Global Warming Can Lead to Depletion of Oxygen by Disrupting Phytoplankton Photosynthesis

21 Pages·2017·1.09 MB·English

Article
Global Warming Can Lead to Depletion of Oxygen
by Disrupting Phytoplankton Photosynthesis:
A Mathematical Modelling Approach
YadigarSekerci1andSergeiPetrovskii2,*
1 DepartmentofMathematics,ArtsandScienceFaculty,AmasyaUniversity,05189Amasya,Turkey;
yadigar.fi[email protected]
2 DepartmentofMathematics,UniversityofLeicester,UniversityRoad,LeicesterLE17RH,UK
* Correspondence:[email protected]
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Received:12May2018;Accepted:25May2018;Published:3June2018
Abstract: Weconsidertheeffectofglobalwarmingonthecoupledplankton-oxygendynamicsinthe
ocean. Thenetoxygenproductionbyphytoplanktonisknowntodependonthewatertemperature
and hence can be disrupted by warming. We address this issue theoretically by considering a
mathematicalmodeloftheplankton-oxygensystem. Themodelisgenericandcanaccountfora
varietyofbiologicalfactors. Wefirstshowthatsustainableoxygenproductionbyphytoplankton
isonlypossibleifthenetproductionrateisaboveacertaincriticalvalue. Thisresultappearstobe
robusttothedetailsofmodelparametrization. Wethenshowthat,oncetheeffectofzooplanktonis
takenintoaccount(whichconsumeoxygenandfeedonphytoplankton),theplankton-oxygensystem
canonlybestableifthenetoxygenproductionrateiswithinacertainintermediaterange(i.e.,nottoo
lowandnottoohigh). Correspondingly,weconcludethatasufficientlylargeincreaseinthewater
temperatureislikelytopushthesystemoutofthesaferange,whichmayresultinoceananoxiaand
evenaglobaloxygendepletion. Wethengeneralizethemodelbytakingintoaccounttheeffectof
environmentalstochasticityandshowthat,paradoxically,theprobabilityofoxygendepletionmay
decreasewithanincreaseintherateofglobalwarming.
Keywords: phytoplankton;oxygendepletion;environmentstochasticity;extinctionprobability
1. Introduction
Global warming, i.e. the tendency of the average Earth surface temperature to increase
with time[1,2], is a major threat to the environment worldwide, which can lead to a number of
adverseconsequencestoecology, agriculture, healthandsociety[3]. Perhapsthebestknownand
widely-discussedresultofclimatechangeistheexpectedincreaseintheoceanlevelbecauseofmelting
ofpolarice,whichwouldleadtofloodingofmanycoastalareas[4,5],henceresultinginhugeeconomic
losses,devastatingeffectsonecologyand,possibly,thelossofhumanlives. Somewhatlesspublicized
in the media, but perhaps even more dangerous is the observed decrease in the concentration of
dissolved oxygen in the ocean [6]. It is predicted that in the worst case scenario, the amount of
oceanoxygencouldfallalmosttwice[7],whichwouldleadtodisastrousconsequencesfortheocean
ecosystemsbysuffocatingmostofmarinelife[8].
Thecurrently-observedoxygendecreaseisattributedtoavarietyofhydrologicalandgeophysical
factors such as lower oxygen solvability in warmer water and slower ocean overturning and
mixing[6,7]. However,adecreaseintheoxygenconcentrationcanalsooccurforadifferentreason,
i.e.,notbecauseofchangesinoxygentransportand/orconsumptionrate,butasaresultofdisruptionin
oxygenproduction. Indeed,oxygenisconsumedthroughseveraldifferentprocesses,e.g.,bybreathing
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Geosciences2018,8,201 2of21
ofmarinefaunaandinbiochemicalreactions(e.g.,remineralization),butismostlyproducedinone
process, i.e., by phytoplankton photosynthesis. (The second largest flux of O emerges from the
2
pyriteregenerationbybacteriaintheocean[9],whichwedonotconsiderhere.) Meanwhile,itisa
well-establishedfactthatthenetproductionofoxygenbymarinephytoplankton(i.e.,thedifference
betweentheamountofoxygenproducedinphotosynthesisduringdaytimeandtheamountconsumed
through breathing during the night-time) depends on the water temperature [10–12]. It has been
observedthattherateofoxygenproductionbysomeplanktonspeciesdecreasemonotonouslyalong
with an increase in the water temperature and even can become negative when the temperature
increasesbyabout5–6◦C:phytoplanktonthenstartconsumingoxygeninsteadofproducingit[13].
