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
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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)D−fR1(R,Q1)g1(u1,u2)u1−fR2(R,Q2)g2(u1,u2)u2 +γ1(Q1)u1+γ2(Q2)u2−ωrR+ωsS,dSdt=(S(0)−S)D−fS1(S,Q1)g1(u1,u2)u1−fS2(S,Q2)g2(u1,u2)u2+ωrR−ωsS,dQ1dt=fR1(R,Q1)g1(u1,u2)+fS1(S,Q1)g1(u1,u2)−μ1(Q1)Q1−γ1(Q1),du1dt=[μ1(Q1)−D]u1,dQ2dt=fR2(R,Q2)g2(u1,u2)+fS2(S,Q2)g2(u1,u2)−μ2(Q2)Q2−γ2(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
One type of the photosynthetic rate of the species
gi(u1,u2)=1zm∫zm0miˆI(z)ai+ˆI(z)dz, | (1.2) |
where
ˆI(z)=Iinexp(−k0z−k1zu1(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 (
gi(u1,u2)=mi(k0+k1u1+k2u2)zmln(ai+Iinai+ˆI(zm))=miln(IinˆI(zm))ln(ai+Iinai+ˆI(zm)), |
where
ˆI(zm)=Iinexp(−k0zm−k1zmu1(t)−k2zmu2(t)). |
We also note that the other type of the photosynthetic rate
gi(u1,u2)=miIai+I, | (1.4) |
with
I(t)=Iinexp(−k0zm−k1zmu1(t)−k2zmu2(t)). |
According to [20], we take the growth rate
μi(Qi)=μmax,iQi−Qmin,iQmax,i−Qmin,i, |
where
μi(Qi)=μ∞,i(1−Qmin,iQi),orμi(Qi)=μ∞,i(Qi−Qmin,i)+Ai+(Qi−Qmin,i)+, |
where
According to [6,16], for
fHi(H,Qi)=ρHi(Qi)HKHi+H. |
Here
ρHi(Qi)=ρhighmax,Hi−(ρhighmax,Hi−ρlowmax,Hi)Qi−Qmin,iQmax,i−Qmin,i, |
for
ρHi(Qi)=ρmax,HiQmax,i−QiQmax,i−Qmin,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,iQi−Qmin,iQmax,i−Qmin,i, |
where
In this whole paper, we always assume that the photosynthetic rate
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
{dRdt=(R(0)−R)D−fR(R,Q)G(u)u+γ(Q)u−ωrR+ωsS,dSdt=(S(0)−S)D−fS(S,Q)G(u)u+ωrR−ωsS,dQdt=fR(R,Q)G(u)+fS(S,Q)G(u)−μ(Q)Q−γ(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)=1zm∫zm0m˜I(z)a+˜I(z)dz=mln(Iin˜I(zm))ln(a+Iina+˜I(zm)), |
where
˜I(z)=Iinexp(−k0z−kzu(t)), 0≤z≤zm, |
or
G(u)=mIa+I, with I(t)=Iinexp(−k0zm−kzmu(t)). |
The feasible domain for system (2.1) takes the form
X={(R,S,Q,u)∈R4+:Q≥Qmin}. | (2.2) |
Then it is easy to show that
Lemma 2.1. Every solution
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
dΘ(t)dt=(R(0)+S(0)−Θ(t))D. | (2.4) |
Thus,
It remains to show that
dV(t)dt=dQ(t)dtQ≤G(0)[fR(R(t),Qmin)+fS(S(t),Qmin)]Q≤12G(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
From (2.4), it is easy to see that
limt→∞Θ(t)=R(0)+S(0), | (2.6) |
and hence,
dQdt≤G(0)[fR(R(0)+S(0),Q)+fS(R(0)+S(0),Q)]+η0−μ(Q)Q−γ(Q), t≥τ0. |
Then
limt→∞Q(t)≤Qη0, |
where
G(0)[fR(R(0)+S(0),Q)+fS(R(0)+S(0),Q)]+η0−μ(Q)Q−γ(Q)=0. |
Thus, solutions of (2.1) are ultimately bounded on
In order to find the species-free equilibrium of system (2.1), which corresponds to the absence of species, we put
{dRdt=(R(0)−R)D−ωrR+ωsS,dSdt=(S(0)−S)D+ωrR−ωsS,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
From Lemma 2.2, the species-free equilibrium of system (2.1), which we label
E0=(R,S,Q,u)=(R∗,S∗,Q∗,0), |
where
fR(R∗,Q∗)G(0)+fS(S∗,Q∗)G(0)−μ(Q∗)Q∗−γ(Q∗)=0 | (2.8) |
The local stability of
J0=(−D−ωrωs0−fR(R∗,Q∗)G(0)+γ(Q∗)ωr−D−ωs0−fS(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=(−D−ωrωsωr−D−ωs). |
Since
Lemma 2.3.
This subsection is devoted to the investigations of persistence of system (2.1).
Theorem 2.4. Assume that
lim inft→∞u(t)≥ξ, provided that u0≠0. |
Further, system (2.1) admits at least one positive equilibrium
Proof. Recall that
X0={(R,S,Q,u)∈X:u>0}, |
and
∂X0:=X∖X0:={(R,S,Q,u)∈X:u=0}. |
It is easy to see that both
Claim:
Since
limt→∞(R(t,P),S(t,P))=(R∗,S∗). |
Then, the equation for
dQdt=fR(R,Q)G(0)+fS(S,Q)G(0)−μ(Q)Q−γ(Q). |
From the theory for asymptotically autonomous semiflows (see, e.g., [19,Corollary 4.3]), it follows that
Let
μ(Q)>μ(Q∗)−η1, ∀ |Q−Q∗|<σ1. | (2.9) |
Claim:
lim supt→∞|Φt(P)−E0|≥σ1, ∀ P∈X0. |
Suppose not. Then there exists a
lim supt→∞|Φt(P)−E0|<σ1. |
Thus, there exists a
|Q(t,P)−Q∗|<σ1, ∀ t≥τ1. |
This and (2.9) imply that
μ(Q(t,P))−D>μ(Q∗)−D−η1=η1, ∀ t≥τ1. |
From this inequality and the fourth equation of (2.1), we have
du(t,P)dt>η1u(t,P), ∀ t≥τ1, |
which shows that
Therefore,
(ˆR,ˆS,ˆQ,ˆu)∈X0. |
Then
{(R(0)−ˆR)D−fR(ˆR,ˆQ)g(ˆu)ˆu+γ(ˆQ)ˆu−ωrˆR+ωsˆS=0,(S(0)−ˆS)D−fS(ˆS,ˆQ)g(ˆu)ˆu+ωrˆR−ωsˆS=0. | (2.10) |
In view of (2.10), we deduce that
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, ∀ Q≥Qmin. | (2.11) |
Then we have the following result:
Theorem 2.5. Suppose
limt→∞(R(t),S(t),Q(t),u(t))=(R∗,S∗,Q∗,0). |
Proof. For
G(0)[fR(R∗,Q)+fS(S∗,Q)]+η−μ(Q)Q=0. | (2.12) |
Recall that
limη→0[μ(Qη)−D]=μ(Q∗)−D and −13[μ(Q∗)−D]>0, |
we may find an
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
{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)D−ωrR+ωsS,dSdt≤(S(0)−S)D+ωrR−ωsS. |
By the comparison arguments and Lemma 2.2, we have
limt→∞(R(t),S(t))≤(R∗,S∗). |
Then there exists a
R(t)≤R∗+σ2, S(t)≤S∗+σ2, ∀ t≥τ2. |
Then it follows from the third equation of (2.1) that
dQdt≤G(0)[fR(R∗+σ2,Q)+fS(S∗+σ2,Q)]−μ(Q)Q, ∀ t≥τ2. | (2.15) |
In view of the first two inequalities of (2.14) and (2.15), we have
dQdt≤G(0)[fR(R∗,Q)+fS(S∗,Q)]+η2−μ(Q)Q, ∀ t≥τ2. |
Using the comparison arguments, we have
limt→∞Q(t)≤Qη2, | (2.16) |
where
Q(t)≤Qη2+σ2, ∀ t≥τ3, |
and hence,
μ(Q(t))≤μ(Qη2+σ2), ∀ t≥τ3. | (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], ∀ t≥τ3. | (2.18) |
By (2.13), (2.18) together with the fourth equation of (2.1), it follows that
dudt=[μ(Q(t))−D]u≤13[μ(Q∗)−D]u, ∀ t≥τ3. | (2.19) |
Since
limt→∞u(t)=0. |
Then
limt→∞(R(t),S(t))=(R∗,S∗). |
Similarly,
dQdt=fR(R∗,Q)G(0)+fS(S∗,Q)G(0)−μ(Q)Q, |
and
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=(R,S,Q1,u1,Q2,u2)=(R∗,S∗,Q∗1,0,Q∗2,0), |
where
fRi(R∗,Q∗i)Gi(0)+fSi(S∗,Q∗i)Gi(0)−μi(Q∗i)Q∗i−γi(Q∗i)=0, i=1,2, |
where
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
{dRdt=(R(0)−R)D−fRi(R,Qi)Gi(ui)ui+γi(Qi)ui−ωrR+ωsS,dSdt=(S(0)−S)D−fSi(S,Qi)Gi(ui)ui+ωrR−ωsS,dQidt=fRi(R,Qi)Gi(ui)+fSi(S,Qi)Gi(ui)−μi(Qi)Qi−γi(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
E1=(R,S,Q1,u1,Q2,u2)=(ˆR1,ˆS1,ˆQ1,ˆu1,ˆQ2,0), |
where
fR2(ˆR1,Q2)g2(ˆu1,0)+fS2(ˆS1,Q2)g2(ˆu1,0)−μ2(Q2)Q2−γ2(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=(R,S,Q1,u1,Q2,u2)=(ˇR2,ˇS2,ˇQ1,0,ˇQ2,ˇu2), |
where
fR1(ˇR2,Q1)g1(0,ˇu2)+fS1(ˇS2,Q1)g1(0,ˇu2)−μ1(Q1)Q1−γ1(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+:Q≥Qmin,i, i=1,2}. |
Then it is easy to show that
Lemma 3.1. Every solution
Assume that
Y0={(R,S,Q1,u1,Q2,u2)∈Y:u1>0 and u2>0}, |
and
∂Y0:=Y∖Y0:={(R,S,Q1,u1,Q2,u2)∈Y:u1=0 or u2=0}. |
Following the ideas in [12,Section 4], we assume that
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
lim supt→∞|Ψ(t)v0−M0|≥δ0, for all v0∈Y0. |
Next, we shall use the strategy in [12,Lemma 4.2] to show the following result:
Lemma 3.3. Let
lim supt→∞dist(Ψ(t)v0,M1)≥δ1, for all v0∈Y0. | (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(ϕ)=μ2(ϕ), ϕ∈B1. |
We may find a
dist(G(ϕ),G(B1))<ϵ1, |
whenever
|G(ϕ)−G(ϕ∗)|=dist(G(ϕ),G(B1))<ϵ1. |
Thus, we have
|μ2(ϕ)−μ2(ϕ∗)|=|G(ϕ)−G(ϕ∗)|<ϵ1, |
whenever
Suppose that (3.5) is not true. Then there exists
lim supt→∞dist(Ψ(t)v0,M1)<δ1. |
This implies that
lim supt→∞dist(Q2(t),B1)<δ1 and lim supt→∞|u2(t)|<δ1. | (3.7) |
From the first inequality of (3.7), we can choose
dist(Q2(t),B1)<δ1, ∀ t≥t1. |
By (3.6), it follows that there exists
|μ2(Q2(t))−μ2(ϕt∗)|<ϵ1, ∀ t≥t1, |
which implies that
μ2(Q2(t))−D>μ2(ϕt∗)−D−ϵ1≥μ2(ˆQmin2)−D−ϵ1=ϵ1, ∀ t≥t1. |
From the sixth equation of (1.1), we have
du2(t)dt=[μ2(Q2(t))−D]u2(t)>ϵ1u2(t), ∀ t≥t1. |
We deduce that
By the same arguments in Lemma 3.3, the following result holds:
Lemma 3.4. Let
lim supt→∞dist(Ψ(t)v0,M2)≥δ2, for all v0∈Y0. |
Now we are in a position to prove the main result of this paper.
Theorem 3.5. Assume that
lim inft→∞ui(t)≥ζ, i=1,2. |
Furthermore, system (1.1) admits at least one (componentwise) positive equilibrium.
Proof. Recall that
M∂:={v0∈∂Y0:Ψ(t)v0∈∂Y0,∀ t≥0}, |
and
Claim:
For any given
(ⅰ) If
(ⅱ) If
(R(t,v0),S(t,v0),Q1(t,v0),u1(t,v0)) |
will eventually enter the global attractor
(ⅲ) If
(R(t,v0),S(t,v0),Q2(t,v0),u2(t,v0)) |
will eventually enter the global attractor
The proof of the claim is complete.
By Lemma 3.2, Lemma 3.3 and Lemma 3.4, it follows that for
lim supt→∞dist(Ψ(t)v0,Mi)≥δi, for all v0∈Y0. |
Note that
(˜R,˜S,˜Q1,˜u1,˜Q2,˜u2)∈Y0. |
Thus,
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
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
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,
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.
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