The Paradox of the Plankton: Species Competition and Nutrient Feedback Sustain Phytoplankton Diversity

Vol. 490: 107–119, 2013 MARINE ECOLOGY PROGRESS SERIES Published September 17 doi: 10.3354/meps10452 Mar Ecol Prog Ser

OPENPEN ACCESSCCESS The paradox of the plankton: species competition and nutrient feedback sustain phytoplankton diversity

Kasia Kenitz1, Richard G. Williams1,*, Jonathan Sharples1,2, Özgür Selsil3 , Vadim N. Biktashev3

1School of Environmental Sciences, University of Liverpool, Liverpool L69 3GP, UK 2National Oceanography Centre, Liverpool, Liverpool L3 5DA, UK 3School of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK

ABSTRACT: The diversity of phytoplankton species and their relationship to nutrient resources are examined using a coupled phytoplankton and nutrient model for a well-mixed box. The phytoplankton community either reaches a competitive exclusion state, where there is an optimal competitor, or the abundance of each phytoplankton species continually varies in the form of repeating oscillations or irregular chaotic changes. Oscillatory and chaotic solutions make up over half of the model solutions based upon sets of 1000 separate model integrations spanning large, moderate or small random changes in the half-saturation coefficient, Kji. The oscillatory or chaotic states allow a greater number of phytoplankton species to be sustained, even for their number to exceed the number of resources after additional species have been injected into the environment. The chaotic response, however, only occurs for particular model choices: when there is an explicit feedback between nutrient supply and ambient nutrient concentration, and when there are physiological differences among species, including cell quota and Kji. In relation to the surface ocean, the nutrient feedback can be viewed as mimicking the diffusive nutrient supply from the nutricline. Inter-species competition might then be important in generating chaos when this diffusive transfer is important, but less likely to be significant when other transport processes sustain surface nutrient concentrations.

KEY WORDS: Plankton paradox · Coexistence · Chaos · Competitive exclusion · Phytoplankton communities

INTRODUCTION number of species coexisting at equilibrium is not expected to exceed the number of resources. For Hutchinson (1961) first posed the paradox of the phytoplankton, the resources can be viewed in terms plankton: Why do so many phytoplankton species of macro nutrients, trace metals and variations in the coexist while competing for a limited number of re - light and temperature environment, such that if 2 sources in a nearly homogeneous environment? For phytoplankton species compete for the same re- example, open ocean and lake surface waters usually source, the most successful competitor is the one sur- contain the order of 1 to 10 dominant phytoplankton viving on the minimum resource (Tilman 1977, species together with many hundreds or more spe- Tilman et al. 1982). This excess in the number of cies at very low concentrations. This high number of phytoplankton species has been explained in terms phytoplankton species appears at odds with the com- of the phytoplankton system not reaching an equilib- petitive exclusion principle (Hardin 1960) where the rium state due to temporal variability, as first specu-

© The authors 2013. Open Access under Creative Commons by *Corresponding author. Email: [email protected] Attribution Licence. Use, distribution and reproduction are un - restricted. Authors and original publication must be credited. Publisher: Inter-Research · www.int-res.com 108 Mar Ecol Prog Ser 490: 107–119, 2013

lated in terms of seasonality by Hutchinson (1961) or rN rN N min⎛ i 1 , , ik⎞ γ i = … (3) spatial variability in the background environment ⎝ KN11i + KNki+ k ⎠ (Richerson et al. 1970). There are many ways in which this temporal and where the subscripts denote the particular species i = spatial variability can be achieved in the real world, 1,…,n and resources j = 1,…,k. In Eq. (1), the nutrient such as from the externally imposed physical vari- concentration, Nj, evolves through a competition be - ability from changes in solar irradiance, weather- tween a source from a nutrient supply and a sink related changes in air-sea forcing and changes in from phytoplankton consumption: the nutrient sup-

mechanical forcing from tides. These changes in ply involves an external supply, Sj, and a feedback physical forcing can then shape the nutrient and light to Nj, for each nutrient j, modulated by the system environment, and affect which phytoplankton species turnover rate, D, referred to as a dilution rate for a are likely to flourish. While this externally imposed chemostat; the sink from the consumption by the sum variability is prevalent, there may also be internally of the phytoplankton species depends on the phyto- γ N induced cyclic behaviour allowing more species to be plankton abundance, Pi, and growth rates, i , for supported than the number of resources (Armstrong each species i and the cell quota, Qji, for each species & McGehee 1980). In particular, Huisman & Weissing i and nutrient j. In Eq. (2), each phytoplankton spe-