Arguably,thismayleadtoawidespreadmarineanoxia.
Wementionherethatthepotentialdisruptionoftheoxygenproductionbyphytoplanktoncan
have negative consequences on a global scale. The oxygen produced inside phytoplankton cells
diffusesintotheoceanwater. Apartofthedissolvedoxygenenterstheatmospherethroughtheocean
surface,thuscontributingtotheglobaloxygenbudget. Itisestimatedthatmorethanonehalfofthe
atmosphericstockofoxygenisproducedintheoceanthroughphytoplanktonphotosynthesis[14].
Correspondingly,shouldtheproductionstop(e.g.,becauseofthenegativeeffectofglobalwarming),
theamountofatmosphericoxygenwouldeventuallyfallmorethantwice,withapparentlydevastating
effectonanimalsandhumans.
Whilstempiricalresearchintothisissueremainsmeagre,ithasrecentlybeenstudiedtheoretically
byconsideringanovelmodelofcoupledplankton-oxygendynamics[15–17]. Themodelissimple
(‘conceptual’), but it has been shown to provide an upper bound for more realistic models [15] so
that,wheneveroxygendepletionandplanktonextinctionhappenintheconceptualmodel,thesewill
necessarilyhappeninamorerealisticmodel. Interestingly,themodelpredictstwodifferenttypes
of oxygen catastrophe as a result of global warming. One of them would be preceded by a slow,
gradualdecreaseintheO concentrationtoverysmallvaluesandisreversibleifglobalwarmingstops
2
andthetemperaturereturnstoitspre-changevalues. Theotherhappenssuddenly,almostwithout
warning;oncetheparameters(e.g.,theO netproductionrateand/orthephytoplanktongrowthrate)
2
surpass certain critical values, the system starts a free fall to a zero-oxygen state. This type of the
catastropheisirreversibleandhencemuchmoredangerous[15].
The above studies left several open questions. Firstly, it remained unclear how robust the
prediction of oxygen depletion in response to a sufficiently large increase in water temperature
is to the details of parametrization of the coupling between phytoplankton and oxygen.
Indeed, model prediction can only be regarded as meaningful if it does not depend strongly on
thespecificchoiceoffunctionalfeedbacks[18–20]. However,thedetailsofthecouplinginareal-world
plankton-oxygensystemarelargelyunknown. Inthispaper,weconsiderseveraldifferentmodelsof
thephytoplankton-oxygensystemandshowthatthepredictionofoxygendepletiondoesnotdepend
onthedetailsofparameterizations. Sincedifferentparameterizationsaccountforsomewhatdifferent
ecologicalsituations,thisbroadensthegeneralityofourresults.
Secondly,previoustheoreticalstudiesontheplankton-oxygendynamicshavemostlybeendone
usingdeterministicmodels[15–17,21–23]. Meanwhile,thehighvariabilityoftheoceanenvironment,
inparticularduetotheinherentlystochasticmarineturbulentflows[24]andweatherconditions[25],
can make such an approach somewhat questionable. Moreover, the process of global warming
itselfexhibitsclearirregularfluctuationsaroundthegeneraltrend[1]. Amorerealisticmodelling
approach should therefore include some form of the environmental stochasticity or noise [26,27].
Remarkably,thesystems’dynamicalpropertiescandiffersignificantlyundertheeffectofnoise[27–29].
Therefore,oursecondgoalinthispaperistoconsiderhowtheearliermodellingpredictionsofoxygen
depletionmaychangeifthemodelincludesenvironmentalnoise.
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2. ParametrizationoftheOxygen-PhytoplanktonSystem
Our conceptual model of coupled phytoplankton-oxygen dynamics is shown in Figure 1.
