MARINE ECOLOGY PROGRESS SERIESMar Ecol Prog Ser

Vol. 490: 107–119, 2013doi: 10.3354/meps10452

Published September 17

INTRODUCTION

Hutchinson (1961) first posed the paradox of theplankton: Why do so many phytoplankton speciescoexist while competing for a limited number of re -sources in a nearly homogeneous environment? Forexample, open ocean and lake surface waters usuallycontain the order of 1 to 10 dominant phytoplanktonspecies together with many hundreds or more spe-cies at very low concentrations. This high number ofphytoplankton species appears at odds with the com-petitive exclusion principle (Hardin 1960) where the

number of species coexisting at equilibrium is notexpected to exceed the number of resources. Forphytoplankton, the resources can be viewed in termsof macro nutrients, trace metals and variations in thelight and temperature environment, such that if 2phytoplankton species compete for the same re -source, the most successful competitor is the one sur-viving on the minimum resource (Tilman 1977,Tilman et al. 1982). This excess in the number ofphytoplankton species has been explained in termsof the phytoplankton system not reaching an equilib-rium state due to temporal variability, as first specu-

© The authors 2013. Open Access under Creative Commons byAttribution Licence. Use, distribution and reproduction are un -restricted. Authors and original publication must be credited.

Publisher: Inter-Research · www.int-res.com

*Corresponding author. Email: ric@liverpool.ac.uk

The paradox of the plankton: species competitionand 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, UK2National 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 resourcesare examined using a coupled phytoplankton and nutrient model for a well-mixed box. The phyto-plankton community either reaches a competitive exclusion state, where there is an optimal com-petitor, or the abundance of each phytoplankton species continually varies in the form of repeat-ing oscillations or irregular chaotic changes. Oscillatory and chaotic solutions make up over halfof the model solutions based upon sets of 1000 separate model integrations spanning large, mod-erate or small random changes in the half-saturation coefficient, Kji. The oscillatory or chaoticstates allow a greater number of phytoplankton species to be sustained, even for their number toexceed 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 explicitfeedback between nutrient supply and ambient nutrient concentration, and when there are phys-iological 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 nutri-cline. Inter-species competition might then be important in generating chaos when this diffusivetransfer is important, but less likely to be significant when other transport processes sustain surface nutrient concentrations.

KEY WORDS: Plankton paradox · Coexistence · Chaos · Competitive exclusion · Phytoplanktoncommunities

OPENPEN ACCESSCCESS

Mar Ecol Prog Ser 490: 107–119, 2013

lated in terms of seasonality by Hutchinson (1961) orspatial variability in the background environment(Richerson et al. 1970).

There are many ways in which this temporal andspatial variability can be achieved in the real world,such as from the externally imposed physical vari-ability from changes in solar irradiance, weather-related changes in air-sea forcing and changes in mechanical forcing from tides. These changes inphysical forcing can then shape the nutrient and lightenvironment, and affect which phytoplankton speciesare likely to flourish. While this externally imposedvariability is prevalent, there may also be internallyinduced cyclic behaviour allowing more species to besupported than the number of resources (Armstrong& McGehee 1980). In particular, Huisman & Weissing(1999, 2001) demonstrate how phyto plankton speciesconsuming an abiotic resource can have a chaotic re-sponse; the phytoplankton abundance of each speciesdoes not reach an equi librium, but instead continuallyevolves in a non-repeating sequence. Alongside thisirregular behaviour, chaos is characterised by a highsensitivity to initial conditions; any differences in ini-tial conditions exponentially increase in time and in-hibit any predictability. With respect to the paradoxof the phytoplankton, the number of phytoplanktonspecies can exceed the number of resources in thesechaotic solutions, subject to there also being a ran -dom injection of spe cies into the environment (Huis-man & Weissing 1999; henceforth HW).

Here, we investigate the conditions for the phyto-plankton community to exhibit chaotic, oscillatoryand competitive exclusion solutions by addressingthe dependence on the nutrient source and the fit-ness between species, as well as how long an inter-mittent addition of new species persists in the phyto-plankton community.

