© 2005 International Council for the Exploration of the Sea
The role of mixotrophy in plankton bloom dynamics, and the consequences for productivity
Department of Biology/Area 2, University of York PO Box 373, York YO10 5YW, England, UK
*Correspondence to A. C. Hammer: tel: +44 1904 328557; fax: +44 1904 328505. e-mail: ah44{at}york.ac.uk.
Mixotrophy (=heterotrophy and photosynthesis by a single individual) is a common phenomenon in aquatic ecosystems, in particular under light- or nutrient-limitation. However, it is not usually considered in mathematical models of biological populations. This paper shows how different types of mixotrophy might be usefully incorporated into a general predator–prey model, and explores the consequences for plankton bloom dynamics and productivity. It is demonstrated, analytically and numerically, that even small levels of type III mixotrophy (a small fraction of the zooplankton also being involved in primary production) have significant effects on a system's equilibrium structure, stability, and short-term dynamics. Type III mixotrophy has a stabilizing effect on the system by reducing its excitability, i.e. its propensity to exhibit blooms. Compared with the non-mixotrophic benchmark, for a phytoplankton bloom to be triggered in a system with type III mixotrophy, a much larger perturbation is necessary. Type II mixotrophy (a small fraction of algae engage in phagotrophy) and type I mixotrophy (equal phagotrophy and phototrophy) are briefly discussed. The potential consequences for productivity are also studied. Our results indicate that the phytoplankton–zooplankton system becomes more productive in the presence of type III mixotrophy.
Keywords: excitable phytoplankton–zooplankton system, mixotrophy, plankton bloom
Received 23 November 2004; accepted 28 February 2005.
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Introduction |
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Traditional models classify species either as producer or as consumer. The research shown here suggests that this may be a significant oversimplification. This is because of the existence of mixotrophic organisms that may act as both producer and consumer. More specifically, mixotrophy is a strategy by which photosynthetic micro-organisms supplement their nutrition by uptake of bacteria/algae (phagotrophy) and/or dissolved organic carbon and, conversely, by which phagotrophic organisms photosynthesize. This nutritional way of life has inherent physiological and ecological advantages, in particular in light-, nutrient-, or particulate-food-limited environments, where it has been found to be quite common (e.g. Berninger et al., 1992; Havskum and Riemann, 1996; Roberts and Laybourn-Parry, 1999; Hammer et al., 2002). The existence of mixotrophs has far-reaching consequences for the population dynamics of prey (e.g. bacteria, through grazing), the dynamics of other grazers within the community attributable to increased competition for prey (e.g. by outcompeting obligatory heterotrophic organisms; Bird and Kalff, 1986), or the mobilization of nutrients within aquatic systems (Boraas et al., 1988). As primarily heterotrophic protists, mixotrophs can account for up to 20% of the primary production (Stoecker et al., 1991), so influencing competition among phytoplankton for limiting resources. Mixotrophy may account for otherwise unexplained deficits or surpluses in the calculation of the primary production of a waterbody. It can also explain plankton blooms under extreme conditions (e.g. Roberts and Laybourn-Parry, 1999; Schumann et al., 2005).
Mixotrophs acting equally as producers and consumers would affect the structure and productivity of microbial foodwebs and their coupling to metazoan production, raising questions regarding the utility of trophic-level definitions, especially in foodweb models. Classical models of a marine ecological system will be inaccurate if they do not account for photosynthesizing phagotrophs and phagotrophic photosynthesizers (Sanders, 1991). However, theoretical insight into this problem (i.e. the integration of mixotrophy into models of the aquatic ecological system) is currently at an early stage. Although models have been used to consider interactions between bacterivorous mixotrophs and bacteria and their impact on the microbial foodweb (linear steady-state model; Thingstad et al., 1996) or nanophytoflagellates (simulation model; Baretta-Bekker et al., 1998), this has rarely been performed for larger mixotrophic species that are non-bacterivorous (e.g. dinoflagellates, such as Prorocentrum and Noctiluca, and ciliates such as Laboea; Stickney et al., 2000; Anderson, 2001; Jost et al., 2004). Stickney et al. (2000) focus their highly complex steady-state model on non-bacterivorous mixotrophs and their impact on phytoplankton/zooplankton biomass and production. In similarly complex simulation models, Anderson (2001) regards mixotrophy as a stabilizing influence at an ecosystem level. In their four-component chemostat model, Jost et al. (2004) investigate the influence of mixotrophic (but primarily photosynthetic) marine dinoflagellates on the existence and stability of equilibria. However, as Jost et al. (2004) point out, steady-state analysis must be applied with care to systems in fluctuating environments. Steady-state models allow comparisons between long-term equilibria, but provide less insight into the short-term dynamics, such as plankton blooms. To our knowledge, this paper sets out for the first time a model of mixotrophy that explicitly incorporates environmental fluctuations and bloom dynamics.