Phytoplankton (u) produce oxygen (c) with the per capita rate Af(c) during the daytime
photosynthesis. Function M(c,u)describestherateofoxygendecreaseduetodifferentprocessessuch
asitsconsumptionbyphytoplanktonduringthenight-time,breathingofmarinefauna,decayinthe
oxygenconcentrationdueto(bio)chemicalreactionsinthewater(e.g.,decayandremineralizationof
detritus)[22,30–34],etc. Thephytoplanktongrowthrateg(c,u)isknowntobecorrelatedwiththerate
ofphotosynthesis[35]. Themechanismsresultinginthiscorrelationarepoorlyunderstood;here,we
assumethatthegrowthratedependsontheamountofavailableoxygen. Thisisinagreementwiththe
generalbiologicalobservationthatalllivingbeings(exceptforthoserelativelyrarespecies,e.g.some
bacteria,themetabolismofwhichisnotbasedonoxygen),animalsandplantsalike,needoxygenfor
theirnormalfunctioning. ThetermQ(c,u)isthephytoplanktonmortalityratethatincludesitsnatural
mortalityanditsconsumptionbyzooplanktonandplanktivorousfish. Notethat,inthegeneralcase,
itmaydependontheoxygenconcentration,e.g.,becausetheefficiencyofzooplanktonfeedingmay
dependontheoxygen(seeSection3).
phytoplankton Q(c,u)
Af(c)u g(c, u)u
oxygen
M(c,u)
Figure1.Flowchartdiagramofmassflowsbetweenthecomponentsofthesystem(quantifiedbythe
growthratesAf(c)uandg(c,u))andlossesofoxygenandphytoplanktonduetotheirflowsoutofthe
system(quantifiedbyM(c,u)andQ(c,u),accordingly);seethedetailsinthetext.
Thecorrespondingmathematicalmodelthatdescribestheprocessesandmassflowsshownin
Figure1isgivenbythefollowingequations:
dc(t)
= Af(c)u−M(c,u), (1)
dt
du(t)
= g(c,u)−Q(c,u), (2)
dt
wherecistheconcentrationofthedissolvedoxygen,uisthedensityofphytoplanktonandtistime.
Function f(c)describesthecombinedeffectoftheoxygenproductioninsidethephytoplanktoncells
and its transport from the cells to the surrounding water. Note that, since the transport through
the cell membrane depends on the O concentration in the vicinity of the cell, f appears to be a
2
monotonously-decreasingfunctionofc;see[17]fordetails. Parameter Atakesintoaccounttheeffect
oftheenvironmentalfactors(suchasthewatertemperature)ontherateofoxygenproductioninside
thephytoplanktoncells. Weassumethatbothplanktonandoxygenaredistributedapproximately
uniformlyintheupperlayeroftheocean(theso-calledapproximationofawell-mixedlayer,cf.[36,37]);
hence,Equations(1)and(2)donotincludespace.
NotethatEquations(1)and(2)areextremelysimple, muchsimplerthan‘realistic’ecological
models[38]orclimatemodels(e.g.,[39]). However,wewanttoemphasizeherethatthemodelgiven
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byEquations(1)and(2)isnotjustamathematicaltoyasitprovidesanupperboundforafamilyof
realisticmarineecosystemmodels;seeSection5fordetails.
Inordertofurtherspecifythemodel,weusethestandardassumptionthatphytoplanktonexhibit
logisticgrowthsothat:
Bcu
g(c,u) = −γu2, (3)
c+c
1
where B is the maximum phytoplankton per capita growth rate in the large oxygen limit, c is a
1
half-saturation constant and γ is the intensity of the intraspecific competition. For function f(c),
assuming that the transport through the cell membrane is described by the Fick law, it can be
parameterizedasfollows[17]:
c
f(c) = 0 , (4)
c+c
0
wherec isaparameter.
0
Specific parametrization of functions M and Q can be different depending on the biological
processestakenintoaccount. Correspondingly,thesystem(1)and(2)canattainsomewhatdifferent
properties,inparticularintermsoftheexistenceandstabilityofitssteadystates. Below,weconsider
severaldifferentparameterizationsof M and Q torevealthat, infact, thequalitativepropertiesof
modelEquations(1)and(2)arerobusttothechoiceof MandQ.
2.1. Model1
We begin with the baseline case where functions M and Q are linear with respect to their
arguments. Although this assumption of linearity is not entirely realistic in the general case,
in particular because it apparently assumes that the per capita amount of oxygen consumed by
phytoplanktonandthepercapitaamountofphytoplanktonconsumedbyitspredatorsareunbounded,
thelinearapproximationofnonlinearpopulationdynamicsmodelsisknowntoworkwellinacertain
parameterrange;see[40],alsoSection2in[41].
Correspondingly,Equation(1)takesthefollowingform(inappropriatelychosendimensionless
variables,cf.[17]):
dc(t) Au
= −mc−uc, (5)
dt c+1
wheretheterm(−mc)accountsforoxygenlossesforreasonsotherthanplanktonbreathing.