MODEL FORMULATION

In this study, the coupled phytoplankton and nutri-ent model of HW is applied for a well-mixed box. Themodel is based on the linear chemostat assumption(Tilman 1977, 1980, Armstrong & McGehee 1980,Huisman & Weissing 1999), where there are n phy to -plankton species, Pi, competing for k resources represented as nutrients, Nj

(1)

(2)

(3)

where the subscripts denote the particular species i =1,…,n and resources j = 1,…,k. In Eq. (1), the nutrientconcentration, Nj, evolves through a competition be -tween a source from a nutrient supply and a sinkfrom phytoplankton consumption: the nutrient sup-ply involves an external supply, Sj, and a feedbackto Nj, for each nutrient j, modulated by the systemturnover rate, D, referred to as a dilution rate for achemostat; the sink from the consumption by the sumof the phytoplankton species depends on the phyto-plankton abundance, Pi, and growth rates, γ iN, foreach species i and the cell quota, Qji, for each speciesi and nutrient j. In Eq. (2), each phytoplankton spe-cies, Pi, grows exponentially depending on the cellgrowth rate, γ iN, and mortality, mi. In Eq. (3), thegrowth rate depends on the maximum growth rate, ri,for each species, modified by the abundance of thelimiting nutrient relative to the half-saturation coeffi-cient, Kji, for each species and resource; note thatfor simplicity the growth rate does not depend oncell quota (as instead applied by Droop 1973). Thechemostat model emulates steady state conditionswhere consumption of a resource is balanced by itsimport, and where maximum growth, resource re -quirements and external supply remain invariant intime.

We firstly consider cases with the same number ofspecies and resources (n = k = 5; Figs. 1 to 5) and sec-ondly where the number of species exceeds the num-ber of resources (n > k = 5; Figs. 6 to 8). The modelparameters and initial conditions follow those of HWunless otherwise stated (see Table S1 in the supple-ment at www.int-res.com/articles/suppl/ m490 p107_supp.pdf).

We now examine the relationship between theabundance of phytoplankton species and nutrients,extending experiments by HW. The model solutionsfor the abundance of phytoplankton species reveal3 different characteristic regimes: (1) competitiveexclusion, when a long-term equilibrium is reachedwhere one or more species dominate and drive theothers to extinction (Fig. 1a); (2) repeating oscilla-tions, when there is a repeating cycle in the abun-dance of each species (Fig. 1b); or (3) chaotic solu-tions, when there are non-repeating changes inspecies abundance (Fig. 1c). These differences startto become apparent over the first 100 d (Fig. 1, leftpanel). The character of the different responses isalso reflected in the nutrient response in the well-mixed box: competitive exclusion leads to steady-

d

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tD S N Q Pj j j jii

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r NK N

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108

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Kenitz et al.: Sustaining phytoplankton diversity

state nutrient concentrations sustained by their nutri-ent source, while oscillations or chaos within thephytoplankton community are associated with peri-odic or irregular fluctuations in the ambient nutrientconcentrations.

In terms of the ‘paradox of the phytoplankton’, boththe repeating oscillations and chaotic solutions are ofinterest as a long-term equilibrium is not reached,part of the explanation suggested by Hutchinson(1961). Taking that view further forward, HW arguedthat a chaotic state enables the number of species toexceed the number of resources.

In our model diagnostics, whether chaos is ob -tained is formally identified using the followingapproaches. Firstly, the temporal changes in phyto-

plankton abundance are illustrated by a trajectory ina phase space, where each dimension represents theabundance of a particular phytoplankton species. Forexample, consider the evolution of 3 arbitrary speciesin a 3-D phase diagram (Fig. 1, right panels): com -petitive exclusion is represented by a single point;repeating oscillations by repeating closed trajecto-ries; and chaotic solutions by irregular and continu-ally changing trajectories. Secondly, the sensitivity toinitial conditions can be estimated by evaluating therate at which 2 points in phase space, initially closetogether, subsequently diverge away from each other.This diagnostic, referred to as the maximal LyapunovExponent (Kantz 1994), is often used to define chaos,identifying when there is an exponential increase in

109

Fig. 1. Phytoplankton community response generated by the model of Huisman & Weissing (1999) displaying sensitivity to thechoice 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 speciesresponses 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)

Mar Ecol Prog Ser 490: 107–119, 2013

the separation of 2 trajectories. Thirdly, we employ abinary test distinguishing chaos from non-chaoticdynamics, referred to as the 0-1 test for chaos,adjusted to detect weak chaos (Gottwald & Mel-bourne 2004, 2009). This technique is the most effi-cient approach when there are many repeated modelintegrations. Further explanation of these methods isprovided in the Supplement.

