b. This pattern of coexistence is somewhat surprising and counterintuitive. At high levels of autocorrelation, we might expect to conserve diversity by decreasing the intensity to counteract the high autocorrelation. However, Fig. 1b shows that the opposite is true. Rather, higher intensities are required to maintain coexistence at high levels of autocorrelation. This general pattern was evident for all combinations of species and disturbance parameters that we examined, and Moloney & Levin (1996) found a similar result using disturbance rate, defined as the average area affected per unit time, instead of intensity.
In each of our example scenarios, the frequency of disturbance occurrence was fixed at 0·5. Of course, varying the frequency may change how autocorrelation interacts with intensity. Miller, Roxburgh & Shea (2011) have shown how frequency and intensity interact at zero autocorrelation. In this study, we have extended this earlier work to include the effects of non-zero autocorrelation. Figure 2 shows how the frequency vs. intensity coexistence region can be combined with the intensity vs. autocorrelation coexistence regions from this study to form a frequency vs. intensity vs. autocorrelation coexistence region in three-dimensional space. With some exceptions (e.g. mowing, for which changing the intensity may have low cost), autocorrelation is the only one of these variables that may be changed or easily controlled at no additional cost.
We have used our disturbance framework to demonstrate that temporal autocorrelation of disturbance occurrence has the potential to alter competitive outcomes and, in cases of coexistence, change population density dynamics. However, it is important to clarify that the purpose of this work is to illustrate the concept of autocorrelation and its possible utility, rather than to provide a direct application to management. In principle, our approach could be used to make recommendations for a specific system and objective, although a new model designed and parameterized specifically for the system in question would be necessary. In some systems, managers may not know enough about the system and its potential dynamics to manipulate disturbance appropriately. In such cases, the use of adaptive management (intervention with a specific plan for learning about the system while managing: Shea et al. 2002; Williams, Szaro & Shapiro 2009) to learn how autocorrelation affects their particular systems would be recommended. Although determining, for example, seed bank survival can be time-consuming and expensive, many land managers do have access to species trait parameters for their communities that could be used in appropriate models.
While we have not restricted the framework to any specific time-scale for this study, managers are often more concerned with short-term results. Adaptive management would also be an effective tool to learn how short-term behaviour may differ from behaviour in the long run. Although the effects of changing autocorrelation may take many years to become apparent in some systems, our results (Fig. 4) show that relatively rapid results can be achieved in annual plant communities. Finally, we note that while competitive exclusion will theoretically occur at any point outside the coexistence region, the rate of exclusion is not uniform, and competitive exclusion will occur at faster rates when the system parameters are farther away from the coexistence region.
Although our work shows that there is large management potential in manipulating autocorrelation of disturbance, there are caveats to our approach. First, our model only describes annual plants. However, in savannas, prairies and alpine meadows, annual species are well represented and amenable to disturbance-based management. Second, while we have presented our results in terms of stochastic disturbance occurrence, we recognize that this may not be tractable for land managers, because using a randomized disturbance occurrence process may cause staffing issues. As we illustrate in Fig. 4, our results are also valid for deterministic disturbance regimes, and thus, control can be achieved via a fixed schedule of disturbance. In a real-world context, the cyclical densities depicted in Fig. 4 may also be utilized to help meet management goals, for instance by applying additional control methods at key points in the cycle. Finally, we note that in our model, competition between species plays a key role, influencing coexistence and community diversity. However, other models of conservation based on disturbance and succession do not explicitly invoke strong species interactions (e.g. Richards, Possingham & Tizard 1999), and understanding how disturbance autocorrelation affects community composition in such models will be an interesting area for future investigation.
We have chosen here to focus on temporal autocorrelation of disturbance occurrence, perhaps the most intuitively obvious form of autocorrelation. However, the scope of autocorrelation in general is much wider than this. For example, many models have provided evidence that spatial environmental autocorrelation can also have an effect on species coexistence (Moloney & Levin 1996; Snyder 2008) and extinction risk (Johst & Drechsler 2003; Vuilleumier et al. 2007). Furthermore, one might consider the potential effects of autocorrelation of other aspects of disturbance, such as extent or timing. The logic used to show that temporal autocorrelation of disturbance occurrence can be changed at no additional cost may be qualitatively extended to include autocorrelation of other disturbance aspects.
The effects of autocorrelation for every aspect of disturbance share the same intriguing factors: the potential to change competitive outcomes and affect diversity maintenance and invader management at a minimal additional cost. Indeed, this work has the potential to provide more options for land managers who are forced to make difficult decisions under tight budget constraints. However, only through a firm commitment to learning how these processes work in their particular field environments can this flexibility be achieved.
We thank members of the Shea laboratory for their helpful comments. This work was supported by NSF grant DEB-0815373 and an NSF REU to K.S.