Here we draw attention to the potential for pelagic bloom-forming cyanobacteria to have substantial effects on nutrient cycling and ecosystem resilience across a wide range of lakes. Specifically, we hypothesize that cyanobacterial blooms can influence lake nutrient cycling, resilience, and regime shifts by tapping into pools of nitrogen (N) and phosphorus (P) not usually accessible to phytoplankton. The ability of many cyanobacterial taxa to fix dissolved N2 gas is a well-known potential source of N, but some taxa can also access pools of P in sediments and bottom waters. Both of these nutrients can be released to the water column via leakage or mortality, thereby increasing nutrient availability for other phytoplankton and microbes. Moreover, cyanobacterial blooms are not restricted to high nutrient (eutrophic) lakes: blooms also occur in lakes with low nutrient concentrations, suggesting that changes in nutrient cycling and ecosystem resilience mediated by cyanobacteria could affect lakes across a gradient of nutrient concentrations. We used a simple model of coupled N and P cycles to explore the effects of cyanobacteria on nutrient dynamics and resilience. Consistent with our hypothesis, parameters reflecting cyanobacterial modification of N and P cycling alter the number, location, and/or stability of model equilibria. In particular, the model demonstrates that blooms of cyanobacteria in low-nutrient conditions can facilitate a shift to the high-nutrient state by reducing the resilience of the low-nutrient state. This suggests that cyanobacterial blooms warrant attention as potential drivers of the transition from a low-nutrient, clear-water regime to a high-nutrient, turbid-water regime, a prediction of particular concern given that such blooms are reported to be increasing in many regions of the world due in part to global climate change.
The majority of cyanobacterial taxa that form blooms in freshwater lakes are capable of fixing N, including Anabaena, Aphanizomenon, Cylindrospermopsis, Gloeotrichia, and Nostoc (Paerl 1988), and some are capable of storing excess N intra-cellularly as cyanophycin (Allen et al. 1984). Interestingly, the few taxa that are not able to fix N (e.g., Microcystis) often dominate hyper-eutrophic lakes, where both N and P may be less limiting than in low-nutrient lakes due to high levels of anthropogenic enrichment from surrounding watersheds (Xu et al. 2010, Paerl and Otten 2013b).
Mechanisms by which cyanobacteria can alter lake P cycling
Cyanobacteria may drive lake P cycling by accessing pools of P that are not generally available to eukaryotic phytoplankton (Fig. 1); these mechanisms are generally less widely appreciated than N fixation, especially outside of limnology. Many cyanobacterial taxa have ecophysiological traits that enable them to access the pool of sediment P, even in oxygenated conditions. These traits include a high-affinity phosphate uptake system activated at low P concentrations (Dignum et al. 2005), the production of extracellular polyphosphatase enzymes to access organic or sediment-bound P (Healey 1982), the capacity for luxury uptake (Healey 1982) and subsequent storage of P in excess of immediate demand (Whitton et al. 1991), and the ability to regulate buoyancy in the water column using gas vesicles (Perakis et al. 1996, Head et al. 1999b). These traits allow certain cyanobacterial taxa to access P stored in lake sediment (Xie 2006) or trapped in the hypolimnion of a stratified lake, sequester it in biological packets, and transport it to surface waters over daily to seasonal time scales (Fig. 1). This cyanobacterial translocation of P may be important in both stratified and well-mixed lakes, although the mechanisms may differ.
At a seasonal time scale, cyanobacteria in both stratified and well-mixed lakes can move P from the lake bottom into surface waters during recruitment from thick-walled dormant cells (akinetes) or from individuals that over-winter actively on or near the lake sediment (arrow 1 in Fig. 1). One example is Gloeotrichia echinulata, which overwinters as akinetes, germinates, and then spends 2–3 weeks in the sediment after germination, accessing sediment P via mechanisms that have not been well studied but are thought to include uptake of interstitial phosphate and excretion of extracellular enzymes to cleave phosphate from organic complexes (Whitton et al. 1991, Istvanovics et al. 1993, Pettersson et al. 1993). Because of its luxury uptake, each G. echinulata colony can contain as much as 81 pg P (Pettersson et al. 1993); this P is then moved into the water column during recruitment (Istvanovics et al. 1993, Pettersson et al. 1993, Tymowski and Duthie 2000). Less actively, entrainment may resuspend colonies, resulting in further upward transport (Mur et al. 1999).