MODEL SENSITIVITY EXPERIMENTS

Sensitivity experiments are now performed tounderstand the different ecosystem response in thewell-mixed box, focussing in turn on the environ-mental control via the nutrient supply, the physiolog-ical control of each species via the cell quota and Kjicoefficient, and the effect of random injections of dif-ferent phytoplankton species.

Environmental control by nutrient supply

The nutrient supply in Eq. (1) includes an externalsupply, DSj, and a feedback term, –DNj, to ambientnutrient concentrations. The external supply and feed -back together act to restore nutrient concentrations,which can be viewed as a crude way of replicatinghow physical processes act to supply nutrients andsustain biological productivity. For example, in a vertical water column, phytoplankton consume inor-ganic nutrients in the euphotic zone, and these inor-ganic nutrients can be resupplied by vertical diffu-sion, acting to transfer nutrients down gradient fromhigh concentrations in the nutricline to the sur -face. This diffusive nutrient supply is given by

, which, applying scale analysis, is typically

(Nsurface – Nnutricline), where κ is the vertical diffu-sivity, Nsurface and Nnutricline are the nutrient concen-trations at the surface and nutricline, separated by avertical spacing Δz. Thus, when Nsurface < Nnutricline,diffusion acts to restore the surface nutrients towardsthe value in the nutricline, reducing the contrastbetween Nsurface and Nnutricline, so acting in a similarmanner to –DNj in the nutrient supply in Eq. (1).

To assess the effect of the nutrient feedback inEq. (1), model experiments are performed with thenutrient supply taking the form D(Sj – αNj) where αranges from 0 to 1 (and otherwise default model para -meters are used; Table S1 in the Supplement). Thefactor α controls the net amount of nutrient suppliedinto the environment, and measures the strength offeedback to the nutrient resource. At weak to moder-

ate feedback (α < 1), there are repeating cycles of asingle species dominating, switching later to a differ-ent single species, and this pattern is progressively repeated (Fig. 2a). Increasing the feedback leads to areduction in the period of each cycle (Fig. 2a,b).

For strong feedback (α ~ 1), there are always time-varying changes in the abundances of the 5 speciesand a chaotic response, when the sequences for theabundances of phytoplankton species do not exactlyrepeat in time (Fig. 2c), as evident in their trajectoriesnot repeating in phase diagrams. Thus, the presenceof –DNj in Eq. (1) fundamentally affects the nature ofthe phytoplankton solutions.

While some form of nutrient feedback is plausiblegiven how diffusion acts to supply nutrients to thesurface, other physical transport processes oftendominate over this diffusive supply, such as entrain-ment at the base of the mixed layer, and the horizon-tal and vertical transport of nutrients (Williams & Follows 2003). Hence, the nutrient feedback acting torestore surface nutrients is unlikely to hold all thetime, possibly varying in an episodic manner, andprobably depending on the physical forcing andbackground circulation. Accordingly, we now con-sider the effect of introducing slight modificationsin –DNj in model experiments using the defaultchaotic parameters.

(1) The nutrient supply, D(Sj – Nj(t)), is now inter-spersed by intermittent periods when there is nofeedback, such that the supply temporarily increasesto DSj for short periods ranging from 10 min to 8 wk(Fig. 3a, shaded). During the intermissions, the phy -toplankton solutions move towards a single speciesdominating at any single time (Fig. 3a, upper panel),rather than 5 species being sustained; this responseis more apparent for prolonged periods withoutrelaxation. After the intermissions, the nutrient sup-ply returns to including the nutrient feedback, andthe phytoplankton solutions return to being chaotic(Fig. 3a). In terms of the nutrient forcing, the nutrientsources for this case with inter missions and thedefault case without intermissions (Fig. 1c) are ini-tially identical, but then differ after the first intermis-sion due to the different evolution of the nutrients(Fig. 3a, lower panel).