Forphytoplankton,thedensity-independentmortalityrate(sothatQislinearwithrespecttou)
notonlytakesintoaccountthenatural,age-relatedmortality,butcanalsoaccountforotherprocesses
resultinginphytoplanktonlosses,inparticulartheeffectofaterminal(lytic)viralinfection[42,43],
whichisknowntobeanimportantfactorcontrollingphytoplanktonabundance[44]. Equation(2)
turnsintothefollowing:
du(t) (cid:18) Bcu (cid:19)
= −γu2 −σu, (6)
dt c+c
1
whereparameterσisthe(constant)mortalityrateofthephytoplankton.
Equations(5)and(6)containseveralparameters. Sinceourmaingoalistorevealhowthesystem
propertiescanchangedependingontherateofoxygenproduction A(whichisknowntodependon
watertemperatureandhenceisaffectedbyglobalwarming),inthispaper,weonlyconsiderexplicitly
theeffectof Aandfixallotherparametersatsomehypotheticalvalues.
For any parameter values, the extinction state (0,0) is a steady state of the system. Having
analysed the direction of the phase flow, it can be shown that it is always stable [17]. In order to
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understandhowtheexistenceofpositive(coexistence)steadystatesdependson A,itisconvenientto
considertheisoclinesofthesystem,whicharegivenbythefollowingequations:
mc(c+1) Bc−σ(c+c )
u = , u = 1 . (7)
I A−c(c+1) II (c+c )γ
1
Figure 2a shows the relative position of the isoclines in the phase plane of the system.
Anyintersectionbetweenthetwoisoclines(i.e.,betweentheredandblackcurves)isasteadystate.
Itisreadilyseenthatthepositivesteadystatesonlyexistif Aisnottoosmall. Incase Aissmaller
thanacertaincriticalvalue,say A ,therearenopositivesteadystates. Thiscanbesummarizedasa
cr
bifurcationcurveshowingthesteadystatevaluesofoxygenvs. A;seeFigure2b. Forany A > A ,
cr
therearetwosteadystates,asshownbythetwobranchesofthecurve(where A correspondstothe
cr
right-mostpointinthedottedredcurve). Itcanbeshown[17]thattheuppersteadystateisstableand
thelowersteadystateisunstable. At A = A ,thetwobranchesmerge.
cr
isoclines
2
2
1.8 first isocline
second isocline 1.8
1.6
1.6
1.4
1.4
1.2
1.2
u 1 c 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0 0.5 1 1.5 2 10 8 6 4 2 0
c A
(a) (b)
Figure2.(a)The(null)-isoclinesoftheoxygen-phytoplanktonsystemforModel1.Blackcurvesshow
theoxygenisocline(u )forA=3.5,8,9,10and11(fromlefttoright,respectively),andtheredcurve
I
showsthephytoplanktonisocline(u ). (b)PositivesteadystatesofEquations(5)and(6)shownas
II
functionsofA.Otherparametersarec =1,B=12.5,γ=2.5,m=1andσ=0.1.
1
These properties of Equations (5) and (6) mean that, for A > A , the system is likely to be
cr
foundinthevicinityoftheupper(stable)steadystate(unlesstheinitialstateisclosetoextinction).
ForA < A ,regardlessoftheinitialstateofthesystem,inthelong-termrun,thesystemwillconverge
cr
totheextinctionstate(0,0). Inthiscase,theextinctionofplanktonandthedepletionofoxygenarethe
onlypossibleresults.
Having considered the properties of the curve in Figure 2b, one readily reveals the generic
responseofthesystemtoadecreaseinA(duetothetemperatureincrease,e.g.,duetoglobalwarming).
Presumably,beforethechange,thesystemisina‘safe’parameterrange,i.e., A > A . Thesystemis
cr
theninitsuppersteadystate,i.e.,atapointontheupperbranchofthebifurcationcurveinFigure2b.
Withadecreasein A, thepointispushedtotheright. Forany A > A , thesystemisinastateof
cr
sustainableoxygenproduction. However,oncethesystemispushedbeyondthebifurcationpoint
A = A ,sustainablefunctioningofthesystemisnotpossibleanymore,andtheextinction/depletion
cr
istheonlypossibleoutcome.