Seasonal recruitment of cyanobacteria can be an important contribution to P cycles in stratified eutrophic lakes, above and beyond the contributions of redox-mediated recycling due to deep-water anoxia. For example, G. echinulata accounted for ∼66% of the total internal P loading in Green Lake, Washington, USA (2.25 mg P m−2 d−1; Barbiero and Welch 1992) and Lake Erken, Sweden (2.4 mg m−2 d−1; Istvanovics et al. 1993). Moreover, water column particulate P in Antermony Loch, Scotland (Pitois et al. 1997) and total P in Lake Peipsi, Estonia (Noges et al. 2004) increased in association with G. echinulata blooms. Other cyanobacterial taxa implicated in P translocation through recruitment include Cylindrospermopsis raciborskii (Istvanovics et al. 2000, 2002), Coelosphaerium naegelianum (Perakis et al. 1996), and Microcystis (Johnston and Jacoby 2003).
At a daily time scale, vertical migration between the upper and lower parts of a stratified water column (arrow 2 in Fig. 1) has been observed for colonial cyanobacteria with strong buoyancy control, including Anabaena (Head et al. 1999b), Aphanizomenon (Trimbee and Harris 1984, Osgood 1988, Barbiero and Welch 1992, Barbiero and Kann 1994, Jacobsen 1994), and Microcystis aeruginosa (Head et al. 1999b). These taxa are able to descend from the warm, well-lit epilimnion to the colder, poorly lit hypolimnion that can contain large amounts of P, especially when bottom waters are anoxic. Once P-replete, colonies return to the well-lit, but nutrient-poor, surface waters using gas vesicles and other mechanisms (Paerl 1988). Because this P would otherwise be released into surface waters only via storm-induced entrainment or seasonal mixing at lake turnover, diel vertical migration by cyanobacteria can provide a key source of P during periods of stable thermal stratification (Osgood 1988). For example, Aphanizomenon flos-aquae has been demonstrated to pump P into the water column during its diurnal migrations, increasing epilimnetic P concentrations when it is abundant (Trimbee and Harris 1984, Osgood 1988, Barbiero and Welch 1992, Barbiero and Kann 1994, Jacobsen 1994); estimated P loading rates for this species range from near those of G. echinulata (2.16–3.56 mg P m−2 d−1; Barbiero and Kann 1994) to 10 times that much (33 mg P m−2 d−1; Osgood 1988).
Finally, some cyanobacteria appear to be particularly effective at liberating bound P in sediments that can then be moved upward in biological packets or through physical processes such as entrainment (arrow 3 in Fig. 1). As with seasonal recruitment, this process can occur in both stratified and well-mixed lakes, and has been studied only in eutrophic systems. For example, Microcystis appears to facilitate the release of sediment-bound P, increasing soluble reactive P in and near the sediments (Brunberg and Bostrom 1992, Brunberg 1995) and in the water column during blooms (Xie et al. 2003, Shi et al. 2004), possibly due to productivity-induced increases in pH (Xie et al. 2003, Xie 2006). In fact, Xie et al. (2003) found that internal P loading by Microcystis during bloom formation could be two to four times as high as the average external loading in Lake Donghu, China (16.6–36.4 mg m−2 d−1 vs. 8.8 mg m−2 d−1).