Here we incorporate mixotrophy into a simple phytoplankton–zooplankton model, to determine how they affect the system's behaviour. Our approach attempts to model marine ecosystems as non-linear dynamic systems to which well understood mathematical techniques such as bifurcational analysis, separation of time scales, and asymptotic expansions, can be applied in a rigorous manner. These techniques have already elucidated the importance of predation (Truscott and Brindley, 1994a; Edwards and Bees, 2001) and climatic forcing (Truscott, 1995; James et al., 2003) in the dynamics of plankton blooms, and the role of functional diversity within ecosystems (Fasham et al., 1999; Pitchford and Brindley, 1999).
Stoecker (1998) distinguishes three general physiological types: type I, the "ideal mixotrophs" that are able to utilize both phototrophy and phagotrophy equally well; type II, the primarily phototrophic phagocytizing "algae", comprising subtypes A, B, and C; and type III, the predominantly heterotrophic photosynthesizing "protozoa", comprising subtypes A and B. Whereas some details regarding subtypes are considered in the Discussion, our principal aim here is to understand how these different types of mixotrophy affect planktonic population dynamics, and to quantify the effect of mixotrophs on primary production and the dynamics of a system. Below we introduce as our basic framework the general excitable predator–prey model for phytoplankton–zooplankton dynamics. Mixotrophy is then modelled as a small perturbation to this predator–prey system, where a small fraction of protozoa engages in primary production (type III mixotrophy), or where a small fraction of algae engages in phagotrophy (type II mixotrophy). The uncommon phenomenon of type I mixotrophy could be viewed as a combination of type II and III mixotrophy. The method to be applied for modelling mixotrophy is similar, in both concept and implementation, to that used by Pitchford and Brindley (1998) when modelling intraguild predation (or functional "cannibalism"). Subsequently, we present the application of this approach to the phenomenon of mixotrophy. This is conceptually straightforward, but its consequences for any non-linear parametrically forced model are non-trivial, and merits detailed study. In the Discussion and summary, we suggest that our results show that a small amount of mixotrophy can have a large effect, both on the model's steady state, and on its response to a seasonally changing environment.
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Incorporating mixotrophy into general predator prey models |
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Consider a population of autotrophic prey P and herbivores Z growing according to the following general equations:
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| (1) |
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represents the efficiency of converting prey biomass to predator biomass: < 1.
Type III mixotrophy involves a small fraction 
(<<1) of Z also being involved in primary production, and therefore competing for light and/or nutrient with P, while also consuming P as in a non-mixotrophic system. The mathematical model then becomes
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= 0 corresponds to the non-mixotrophic predator–prey situation (Equations 1 and 2), while = 1 corresponds (unrealistically) to P and Z being equally photosynthetic. This description of mixotrophy as an "order perturbation" from the standard predator–prey system allows its consequences, for both steady states and seasonal dynamics, to be quantified. Whereas observations show that the proportion of mixotrophs to zooplankton typically remains small (e.g. Sanders, 1995; Tamigneaux et al., 1997), the assumption that is fixed is best regarded as a modelling expedient; the consequences of relaxing this assumption are discussed in the Discussion section.
To provide a concrete example, the remainder of this paper concentrates on the excitable plankton system proposed by Truscott and Brindley (1994b), where A(P) describes a simple logistic limitation on growth (competition among algae for light and/or nutrients), g(P) is a Holling type III predation function, assuming the predators' response to low prey density is very small, as is usually associated with active hunting, and µ(Z) is regarded as a constant. The equations summarizing the non-mixotrophic predator–prey dynamics are
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governs how quickly that maximum is attained as prey densities increase. This system exploits the inherent differences in the growth rates of phyto- and zooplankton, very naturally introducing time scales and measures for bloom triggering and duration. For the excitability in the phytoplankton population to manifest itself, the time scale for P change must be much faster than the time scale for Z change. In this model, this property is a direct consequence of the assumption that the conversion efficiency << 1.
Introducing type III mixotrophy into Equations (5) and (6) yields:
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Results |
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Analysis of steady states
First, consider system (Equations 5 and 6), equivalent to Equations (7) and (8), but with
= 0. The stability and dynamic properties of this model are dealt with in detail in Truscott and Brindley (1994b). The system typically has non-trivial null-clines (curves upon which either dP/dt = 0 or dZ/dt = 0), as shown in Figure 1; a sigmoidal P null-cline given by
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The stability of the steady state (P*, Z*) is governed by the gradient (dZ/dP) of the P null-cline; following Truscott and Brindley (1994b), it is assumed that (P*, Z*) lies to the left of the lower turning point of the P null-cline, thus ensuring its linear stability. This stable equilibrium represents a system's quiescent non-bloom steady state, i.e. a balance of predator and prey towards which the populations would be attracted in the absence of any other external influence.
Now consider type III mixotrophy, i.e. fix > 0. Expressing the non-trivial P null-cline as a perturbation to the P null-cline with = 0 (see Pitchford and Brindley, 1998) shows that, for a given P:
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= 0.01), the shift downwards is at most 0.45%, i.e. mixotrophy does not significantly change either the geometry or the location of the P null-cline. This fact is important, because it is the shape of the P null-cline that is crucial to the system's excitability and thereby to its usefulness in describing plankton bloom dynamics.
The mixotrophic Z null-cline can similarly be written as 
, where B is a negative constant given by
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. As the excitable paradigm assumes (for good biological and ecological reasons) that is small, this suggests a substantial change in the Z null-cline; a small amount of mixotrophy is amplified by a factor of 1/, with large consequences for the model's equilibrium. For example, using the parameter values in Truscott and Brindley (1994b) and assuming 1% mixotrophy ( = 0.01) causes an 18% shift to the left of the equivalent Z null-cline in the = 0 system.
This simple mathematical treatment immediately yields the principal result, namely that, in general, type III mixotrophy reduces the equilibrium value of the prey and increases that of the predator, and moreover that a small amount of mixotrophy can have a large effect. Figure 1 illustrates this graphically; the non-trivial null-clines of the = 0 system are shown as solid lines, and those for = 0.01 as dotted lines, and the = 0.01 equilibrium is denoted (PM, ZM).
That mixotrophy shifts the equilibrium leftwards and upwards along the stable part of the P null-cline, so making it less likely to undergo a bifurcation from stable to unstable equilibrium, suggests a stabilizing effect of type III mixotrophy. A more rigorous demonstration of this assertion is provided below.
The effect of mixotrophy on bloom dynamics
Although the above reasoning shows that mixotrophy can be important in determining steady states, ecosystems (and plankton systems in particular) are seldom truly at equilibrium. Indeed, the strength of the original excitable model without mixotrophy (Equations 5 and 6) is its ability to describe and quantify blooms triggered by environmental change.
A bloom may be triggered in system (Equations 5 and 6) by any change in parameters or functions that allow P to escape grazing control by Z for a sufficient time. The most realistic way for a bloom to be triggered is an increase in r, corresponding to an increase in temperature, light, nutrient, or any of the other factors stimulating phytoplankton growth. Suppose the original system (Equations 5 and 6) is at equilibrium, and then r is increased; this does not affect the Z null-cline, but effectively "lifts" the P null-cline, and shifts the Z coordinate away from the stable state (Truscott, 1995). If this lifting is sufficiently rapid and large, then the solution is unable to track the equilibrium point (which remains linearly stable throughout), and a bloom trajectory is followed. In this case, there is no classical bifurcation of the system's steady state, but asymptotic analysis can be applied to indicate when an increase in r would be expected to cause blooms, as presented in detail in Truscott (1995). To model an increase in the rate of production for phytoplankton, the parameter r was taken as a function of time (Truscott and Brindley, 1994b):
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Figure 2a illustrates the response of the original P–Z system (Equations 5 and 6) to forcing in r. In the graph, r is increased from 0.4 to 0.6 at a constant rate, with the system starting off at its equilibrium. As dr/dt is increased, the response of the system undergoes a marked qualitative change, from a slow change of equilibrium position to excitable behaviour. The onset of this transition occurs over a narrow range of dr/dt, between 0.0031 and 0.0032 day–2; any value of dr/dt exceeding 0.0032 day–2 (for these parameter values) would produce a response resembling Figure 2a.
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Including a small amount of mixotrophy in the model alters its response to forcing dramatically. Figure 2b shows the response of a system identical in every way to that in Figure 2a, except for the addition of 1% mixotrophy: the forcing causes a small change in the levels of P and Z, but there is definitely no bloom. Further numerical investigation shows that the 1% mixotrophy model can exhibit blooms akin to Figure 2a, but only when dr/dt exceeds 0.025 day–2, i.e. when the forcing is an order of magnitude larger than that required for Equations (5) and (6).
It could be argued that this is an unfair comparison: Equations (5) and (6), and Equations (7) and (8) with = 0.01, are shown above to have different equilibria, and blooms are a phenomenon involving large-scale excursions from equilibrium. This argument can be countered by tuning the value of µ in Equations (7) and (8) so that the unforced equilibrium is identical to that in Equations (5) and (6). This was achieved numerically by using Equations (11) and (12), solving for µ by requiring an equilibrium at (P*, Z*) defined by Equations (9) and (10). This "fair comparison" between systems changes the result quantitatively but not qualitatively: the transition into bloom dynamics occurs when dr/dt exceeds 0.0037 day–2, so that 1% mixotrophy requires a 16% higher minimum "activating" rate of change for the phytoplankton reproduction rate. In short, mixotrophy stabilizes the system significantly.
Figure 3 shows a graph of the response of non-mixotrophic (Equations 5 and 6) and mixotrophic (Equations 7 and 8) systems as dr/dt is varied through the threshold value, using the "fair comparison" version of Equations (7) and (8), i.e. choosing µ so that the unforced equilibrium is invariant. The response is taken as the maximum value of P during the forcing period. The jump between subthreshold and superthreshold behaviour is clearly very abrupt in both systems, but it occurs at a higher dr/dt in the system with 1% mixotrophy. The numerical simulations in Figures 2 and 3 show that type III mixotrophy does not necessarily affect a model's excitability, but that it does make it significantly less likely to exhibit blooms.
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On the basis of these findings, it is now of value to assess the role of mixotrophy in a system's productivity, and thereby its consequences for modelling carbon budgets. For type III mixotrophy, any measure of total carbon flux through a system needs to account for the extra mixotrophic primary production. The primary production in such a mixotrophic system can thus be calculated as
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The principal conclusions from this study are as follows: type III mixotrophy has a stabilizing effect on the system by reducing its propensity to exhibit blooms. In the example, 1% type III mixotrophy decreases the original equilibrium P value by 18%, so making a bifurcation less likely. The basic mechanisms of excitability are not changed by considering mixotrophy in a system, but significantly higher values of dr/dt (i.e. a larger forcing strength) are necessary to generate plankton bloom responses. Type III mixotrophy can have a significant effect on the size and the timing of a bloom, and can also increase the total primary production over the duration of a bloom.
In the light of these results, the effect on plankton dynamics of type III mixotrophy is significant and deserves careful attention.
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Discussion |
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This paper develops a first simple non-linear dynamic model, investigating the influence of mixotrophy on plankton dynamics, especially under bloom situations. Most models use the basic tenet "zooplankton eats phytoplankton". Any variations, such as mixotrophy or intraguild predation, are very rarely considered. The method applied to modelling mixotrophy is similar to that used by Pitchford and Brindley (1998) when modelling intraguild predation (or functional "cannibalism"). As cannibalism typically accounts for <20% of predation, one can analyse a cannibalistic system as a perturbation (involving a small parameter
) of a simpler, well understood, non-cannibalistic predator–prey system. In systems where there is a large difference in population growth rates between producers and consumers, a small level (O()) of cannibalism in the consumers can have a large (O(1)) effect on a system's equilibrium structure and stability properties. In other words, a small level of cannibalism can have a large effect, particularly in relation to a system's response to parametric forcing. In analogy to the model of functional cannibalism (Pitchford and Brindley, 1998), we expected and found that even small levels of mixotrophy have significant effects on a system's equilibrium structure and stability. The general results demonstrate that the simple model of type III mixotrophy formulated above possesses useful properties. The integration of type III mixotrophy into P–Z models would be of special importance in oligotrophic waterbodies under spring bloom conditions, where phytoplankton growth is boosted by an increase in light intensity and temperature. At the same time, conditions for zooplankton are less optimal because of a lack in particulate food. In this situation, one might expect parts of the zooplankton community to be type III mixotrophs, either subtype A, where individuals photosynthesize in order to obtain carbon when prey is limiting, or more commonly, B, where individuals supplement their carbon budget by harboured endosymbionts. (Equations (3) and (4) strictly describe subtype IIIA. For subtype IIIB one could modify the model slightly, e.g. by parameterizing a dependence on the grazing function g(P) in the second term of Equation (4). Such a complication of the model is beyond the scope of this paper, and would rely on additional, species-specific, data. It is unlikely that such modifications would change the general behaviour of the model, i.e. the effect of mixotrophy on stability and bloom formation.) High biomasses of non-bacterivorous mixotrophic ciliates were found during spring bloom in the North Atlantic, and in subtropical oligotrophic lakes, they constituted 3–20% of the total photosynthetic biomass (Stoecker et al., 1991). From the results above, prevalence of type III mixotrophy would result, under these conditions, in a delayed and less pronounced spring bloom, i.e. a stabilizing effect on the phytoplankton community. This is broadly consistent with the simulation results of Anderson (2001), but the results presented here apply to both constant and time-varying environments, making them germane to discussions of plankton blooms and productivity. To our knowledge, field data on bloom events with/without type III mixotrophy do not yet exist. Mixotrophy can have a significant impact on calculations of primary production; in our model, even 1% of type III mixotrophy leads to a substantial increase in primary production with the onset of phytoplankton bloom formation (Figure 4). This indicates that the system becomes more productive with introducing type III mixotrophy, which is not consistent with the previous steady-state modelling study of Stickney et al. (2000), but is consistent with various suggestions about the positive influence of mixotrophy on productivity (Sanders, 1991; Baretta-Bekker et al., 1998; Stoecker, 1998).
The assumption that is fixed artificially constrains the relative proportion of mixotrophs within a herbivore population. In practice, this proportion would be expected to change over time. Provided such changes occur more slowly than the processes governing bloom triggering (i.e. a few days), the analysis presented here remains valid. The constant assumption can be relaxed by considering each class separately, resulting in a three-component system for P (phytoplankton), Z1 (mixotrophs), and Z2 (non-mixotrophs). Such models would be expected to retain their excitability, i.e. their propensity to exhibit blooms under appropriate forcing, provided there still exists some non-bloom (quasi-) equilibrium state (Pitchford and Brindley, 1999). However, to ensure the existence of such an equilibrium, further model assumptions and parameters are necessary, which in turn require detailed data relating to a particular system. Where these data exist, the explicit modelling of mixotroph species will provide valuable insight into population regulation and ecosystem function.