Now, we are going to check whether the properties of the system and its bifurcation curve
(inparticular,theexistenceofthecriticalvalue A andthedepletioncatastrophefor A < A )remain
cr cr
similarfordifferentassumptionsleadingtodifferent MandQ.
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2.2. Model2
Now,wetakeintoaccountthat,inthecasewherethephytoplanktonmortalitytermQislinked
to itsconsumption by aspecies from a highertrophiclevel, e.g.,zooplankton, the Holling TypeII
density-dependentpredatorresponseismorerealisticthanalinearresponse. Consideringasabove M
tobealinearfunctionwithrespecttoitsarguments,Equations(1)and(2)takethefollowingform:
dc(t) Au
= −mc−uc, (8)
dt c+1
du(t) (cid:18) Bcu (cid:19) uv
= −γu2 − , (9)
dt c+c u+h
1
where v is the zooplankton density (regarded as a parameter; a more general model where v is a
dynamicalvariablewillbeconsideredinSection3).
TheisoclinesofEquations(8)and(9)are,respectively,asfollows:
(cid:16) (cid:17)
c(c+1) c1 u+vh +γu
uI = A−c(c+1), cII = B−(cid:16) v +γu(cid:17). (10)
u+h
Figure3ashowstherelativepositionoftheisoclinesgivenbyEquations(10)inthephaseplaneof
Equations(8)and(9),andFigure3bshowsthecorrespondingbifurcationcurve. Itisreadilyseenthat
thepropertiesofthesystemareessentiallythesameasforModel1,andhence,allconclusionsabout
thesystempropertiesareexactlythesameasforModel1.
isoclines
4 2.5
3.5
2
3
2.5
1.5
u 2 c
1.5 1
1
0.5
0.5 first isocline
second isocline
0 0
0 0.5 1 1.5 2 2.5 3 3.5 10 8 6 4 2 0
c A
(a) (b)
Figure3.(a)The(null-)isoclinesoftheoxygen-phytoplanktonsystemforModel2.Blackcurvesshow
theoxygenisocline(u )forA=3.5,8,9,10and11(fromlefttoright,respectively),andtheredcurve
I
showsthephytoplanktonisocline(c ). (b)PositivesteadystatesofEquations(8)and(9)shownas
II
functionsofA.Here,v=0.3,andotherparametersarethesameasinFigure2.
2.3. Model3
We keep M(c,u) as a bilinear function. For function Q, we combine the density-dependent
HollingTypeIIresponsetoaccountfortheeffectofpredationwiththedensity-independentlinear
responsetoaccountforthenaturalmortalityandtheeffectofviruses. Equations(1)and(2)takethe
followingform:
dc(t) Au
= −mc−uc, (11)
dt c+1
du(t) (cid:18) Bcu (cid:19) uv
= −γu2 − −σu. (12)
dt c+c u+h
1
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Notethatinthespecialcasev =0,Equations(11)and(12)coincidewithModel1.
TheisoclinesofEquations(11)and(12)areasfollows:
(cid:16) (cid:17)
c(c+1) c1 u+vh +σ+uγ
uI = A−c(c+1), cII = B−(cid:16) v +σ+uγ(cid:17). (13)
u+h
Figure4showstherelativepositionoftheisoclinesandthecorrespondingbifurcationdiagram
(the steady state values of oxygen as a function of parameter A). Thus, the system properties are
qualitativelythesameasinModels1and2.
isoclines
4 2.5
3.5
2
3
2.5
1.5
u 2 c
1.5 1
1
0.5
0.5 first isocline
second isocline
0 0
0 0.5 1 1.5 2 2.5 3 10 8 6 4 2 0
c A
(a) (b)
Figure4.(a)The(null-)isoclinesoftheoxygen-phytoplanktonsystemforModel3.Blackcurvesshow
theoxygenisocline(u )forA=3.5,8,9,10and11(fromlefttoright,respectively),andtheredcurve
I
showsthephytoplanktonisocline(c ).(b)PositivesteadystatesofEquations(11)and(12)shownas
II
functionsofA.Here,v=0.3andh=0.5,andotherparametersarethesameasinFigure2.
2.4. Model4
Now, we consider Holling Type II predation for the consumption of phytoplankton
(e.g.,byzooplankton)andneglectthelinearterminQ,thusassumingthatthephytoplanktonlosses
duetotheirpredationaremoreimportantthanlossesduetotheirnaturalmortalityorviralinfection.