The goals of this paper are to draw attention to the potential for cyanobacteria to have profound effects on lake nutrient cycling and ecosystem resilience and begin to explore these effects using a simple model of coupled N and P cycling. We demonstrate how some cyanobacteria can influence nutrient transport, lake trophic state, and resilience via biological alterations of N and P cycles, potentially amplifying the effects of redox-mediated internal P loading, anthropogenic activities in watersheds, and climate warming. Moreover, because cyanobacteria can reach high densities even in lakes with relatively modest total P concentrations (Downing et al. 2001, Ernst et al. 2009, Winter et al. 2011, Carey et al. 2012a), they may drive nutrient dynamics and resilience in lakes across a range of trophic states and stratification regimes.
Cyanobacteria and Lake Regime Shifts
The idea that cyanobacteria have the potential to drive lake N and P cycling complements and extends current literature about resilience and regime shifts in lakes. Here, we define “resilience” as the ability of an ecosystem to maintain itself in the same general state despite perturbation (sensu Holling 1973) and a “regime shift” as a major reorganization in ecosystem structure and/or function (e.g., Scheffer et al. 2001) linked to changes in the dominant feedbacks controlling system functions (e.g., Scheffer et al. 2001, Carpenter 2003, Scheffer 2009). Changes associated with regime shifts can be summarized in three steps: first, stabilizing feedbacks that keep a system in its current state weaken; second, stabilizing feedbacks are replaced by switching feedbacks that drive the system to a new state; and third, stabilizing feedbacks for the new state are established, thereby inhibiting restoration to the original state.
For example, low-nutrient lakes remain clear in large part because there is strong nutrient limitation of phytoplankton growth (Wetzel 2001). However, hypolimnetic anoxia creates a difficult-to-reverse switching feedback during the transition to a turbid, eutrophic regime (Nurnberg 1984, Carpenter 2003): when bottom waters go anoxic, the large pool of stored sediment P is released from complexes with metals. This P mixes upward, decreasing P limitation and increasing phytoplankton productivity (blooms). Later, after phytoplankton blooms senesce, they sink to the bottom and decompose, reducing oxygen still further and releasing yet more P (e.g., Carpenter and Cottingham 1997, Cooke et al. 2005, Smith and Schindler 2009). Hypolimnetic anoxia can also become a stabilizing feedback maintaining the high-nutrient state (Carpenter 2003, Havens 2008), as can positive feedbacks resulting from the organic matter generated by cyanobacteria (e.g., Huisman et al. 2005, Paerl and Otten 2013a, b).
Building from this conceptual framework, we hypothesize that some cyanobacteria have the potential to serve as a switching feedback that decreases the resilience of low-nutrient lakes by tapping into pools of N and P that would not be otherwise available (Boxes 1 and 2). These “new” nutrients may alleviate some degree of nutrient limitation in the water column, thereby facilitating the growth of other phytoplankton whose later decomposition may promote hypolimnetic anoxia. For example, we know from eutrophic lakes that the large colonial cyanobacterium Gloeotrichia echinulata increases internal P loading (Barbiero and Welch 1992, Istvanovics et al. 1993, Pettersson et al. 1993) and can fix N (Stewart et al. 1967, Vuorio et al. 2006, 2009), suggesting that this species might stimulate phytoplankton by increasing nutrient availability. Consistent with this expectation, G. echinulata stimulates other phytoplankton in the laboratory (Carey and Rengefors 2010) and in nutrient-limited mesocosms (Carey et al. 2014a, b). Moreover, laboratory experiments suggest that large populations of G. echinulata may increase water column N and P (Carey et al. 2014b), as observed in more nutrient-rich lakes (Pitois et al. 1997, Noges et al. 2004). Further, paleoecological studies of Long Pond in Maine show increases in markers of eutrophication coincident with increases in G. echinulata, but without clearly corresponding increases in water column P (H. A. Ewing et al., unpublished data). Together, these lines of evidence suggest the potential for G. echinulata to initiate positive feedback loops that accelerate eutrophication.