There are other types of mixotrophy, as mentioned in the Introduction. Type II mixotrophy, i.e. phagotrophy in "algae", comprising subtypes A (feed when DIN is limiting), B (feed when a trace organic growth factor is limiting), and C (feed when light is limiting), can be considered in a general way in simple P–Z models independently from environmental conditions and irrespective of subtype. This is achieved by allowing a fraction of the P population to behave phagotrophically, using exactly the same approach as we do above, resulting in the following system:
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represents conversion efficiency for P, the phagotrophic part of the algae. In practice, this would be very similar to . 
describes the grazing by P on (P – P), where ß is the encounter rate. There is evidence that the functional response of some suspension-feeding mixotrophic protozoa follows a Holling type I functional response (Hansen and Nielsen, 1997), i.e. no active search for food, uptake simply based on encounter. ß can be calculated from MacKenzie and Kiorboe (1995), which yields ß << 1 (10–5 l day–1 ind–1), giving good agreement with the model of Saiz et al. (2003). As ß is miniscule, the P null-cline barely shifts, and as clearly the Z null-cline is unaffected, the conclusion is that type II mixotrophy has no influence on stability, dynamics, bloom triggering, and productivity. This is consistent with the findings of Stickney et al. (2000), which suggests that type II mixotrophs resemble phytoplankton more closely than do the other types and thus that the replacement of phytoplankton by type II mixotrophs has less impact on the system than in the other cases. This result contrasts with that of Jost et al. (2004), who found a stabilizing effect of type II mixotrophy within their model. This apparent contradiction serves to illustrate the importance of considering models at a variety of time scales and in non-constant environments; both modelling approaches are "correct", but the methods described here are more relevant to transient phenomena, such as plankton blooms. This study has excluded bacteria as a source of nutrients for type II mixotrophs. While this assumption is justified for larger mixotrophic species, which do not feed significantly on bacteria, bacterivory among phytoflagellates (type II mixotrophy) is widespread (e.g. small photosynthetic flagellates; Havskum and Riemann, 1996; Hammer, 2003). Further, in eutrophic environments, it is also necessary to consider the interaction of osmotrophy (Lewitus and Kana, 1995; Hammer, 2003) with phagotrophy in mixotrophic phytoflagellates. Future research intends to include bacteria and DOC in the model and to study the impact of bacterivorous/osmotrophic type II mixotrophs on ecosystem dynamics and carbon budgets.
Type I mixotrophy is the "ideal" mixotrophy, where organisms can grow equally well as phototrophs and as phagotrophs. This nutritional mode can be modelled as a combination of types II and III mixotrophy, whereby the approach would have the merit of remaining mathematically tractable. Based on available data, however, mixotrophs that fit type I are rare, so the relevance of such models of type I mixotrophy at an ecosystem scale, and therefore in relation to carbon budgets, is expected to be limited (Stoecker, 1998).
To summarize, (i) it is possible to model the effects of mixotrophy in general predator–prey models in a relatively straightforward way; (ii) using an asymptotic method regarding the degree of mixotrophy as a small parameter, the effects on a system's equilibrium and its response to forcing can be addressed analytically and numerically; (iii) including small amounts of type III mixotrophy in an excitable predator–prey model can have a significant stabilizing effect, whereas the effects of small amounts of type II mixotrophy are likely to be negligible.
The models of mixotrophy presented here show the fundamental mechanisms, i.e. expose the key processes at work, and must not be confused with quantitative simulations. Our principal interest was to study the impact of mixotrophy on a simple plankton model under dynamic conditions, in order to understand how the introduction of mixotrophy affects phytoplankton and zooplankton biomass, production under dynamic conditions, and phytoplankton bloom formation. As we have seen, the blurring of trophic levels promotes stability.
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Acknowledgements |
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This work was supported by a grant from the German Academy of Natural Scientists Leopoldina, BMBF–LPD 9901/8–100 to the first author. We are grateful to Angela E. Douglas, Richard Law, and two anonymous referees for valuable comments on the draft text.
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