Fortherateofoxygenconsumptionbymarinefloraandfauna,wechangethebilineartermucinMto
amorerealisticMonod-typeresponse,whichaccountsforthegeneralobservationthatthepercapita
rateofconsumptionofoxygenisbounded. Equations(1)and(2)takethefollowingform:
dc(t) Au uc
= − −mc, (14)
dt c+1 c+c
2
du(t) (cid:18) Bcu (cid:19) uv
= −γu2 − . (15)
dt c+c u+h
1
TheisoclinesofEquations(14)and(15)are,respectively,asfollows:
(cid:16) (cid:17)
uI = A(cc(+c+c21))−(cc+(cc+2)1), cII = Bc1−(cid:16)u+vvh ++γγuu(cid:17). (16)
u+h
Figure5showstherelativepositionoftheisoclinesandthecorrespondingbifurcationdiagramof
Equations(14)and(15). Thesystempropertiesthereforeremainthesameasabove.
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isoclines
5
4
4.5
3.5
4
3 3.5
2.5 3
u 2 c 2.5
2
1.5
1.5
1
1
0.5 fsierscto insdo cislionceline 0.5
0 0
0 1 2 3 4 5 10 8 6 4 2 0
c A
(a) (b)
Figure5.(a)The(null-)isoclinesoftheoxygen-phytoplanktonsystemforModel4.Blackcurvesshow
theoxygenisocline(u )forA=3.5,8,9,10and11(fromlefttoright,respectively),andtheredcurve
I
showsthephytoplanktonisocline(c ).(b)PositivesteadystatesofEquations(14)and(15)shownas
II
functionsofA.Herec =0.5,otherparametersarethesameasinFigure4.
2
2.5. Model5
Forthismodel,weusethesamefunctionMasinModel4. ForfunctionQ,wekeeptheHolling
TypeIIpredationtermandaddthelinearterm,henceassumingthatphytoplanktonlossesoccurduetoa
combinedactionofthedensity-dependentpredation,virusesandnaturalmortality.Equations(1)and(2)
nowtakethefollowingform:
dc(t) Au uc
= − −mc, (17)
dt c+1 c+c
2
du(t) (cid:18) Bcu (cid:19) uv
= −γu2 − −σu. (18)
dt c+c u+h
1
Theisoclinesofthesystemareasfollows:
(cid:16) (cid:17)
uI = A(cc(+c+c21))−(cc+(cc+2)1) (I), cII = Bc1−(cid:16)u+vvh ++σσ++uuγγ(cid:17). (19)
u+h
Figure6showstherelativepositionoftheisoclinesandthecorrespondingbifurcationdiagram. It
isreadilyseenthatthesystempropertiesarethesameasabove.
2.6. Model6
Weconsiderthesamefunction MasinModels4and5(i.e.,acombinationofthelineartermand
Monodkinetics). Forthelossesofphytoplankton,weneglectthedensity-dependentpredationand
onlyconsiderthelinearterm. Thiscorrespondtoasituationwherethephytoplanktonabundanceis
mostlycontrolledbyitsnaturalmortalityandpossiblyviruses,butnotbypredation. Equations(1)
and(2)thentakethefollowingform:
dc(t) Au uc
= − −mc, (20)
dt c+1 c+c
2
du(t) (cid:18) Bcu (cid:19)
= −γu2 −σu. (21)
dt c+c
1
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Theisoclinesofthesystemareasfollows:
c(c+1)(c+c ) Bc−σ(c+c )
u = 2 , u = 1 . (22)
I A(c+c )−c(c+1) II γ(c+c )
2 1
Figure7showstherelativepositionoftheisoclinesandthesteadystatevaluesofoxygenasa
functionofparameter A(bifurcationdiagram). Thepropertiesarethesameasinthepreviousmodels.
isoclines
5
4
4.5
3.5
4
3 3.5
2.5 3
u 2 c 2.5
2
1.5
1.5
1
1
0.5 fsierscto insdo cislionceline 0.5
0 0
0 1 2 3 4 5 10 8 6 4 2 0
c A
(a) (b)
Figure6.(a)The(null-)isoclinesoftheoxygen-phytoplanktonsystemforModel5.Blackcurvesshow
theoxygenisocline(u )forA=3.5,8,9,10and11(fromlefttoright,respectively),andtheredcurve
I
showsthephytoplanktonisocline(c ).(b)PositivesteadystatesofEquations(17)and(18)shownas
II
functionsofA.Here,c =0.5,otherparametersarethesameasinFigure4.
