Phytoplankton competition in deep biomass maximum

1.3 MB·English

by  Alexei B. Ryabov

Theor Ecol manuscript No.
(will be inserted by the editor)
Phytoplankton competition in deep biomass maximum
Alexei B. Ryabov
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Received: 10 January 2011 /Accepted: 4January 2012

n
a
J Abstract Resource competition in heterogeneous en- 1 Introduction

5 vironments is still an unresolved problem of theoret-
2
ical ecology. In this article I analyze competition be- Primary production forms the basis of metabolic ac-

tween two phytoplankton species in a deep water col- tivityoftheocean.Distinctphytoplanktongroupscon-
]
E umn, where the distributions of main resources (light tributedifferentlyinthesequestrationofCO (Frankig-
2
P and a limiting nutrient) have opposing gradients and noulle et al. 1994; Smetacek 1999), production of oxy-
o. co-limitation by both resources causes a deep biomass gen (Falkowski and Isozaki 2008), support of marine
i maximum.Assumingthatthespecieshaveatrade-offin food webs (Christoffersen 1996), etc. A shift in the
b
- resource requirements and the water column is weakly species composition may dramatically affect function-
q
mixed,I apply the invasionthresholdanalysis(Ryabov ingofthewholeecosystem(Waltheretal.2002;Cermen˜o
[
and Blasius 2011)to determine relations between envi- etal.2008;PaerlandHuisman2009).However,inspite

1 ronmental conditions and phytoplankton composition. oftheprincipalroleofresourcecompetitioninthecom-
v
Although species deplete resources in the interior of munity structuring, the conditions of coexistence and
1
the water column, the resource levels at the bottom competitiveexclusioninspatiallyvariableenvironments
1
2 and surface remain high. As a result, the slope of re- still remain largely unknown.
5 sources gradients becomes a new crucial factor which, Theclassicaltheory,advancedbyMacArthur(1972),
.
1 rather than the local resource values, determines the Le´on and Tumpson (1975), and Tilman (1980, 1982),
0 outcome of competition. The value of resource gradi- analysesresourcecompetitioninuniformenvironments
2
ents nonlinearly depend on the density of consumers. andshowsthatstationarycoexistenceoftwospecieson
1
: This leads to complex relationships between environ- two resources is possible only if growth of each species
v mental parameters and species composition. In partic- is finally restricted by its most limiting resource. The
i
X ular, it is shown that an increase of both the incident sameresultsholdforcompetitioninamixedwatercol-
r lightintensityorbottomnutrientconcentrationsfavors umn where light exponentially decreases with depth
a
the bestlightcompetitors,while anincreaseofthetur- (Huisman and Weissing 1995; Diehl 2002). However,
bulent mixing or background turbidity favors the best in weakly-mixed systems these conditions may not be
nutrientcompetitors.Theseresultsmightbeimportant met.Competitorscanbesimultaneouslylimitedbytwo
for prediction of species composition in deep ocean. or more resources, if their favorable habitats are sur-
rounded by areas lacking these resources.
Keywords deep chlorophyll maximum · resource For instance, in deep oligotrophic aquatic systems
competition · water column · invasion threshold the lightintensity reduceswithdepth, whereasconcen-
trations of nutrients typically have opposing gradients.
As a consequence, species with distinct resource re-
A.B. Ryabov
quirementscanhavemaximaofproductionatdifferent
ICBM,Universityof Oldenburg,26111 Oldenburg,Germany
depths, which potentially decreases niche overlaps and
Tel.: [phone]
Fax:[phone] increasesbiodiversity(Chesson1990,2000).However,a
E-mail: [email protected] general extension of the competition theory to contin-
2 Alexei B. Ryabov
uous spatially variable habitats leads to difficult math- ∂N n ∂2N
= − α µ (N,I)P +D , (2)
ematical problems (Grover 1997) and was addressed ∂t i i i ∂z2
i=1
mainly in the mathematical literature (Hsu and Walt- X
where P (z,t) is the population density of the phyto-
man 1993). The analysis of this problem from the eco- i
plankton species i at depth z and time t, µ (I,N) is
logical point of view (Huisman et al. 1999; Yoshiyama i
the growth rate, which depends on the local values of
et al. 2009;Dutkiewicz et al. 2009;Ryabov et al. 2010;
the light intensity, I(z,t), and nutrient concentration,
RyabovandBlasius2011)isstillfarfromcompleteand
N(z,t), m is the mortality rate, D is the turbulent
further research which would connect results for uni- i
diffusivity, and α is the nutrient content of a phyto-
form and spatially variable systems is required. i
plankton cell.
