Bulai, Iulia Martina

Venturino, Ezio

English

Bulai, IULIA MARTINA

Institutional Research Information System University of Turin

Proceedings of the 16th International Conferenceon Computational and Mathematical Methodsin Science and Engineering, CMMSE 20164–8 July, 2016.Competition between algae and fungi in a lake: amathematical modelIulia Martina Bulai1 and Ezio Venturino11 Dipartimento di Matematica “Giuseppe Peano”, Universita` di Torino,via Carlo Alberto 10, 10123 Torino, Italyemails: iuliam@live.it, ezio.venturino@unito.itAbstractIn this paper a mathematical model for handling water pollution is introduced. Weassume that algae and fungi are in competition for resources that come from wastewater.Both algae and fungi need dissolved oxygen (DO) for their biologial process of growth.But there is a difference, indeed algae produce it too and in a higher quantity than theone they use. It is shown that if the coexistence equilibrium exists, it is stable withoutadditional conditions. If the competition rate between algae and fungi is not high fora chosen set of parameters the stability of the coexistence equilibrium is reached evenwithout an external constant input of DO in the system.Key words: mathematical model, algae, fungi, competition, wastewaterMSC 2000: AMS codes (optional)1 IntroductionAlgae are important in a lake, they can improve the quality of the aquatic ecosystem. Underright conditions such as adequate nutrients (mostly phosphorus, but nitrogen is importanttoo) they grow. The nutrients that are present in the wastewater can derive from agriculturaland/or industrial discharges. Fungi can be used for biodegradation of organic pollutant in awaterbody, and they grow using the nutrients obtained from the biodegradation, [1]. Somemathematical models in the literature study the behavior of algae biomass in a waterbodyin the presence of organic pollutants, [4, 5]. In [2, 3] the case of fungi has been addressed.In this paper we want to study what happens when both algae and fungi are present inthe same waterbody, for example in a lake. Furthermore we suppose that they are incompetition for the resources coming from the pollutants.c©CMMSE ISBN: 978-84-608-6082-2261Competition between algae and fungi in a lake2 The mathematical modelIn this paper we introduce a mathematical system that models the behavior of algae andfungi in a waterbody. The waterbody considered could be nutrient-rich waters, like mu-nicipal wastewater or some industrial effluents. Both algae and fungi can feed on thesewastes and therefore purify the water, while also producing a biomass suitable for biofuelsproduction. Thus algae and fungi are in competition for food, since both share the sameresources. Further, fungi as well as algae need DO to thrive but we assume that the algae’sproduction and input of DO into the system is much larger than their own use for theirgrowth.The model consists of three equations that describe the time evolution of the algaepopulation, the fungi population and the DO respectively. The model, in which all theparameters are nonnegative, reads:dAdt= rAA− aAA− bAA2 − cAF (1)dFdt=hOFk + kOO− aFF − bFF 2 − cAFdOdt= qO + gA− aOO − f hOFk + kOO.In the first equation algae grow at a constant rate rA and are washed out at a constant rateaA. We assume that algae are in competition among themselves at a constant rate bA andalso experience interspecific