2
isoclines
5 5
4.5 4.5
4 4
3.5 3.5
3 3
u 2.5 c 2.5
2 2
1.5 1.5
1 1
first isocline
0.5 second isocline 0.5
0 0
0 1 2 3 4 5 10 8 6 4 2 0
c A
(a) (b)
Figure7.(a)The(null-)isoclinesoftheoxygen-phytoplanktonsystemforModel6.Blackcurvesshow
theoxygenisocline(u )forA=3.5,8,9,10and11(fromlefttoright,respectively),andtheredcurve
I
showsthephytoplanktonisocline(u ).(b)PositivesteadystatesofEquations(20)and(21)shownas
II
functionsofA.ParametersarethesameasinFigure6.
2.7. Model7
Thismodelconsidersthemostgeneralcase(withinthegivenframework),whichcombinesallthe
specificcasesabove. Function Mnowtakesintoaccounttheoxygenlossesduetoitsconsumption
byphytoplankton(duringthenight-time)andduetobreathingbymarinefauna(e.g.,zooplankton),
bothtermsdescribedbyMonodkinetics. Italsoincludesthelineartermtoaccountforoxygenlosses
duetootherreasons,whichincludelossesduetooxygendiffusionthroughtheoceansurface. Function
Q includesboththelinearterm(thattakesintoaccountthenaturalmortalityandthevirulencein
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the case of a viral infection) and Holling Type II predation. Equations (1) and (2) then take the
followingform:
dc(t) Au uc νcv
= − − −mc, (23)
dt c+1 c+c c+c
2 3
du(t) (cid:18) Bcu (cid:19) uv
= −γu2 − −σu. (24)
dt c+c u+h
1
Inthespecialcasev =0,Equations(23)and(24)coincidewithModel6.
TheisoclinesofEquations(23)and(24)are,respectively,asfollows:
(cid:16) (cid:17)
uI = c+Ac1+−cν+cc+cvc3c2, cII = Bc1−(cid:16)uu+v+vhh++σσ++uuγγ(cid:17). (25)
Figure 8 shows the relative position of the isoclines in the phase plane of the system and the
correspondingbifurcationdiagram,whichareapparentlysimilartothoseinthepreviousmodels.
isoclines
5 5
4.5 4.5
4 4
3.5 3.5
3 3
u 2.5 c 2.5
2 2
1.5 1.5
1 1
first isocline
0.5 second isocline 0.5
0 0
0 1 2 3 4 5 10 8 6 4 2 0
c A
(a) (b)
Figure8.(a)The(null-)isoclinesoftheoxygen-phytoplanktonsystemforModel7.Blackcurvesshow
theoxygenisocline(u )forA=3.5,8,9,10and11(fromlefttoright,respectively),andtheredcurve
I
showsthephytoplanktonisocline(c ).(b)PositivesteadystatesofEquation(23)and(24)shownas
II
functionsofA.Here,ν=0.01andc =1,andotherparametersarethesameasinFigure6.
3
A brief inspection of the bifurcation diagrams obtained for Models 1–7 immediately reveals
thattheyarequalitativelythesame;inparticular,theyallpredictthatsustainablefunctioningofthe
plankton-oxygensystemcanonlyoccuriftherateofoxygenproductionisnottoolow. Asummary
oftheprocessestakenintoaccountbydifferentmodelsisgiveninTable1. Itisreadilyseenthatthe
criticalbehaviour(disappearanceofthepositivesteadystates)inresponsetoadecreasein Aisshown
byallmodels,althoughtheparametrizationoffunctions MandQvariessignificantly. Wetherefore
concludethatthisisarelativelygeneralpropertyofthephytoplankton-oxygendynamicsandnotan
artefactofaspecificparametrization. Inthecasethattherateofoxygenproductiondecreaseswith
anincreaseinthewatertemperature(aswasobservedforsomephytoplanktonspecies[10,11,13]),
thegenericEquations(1)and(2)predictanecologicalcatastropheresultinginoxygendepletionand
planktonextinctionwhenthetemperatureexceedsacertaincriticalvalue.
Arguably,fromtherangeofmodelsconsideredabove,themostrealisticisModel7,asitaccounts
forthelargestnumberofrelevantfactors. Thus,wewilluseitintherestofthepaper.