In this paper, we consider a competition model between $n$ species in a
chemostat including both monotone and non-monotone response functions, distinct
removal rates and variable yields. We show that only the species with the
lowest break-even concentration survives, provided that additional technical
conditions on the growth functions and yields are satisfied. LaSalle's
extension theorem of the Lyapunov stability theory is the main tool.Comment: 7 page

Sari, Tewfik

English

arXiv

7 pagesIn this paper, we consider a competition model between $n$ species in a chemostat including both monotone and non-monotone response functions, distinct removal rates and variable yields. We show that only the species with the lowest break-even concentration survives, provided that additional technical conditions on the growth functions and yields are satisfied. LaSalle's extension theorem of the Lyapunov stability theory is the main tool

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HAL Id: hal-00459923https://hal.archives-ouvertes.fr/hal-00459923Preprint submitted on 25 Feb 2010HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Global dynamics of the chemostat with variable yieldsTewfik SariTo cite this version:Tewfik Sari. Global dynamics of the chemostat with variable yields. 2010. ￿hal-00459923￿GLOBAL DYNAMICS OF THE CHEMOSTAT WITH VARIABLEYIELDSTEWFIK SARIAbstract. In this paper, we consider a competition model between n speciesin a chemostat including both monotone and non-monotone response functions,distinct removal rates and variable yields. We show that only the specieswith the lowest break-even concentration survives, provided that additionaltechnical conditions on the growth functions and yields are satisfied. LaSalle’sextension theorem of the Lyapunov stability theory is the main tool.1. IntroductionIn this paper we study the global dynamics of the following model of the chemo-stat in which n populations of microorganisms compete for a single growth-limitingsubstrate:S′(t) = D[S0 − S(t)]−∑ni=1 fi(S(t))xi(t)x′i(t) = [pi(S(t)) −Di]xi(t), i = 1 · · ·n,(1.1)where S(0) ≥ 0 and xi(0) > 0, i = 1 · · ·n and S0, D and Di are positive constants.In these equations, S(t) denotes the concentration of the substrate at time t; xi(t)denotes the concentration of the ith population of microorganisms at time t; fi(S)represents the uptake rate of substrate of the ith population; pi(S) represents theper-capita growth rate of the ith population and so the function yi(S) defined byyi(S) =pi(S)fi(S)is the growth yield; S0 and D denote, respectively, the concentration of substrate inthe feed bottle and the flow rate of the chemostat; each Di represents the removalrate of the ith population. We make the following assumptions on the functions piand fi:• pi, fi : R+ → R+ are continuous,• pi(0) = fi(0) = 0 and for all S > 0, pi(S) > 0 and fi(S) > 0,For general background on model (1.1), in the constant yield case yi(S) = Yi,the reader is referred to the monograph of Smith and Waltman [13]. FollowingButler and Wolkowicz [3], we make the following assumptions on the form of theresponse functions pi: there exist positive extended real numbers λi and µi withDate: February 25, 2010.2000 Mathematics Subject Classification. 