(2) The model solutions are altered if the nutrientsupply is adjusted to D(Sj –

~Nj), where

~Nj represents

the past record of forcing based upon the defaultNj(t) record (shown to trigger the chaotic response inFig. 1c with α = 1), but now including prescribedintermissions. After the first intermission, the lack ofany interactive nutrient feedback leads to the phyto-plankton solutions changing from being chaotic and

∂∂

∂∂( )zNzκ

− κΔz2

110

Kenitz et al.: Sustaining phytoplankton diversity

evolving to a single species dominating (Fig. 3b); thedominant species can alternate in time with a periodlengthening with every cycle, referred to as hetero-clinic cycles (Huisman & Weissing 2001). The nutri-ent source in this case and the default are nearlyidentical (Fig. 3b, lower panel), but the lack of anyinteractive adjustment prevents the chaotic solutionsbeing sustained. Thus, the presence of the inter -active feedback is crucial for the chaotic solutions toemerge and persist.

(3) Given the importance of the nutrient feedback,the effect of a slight delay is now introduced into thenutrient supply (an arbitrary lag of 1 day), so that thesupply term becomes D(Sj – Nj(t – 1 day)). The nutri-ent supply retains the interactive feedback, althoughthe lag implies that the nutrient supply is not exactly

the same as in the chaotic case (1) (Fig. 3a). However,including the temporal lag does not significantly alterthe character of the solutions: chaos is either sustainedor moves to multiple-period oscillations (Fig. 3c) withall 5 species persisting and varying in time.

In summary, the chaotic nature for the abundanceof the phytoplankton species is reliant on there beinga feedback to the nutrient concentration: an absent ortoo weak feedback leads to competitive exclusion oroscillatory changes in the dominant phytoplanktonspecies, which sustain fewer species at any particulartime. In partial accord with this viewpoint, chemostatlaboratory experiments find that the communityresponse is sensitive to nutrient supply rates (Beckset al. 2005), where the nutrient supply is modelledwith feedback terms as in Eq. (1).

111

Fig. 2. Phytoplankton community response to the nutrient supply, D(Sj – αNj) (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)

Mar Ecol Prog Ser 490: 107–119, 2013

Physiological choices

Physiological traits and related trade-offsdefine the ecological niche of species andaffect their survival ability. The effect ofmodifying the choice of cell quota and Kjiis now assessed on the phytoplankton community structure.

Cell quota

In a similar manner to the way the nutri-ent relaxation is investigated, the cellquota, Qji, is assumed either (1) to be thesame for all species and alter in the samemanner for each resource, or (2) to vary in adifferent manner for each species andresource (following HW):

(4)

where the values in the matrix for Qji arefor each resource j in the rows and for eachspecies i in the columns, and β varies from0 to 1; other model parameters are thedefault (Table S1). A choice of β to 0 repre-sents the same cell quota for all species,while β to 1 is representative of HW withan increase in the contrast in cell quota fora particular resource for each species.When the cell quota is identical for eachspecies, there is competitive exclusion(Fig. 4a) and the fittest species has thelowest re quirement for the limiting re -source (Til man 1977). When moderatechanges in cell quota are chosen, there areoscillations in the phy toplankton response(Fig. 4b). When large contrasts in cellquota are chosen for each species, thereare chaotic fluctuations in the con cen -trations of each phytoplankton species(Fig. 4c), allowing the coexistence of all5 species.

Qji

. . . . .

. . . . .=

0 04 0 04 0 04 0 04 0 040 08 0 08 0 08 0 08 0 080.. . . . .. . . . . .. .

10 0 10 0 10 0 10 0 100 03 0 03 0 03 0 03 0 030 07 0 007 0 07 0 07 0 07

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(a) Feedback with breaks

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Species 1Species 2Species 3Species 4Species 5

Fig. 3. Phytoplankton species abundance (upper panels) and nutrientsource (lower panels) versus time with the modified nutrient supply: (a)nutrient source with feedback and intermittent disruptions (grey shad-ing) lasting 10 min (Day 200), 3 wk (Day 300) and 8 wk (Day 500), whenthe default nutrient feedback is temporarily removed, DSj; (b) nutrientsource without feedback defined by the record of the default nutrientsource (as in Fig. 1c) including intermissions (as in Panel a); (c) nutrientsource with lagged feedback, where nutrient supply depends on the nu-trient concentration from the previous day, D (Sj – Nj (t – 1 day)). In eachcase, the time series of the nutrient source for resource 1 (black line) iscompared with that for the default source term (dashed red line) in the

bottom panels. For abbreviations see ‘Model formulation’

