Research article Special Issues

Mathematical analysis of a chemostat system modeling the competition for light and inorganic carbon with internal storage

  • This paper investigates a mathematical model of competition between two species for inorganic carbon and light in a well-mixed water column. The population growth of the species depends on the consumption of two substitutable forms of inorganic carbon, "CO2" (dissolved CO2 and carbonic acid) and "CARB" (bicarbonate and carbonate ions), which are stored internally. Besides, uptake rates also includes self-shading by the phytoplankton population, that is, an increase in population density will reduce light available for photosynthesis, and thereby reducing further carbon assimilation and population growth. We also incorporate the fact that carbon is lost by respiration, and the respiration rate is assumed to be proportional to the size of the transient carbon pool. Then we study the extinction and persistence of a single-species system. Finally, we show that coexistence of the two-species system is possible, depending on parameter values, and both persistence of one population.

    Citation: Fu-Yuan Tsai, Feng-BinWang. Mathematical analysis of a chemostat system modeling the competition for light and inorganic carbon with internal storage[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 205-221. doi: 10.3934/mbe.2019011

    Related Papers:

    [1] Ling Xue, Sitong Chen, Xinmiao Rong . Dynamics of competition model between two plants based on stoichiometry. Mathematical Biosciences and Engineering, 2023, 20(10): 18888-18915. doi: 10.3934/mbe.2023836
    [2] Xiong Li, Hao Wang . A stoichiometrically derived algal growth model and its global analysis. Mathematical Biosciences and Engineering, 2010, 7(4): 825-836. doi: 10.3934/mbe.2010.7.825
    [3] Zhun Han, Hal L. Smith . Bacteriophage-resistant and bacteriophage-sensitive bacteria in a chemostat. Mathematical Biosciences and Engineering, 2012, 9(4): 737-765. doi: 10.3934/mbe.2012.9.737
    [4] Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay . Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences and Engineering, 2014, 11(4): 877-918. doi: 10.3934/mbe.2014.11.877
    [5] Junjing Xiong, Xiong Li, Hao Wang . The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium. Mathematical Biosciences and Engineering, 2019, 16(4): 2717-2737. doi: 10.3934/mbe.2019135
    [6] Zhenyao Sun, Da Song, Meng Fan . Dynamics of a stoichiometric phytoplankton-zooplankton model with season-driven light intensity. Mathematical Biosciences and Engineering, 2024, 21(8): 6870-6897. doi: 10.3934/mbe.2024301
    [7] Yuanshi Wang, Hong Wu . Transition of interaction outcomes in a facilitation-competition system of two species. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1463-1475. doi: 10.3934/mbe.2017076
    [8] Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari . Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences and Engineering, 2016, 13(4): 631-652. doi: 10.3934/mbe.2016012
    [9] Yang Kuang, Kaifa Wang . Coexistence and extinction in a data-based ratio-dependent model of an insect community. Mathematical Biosciences and Engineering, 2020, 17(4): 3274-3293. doi: 10.3934/mbe.2020187
    [10] Xinyan Chen, Zhaohui Jiang, Qile Tai, Chunshan Shen, Yuan Rao, Wu Zhang . Construction of a photosynthetic rate prediction model for greenhouse strawberries with distributed regulation of light environment. Mathematical Biosciences and Engineering, 2022, 19(12): 12774-12791. doi: 10.3934/mbe.2022596
  • This paper investigates a mathematical model of competition between two species for inorganic carbon and light in a well-mixed water column. The population growth of the species depends on the consumption of two substitutable forms of inorganic carbon, "CO2" (dissolved CO2 and carbonic acid) and "CARB" (bicarbonate and carbonate ions), which are stored internally. Besides, uptake rates also includes self-shading by the phytoplankton population, that is, an increase in population density will reduce light available for photosynthesis, and thereby reducing further carbon assimilation and population growth. We also incorporate the fact that carbon is lost by respiration, and the respiration rate is assumed to be proportional to the size of the transient carbon pool. Then we study the extinction and persistence of a single-species system. Finally, we show that coexistence of the two-species system is possible, depending on parameter values, and both persistence of one population.


    It has been known that organic nutrients (e.g., nitrogen and phosphorus), light, and inorganic carbon are the important factors that affect the growth of phytoplankton. However, previous competition theory only focused on the interaction between the species and nutrients/light (see, e.g., [7,9]), and neglected the role of inorganic carbon. This is probably due to the complexities including the biochemistry of carbon acquisition by phytoplankton and the geochemistry of inorganic carbon in the ecosystem [10,20]. In the Supplementary Information of [20], the authors proposed a system of ODEs modeling the competition of the species for inorganic carbon and light in a well-mixed water column. Dissolved CO2 and carbonic acid are regarded as one resource (denoted as "CO2"), and bicarbonate and carbonate ions are regarded as another (denoted as "CARB"). The resources "CO2" and "CARB" are stored internally, and they are substitutable in their effects on algal growth [17,20]. On the other hand, uptake rates also includes self-shading by the phytoplankton population, namely, an increase in population density will reduce light available for photosynthesis, and thereby suppressing further carbon assimilation and population growth [20].

