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A model for an aquatic ecosystem HanLiQiao,EzioVenturino DepartmentofMathematics“G.Peano”,UniversityofTorino,viaCarloAlberto10,I–10123Torino,Italy Abstract 7 An ecosystem made of nutrients, plants, detritus and dissolved oxygen is presented. Its equilibria are established. 1 0 Sufficient conditions for the existence of the coexistence equilibrium are derived and its feasibility is discussed in 2 everydetail. n Keywords: aquaticplants,dissolvedoxygen,nutrientrecycling a J 3 2 1. Introduction ] Eutrophicationis the large increase of aquatic plants due to the discharge in waterbodies of chemical elements E suchasnitrogenandphosphorus.Deadaquaticplantsgeneratedetritus,exploitinglargeamountofdissolvedoxygen P for its decomposition, and therefore endangeringaquatic life and fish survival. We introduce here a mathematical . o modelforsuchanecosystemandinvestigateitsbehavior. i b - q 2. Modelformulation [ LetN, DandOrespectivelydenotethenutrients,detritusanddissolvedoxygenamountinthewater. Further,let 1 P denote the aquatic plants population. The nutrients N are assumed to be dischargedinto the lake at rate q; they v alsoaregeneratedbythereconversionofdecomposeddetritus,atrateu,afraction0< f <1ofwhichbecomesnew 1 3 nutrient.Thisprocessinvolvesalsotheusageofdissolvedoxygen,andthereforethetermrepresentingitmustcontain 9 theproductofboththerequiredquantities,OandD. Inaddition,nutrientscanbewashedawayfromthebodywater 6 byemissaries,atrateaandtherearefurtherlossesduetonutrientsuptakebytheplants,thatneedthemforsurviving. 0 Theuptakerateisdenotedbyb. Theplantsreproducebecausetheyassumenutrients,thisgrazingbeingexpressedby . 1 theproductNP,butonlyafraction0 < e < 1oftheassumednutrientscontributetoincreasetheplantbiomass. The 0 plantshavea naturalmortalityrate m, andexperienceintraspecificcompetitionatrate c. Thedetritusoriginatesby 7 deadplants. Aproportion0 < g < 1oftheplantsthatdiesinktothebottomofthewaterbodyandbecomedetritus. 1 : Inpartthelattermaybewashedawaybyoutgoingstreams,atratew;inaddition,usingdissolvedoxygen,detritusat v rateumaybedecomposedbymicroorganismsintoitsbasicelements,whichcontributetotheformationofnutrients. i X Finally,dissolvedoxygenmaybecontainedintothestreamsflowingintothelake,weassumethistooccuratrateh, r butit is also producedby the aquatic plantsthroughphotosynthesis, at rate s. Because most of aquaticplantsfloat a onthesurfaceofalake, largeamountoftheproducedoxygenmaygointotheair, justifyingthefactthatthe rate s maybesmall. Inaddition,therecouldbeashadingeffect,forwhichonlytheplantsnearthesurfacegetenoughlight, whiletheonesinthelowerlayersofwaterareinpartshadedfromdirectsunlightbythetopmostones. Inorderfor photosynthesistobeviable,plantsneedclorophyllandconsumenutrientsforitsproduction. Thus,inthemodelwe mustmodelthisprocessviathebilinearterm PN. Further,thereisanaturalupperlimitontheiroxygenproduction capacity, that producesa saturation limit, a fact that is expressed by using a Holling type II functionto modelit, k beingthehalfsaturationconstant.Inaddition,dissolvedoxygenisalsowashedawayatratev. Otherlossesaredueto thedecompositionofdetritusintoitsbasiccomponents,aprocessthatwesaidrequiresmicroorganismsthatconsume 1 oxygen,atratez. Withtheseassumptions,themodeltakesthefollowingform: dN dP =q+ fuOD−aN−bNP, =ebNP−mP−cP2, (1) dt dt dD dO sPN =g(mP+cP2)−wD−uOD, =h−vO+ −zOD dt dt k+PN 3. Boundednessofthesystem’strajectories Wenowevaluatetheregionofattractionforallsolutionsinitiatinginthepositiveoctant. LetT = N+P+D+O. Addingalltheequilibriumequations,weobtain d PN T =q+h−aN−m(1−g)P−c(1−g)P2−b(1−e)NP−u(1− f)OD−zOD−wD−vO+s . dt k+PN Droppingsomeofthenegativeterms,takingintoaccountthat0≤e, f,g≤1andthatthelastfractionissmallerthan oneanddefiningβ =min{a,m(1−g),w,v}andH =q+h+s,theabovedifferentialequationsbecomesthefollowing m differentialinequality d T ≤q+h+s−aN−m(1−g)P−wD−vO≤ H−β T. m dt ThesolutionoftheassociateddifferentialequationisT = Hβ−1(1−exp(−β t)+T(0)exp(−β t). Thusweobtainthe m m m finalestimatethatdefinestheregionofattractionforallsolutionsinitiatingintheinteriorofthepositiveorthant: T(t)≤max{Hβ−1,T(0)}. m 4. System’sequilibria Themodel(1)hasonlytwoequilibria:theenvironmentcontainingonlyoxygenandnutrients,E =(qa−1,0,0,hv−1), 3 andthesysteminwhichallthecomponentsareatnonzerolevelE∗ =(N∗,P∗,D∗,O∗). Inadditiontothegeneralcase, thereare severalparticularcases, thatarisewhenoneorbothofthe exogenousinputsare blocked: E = (0,0,0,0) 0 arisingwithq=0,h=0,E =(qa−1,0,0,0)wheneverh=0,E =(0,0,0,hv−1)obtainedforq=0. 1 2 Feasibilityofeach E, i = 0,...,3isobvious. Toanalyzecoexistence,fromthesecondequilibriumequationof i (1) N∗ =(m+cP)(eb)−1≥0, NP=P(m+cP)(eb)−1 (2) areobtained.Nowsubstitutingintothefirstequilibriumequation,wefind OD=Ψ(P)(befu)−1, Ψ(P)≡bcP2+(ac+bm)P+am−beq. (3) ThethirdequilibriumequationnowgivesD∗ =[g(mP+cP2)−uOD]w−1. Using(3),wehaveexplicitly: 1 D∗ = Φ(P), Φ(P)≡AP2+BP+C, A=bc(efg−1)<0, B=befgm−(ac+bm)=bm(efg−1)−ac<0. (4) befw C =beq−am. Now,thesignofODonlydependsonΨ(P). From(3),whenC > 0,(4),toguaranteeΨ(P) ≥ 0,if P isitspositive 2 root, we need P ≥ P . WhenC < 0 instead, Ψ(P) ≥ 0 forevery P ≥ 0. To impose D ≥ 0, we study the concave 2 parabolaΦ(P),withroots P ≤ P . ForC > 0, wehavethatΦ(P) ≥ 0for0 ≤ P ≤ P , goingdownfrom(0,C)to 3 4 4 (P ,0).From(3),(4): 4 OD wΨ(P) O (P)= = . (5) 1 D u Φ(P) 2 Toguaranteeitsnonnegativity,wecombinetheaboveresultsforΨ(P)andΦ(P),togetthefollowingconditionand thecorrespondingnonnegativityinterval: (a)C >0, B<0, P ≤P≤ P . (6) 2 4 Letχ(P)=behk+m(h+s)P+c(h+s)P2>0,Π(P)=bek+mP+cP2>0,Γ(P)=Q+MP+LP2,Q=befvw+zC, M = zB < 0, L = zA < 0andthediscriminantofΓbe∆Γ = M2−4LQ > M2. EliminatingNPfrom(2),thefourth equilibriumequationinsteadgives: O2(P)=hh+sPN∗(k+PN∗)−1i(v+zD∗)−1 =befwχ(P)[Π(P)Γ(P)]−1, (7) TofindsufficientconditionsfortheintersectionofO andO forP≥0,westudyeachfunctionseparately. 1 2 O is positive if conditions (6) hold. In the case (a) O increases from a zero at P = P toward the vertical 1 1 2 asymptoteatP= P . 4 The sign of O is instead determined just by the one of Γ. In principle, there is the following possible case 2 guaranteeingthatthe concaveparabolaΓ is positive. (1) Q > 0, M < 0 which impliesa decreasing branchof this parabolaliesinthefirstquadrantbetweentheverticalaxisanditspositiverootγ+,joining(0,Q)with(γ+,0). Correspondingly,thereisapossiblecaseforO beingnonnegativewhenP≥0:itraisesupfromapositiveheight 2 attheorigintotheverticalasymptoteatP=γ+. We now combine O and O to find an intersection in the first quadrant. (1) is compatible with (a). Sufficient 1 2 conditionsfor(P∗,O∗)tolieinthefirstquadrantis (1)−(a): Q>0, B<0, C >0, P2 < P∗ <P4 <γ+. (8) 5. Stability LetJbetheJacobianof(1)andletJ denotethematrixMevaluatedateachequilibriumE: i i −a−bP −bN fuO fuD   J = (ke+sb0PkPPN)2 ebgNm(−k++smkPN2Ng−)2c2PcP −u−O0zO−w −v−−u0DzD  (9) For J the eigenvalues are all negative, namely −a, −m, −w, −v. Therefore, E is stable. J has the following 0 0 1 eigenvalues −a, (beq − am)a−1, −w, −v. From the condition (8), C = beq − am > 0 we can obtain that if the equilibriumE∗exists,E becomesasaddlepoint. J hasfournegativeeigenvalues−a,−m,−huv−1−w,−vsothatE 1 2 2 isalsoalwayslocallyasymptoticallystable.TheeigenvaluesattheequilibriumE are−a,(beq−am)a−1,−huv−1−w, 3 −vandagainthisequilibriumisasaddlepointwhenE∗ exists. ThestabilityofthecoexistencepointE∗ isexamined numerically. 