In this article I analyze competition between two
The light intensity decreases with depth owing to
phytoplankton species in a deep weakly-mixed water
the absorption of light by water and phytoplankton
column, assuming that limitation by light in deep lay-
biomass (Kirk 1994)
ers and limitation by nutrients at shallowdepths cause
deepchlorophyllorbiomassmaxima(Holm-Hansenand z n
I(z)=I exp −K z− k P (ξ,t)dξ , (3)
Hewes 2004; Kononen et al. 2003), which are a wide in bg i i
spread phenomenon in oligotrophic basins (Abbott et " Z0 Xi=1 #
al. 1984; Karl and Letelier 2008). The location of a fa- where Iin is the intensity of incident light, Kbg is the
vorable layer in such systems is not fixed, rather it de- water turbidity and ki is the attenuation coefficient of
pends on initial and boundary conditions, the stage of phytoplankton cells.
the relaxation process, etc. (Klausmeier and Litchman Assumethatbothresourcesareessential(vonLiebig’s
2001; Yoshiyama and Nakajima 2002; Ryabov et al. lawofminimum) andthe resourcelimitationofgrowth
2010).Furthermore,a species, establishing at a certain can be parametrized by the Monod kinetics (Turpin
depth,changesresourcedistributionsandmayaffectall 1988), then the growth rate of species i follows
other species throughout the water column. Thereby N I
µ (N,I)=µ min , , (4)
this species acts as an ecosystem engineer, modifying i max,i
H +N H +I
its nutrient and light environment. (cid:26) N,i I,i (cid:27)
whereµ isthemaximalgrowthrate,andH and
For the analysis of competition in such a system max,i N,i
H arethe half-saturationconstants,whichdefine the
Ryabovand Blasius (2011)recently introducedthe no- I,i
species resources requirements.
tion of an invasion threshold, defined as a line (in case
The phytoplankton cells cannot diffuse across the
of two limiting resources) or a hypersurface (in gen-
surface and bottom of the water column
eral)inspaceofresourcerequirements,whichseparates
∂P(z,t) ∂P(z,t)
the species that can grow in the presence of a resident =0 , =0 .
species from those that cannot grow. The form and lo- ∂z (cid:12)z=0 ∂z (cid:12)z=ZB
(cid:12) (cid:12)
cationofinvasionthresholdsdependonthecharacteris- The surfa(cid:12)ce is also impenetrable(cid:12) for the nutrient, and
(cid:12) (cid:12)
ticsofcompetitorsaswellasontheenvironmentalcon- the nutrient concentration at the bottom is constant
ditions. The investigation of these dependences in the
∂N(z,t)
phytoplankton model reveals conditions that favor dif- =0 , N(ZB)=NB .
∂z
ferentcompetitors and canexplainshifts in the species (cid:12)z=0
(cid:12)
composition caused by environmental changes. Fig.1(cid:12)showstypicalequilibriumresourceandbiomass
(cid:12)
distributions in this model. The yellow shaded area in-
dicates the production layer where both resources are
availableandthegrowthrateexcessesmortality,µ(N,I)>
2 Model m. In a non-uniform system the total net growth on a
favorable area should be large enough to compensate
Competitionbetweentwophytoplanktonspeciesforlight for losses into adjacent unfavorable areas (Ryabov and
andalimitingnutrient(e.g.,nitrogenorphosphorus)in Blasius 2008). Denote the depths, which confine this
a water column can be modeled in terms of a nonlocal area, as z if N limits species growth and z if light
N I
systemofcoupledreaction-diffusionequations(Radach limits species growth. The resource availability reaches
andMaier-Reimer1975;Jamartetal.1977;Klausmeier at these depths the critical values
and Litchman 2001; Huisman et al. 2006) m m
∗ i ∗ i
N = H , I = H , (5)
i µ −m N,i i µ −m I,i
∂P ∂2P max,i i max,i i
i = µ (N,I)P −m P +D i , (1)
∂t i i i i ∂z2 at which growth equals mortality. The location of the
productionlayerdependsonthecriticalresourcevalues
Phytoplanktoncompetition indeep biomass maximum 3
(a) Light, µmol photons m-2s-1 (b) Light, µmol photons m-2s-1 (c)
0 50 100 150 1 10 100
20 20
h, m zN Production 30
pt z layer
e I
D
40 40
Succesful
invasion
50 50
0 1 2 3 10-4 10-2 100
Nutrient, mmol/m3 Nutrient, mmol/m3
0 10 20
Plankton, 107cells/m3
Fig. 1 (a) Distributions of the nutrient concentration (black), light intensity (gray), and phytoplankton biomass (green) in
themodelEqn.1-4.(b)Distributionsofthenutrientandlight(solidlines)andexponentialfitting(dashedlines)accordingto
Eqn.7and8 plotted inthelogarithmic scale. Theyellowshadedarea istheproduction layer, wherethegrowth rate excesses
mortality since N > N∗ and I > I∗. (c) Zero net growth isoclines of species 1 and species 2 (green and red dashed lines,
respectively);invasionthreshold(bluelines)inthepresenceofspecies1astheresidentundertheconditionslistedinTable1.