    It was known that pH and alkalinity are two main factors in the modeling of inorganic carbon [20]. The consumption terms for "CO2" and "CARB" used in [20] include computations of feedbacks that arise from changes in pH and alkalinity during algal growth. In the recent work [17], the authors ignore these latter feedbacks and assume that the parameters in the system are constants, simplifying the complex processes of "CO2" and "CARB" involved. Incorporating the simplifications used in [17], we modify the model presented in the Supplementary Information of [20] and we shall investigate the following chemostat-type model with internal storage:

    {dRdt=(R(0)R)DfR1(R,Q1)g1(u1,u2)u1fR2(R,Q2)g2(u1,u2)u2                    +γ1(Q1)u1+γ2(Q2)u2ωrR+ωsS,dSdt=(S(0)S)DfS1(S,Q1)g1(u1,u2)u1fS2(S,Q2)g2(u1,u2)u2+ωrRωsS,dQ1dt=fR1(R,Q1)g1(u1,u2)+fS1(S,Q1)g1(u1,u2)μ1(Q1)Q1γ1(Q1),du1dt=[μ1(Q1)D]u1,dQ2dt=fR2(R,Q2)g2(u1,u2)+fS2(S,Q2)g2(u1,u2)μ2(Q2)Q2γ2(Q2),du2dt=[μ2(Q2)D]u2,R(0)0, S(0)0, ui(0)0, Qi(0)Qmin,i, i=1,2. (1.1)

    Here R(t) and S(t) denote the concentrations of "CO2" and "CARB" in the chemostat at time t, respectively. ui(t) denotes the concentration of species i at time t. Qi represents the amount of cell quota of resources R and S per individual of species i at time t. μi(Qi) is the growth rates of species i as a function of cell quota Qi. fRi(R,Qi) (fSi(S,Qi)) is the per capita uptake rate of species i as a function of resource concentration R (S) and cell quota Qi. D is the dilution rate of the chemostat. Each nutrient is supplied at the rate D, and both input concentrations are R(0) and S(0) respectively. Qmin,i denotes threshold cell quota below which no growth of species i occurs. γi(Qi) represents the respiration rate of species i as a function of cell quota Qi. gi(u1,u2) stands for the photosynthetic rate of the species i as a function of u1(t) and u2(t). Following the ideas of model simplifications in [17], we also assume that carbonic acid loses a proton to become bicarbonate at the rate ωr, and the rate of the reverse reaction is denoted by ωs.

    One type of the photosynthetic rate of the species i, gi(u1,u2), takes

    gi(u1,u2)=1zmzm0miˆI(z)ai+ˆI(z)dz, (1.2)

    where mi and ai are the maximal growth rate and half saturation constant of species i, respectively, and

    ˆI(z)=Iinexp(k0zk1zu1(t)k2zu2(t)). (1.3)

    Here we have assumed that the light intensity at each depth is described by Lambert-Beer law [11,14], which states that the amount of light absorbed is proportional to the light intensity (Iin), but decreases with the depth in the water column (z), the background turbidity of the water itself (k0), the specific light attenuation coefficients of the competing species (ki), and the population densities of the species (ui). Assume that zm is the total depth of the water column. Substituting (1.3) into (1.2), then the photosynthetic rate gi(u1,u2) becomes

    gi(u1,u2)=mi(k0+k1u1+k2u2)zmln(ai+Iinai+ˆI(zm))=miln(IinˆI(zm))ln(ai+Iinai+ˆI(zm)),

    where

    ˆI(zm)=Iinexp(k0zmk1zmu1(t)k2zmu2(t)).

    We also note that the other type of the photosynthetic rate gi(u1,u2) takes the form

    gi(u1,u2)=miIai+I, (1.4)

    with

    I(t)=Iinexp(k0zmk1zmu1(t)k2zmu2(t)).

    According to [20], we take the growth rate μi(Qi) as follows

    μi(Qi)=μmax,iQiQmin,iQmax,iQmin,i,

    where μmax,i is the maximum specific growth rate of species i; Qmin,i is the minimum cellular carbon content required for growth of species i; Qmax,i is the maximum cellular carbon content of species i. From [3,4,5], for i=1,2, the growth rate μi(Qi) can also take the forms :

    μi(Qi)=μ,i(1Qmin,iQi),orμi(Qi)=μ,i(QiQmin,i)+Ai+(QiQmin,i)+,

    where (QiQmin,i)+ is the positive part of (QiQmin,i) and μ,i is the maximal growth rate at infinite quotas (i.e., as Qi) of the species i.

    According to [6,16], for H=R, S and i=1,2, the uptake rate fHi(H,Qi) takes the form:

    fHi(H,Qi)=ρHi(Qi)HKHi+H.

    Here H=R, S represents the concentration of the extracellular resource; ρHi(Qi) represents the maximal uptake rate of the species i; KHi is the half-saturation constant, the resource concentration at which uptake rate is half of the maximal rate. The maximal resource uptake rate, ρHi(Qi), is a decreasing, linear function of quota [8,16], which is defined by

    ρHi(Qi)=ρhighmax,Hi(ρhighmax,Hiρlowmax,Hi)QiQmin,iQmax,iQmin,i,

    for Qmin,iQiQmax,i. In other words, the maximal rate of resource uptake, ρHi(Qi), varies between upper and lower bounds, ρhighmax,Hi and ρlowmax,Hi, respectively, while quota varies between lower and upper bounds, Qmin,i and Qmax,i, respectively (see, e.g., [8]). Cunningham and Nisbet [1,2] took ρHi(Qi) to be a constant. The authors in [20] put ρhighmax,Hiρmax,Hi, ρlowmax,Hi=0, then ρHi(Qi) becomes

    ρHi(Qi)=ρmax,HiQmax,iQiQmax,iQmin,i.

    System (1.1) also includes the fact that carbon is lost by respiration. The respiration rate is proportional to the size of the transient carbon pool [20]:

    γi(Qi)=γmax,iQiQmin,iQmax,iQmin,i,

    where Qmin,iQiQmax,i, and γmax,i is the maximum respiration rate of species i.

