The objective of this study is to analyze a chemostat model of very simple type with the Haldane expression of growth rate and a variable yield coefficient. The proposed modified model is analyzed qualitatively and quantitatively. Analytic conditions for stability and optimality are determined for washout and no washout equilibrium solutions. One of the main focuses of the study is to determine parameter values for which Hopf Bifurcations occur in a bioreactor. It has been shown that the maximum stable non - washout equilibrium exits at a residence time under suitable parameter values. Hopf bifurcation is observed at three different conditions of the parameters.
Researchers have been studying extensively to improve product yield in chemical reactors or bio-reactors for many years. To gain desired scale of production, it is necessary to find effective engineering instruments and mechanism in a bio-reactor. One such engineering instrument is application of oscillatory external force. Since implementation of oscillatory external force is costly, it has very limited use in industry [
Our aim is to investigate a microbial system, the process of using systems biology to understand microbes and their environment, where growth of the cell mass (X) depends on substrate species (S). In this study, the specific growth rate is taken from the Haldane kinetic model with variable yield coefficient. We are studying a simple chemostat model to optimize the cell mass concentration as a function of residence time in a reactor and similar kind of study of microbial systems can be found in [
The microbial system which we consider in this study is represented by the following Equations (1)-(4)
V d S d t = F ( S 0 − S ) − V X μ ( S ) Y ( S ) (1)
V d X d t = F ( X 0 − X ) + V X μ ( S ) (2)
with the specific growth rate
μ ( S ) = μ m S ( t ) K s + S + S 2 K i (3)
and a Monod expression with variable Yield Coefficients
Y ( S ) = α + β S , ( α , β > 0 ) (4)
where F is the flow rate (l·hr−1), Ks is the half-saturation constant (g·l−1), Ki is the inhibitory constant (g·l−1), V is the volume (l), X0 is the initial cell mass concentration (g·l−1), S0 is the initial substrate concentration (g·l−1), μ m is the maximum specific growth rate (/h), α(-) and β(g·l−1) are constant yield coefficients.
To make the above model dimensionless, we use the transformations S * = S K s , X * = X α K s , and t * = μ m t and we get the following dimensionless model containing the parameters S 0 * , X 0 * , τ * , β * , and γ * .
d S * d t * = S 0 * − S * τ * − S * X * ( 1 + S * + γ * S * 2 ) ( 1 + β * S * ) (5)
d X * d t * = X 0 * − X * τ * + S * X * 1 + S * + γ * S * 2 (6)
where τ * = V μ m F , β * = β K s α , and γ * = K s K i .
We consider a chemostat setup consists of a sterile feed means that X 0 * will be zero and assume τ * (residence time) as our primary bifurcation parameter. Our secondary bifurcation parameters are S 0 * (substrate concentration in the feed), β * (dimensionless yield coefficient) which depends on specific microbial system and hence not so flexible for tuning, and γ * (dimensionless inhibitory constant). We will denote the dimensional and dimensionless variables interchangeably by the same names and notations since they are connected by one to one relation. For example, “the yield coefficient” and “the dimensionless yield coefficient” will be denoted by the same notation β .
The system of differential Equations (5) and (6) has one washout steady state solution ( S 0 , X 0 ) , where
S 0 = S 0 , X 0 = 0 (7)
and two no washout steady state solutions ( S 1 , X 1 ) and ( S 2 , X 2 ) where
S 1 = − 1 − − 4 γ + ( − 1 + τ ) 2 + τ 2 γ X 1 = γ ( 1 + 2 S 0 γ + − 4 γ + ( − 1 + τ ) 2 − τ ) 2 γ 2 + β ( 1 + − 4 γ + ( − 1 + τ ) 2 − τ ) ( − 1 + τ ) 2 γ 2 − β γ ( − 2 + S 0 + S 0 − 4 γ + ( − 1 + τ ) 2 − S 0 τ ) 2 γ 2 (8)
and
S 2 = − 1 + − 4 γ + ( − 1 + τ ) 2 + τ 2 γ X 2 = γ ( 1 + 2 S 0 γ − − 4 γ + ( − 1 + τ ) 2 − τ ) 2 γ 2 − β ( − 1 + − 4 γ + ( − 1 + τ ) 2 + τ ) ( − 1 + τ ) 2 γ 2 + β γ ( 2 − S 0 + S 0 − 4 γ + ( − 1 + τ ) 2 + S 0 τ ) 2 γ 2 (9)
If τ ≥ 1 + 2 γ , then the substrate components, S 1 and S 2 , in (8) and (9) will be real and positive. Conditions for the cell mass concentration component, X 1 , in (8) to be real and non-negative are γ > 1 S 0 2 and τ ≥ 1 + 2 γ , and for the cell mass concentration, X 2 , in (9) to be real and non-negative are γ > 1 S 0 2 and 1 + 2 γ ≤ τ < 1 + S 0 + S 0 2 γ S 0 . These conditions for the cell mass concentration lead us to conclude that both no washout steady state solutions are physically meaningful whenever γ > 1 S 0 2 and 1 + 2 γ ≤ τ < 1 + S 0 + S 0 2 γ S 0 . We need to study the effect of variation of the residence time on the reactor’s steady state performance. That means, we need to investigate the relationship between the cell mass concentration X and the residence time τ in the no washout state. Specially, we want to determine the value of τ at which X will have optimum values.
