Introduction

In the last three decades, one of the most widely reported phenomena to occur on Caribbean reefs is the general fall in the cover of living coral and rise in macroalgae (Kramer 2003; Cote et al. 2005). The trend is by no means ubiquitous with exceptional, coral-dominated reefs being found either in isolated patches (Idjadi et al. 2006) or representing depth strata of entire reef systems (e.g. Bruckner and Bruckner 2003). However, the prevailing ecological balance between macroalgae and corals has changed for a variety of reasons, some of which continue to be debated (Aronson and Precht 2006; Burkepile and Hay 2006; Littler et al. 2006; Mumby and Steneck 2008). Major factors driving the bloom in macroalgae include the loss of corals by disease (Bythell and Sheppard 1993), hurricanes (Bythell et al. 1993) and coral bleaching (Kramer et al. 2003); reductions in top-down grazer controls of algae after the die-off of the urchin Diadema antillarum (Lessios et al. 1984) and overfishing of herbivorous fishes (Hughes 1994) and local deterioration of water quality (Littler et al. 1993).

The change in reef community structure in favour of macroalgae is an example of a phase shift (Done 1992). The stability of this phase shift is a matter of great concern, because macroalgal-dominated reefs have been viewed (by some) to represent a stable community state of coral reefs (Hatcher 1984; Knowlton 1992; Bellwood et al. 2004; McManus and Polsenberg 2004). Specifically, if macroalgal dominance is an alternate state of the ecosystem to coral dominance, and if both states are stable, then it could be challenging to reverse the trajectories of many Caribbean reefs and facilitate a rise in living coral. Alternate states of the system become stable because they are reinforced by self-perpetuating positive or negative feedback mechanisms (Holling 1973; Scheffer and Carpenter 2003; Mumby and Steneck 2008). In the case of a shift to a coral-depleted state, for example, macroalgal levels can become high enough to limit coral recruitment, thereby starting a bottleneck in the coral population (Fig. 1). Once adult corals are lost, either because of disturbance or chronic mortality, coral cover is likely to decline because the bottleneck constrains the rate at which adults are replaced. The fall in coral cover liberates new space for macroalgal colonisation and essentially decreases the intensity of grazing (i.e. the rate at which each patch of substratum is regrazed tends to decline as available area increases). The drop in grazing intensity allows macroalgal cover to increase further, thus strengthening the bottleneck in the coral population, and further reinforcing the feedback (Fig. 1).

Fig. 1
figure 1

Schematic representation of positive and negative feedback loops on coral reefs (source: Mumby and Steneck 2008, used with permission)

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Whether coral- and macroalgal-dominated reefs represent alternate stable states of the system has not been proven. An alternative scenario is that the phase shift represents a linear continuum of reef state without feedback mechanisms. If that is the case, then a reduction in macroalgal cover may facilitate a rise in coral cover fairly easily. Indeed, in their review of alternate stable states, Petraitis and Dudgeon (2004) raise the possibility that coral–algal phase shifts may not necessarily involve alternative stable states. The authors point out that the increase in coral recruitment observed in Jamaica, following the local recovery of Diadema and reduction in macroalgae (Edmunds and Carpenter 2001), might imply a simple phase–shift relationship, namely, that the return to coral dominance could be a simple phase shift brought about by an increase in grazing. Further, there is widespread evidence that macroalgal cover and grazing are strongly negatively related in the Caribbean (Williams and Polunin 2000; Kramer 2003; Newman et al. 2006; Mumby et al. 2007b) and that reduced macroalgal cover is associated with elevated densities of juvenile corals (Edmunds and Carpenter 2001; Mumby et al. 2007a). Therefore, given this evidence, do reefs exhibit alternate stable states or are phase shifts easily reversible phenomena?

Determining the stability properties of coral reefs from field experiments is almost prohibitively difficult, largely because of ethical considerations in performing large-scale manipulations and the logistical constraint that many coral species have long generation times (Petraitis and Dudgeon 2004). In a recent article, we explored the stability of Caribbean coral populations by using a mechanistic ecological model (Mumby et al. 2007a). The model included many important ecological processes, and each was parameterised from empirical data. The model appeared to behave reasonably as it was found to fit an independent field dataset from Jamaica from the 1970s to the 1990s. We concluded that Caribbean coral populations exhibit two alternative stable states under certain physical and biological conditions. Stability was inferred because each state formed a zone of attraction that resisted certain fluctuations in grazing or the strength of coral–algal competition.