6. Numericalsimulations TheequilibriumE∗ isfeasibleandstablefortheparametervalues: q= 0.5, f = 0.5,u= 0.5,a= 0.05,b= 0.41, e = 0.9, m = .095, c = 0.08, g = 0.9, w = 0.013, h = 0.3, v = 0.08, s = 0.02, k = 0.02, z = 0.025, obtaining N∗ =0.7801,P∗ =2.4107,D∗ =0.3436,O∗ =3.6098. Letting the input rate of nutrients vary, q, from Figure ?? we observe that with increasing q, the densities of nutrients,aquaticplantsanddetritusincrease.Infact,forq=0theirdensitiesbecomezerowithinashortperiod.The oppositeresultsoccurinstead fordissolvedoxygen,the concentrationof whichincreaseswith decreasingq. When q=0,dissolvedoxygenattainsitsmaximumconcentration. Figure??depictsinsteadtheeffectofrateofconversionofdetritusintonutrientsonthecoexistenceequilibrium E∗. AsimilarsituationasforFigure??arises. Withincreasing f,thedensitiesofnutrients,aquaticplantsanddetritus increase,whiletheconcentrationofdissolvedoxygendecreases. 3 7. Discussion The ecosystem could disappear, as the equilibrium E is locally asymptotically stable, not surprisingly. In fact 0 a pond lacking sufficient inputs becomes dry, and all its componentsdisappear. The no-life equilibrium E is also 2 always stably achievable. E and E can be rendered stable if C < 0. Rewritten as bem−1 < aq−1 it means that 1 3 thereducedplantsnatality,i.e. theratioofnewplantsovertheirmortality,shouldbeboundedabovebythereduced nutrientsdepletionrate, i.e. the ratiooftheirwashoutrate tothe netinputrate. Hence, morenutrientsare supplied intothe systemthan theyare usedforplantreproduction. Insuchcase the nutrients-onlyequilibriumandthepoint withonlynutrientsanddissolvedoxygenarestable. Density of Nutrients0001111.......124682468 qqq===000..58 Density of Aquatic plants123...123555 qqq===000..58 Density of Detritus0000000.......2345678 qqq===000..58 Concentration of DO33333.....456789 qqq===000..58 0.2 0.5 0.1 3.4 00 50 Time/day 100 150 0 0 50 Time/day 100 150 00 50 Time/day 100 150 3.30 50 Time/day 100 150 Figure1: Variationsofnutrients,aquaticplants,detritusanddissolvedoxygenasfunctionsofq. Areductionofthe exogeousinputofnutientslowerstheequilibriumvaluesofalltheecosystemcomponentsbutfordissolvedoxygen. 2 333...246 00..78 fff===000..58 3.49 fff===000..58 Density of Nutrients1.15 fff===000..58 Density of Aquatic plants2222....32468 fff===000..58 Density of Detritus0000....3456 Concentration of DO3333....5678 ff==00.5 2 1.8 0.2 3.4 f=0.8 0.5 0 50 100 150Time/da2y00 250 300 350 1. 60 50 Time/day 100 150 0.1 0 50 100 150Time/da2y00 250 300 350 3.30 50 Time/day 100 150 Figure2:Variationsofnutrients,aquaticplants,detritusanddissolvedoxygenasfunctionsof f. Reducingtheconver- sionrateofdetritusintonutrientsenhancesdissolvedoxygenbuthindersnutrients,aquaticplantsandunexpectedly alsodetritusitself. 1.28 22..389 hhh===000...135 11..112 67 Density of Nutrients111...1246 hhh===000...135 Density of Aquatic Plants22222.....34567 Density of Detritus00000.....56789 hhh===000...135 Concebtration of DO345 hhh===000...135 0.8 22..12 00..34 2 0 50 100 150Time/da2y00 250 300 350 20 50 Time/day 100 150 0.2 0 50 Time/day 100 150 1 0 50 100 150Time/da2y00 250 300 350 Figure3: Variationsofnutrients,aquaticplants,detritusanddissolvedoxygenasfunctionsofh. Whilenutrientsand aquaticplantsarescarcelyaffectedbychangesinthisparameter,forsmallervaluesofhwefindasharpincreasein detritusand a correspondingsensible decrease in the dissolvedoxygento verylow concentrations. To maintain an inflowofdissolvedoxygenappearsthustobenecessarytokeepthewaterbodyhealthy. Acknowledgements Thisworkhasbeenpartiallysupportedbytheprojects“Metodinumericiinteoriadellepopolazioni”and“Metodi numericinellescienzeapplicate”oftheDipartimentodiMatematica“GiuseppePeano”oftheUniversita`diTorino. 4 References [1] A.K.Misra,NonlinearAnalysis:ModellingandControl,12(4),511–524(2007). [2] A.K.Misra,NonlinearAnalysis:ModellingandControl,15(2),185–198(2010). [3] A.K.Misra,AppliedMathematicsandComputation,217,8367–8376(2011). 5