92A15, 92A17, 34C15, 34C35.Key words and phrases. chemostat, competitive exclusion principle, Lyapunov function, globalasymptotic stability, variable yield model.12 TEWFIK SARIλi ≤ µi ≤ +∞ such thatpi(S) < Di if S /∈ [λi, µi]pi(S) > Di if S ∈]λi, µi[.Hence there are at most two values of S, S = λi and S = µi, called the break-even concentrations, satisfying the equation pi(S) = Di. We adopt the conventionµi = ∞ if this equation has only one solution and λi = ∞ if it has no solution.The global analysis of this model was considered by Hsu, Hubbell and Waltman[5], in the Monod case [9] when the response functions are of Michaelis-Mentenform,pi(S) =aiSbi + S, (1.2)and the yields are constant yi(S) = Yi, and Di = D for i = 1 · · ·n. The authorsshowed that only the species with the lowest break-even concentration survives.Thus the competitive exclusion principle (CEP) holds: only one species survives,namely the species which makes optimal use of the resources. Hsu [4] applied aLyapunov-LaSalle argument to give a simple and elegant proof of the result in [5]for the case of different removal rates Di. The Lyapunov function VH used by HsuisVH =∫ Sλ1σ − λ1σdσ + c1∫ x1x∗1ξ − x∗1ξdξ +n∑i=2cixi, (1.3)whereci =1Yiaiai −Di, i = 1 · · ·n, and x∗1 = DY1S0 − λ1D1.CEP has been proved under a variety of hypothesis [2, 3, 8, 14, 15]. For a surveyon the contribution of each paper the reader may consult the introduction of thepapers [7, 8, 12]. The variable yield case was considered, for n = 1 and n = 2by Pilyugin and Waltman [10], with a particular interest to linear and quadraticyields, and by Huang, Zhu and Chang [6]. The model (1.1) was considered byArino, Pilyugin and Wolkowicz [1] and Sari and Mazenc [12].In this paper we generalize [14] by allowing variable yields and we generalize[1] by allowing multi species. We further extend [12] by providing less restrictiveassumptions on the system. For biological motivations concerning the dependanceof the yields on the substrate, the reader is refeered to [1, 10] and the referencestherein.2. Analysis of the modelIt is known (see Theorem 4.1 [1]) that the non-negative cone is invariant underthe flow of (1.1) and all solutions are defined and remain bounded for all t ≥ 0. Sys-tem (1.1) can have many equilibria: the washout equilibrium E0 = (S0, 0, · · · , 0),which is locally exponentially stable if and only if for all i = 1 · · ·n, S0 /∈ [λi, µi]and the equilibria E∗i and E∗∗i where all component of E∗i and E∗∗i vanish exceptfor the first and the (i + 1)th, which areS = λi, xi = x∗i := Fi(λi), for E∗iandS = µi, xi = x∗∗i := Fi(µi), for E∗∗iGLOBAL DYNAMICS OF THE CHEMOSTAT WITH VARIABLE YIELDS 3respectively, whereFi(S) = DS0 − Sfi(S).The equilibrium E∗i lies in the non-negative cone if and only if λi ≤ S0. If λi < λjfor all i 6= j and F ′i (λi) < 0 then it is locally exponentially stable. It coalesces withE0 when λi = S0. The equilibrium E∗∗i lies in the non-negative cone if and only ifµi ≤ S0 and is locally exponentially unstable if it exists. Its coalesces with E0 whenµi = S0. Besides these equilibria, the system (1.1) can have a continuous set ofnon-isolated equilibria in the non-generic cases where two or more of the break-evenconcentrations are equal. In what follows we assume, thatλ1 < λ2 ≤ · · · ≤ λn, and λ1 < S0 < µ1. (2.1)Hence E0 is locally exponentially unstable and the equilibrium E∗1 lies in the non-negative cone. It is locally exponentially stable if and only ifF ′1(λ1) < 0 ⇐⇒ f1(λ1) + f′1(λ1)(S0 − λ1) > 0. (2.2)We consider the global asymptotic stability of E∗1 .Before presenting the results, we need the following lemma,Lemma 2.1. The solutions S(t), xi(t), i = 1 · · ·n of (1.1) with positive initialconditions are positive and bounded, and if λi < S0 < µi for some i = 1 · · ·n, thenS(t) < S0 for all sufficiently large t.Proof. The proof is similar to the proof of Lemma 2.1 in [14] obtained for the model(1.1) in the case where the yields are constant. Theorem 2.2. Assume that(1) λ1 < λ2 ≤ · · · ≤ λn, and λ1 < S0 < µ1.