(1999, 2001) demonstrate how phytoplankton species cies, Pi, grows exponentially depending on the cell γ N consuming an abiotic resource can have a chaotic re- growth rate, i , and mortality, mi. In Eq. (3), the sponse; the phytoplankton abundance of each species growth rate depends on the maximum growth rate, ri, does not reach an equi librium, but instead continually for each species, modified by the abundance of the evolves in a non-repeating sequence. Alongside this limiting nutrient relative to the half-saturation coeffi- irregular behaviour, chaos is characterised by a high cient, Kji, for each species and resource; note that sensitivity to initial conditions; any differences in ini- for simplicity the growth rate does not depend on tial conditions exponentially increase in time and in- cell quota (as instead applied by Droop 1973). The hibit any predictability. With respect to the paradox chemostat model emulates steady state conditions of the phytoplankton, the number of phytoplankton where consumption of a resource is balanced by its species can exceed the number of resources in these import, and where maximum growth, resource re - chaotic solutions, subject to there also being a ran - quirements and external supply remain invariant in dom injection of spe cies into the environment (Huis- time. man & Weissing 1999; henceforth HW). We firstly consider cases with the same number of Here, we investigate the conditions for the phyto- species and resources (n = k = 5; Figs. 1 to 5) and sec- plankton community to exhibit chaotic, oscillatory ondly where the number of species exceeds the num- and competitive exclusion solutions by addressing ber of resources (n > k = 5; Figs. 6 to 8). The model the dependence on the nutrient source and the fit- parameters and initial conditions follow those of HW ness between species, as well as how long an inter- unless otherwise stated (see Table S1 in the supple- mittent addition of new species persists in the phyto- ment at www.int-res.com/articles/suppl/ m490 p107_ plankton community. supp.pdf). We now examine the relationship between the abundance of phytoplankton species and nutrients, MODEL FORMULATION extending experiments by HW. The model solutions for the abundance of phytoplankton species reveal In this study, the coupled phytoplankton and nutri- 3 different characteristic regimes: (1) competitive ent model of HW is applied for a well-mixed box. The exclusion, when a long-term equilibrium is reached model is based on the linear chemostat assumption where one or more species dominate and drive the (Tilman 1977, 1980, Armstrong & McGehee 1980, others to extinction (Fig. 1a); (2) repeating oscilla- Huisman & Weissing 1999), where there are n phyto - tions, when there is a repeating cycle in the abun- plankton species, Pi, competing for k resources dance of each species (Fig. 1b); or (3) chaotic solu- represented as nutrients, Nj tions, when there are non-repeating changes in species abundance (Fig. 1c). These differences start dN n j = DS()−− N Qγ N P , jjk=…1,, (1) to become apparent over the first 100 d (Fig. 1, left dt jj∑ i=1 jii i panel). The character of the different responses is dP also reflected in the nutrient response in the well- i = Pm(),γ N − i=…1,, n(2) dt i i i mixed box: competitive exclusion leads to steady- Kenitz et al.: Sustaining phytoplankton diversity 109

Fig. 1. Phytoplankton community response generated by the model of Huisman & Weissing (1999) displaying sensitivity to the choice of the half-saturation coefficient. The model responses incorporate (a) competitive exclusion, (b) oscillations and (c) chaos, generated with K4,1 = 0.20, K4,1 = 0.40 and K4,1 = 0.30 respectively, where K is the half-saturation coefficient. The species responses are shown for the initial period of 100 d and over 1000 d (left and central panels), and their phase diagrams are from 500 to 5000 d (right panels) state nutrient concentrations sustained by their nutri- plankton abundance are illustrated by a trajectory in ent source, while oscillations or chaos within the a phase space, where each dimension represents the phytoplankton community are associated with peri- abundance of a particular phytoplankton species. For odic or irregular fluctuations in the ambient nutrient example, consider the evolution of 3 arbitrary species concentrations. in a 3-D phase diagram (Fig. 1, right panels): com - In terms of the ‘paradox of the phytoplankton’, both petitive exclusion is represented by a single point; the repeating oscillations and chaotic solutions are of repeating oscillations by repeating closed trajecto- interest as a long-term equilibrium is not reached, ries; and chaotic solutions by irregular and continu- part of the explanation suggested by Hutchinson ally changing trajectories. Secondly, the sensitivity to (1961). Taking that view further forward, HW argued initial conditions can be estimated by evaluating the that a chaotic state enables the number of species to rate at which 2 points in phase space, initially close exceed the number of resources. together, subsequently diverge away from each other. In our model diagnostics, whether chaos is ob - This diagnostic, referred to as the maximal Lyapunov tained is formally identified using the following Exponent (Kantz 1994), is often used to define chaos, approaches. Firstly, the temporal changes in phyto- identifying when there is an exponential increase in 110 Mar Ecol Prog Ser 490: 107–119, 2013

the separation of 2 trajectories. Thirdly, we employ a ate feedback (α < 1), there are repeating cycles of a binary test distinguishing chaos from non-chaotic single species dominating, switching later to a differ- dynamics, referred to as the 0-1 test for chaos, ent single species, and this pattern is progressively adjusted to detect weak chaos (Gottwald & Mel- repeated (Fig. 2a). Increasing the feedback leads to a bourne 2004, 2009). This technique is the most effi- reduction in the period of each cycle (Fig. 2a,b). cient approach when there are many repeated model For strong feedback (α ~ 1), there are always time- integrations. Further explanation of these methods is varying changes in the abundances of the 5 species provided in the Supplement. and a chaotic response, when the sequences for the abundances of phytoplankton species do not exactly repeat in time (Fig. 2c), as evident in their trajectories MODEL SENSITIVITY EXPERIMENTS not repeating in phase diagrams. Thus, the presence

of –DNj in Eq. (1) fundamentally affects the nature of Sensitivity experiments are now performed to the phytoplankton solutions. understand the different ecosystem response in the While some form of nutrient feedback is plausible well-mixed box, focussing in turn on the environ- given how diffusion acts to supply nutrients to the mental control via the nutrient supply, the physiolog- surface, other physical transport processes often ical control of each species via the cell quota and Kji dominate over this diffusive supply, such as entrain- coefficient, and the effect of random injections of dif- ment at the base of the mixed layer, and the horizon- ferent phytoplankton species. tal and vertical transport of nutrients (Williams & Follows 2003). Hence, the nutrient feedback acting to restore surface nutrients is unlikely to hold all the Environmental control by nutrient supply time, possibly varying in an episodic manner, and probably depending on the physical forcing and The nutrient supply in Eq. (1) includes an external background circulation. Accordingly, we now con- supply, DSj, and a feedback term, –DNj, to ambient sider the effect of introducing slight modifications nutrient concentrations. The external supply and feed - in –DNj in model experiments using the default back together act to restore nutrient concentrations, chaotic parameters. which can be viewed as a crude way of replicating (1) The nutrient supply, D(Sj – Nj(t)), is now inter- how physical processes act to supply nutrients and spersed by intermittent periods when there is no sustain biological productivity. For example, in a feedback, such that the supply temporarily increases vertical water column, phytoplankton consume inor- to DSj for short periods ranging from 10 min to 8 wk ganic nutrients in the euphotic zone, and these inor- (Fig. 3a, shaded). During the intermissions, the phy - ganic nutrients can be resupplied by vertical diffu- toplankton solutions move towards a single species sion, acting to transfer nutrients down gradient from dominating at any single time (Fig. 3a, upper panel), high concentrations in the nutricline to the sur - rather than 5 species being sustained; this response face. This diffusive nutrient supply is given by is more apparent for prolonged periods without ∂ ∂N ∂z ()κ ∂z , which, applying scale analysis, is typically relaxation. After the intermissions, the nutrient sup- κ κ − Δz 2 (Nsurface – Nnutricline), where is the vertical diffu- ply returns to including the nutrient feedback, and sivity, Nsurface and Nnutricline are the nutrient concen- the phytoplankton solutions return to being chaotic trations at the surface and nutricline, separated by a (Fig. 3a). In terms of the nutrient forcing, the nutrient vertical spacing Δz. Thus, when Nsurface < Nnutricline, sources for this case with inter missions and the diffusion acts to restore the surface nutrients towards default case without intermissions (Fig. 1c) are ini- the value in the nutricline, reducing the contrast tially identical, but then differ after the first intermis- between N surface and Nnutricline, so acting in a similar sion due to the different evolution of the nutrients manner to –DNj in the nutrient supply in Eq. (1). (Fig. 3a, lower panel). To assess the effect of the nutrient feedback in (2) The model solutions are altered if the nutrient ~ ~ Eq. (1), model experiments are performed with the supply is adjusted to D(Sj – Nj), where Nj represents α α nutrient supply taking the form D(Sj – Nj) where the past record of forcing based upon the default ranges from 0 to 1 (and otherwise default model para- Nj(t) record (shown to trigger the chaotic response in meters are used; Table S1 in the Supplement). The Fig. 1c with α = 1), but now including prescribed factor α controls the net amount of nutrient supplied intermissions. After the first intermission, the lack of into the environment, and measures the strength of any interactive nutrient feedback leads to the phyto- feedback to the nutrient resource. At weak to moder- plankton solutions changing from being chaotic and Kenitz et al.: Sustaining phytoplankton diversity 111

evolving to a single species dominating (Fig. 3b); the the same as in the chaotic case (1) (Fig. 3a). However, dominant species can alternate in time with a period including the temporal lag does not significantly alter lengthening with every cycle, referred to as hetero- the character of the solutions: chaos is either sustained clinic cycles (Huisman & Weissing 2001). The nutri- or moves to multiple-period oscillations (Fig. 3c) with ent source in this case and the default are nearly all 5 species persisting and varying in time. identical (Fig. 3b, lower panel), but the lack of any In summary, the chaotic nature for the abundance interactive adjustment prevents the chaotic solutions of the phytoplankton species is reliant on there being being sustained. Thus, the presence of the inter - a feedback to the nutrient concentration: an absent or active feedback is crucial for the chaotic solutions to too weak feedback leads to competitive exclusion or emerge and persist. oscillatory changes in the dominant phytoplankton (3) Given the importance of the nutrient feedback, species, which sustain fewer species at any particular the effect of a slight delay is now introduced into the time. In partial accord with this viewpoint, chemostat nutrient supply (an arbitrary lag of 1 day), so that the laboratory experiments find that the community supply term becomes D(Sj – Nj(t – 1 day)). The nutri- response is sensitive to nutrient supply rates (Becks ent supply retains the interactive feedback, although et al. 2005), where the nutrient supply is modelled the lag implies that the nutrient supply is not exactly with feedback terms as in Eq. (1).

α Fig. 2. Phytoplankton community response to the nutrient supply, D(Sj – Nj) (for abbreviations see ‘Model formulation’), with varying levels of feedback: (a) weak (α = 0.2), (b) moderate (α = 0.6), and (c) strong (α = 0.8). Time series plots are for the initial 2000 d (left panels) and phase plots over the later 7000 to 10 000 d (right panels) 112 Mar Ecol Prog Ser 490: 107–119, 2013

(a) Feedback with breaks Physiological choices 60 Physiological traits and related trade-offs

40 define the ecological niche of species and affect their survival ability. The effect of

modifying the choice of cell quota and Kji 20 is now assessed on the phytoplankton

Species abundance Species community structure. 0 0100 200 300 400 500 600 700 800 900 1000 ) –3 1.6 Cell quota

x 10 1.4 In a similar manner to the way the nutri- 1.2 0100 200 300 400 500 600 700 800 900 1000 ent relaxation is investigated, the cell Source ( Source Days quota, Qji, is assumed either (1) to be the (b) No interactive feedback same for all species and alter in the same 140 manner for each resource, or (2) to vary in a 120 different manner for each species and 100 resource (following HW): 80 Species 1 60 Species 2 Species 3 ⎛ 004..... 004 004 004 004 ⎞ 40 Species 4 ⎜ 008..... 008 008 008 008 ⎟ 20 Species 5 Species abundance Species ⎜ ⎟ Q = 0...... 10 0 10 0 10 0 10 0 10 + 0 ji ⎜ ⎟ 0100 200 300 400 500 600 700 800 900 1000 ⎜ 003...... 003 003 003 003 ⎟

) ⎜ ⎟

–3 1.6 ⎝ 007.. 0007 0... 07 0 07 0 07 ⎠ (4) x 10 1.4 ⎛ 00.... 00 003 00 0.0 ⎞ ⎜ 00..... 00 00 002 00 ⎟ 1.2 ⎜ ⎟ 0100 200 300 400 500 600 700 800 900 1000 β 00..... 00 00 00 004 Source ( Source ⎜ ⎟ Days ⎜ 002000000. ...000. ⎟ (c) Lagged feedback ⎜ ⎟ ⎝ 00..... 002 00 00 00 ⎠ 60 where the values in the matrix for Qji are for each resource j in the rows and for each 40 species i in the columns, and β varies from 0 to 1; other model parameters are the 20 default (Table S1). A choice of β to 0 repre-

Species abundance Species sents the same cell quota for all species, 0 β 0100 200 300 400 500 600 700 800 900 1000 while to 1 is representative of HW with

) an increase in the contrast in cell quota for –3 1.6 a particular resource for each species. x 10 1.4 When the cell quota is identical for each species, there is competitive exclusion 1.2 0100 200 300 400 500 600 700 800 900 1000 Source ( Source (Fig. 4a) and the fittest species has the Days lowest re quirement for the limiting re - Fig. 3. Phytoplankton species abundance (upper panels) and nutrient source (Til man 1977). When moderate source (lower panels) versus time with the modified nutrient supply: (a) changes in cell quota are chosen, there are nutrient source with feedback and intermittent disruptions (grey shad- ing) lasting 10 min (Day 200), 3 wk (Day 300) and 8 wk (Day 500), when oscillations in the phytoplankton response the default nutrient feedback is temporarily removed, DSj; (b) nutrient (Fig. 4b). When large contrasts in cell source without feedback defined by the record of the default nutrient quota are chosen for each species, there source (as in Fig. 1c) including intermissions (as in Panel a); (c) nutrient are chaotic fluctuations in the con cen - source with lagged feedback, where nutrient supply depends on the nu- trations of each phytoplankton species trient concentration from the previous day, D(Sj – Nj (t – 1 day)). In each case, the time series of the nutrient source for resource 1 (black line) is (Fig. 4c), allowing the coexistence of all compared with that for the default source term (dashed red line) in the 5 species. bottom panels. For abbreviations see ‘Model formulation’ Kenitz et al.: Sustaining phytoplankton diversity 113

β Fig. 4. Phytoplankton community response to changes in cell quota, Qji: (a) nearly uniform cell quota for each species, = 0.2; (b) moderate contrasts in cell quota for each species, β = 0.6; and (c) strong contrasts in cell quota, β = 0.7. The temporal adjustment is shown over the first 100 and 1000 d (left and middle panels) and corresponding phase plots (right panels) for abundance of species 1, 3 and 5 for 500 to 5000 d

Half-saturation coefficient where ki are randomly generated numbers, such that k1 < k2 < k3 < k4 < k5 and the values in the matrix are The sensitivity to the half-saturation coefficient, Kji, for each resource j in the rows and for each species i is investigated by varying the values for each species in the columns. Three separate sets of simulations and resource, but in an ordered manner so that each are included, with ki randomly chosen (retaining the of the species is the optimal competitor for one of the above structure and ordering) within the intervals 0.2 resources: to 0.23, 0.2 to 0.5, and 0.1 to 1. In each set, the model

⎛ kkkkk54321⎞ was integrated 1000 times over 50000 days and all ⎜ kkkkk⎟ solutions were identified using the 0-1 Test for Chaos ⎜ 15432⎟ (5) (see the Supplement). K ji = ⎜ kkkkk21543⎟ ⎜ ⎟ At any particular time, the solutions take the form ⎜ kkkkk32154⎟ ⎜ ⎟ of either competitive exclusion involving a single ⎝ k443215kkkk⎠ 114 Mar Ecol Prog Ser 490: 107–119, 2013

dominant species (Fig. 5, blue), oscillations with a intermediate species with the other competitors (Fig. 5). repeating cycle in species abundance or irregular For the intermediate competitor, k3, compared with chaos, both involving all 5 species (Fig. 5, green and the 2 strongest competitors, k1 and k2, competitive ex- red respectively). For competitive exclusion, the clusion is the most likely response when species are of dominant species might alter and be replaced by comparable fitness, but alters to chaos and then oscil- another species, taking the form of heteroclinic lations with greater contrasts in the strength of these cycles (as shown earlier in Fig. 3b); the resulting competitors (Fig. 5, left panels). Hence, the more com- ordered sequence is a consequence of each species petitive the intermediate competitor, the more chance being the optimal competitor for a different resource. of there being an optimal competitor and obtaining A pattern in the different model responses is evi - competitive exclusion, while a weaker intermediate dent when comparing the competitive ability of the competitor encourages chaos or oscillations.

(a) Large changes in half-saturation coefficient, Kji 1 1

0.7 0.7 3 3 k k Chaos 0.4 Oscillations 0.4 Competitive exclusion 0.1 0.1 0.1 0.4 0.7 1 0.1 0.4 0.7 1 k2 k5

(b) Moderate changes in half-saturation coefficient, Kji 0.5 0.5

0.4 0.4 3 3 k k

0.3 0.3

0.2 0.2 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 k2 k5

0.22 0.22 3 3 k k

0.21 0.21

0.2 0.2 0.2 0.21 0.22 0.23 0.2 0.21 0.22 0.23 k2 k5

Fig. 5. The phytoplankton community response to randomly assigned half-saturation coefficient, Kji, within prescribed bounds for 1000 model integrations, each lasting 50 000 d. The model responses include competitive exclusion with 1 dominant species at any time (blue), oscillations (green) and chaos (red). Illustrated are the relationships between different Kji for (a) large, (b) moderate and (c) small contrasts. The model solutions for a strong versus intermediate competitor, k2 versus k3 are shown in the left panels, and a weak versus intermediate competitor, k5 versus k3 in the right panels Kenitz et al.: Sustaining phytoplankton diversity 115

The other side of this response is that comparing 10 d), starting at Day 90 and persisting for 1 yr. The the intermediate competitor, k3, with the 2 weakest additional species have their cell traits stochastically competitors, k4 and k5, leads to the reversed pattern determined for each model integration, with Kji cho- (Fig. 5, right panels): a similar fitness of the 3 species sen within the interval 0.2 to 0.5, and Qji within the favours oscillations, while increased contrasts gener- in terval 0.01 to 0.1. These biological parameters were ally lead to chaos and eventually are more likely to assigned for each species and resource either in a lead to competitive exclusion. Indeed, the more simi- random manner, or assuming a negative relation be - lar the intermediate competitor is to the weaker spe- tween fitness and cell quota (scenarios 1 and 3 of Huis- cies, the more the intermediate competitor differs from man et al. 2001, respectively); however, the long- the strong competitors, which explains the reversed term character of the model results turned out not to pattern. No regular structure is evident when Kji are be sensitive to these scenarios. compared for strong versus weak competitors. The model state prior to the invasion is our default

When the perturbations in Kji are in a very narrow choice: 5 species competing for 5 resources, so that range (0.2 to 0.23), competitive exclusion is the dom- competition theory predicts that up to 5 different spe- inant response, occurring over 47% of the parameter cies should be sustained for a long-term equilibrium. space, while oscillations occur in 17% and chaotic To sample the different characteristic regimes, the solutions in 14% of parameter space (Table 1); the model experiments are repeated for a range of choices remaining 22% of solutions are not distinguished for Kji: obtaining (1) chaos with the default Kji matrix, between oscillations and chaos. When the perturba- (2) single-period oscillations with K5,4 = 0.37, and (3) tions in Kji are in a larger range (0.1 to 1.0), competi- competitive exclusion with K2,4 = 0.20; with other- tive exclusion reduces to 19% of parameter space wise default choices for the rest of Kji for all 3 cases. and instead oscillations increase to 45% and chaos to In the chaotic case, the number of phytoplankton

17% of parameter space. Hence, when Kji of interme- species exceeds the number of resources over the diate and strong competitors are close together, there length of the integration of 2500 d (Fig. 6a, panel [i]). is more chance of identifying the optimal competitor Chaotic fluctuations then allow the number of spe- and obtaining competitive exclusion. cies to exceed the number of resources, referred to as ‘supersaturation’, in our integrations supporting 20 to 30 species within 3 months from the last input of Random injection of phytoplankton species new species (Fig. 7a). The number of coexisting species gradually reduces to 10–15 surviving species We next investigate the response of the model to an after 1 yr, and decreases further to less than 5 after intermittent ‘injection’ of new species, replicating 2 yr for the majority of the model compilations. The how ocean circulation leads to the transport and dis- chaotic fluctuations can sometimes abruptly diminish persal of phytoplankton species. (Fig. 6a, panel [ii]), without any intermittent disrup- To investigate this species injection and the longer- tion prior to the event. Thus, the fittest competitors term community response, an ‘invasion approach’ is persist, while the weaker species progressively be - applied broadly following Huisman et al. (2001): come extinct. During the process of introducing more additional species are introduced with 3 new species species, there is more chance for an optimal compe - with initial abundance Pi = 0.1, typically introduced titor to be identified and so there is less chance for every 30 d (with random deviations of a maximum of chaos and oscillations to emerge. Oscillatory solutions lead to a broadly similar re - sponse to chaotic solutions: there is a supersaturation Table 1. Different phytoplankton community responses (%) for 3 separate sets of 1000 model integrations, each with in the number of species, which gradually declines in a different range of randomly generated half-saturation time, as illustrated for 1-period oscillations (Figs. 6b coefficient, Kji. For a proportion of model simulations, & 7b) and also obtained for 2-period oscillations (not community behaviour could not be distinguished between shown). oscillations and chaos In the case of competitive exclusion, the community is already dominated by an optimal competitor Kji range Competitive Oscilla- Chaos Oscillations exclusion tions or chaos and so there is a very weak, short-lived response to an injection of additional species (Fig. 6c). Supersat- 0.2–0.23 47 17 14 22 uration is only sustained for a brief 6-month period 0.2–0.5 32 40 12 16 0.1–1.0 19 45 17 19 after the last input of additional species, swiftly re - turning to fewer than 5 coexisting species (Fig. 7c). 116 Mar Ecol Prog Ser 490: 107–119, 2013

(a) Chaos (i) 60 30 40 20

20 10

0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 (ii) 60 30 40

Species abundance 20 Number of survivals

20 10

0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500

(b) Oscillations 80 30 60 20 40 10 20

0 Number of survivals Species abundance 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500

Species 1 20 50 Species 2 Species 3 10 Species 4 Species 5

Species abundance 0 0 0 500 1000 1500 2000 2500 Number of survivals 0 500 1000 1500 2000 2500 Day Day Fig. 6. Phytoplankton species abundance (left panels) and number of survival species (right panels) versus time after 12 intermittent injections of 3 additional species (starting from Day 90 to Day 450, vertical line), depending on whether there is (a) chaos, shown for 2 examples (i) and (ii) with different randomly generated species, (b) oscillations or (c) competitive exclusion. For all cases, there is the same timing of species injections, with the final input indicated by the vertical dotted line. Survival species are defined by the abundance greater than an arbitrary small value of 0.0001. The horizontal dashed line shows the maximum species number predicted from resource competition theory for an equilibrium state

Hence, none of the species added to the system are fit petition theory predicting that at equilibrium the enough to outcompete the optimal competitor once it number of species cannot exceed the number of lim- is strongly established in the community. iting resources. He suggested that this ‘paradox of In summary, chaos and oscillations support a com- the phytoplankton’ and inconsistency with competi- parable number of species, exceeding the number of tion theory might be reconciled by the phytoplankton resources for as long as 2 yr after the last input of new community not being at equilibrium. species, while competitive exclusion usually sustains There are a variety of explanations as to why an a lower number of species than expected from the equilibrium state for the phytoplankton community resource competition theory (Fig. 8). might not be achieved, possibly reflecting a response to the spatial and temporal heterogeneity in the physical environment, or instead an ecological response in- DISCUSSION volving inter-species competition. Phytoplankton species typically have a doubling timescale of 2 to 5 d, Hutchinson (1961) first questioned why so many and competitive exclusion might be expected to occur different phytoplankton species persist, given com- over the order of 10 generations, suggesting a time Kenitz et al.: Sustaining phytoplankton diversity 117

(a) Chaos span for equilibrium to be reached of typi- 40 cally 1 to 2 months (Reynolds 1995). On this timescale, the ocean surface boundary layer 30 5 yr is strongly forced by the passage of atmo - 20 2 yr spheric weather systems, modifying the 1.5 mo 1 yr 6 mo 3 mo convection and mixing within the surface 10 boundary layer and the light received by phytoplankton. In addition, in coastal seas, 0 0 5 10 15 20 25 30 35 the surface boundary layer is modified by the spring-neap changes in the tides. Given (b ) Oscillations 40 this temporal variability in the physical forc- 5 yr ing, there are 2 limits leading to relatively 30 low phytoplankton diversity: (1) if there is 2 yr severe forcing, such as involving a sustained 20 1 yr period of no light or nutrient supply followed 3 mo 1.5 mo by an onset of favourable conditions, then 10 6 mo

Model runs (%) the phytoplankton species with the fastest 0 growth rate dominates; and conversely, (2) 0 5 10 15 20 25 30 35 persistent conditions lead to the optimal (c ) Competitive exclusion competitors flourishing for a stable envi- 70 ronment. Hence, the maximum diversity in 60 5 yr phytoplankton species is expected between 50 these 2 limits, referred to as the intermediate 2 yr 40 1 yr disturbance hypothesis, which was applied 30 6 mo by Connell (1978) for tropical rainforests 3 mo 20 1.5 mo and coral reefs, discussed for phytoplankton 10 by Padisák (1995) and Rey nolds (1995), and 0 used to explain observed changes in the 0 5 10 15 20 25 30 35 Number of survival species phytoplankton community for a shallow eu- Fig. 7. Number of species sustained at a particular time after the last in- trophic lake (Weithoff et al. 2001). Thus, the jection of species, for 1000 model integrations, with each model compila- physical forcing might induce continual tem- tion generated with a different set of random cell traits of injected spe- poral and spatial changes in the environment, cies. Each set of 1000 runs is induced with different choices of the which the phytoplank ton community is con- half-saturation coefficients, Kji for the initial 5 species, which leads to (a) tinually adjusting to, such that competitive chaos (with default K ), (b) 1-period oscillations (with K = 0.37) and (c) ji 5,4 exclusion is not reached. competitive exclusion (with K2,4 = 0.20). The dashed line indicates the maximum of 5 species surviving on 5 resources predicted for equilibrium An alternative view to this physically in - by the resource competition theory duced heterogeneity is that there may be

Fig. 8. Mean number of species sustained throughout 1000 model integrations after the last pulse of extra species (indicated by vertical, dotted line). Each set of 1000 runs is induced with different choices of the half-saturation coefficients, Kji, for the initial 5 species, which leads to (a) chaos (red line, with default Kji), (b) 1-period oscillations (green line, with K5,4 = 0.37 and (c) competitive exclusion (blue line, with K2,4 = 0.20). SD is indicated by the corresponding shaded regions. The horizontal dashed line specifies the maximum of 5 species coexisting on 5 resources based on the resource competition theory 118 Mar Ecol Prog Ser 490: 107–119, 2013

more phytoplankton variability due to inter-species included from the wider environment, we find that competition for resources, as advocated by Huisman if there are oscillations or chaotic solutions, then a & Weissing (1999, 2001). Rather than a single or a few short-term injection of species leads to a long-term species dominating as in competitive exclusion, the sustenance of more species than resources. In both phytoplankton community can continually vary in cases, there is a very similar response with super - the form of repeating oscillations or chaotic changes saturation in the number of species. In contrast, when in the abundance of different species. there is a competitive exclusion, an additional injec- Whether the model solutions lead to competitive tion of species only leads to a short-lived excess of exclusion, oscillations or chaos turns out to be sensi- species, which quickly die away. Thus, given a ran- tive to the cell physiology and nutrient requirements. dom injection of species, both oscillatory and chaotic Competition between species of similar fitness is solutions help sustain more phytoplankton species most likely to lead to competitive exclusion, with the than resources. optimal competitor having the lowest requirement In our model experiments, the emergence of chaos for a resource (Tilman 1977). Including competition versus oscillations is very sensitive to whether a between species with variability in cell physiology nutrient feedback is included. When the feedback is and nutrient requirement via cell quota does not lead strong, chaotic solutions emerge, but when the feed- to an optimal competitor emerging, and instead back is weak or absent then the solutions switch to favours oscillatory or chaotic behaviour. The detailed oscillations or competitive exclusion. A choice of response often turns out to be controlled by the nutri- strong feedback acting to restore nutrients is appro- ent requirement of the intermediate species com- priate for the way a chemostat operates or for a pared with that of the other species. In our sets simple 1-dimensional problem, such as how vertical of 1000 model experiments with different ranges diffusion acts to supply nutrients down-gradient to in half-saturation coefficient (Table 1), competitive the euphotic zone and sustain productivity. However, exclusion occurs for 19% of the integrations if there is a question as to the extent that the nutrient there are large contrasts in half-saturation coeffi- feedback always holds in the open ocean. The nutri- cient, and increases to 47% of the integrations if ent supply to the euphotic zone is affected by a wide there are small contrasts in half-saturation coefficient range of physical processes, including convection, (Table 1), reflecting the increased chance of identi - entrainment at the base of the mixed layer, and hori- fying an optimal competitor with small contrasts in zontal and vertical transport by the gyre, eddy and half saturation. In turn, a combination of oscillations basin scale overturning circulations (Williams & Fol- and chaos then occur for at least half of the model lows 2003). These processes can either enhance or integrations. inhibit biological productivity. For example, wind- A particular criticism of whether inter-species com- driven upwelling induces productive surface waters petition explains the paradox of the plankton is that over subpolar gyres, while wind-driven downwelling chaotic solutions might be an unusual occurrence, as induces oligotrophic surface waters over subtropical suggested by model experiments initialised with ran- gyres. These physical processes are unlikely to always domly assigned characteristics for the phytoplankton provide a nutrient feedback to sustain inter-species (Schippers et al. 2001). However, this conclusion is driven chaos. There may be some regimes, particu- challenged by Huisman et al. (2001) who argue that a larly physically isolated cases, when species compe- different response is obtained if additional phyto- tition might induce chaos, such as in the deep chloro- plankton species are injected at different times and phyll maximum in oligotrophic gyres during the a wider range of physiological choices are made. summer when there is weak mixing (Huisman et al. Our model diagnostics support the view of Huisman 2006). Elsewhere, phytoplankton diversity is proba- & Weissing (1999, 2001) that chaos can emerge in a bly determined by a combination of inter-species well-mixed box through inter-species competition competition and the effects of spatial and temporal for phytoplankton communities. Indeed, a long-term variations in physical forcing. For example, phyto- laboratory mesocosm experiment which monitored plankton diversity is enhanced in western boundary the plankton community twice a week for 2300 d currents and gyre boundaries by the combination of revealed chaotic fluctuations in phytoplankton spe- transport, lateral mixing and dispersal, as shown by cies abundances (Benincà et al. 2008), consistent Barton et al. (2010) and Follows et al. (2007). with a lack of predictability beyond 15 to 30 d. The sensitivity of our phytoplankton solutions to With respect to how many phytoplankton species the coupling between phytoplankton species and the are supported when transport and dispersal are abiotic resource is perhaps analogous to how pre- Kenitz et al.: Sustaining phytoplankton diversity 119

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Editorial responsibility: Christine Paetzold, Submitted: July 27, 2012; Accepted: June 20, 2013 Oldendorf/Luhe, Germany Proofs received from author(s): September 2, 2013