Kenitz et al.: Sustaining phytoplankton diversity

Half-saturation coefficient

The sensitivity to the half-saturation coefficient, Kji,is investigated by varying the values for each speciesand resource, but in an ordered manner so that eachof the species is the optimal competitor for one of theresources:

(5)

where ki are randomly generated numbers, such thatk1 < k2 < k3 < k4 < k5 and the values in the matrix arefor each resource j in the rows and for each species iin the columns. Three separate sets of simulationsare included, with ki randomly chosen (retaining theabove structure and ordering) within the intervals 0.2to 0.23, 0.2 to 0.5, and 0.1 to 1. In each set, the modelwas integrated 1000 times over 50000 days and allsolutions were identified using the 0-1 Test for Chaos(see the Supplement).

At any particular time, the solutions take the formof either competitive exclusion involving a single

K

k k k k k

k k k k k

k k k k k

k k k k k

k

ji =

5 4 3 2 1

1 5 4 3 2

2 1 5 4 3

3 2 1 5 4

44 3 2 1 5k k k k

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

113

Fig. 4. Phytoplankton community response to changes in cell quota, Qji: (a) nearly uniform cell quota for each species, β = 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

Mar Ecol Prog Ser 490: 107–119, 2013

dominant species (Fig. 5, blue), oscillations with arepeating cycle in species abundance or irregularchaos, both involving all 5 species (Fig. 5, green andred respectively). For competitive exclusion, thedominant species might alter and be replaced byanother species, taking the form of heterocliniccycles (as shown earlier in Fig. 3b); the resultingordered sequence is a consequence of each speciesbeing the optimal competitor for a different resource.

A pattern in the different model responses is evi -dent when comparing the competitive ability of the

in termediate species with the other competitors (Fig. 5).For the intermediate competitor, k3, compared withthe 2 strongest competitors, k1 and k2, competitive ex-clusion is the most likely response when species are ofcomparable fitness, but alters to chaos and then oscil-lations with greater contrasts in the strength of thesecompetitors (Fig. 5, left panels). Hence, the more com-petitive the intermediate competitor, the more chanceof there being an optimal competitor and obtainingcompetitive exclusion, while a weaker intermediatecompetitor encourages chaos or oscillations.

114

0.1 0.4 0.7 10.1

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Chaos

0.1 0.4 0.7 10.1

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1

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k 3

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k 3

k2 k5

k2 k5

k2 k5

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

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

(c) Small changes in half-saturation coefficient, Kji

Fig. 5. The phytoplankton community response to randomly assigned half-saturation coefficient, Kji, within prescribed boundsfor 1000 model integrations, each lasting 50 000 d. The model responses include competitive exclusion with 1 dominant spe-cies 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

The other side of this response is that comparingthe intermediate competitor, k3, with the 2 weakestcompetitors, k4 and k5, leads to the reversed pattern(Fig. 5, right panels): a similar fitness of the 3 speciesfavours oscillations, while increased contrasts gener-ally lead to chaos and eventually are more likely tolead to competitive exclusion. Indeed, the more simi-lar the intermediate competitor is to the weaker spe-cies, the more the intermediate competitor differs fromthe strong competitors, which explains the reversedpattern. No regular structure is evident when Kji arecompared for strong versus weak competitors.

When the perturbations in Kji are in a very narrowrange (0.2 to 0.23), competitive exclusion is the dom-inant response, occurring over 47% of the parameterspace, while oscillations occur in 17% and chaoticsolutions in 14% of parameter space (Table 1); theremaining 22% of solutions are not distinguishedbetween oscillations and chaos. When the perturba-tions in Kji are in a larger range (0.1 to 1.0), competi-tive exclusion reduces to 19% of parameter spaceand instead oscillations increase to 45% and chaos to17% of parameter space. Hence, when Kji of interme-diate and strong competitors are close together, thereis more chance of identifying the optimal competitorand obtaining competitive exclusion.

Random injection of phytoplankton species

We next investigate the response of the model to anintermittent ‘injection’ of new species, replicatinghow ocean circulation leads to the transport and dis-persal of phytoplankton species.