(2) There exist constants αi > 0 for each i ≥ 2 satisfying λi < S0 such thatmax0<S<λ1gi(S) ≤ αi ≤ minλi<S<ρigi(S), i ≥ 2. (2.3)wheregi(S) =fi(S)f1(λ1)p1(S)−D1pi(S)−DiS0 − λ1S0 − S. (2.4)(3) The functionF (S) =f1(S)S0 − S. (2.5)satisfiesF (S) < F (λ1) if S ∈]0, λ1[, F (S) > F (λ1) if S ∈]λ1, S0[. (2.6)Then the equilibrium E∗1 is globally asymptotically stable for system (1.1) with re-spect to the interior of the positive cone.Proof. From Lemma 2.1 it follows that there is no loss of generality in restrictingour attention to 0 ≤ S < S0. Consider the function V = V (S, x1, · · · , xn) as followsV =∫ Sλ1(p1(σ) −D1)(S0 − λ1)f1(λ1)(S0 − σ)dσ +∫ x1x∗1ξ − x∗1ξdξ +n∑i=2αixi (2.7)4 TEWFIK SARIwhere αi are the positive constants satisfying (2.3). The function V is continuouslydifferentiable for 0 < S < S0 and xi > 0 and positive except at the point E∗1 . Thederivative of V along the trajectories of (1.1) is given byV ′ = x1[p1(S)−D1]F (λ1)− F (S)F (λ1)+n∑i=2xihi(S)wherehi(S) = [pi(S)−Di][αi − gi(S)]and gi(S) and F (S) are given by (2.4) and (2.5) respectively. First, note that,using (2.1) and (2.6), the first term of the above sum is always non-positive for0 < S < S0 and equals 0 for S ∈]0, S0[ if and only if S = λ1 or x1 = 0. IfS ∈ [λ1, λi] then pi(S) < Di and p1(S) > D1 so that gi(S) < 0 < αi for any choiceof αi > 0. Similarly if µi < S0 and S ∈ [µi, S0] then pi(S) < Di and p1(S) > D1 sothat gi(S) < 0 < αi for any choice of αi > 0. On the other hand, if S ∈ [0, λ1] thenpi(S) < Di and, using (2.3), gi(S) ≤ αi so that hi(S) < 0. Finally, if S ∈ [λi, ρi]then pi(S) > Di and gi(S) ≥ αi so that hi(S) < 0. Thus hi(S) < 0 for everyS ∈]0, S0[, provided that the numbers αi satisfy (2.3). Hence V′ ≤ 0 and V ′ = 0if and only if xi = 0 for i = 1 · · ·n or S = λ1 and xi = 0 for i = 2 · · ·n. We useLaSalle theorem (the details are as in the proof of Theorem 2.3 in [14]), the ω-limitset of the trajectory is E∗1 . Condition (2.6) is satisfied in the particular case where F ′(S) > 0 for 0 < S < S0.For S = λ1 we obtain the condition (2.2) of local exponential stability of E∗1 .3. ApplicationsIn this section we show how Theorem 2.2 extends results in [1, 12, 14].3.1. One species. In the case n = 1 the equations take the formS′ = D(S0 − S)− x1f1(S)x′1 = [p1(S)−D1]x1(3.1)Corollary 3.1. If λ1 < S0 < µ1 and 1−f1(S)(S0−λ1)f1(λ1)(S0−S)has exactly one sign changefor S ∈ (0, S0) then E∗1 is globally asymptotically stable with respect to the interiorof the positive quadrant.Proof. The result follows from Theorem 2.2 since the hypothesis on the change ofsign is is equivalent to (2.6). In the case when n = 1 the condition (2.3) is obviouslysatisfied. Corollary 3.1 was obtained by Arino, Pilyugin and Wolkowicz (see [1], Theorem2.11). These authors used the following Lyapunov functionVAPW =∫ Sλ1(p1(σ) −D1)(S0 − λ1)f1(λ1)(S0 − σ)dσ +∫ x1x∗1ξ − x∗1ξdξ.Notice that the Lyapunov function (2.7) we use reduces to the function VAPW inthe case when n = 1.GLOBAL DYNAMICS OF THE CHEMOSTAT WITH VARIABLE YIELDS 53.2. Constant yields. In the case when the yields are constant, yi(S) = Yi, theequations take the formS′ = D[S0 − S]−∑ni=1pi(S)Yixix′i = [pi(S)−Di]xi, i = 1 · · ·n,(3.2)Corollary 3.2. Assume that(1) λ1 < λ2 ≤ · · · ≤ λn, and λ1 < S0 < µ1.