Thebluesolidlinewascalculatednumerically,bytestofmorethan5000invaderswithdifferenthalf-saturationconstants;the
bluedashed lineis afirstorder approximation withslopeγ1 =cI,1/cN,1 inthelog-log scale (Eq.15)
andenvironmentalparameters.Inthepresentarticle,to 3 Competition in a spatially variable
focusontheinfluenceofresourcegradients,theparam- environment
eters are chosen to reproduce deep biomass maxima:
the production layer is located in deep layers and the Assume that the competitors trade off in resource re-
biomassdensityvanishesatthebottomandsurface(see quirements: species 1 is a better nutrient competitor
Table 1 for parameters). and species 2 is a better competitor for light (Fig. 1c),
and consider the invasion of species 2, assuming that
species 1 has reached an equilibrium distribution.
The invasion analysis is applied to determine out-
comes of competition. Initially one species (the resi-
dent) grows alone during a sufficiently long time and 3.1 Invasion threshold
then another species (the invader) can grow from a
small biomass density. Two species coexist if each of Although the resident species depletes resources in the
them can invade in the presence of its competitor. For middle of the water column, the light intensity at the
thegivenmodelparametersthedistributionsofbiomass surface and nutrient concentrations at the bottom are
and nutrients approach to equilibrium after approxi- still high (Fig. 1a, b). Due to a difference in resource
mately500simulationdays.Tomakesurethatthe res- requirements the maximum of production of species 2
identisatequilibriumthistimeintervalisincreasedup can be shifted towards, for instance, the nutrient sup-
to 2000 days, after which an invader can grow and the ply, and even a strong requirement for high nutrient
systemissimulatedforfurther18000daystoobtainthe concentrationscan be compensated by adaptation to a
final competition outcome. lowlightintensityandvice versa. As aresult,the inva-
sionthresholdtakestheshapeofacurve(thebluesolid
lineinFig.1ciscalculatedbyinvasionofapproximately
The initial distribution of nutrients change linearly 5000 species with different half-saturation constants).
from0 atthe surfaceto N atthe bottom.The phyto- Theshapeofinvasionthresholdsdependsonthere-
B
plankton species have initially uniform distribution of source distribution. However, to find this dependence
small density, P = 100 cells/m3. For the numerical so- we should find the principal eigenvalues corresponding
lutionthepartialdifferentialequationswerediscretized totheboundaryproblemofEqs.1–2(HsuandWaltman
onagridof0.25m.Theresultingsystemofordinarydif- 1993; Grover 1997; Ryabov and Blasius 2011), which
ferential equations was solved by the CVODE package canbe solvedin generalonly numerically(Troostetal.
(http://www.netlib.org/ode)usingthebackwarddiffer- 2005).However,asimple analysiscanbe performedfor
entiation method. competitorswithcloseresourcerequirements.Thenthe
4 Alexei B. Ryabov
maximum production of invaders occurs in the vicinity layer,m≪µ ,sothatthe biomasscandiffuse with-
max
of the resident production layer (Fig. 1c) and it is pos- outessentiallosses.Then,accordingtoEq.3,theabso-
sible to express the growth rate of invaders in terms of lute value of the logarithmic gradient of light intensity
the growth rate of the resident. is constant and equals
dlnI(z)
c =− =K +kP . (6)
I bg 0
dz
3.2 Resource distribution
Thus, the light distribution can be approximated as
Due tothediffusionofcellsanessentialpartofthe res- I(z)=I∗e−cI(z−zI) . (7)
ident biomass is located outside the production layer.
To find the nutrient distribution note that accord-
Consequently the resources below and above this layer e
ing to Eqn. 5 in the limit of small mortality the criti-
are depleted to values which are even smaller than the
critical values of the resident N∗ and I∗. For instance, cal nutrient concentrations are also small, N∗ ≪ HN,
1 1
therefore the growth rate close to the depth z can be
thenutrientconcentrationabovethebiomassmaximum N
canbefewordersofmagnitudesmallerthanN∗(Fig.1b). linearized as µN(N) ≈ µmaxN/HN. Substituting this
1
expression into Eq. 2 we obtain in equilibrium
Therefore, the resident species shapes resource gradi-
ents both within and outside of its production layer. N d2N
−αµ P +D =0 .
The shape of resourcedistributions in this areawill maxH 0 dz2
N
play a key role for the further analysis.It is commonly
Asolutiontothisequation,thatmonotonicallyincreases
assumed, that the light intensity is exponentially dis-
with depth, gives the equilibrium nutrient distribution
tributed, while nutrient concentrations can be fitted
in the vicinity of the depth z ,
by a line (see the gray and black line, respectively, in N
Fig. 1a). However, the linear part of the nutrient pro- N(z)=N∗ecN,1(z−zN) (8)
file is typically observed in the deep layers below the
with the logarithmic gradient
biomassmaximum,where the lightratherthan the nu- e
trient limitation determines species growth. Similarly, dlnN(z) αµmaxP0
c = = . (9)
N
above the production layer the growth is nutrient lim- dz DH
r N
ited and the net growth rate is negative independently Thusinthevicinityoftheproductionlayerboththe
of the shape of the nutrient distribution. The shape light and nutrient distribution have exponential shape,
of resource distributions is crucial within the produc- which implies that the logarithmic resource gradients
tion layer, where the local net growth rate strongly have small variations with depth.
correlates with local resource availability. As shown in
Fig.1b,inthisareaboththelightandnutrientdistribu-
tions closely follow exponential dependences (straight 3.3 Approximate calculation of the invasion thresholds
dashed lines on the logarithmic scale).