    In this whole paper, we always assume that the photosynthetic rate gi(u1,u2) takes the form in (1.2) or (1.4); the functions μi(Qi), fHi(H,Qi), and γi(Qi) satisfy the following assumptions: (i=1,2 and H=R, S)

    (C1) μi(Qi) is continuously differentiable for QiQmin,i, μi(Qmin,i)=0, μi(Qi)0 and μi(Qi)>0 for QiQmin,i.

    (C2) fHi(H,Qi) is continuously differentiable for H>0 and QiQmin,i, fHi(0,Qi)=0, fHi(H,Qi)0, fHi(H,Qi)H>0 and fHi(H,Qi)Qi0 for H>0 and QiQmin,i.

    (C3) γi(Qi) is continuously differentiable for QiQmin,i, γi(Qmin,i)=0, γi(Qi)0 and γi(Qi)>0 for QiQmin,i.

    The rest of the paper is organized as follows. Section 2 is devoted to the study of the single population model. In Section 3, we shall investigate the possibility of coexistence of the two competing species system (1.1). A brief discussion section completes this paper.

    In this section, we first investigate the extinction and persistence of the single population model. Mathematically, it simply means that we remove equations of Q2 and u2 from (1.1). In order to simplify notation, all subscripts are dropped in the remaining equations and the single population model takes the form:

    {dRdt=(R(0)R)DfR(R,Q)G(u)u+γ(Q)uωrR+ωsS,dSdt=(S(0)S)DfS(S,Q)G(u)u+ωrRωsS,dQdt=fR(R,Q)G(u)+fS(S,Q)G(u)μ(Q)Qγ(Q),dudt=[μ(Q)D]u,R(0)0, S(0)0, u(0)0, Q(0)Qmin. (2.1)

    The specific growth rate G(u) takes the form

    G(u)=1zmzm0m˜I(z)a+˜I(z)dz=mln(Iin˜I(zm))ln(a+Iina+˜I(zm)),

    where

    ˜I(z)=Iinexp(k0zkzu(t)), 0zzm,

    or

    G(u)=mIa+I, with I(t)=Iinexp(k0zmkzmu(t)).

    The feasible domain for system (2.1) takes the form

    X={(R,S,Q,u)R4+:QQmin}. (2.2)

    Then it is easy to show that X is positively invariant for system (2.1). Next, we study the boundedness of solutions of (2.1):

    Lemma 2.1. Every solution (R(t),S(t),Q(t),u(t)) of system (2.1) exists globally. Furthermore, solutions of (2.1) are ultimately bounded on X.

    Proof. By the continuation theorem, it suffices to prove that the solution of system (2.1) is bounded on finite time intervals. Let

    Θ(t)=R(t)+S(t)+u(t)Q(t). (2.3)

    Then Θ(t) satisfies

    dΘ(t)dt=(R(0)+S(0)Θ(t))D. (2.4)

    Thus, Θ(t) is bounded on finite time intervals. This fact and the positivity of the solution imply that R(t), S(t), and u(t)Q(t) are bounded on finite time intervals. Since Q(t)Qmin, we can also deduce that u(t) is bounded on finite time intervals.

    It remains to show that Q(t) is bounded on finite time intervals. Inspired by the ideas in [15,Proposition 3], we will investigate the dynamics of the variable V(t)=12(Q(t))2. By the monotonicity of fN for N=S,R, the inequality x12(1+x2), and the fact that QQmin, it is easy to see that V(t) satisfies

    dV(t)dt=dQ(t)dtQG(0)[fR(R(t),Qmin)+fS(S(t),Qmin)]Q12G(0)[fR(R(t),Qmin)+fS(S(t),Qmin)][1+Q2]=G(0)[fR(R(t),Qmin)+fS(S(t),Qmin)][12+V]. (2.5)

    Since R(t) and S(t) in (2.5) are bounded on finite time intervals, we see that V(t) is bounded on finite time intervals, and hence, so is Q(t). Thus, every solution of system (2.1) exists globally.

    From (2.4), it is easy to see that

    limtΘ(t)=R(0)+S(0), (2.6)

    and hence, Θ(t) is ultimately bounded. This together with the positivity of the solution, we deduce that R(t), S(t), and u(t)Q(t) are ultimately bounded. Using the fact Q(t)Qmin, we see that u(t) is ultimately bounded. Finally, we show that Q(t) is ultimately bounded. From (2.3), (2.6), and the third equation of (2.1), it follows that there exist τ0>0 and η0>0 such that

    dQdtG(0)[fR(R(0)+S(0),Q)+fS(R(0)+S(0),Q)]+η0μ(Q)Qγ(Q), tτ0.

    Then

    limtQ(t)Qη0,

    where Qη0 is the positive root uniquely determined by

    G(0)[fR(R(0)+S(0),Q)+fS(R(0)+S(0),Q)]+η0μ(Q)Qγ(Q)=0.

    Thus, solutions of (2.1) are ultimately bounded on X since η0>0 is independent of initial values.

    In order to find the species-free equilibrium of system (2.1), which corresponds to the absence of species, we put u=0 in (2.1). Then we consider the following system

    {dRdt=(R(0)R)DωrR+ωsS,dSdt=(S(0)S)D+ωrRωsS,R(0)0, S(0)0. (2.7)

    It is easy to see that (2.7) is a cooperative/monotone system (see, e.g., [18]), and

    (R,S):=(DR(0)+ωsR(0)+ωsS(0)D+ωr+ωs,DS(0)+ωrS(0)+ωrR(0)D+ωr+ωs)

    is the unique equilibrium for (2.7). For a monotone dynamical system, the unique steady state is globally asymptotically stable if and only if every forward orbit has compact closure (see [13,Theorem D]). By Lemma 2.1 and the above discussions, we have the following results:

    Lemma 2.2. The unique equilibrium (R,S) is globally asymptotically stable for (2.7) in R2.