(2) There exist constants αi > 0 for each i ≥ 2 satisfying λi < S0 such thatmax0<S<λ1gWLi (S) ≤ αi ≤ minλi<S<ρigWLi (S), i ≥ 2. (3.3)wheregWLi (S) =pi(S)D1p1(S)−D1pi(S)−DiS0 − λ1S0 − S. (3.4)Then the equilibrium E∗1 is globally asymptotically stable for system (3.2) with re-spect to the interior of the positive cone.Proof. In the case when the yields are constant, we havegi(S) =Y1YigWLi (S).Hence the conditions (3.3) imply the conditions (2.3) with constants αi replacedby αiY1/Yi. On the other hand F (S) =p1(S)(S0−S)Y1. Thus, assumption (2.6) followsfrom hypothesis λ1 < S0 < µ1. The result follows from Theorem 2.2. Corollary 3.2 was obtained by Wolkowicz and Lu (see [14], Theorem 2.3). Theseauthors used the following Lyapunov functionVWL =∫ Sλ1(p1(σ)−D1)(S0 − λ1)D1(S0 − σ)dσ +1Y1∫ x1x∗1ξ − x∗1ξdξ +n∑i=2αiYixi.Notice that the Lyapunov function (2.7) we use satisfies V = Y1VWL in the casewhen the yields are constant.3.3. Variable yields. In [12], another Lyapunov function is proposed in the casewhen the yields are variable, leading to the following result.Corollary 3.3. Assume that(1) λ1 < λ2 ≤ · · · ≤ λn, and λ1 < S0 < µ1.(2) There exist constants αi > 0 for each i ≥ 2 such thatmax0<S<λ1gSMi (S) ≤ αi ≤ minλi<S<ρigSMi (S), i ≥ 2. (3.5)wheregSMi (S) =fi(S)f1(S)p1(S)−D1pi(S)−Di. (3.6)(3) The function F (S) =f1(S)S0 − SsatisfiesF (S) < F (λ1) if S ∈]0, λ1[, F (S) > F (λ1) if S ∈]λ1, S0[. (3.7)Then the equilibrium E∗1 is globally asymptotically stable for system (1.1) with re-spect to the interior of the positive cone.6 TEWFIK SARIProof. The condition (3.7) is the same as the condition (2.6). On the other handwe havegi(S) =S0 − λ1f1(λ1)F (S)gSMi (S).By (3.5) and (3.7) we have (2.3) with appropriate constants αi. The result followsfrom Theorem 2.2. Corollary 3.3 was obtained by Sari and Mazenc (see [12], Theorem 2.2). Theseauthors used the following Lyapunov functionVSM =∫ Sλ1p1(σ) −D1f1(σ)dσ +∫ x1x∗1ξ − x∗1ξdξ +n∑i=2αixiwhere αi, i = 2 · · ·n are positive constants satisfying (3.5). In the case when theresponse functions are of Michaelis-Menten form (1.2) and the yields are constantthe Lyapunov function VSM reduces to the Lyapunov function VH (1.3) useb byHsu [4]. Notice that the Lyapunov function (2.7) we use is not proportional to thefunction VSM .4. DiscussionIn this paper we considered a mathematical model (1.1) of n species of microor-ganisms in competion in a chemostat for a single resource. The model incorporatesboth monotone and non-monotone response functions, distinct removal rates andvariable yields. We demonstrated that the CEP holds for a large class of growthfunctions and yields.Even with constant yields, the problem is not yet completely solved: the CEPholds for a large class of growth functions [2, 3, 4, 8, 14, 15] but an important openquestion remains: is the CEP true assuming only that the fi are monotone with norestriction on the Di ? This major open problem remains unresolved after morethan thirty years [7]. However, in the case of constant yields numerical simulationsof model (1.1) have only displayed competitive exclusion.In the case where the yields are constant, it is known [3] that the CEP holdsprovided that Di = D for all i, the set Q =⋃i∈N ]λi, µi[ is connected, and S0 ∈ Q,where N = {i : λi < S0}. Wolkowicz and Lu [14] conjectured that this resultcan be extented to the case of different removal rates. Under condition (2.1), it isclear that the set Q is connected, and S0 ∈ Q. The condition λ1 < λi for i 6= 1can be stated without loss of generality, by labelling the populations such that theindex i = 1 corresponds to the lowest break-even concentration, but the conditionλ1 < S0 < µ1 in (2.1) cannot be stated without loss of generality. If µ1 < S0,it is not possible to show the CEP