Thus, a handful of studies provide evidence that at least one cyanobacterial taxon may contribute to positive feedbacks that enhance eutrophication, but we lack empirical studies quantifying the general role of many different types of cyanobacteria in N and P cycling, especially in low-nutrient lakes. Are cyanobacteria important early in the process of eutrophication, in that they help drive a low-nutrient system to the eutrophic state? Does P translocation by cyanobacteria in mesotrophic lakes interfere with management strategies that attempt to limit P availability by keeping external loading low or by maintaining an oxic hypolimnion?
Switching Feedbacks: Explorations with a Simple Model
Our model describes changes in the pools of N and P in the part of the water column occupied by phytoplankton and allows us to explore the factors that are likely to alter lake trophic state, resilience and regime shifts. Two state variables, N and P, represent the total pools of nitrogen and phosphorus, respectively, including all forms of the nutrients (e.g., available, dissolved organic, and particulate organic [stored in bodies]). Changes in N and P are a function of external inputs, internally mediated P recycling and N fixation, and a loss term (Fig. 2), with N fixation tied explicitly to P concentration (see Eq. 2).
where IP is the external input of P (inflow + deposition in Fig. 2), s is the output rate (sP is outflow + sedimentation in Fig. 2), and the third term represents internally driven recycling of sediment P: r is the maximum P recycling rate, m indicates the P concentration at which P recycling reaches half the maximum rate (and at which it is increasing most rapidly), and q is a shape parameter that controls how quickly P recycling reaches its maximum (q ≥ 2).
Using this framework, we considered two mechanisms by which cyanobacteria could affect the P cycle. First, we considered cyanobacteria that follow arrow 1 in Fig. 1 as increasing the inputs of P represented by the parameter IP: their access to sediment P is generally independent of water column P concentrations and makes P available that would not normally be available in surface waters, especially in the well-oxygenated, low-nutrient systems in which redox-mediated recycling of sediment P occurs rarely, if at all. Alternatively, we considered cyanobacterial mobilization of P in arrows 2 and 3 in Fig. 1 as altering recycling rates depending on water column P concentration. In this case, cyanobacteria could increase r, the maximum P recycling rate, or decrease m, the P concentration at which P recycling reaches half of that maximum rate. We considered each of these possibilities in evaluating the potential effects of cyanobacteria on model behavior.
For N, we incorporated both stoichiometric constraints on N fixation and patterns in cyanobacterial abundance and N fixation across lakes (e.g., Smith 1983, Downing et al. 2001), such that
where IN is the external input of N (inflow + deposition in Fig. 2); s is the output rate for losses due to outflow, sedimentation, and denitrification; f is the maximum N fixation rate; and k and α are shape parameters with the requirements that k > 0 and α > 0. For this initial model, we assumed that cyanobacterial addition of “new” N via N fixation increases linearly with P concentration, since P is needed to build the enzymes required in N fixation (Elser et al. 1996). This formulation is simpler than the asymptotic association reported by Jankowski et al. (2012) but provides a tractable starting point. We also assumed, based on cyanobacterial abundance patterns (Smith 1983, Gophen et al. 1999), that N fixation rates follow a logistic function of the N:P ratio (by mass) in which N fixation is highest at low N:P ratios, decreases as N:P ratios begin to increase, and declines to zero at high N:P ratios. As with the Carpenter et al. (1999) model of P dynamics, this equation is just a starting point: we recognize that N fixation is controlled by many different environmental conditions (e.g., Wetzel 2001, Forbes et al. 2008) and that this simple formulation ignores benthic N transformations, including N fixation (e.g., Reuter and Axler 1992) and denitrification (e.g., Seitzinger et al. 2006). We assume that denitrification losses are relatively small, and thus use the same output rate s for both P and N for the model runs presented here. This is likely reasonable for low-nutrient conditions, but an oversimplification for lakes with higher nutrient concentrations.