    From Lemma 2.2, the species-free equilibrium of system (2.1), which we label E0, is given by

    E0=(R,S,Q,u)=(R,S,Q,0),

    where Q satisfies

    fR(R,Q)G(0)+fS(S,Q)G(0)μ(Q)Qγ(Q)=0 (2.8)

    The local stability of E0 is determined by the Jacobian matrix of (2.1) at E0, denoted by

    J0=(Dωrωs0fR(R,Q)G(0)+γ(Q)ωrDωs0fS(S,Q)G(0)fR(R,Q)RG(0)fS(S,Q)SG(0)j33[fR(R,Q)+fS(S,Q)]G(0)000μ(Q)D),

    where

    j33=[fR(R,Q)Q+fS(S,Q)Q]G(0)[μ(Q)+μ(Q)Q]γ(Q)<0.

    It is easy to see that the eigenvalues of J0 are j33, μ(Q)D, together with the eigenvalues of the following matrix

    ˜J0=(DωrωsωrDωs).

    Since j33<0 and the eigenvalues of ˜J0 are both negative, we see that the sign of μ(Q)D determines the stability of E0. That is, E0 is locally asymptotically stable if μ(Q)D<0, and unstable if μ(Q)D>0. Thus, we have proved the following result concerning with the local stability of E0:

    Lemma 2.3. E0 is locally asymptotically stable if μ(Q)D<0, and unstable if μ(Q)D>0.

    This subsection is devoted to the investigations of persistence of system (2.1).

    Theorem 2.4. Assume that (R(t),S(t),Q(t),u(t)) is the unique nonnegative solution of system (2.1), for all t[0,), with the initial value (R0,S0,Q0,u0)X. If μ(Q)D>0, then system (2.1) is uniformly persistent in the sense that there exists a ξ>0 such that

    lim inftu(t)ξ, provided that u00.

    Further, system (2.1) admits at least one positive equilibrium (ˆR,ˆS,ˆQ,ˆu).

    Proof. Recall that X is defined in (2.2). Let

    X0={(R,S,Q,u)X:u>0},

    and

    X0:=XX0:={(R,S,Q,u)X:u=0}.

    It is easy to see that both X and X0 are positively invariant for system (2.1), and X0 is relatively closed in X. Furthermore, system (2.1) is point dissipative (see Lemma 2.1). Let Φ(t):XX be the solution maps associated with system (2.1). Set ˜M:={PX0:Φ(t)PX0,  t0} and ˜ω(P) be the omega limit set of the orbit ˜O+(P):={Φ(t)P:t0}. We show the following claim.

    Claim: ˜ω(P)={E0},  P˜M.

    Since P˜M, we have Φ(t)PX0,  t0. Thus, u(t,P)=0,  t0. In view of the first two equations of (2.1), it follows that (R(t,P),S(t,P)) satisfies (2.7),  t0. Then Lemma 2.2 implies that

    limt(R(t,P),S(t,P))=(R,S).

    Then, the equation for Q(t) in (2.1) is asymptotic to

    dQdt=fR(R,Q)G(0)+fS(S,Q)G(0)μ(Q)Qγ(Q).

    From the theory for asymptotically autonomous semiflows (see, e.g., [19,Corollary 4.3]), it follows that limtQ(t,P)=Q, where Q is given in (2.8). Hence, the claim is proved.

    Let η1:=12(μ(Q)D)>0. Then it follows from the continuity of μ(Q) that there exists σ1>0 such that

    μ(Q)>μ(Q)η1,  |QQ|<σ1. (2.9)

    Claim: E0 is a uniform weak repeller for X0 in the sense that

    lim supt|Φt(P)E0|σ1,  PX0.

    Suppose not. Then there exists a PX0 such that

    lim supt|Φt(P)E0|<σ1.

    Thus, there exists a τ1>0 such that

    |Q(t,P)Q|<σ1,  tτ1.

    This and (2.9) imply that

    μ(Q(t,P))D>μ(Q)Dη1=η1,  tτ1.

    From this inequality and the fourth equation of (2.1), we have

    du(t,P)dt>η1u(t,P),  tτ1,

    which shows that limtu(t,P)=, a contradiction.

    Therefore, E0 is isolated in X and ˜Ws(E0)X0=, where ˜Ws(E0) is the stable set of E0 (see [21]). Since Φt:XX is point dissipative and compact, we conclude from [21,Theorem 1.1.3] that there exists a global attractor A for Φt in X. By [21,Theorem 1.3.1] on strong repellers, Φt:XX is uniformly persistent with respect to (X0,X0). It follows from [21,Theorem 1.3.6] that there exists a global attractor A0 for Φt in X0 and Φt admits at least one fixed point

    (ˆR,ˆS,ˆQ,ˆu)X0.

    Then ˆQQmin>0, ˆu>0, and (ˆR,ˆS) satisfies

    {(R(0)ˆR)DfR(ˆR,ˆQ)g(ˆu)ˆu+γ(ˆQ)ˆuωrˆR+ωsˆS=0,(S(0)ˆS)DfS(ˆS,ˆQ)g(ˆu)ˆu+ωrˆRωsˆS=0. (2.10)

    In view of (2.10), we deduce that ˆR>0, ˆS>0. Thus, (ˆR,ˆS,ˆQ,ˆu) is a positive steady-state solution for (2.1). We complete the proof.