competition with fungi at rate c.In the second equation the fungi’s growth depends on the presence of DO. They arewashed out at rate aF . The intraspecific competition occurs at rate bF while c denotes therate of the interspecific competition with the algae population.The third equation shows the evolution in time of DO. We assume that it is suppliedfrom external sources at rate qO, but a part of it comes from the algae own production atrate g. We take further into account its washing out, at rate aO and its depletion due to itsassumption by fungi at rate f  1.3 The qualitative analysis of the modelFirst of all to find the equilibrium points of the model, we need to solve the system obtainedby setting the right hand side of (1) to zero,⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩A(rA − aA − bAA− cF ) = 0F(hOk + kOO− aF − bFF − cA)= 0qO + gA− aOO − f hOFk + kOO= 0.(2)c©CMMSE ISBN: 978-84-608-6082-2262Iulia Martina Bulai, Ezio VenturinoFurther, for the stability analysis, we need to calculate the Jacobian matrix of the system(1)J =⎡⎢⎢⎢⎣rA − aA − 2bAA− cF −cA 0−cF −aF − 2bFF − cA+ hOk + kOOhkF(k + kOO)2g − fhOk + kOO−aO − fhkF(k + kOO)2⎤⎥⎥⎥⎦ .(3)Solving (2) we obtain the analytic expression of three equilibrium points. In addition,we prove that two other equilibria exist. We also show that all these points are conditionallylocally asymptotically stable, while the coexistence equilibrium is stable if it is feasible.Proposition 1. The trivial equilibrium point, E0 = (0, 0, 0), exists ifqO = 0. (4)Furthermore, it is stable if the following condition holds:rA < aA. (5)Proof. For A = F = O = 0 in the system (2) we get that E0 exists if qO = 0. Thecharacteristic polynomial associated to the matrix (3) evaluated at E0 isdet(J − μ) = (rA − aA − μ)(−aF − μ)(−aO − μ) = 0.To have the stability of E0 all the eigenvalues shoud be negative thus the condition (5) musthold.Proposition 2. The fungi-and-algae-free point E1 =(0, 0, qOa−1O)exist always. It isstable if the following conditions hold:rA < aA andhqOkaO + kOqO< aF . (6)Proof. In fact for A = F = 0 in (2) from the last equation we get O = qOa−1O . While thecharacteristic polynomial associated to E1 isdet(J − μ) = (rA − aA − μ)(−aO − μ)(hqOkaO + kOqO− aF − μ)= 0.To have all the eigenvalues negative the conditions (6) must hold.c©CMMSE ISBN: 978-84-608-6082-2263Competition between algae and fungi in a lakeFigure 1: The equilibrium E1 is stably achieved with the parameter values rA = 10.1273,aA = 16.93072, bA = 11.7382, c = 19.3012, h = 1.61592, k = 0.454245, kO = 5.87845,aF = 2.55798, bF = 0.0344478, qO = 2, g = 3.63317, aO = 5.41771, f = 1.In Figure 1 one can see that for a chosen set of parameters the equilibrium E1 is stablyachieved.Proposition 3. The fungi-free equilibrium E2 =(rA − aAbA, 0,qObA + g(rA − aA)bAaO)isfeasible ifrA > aA (7)and it is stable ifhO2k + kOO2< aF + cA2 (8)hold.Proof. If F = 0 in the system (2) we get⎧⎪⎨⎪⎩rA − aA − bAA = 0F = 0qO + gA− aOO = 0.