The exponential shape of resource distributions is Togainaninsightintothedependenceofinvasionthresh-
a crucial point of the following theory. For this rea- olds on resource distributions assume that the resident
son, it is important to note that this shape is not just (species 1) and invader (species 2) differ only in their
an artifact of a specific model, but it was also found half-saturation constants, HN,i and HI,i, but are oth-
in field observations. For instance, Karl and Letelier erwise identical, i.e. µmax,1 = µmax,2, m1 = m2, and
(2008) clearly demonstrate exponential dependence of D1 =D2, (see RyabovandBlasius (2011)fora general
nutrientconcentrationsintheareaofnutrientconsump- approach).
tion. The emergenceof suchdistributions canhave dif- An invader of small initial density has a vanishing
ferent nature. In particular, it is easy to show that it influence on the resources, therefore the possibility of
emerges, when the biomass variation within the pro- invasiondependsonlyonitsgrowthrateintheresource
duction layer is small. distribution shaped by the resident, µ2(N1(z),I1(z)).
Suppose that the phytoplanktondensity canbe ap- Considerthedifferencebetweenthenutrientlimitations
proximatedby a rectangulardistribution with the con- ofthe residentandinvader.TheMonodkineeticsceanbe
stant density, presented in the form
1 zI N1(z) N1(z)/HN,2 N1(z)
P0 = zI −zN ZzN P(z)dz N1(z)+HN,2 = (N1(z)/HN,2)+1 =µN HN,2 ! ,(10)
e e e
within andin the vicinity ofthe productionlayer.This whichshowsthatthenutrientlimitationofgrowth,µ ,
N
e e
implies a small mortality level outside the production depends only on the ratio N /H .
1 N,2
e
Phytoplanktoncompetition indeep biomass maximum 5
Light intensity I
(a) (b) (c) (d)
0 0 0 0
20 20 20 20
40 40 zN 40 zN 40 zN
60 60 60 60
z 80 zI 80 zI 80 zI 80
zI
100 100 100 100
120 120 120 120
Species 1 Species 1 Species 1 Species 1
140 140 140 140
Species 2 Species 2 Species 2 Species 2
0.0 0.5 1.0 0.00 0.02 0.04 0.00 0.02 0.04 0.00 0.02 0.04
I/(I+HI) (N, I) (N, I) (N, I)
Fig. 2 Schematic profiles of resource limitation of species growth, assuming that resource distributions are exponentially
shaped. (a) If I(z) ∝ e−cIz (gray straight line on a logarithmic scale) then a difference in half-saturation constants of the
species results in a parallel translation of the Monod function by ∆zI =−c−I1ln(HI,2/HI,1) along z-axis. In Figs.b-d solid
lines show growth rates of each species, µ(N,I). The limitations of growth by particular resources, µN(N) and µI(I), are
shown as dashed and dot-dashed lines respectively in the regions where they do not coincide with µ(N,I). The black dashed
lineshows the mortality rate, m. (b)∆zI =∆zN, parallel translation of the growth rate profiles, leads to the same total net
growth of the species. In (c) and (d) either ∆zI < ∆zN or ∆zI > ∆zN and the net total growth rate of species 2 is either
smaller orlarger, respectively,than thetotal netgrowth of species 1.
However, the mathematical identity ∆z has the opposite sign, because the light intensity
I
ecz ec(z−∆z) 1 H has an inverse gradient. Assuming that the invader is
2
= , where ∆z = ln , (11) a better light competitor (H < H ), we obtain
H H c H I,2 N,1
2 1 1
that ∆z is positive and the profile of light limitation
I
showsthatdivisionofanexponentialfunctionbydiffer-
is shifted downwards (see Fig. 2a).
ent constants H and H is equivalent to a shift ∆z in
1 2
The values ∆z and ∆z show changes in the net
position along the z-axis. Thus, using the exponential I N
growtharisingfrombetteradaptationtooneoranother
approximation for the nutrient distribution, Eq. 8, we
resource,andthevalue∆=∆z −∆z definesthedif-
obtain N I
ferenceinthesizeoftheresidentandinvader’sfavorable
N (z) N (z−∆z )
1 = 1 N , (12) habitats. If boundary effects on biomass distributions
H H
N,2 N,1 are negligible, then the fate of the invader species de-
f f
and the nutrient limitation of invader growth can be pends solely on the sign of ∆. Indeed, if ∆ = 0 then
expressed through that of the resident the distinct resource requirements of these species re-
sult in a parallel translation of the growth rate profile
N (z) N (z−∆z )
µ 1 =µ 1 N , alongthez-axis(Fig.2b).Sincetheresidentspecieshas
N N
H H
N,2 ! N,1 ! zerototalgrowthinequilibrium,thesameholdsforthe
f f
1 HN,2 invader and its population cannot establish. If ∆ > 0
where ∆z = ln . (13)
N
c H then the invader habitat is even smaller, consequently
N,1 N,1
Therefore,the profilesofnutrientlimitationofresident the invader total net growth will be negative (Fig. 2c).