    In this subsection, we neglect the effect of respiration and investigate the extinction of system (2.1). Putting the respiration rate to be zero,

    γ(Q)0,  QQmin. (2.11)

    Then we have the following result:

    Theorem 2.5. Suppose (R(t),S(t),Q(t),u(t)) is the unique nonnegative solution of system (2.1), for all t[0,), with the initial value (R0,S0,Q0,u0)X. Assume that (2.11) holds. If μ(Q)D<0, then system (2.1) is washed out, that is,

    limt(R(t),S(t),Q(t),u(t))=(R,S,Q,0).

    Proof. For η>0, we assume that Qη is the unique root of

    G(0)[fR(R,Q)+fS(S,Q)]+ημ(Q)Q=0. (2.12)

    Recall that Q is the unique root of (2.8). Since

    limη0[μ(Qη)D]=μ(Q)D and 13[μ(Q)D]>0,

    we may find an η2>0 such that

    μ(Qη2)D<[μ(Q)D]+13[μ(Q)D]=23[μ(Q)D]. (2.13)

    On the other hand, by the continuity, we may find a σ2>0 such that

    {fR(R+σ2,Q)<fR(R,Q)+η22G(0),fS(S+σ2,Q)<fS(S,Q)+η22G(0),μ(Qη2+σ2)<μ(Qη2)+13[μ(Q)D] (2.14)

    In view of the assumption (2.11) and the first two equations of system (2.1), it follows that

    {dRdt(R(0)R)DωrR+ωsS,dSdt(S(0)S)D+ωrRωsS.

    By the comparison arguments and Lemma 2.2, we have

    limt(R(t),S(t))(R,S).

    Then there exists a τ2>0 such that

    R(t)R+σ2, S(t)S+σ2,  tτ2.

    Then it follows from the third equation of (2.1) that

    dQdtG(0)[fR(R+σ2,Q)+fS(S+σ2,Q)]μ(Q)Q,  tτ2. (2.15)

    In view of the first two inequalities of (2.14) and (2.15), we have

    dQdtG(0)[fR(R,Q)+fS(S,Q)]+η2μ(Q)Q,  tτ2.

    Using the comparison arguments, we have

    limtQ(t)Qη2, (2.16)

    where Qη2 is the unique root of (2.12) with η=η2. Thus, we may find a τ3>0 such that

    Q(t)Qη2+σ2,  tτ3,

    and hence,

    μ(Q(t))μ(Qη2+σ2),  tτ3. (2.17)

    In view of the third inequality of (2.14) and (2.17), we have

    μ(Q(t))μ(Qη2)+13[μ(Q)D],  tτ3. (2.18)

    By (2.13), (2.18) together with the fourth equation of (2.1), it follows that

    dudt=[μ(Q(t))D]u13[μ(Q)D]u,  tτ3. (2.19)

    Since μ(Q)D<0 and (2.19), we have

    limtu(t)=0.

    Then (R(t),S(t)) is asymptotic to (2.7). By Lemma 2.2 and the theory for asymptotically autonomous semiflows (see, e.g., [19,Corollary 4.3]), it follows that

    limt(R(t),S(t))=(R,S).

    Similarly, Q(t) is asymptotic to

    dQdt=fR(R,Q)G(0)+fS(S,Q)G(0)μ(Q)Q,

    and limtQ(t)=Q, where Q is given in (2.8) with γ(Q)0. We complete the proof.

    In this section, we shall concentrate on the study of coexistence of system (1.1). The subsequent discussions will reveal that two semi-trivial steady-state solutions of system (1.1), corresponds to the presence of one of the species and the absence of the other species, are not necessarily unique. This makes our analysis more difficult. Fortunately, we can adopt the ideas developed in [12,Section 4] to overcome this difficulty.

    The trivial steady-state solution of (1.1), labeled E0, corresponds to the absence of both species. It is given by

    E0=(R,S,Q1,u1,Q2,u2)=(R,S,Q1,0,Q2,0),

    where (R,S) is given in Lemma 2.2, and Qi satisfies

    fRi(R,Qi)Gi(0)+fSi(S,Qi)Gi(0)μi(Qi)Qiγi(Qi)=0, i=1,2,

    where G1(u1)=g1(u1,0) and G2(u2)=g2(0,u2).

    In order to determine the semi-trivial steady-state solutions of (1.1), we need the following single population system associated with the growth of species i:

    {dRdt=(R(0)R)DfRi(R,Qi)Gi(ui)ui+γi(Qi)uiωrR+ωsS,dSdt=(S(0)S)DfSi(S,Qi)Gi(ui)ui+ωrRωsS,dQidt=fRi(R,Qi)Gi(ui)+fSi(S,Qi)Gi(ui)μi(Qi)Qiγi(Qi),duidt=[μi(Qi)D]ui,R(0)0, S(0)0, ui(0)0, Qi(0)Qmin,i, i=1,2. (3.1)

    Next, we shall summarize the result of system (3.1). By Theorem 2.4, for i=1,2, system (3.1) admits at least one positive equilibrium and we may assume that A0iIntR4+ is a global attractor of the semiflows generated by system (3.1), under the condition that μi(Qi)D>0. One of the semi-trivial steady-state solutions of (1.1), labeled E1, corresponds to the presence of species 1 and the absence of species 2. It is given by

    E1=(R,S,Q1,u1,Q2,u2)=(ˆR1,ˆS1,ˆQ1,ˆu1,ˆQ2,0),

    where (ˆR1,ˆS1,ˆQ1,ˆu1)A01 is a positive equilibrium of system (3.1) with i=1, which is not necessarily unique. Here, ˆQ2=ˆQ2(ˆR1,ˆS1,ˆQ1,ˆu1) is the root of

    fR2(ˆR1,Q2)g2(ˆu1,0)+fS2(ˆS1,Q2)g2(ˆu1,0)μ2(Q2)Q2γ2(Q2)=0. (3.2)