To investigate this species injection and the longer-term community response, an ‘invasion approach’ isapplied broadly following Huisman et al. (2001):additional species are introduced with 3 new specieswith initial abundance Pi = 0.1, typically introducedevery 30 d (with random deviations of a maximum of

10 d), starting at Day 90 and persisting for 1 yr. Theadditional species have their cell traits stochasticallydetermined for each model integration, with Kji cho-sen within the interval 0.2 to 0.5, and Qji within thein terval 0.01 to 0.1. These biological parameters wereassigned for each species and resource either in arandom manner, or assuming a negative relation be -tween fitness and cell quota (scenarios 1 and 3 of Huis -man et al. 2001, respectively); however, the long-term character of the model results turned out not tobe sensitive to these scenarios.

The model state prior to the invasion is our defaultchoice: 5 species competing for 5 resources, so thatcompetition theory predicts that up to 5 different spe-cies should be sustained for a long-term equilibrium.To sample the different characteristic regimes, themodel experiments are repeated for a range of choicesfor Kji: obtaining (1) chaos with the default Kji matrix,(2) single-period oscillations with K5,4 = 0.37, and (3)competitive exclusion with K2,4 = 0.20; with other-wise default choices for the rest of Kji for all 3 cases.

In the chaotic case, the number of phytoplanktonspecies exceeds the number of resources over thelength of the integration of 2500 d (Fig. 6a, panel [i]).Chaotic fluctuations then allow the number of spe-cies to exceed the number of resources, referred toas ‘supersaturation’, in our integrations supporting20 to 30 species within 3 months from the last input ofnew species (Fig. 7a). The number of coexisting spe-cies gradually reduces to 10–15 surviving speciesafter 1 yr, and decreases further to less than 5 after2 yr for the majority of the model compilations. Thechaotic fluctuations can sometimes abruptly diminish(Fig. 6a, panel [ii]), without any intermittent disrup-tion prior to the event. Thus, the fittest competitorspersist, while the weaker species progressively be -come extinct. During the process of introducing morespecies, there is more chance for an optimal compe -titor to be identified and so there is less chance forchaos and oscillations to emerge.

Oscillatory solutions lead to a broadly similar re -sponse to chaotic solutions: there is a supersaturationin the number of species, which gradually declines intime, as illustrated for 1-period oscillations (Figs. 6b& 7b) and also obtained for 2-period oscillations (notshown).

In the case of competitive exclusion, the commu-nity is already dominated by an optimal competitorand so there is a very weak, short-lived response toan injection of additional species (Fig. 6c). Supersat-uration is only sustained for a brief 6-month periodafter the last input of additional species, swiftly re -turning to fewer than 5 coexisting species (Fig. 7c).

115

Kji range Competitive Oscilla- Chaos Oscillations exclusion tions or chaos

0.2–0.23 47 17 14 220.2–0.5 32 40 12 160.1–1.0 19 45 17 19

Table 1. Different phytoplankton community responses (%)for 3 separate sets of 1000 model integrations, each witha different range of randomly generated half-saturation coefficient, Kji. For a proportion of model simulations, community behaviour could not be distinguished between

oscillations and chaos

Mar Ecol Prog Ser 490: 107–119, 2013

Hence, none of the species added to the system are fitenough to outcompete the optimal competitor once itis strongly established in the community.

In summary, chaos and oscillations support a com-parable number of species, exceeding the number ofresources for as long as 2 yr after the last input of newspecies, while competitive exclusion usually sustainsa lower number of species than expected from theresource competition theory (Fig. 8).

DISCUSSION

Hutchinson (1961) first questioned why so manydifferent phytoplankton species persist, given com-

petition theory predicting that at equilibrium thenumber of species cannot exceed the number of lim-iting resources. He suggested that this ‘paradox ofthe phytoplankton’ and inconsistency with competi-tion theory might be reconciled by the phytoplanktoncommunity not being at equilibrium.