The present study aims to reconcile the potential paradox that a mechanistic model of Caribbean reefs finds coral-depauperate states to be insensitive to certain changes in grazing, whereas a wealth of empirical and experimental evidence suggests that macroalgal cover is strongly negatively correlated with grazing (Lewis 1986; Williams and Polunin 2000; Kramer 2003; Mumby et al. 2006). In other words, how can a modelled ‘macroalgal-dominated, coral-depauperate state’ resist changes in grazing, whereas empirical studies show that macroalgal cover is directly related to grazing? Furthermore, does the recovery of corals after modest Diadema recovery imply that Caribbean reefs exhibit easily-reversible phase shifts rather than alternate stable states as mooted by Petraitis and Dudgeon (2004)? Resolving these issues will help clarify whether the suggestion of alternate stable states is consistent with field observations for Caribbean systems.

Methods

A simulation model was originally designed to represent mid-depth Montastraea-dominated forereefs which typically harbour the highest biomass and diversity of reef organisms (Mumby et al. 2008). Since white band disease has depleted populations of large, branching corals (Aronson and Precht 2001), stylised massive growth forms of coral were simulated together with rates of recruitment, growth, reproduction and mortality. The model is a square lattice of 2,500 cells each of which approximates 0.25 m2 of reef, and can be occupied by a mixture of living and dead substrata (Table 1). Although the reef has a continuous (toroidal) lattice of 2,500 cells, the lattice structure merely helps define probabilistic rules of coral recruitment and vegetative algal growth. Individual cells comprise multiple coral colonies and algal patches so that interactions occur at colony scales as they do in situ. Corals can recruit to individual patches of cropped algae (which in this model harbour little sediment) but not macroalgae. Macroalgae have a 70% chance of becoming established if dead coral (cropped algae) is not grazed for 6 months (mostly based on Dictyota), and this increases to 100% probability after 12 months of no grazing (mostly based on Lobophora). Rates of algal dynamics were acquired from detailed centimetre-resolution observations of algal dynamics with and without grazing (Box 2008). Once established, macroalgae also spread vegetatively over cropped algae (de Ruyter van Steveninck and Breeman 1987).

Table 1 Contents of individual cells (0.25 m2) within the model
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The modelling of grazing is carried out at two scales. First, the simulation model requires an overall grazing impact, which defines the proportion of reef effectively maintained in a cropped algal state during a 6-month iteration of the model. In other words, given the overall balance between algal production and grazing, the grazing impact is the proportion of a reef that is grazed sufficiently intensively that algae are kept in a cropped state rather than developing into a macroalgal canopy within a 6-month period. A ‘6 monthly’ value of grazing impact is needed for model implementation because many of the parameters, such as the vegetative growth of corals and macroalgae, are derived at monthly scales that are many orders of magnitude greater than the instantaneous grazing rate of fishes (seconds). Grazing is spatially constrained (Williams et al. 2001; Mumby 2006; Mumby et al. 2006). An unfished community of parrotfishes grazes a maximum of 40% of the seabed per 6-month time interval. The dynamic basis of this grazing threshold is poorly understood seeing as most measures of grazing take place at scales of seconds and most studies of algal dynamics take place on monthly scales. Nonetheless empirical studies (Mumby 2006), and experimental results scaled to the complex forereef (Williams et al. 2001), have identified a grazing limit of 30–40% of the reef during 6 months. This value allows for a numeric positive response by parrotfish after severe coral mortality events during which colonisation space for algae increases dramatically. For example, an increase in parrotfish biomass over 5 years maintained the cover of cropped algae at Glovers Reef at 30–40% after Hurricane Mitch caused extensive coral mortality and liberated new settlement space for macroalgae (Mumby et al. 2005) (i.e. grazing impact remained at around 30% 6 month−1 even though coral cover dropped from around 60–20%, suggesting that the approach is robust during phase changes). The reasons for this are not fully understood. All parrotfish species graze algal turfs and in doing so constrain the colonisation and vegetative growth of macroalgae. Direct removal of macroalgae occurs through the grazing of larger sparisomid species (Mumby 2006).

Minimum values of grazing (5% 6 month−1) were estimated using a second, instantaneous model of grazing behaviour parameterised for the size distribution of parrotfishes from a heavily fished site in Jamaica (Mumby et al. 2006). This second model measures grazing behaviour which is the instantaneous grazing behaviour of a parrotfish species or community measured as a percentage of horizontal reef area grazed per hour and is sensitive to species, length and life phase (Mumby et al. 2006). Grazing affects all algal classes such that macroalgae are converted to the class ‘cropped algae’ and grazed cropped algae are maintained in the same state. This parameterization is appropriate for reefs in which the dominant macroalgal genus is Dictyota and where sparisomids dominate the grazer community, such as in Belize. The most common sparisomids (Sparisoma viride and Sparisoma aurofrenatum) both feed extensively on this macroalga.

Competitive interactions between corals and macroalgae reduce the growth rate of each taxon. Macroalgae have an ability to overgrow corals but are not a major source of coral mortality in the model (Nugues and Bak 2006). Corals are subjected to size-dependent partial-colony mortality (including parrotfish lesions and minor disease) and whole-colony mortality. Coral recruits experience additional mortality from parrotfish predation (Box and Mumby 2007) but reach an escape in size at 5 cm2 diameter (Meesters et al. 1997). All simulations assume no stock–recruitment relationship and corals recruit at maximum levels irrespective of stock size (i.e. up to 4 per 0.25 m2). Previous studies with the model have found it to be insensitive to the type of stock–recruitment relationship used, largely because of high post-settlement mortality (Mumby 2006). Individual cells in the lattice are updated in random sequence. All parameters are fitted from empirical studies (Table 2). Sensitivity analyses reveal that model predictions are robust to stochastic variation in parameter values and able to represent long-term fluctuations in the health of reefs in Jamaica for which data were available (Table 3).

Table 2 Basic parameterisation of simulation model for mid-shelf reefs without significant sediment input and sparisomid-dominated grazing
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Table 3 Sensitivity of model, defined as difference in threshold grazing at 10% live coral cover between standard parametrisation and that with adjusted parameter
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Model experiments

In order to illustrate the key dynamics of corals and macroalgae, a simple simulation was implemented in which coral cover was set initially at 35% (with an equal mix of brooders and spawners) and macroalgal cover at 10%. A total of 36 reefs were created, each having a unique but fixed level of grazing impact ranging from 5% 6 month−1 to 40% 6 month−1. Sources of acute disturbance were removed although chronic levels of background partial-colony mortality, whole-colony mortality and algal overgrowth of coral were free to occur. In essence, the simulation represents the natural trajectory of reefs between acute disturbance events, i.e. when free of further acute disturbance does the reef degrade, remain at equilibrium or become healthier?

In order to explore whether the model, which exhibits alternate stable states, can reproduce the observed continuous negative correlation between macroalgal cover and grazer impact (Williams and Polunin 2000), a simple experiment was conducted. Taking the range of coral cover reported by Williams and Polunin (2000) (6–36%), the dynamics of a similar set of reefs, with starting cover ranging from 5% to 35% in 5% increments, were simulated for a full range of fish grazing levels (i.e. 7 levels of initial coral cover × 36 levels of grazing = 252 reefs). After a 10-year period, a reef was selected at random for each level of fish grazing and the corresponding cover of macroalgae determined (e.g. for grazing level 9% 6 month−1, macroalgal cover might have been extracted for the reef initialised with 5% coral cover). This approach avoided introducing a systematic bias between coral cover and grazing impact on macroalgal cover. The approach is also conceptually similar to a field study that randomly (or pseudo-randomly) samples a number of reefs, each on their own trajectories, but within some initial range of possible starting conditions. The strength and direction of relationship between grazing impact and macroalgal cover were then evaluated using the Pearson Product-Moment correlation coefficient. The entire process was simulated 100 times, and the mean correlations and inter-quartile ranges are comparable to published data from Williams and Polunin (2000). Note that absolute values of macroalgal cover were not compared between model and field data because additional field data would be required to fully parameterize the model for site-specific conditions such as sediment level and the cover of other taxa such as sponges, and such data were not available.

Results and discussion

Alternate stable states and the equilibrial dynamics of corals and macroalgae

Equilibrial dynamics of corals

The first model experiment tracked the trajectories over time of 36 reefs, each initialised with the same coral cover (35%) but at various levels of fish grazing. A striking pattern in reef trajectories developed over time as a function of grazing impact (Fig. 2a, b). Those reefs with grazing of <20% 6 month−1 exhibited a gradual decline in coral cover and the rate of decline increased at lower levels of grazing impact (Fig. 2a). A contrasting pattern of coral cover occurred once grazing levels exceeded 20% 6 month−1. Here, coral cover increased and did so at an increasingly faster rate as grazing levels rose (Fig. 2a). Thus, reefs with higher grazing exhibited recovery (coral colonisation and growth outweighing mortality), whereas reefs at lower grazing exhibited degradation (mortality outweighing processes of colonisation and growth). The reef at the fulcrum or threshold level of grazing (around 20% 6 month−1) remained at the initial coral cover of 35% and was therefore at equilibrium. Viewing these dynamics allows some terminology to be assigned. Transient dynamics are those that occur as the reef moves towards a stable equilibrium (transients are shown in circles and equilibrial dynamics as squares in Fig. 2). The equilibrium at 35% coral cover and grazing of 20% 6 month−1 is considered to be unstable because reefs slightly to either side of this point are attracted towards either high or low coral cover. Indeed, most reefs eventually reach one of two possible stable equilibria in the absence of acute disturbance; a high coral cover state or a coral-depauperate state.

Fig. 2
figure 2

Temporal trajectories of reefs under chronic disturbance and varying fish grazing starting from an initial coral cover of 35%. Transient cover is represented using circular markers and stable equilibria by squares. Plots shown simultaneously for a coral cover and b macroalgal cover

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The example given here identified an unstable equilibrium (threshold) occurring at a coral cover of 35% and a grazing level of 20% 6 month−1. In order to locate other thresholds, the simulation model was run for 200 years for every combination of coral cover and grazing. For many levels of grazing, three equilibria emerged; a stable coral-depauperate state, an unstable equilibrium at intermediate coral cover and a stable coral-rich state (shown in detail in Fig. 3 which identifies two additional unstable equilibria to that identified in Fig. 2). Unstable equilibria were identified where coral cover remained within two percentage units of the initial value during the 200-year simulation (Fig. 3), thereby discriminating unstable equilibria from non-equilibrium points nearby. For the range of grazing associated with herbivorous fish on exposed forereefs (i.e. 5–40% 6 month−1), unstable equilibria fell along a diagonal line from an upper left bifurcation (>70% coral cover and grazing 5% 6 month−1) to a bottom right bifurcation, located at low coral cover (<5%) and a relatively high grazing level of 30% 6 month−1 (all equilibria shown in Fig. 4a). The two bifurcations delimit a zone of system instability, termed the ‘bifurcation fold’, such that corals can either exhibit recovery trajectories, remain constant, or exhibit decline; the outcome depends on whether the reef sits above, on, or below the threshold, respectively. Thus, a single level of grazing (e.g. 15% 6 month−1) can be associated with either reef recovery, stasis, or decline, the direction being determined by coral cover (Fig. 4a). Such bistability, or the existence of two possible community states from a single level of a process (grazing) is the very essence of alternate states (Petraitis and Dudgeon 2004). Such systems exhibit hysteresis; the level of grazing needed to maintain the coral-dominated system differs dramatically from that needed to return to coral dominance once the system has flipped. In order to view this graphically (Fig. 4a), consider a reef at 70% coral cover and grazing impact of 10% 6 month−1. The reef lies above the threshold unstable equilibrium and is stable. Now imagine that a mass coral bleaching event causes a sudden reduction in coral cover to a level below the threshold, such as 20%. The reef will now begin to shift towards a coral-depauperate state unless grazing levels can be elevated from 10% 6 month−1 to approximately 30% 6 month−1 to cross the threshold. Thus, a very different level of grazing is needed to reverse the loss of coral.

Fig. 3
figure 3

Identification of system equilibria for three levels of starting coral cover; 55% (black squares), 35% (green circles) and 15% (blue diamonds). Stable equilibria are denoted using solid markers and unstable equlibria using open markers. Unstable equilibria lie between two alternate stable states. In order to aid inspection of the graph, the stable equilibria for each initial coral cover have been offset artificially above one another, whereas in reality all stable equilibria fall at the level indicated for 15% cover

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Fig. 4
figure 4

Positions of stable (solid squares) and unstable (open circles) equilibria for Montastraea reefs at 7–15 m depth with high productivity and little sediment input. Equilibria determined after 200-year simulations and shown simultaneously for corals (a) and macroalgae (b). Colour coding of stable equilibria denotes the corresponding states of coral and macroalgae

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In short, the model implies that two stable alternate states of the system exist if grazing levels lie between 5% 6 month−1 and 30% 6 month−1 (Mumby et al. 2007a). Once grazing exceeds the upper bifurcation (30% 6 month−1), the system has only one stable equilibrium at high coral cover. It should, of course, be borne in mind that acute disturbances will prevent many reefs ever reaching that equilibrium, but the trend would tend to be one of recovery between successive disturbances.

A key point evident from Fig. 4a is that the locations of unstable equilibria covary with coral cover and grazer impact. This is because macroalgal cover is driven by grazing intensity, and grazing intensity depends on both the community composition of herbivorous fish (what is termed grazing impact in the model) and the area of available substrate. Thus, as the cover of corals or other ungrazable substrata increase, grazing intensity will tend to intensify and constrain macroalgal cover even if the number of herbivores remains unchanged (Williams et al. 2001). It is for this reason that unstable equilibria covary with coral cover and grazing impact (Fig. 4); coral cover and grazer biomass interact to determine grazing intensity, macroalgal cover and coral recruitment.

Unstable equilibria form threshold states of the system around which the trajectory can be one of recovery or decline. It is important to bear in mind that grazing is plotted on the x-axis for convenience and that the impact of many other ecosystem processes, such as nutrification, is to alter the shape of the bifurcation fold and the positions of thresholds. For example, if algal growth rates were elevated by nutrient flux, threshold values would tend to move towards the upper right end of the plot (see sensitivity analysis, Table 3).

Equilibrial dynamics of macroalgae

The trajectories of macroalgae during the first simulation experiment largely mirrored those of corals, albeit moving in the opposite direction (Fig. 2b). Once again, an unstable equilibrium occurred associated with a coral cover of 35% and grazing impact of 20% 6 month−1. However, the equilibria of macroalgae differed to corals in one important respect. For the ‘macroalgal-dominated’ or ‘coral-depauperate’ equilibrium, macroalgal cover varied linearly with grazing, whereas coral cover did not (Fig. 4b). For example, whenever a reef started out below the unstable equilibrium, the final equilibrial coral cover was vanishingly low irrespective of grazing level. However, unlike corals, the equilibrial macroalgal cover began to change linearly as grazing increased and eventually crossed a threshold beyond which it became excluded by corals (Fig. 2b). Thus, the key difference between the stable equilibria of corals and macroalgae is that the coral-depauperate state is insensitive to changes in grazing (at least until a threshold is crossed), whereas the macroalgal-dominated state changes linearly with grazing until a threshold is crossed. The reason for the difference is that there is insufficient coral in the coral-depauperate state to limit macroalgae. In this case, the cover of macroalgae is set by the balance of grazing and algal production (i.e. what is termed grazing impact in the model).

Transient dynamics of corals and macroalgae

In order to complete the description of equilibrial dynamics of corals and macroalgae, it is important to distinguish the rates of transient and equilibrial dynamics between taxa. Coral generation times typically involve decadal to century time scales (Hudson 1981) and therefore increases in coral cover occur slowly (Fig. 1a). Unlike corals, macroalgal dynamics occur on two inherent time scales. Macroalgae themselves exhibit fast transient dynamics with generation times on a scale of weeks to months (de Ruyter van Steveninck and Breeman 1987). Such fast dynamics allow macroalgae to equilibrate rapidly (on the order of months) to the present coral cover and grazing regime (Williams et al. 2001). However, a second, slow dynamic tracks the availability of settlement substratum and follows changes in coral cover. The equilibria plotted for corals and macroalgae in Fig. 4 reflect the long-term equilibria of the system, determined at the slow time scale over which corals reach equilibria.

Reconciling alternative stable states with a linear empirical relationship between grazing and macroalgal cover

Low coral cover is the key to reconciling model predictions of alternative stable states with linear empirical relationships between grazing and macroalgal cover. Given the recent history of intense disturbance in the Caribbean, the coral cover on many reefs today is low and typically <20% (Kramer 2003). If herbivorous fish can only graze approximately 40% of the reef (Williams and Polunin 2000; Mumby 2006) then settlement substratum is rarely going to be a limiting resource for macroalgae and the fast dynamics of algae enable them to equilibrate rapidly to the available settlement substratum and grazing regime, allowing negative correlations to emerge between grazing and macroalgal cover.

A model experiment simulated the transient dynamics of coral and macroalgal cover for low initial coral covers (5–35%) and a variety of fish grazing levels (Fig. 5). Random resampling of reefs for each level of grazing (see, for example, Fig. 6) led to a mean negative correlation between macroalgal cover and grazing of −0.74 (P < 0.001) which compares favourably to the value of −0.82 found by Williams and Polunin (2000). Moreover, the mean inter-quartile range of macroalgal cover from samples of 36 reefs (result = 25% cover) matched the variation reported by Williams and Polunin (2000). In short, a model coral ecosystem that displays alternate stable states was able to reproduce the continuous negative relationships between macroalgal cover and grazing found in the field. Such empirical relationships are therefore consistent with the existence of multiple stable states on Caribbean reefs.

Fig. 5
figure 5

Model-derived transient macroalgal cover after 10 years of simulations as a function of grazing and initial coral cover

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Fig. 6
figure 6

Example of relationship between grazing and macroalgal cover for reefs under nonequilibrial conditions (i.e. after just 10 years of simulations with initial coral cover = C, where 5% ≤ C ≥ 35%). For each level of grazing impact a reef was chosen at random to plot macroalgal cover

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The model predicts that macroalgal cover can vary considerably within a coral-depauperate state. This is an important outcome because ‘coral-depauperate’ reefs do not necessarily have to be totally dominated by macroalgae, nor are they totally devoid of coral. The key feature of a stable coral-depauperate reef state is that coral recruitment rates are not high enough to maintain the population (Fig. 1). However, juvenile corals do exist in this coral-depauperate state and their density increases with grazing (Fig. 7), which also explains the positive relationship between the rate of increase of transient coral cover and grazing observed in Fig. 2a. Indeed, the model prediction of a positive correlation between grazing impact and recruitment density (Fig. 7) is borne out empirically by a study of coral recruitment along a gradient of parrotfish grazing (Mumby et al. 2007b).

Fig. 7
figure 7

Relationship between fish grazing impact and the density of juvenile corals (colony area <60 cm2) under nonequilibrial time scales of 10 years and at 5% initial coral cover

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Empirical surveys of macroalgal cover may discover even lower levels of cover than those found by the model for a variety of reasons. These include very recent physical disturbance, seasonal reductions in macroalgal cover (Hoyt 1927), high cover of substrates that are not grazed (e.g. sponges), which serve to intensify grazing in the same manner as live coral, and the occurrence of mobile sediment that can prevent the colonisation of both corals and algae. The latter process may be particularly important in hardbottom habitats where both coral and macroalgal cover are low; arguably, such areas are not reefs at all.

Given the variability in macroalgal cover that can occur during a coral-depauperate stable state, it would perhaps avoid confusion in future if the scientific community adopted the term ‘coral-depauperate state’ rather than ‘macroalgal-dominated state’.

Reconciling phase shift reversal after Diadema recovery with alternate stable states

The coral bifurcation plot (Fig. 4a) must be revisited to answer the question raised by Petraitis and Dudgeon (2004) that recovery of Diadema could, in principle, indicate a simple phase–shift relationship with grazing rather than one that involves alternate stable states. The plot implies that increases in grazing within the bifurcation fold will not necessarily reverse the coral–algal phase shift. For example, increases in grazing from 5% 6 month−1 to 20% 6 month−1 on a reef with only 10% coral cover will lead to a modest reduction in algal cover but not enable a recovery of the coral population because the reef would remain below the unstable equilibrium. However, if grazing increased beyond the bifurcation fold (i.e. to at least 35% 6 month−1) algal cover would be sufficiently reduced that coral recruitment would attain levels necessary to allow a gradual increase in coral cover and eventually reverse the phase shift. The issue, therefore, is how the change in grazing relates to the position of critical thresholds. In a recent article, a modest recovery of Diadema in a system lacking many herbivorous fish was posited to shift grazing levels to at least 45% 6 month−1 and therefore enables corals to recover (Mumby et al. 2007a). Thus, a recovery of coral after a return of Diadema is fully consistent with the existence of alternate stable states: urchin recovery increases grazing so dramatically that the system moves beyond the zone of bistability.

Implications for reef studies

A mechanistic model of Caribbean reef dynamics that exhibits bistability has been found to be consistent with field observations of coral cover, macroalgal cover, grazing and patterns of coral recruitment: the low cover of coral on many Caribbean reefs allows simple linear relationships to emerge between these variables over short, transient time scales. Predictions of the ecological model are also consistent with observed reef state in those relatively rare situations in which coral cover and fish grazing are presently high, such as in Bonaire: the model predicts that macroalgae will be virtually excluded under these conditions (Fig. 4a), which is borne out by field observations from shallow (<20 m) parts of these reefs (Bruggemann et al. 1994; Bruckner and Bruckner 2003).

While use of a mechanistic model does not prove that Caribbean reefs can exhibit bistability under certain levels of fish grazing, there is as yet no field study that has rigorously tested the existence of multiple stable states on a coral reef. Perhaps the best hopes for testing the existence of bistability include improving mechanistic models, finding direct evidence of positive and negative feedbacks and interference competition (Grimm and Wissel 1997) and the use of field monitoring to identify the location of critical thresholds.

Finding field evidence for the existence of critical thresholds will be difficult for a number of reasons. Chief amongst these is that the underlying chronic dynamics of reefs—and therefore the location of the bifurcation fold and thresholds—will vary from one physical environment to another. For example, changes in algal productivity and sediment scour will strongly influence the location of thresholds yet such information is rarely quantified during monitoring or census projects. Moreover, many monitoring and census projects focus on measuring state variables (e.g. coral cover) yet data are also needed on rates of processes (e.g. grazing, productivity). Having said that, large repositories of monitoring data may prove to be invaluable in identifying thresholds, once the process variables have been quantified and categorised among sites. This is because reef trajectories differ either side of a threshold and it is these trajectories that must be observed. This can only be done through careful monitoring with high statistical power. Sites exhibiting positive trajectories of coral cover have physical and biological properties that place them above a threshold, whereas reefs exhibiting continued decline in the absence of acute disturbance (including disease outbreaks) lie below a threshold.

The importance of statistical power can be appreciated by considering the rates at which reefs might recover. In an average project lifetime of 5 years, changes in coral cover from a coral-depleted reef would be small, even if grazing levels were manipulated massively through use of a reserve or after urchin recovery (Fig. 8). In the example presented for a reef starting at 15% coral cover (Fig. 8), the maximum change in coral cover expected under chronic disturbance would be <6%, some seven times less than the change that could occur in macroalgae (though the rate of change in coral cover in situ will also be affected by local recruitment rates and the size distributions of corals). Given the slow pace of recovery of many Caribbean reefs (Connell 1997), the variable nature of macroalgal cover at a coral-depauperate state, and the variety of disturbance trajectories on which reefs lie which reinforce non-equilibrial dynamics, it should also be borne in mind that meta-analyses of coral cover are unlikely to provide definitive evidence for or against alternate stable states because reefs do not necessarily cluster into coral-rich and coral-depauperate states on transient time scales (cf. Bruno et al. 2009).

Fig. 8
figure 8

Effect of grazing on the relative rate of change in the cover of corals versus macroalgae during a management time frame of 5 years. Coral cover started at 15% in all scenarios and either exhibited chronic decline, stasis or limited recovery

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Understanding the mechanisms of hysteresis and locations of bifurcation folds will provide improved guidance for reef management (Nystrom et al. 2008). The existence of stable states means that reefs become more difficult to restore as their health deteriorates (Mumby et al. 2007a) and modest changes to the system, such as minor increases in grazing or reductions to nutrient flux, might not necessarily precipitate the expected benefits to coral populations. A better understanding of thresholds will enable us to assess the cost-effectiveness of potential management interventions and identify those interventions that offer the greatest shift to or from a threshold for a given cost.