    Inspired by the arguments in [12,Section 4], we assume that

    ˆQmin2=inf{ˆQ2(ˆR1,ˆS1,ˆQ1,ˆu1):(ˆR1,ˆS1,ˆQ1,ˆu1)A01}. (3.3)

    The other semi-trivial steady-state solution of (1.1), labeled E2, corresponds to the presence of species 2 and the absence of species 1. It is given by

    E2=(R,S,Q1,u1,Q2,u2)=(ˇR2,ˇS2,ˇQ1,0,ˇQ2,ˇu2),

    where (ˇR2,ˇS2,ˇQ2,ˇu2)A02 is a positive equilibrium of system (3.1) with i=2, which is not necessarily unique. Here, ˇQ1=ˇQ1(ˇR2,ˇS2,ˇQ2,ˇu2) is the root of

    fR1(ˇR2,Q1)g1(0,ˇu2)+fS1(ˇS2,Q1)g1(0,ˇu2)μ1(Q1)Q1γ1(Q1)=0.

    Similarly, we assume

    ˇQmin1=inf{ˇQ1(ˇR2,ˇS2,ˇQ2,ˇu2):(ˇR2,ˇS2,ˇQ2,ˇu2)A02}. (3.4)

    The feasible domain for system (1.1) takes the form

    Y={(R,S,Q1,u1,Q2,u2)R6+:QQmin,i, i=1,2}.

    Then it is easy to show that Y is positively invariant for system (1.1). By the similar arguments in Lemma 2.1, we can show the following result:

    Lemma 3.1. Every solution (R(t),S(t),Q1(t),u1(t),Q2(t),u2(t)) of system (1.1) exists globally. Furthermore, solutions of (1.1) are ultimately bounded on Y.

    Assume that Ψ(t):YY is the semiflow associated with system (1.1). Let

    Y0={(R,S,Q1,u1,Q2,u2)Y:u1>0 and u2>0},

    and

    Y0:=YY0:={(R,S,Q1,u1,Q2,u2)Y:u1=0 or u2=0}.

    Following the ideas in [12,Section 4], we assume that M0={E0},

    M1={(ˆR1,ˆS1,ˆQ1,ˆu1,ˆQ2,0)Y:(ˆR1,ˆS1,ˆQ1,ˆu1)A01 and ˆQ2  is defined by (3.2)},

    and

    M2={(ˇR2,ˇS2,ˇQ1,0,ˇQ2,ˇu2)Y:(ˇR2,ˇS2,ˇQ2,ˇu2)A02 and ˇQ1  is defined by (3.4)}.

    One can easily to use "the method of proof by contradiction" to deduce the following result:

    Lemma 3.2. Let μi(Qi)D>0, for some i{1,2}. Then M0 is a uniform weak repeller in the sense that there exists a δ0>0 such that

    lim supt|Ψ(t)v0M0|δ0, for all v0Y0.

    Next, we shall use the strategy in [12,Lemma 4.2] to show the following result:

    Lemma 3.3. Let μi(Qi)D>0, for each i{1,2}. If μ2(ˆQmin2)D>0, then M1 is a uniform weak repeller in the sense that there exists a δ1>0 such that

    lim suptdist(Ψ(t)v0,M1)δ1, for all v0Y0. (3.5)

    Proof. Let

    B1={ˆQ2=ˆQ2(ˆR1,ˆS1,ˆQ1,ˆu1):(ˆR1,ˆS1,ˆQ1,ˆu1)A01 and ˆQ2  is defined by (3.2)}.

    Then

    ˆQmin2=inf{ˆQ2(ˆR1,ˆS1,ˆQ1,ˆu1):ˆQ2(ˆR1,ˆS1,ˆQ1,ˆu1)B1}.

    Setting

    ϵ1=12[μ2(ˆQmin2)D]>0.

    Define G:B1R by

    G(ϕ)=μ2(ϕ), ϕB1.

    We may find a δ1>0 such that

    dist(G(ϕ),G(B1))<ϵ1,

    whenever ϕR with dist(ϕ,B1)<δ1. Since B1 is compact, it follows that for any ϕR with dist(ϕ,B1)<δ1, there exists ϕB1 with ϕ depending on ϕ such that

    |G(ϕ)G(ϕ)|=dist(G(ϕ),G(B1))<ϵ1.

    Thus, we have

    |μ2(ϕ)μ2(ϕ)|=|G(ϕ)G(ϕ)|<ϵ1,

    whenever ϕR with dist(ϕ,B1)<δ1.

    Suppose that (3.5) is not true. Then there exists v0Y0 such that

    lim suptdist(Ψ(t)v0,M1)<δ1.

    This implies that

    lim suptdist(Q2(t),B1)<δ1  and  lim supt|u2(t)|<δ1. (3.7)

    From the first inequality of (3.7), we can choose t1>0 such that

    dist(Q2(t),B1)<δ1,  tt1.

    By (3.6), it follows that there exists ϕtB1 such that

    |μ2(Q2(t))μ2(ϕt)|<ϵ1,  tt1,

    which implies that

    μ2(Q2(t))D>μ2(ϕt)Dϵ1μ2(ˆQmin2)Dϵ1=ϵ1,  tt1.

    From the sixth equation of (1.1), we have

    du2(t)dt=[μ2(Q2(t))D]u2(t)>ϵ1u2(t),  tt1.

    We deduce that limtu2(t)= since ϵ1>0 and u2(t1)>0. This contradicts the second inequality of (3.7) and we complete the proof.