There are a variety of explanations as to why anequilibrium state for the phytoplankton communitymight not be achieved, possibly reflecting a responseto the spatial and temporal heterogeneity in the physi-cal environment, or instead an ecological re sponse in-volving inter-species competition. Phytoplankton spe-cies typically have a doubling timescale of 2 to 5 d,and competitive exclusion might be expected to occurover the order of 10 generations, suggesting a time

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(a) Chaos

(b) Oscillations

(c) Competitive exclusion

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Fig. 6. Phytoplankton species abundance (left panels) and number of survival species (right panels) versus time after 12 inter-mittent 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. Survivalspecies 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

Kenitz et al.: Sustaining phytoplankton diversity

span for equilibrium to be reached of typi-cally 1 to 2 months (Reynolds 1995). On thistimescale, the ocean surface boundary layeris strongly forced by the passage of atmo -spheric weather systems, modifying the convection and mixing within the surfaceboundary layer and the light received byphytoplankton. In addition, in coastal seas,the surface boundary layer is modified bythe spring-neap changes in the tides. Giventhis temporal variability in the physical forc-ing, there are 2 limits leading to relativelylow phytoplankton diversity: (1) if there issevere forcing, such as involving a sustainedperiod of no light or nutrient supply followedby an onset of favourable conditions, thenthe phytoplankton species with the fastestgrowth rate dominates; and conversely, (2)persistent conditions lead to the optimalcompetitors flourishing for a stable envi -ronment. Hence, the maximum diversity inphytoplankton species is expected betweenthese 2 limits, referred to as the intermediatedisturbance hypothesis, which was appliedby Connell (1978) for tropical rainforestsand coral reefs, discussed for phytoplanktonby Padisák (1995) and Rey nolds (1995), andused to explain observed changes in thephytoplankton community for a shallow eu-trophic lake (Weithoff et al. 2001). Thus, thephysical forcing might in duce continual tem-poral and spatial changes in the environment,which the phytoplank ton community is con-tinually adjusting to, such that competitiveexclusion is not reached.

An alternative view to this physically in -duced heterogeneity is that there may be

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Fig. 7. Number of species sustained at a particular time after the last in-jection of species, for 1000 model integrations, with each model compila-tion generated with a different set of random cell traits of injected spe-cies. Each set of 1000 runs is induced with different choices of thehalf-saturation coefficients, Kji for the initial 5 species, which leads to (a)chaos (with default Kji), (b) 1-period oscillations (with K5,4 = 0.37) and (c)competitive exclusion (with K2,4 = 0.20). The dashed line indicates themaximum of 5 species surviving on 5 resources predicted for equilibrium

by the resource competition theory

Fig. 8. Mean number of species sustained throughout 1000 model integrations after the last pulse of extra species (indicated byvertical, dotted line). Each set of 1000 runs is induced with different choices of the half-saturation coefficients, Kji, for the initial5 species, which leads to (a) chaos (red line, with default Kji), (b) 1-period oscillations (green line, with K5,4 = 0.37 and (c) com-petitive 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

Mar Ecol Prog Ser 490: 107–119, 2013

more phytoplankton variability due to inter-speciescompetition for resources, as advocated by Huisman& Weissing (1999, 2001). Rather than a single or a fewspecies dominating as in competitive exclusion, thephytoplankton community can continually vary inthe form of repeating oscillations or chaotic changesin the abundance of different species.

Whether the model solutions lead to competitiveexclusion, oscillations or chaos turns out to be sensi-tive to the cell physiology and nutrient requirements.Competition between species of similar fitness ismost likely to lead to competitive exclusion, with theoptimal competitor having the lowest requirementfor a resource (Tilman 1977). Including competitionbetween species with variability in cell physiologyand nutrient requirement via cell quota does not leadto an optimal competitor emerging, and insteadfavours oscillatory or chaotic behaviour. The detailedresponse often turns out to be controlled by the nutri-ent requirement of the intermediate species com-pared with that of the other species. In our setsof 1000 model experiments with different rangesin half-saturation coefficient (Table 1), competitiveexclusion occurs for 19% of the integrations ifthere are large contrasts in half-saturation coeffi-cient, and increases to 47% of the integrations ifthere are small contrasts in half-saturation coefficient(Table 1), reflecting the increased chance of identi -fying an optimal competitor with small contrasts inhalf saturation. In turn, a combination of oscillationsand chaos then occur for at least half of the modelintegrations.

A particular criticism of whether inter-species com-petition explains the paradox of the plankton is thatchaotic solutions might be an unusual occurrence, assuggested by model experiments initialised with ran-domly assigned characteristics for the phytoplankton(Schippers et al. 2001). However, this conclusion ischallenged by Huisman et al. (2001) who argue that adifferent response is obtained if additional phyto-plankton species are injected at different times anda wider range of physiological choices are made.Our model diagnostics support the view of Huisman& Weissing (1999, 2001) that chaos can emerge in awell-mixed box through inter-species competitionfor phytoplankton communities. Indeed, a long-term laboratory mesocosm experiment which monitoredthe plankton community twice a week for 2300 drevealed chaotic fluctuations in phytoplankton spe-cies abundances (Benincà et al. 2008), consistentwith a lack of predictability beyond 15 to 30 d.

With respect to how many phytoplankton speciesare supported when transport and dispersal are

included from the wider environment, we find thatif there are oscillations or chaotic solutions, then ashort-term injection of species leads to a long-termsustenance of more species than resources. In bothcases, there is a very similar response with super -saturation in the number of species. In contrast, whenthere is a competitive exclusion, an additional injec-tion of species only leads to a short-lived excess ofspecies, which quickly die away. Thus, given a ran-dom injection of species, both oscillatory and chaoticsolutions help sustain more phytoplankton speciesthan resources.

In our model experiments, the emergence of chaosversus oscillations is very sensitive to whether anutrient feedback is included. When the feedback isstrong, chaotic solutions emerge, but when the feed-back is weak or absent then the solutions switch tooscillations or competitive exclusion. A choice ofstrong feedback acting to restore nutrients is appro-priate for the way a chemostat operates or for a simple 1-dimensional problem, such as how verticaldiffusion acts to supply nutrients down-gradient tothe euphotic zone and sustain productivity. However,there is a question as to the extent that the nutrientfeedback always holds in the open ocean. The nutri-ent supply to the euphotic zone is affected by a widerange of physical processes, including convection,entrainment at the base of the mixed layer, and hori-zontal and vertical transport by the gyre, eddy andbasin scale overturning circulations (Williams & Fol-lows 2003). These processes can either enhance orinhibit biological productivity. For example, wind-driven upwelling induces productive surface watersover subpolar gyres, while wind-driven downwellinginduces oligotrophic surface waters over subtropicalgyres. These physical processes are unlikely to alwaysprovide a nutrient feedback to sustain inter-speciesdriven chaos. There may be some regimes, particu-larly physically isolated cases, when species compe-tition might induce chaos, such as in the deep chloro-phyll maximum in oligotrophic gyres during thesummer when there is weak mixing (Huisman et al.2006). Elsewhere, phytoplankton diversity is proba-bly determined by a combination of inter-speciescompetition and the effects of spatial and temporalvariations in physical forcing. For example, phyto-plankton diversity is enhanced in western boundarycurrents and gyre boundaries by the combination oftransport, lateral mixing and dispersal, as shown byBarton et al. (2010) and Follows et al. (2007).

The sensitivity of our phytoplankton solutions tothe coupling between phytoplankton species and theabiotic resource is perhaps analogous to how pre -

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Kenitz et al.: Sustaining phytoplankton diversity

dator–prey cycles and their chaotic solutions are sen-sitive to the nature of their coupling. For example,coupling of the predator–prey cycles through com-petition between predators for all prey species leadsto predator abundance increasing in phase with theprey, while coupling the cycles of specialist predatorsleads to the opposite response of prey species de -clining with increasing predator abundance (Van -dermeer 2004, Benincà et al. 2009). The strength ofpre dator–prey interactions also affects whether com-petitive exclusion, oscillatory or chaotic responsesoccur (Vandermeer 1993, 2004). Overabundant preycan even destabilize the ecosystem, leading to largeamplitude cycles of predator populations (Rosen-zweig 1971).

Returning to the question of how the diversity ofthe phytoplankton community is sustained, as origi-nally posed by Hutchinson (1961), there are 2 appar-ently contrasting views: the effect of spatial and tem-poral variability in forcing, and the inter-speciescompetition view. However, both viewpoints involvemechanisms preventing the optimal competitor dom-inating and leading to an equilibrium state, eitherachieved via the physical disturbance of the environ-ment or by a transient flourishing of sub-optimal com -petitors as part of oscillatory and chaotic solutions.

Acknowledgements. K.K. was supported by a UK NERC stu-dentship. We thank A. Barton and M. Begon for insightfuldiscussions, and 3 anonymous referees for their constructiveadvice and feedback.

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Editorial responsibility: Christine Paetzold, Oldendorf/Luhe, Germany

Submitted: July 27, 2012; Accepted: June 20, 2013Proofs received from author(s): September 2, 2013

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