From Equation (8), we can show that d X d τ = 0 at
τ max = − 2 β + 4 β 2 + 2 S 0 β 2 + γ − 2 S 0 β γ + S 0 2 β 2 γ 2 β ( − 1 + S 0 β ) (10)
subject to
S 0 > 1 β (11)
and
1 S 0 2 < γ < 4 β 2 1 − 2 S 0 β + S 0 2 β 2 (12)
X ″ ( τ ) will be negative if critical point τ max in (10) belongs to the following interval
1 + 2 γ < τ max < Root [ β − 3 β 2 − S 0 β 2 − 6 β γ + 16 β 2 γ + 6 S 0 β 2 γ + γ 2 − 2 S 0 β γ 2 + S 0 2 β 2 γ 2 + ( − 3 β + 6 β 2 + 3 S 0 β 2 + 6 β γ − 6 S 0 β 2 γ ) # 1 + ( 3 β − 3 β 2 − 3 S 0 β 2 ) # 1 2 + ( − β + S 0 β 2 ) # 1 3 & , 3 ]
where the upper bound of τ max depends on β , γ , and S 0 . For example, if we take β = 2 , γ = 0.9 , S 0 = 2 , we get an upper bound of τ max at 3.313471. The τ max does not exceed the upper bound when
1 S 0 2 < γ < 4 S 0 2 (13)
or
γ > 4 S 0 2 and β < 2 γ ( − 4 + S 0 2 γ ) 2 + S 0 γ − 4 + S 0 2 γ (14)
Therefore, a local maximum value of X has occurred at τ max .
From Equation (9), we can show that d X d τ = 0 at τ max , subject to the restriction that S 0 > 1 β and 4 β 2 1 − 2 S 0 β + S 0 2 β 2 . X ″ ( τ ) will be negative if the critical point τ max is satisfied the following conditions.
γ > 4 S 0 2 and β > 2 γ ( − 4 + S 0 2 γ ) 2 + S 0 γ − 4 + S 0 2 γ (15)
Therefore, X has a local maximum value at τ max .
Jacobian Matrix: At the washout steady state solution (7), the system (5) and (6) has the following Jacobian matrix which has the eigenvalues − 1 τ and S 0 1 + S 0 + S 0 2 γ − 1 τ .
J = ( − 1 τ − S 0 ( 1 + S 0 β ) ( 1 + S 0 + S 0 2 γ ) 0 S 0 1 + S 0 + S 0 2 γ − 1 τ ) (16)
Therefore, if τ < 1 S 0 + 1 + S 0 γ then the washout steady state solution will be stable.
Jacobian Matrix: At the washout steady state solution (8), the system (5) and (6) has the following Jacobian matrix
J = ( J 11 J 12 J 21 0 )
where
J 11 = 8 γ 2 ( − 4 S 0 γ 2 + S 0 γ ( − 1 + τ ) ( − 1 + τ − ω ) − 2 γ ( 2 − 3 τ + ω ) ) τ 2 ( 2 γ + β ( − 1 + τ − ω ) ) 2 ( 1 − τ + ω ) 2
+ 8 γ 2 ( ( 1 − 3 τ + 2 τ 2 ) ( 1 − τ + ω ) ) τ 2 ( 2 γ + β ( − 1 + τ − ω ) ) 2 ( 1 − τ + ω ) 2 + 8 β γ ( ( − 1 + τ ) 2 ( − 2 + 5 τ ) ( 1 − τ + ω ) + 2 γ 2 ( − 4 + S 0 ( 4 − 5 τ + 2 ω ) ) ) τ 2 ( 2 γ + β ( − 1 + τ − ω ) ) 2 ( 1 − τ + ω ) 2 + 8 β γ ( γ ( 10 + 19 τ 2 + S 0 ( 2 − 5 τ + 3 τ 2 ) ) ) τ 2 ( 2 γ + β ( − 1 + τ − ω ) ) 2 ( 1 − τ + ω ) 2 + 8 β γ 2 ( ( − 1 + τ − ω ) + 6 ω − τ ( 29 + 9 ω ) ) τ 2 ( 2 γ + β ( − 1 + τ − ω ) ) 2 ( 1 − τ + ω ) 2
+ 8 β 2 ( 4 S 0 γ 3 + ( − 1 + τ ) 3 ( − 1 + 3 τ ) ( 1 − τ + ω ) ) τ 2 ( 2 γ + β ( − 1 + τ − ω ) ) 2 ( 1 − τ + ω ) 2 + 8 β 2 ( − γ ( − 1 + τ ) ( − 6 + 20 τ − 14 τ 2 − 4 ω + 8 τ ω ) ) τ 2 ( 2 γ + β ( − 1 + τ − ω ) ) 2 ( 1 − τ + ω ) 2 + 8 β 2 ( − γ ( − 1 + τ ) ) ( S 0 ( 1 − 3 τ + 2 τ 2 ) ( 1 − τ + ω ) ) τ 2 ( 2 γ + β ( − 1 + τ − ω ) ) 2 ( 1 − τ + ω ) 2 + 8 β 2 ( + γ 2 ( 2 ( 4 − 6 τ + ω ) + S 0 ( − 5 − 8 τ 2 − 3 ω + τ ( 13 + 4 ω ) ) ) ) τ 2 ( 2 γ + β ( − 1 + τ − ω ) ) 2 ( 1 − τ + ω ) 2 (17)
J 12 = − 2 γ τ ( 2 γ + β ( − 1 + τ − ω ) ) (18)
J 21 = ( 4 γ + ( − 1 + τ ) ( 1 − τ + ω ) ) ( β γ ( 2 + S 0 ( − 1 + τ − ω ) ) ) γ τ 2 ( 1 − τ + ω ) 2 + ( γ τ 2 ( 1 − τ + ω ) 2 ) ( β ( − 1 + τ ) ( 1 − τ + ω ) + γ ( 1 + 2 S 0 γ − τ + ω ) ) γ τ 2 ( 1 − τ + ω ) 2 (19)
and ω = − 4 γ + ( − 1 + τ ) 2 .
By following the restrictions (11), (12), (13) and (14), it can be shown that Tr ( J ) = J 11 is negative and Det ( J ) = − J 21 J 12 is positive at the critical point τ max . Hence, we can make a conclusion that the steady state solution (8) is stable at τ max and has practical importance to study.
If τ = 1 + 2 γ or τ = 1 + S 0 + S 0 2 γ S 0 , then D e t ( J ) = J 12 J 21 = 0 and Jacobian matrix has a zero eigenvalue. The required conditions to have double zero eigenvalues of the Jacobian matrix are
Det ( J ) = J 12 J 21 = 0 and Tr ( J ) = J 11 + J 22 = 0.
It can be shown that the conditions to have double zero eigenvalues are satisfied when
S 0 = 1 β + 2 γ and 1 + 2 γ = τ
Here S 0 is positive since β > 0 and γ > 0 . This positivity of S 0 in the feed immediately implies that double zero eigenvalues can occur.