by the methods that we used. To the best ofour knowledge, in the case of different removal rates and non-monotone responsefunctions, the CEP has been proved only under the assumption S0 < µ1 [8, 14, 15].However, Rapaport and Harmand [11] considered the case of two populations andproposed conditions on the growth functions such that the CEP holds under thecondition µ1 < S0. It should be interesting to extend their methods to more generalcases. We leave this problem for future investigations.Our result concern also the case of variable yields, for which it is known [1, 6, 10]that more exotic dynamical behaviours, including limit cycles, are possible. ThusGLOBAL DYNAMICS OF THE CHEMOSTAT WITH VARIABLE YIELDS 7in the case of variable yields, it is of great importance to have criteria ensuring theglobal convergence to an equilibrium with at most one surviving species.References[1] J. Arino, S.S. Pilyugin, G.S.K. Wolkowicz, Considerations on yield, nutrient uptake,cellular growth, and competition in chemostat models. Canadian Applied Mathematics Quar-terly 11, 2 (2003), 107-142.[2] R.A. Armstrong, R. McGehee, Competitive exclusion. Amer. Natur., 115 (1980), 151-170.[3] G.J. Butler, G.S.K. Wolkowicz, A mathematical model of the chemostat with a generalclass of functions describing nutrient uptake. SIAM Journal on Applied Mathematics 45(1985), 138-151.[4] S.B. Hsu, Limiting behavior for competing species. SIAM Journal on Applied Mathematics34 (1978), 760-763.[5] S.B. Hsu, S.P. Hubbell, P. Waltman, A mathematical theory for single nutrient compe-tition in continuous culture of micro-organisms. SIAM Journal on Applied Mathematics 32(1977), 366-383.[6] X. Huang, L. Zhu, E.H.C. Chang, Limit cycles in a chemostat with variable yields andgrowth rates. Nonlinear Analysis, Real World Applications, 8, (2007), 165-173.[7] P. de Leenheer, B. Li, H.L. Smith, Competition in the chemostat: some remarks. Can.Appl. Math. Q. 11 (2003), no. 3, 229-248.[8] B. Li, Global asymptotic behavior of the chemostat : general response functions and differ-ential removal rates. SIAM Journal on Applied Mathematics 59 (1998), 411-422.[9] J. Monod, La technique de culture continue. Théorie et applications. Ann. Inst. Pasteur, 79(1950), 390-410[10] S.S. Pilyugin, P. Waltman, Multiple limit cycles in the chemostat with variable yields.Mathematical Biosciences 182, (2003), 151-166.[11] A. Rapaport, J. Harmand, Biological control of the chemostat with nonmonotone responseand different removal rates. Mathematical Biosciences and Engineering 5, no. 3 (2008), 539-547.[12] T. Sari, F. Mazenc, Global dynamics of the chemostat with different removal rates andvariable yields (2009). http://hal.archives-ouvertes.fr/hal-00418676/fr/[13] H.L. Smith, P. Waltman, The Theory of the Chemostat, Dynamics of Microbial Competi-tion. Cambridge University Press, 1995.[14] G.S.K. Wolkowicz, Z. Lu, Global dynamics of a mathematical model of competition in thechemostat : general response functions and differential death rates. SIAM Journal on AppliedMathematics 52 (1992), 222-233.[15] G.S.K. Wolkowicz, H. Xia, Global asymptotic behavior of a chemostat model with discretedelays. SIAM Journal on Applied Mathematics 57 (1997), 1019-1043.Laboratoire de Mathématiques, Informatique et Applications, Université de HauteAlsace, 4 rue des frères Lumière, 68093 Mulhouse, and EPI MERE INRIA-INRA, UMRMISTEA, 2, pl. Viala, 34060 Montpellier, FranceE-mail address: Tewfik.Sari@uha.fr

Global dynamics of the chemostat with variable yields

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