In parameterizing these equations, we assumed that only one cyanobacterial species is dominant or that the whole assemblage of cyanobacterial species responds similarly across a range of P and N:P ratios, even if the relative abundance of individual taxa within the assemblage changes. We first generated plausible ranges for each parameter based on the literature wherever possible, and in the absence of published studies for some parameters, used data from unpublished sources (Table 1).
Table 1. Model parameterization.
Here, we describe the highlights of our model explorations as they relate to potential cyanobacterial effects on lakes, particularly the ability of cyanobacteria to facilitate a shift from a low-nutrient to a high-nutrient state. Most of our simulation studies began from the perspective of a lake with low-to-moderate nutrient conditions experiencing increased cyanobacterial abundance (the “default” values in Table 1), and we report findings primarily from ranges of parameters where small, realistic changes in the values—comparable to those that might be produced by changing cyanobacterial abundance—strongly influence equilibrial outcomes (Appendix). Although we focus on equilibrium dynamics, we recognize that both transient dynamics and stochasticity are likely to be important in lakes. Moreover, the relative importance of cyanobacteria-mediated nutrient cycling for any particular lake will depend on characteristics such as morphometry, stratification, and water residence time. Further refinement of the model would be needed to simulate conditions in any particular lake or to capture the entire array of possible processes.
Consistent with the empirical evidence that cyanobacteria can drive lake N and P cycling (Boxes 1 and 2), changes in several model parameters associated with cyanobacteria (e.g., IP, r, m, and f) can alter the number, location, and stability of the equilibrium N and P concentrations. For example, the number and location(s) of stable equilibria depend strongly on the two parameters affecting P recycling, r (Figs. 3 and 4) and m (Fig. 4), consistent with Carpenter et al. (1999). As the maximum P recycling rate r increases (or the inflection point m decreases), the system moves from one stable low-nutrient equilibrium, to two stable equilibria separated by an unstable equilibrium, to one stable high-nutrient equilibrium (Fig. 3). In the region of parameter space where there are two stable equilibria, increases in the maximum P recycling rate r (and decreases in the inflection point m) decrease the resilience of the low-nutrient equilibrium (Fig. 4). Hence, to the extent that cyanobacteria have the ability to increase the maximum P recycling rate or decrease the lake P concentration at which recycling reaches half of its maximum value, they can erode the stability of the low-nutrient equilibrium and make it more likely that a lake might come to the point of substantial internal P loading.
If we focus on the role of cyanobacteria in increasing the P inputs to a lake (IP), then equilibrium locations change within the constraints set by the recycling characteristics described by r and m: nutrient concentrations for both stable equilibria increase and the unstable equilibrium moves closer to the low-nutrient equilibrium (Fig. 3). Thus, even without extensive P recycling from the sediment, the resilience of the low-nutrient equilibrium can be eroded by cyanobacterial P loading. Increases in IP also alter the critical values of the maximum P recycling rate r for which multiple equilibria occur (Fig. 3). As such, cyanobacteria that influence P inputs through diurnal migrations (arrow 2 in Fig. 1) or seasonal recruitment from dormant benthic stages to the water column (arrow 1 in Fig. 1) may increase the sensitivity of a low-nutrient system to a recycling-driven regime change, even before anoxia-driven internal P loading occurs.
In addition to enhancing switching feedbacks and so contributing to a shift to a more eutrophic regime by altering IP, r, or m, cyanobacteria-mediated shifts in P cycling also increase the resilience of the eutrophic equilibrium, making a shift back to the low-nutrient equilibrium (i.e., lake restoration) more difficult (Fig. 4). At the high-nutrient equilibrium, reversal to the low-nutrient state requires parameter changes that can be difficult to effect in nature without substantial cost and effort, similar to management activities to remove nutrients by increasing sedimentation or outflow (s) (e.g., alum or flocculation treatment; Cooke et al. 2005), decreasing external inputs of N and P (IN and IP), or decreasing internal loading (e.g., by removing cyanobacteria or oxygenating bottom water to prevent anoxia-driven P release). This model finding is consistent with empirical studies demonstrating that cyanobacteria can enhance the resilience of the eutrophic regime by creating stabilizing feedbacks that can maintain internal nutrient loading, even if external nutrient loading is curtailed by management activities (e.g., Fogg 1969, Reynolds and Walsby 1975, Huisman et al. 2005, Reynolds 2006). In fact, numerous authors have noted that a better understanding of the effects of cyanobacteria on nutrient feedbacks is essential for the development of more effective strategies for lake restoration (e.g., Scheffer et al. 1993, Jeppesen et al. 1998, Xie et al. 2003, Havens 2008).
In considering the effects of cyanobacteria on N cycling, we found that adjusting parameters associated with N fixation does not change the number of equilibria, although these parameters do change the equilibrium concentrations of N (e.g., Appendix) and the resilience of the stable equilibria in the regions of parameter space where multiple equilibria are possible (Fig. 5). For example, increases in the maximum N fixation rate f decrease the slope of the N isocline, tilting the diagonal line in Fig. 3 ever further to the right, towards horizontal. A more horizontal N isocline increases the N concentration at equilibrium but leaves the location of the P equilibrium unchanged. In the region of parameter space where multiple equilibria are possible (based on the P recycling parameters r or m as described above), the resilience of both stable equilibria increases with the maximum N fixation rate f, although the resilience of the high-nutrient equilibrium is much higher and increases to a greater degree than that of the low-nutrient equilibrium (Fig. 5). Therefore, as with P recycling, N fixation resulting from cyanobacteria in lakes that are N-limited may make restoration of high-nutrient lakes more difficult. More work is needed, however, to capture other complexities in the N cycle, including how benthic N fixation or denitrification might alter the locations and stability of the high-nutrient equilibrium.
Synthesis and Testable Hypotheses
This simple model suggests that the ability of cyanobacteria to alter N and P cycling could influence not only lake N and P concentrations, but also the range of trophic states likely to occur, the proximity of a system to a threshold that leads to a regime shift, and the tendency of a lake to remain in either a high- or low-nutrient condition. As such, it draws our attention to several testable hypotheses about the influence of cyanobacteria on lake nutrient cycling and resilience, especially in the lower-nutrient lakes for which the model was parameterized.
Cyanobacteria-mediated modification of N and P cycling can facilitate a shift from a low-nutrient to a high-nutrient state, depending on the species present and the interaction between the characteristics of those taxa and other lake characteristics. We hypothesize that cyanobacterial taxa that can both fix N and acquire P are particularly likely to have strong effects on trophic state and resilience. In addition to Gloeotrichia echinulata, at least three other colonial taxa in the Nostocales are known to have life histories that may allow them to initiate or sustain positive feedback loops that lead to eutrophication by bringing both N and P into the water column: Anabaena (Thompson et al. 1994), Aphanizomenon (Barbiero and Welch 1992, Barbiero and Kann 1994, Jacobsen 1994), and Cylindrospermopsis (Istvanovics et al. 2002). Like G. echinulata, their germination from sediment resting stages is dependent upon the high light conditions most commonly found in low-nutrient systems (Barbiero and Kann 1994, Karlsson-Elfgren et al. 2003, Carey et al. 2008), suggesting that these taxa may have similar effects to G. echinulata in increasing nutrients and phytoplankton in the water column in low-nutrient lakes. However, other taxa may exert a stronger effect on either N or P. For example, Microcystis has been strongly implicated in increasing P concentrations in the water column in shallow eutrophic lakes (Xie et al. 2003, Xie 2006), but does not fix N. It seems likely, therefore, that the effects of cyanobacteria on N and P cycling in a particular lake ecosystem will depend on such factors as the degree of limitation by N versus P and the cyanobacterial taxa present.
Systems with long residence times or slower sedimentation rates (i.e., smaller output rates, s) are more susceptible to cyanobacteria-mediated increases in the maximum P recycling rate, r (Fig. 6). When the output rate s is low, there is a narrow range of maximum P recycling rates associated with the low-nutrient equilibrium (the blue area in Fig. 6). Further, if the maximum P recycling rate increases through the cyanobacterial activities described in Box 2 without a substantial increase in output rate (s), the model suggests that a lake is far more likely to end up at the high-nutrient equilibrium. Although the sedimentation rate might be expected to increase to some degree with higher productivity, the output rate s is also constrained by lake water residence time, suggesting that lakes with longer residence times are more susceptible to regime shifts facilitated by cyanobacteria that can increase the maximum P recycling rate (Fig. 6). This hypothesis could be tested through comparative studies across lakes that differ in their residence times but have similar nutrient conditions and cyanobacterial populations.
Since this model, as with all other models, is an abstraction that does not represent the full complexity of lake ecosystems, it is illustrative of processes and interactions. Nevertheless, more work can be done to establish the generality of these findings and the conditions under which we expect cyanobacteria to be important in nature. For example, applications to real-world systems that may never come to equilibrium and exploration of alternative, more complex models—for example, those that provide a more detailed description of the N cycle or incorporate hydrodynamics—are logical next steps.
One particularly fruitful area for further inquiry may be linking an open-source hydrodynamic model (e.g., the General Lake Model, Hipsey et al. 2013) to a simple N and P model such as ours. With a model that couples hydrodynamics, N, and P, we could begin to explore hypotheses about the particular conditions that may affect the relative importance of cyanobacteria-mediated changes in nutrient cycling for any particular lake. For example, our literature review (Box 2) suggests that the relative importance of P translocation by cyanobacteria depends on lake stratification and morphometry. Stratified oligotrophic lakes in which water column productivity is strongly nutrient-limited and the hypolimnion is well-oxygenated are likely to be more susceptible to changes in cyanobacteria-driven internal P loading than lakes with anoxic hypolimnia, as suggested by the strong sensitivity of trophic state to small changes in P recycling rates (Fig. 3). Moreover, cyanobacterial transport of nutrients via diel vertical migration (arrow 2 in Fig. 1) may be most important during periods of stable thermal stratification, whereas benthic species that mobilize sediment P (arrow 3 in Fig. 1) may be most important in well-mixed lakes. Finally, P translocation by seasonal recruitment from sediment resting stages to the water column (arrow 1 in Fig. 1) seems most likely in lakes with an ample area of shallow, relatively warm, well-lit sediments (Head et al. 1999a), even if the maximum depth is quite deep. These hypotheses can be tested in models and with rigorous, cross-lake comparisons.
Additional Research Opportunities and Practical Challenges
More broadly, the interaction between internal and external drivers of lake regime shifts poses some of the most pressing and challenging questions at the intersection of limnology and lake ecosystem management. Management, remediation, and policy actions must be tailored to address both external and internal drivers of ecosystem change and the interactions among drivers that may act across spatial and temporal scales. For example, catchment P loading is a local-to-regional external driver that can be controlled through local regulations (Brookes and Carey 2011), while atmospheric deposition of N and other pollutants is a regional problem whose sources can be many kilometers away (Weathers et al. 2006, Holtgrieve et al. 2011) and hence may require national or international regulation. Given that external P loading can, in theory, be controlled on a catchment basis, reductions in P loading are relatively easy targets for controlling cyanobacterial biomass (Brookes and Carey 2011) and thus cyanobacteria-driven eutrophication. However, the reality of drivers beyond the catchment—such as global climate change—and the importance of within-lake loading due to both hypolimnetic anoxia and cyanobacteria pose complex challenges for management that necessitate multi-faceted monitoring of both in-lake and external contributors to nutrient loading. Given the potential for cyanobacteria to erode the resilience of the low-nutrient state, increases in their biomass should be a warning sign that lake managers monitor closely.