andinvadergrowthhavethesameshape,butareshifted By contrast,if ∆<0,the invader habitat is largerand
by ∆z with respect to each other. If H > H , itcaninvadethe systembecauseits totalproductionis
N N,2 N,1
i.e. the invader needs higher nutrient concentrations, positive (Fig. 2d).
then ∆z is positive and the invader nutrient limita- According to Eqn. 5, when the growth and mor-
N
tion profile is shifted downwards. Performing the same tality rates are equal, the ratio of half-saturation con-
calculations for the light limitation we obtain stants equals the ratio of critical resource values (e.g.,
∗ ∗
H /H = I /I ), and the invasibility threshold can
I (z) I (z−∆z ) I,2 I,1 2 1
µ 1 = µ 1 I , be presented in the form
I I
H H
I,2! I,1 !
e e ∗ ∗
1 H 1 N 1 I
where ∆z = − ln I,2 . (14) ∆= ln 2 + ln 2 <0 . (15)
I c H c N∗ c I∗
I,1 I,1 N,1 1 I,1 1
6 Alexei B. Ryabov
Fig. 3 (Leftpanel)Competition outcome independenceof theslopeof invasionthresholdsof species1(green)andspecies2
(red). The gray dashed lineshows the critical slope, γcr. (a) γ1,2 <γcr – species 2 wins, (b) γ2 <γcr <γ1 – bistability,(c)
γ1<γcr <γ2 –coexistence, (d)γ1,2>γcr –species1wins.(Rightpanel,e)All outcomes can berepresented independence
on γ1 and 1/γ2.
This inequality defines a first order approximation of same time the invasion threshold of species 2 should
the invasion threshold and has a straightforward geo- have a steeper slope (γ > γ ), and therefore c
2 cr N,2
metricalinterpretation:species2caninvadeinthepres- should be relatively small. In other words, this species
∗ ∗
ence of species 1 if its critical resource values (N , I ) shouldnotdiminishtheupwardnutrientfluxtoomuch.
2 2
liebelowalinewithslopeγ =c /c passingthrough Thus,weobtainageneralrulethatforcoexistenceeach
1 I,1 N,1
∗ ∗
the point (N , I ) in the double-logarithmic resource speciesshouldshape resourcedistributions witha rela-
1 1
plane(bluedashedlineinFig.1c)andtherangeofpos- tively smaller gradient of its most limiting resource.
sible invaders is determined by the ratio of logarithmic
It is convenient to represent all possible outcomes
resource gradients shaped by the resident.
of competition in dependence of γ and 1/γ (Fig. 3e),
The value of γ depends on the characteristics of 1 2
whichreflecttheabilityofaresidenttoshapeastronger
the resident, and species 2 growing alone will shape a
gradient of its less limiting resource and quantify the
distinct resource distribution with different ratio γ =
2
∗ dominance of the species over its competitor. Then co-
c /c .Withoutlossofgenerality,assumethatN >
I,2 N,2 2
∗ ∗ ∗ existence is possible if the mutual dominance of both
N and I < I , and denote by γ the slope (taken
1 2 1 cr
species is small, whereas large values of γ and 1/γ
with opposite sign) of a straight line passing through 1 2
lead to bistability.
the two critical resource points (gray dashed lines in
Figs. 3a–d),
To get another perspective on the role of resource
∗ ∗
lnI /I
γ =− 2 1 . (16) gradients, consider invasion in the presence of a resi-
cr lnN2∗/N1∗ dent, which has shaped a large gradient of the nutri-
Then,combining Eq.15andits counterpartfor species ent distribution and a small gradient of light intensity,
2, we obtain the outcomes of spatial resource competi- c ≫ c . According to Eq. 14, the change in the
N,1 I,1
tion. If γ < γ < γ , then both species can invade area of light limitation, ∆z , approaches infinity when
1 cr 2 I
the monoculture of each other and can thus coexist c → 0, thus even a small difference in light require-
I,1
∗ ∗
(Fig. 3c). In the opposite case, γ > γ > γ none of ment, ln(I /I )<0, canlead to a large increaseof the
1 cr 2 2 1
the two species can invade, which leads to alternative favorable area. Therefore, a better adaptation to this
stable states (Fig. 3b). Finally, one species dominates resource is very efficient. By contrast, a large nutrient
if γ >γ or γ <γ (Fig. 3a, d). gradientc makes competition muchharder,because
1,2 cr 1,2 cr N,1
Considerindetailtheconditionsofcoexistence.Fig.3c the areas of the resident and invader nutrient limita-
showsthatthebestnutrientcompetitor(species1,green) tion will almost coincide (∆z ∝ 1/c → 0, unless
N N,1
should have a relatively shallow slope (γ < γ ) of its N∗ ≪ N∗ or N∗ ≫ N∗, see Eq. 13). In the limit case
1 cr 2 1 2 1
invasion threshold, therefore this species should shape c →∞,competitionforthenutrientbecomesimpos-
N,1
a resource distribution with a relatively small gradient sible because the habitat is virtually divided into two
of light intensity, c . This will provide more solar ra- parts:withverylownutrientconcentrations(allspecies
I,1
diation for species 2 (red), which, owing to its stronger arenutrient limited) andvery highconcentrations(the
nutrientlimitation,has anicheindeeper layers.Atthe nutrient limitation plays no role).
Phytoplanktoncompetition indeep biomass maximum 7
Fig. 4 Geometrical method to project the outcome of spatial resource competition in dependence of the nutrient concen-
tration at the bottom, NB. Top panel: shift in the competition outcome between species 1 (green) and species 2 (red) in the
phytoplankton model caused by an increase of NB. Bottom panel illustrates this shift as the result of a counter-clockwise
rotation of the invasion thresholds. The slopes, γ1 and γ2, of the invasion thresholds in the bottom panel were calculated
numerically for amonoculture of species 1and 2.
4 The influence of environmental parameters of D favors to the better nutrient competitor (Fig. 5).
on the competition outcome To explain these differences consider the influence of
theseparametersontheresourcegradientsandtheslope
Inthissectiontheinvasionthresholdanalysisisapplied of invasion thresholds.
toexplainshiftsinthespeciescompositionpredictedby
More nutrients at the bottom give rise to a larger
model Eqn. 1–4. First two examples demonstrate how
andsharperbiomassdistributionoftheresidentspecies,
changes in the competition outcome can be visualized
which in turn yields a steeper gradient of the nutrient
as rotation of invasion thresholds and then changes in
concentration.As a result, with an increase of N , the
B
the species compositionare consideredinthe (N ,I )
B in low light adapted species 2 “shades” the nutrient flux
and (D,K ) planes.
bg more strongly and hinders invasion of a better nutri-
ent competitor species 1, which, in consequence of its
lightlimitation,occupieshigherlayers.Becausealarger
4.1 Shifts in the species composition with D and N
B
nutrientgradientc leads to smaller values ofγ ,this
N,i i
transitioncanberepresentedasacounter-clockwisero-
Compare changes in the composition of phytoplank-
tation of the invasion thresholds in the resource plane
ton species, caused by an increase of turbulent mixing,
(the bottom panel in Fig. 4).
D,andbottomnutrientconcentration,N .Boththese
B
parameters increase the nutrient availability through- By contrast, with an increase of mixing, D, the
out the water column. In a homogeneous environment biomass distribution becomes wider and the nutrient
highernutrientconcentrationswouldprovidebettercon- gradient decreases (Fig. 5). As a result, more nutri-
ditionsforthebetterlightcompetitor(species2).How- ents become available for species 1, which finally wins
ever, surprisingly, in the phytoplankton model these the competition. In the resource plane this change can
changes have opposite effect. While an increase of N be represented as a clockwise rotation of the invasion
B
favors the better light competitor (Fig. 4), an increase thresholds.
8 Alexei B. Ryabov
N* N* N* N* N* N*
1 2 1 2 1 2
101 101 101
I*
I* I* I*
1 1 1
I* I* I*
2 2 2
10-2 N* 10-2 N* 10-2 N*
Fig. 5 Thesame as inFig. 4 butin relation todiffusivity.NB =2 mmol nutrientm−3
4.2 Numerical simulations This approach can be extended to include other
model parameters. For instance, Figs. 6d–f show that
an increase of both the turbulent diffusivity, D, and
Thevalueγ quantifiesthelikelihoodofinvasionandcan background turbidity, K increases the slopes of the
bg
beusedderivethecompetitionoutcomefromtheresults invasionthresholdsandthe competitionoutcomeshifts
obtained for each competitor alone. Consider species 1 from the dominance of species 2 through the range of
(thebetternutrientcompetitor)astheresident.Fig.6a coexistencetothedominanceofspecies1.Inthisfigure
shows the values of arctanγ1 in the (Iin,NB) plane, again, the range of species coexistence predicted from
whichwerenumericallycalculatedfromequilibriumre- simulation of one-species model (the area between two
sourcedistributionsinthepresenceofthis species.The black lines in Fig. 6f) is in a good agreement with the
intensity of red color decreases with γ1 and character- results obtained in two-species model (the blue area).
izes the ability of a better light competitor (species 2)
to invade. Contrary, the dominance of species 1 over
species 2 increases with γ. The black line shows the 5 Discussion
level γ = γ . In the region to the right of this line
1 cr
γ1 < γcr and therefore, species 2 can invade. In a sim- 5.1 Invasion thresholds
ilar manner, Fig. 6b shows the slope of the invasion
threshold, arctanγ , in the presence of species 2 alone. In a uniform system a species can increase its biomass
2
Here,however,thecolorintensityincreaseswithγ and if its resource requirements are lower than the present
2
∗
characterizesmorefavorableconditionsforabetternu- level of ambient resources (R -rule). This rule, how-
trient competitor. The black line is the boundary of ever, has to be generalized for spatially variable envi-
the range γ > γ , where species 1 can invade. An ronments, where on the one hand the size of favorable
2 cr
intersection of these ranges shows the range of coexis- area becomes a crucial factor, and on the other hand
tence, in which γ < γ < γ and each species can in- the resource heterogeneity provides an opportunity to
1 cr 2
vade inthe presenceofits competitor.Fig.6ccompares compensate a lack of one resource by superfluous con-
thisrange,basedontheone-speciesmodeling(between centrations of another. To extend the competition the-
black lines), with the range of coexistence obtained by ory to nonuniform systems Ryabov and Blasius (2011)
∗
two-species modeling (the blue area). recently suggestedto replacethe R -rule by the notion
Phytoplanktoncompetition indeep biomass maximum 9
(a) (b) (c)
arctanγ arctanγ logB1
103 1.51 103 1.52 103 B2
γ <γ 2
1 cr
Species2
I Scapne cinievsad2e 1 Scapne cinievsad1e 1 Coex wins 1
in γ2>γcr isten 0
c
Dominance of species 1 0.5 Doof mspineacinecse 2 0.5 Species1wins e --21
102 0 102 0 102
100 101 100 101 100 101
(d) N (e) N (f) N
B B B
1.5 1.5
γ >γ Species1 Species1 2
2 cr can invade wins
Kb1g0-1 γ <γ Doofm iSsnppaeencciceeiess1 2 01.5 10-1 Doof msipneacnicees 2 01.5 10-1 Swpinescies2 Coexistence -011
1 cr can invade
-2
10-2 0 10-2 0 10-2
10-1 100 10-1 100 10-1 100
D D D
Fig. 6 Invasion analysis in the phytoplankton model. (Top panel) The slope of the invasion thresholds as a function of
NB and Iin numerically calculated for a monoculture of (a) species 1 and (b) species 2. The color intensity changes with
arctanγi andcharacterizes thelikelihoodofinvasionof(a)betterlightcompetitors (decreaseswithγ)and(b)betternutrient
competitors (increases with γ). Black lines shows the boundary of the parameter range where the competitor can invade. (c)
Compares theboundaryofthecoexistencerangeobtainedfromresultsformonocultures (Figs.aandb)andfromtwo-species
modeling. The color shows the logarithm of the biomass ratio (logB1/B2) after 20000 simulation days. (Bottom panel) The
same analysisindependenceon D and Kbg.
of an invasion threshold, which is defined as the maxi- Thiseffectaltersthemechanismofcoexistence.Ina
malrequirementsofsuccessfulinvadersinthepresences uniform system two species can coexist if each of them
of a resident species in equilibrium. If the critical val- mostly consumesits mostlimiting resource,andfinally
∗ ∗
ues (N ,I ) of a species lie below the threshold, then becomes self-limited by this resource. If however, the
the species can invade. In a nonuniform environment favorable area is bounded by resource availabilities, a
the shape of invasion thresholds can be complex and somewhat opposite rule can be formulated: the mono-
nonlinear.However,anapproximatedtechnique canbe culture of each species should shape resource distribu-
developed for competition between species with close tionswitharelativelysmallergradientofitsmostlimit-
resource requirements. For these species the invasion ingresource.Theneachcompetitormaybenefitfromits
threshold can be approximated by a strait line on a adaptationtoa specific resource.As shownin (Ryabov
double-logarithmic scale and the slope of this line (de- and Blasius 2011), for phytoplankton competition this
terminedbytheratiooflogarithmicresourcegradients) difference is even more striking, because γ grows with
becomes a critical determinant of the competition out- lightattenuationcoefficient,k,anddecreasewithnutri-
come. entcontentofcells,α.Thus,forcoexistenceindeeplay-
ersofawatercolumnalow-light/high-nutrientadapted
species should have a smaller value of α, so that the
The dependence of invasion thresholds on resource nutrient is available also for species in the upper lay-
gradients,ratherthanonthelocalresourcevalues,leads ers.Conversely,thespecieswithhigh-light/low-nutrient
to new rules for invasionand coexistence. In particular adaptation should have a smaller coefficient k, thereby
a large value of γ means that the invasion threshold minimizing the light shading. Note that similar corre-
approaches to a vertical line, which favors to good nu- lations between consumption rates and resource adap-
trientcompetitors(seeFig.3).Bycontrast,ifγ issmall tationwererecentlyfoundinthe experimentalanalysis
thentheinvasionthresholdsisclosetoahorizontalline, ofcompetitionforlightandphosphorus(Passargeetal.
therefore good light competitors are in more favorable 2006).
conditions.