Jacobian Matrix: At the washout steady state solution (9), the system (5) and (6) has the following Jacobian matrix
J = ( J 11 J 12 J 21 0 )
where
J 11 = − 8 γ 2 ( 4 S 0 γ 2 + ( 1 − 3 τ + 2 τ 2 ) ( − 1 + τ + ω ) ) τ 2 ( − 1 + τ + ω ) 2 ( 2 γ + β ( − 1 + τ + ω ) ) 2 + − 8 γ 2 ( γ ( S 0 ( − 1 + τ ) ( − 1 + τ + ω ) + 2 ( − 2 + 3 τ + ω ) ) ) τ 2 ( − 1 + τ + ω ) 2 ( 2 γ + β ( − 1 + τ + ω ) ) 2 − 8 β γ ( ( − 1 + τ ) 2 ( − 2 + 5 τ ) ( − 1 + τ + ω ) ) τ 2 ( − 1 + τ + ω ) 2 ( 2 γ + β ( − 1 + τ + ω ) ) 2 − 8 β γ ( γ ( 10 − 29 τ + 19 τ 2 − 6 ω + 9 τ ω + S 0 ( 2 − 5 τ + 3 τ 2 ) ( − 1 + τ + ω ) ) ) τ 2 ( − 1 + τ + ω ) 2 ( 2 γ + β ( − 1 + τ + ω ) ) 2 − 8 β γ ( + 2 γ 2 ( 4 + S 0 ( − 4 + 5 τ + 2 ω ) ) ) τ 2 ( − 1 + τ + ω ) 2 ( 2 γ + β ( − 1 + τ + ω ) ) 2
+ 8 β 2 ( − 4 S 0 γ 3 + ( − 1 + τ ) 3 ( − 1 + 3 τ ) ( − 1 + τ + ω ) ) τ 2 ( − 1 + τ + ω ) 2 ( 2 γ + β ( − 1 + τ + ω ) ) 2 − 8 β 2 ( − γ ( − 1 + τ ) ( 6 + 14 τ 2 − 4 ω + S 0 ( 1 − 3 τ + 2 τ 2 ) ( − 1 + τ + ω ) ) ) τ 2 ( − 1 + τ + ω ) 2 ( 2 γ + β ( − 1 + τ + ω ) ) 2 + 8 β 2 ( − γ ( − 1 + τ ) 4 τ ( − 5 + 2 ω ) ) τ 2 ( − 1 + τ + ω ) 2 ( 2 γ + β ( − 1 + τ + ω ) ) 2 − 8 β 2 ( γ 2 ( 2 ( − 4 + 6 τ + ω ) + S 0 ( 5 + 8 τ 2 − 3 ω + τ ( − 13 + 4 ω ) ) ) ) τ 2 ( − 1 + τ + ω ) 2 ( 2 γ + β ( − 1 + τ + ω ) ) 2 (20)
J 12 = − 2 γ τ ( 2 γ + β ( − 1 + τ + ω ) ) (21)
J 21 = ( 4 γ − ( − 1 + τ ) ( − 1 + τ + ω ) ) ( γ ( 1 + 2 S 0 γ − τ − ω ) ) γ τ 2 ( − 1 + τ + ω ) 2 − ( 4 γ − ( − 1 + τ ) ( − 1 + τ + ω ) ) β ( − 1 + τ ) ( − 1 + τ + ω ) γ τ 2 ( − 1 + τ + ω ) 2 + ( 4 γ − ( − 1 + τ ) ( − 1 + τ + ω ) ) ( β γ ( 2 + S 0 ( − 1 + τ + ω ) ) ) γ τ 2 ( − 1 + τ + ω ) 2 (22)
and ω = − 4 γ + ( − 1 + τ ) 2 .
By following the condition (15), the Tr ( J ) is never negative and the Det ( J ) is never positive at the critical point. Therefore, the maximum steady state solution (9) is not stable.
If τ = 1 + 2 γ or τ = 1 + S 0 + S 0 2 γ S 0 , then D e t ( J ) = J 12 J 21 = 0 and Jacobian matrix has a zero eigenvalue. But the later value of τ is not relevant. The required conditions to have double zero eigenvalues of the Jacobian matrix are
Det ( J ) = J 12 J 21 = 0 and Tr ( J ) = J 11 + J 22 = 0.
It can be shown that the conditions to have double zero eigenvalues are satisfied when
S 0 = 1 β + 2 γ and 1 + 2 γ = τ .
Here S 0 is positive since β > 0 and γ > 0 . This positivity of S 0 in the feed immediately implies that double zero eigenvalues can occur.
A Hopf bifurcation will take place if J 11 = 0 and J 12 J 21 < 0 . J 12 J 21 will be negative when
τ > 1 + 2 γ and 1 + 2 S 0 γ + γ − 4 γ + ( − 1 + τ ) 2 γ 2 − τ > 0. (23)
These conditions are equivalent to the following three cases. In each case, the values of the residence time correspond to the root of J 11 = 0 at which Hopf bifurcations occur .