(9)From the first equation of (9) it followsA =rA − aAbA.Thus for the nonnegativity of the algae population, (7) must hold. From the third equationinstead we get the equilibrium value of the oxygen.O =qObA + g(rA − aA)bAaO.c©CMMSE ISBN: 978-84-608-6082-2264Iulia Martina Bulai, Ezio VenturinoThe characteristic polynomial associated to E1 is once again easily obtained,det(J − μ) = (−rA + aA − μ)(−aO − μ)(hO2k + kOO2− aF − cA2 − μ)= 0,as well as its eigenvaluesμ1 = −ra + aA < 0μ2 = −aO < 0μ3 =hO2k + kOO2− aF − cA2Requiring μ3 < 0 we get (8).In Figure 2 we show that for a chosen set of parameters the stability of the fungi-freeequilibrium, E2, is attained.Figure 2: The equilibrium E2 is stable for the parameter values rA = 8.90096, aA = 4.14886,bA = 1.62848, c = 0.0691916, h = 11.8402, k = 1.92602, kO = 16.7554, aF = 18.7713,bF = 15.4976, qO = 69.5303, g = 13.847, aO = 19.037, f = 1.Proposition 4. The algae-free point is in fact a set of multiple equilibria, namely(0, F3, O3), (0, F4, O4) and (0, F5, O5). Of them, only one is feasible, ifO >aFkh− aFkO , (10)and it is stable ifrA < aA + cF3. (11)c©CMMSE ISBN: 978-84-608-6082-2265Competition between algae and fungi in a lakeProof. Part 1: existenceFor A = 0 the system (2) becomes⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩A = 0hOk + kOO− aF − bFF = 0qO − aOO − f hOFk + kOO= 0.(12)Solving the second equation with respect to F we getF =O(h− aFkO)− aFkbF (k + kOO). (13)Condition (10) arises by requiring the positivity of the expression (13). Note that theopposite case obtained when h− aFkO < 0, cannot arise,O <aFkh− aFkO ,because from it, O < 0 follows, which is impossible.Substituting the expression (13) for F into the third equation of the system (12) weobtain the following third degree equation in OaO3 + bO2 + cO + d = 0, (14)witha = −aObFk2O < 0b = qObFk2O − 2aObFkkO − fh2 + fhaFkOc = 2qObFkkO − aObFk2 + fhaFkd = qObFk2 > 0.Since a < 0 and d > 0 by Descartes’ rule of signs the third degree polynomial (14) in O hasat least one positive root. We are able to show that there is exactly one such root. In thenext Table the only four possible cases are summarized.Cases a b c d number of real positive roots1) − + − + 3 (impossible)2) − + + + 13) − − − + 14) − − + + 1c©CMMSE ISBN: 978-84-608-6082-2266Iulia Martina Bulai, Ezio VenturinoThe first case is impossible, in fact assuming that b > 0 and c < 0 we findaObFk +fh2kO< qObFkO − aObFk + fhaF < −qObFkO.But this is a contradiction, because the term in the middle should be less than a negativeterm (on the right) and greater than a positive one (on the left side).Thus there is only one positive equilibriumE3 =(0,O3(h− aFkO)− aFkbF (k + kOO3), O3)if O3 >aFkh− aFkO ,with O3 the real positive root of (14).Part 2: stabilityTo study the stability of the equilibrium point we evaluate the Jacobian matrix (3) at E3.The resulting characteristic polynomial isdet(J − μ) = (rA − aA − cF3 − μ){μ2 +(aF + 2bFF3 + aO +fhkF3(k + kOO3)2+− hO3k + kOO3)μ+(aO +fhkF3(k + kOO3)2)(aF + 2bFF3)− aOhO3k + kOO3}= 0.The eigenvalue μ1 = rA−aA− cF3 is negative if (11) holds, while the roots of the quadraticpolynomial in μ are negative with no further conditions. It turns out that both coefficientsof the terms of the two lowest degrees in μ are positive. In fact, substituting F3, (13), forthe coefficient of μ we get(aF +2hO3k + kOO3− 2(O3aFkO + aFk)k + kOO3+ aO +fhkF3(k + kOO3)2− hO3k + kOO3)=(aF +hO3k + kOO3− 2aF + aO + fhkF3(k + kOO3)2)=O3(h− aFkO)− aFkbF (k + kOO3)+ aO +fhkF3(k + kOO3)2= F3 + aO +fhkF3(k + kOO3)2> 0.Similarly, for the constant term, by dividing by aO, denoting by H is a positive term, wehave:aF + 2bFF3 − hO3k + kOO3+H > 0.Figure 3 shows that for a chosen set of parameters the algae-free equilibrium, E3, isstably achieved.c©CMMSE ISBN: 978-84-608-6082-2267Competition between algae and fungi in a lakeFigure 3: The equilibrium E3 is stabile for the parameters rA = 4.85571, aA = 13.2729,bA = 11.6077, c = 12.7024, h = 11.4803, k = 2.3815, kO = 1.11246, aF = 1.40551,bF = 1.95987, qO = 65.3973, g = 16.7907, aO = 15.374, f = 1.For the coexistence equilibrium point we have the following resultProposition 5 There exists at least one feasible coexistence equilibrium E4 = (A∗, F ∗, O∗)if the following three conditions hold:rA − aA − bAA∗ > 0, bF bA − c2 > 0, (aF c+ bF )(k + kOO∗)(rA − aA) > chO∗ (15)and whenever it exists, it is stable.Proof. To find the conditions for the existence of the coexistence equilibrium point fromthe first equation of the system (2) we getF =rA − aA − bAAcand substitute it into the remaining two equations. We solve these two equations withrespect to A and we match the resulting expressionsA =(aF c+ bF )(rA − aA)(k + kOO)− chO(k + kOO)(bF bA − c2) =fhO(rA − aA) + c(aOO − qO)(k + kOO)cg(k + kOO) + fhbAO.Thus, we now have the following cubic polynomial in O:a1O3 + b1O2 + c1O + d1 = 0, (16)c©CMMSE ISBN: 978-84-608-6082-2268Iulia Martina Bulai, Ezio Venturinowitha1 = −aOk2O(bF bA − c2) < 0 for bF bA − c2 > 0b1 = kO(fhc+ bFkOg)(rA − aA) + kO(kOqO − 2aOk)(bF bA − c2) ++(aFkO − h)(fhbA + ckOg)c1 = k(fhc+ 2bFkOg)(rA − aA) + k(2kOqO − aOk)(bF bA − c2) ++chg(2aFkO − h) + aFkhbAd1 = aF ck2g + k2qO(bF bA − c2) + bFk2g(ra − aA) > 0.Since for bF bA − c2 > 0, a1 < 0 and d1 > 0 the polynomial (16) has at least one positiveroot O∗ by the Descartes’ rule of signs. For the feasibility of the equilibrium we need tohave F ∗ > 0 and A∗ > 0, providing thus the first and the third conditions in (8).To study the stability, the characteristic polynomial of (3) evaluated at E4 gives thecubic equationdet(J − μ) = μ3 +Rμ2 + Sμ+ P = 0withR =(bAA∗ + aO + bFF ∗ +fhkF ∗(k + kOO∗)2)> 0S =(A∗F ∗(bAbF − c2) + bFaOF ∗ + fhkF∗(bAA∗ + bFF ∗)(k + kOO∗)2+h2kF ∗O∗(k + kOO∗)3)> 0P =((bF bA − c2)(aOA∗F ∗ +fhkF ∗2A∗(k + kOO∗)2)+bAA∗h2kF ∗O∗(k + kOO∗)3+cghkA∗F ∗(k + kOO∗)2)> 0.For bF bA − c2 > 0, which holds by feasibility, the three eigenvalues are negative. Thus E4is stable whenever it is feasible.In Figure 4 we show that the coexistence equilibrium is stable for a selected set ofparameter values.In the next Table we summarize the feasibility conditions for the five equilibrium pointsof model (1).In Figure 5 for a chosen set of parameters and the same initial conditions changingthe value of qO, the constant input of DO in the system, we obtain the stability of thecoexistence equilibrium, E4 on the left and of E2 on the right. Thus, starting from thecoexistence equilibrium, by decreasing the rate qO at which oxygen is supplied into thesystem, we can obtain the fungi-free equilibrium.c©CMMSE ISBN: 978-84-608-6082-2269Competition between algae and fungi in a lakeFigure 4: Two possible configurations for the equilibrium E4, qO = 0 and qO = 1. It isstable in the following cases. Left: qO = 0, (1.20, 0.30, 0.55) is achieved for the parametersrA = 1, aA = 0.001, bA = 0.8, c = 0.1224, h = 1.1, k = 1, kO = 1, aF = 0.001, bF = 0.8,qO = 0, g = .1, aO = 0.001, f = 1. Right: qO = 1, (1.08, 1.08, 10.15) is obtained for theparameters rA = 1, aA = 0.001, bA = 0.8, c = 0.1224, h = 1.1, k = 1, kO = 1, aF = 0.001,bF = 0.8, qO = 1, g = .1, aO = 0.001, f = 1.Eq. Feasibility conditions Stability conditionsE0 q0 = 0 rA < aAE1 none rA < aA andhqOkaO + kOqO< aFE2 rA > aAhO2kaO + kOO2< aF + cA2E3 O3 >aFkh− aFkO rA < aA + cF3E4 rA − aA − bAA∗ > 0, bF bA − c2 > 0(aF c+ bF )(k + kOO∗)(rA − aA) > chO∗ none4 Conclusions and future workA three dimensional, nonlinear mathematical model has been introduced and analysed.In addition to the trivial equilibrium, four additional equilibrium points have been found.Their stability was been completely analysed. For a chosen set of parameters with the sameinitial conditions we get the stability of the coexistence equilibrium, E4, both in the absence,qO = 0, and with full, qO = 1, external oxygen supply, Figure 4. Thus the constant inputof DO is not necessarily needed if the parameters are chosen appropriately, to have a viablec©CMMSE ISBN: 978-84-608-6082-2270Iulia Martina Bulai, Ezio VenturinoFigure 5: Left: the equilibrium E4 is stable for qO = 30; rA = 19.6445, aA = 1.73234,bA = 11.5828, c = 3.34324, h = 16.676, k = 12.4305, kO = 0.517493, aF = 3.04963,bF = 1.3597, qO = 30, g = 11.5323, f = 0.835718, aO = 5.42261 Right: the equilibriumE2 at the stability qO = 20 for rA = 19.6445, aA = 1.73234, bA = 11.5828, c = 3.34324,h = 16.676, k = 12.4305, kO = 0.517493, aF = 3.04963, bF = 1.3597, qO = 20, g = 11.5323,f = 0.835718, aO = 5.42261.system. In fact algae contribution of DO to the system is enough for the fungi utilization.The simulations of Figure 5 instead show that the DO concentration should not drop belowa critical threshold, because in such situation the fungi may disappear. Such a loss wouldbe detrimental for the ecosystem.One of the hypothesis of the model is the competition for food between algae and fungi,but in an indirect way the results indicate that algae help the fungi growth by producingDO.In our future research we will compare the model introduced here (1) with another onein which the nutrient equation is also considered, as follows:dAdt=hANAkA + kNN− eAA−mAA2 (17)dFdt= k(N,O)F − eFF −mFF 2dNdt= qN − r hANAkA + kNN− sk(N,O)F − eNNdOdt= qO + gAA− eOO − cOk(N,O)Fwithk(N,O) =k1NOk2 + k3N + k4O + k3k4NO.c©CMMSE ISBN: 978-84-608-6082-2271Competition between algae and fungi in a lakeAcknowledgmentsThis work has been partially supported by the projects “Metodi numerici in teoria dellepopolazioni” and “Metodi numerici nelle scienze applicate” of the Dipartimento di Matem-atica “Giuseppe Peano” of the Universita` di Torino.References[1] A. Anastasi, F. Spina, A. Romagnolo, V. Tigini, V. Prigione, G.C. Varese, Integratedfungal biomass and activated sludge treatment for textile wastewaters bioremediation,Bioresour Technol. 123: 106-111 (2012).[2] I.M. Bulai, E. Venturino, Biodegradation of organic pollutants in a water body, Journalof Mathematical Chemistry 1-17 (2016).[3] A. Goyal, R. Sanghi, A.K. Misra, J.B. Shukla, Modeling and analysis of the removalof an organic pollutant from a water body using fungi, Appl. Math. Model. 38 (2014)4863–4871.[4] A.K. Misra, Modeling the depletion of dissolved oxygen in a lake due to submergedmacrophytes, Nonlinear Analysis: Modelling and Control 15(2) (2010) 185–198.[5] Han Li Qiao, E. Venturino, A model for an aquatic ecosystem, ICNAAM 2015, toappear in AIP Conference Proceedings.c©CMMSE ISBN: 978-84-608-6082-2272

Competition between algae and fungi in a lake: a mathematical model

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Competition between algae and fungi in a lake: a mathematical model

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