10 Alexei B. Ryabov
The dependence of competition outcome on envi- approximation which is valid for species with close re-
ronmentalparametersinthe phytoplanktonmodelalso source requirements. The larger the interval where re-
reveals a number of intriguing results and shows that sourceschangeexponentially,the largerthe segmentof
the intuition basedonhomogeneousmodelsmay failin the invasion threshold which follows the linear depen-
analysisofheterogeneoussystems.Forinstance,Figs.6a- dence in log-log scale. Often the resource level changes
c show that an increase of both the incident light in- exponentially for few and more orders of magnitude
tensity,I ,andnutrientconcentrationsatthe bottom, (Ryabov & Blasius 2011). In this case the invasibility
in
N , decreases γ and favors to the best light competi- criterion Eq. 15 is applicable, if differences in species
B
tor (species 2). This is because both factors lead to half-saturation constants have the same or smaller or-
a sharper biomass maximum and hence to a steeper der.
nutrient gradient. Recall that in a uniform system an Second, it was assumed that the production layers
increaseofI wouldalwaysfavortoagoodnutrientcom- are confined by the resource availability and located
petitor, while an increase of N would favor to a good sufficientlyfarfromtheboundaries.Theboundariesef-
light competitor. fect species distributions and survival conditions. For
Further, Figs. 6d-f show that anincrease of the wa- instance, an impenetrable boundary is a favorable fac-
ter turbidity, Kbg, orturbulent diffusivity, D,increases torforspeciessurvival,becausethecelldiffusivefluxre-
γ andfavorstothebestnutrientcompetitor(species1). flectsfromtheboundaryandmorecellscanreturninto
Althoughleadingto the sameresults,these changesin- the production layer (Cantrell and Cosner 1991). The
volvedifferentmechanisms:anincreaseofKbg increases boundaries also influence the species spatial segrega-
theratioγ =cI/cN viaanincreaseofthelightgradient, tion. The species, which canoccupy different depths in
cI, whereas an increase of D leads to the same result the deep layers, have to compete locally if the biomass
because the nutrient gradient cN becomes smaller. maximum occurs in the benthic or surface layer,which
These results might have important outcomes for strengthens the interspecific interactions. All these ef-
field research. For instance, a decrease of NB or D de- fects have a profound impact on species composition
creases the nutrient availability in the water column, and can even reverse the outcome of competition, will
however, leads to opposite effects on the competition be published elsewhere.
outcome.Thus,ashiftinthespeciescompositioncaused Third, to focus on the role of resource gradients
by higher stratification of the ocean waters can be op- it was assumed that competitors have the same maxi-
positetothatcausedbyareductionofnutrientlevelsin malgrowthrates,mortalitiesanddispersal.Inthis set-
deep layers. This effect can possibly explain both pos- tings an invasion threshold always passes through the
itive and negative correlations between nutrient con- resident’s (N∗, I∗) values. A more general approach
centrations and the abundance of high-light adapted (RyabovandBlasius2011)shows,however,thatifthere
species observed along environmental gradients in the is any difference in these parameters then the invasion
Atlantic Ocean (Johnson et al. 2006). threshold can be shifted towards higher or lower re-
source requirements. As a result an invader with, for
instance,higherµ caninvadeevenifithashigherre-
5.2 Model assumptions max
quirementsofbothresourcesandtwospeciescanstably
coexistviaapositivecorrelationinthemaximalgrowth
To derive the invasibility criterion (Eq. 15), an “ideal”
rateandresourcerequirements(so-calledgleaner-opportunist
system was considered. It was assumed that the re-
trade-off).
sourcesareexponentiallydistributed,thebiomassmax-
imum is located in the deep layer and µ =µ .
max,1 max,2
Under these assumptions the analysis of competition
becomes fairly simple, because the invasion threshold 5.3 Comparison with other models
follows a straight line with slope γ in double logarith-
mic space. Now consider a more general situation in There are few approaches suggested to describe phy-
which these assumptions do not hold. toplankton competition in a water column. These ap-
First, if the resource distributions are not expo- proached can be classified based on the assumed in-
nentially shaped then the invasion threshold takes the tensity of water mixing. Namely, Huisman and Weiss-
shape of a curve (blue solid line in Fig. 1c). However, ing (1994, 1995) consider competition in well-mixed
∗ ∗
a tangent line to this curve at the point (N , I ) is systems; by contrast, Yoshiyama et al. (2009) suggest
1 1
exactly the linearly approximated invasion threshold an approach for poorly mixed environments; finally,
(blue dashed line). Thus, if resources deviate from ex- RyabovandBlasius(2011)andthe presentpapercom-
ponential distributions, Eq. 15 provides a first order plement these studies for the case of moderate mixing.