Case I: 0 < S 0 ≤ 2 , γ > 1 S 0 2 , and τ ≥ 1 + 2 γ . If we take S 0 = 1.7 , γ = 2 , and β = 5.25 , then J 11 = 0 at τ = 4.20144 (
A Degenerate Hopf bifurcation occurs at a value of residence time τ where two Hopf points resulting in a single point by annihilating each other which is known as H21 degeneracy and it occurs when J 11 = 0 and d J 11 d τ = 0 [
In this case, it is possible to show that when J 11 = 0 , d J 11 d τ vanishes for suitable values of γ and β . For example, if we take γ = 2 , β = 5.25 , then d J 11 d τ vanishes at ( S 0 , τ ) ≈ ( 1.53265,3.90678 ) . Therefore, if the substrate concentration S 0 is adequately small ( S 0 < 1.53265 ) or ( S 0 < 2.68214 g ⋅ l − 1 ) natural oscillations are not possible to occur for γ = 2 , β = 5.25 .
Case II: S 0 > 2 , 1 S 0 2 < γ ≤ 1 4 , and τ ≥ 1 + 2 γ . Let S 0 = 3 , γ = 8 36 , β = 5.25 , then J 11 = 0 at τ = 2.281137 and τ = 2.467646 (
For this situation, it is possible to show that when J 11 = 0 , d J 11 d τ vanishes for appropriate values of γ and β . For example, if γ = 8 36 , β = 5.25 , then d J 11 d τ vanishes at ( S 0 , τ ) ≈ ( 2.97798,2.37231 ) . Therefore, if the substrate concentration S 0 is sufficiently small ( S 0 < 2.97798 ) or ( S 0 < 5.21147 g ⋅ l − 1 ) natural oscillations are not possible for γ = 8 36 , β = 5.25 .
Case III: S 0 > 2 , 4 γ > 1 , and τ ≥ 1 + 2 γ . Let S 0 = 3 , γ = 1 2 , β = 5.25 , then J 11 = 0 at τ = 3.181914 and τ = 2.414296 (
In this case, it is possible to show that when J 11 = 0 , d J 11 d τ vanishes for appropriate values of γ and β . For example, if we take γ = 1 2 , β = 5.25 , then
d J 11 d τ vanishes at ( S 0 , τ ) ≈ ( 2.43946 , 2.68646 ) . Therefore, if the substrate concentration S 0 is small enough ( S 0 < 2.43946 ) or ( S 0 < 4.26906 g ⋅ l − 1 ) natural oscillations are not possible for γ = 1 1 , β = 5.25 .
In all cases above, we have used parameter values K s = 1.75 g − 1 , α = 0.01 , β = 0.031 g − 1 , μ m = 0.3 h − 1 , and β * = 5.25 from [
The work reported in [
We have studied, analytically, a simple chemostat model in a flow reactor with a variable yield coefficient in which the growth rate is taken to be a Haldane expression. In this study, three steady-state solutions have been discussed, which characterize no washout and washout circumstances in the closed photobioreactor. Under suitable parameter values, a stable steady-state solution attains its maximum value. We also corroborated the parameter ranges for the model with variable yield coefficient and growth rate, which describes natural oscillations in the chemostat. In all three cases of Hopf bifurcation analysis, we have found that natural oscillations can be achieved at significantly lower values of the substrate concentrations. In cases I, II, and III, the value is S 0 > 2.68214 g ⋅ l − 1 , S 0 > 5.21147 g ⋅ l − 1 , and S 0 > 4.26906 g ⋅ l − 1 respectively. A previous study reported that natural oscillation is achieved at a substrate concentration S 0 > 6.84203 g ⋅ l − 1 and results claim that natural oscillations are possible for sufficiently large values of substrate concentration ( S 0 > 10 g ⋅ l − 1 ) but the conditions of natural oscillations were not reported.
The authors declare no conflicts of interest regarding the publication of this paper.
Islam, Md.S., Hossain, T. and Islam, M.S. (2022) Dynamics and Hopf Bifurcation Analysis of a Chemostat Model with Modified Growth Rate and Variable Yield Coefficient. Open Journal of Modelling and Simulation, 10, 417-427. https://doi.org/10.4236/ojmsi.2022.104023