    By the same arguments in Lemma 3.3, the following result holds:

    Lemma 3.4. Let μi(Qi)D>0, for each i{1,2}. If μ1(ˇQmin1)D>0, then M2 is a uniform weak repeller in the sense that there exists a δ2>0 such that

    lim suptdist(Ψ(t)v0,M2)δ2, for all v0Y0.

    Now we are in a position to prove the main result of this paper.

    Theorem 3.5. Assume that (R(t),S(t),Q1(t),u1(t),Q2(t),u2(t)) is the unique solution of (1.1) with the initial value (R(0),S(0),Q1(0),u1(0),Q2(0),u2(0))Y. Let μi(Qi)D>0,  i=1,2, μ2(ˆQmin2)D>0, and μ1(ˇQmin1)D>0. Then system (1.1) is uniformly persistent with respect to (Y0,Y0) in the sense that there is a positive constant ζ>0 such that if u1(0)0 and u2(0)0, we have

    lim inftui(t)ζ, i=1,2.

    Furthermore, system (1.1) admits at least one (componentwise) positive equilibrium.

    Proof. Recall that Ψ(t):YY is the semiflow associated with system (1.1). It is easy to see that Ψ(t)Y0Y0. Since solutions of the system (1.1) are ultimately bounded (see Lemma 3.1), it follows that Ψ(t) is point dissipative and compact, and hence, Ψ(t) admits a global attractor (see, e.g., [21,Theorem 1.1.3]). Let

    M:={v0Y0:Ψ(t)v0Y0, t0},

    and ω(v0) be the omega limit set of the orbit O+(v0):={Ψ(t)v0:t0}.

    Claim: v0Mω(v0)M0M1M2.

    For any given v0:=(R0,S0,Q01,u01,Q02,u02)M, we have v0Y0 and Ψ(t)v0Y0,  t0. We discuss the following three subcases:

    (ⅰ) If u01=0, u02=0, then we have u1(t,v0)=0 and u2(t,v0)=0,  t0. Thus, it is easy to see that limtΨ(t)v0=E0.

    (ⅱ) If u010, u02=0, then we have u1(t,v0)>0 and u2(t,v0)=0,  t0. Then (R(t,v0),S(t,v0),Q1(t,v0),u1(t,v0)) satisfies system (3.1) with i=1. Since μ1(Q1)D>0, it follows from Theorem 2.4 that

    (R(t,v0),S(t,v0),Q1(t,v0),u1(t,v0))

    will eventually enter the global attractor A01IntR4+, and hence, Ψ(t)v0 will eventually enter M1.

    (ⅲ) If u01=0, u020, then we have u1(t,v0)=0 and u2(t,v0)>0,  t0. Then (R(t,v0),S(t,v0),Q2(t,v0),u2(t,v0)) satisfies system (3.1) with i=2. Since μ2(Q2)D>0, it follows from Theorem 2.4 that

    (R(t,v0),S(t,v0),Q2(t,v0),u2(t,v0))

    will eventually enter the global attractor A02IntR4+, and hence, Ψ(t)v0 will eventually enter M2.

    The proof of the claim is complete.

    By Lemma 3.2, Lemma 3.3 and Lemma 3.4, it follows that for i=0,1,2, Mi is a uniform weak repeller for Y0 in the sense that there exists δi>0 such that

    lim suptdist(Ψ(t)v0,Mi)δi, for all v0Y0.

    Note that M0, M1, and M2 are pairwise disjoint, compact and isolated invariant sets for Ψ(t) in Y0. Further, each Mi is isolated in Y and Ws(Mi)Y0=, where Ws(Mi) is the stable set of Mi (see [21]). It is easy to see that no subsets of M0, M1, and M2 forms a cycle in Y0. By [21,Theorem 1.3.1] on strong repellers, Ψ(t):YY is uniformly persistent with respect to (Y0,Y0). It follows from [21,Theorem 1.3.6] that there exists a global attractor ˆA0 for Ψ(t) in Y0 and Ψ(t) has at least one fixed point

    (˜R,˜S,˜Q1,˜u1,˜Q2,˜u2)Y0.

    Thus, (˜R,˜S,˜Q1,˜u1,˜Q2,˜u2) is a positive steady-state solution for system (1.1). This completes the proof.

    In this paper, we study the chemostat-type system (1.1) modeling the interactions of two species competing for "CO2" (dissolved CO2 and carbonic acid), "CARB" (bicarbonate and carbonate ions), and light in a spatially homogeneous water column. Our mathematical model presented in this paper is inspired by the recent works [17,20]. In fact, system (1.1) is a modified version of the model in the Supplementary Information of [20], where the specific growth rate of the competing species i depends on their stored cellular carbon content (quota) Qi. The dynamics of Qi can be affected by the uptake rates of inorganic carbon, "CO2" (R(t)) and "CARB" (S(t)), photosynthetic activity (gi(u1,u2)), and respiration (γi(Qi)). The resources "CO2" and "CARB" are substitutable in their effects on algal growth, which also involves a very complex processes. In order to make our system (1.1) analytically tractable, we have adopted the ideas in [17] to assume that carbonic acid loses a proton to become bicarbonate at the rate ωr, and the rate of the reverse reaction is denoted by ωs, which simplifies the complex processes of "CO2" and "CARB" involved.