We thank Nina Caraco, Jon Cole, Sam Fey, Laurie Griesinger, Cristina Herren, Elizabeth Traver, Nelson Hairston Jr., Alex Flecker, Emil Rydin, Jessica Trout-Haney, Peter Leavitt, Mike Vanni, Lars-Anders Hansson, and Sabrina Hepburn for helpful discussions; Amanda Lindsey for data support; and Chris Klausmeier for Mathematica tips. Tony Ives, Mireia Bartrons, Sam Fey, and four anonymous reviewers provided comments on earlier drafts. We appreciate financial support from the U.S. National Science Foundation, including DEB-0749022 to K. L. Cottingham., K. C. Weathers, and H. A. Ewing; EF-0842267 to K. L. Cottingham; EF-0842112 to H. A. Ewing; EF-0842125 to K. C. Weathers; DEB-1010862 to C. C. Carey and Nelson G. Hairston, Jr.; and a Graduate Research Fellowship to C. C. Carey. Our work has also been supported by Dartmouth, Bates, and the American-Scandinavian Foundation.
Analytical evaluations of the model and simulation studies
The form of the equations for phosphorus (P) and nitrogen (N) cycling given in the text are as follows:
where IP and IN are the external input of P and N, respectively (inflow + deposition in Fig. 2); s is the output rate (outflow + sedimentation in Fig. 2); r is the maximum P recycling rate; f is the maximum N fixation rate; and m, q, k and α are shape parameters with the requirements that m > 0, q ≥ 2, k > 0 and α > 0.
The non-linearities in the equations for both P and N make it difficult to solve for equilibrium nutrient concentrations analytically. However, we can put bounds on both equilibria. In the equation for dP/dt, the term rPq(mq + Pq)−1 cannot become greater than r, and therefore any equilibrium value of P must be less than (IP + r)/s. Within the N equation, the logistic function for N fixation means that dN/dt is bounded between IN − sN (when there is no N fixation due to a high N:P ratio) and IN − sN + fP (when N fixation is at the maximum rate). This means that the equilibrium concentration for N will be between IN/s and (IN + fP)/s; given the restriction on P—that it must be less than (IP + r)/s—we can show that N cannot exceed [IN + (f/s)(IP + r)]/s. These expressions make it clear that external loading directly, and linearly, increases the bounds on the equilibrium values. Higher levels of N fixation and P recycling also increase equilibrium value bounds linearly, and higher levels of outputs decrease equilibrium value bounds.
For the figures shown in the main text, we used Wolfram Mathematica v8 to visualize potential equilibria and their stability using ContourPlot and to estimate equilibrium locations using the Solve function. Isoclines for Fig. 3 in the main text were generated with code in the Mathematica notebook “Figure 3.nb”. Equilibria in Figs. 4 and 5 were generated with code structured as in “Demo code for Figures 4 and 5.nb”, but modified for the particular parameter values of interest. Isoclines for Fig. 6 were generated using “Figure 6.nb”. All three Mathematica notebooks are available as a Supplement.
To complement these analytical explorations, we visualized model dynamics through time for a wide range of parameter values and initial conditions using simulations in MATLAB 188.8.131.529 (R2010a) with an adaptive step-size algorithm (ode45) and a variety of initial conditions. These studies were not used to generate figures for this paper, but were essential to exploring model behavior. Code for these simulations was built using the core program “runmodel.m”, calling the subprogram “derivsub_ratio.m”, with different versions again tuned to the parameters of interest; both programs are included in the Supplement. Programs were typically run for at least four different initial conditions to confirm convergence of outcomes.
Mathematica notebooks and Matlab programs used to analyze our coupled model of nitrogen and phosphorus cycling, as explained in the Appendix (Ecological Archives, http://dx.doi.org/10.1890/ES14-00174.1.sm).