    Solutions of both the two-species system (1.1) and its single-species sub-system (2.1) follow mass conservation laws, and are eventually bounded (see Lemma 2.1 and Lemma 3.1). Persistence of a single species depends on the sign of μ(Q)D (see Theorem 2.4), where Q is given in (2.8). Biologically, Q represents the quota that a species can obtain when the inorganic carbon concentration is at its long-term upper bound (R,S), which is the unique equilibrium for system (2.7) governing the available inorganic carbon in a species-free habitat. Then Theorem 2.4 states that the species can persist if the quota Q exceeds the quota ˆQ required for growth to balance losses (i.e., μ(ˆQ)=D). Thus, the persistence criterion (i.e., μ(Q)D>0) summarizes the characteristics of carbon uptake, the growth rate, light availability, and the respiration rate. If the quota Q is less than the quota ˆQ, then we can show that the species population is washed out of the habitat (see Theorem 2.5), where we have ignored the effect of respiration (see the specific assumption (2.11)), due to a technical reason.

    In Theorem 2.4, we only show that the single-species model (2.1) admits at least one positive equilibrium if the species can persist by using the theory of uniform persistence. The uniqueness and global stability of positive equilibrium for (2.1) are still open if no extra assumptions are imposed. Thus, two semi-trivial steady-state solutions of the two-species system (1.1), corresponds to the presence of one of the species and the absence of the other species, are not necessarily unique. This makes the investigation of coexistence for the two-species system (1.1) more difficult. Inspired by [12,Section 4], we first define two suitable parameters, ˆQmin2 and ˇQmin1 (see 3.3 and 3.4), then we are able to show that the compact attractor M1 (on the boundary u2=0), and the compact attractor M2 (on the boundary u1=0) are uniform weak repellers for two-species system (1.1) (see Lemma 3.3 and Lemma 3.4) under appropriate conditions depending on ˆQmin2 and ˇQmin1, respectively. Finally, we are able to show that system (1.1) is uniformly persistent, and it admits at least one coexistence (componentwise positive) steady-state solution (see Theorem 3.5) when the trivial steady-state solution, the compact set M1, and the compact set M2 are all invasible. From biological viewpoints, invasibility will depend on whether the missing competitor obtains sufficiently large quotas (ˆQmin2 or ˇQmin1) to permit a growth rate that exceeds the loss to dilution (D). Then robust coexistence occurs when there is mutual invasibility of both M1 and M2.

    Research of F.-B. Wang is supported in part by Ministry of Science and Technology, Taiwan; and National Center for Theoretical Sciences (NCTS), National Taiwan University; and Chang Gung Memorial Hospital (CRRPD3H0011, BMRPD18 and NMRPD5F0543). We would like to express our thanks to Professor Sze-Bi Hsu for the helpful discussions in this paper.

    All authors declare no conflicts of interest in this paper.



    [1] A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theoret. Biol., 84 (1980), 189–203.
    [2] A. Cunningham and R. M. Nisbet, Transient and Oscillation in Continuous Culture, in Mathematics in Microbiology, M. J. Bazin, ed., Academic ress, New York, 1983.
    [3] M. Droop, Vitamin B12 and marine ecology. IV. The kinetics of uptake, growth and inhibition in Monochrysis Lutheri, J. Mar. Biol. Assoc. UK, 48 (1968), 689–733.
    [4] M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264–272.
    [5] M. Droop, The nutrient status of algal cells in continuous culture, J. Mar. Biol. Assoc. UK, 54 (1974), 825–855.
    [6] J. P. Grover, Constant- and variable-yield models of population growth: Responses to environmental variability and implications for competition, J. Theoret. Biol., 158 (1992), 409–428.
    [7] J. P. Grover, Resource Competition, Chapman and Hall, London, 1997.
    [8] J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, The American Naturalist, 178 (2011), E 124–E 148.
    [9] S.-B Hsu and C. J. Lin, Dynamics of two phytoplankton Species Competing for light and nutrient with internal storage, Discrete Cont. Dyn. S, 7 (2014), 1259–1285.
    [10] S. B. Hsu, K. Y. Lam and F. B. Wang, Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat, J. Math. Biol., 75 (2017), 1775–1825.
    [11] J. Huisman, P. v. Oostveen and F. J.Weissing, Species dynamics in phytoplankton blooms: incomplete mixing and competition for light, The American Naturalist, 154 (1999), 46–67.
    [12] S. B. Hsu, F. B.Wang, and X. Q. Zhao, A reaction-diffusion model of harmful algae and zooplankton in an ecosystem, J. Math. Anal. Appl., 451 (2017), 659–677.
    [13] J. Jiang, On the global stability of cooperative systems, Bull London Math. Soc., 26 (1994), 455– 458.
    [14] J. T. O. Kirk, Light and photosynthesis in aquatic ecosystems, 2nd edition, Cambridge University Press, Cambridge, 1994.
    [15] P. D. Leenheer, S. A. Levin, E. D. Sontag and C. A. Klausmeier, Global stability in a chemostat with multiple nutrients, J. Math. Biol., 52 (2006), 419–438.
    [16] F. M. M. Morel, Kinetics of nutrient uptake and growth in phytoplankton, J. Phycol., 23 (1987), 137–150.
    [17] H. Nie, S. B. Hsu and J. P. Grover, Algal Competition in a water column with excessive dioxide in the atmosphere, J. Math. Biol., 72 (2016), 1845–1892.
    [18] H. L. Smith, Monotone Dynamical Systems:An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.
    [19] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755–763.
    [20] D. B. V. deWaal, J. M. H. Verspagen, J. F. Finke, V. Vournazou, A. K. Immers,W. E. A. Kardinaal, L. Tonk, S. Becker, E. V. Donk, P. M. Visser and J. Huisman, Reversal in competitive dominance of a toxic versus non-toxic cyanobacterium in response to rising CO2, ISME J., 5 (2011), 1438–1450.
    [21] X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4377) PDF downloads(643) Cited by(0)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog