Optimal Management during the Microorganism Culture Based on the Continuous Purifying Effort

Abstract

This paper deals with the problem of selective harvesting in a chemostat model. Here, we have taken the purifying effort as a dynamic variable and tax as a control instrument. The existence of the possible steady states along with their globally stable equilibrium is discussed. The optimal tax policy is also discussed with the help of Pontryagin's maximum principle. Finally, numerical examples are taken to illustrate some of the key results.

1. Introduction

The chemostat is an important laboratory apparatus used to culture microorganisms [1–4]. Species grow in continuously stirred homogenous fermenters which are fed continuously by a nutrient and the cells are drawn off continuously. Such models have applications in ecology to model biological behavior of a simple lake and in biotechnology to model bioreaction in commercial bioreactors. Predictions based on parameters in the model that can be measured have been tested experimentally and outcomes have shown to agree rather well with the theory. Therefore, it has been extensively used in agriculture and many industrial applications (i.e., pharmaceuticals, nutraceuticals, hydrogen production, and waste treatment).

It is well known that extracting microorganisms has a strong impact on the dynamic behavior of chemostat, and microorganism resources in the chemostat are usually harvested with the purpose of achieving the economic interest. Achieving a high productivity in a bioreactor plays a crucial role in determining the economics of bulk biochemical products such as ethanol. For the purpose of continuously culturing the microorganism and reaching the maximum profit, it is necessary to establish a constructive management of commercial extraction of the microorganism resources. The techniques and issues associated with the bioeconomic exploitation have been discussed in detail by Clark [5]. Taxation and market price usually are considered as possible factors affecting the producers' profit. However, because of the economic flexibility of the Taxation [5], economists are particularly attracted to taxation because a competitive system can be better maintained under taxation rather than other regulatory methods.

Recently, there has been a considerable interest in the modeling of harvesting of biological resources [6–9]. The harvest effort is considered to be a dynamic variable; several kinds of harvesting policies are utilized to study the dynamical behavior of the model system. Furthermore, the optimal harvesting dynamical behavior of a harvested prey-predator model policies with taxation is also discussed. However, to the authors' best knowledge, there is no attempt focusing on discussing the optimal tax policy in the chemostat. In this paper, an extracted chemostat model is established in Section 2. The stability analysis of the equilibria is performed in Section 3. Furthermore, an optimal extracting policy for microorganism is also discussed in Section 4.

2. Model Formulation and Stability of the Equilibria

The general model of continuously culturing microorganism in a chemostat is given by the following form of differential equations [10]: 𝑑𝑆𝑆𝑑𝑡=𝐷⁰−𝑆−𝜇𝑆𝑥𝛿𝐾_𝑠,+𝑆𝑑𝑥=𝑑𝑡𝜇𝑆𝑥𝐾_𝑠+𝑆−𝐷𝑥, (2.1) where 𝑆(𝑡) denotes the concentration of the substrate and 𝑥(𝑡) denotes the concentration of the microorganism in the chemostat at time 𝑡. 𝐷 is the dilution rate of the chemostat. 𝑆⁰ is the concentration of the input substrate. The constant 𝛿 is the yield constant. 𝜇 is called the maximal specific growth rate of the microorganisms. 𝐾_𝑠 is the self-saturation constant. In practice, when the microorganism has been cultured, it should be purified from the vessel. Hence, we suppose 𝐸(𝑡) represents the harvest (or extraction) effort in order to purify the microorganism. To conserve the resource, the regulatory agency imposes a tax 𝜏>0 per unit biomass (𝜏<0 denotes the subsidies given to the harvest (or extraction) effort). Based on the above aspects, the model can be governed by the following differential equations: 𝑑𝑆𝑆𝑑𝑡=𝐷⁰−𝑆−𝜇𝑆𝑥𝛿𝐾_𝑠,+𝑆𝑑𝑥=𝑑𝑡𝜇𝑆𝑥𝐾_𝑠+𝑆−𝐷𝑥−𝐸𝑥,𝑑𝐸,𝑑𝑡=𝑘𝐸𝑝−𝜏𝑥−𝑐 (2.2) where 𝑝 is the price of the unit harvest effort and 𝑐 is the cost of the unit effort. Other parameters are the same as system (2.1).

By simple computation, we obtain that system (2.2) has a trivial equilibrium 𝑃₀(𝑆⁰,0,0) and 𝑃₁(𝐾_𝑠𝐷/(𝜇−𝐷),𝛿(𝑆⁰−𝐾_𝑠𝐷/(𝜇−𝐷)),0), where 𝑃₁ exists if 𝑆⁰>𝐾_𝑠𝐷/(𝜇−𝐷). System (2.2) has a positive equilibrium 𝑃^∗₂(𝑆^∗,𝑥^∗,𝐸^∗) if 𝑆^∗>𝐾_𝑆𝐷/(𝜇−𝐷), where 𝑆^∗=(−(𝜇𝑐/((𝑝−𝜏)𝐷𝛿)+𝐾_𝑠−𝑆⁰)+√Δ)/2,  Δ=(𝜇𝑐/((𝑝−𝜏)𝐷𝛿)+𝐾_𝑠−𝑆⁰)²+4𝐾_𝑠𝑆⁰, 𝑥^∗=𝑐/(𝑝−𝜏), 𝐸^∗=𝜇𝑆^∗/(𝐾_𝑆+𝑆^∗)−𝐷. Owing to 𝜕𝑆^∗/𝜕𝜏<0, there exists a maximum value 𝜏_max such that 0≤𝜏≤𝜏_max. It provides the range of tax for the existence of interior equilibrium, which is of inspiration for people to regulate the extraction effort by means of economic instrument. Furthermore, it is easy to show the positivity and boundedness of solutions of the model system (2.2).

Remark 2.1. It is biologically meaningful to interpret the positivity and boundedness of solutions of the model system (2.2). Since the component (𝑆(𝑡),𝑥(𝑡)) of the solution of system (2.2) represents the relation between the microorganism and substrate, the positivity of solutions reflects the survival of microorganism in the cultured vessel and the boundedness of solutions reveals a natural restriction to growth as a consequence of limited resources. Furthermore, due to the limitation of the microorganism, the extraction effort cannot increase without any restriction.
Next, we begin to analyze the stability of the equilibria.
The characteristic equation of equilibrium 𝑃₀ is (𝜆+𝐷)(𝜆+𝑘𝑐)𝜆+𝐷−𝜇𝑆⁰𝐾_𝑆+𝑆⁰=0; (2.3) obviously, (2.3) has two negative roots 𝜆=−𝐷,  𝜆=−𝑘𝑐. The stability of the equilibrium 𝑃₀(𝑆⁰,0,0) is determined by 𝜆=𝜇𝑆⁰/(𝐾_𝑆+𝑆⁰)−𝐷. Therefore, we have the following theorem.

Theorem 2.2. If 𝜇𝑆⁰/(𝐾_𝑆+𝑆⁰)<𝐷 holds, then the microorganism-free equilibrium 𝑃₀(𝑆⁰,0,0) is stable. It is unstable if 𝜇𝑆⁰/(𝐾_𝑆+𝑆⁰)>𝐷.
The characteristic equation of equilibrium 𝑃₁ is 𝜆²+𝐷+𝜇𝐾_𝑠𝑥_∗𝛿𝐾_𝑆+𝑆_∗𝜆+𝜇𝐾_𝑆𝑥∗𝛿𝐾_𝑠+𝑆_∗³𝜆−𝑘𝑝−𝜏𝑥_∗−𝑐=0. (2.4) The stability of the equilibrium 𝑃₁ is determined by 𝜆²+𝐷+𝜇𝐾_𝑠𝑥_∗𝛿𝐾_𝑆+𝑆_∗𝜆+𝜇𝐾_𝑆𝑥∗𝛿𝐾_𝑠+𝑆_∗³=0, (2.5) and 𝜆=𝑘((𝑝−𝜏)𝑥_∗−𝑐). According to the relation between roots and coefficients, all the eigenvalues of system (2.5) has negative real parts. If 𝜆=𝑘((𝑝−𝜏)𝑥_∗−𝑐)=𝑘((𝑝−𝜏)𝛿(𝑆⁰−𝐾_𝑆𝐷/(𝜇−𝐷))−𝑐)<0, that is, (𝑝−𝜏)𝛿𝑆⁰<(𝑝−𝜏)𝛿𝐾_𝑆𝐷/(𝜇−𝐷)+𝑐, then the equilibrium 𝑃₁ is locally stable.

Theorem 2.3. If (𝑝−𝜏)𝛿𝑆⁰<(𝑝−𝜏)𝛿𝐾_𝑆𝐷/(𝜇−𝐷)+𝑐 holds, then the equilibrium 𝑃₁ is globally asymptotically stable.

Proof. The local stability has been proved above. Next, we prove the attractivity. Construct a Lyapunov function 𝑉(𝑆,𝑥,𝐸)=𝑐₁^𝑆_𝑆∗𝜂−𝑆_∗𝜂𝑑𝜂+𝑐₂^𝑥_𝑥∗𝜂−𝑥_∗𝜂𝑑𝜂+𝑐₃𝐸(𝑡), (2.6) where the nonnegative constants 𝑐_𝑖(𝑖=1,2,3) will be determined later.
We compute the derivative 𝑉(𝑆,𝑥,𝐸) along the system (2.2) 𝑑𝑉=𝑐𝑑𝑡¹𝑆−𝑆_∗𝑆𝑑𝑆+𝑐𝑑𝑡²𝑥−𝑥_∗𝑆𝑑𝑥𝑑𝑡+𝑐₃𝑑𝐸𝑑𝑡=−𝑐₁𝑆−𝑆_∗²𝑆+𝑐₁𝜇𝑆−𝑆_∗𝑆𝐾_𝑠𝑆_∗𝑥_∗−𝑆𝑥−𝑆^∗𝑆𝑥−𝑥_∗𝐾_𝑆+𝑆𝐾_𝑆+𝑆_∗+𝑐₂𝜇𝐾_𝑆𝑆−𝑆_∗𝑥−𝑥_∗𝐾_𝑆+𝑆𝐾_𝑆+𝑆_∗−𝑐₂𝐸𝑥−𝑥_∗+𝑐₃𝑘𝐸𝑝−𝜏𝑥−𝑥_∗+𝑐₃𝑘𝐸𝑝−𝜏𝑥_∗𝑐−𝑐=−¹𝑆−𝑆^∗2𝑆−𝐾_𝑆𝜇𝑥_∗𝑐₁𝑆−𝑆_∗²𝐾_𝑆+𝑆𝐾_𝑆+𝑆_∗−𝜇𝑐₁𝑆−𝑆_∗𝑥−𝑥_∗𝛿𝐾_𝑆+𝑐+𝑆²𝜇𝐾_𝑆𝐾_𝑆+𝑆^∗𝑥−𝑥_∗𝑆−𝑆_∗𝐾_𝑆+𝑆−𝑐₂𝐸𝑥−𝑥_∗+𝑐₃𝑘𝐸𝑝−𝜏𝑥−𝑥_∗+𝑐₃𝑘𝐸𝑝−𝜏𝑥_∗−𝑐; (2.7) let 𝑐₁=𝛿(𝜇−𝐷)/𝜇,𝑐₂=1,𝑐₃=1/𝑘(𝑝−𝜏), we can obtain 𝑑𝑉/𝑑𝑡<0 for 𝑝𝛿𝑆⁰<𝑝𝛿𝐾_𝑆𝐷/(𝜇−𝐷)+𝑐.
Hence, we obtain the equilibrium 𝑃₁ which is globally asymptotically stable for (𝑝−𝜏)𝛿𝑆⁰<(𝑝−𝜏)𝛿𝐾_𝑆𝐷/(𝜇−𝐷)+𝑐.
The proof is completed.

Next, we consider the stability of the positive equilibrium.

Theorem 2.4. If 𝜇𝑆^∗/(𝐾_𝑆+𝑆^∗)>𝐷 holds, the positive equilibrium 𝑃₂(𝑆^∗,𝑥^∗,𝐸^∗) is globally asymptotically stable, where 𝑆^∗=(−(𝜇𝑐/((𝑝−𝜏)𝐷𝛿)+𝐾_𝑠−𝑆⁰)+√Δ)/2, Δ=(𝜇𝑐/((𝑝−𝜏)𝐷𝛿)+𝐾_𝑠−𝑆⁰)²+4𝐾_𝑠𝑆⁰.

Proof. The characteristic equation of equilibrium 𝑃₂ is 𝜆³+𝑎₁𝜆²+𝑎₂𝜆+𝑎₃=0, (2.8) where 𝑎₁=𝐷+𝜇𝐾_𝑆𝑥^∗/𝛿(𝐾_𝑆+𝑆^∗)², 𝑎₂=𝜇²𝐾_𝑆𝑆^∗𝑥^∗/𝛿(𝐾_𝑆+𝑆^∗)³+𝑘(𝑝−𝜏)𝐸^∗𝑥^∗, 𝑎₃=(𝐷+𝜇𝐾_𝑆𝑥^∗/𝛿(𝐾_𝑆+𝑆^∗)²)𝑘(𝑝−𝜏)𝐸^∗𝑥^∗.
According to the Routh-Hurwitz criterion [11], all the roots of the characteristic equation have negative real parts for the above cubic equation and the following criteria 𝑎₁>0,𝑎₁𝑎₂−𝑎₃>0 should be satisfied. Hence, the equilibrium 𝑃₂(𝑆^∗,𝑥^∗,𝐸^∗) is locally asymptotically stable.
Define a function 𝑉(𝑆,𝑥,𝐸)=𝑐₁^𝑆_𝑆∗𝜂−𝑆^∗𝜂𝑑𝜂+𝑐₂^𝑥_𝑥∗𝜂−𝑥^∗𝜂𝑑𝜂+𝑐₃^𝐸_𝐸∗𝜂−𝐸^∗𝜂𝑑𝜂, (2.9) where the nonnegative constants 𝑐_𝑖(𝑖=1,2,3) are positive. We compute the derivative 𝑉(𝑆,𝑥,𝐸) along the system (2.2) 𝑑𝑉=𝑐𝑑𝑡¹𝑆−𝑆^∗𝑆𝑑𝑆+𝑐𝑑𝑡²𝑥−𝑥^∗𝑆𝑑𝑥+𝑐𝑑𝑡³𝐸−𝐸^∗𝐸𝑑𝐸𝑑𝑡=−𝑐₁𝐷𝑆−𝑆^∗2𝑆+𝑐₁𝜇𝑆−𝑆^∗𝑆𝐾_𝑠𝑆^∗𝑥^∗−𝑆𝑥−𝑆^∗𝑆𝑥−𝑥^∗𝐾_𝑆+𝑆𝐾_𝑆+𝑆^∗+𝑐₂𝜇𝐾_𝑆𝑆−𝑆^∗𝑥−𝑥^∗𝐾_𝑆+𝑆𝐾_𝑆+𝑆^∗−𝑐₂𝐸−𝐸^∗𝑥−𝑥^∗+𝑐₃𝑘𝐸−𝐸^∗𝑝−𝜏𝑥−𝑥^∗𝑐=−¹𝐷𝑆−𝑆^∗2𝑆−𝐾_𝑆𝜇𝑥^∗𝑐₁𝑆−𝑆^∗2𝐾_𝑆+𝑆𝐾_𝑆+𝑆^∗−𝜇𝑐₁𝑆−𝑆^∗𝑥−𝑥^∗𝛿𝐾_𝑆+𝑐+𝑆²𝜇𝐾_𝑆𝐾_𝑆+𝑆^∗𝑥−𝑥^∗𝑆−𝑆^∗𝐾_𝑆+𝑆−𝑐₂𝐸−𝐸^∗𝑥−𝑥^∗+𝑐₃𝑘𝐸−𝐸^∗𝑝−𝜏𝑥−𝑥^∗. (2.10) Let 𝑐₁=𝛿𝐾_𝑆/(𝐾_𝑆+𝑆^∗), 𝑐₂=1, 𝑐₃=1/𝑘(𝑝−𝜏), we can obtain 𝑑𝑉/𝑑𝑡<0. Therefore, the positive equilibrium 𝑃₂(𝑆^∗,𝑥^∗,𝐸^∗) is globally asymptotically stable.

3. Optimal Extraction Policy during the Bioprocess

Fermentation technology is a response for producing the majority of bioproducts. Since some substrates of bioprocess are expensive, it is important to optimize the process to maximize the desired products and profits. The objective of the regulatory agency is to maximize the total discounted net revenues that the factory derives from the microorganism fermentation. Symbolically, this objective amounts to maximizing the present value 𝐽 of a continuous time-stream of revenues given by 𝐽=^∞₀𝑒^−𝛿𝑡𝑝𝑥−𝑐𝐸𝑑𝑡, (3.1) where 𝛿 denotes the instantaneous annual rate of discount. 𝑐 is the extraction cost per unit effort. 𝑝 is the price per unit biomass of 𝑥. To solve this optimization problem, we utilize Pontryagin’s maximal principle [12].

We treat 𝜏 as the control variable and wish to determine a suitable effort which maximizes 𝐽 subject to the system (2.2) and the control constraints 𝜏_min≤𝜏(𝑡)≤𝜏_max. (3.2)𝜏_min and 𝜏_max represent a feasible upper and lower limit of tax for the harvest effort, respectively. Specially 𝜏_min<0 implies that subsidies have the effect of increasing the rate of expansion of the extraction.

The Hamiltonian function is given by 𝐻=𝑒^−𝛿𝑡𝑝𝑥−𝑐𝐸+𝜆₁𝐷𝑆⁰−𝐷𝑆−𝜇𝑆𝑥𝛿𝐾_𝑆+𝑆+𝜆₂𝜇𝑆𝑥𝐾_𝑆+𝑆−𝐷𝑥−𝐸𝑥+𝜆₃𝑘𝐸𝑝−𝜏(𝑡)𝑥−𝑐, (3.3) where 𝜆_𝑖(𝑡)(𝑖=1,2,3) are additional unknown functions called the adjoint variables. The Hamiltonian (3.3) must be maximized for 𝜏∈[𝜏_min,𝜏_max]. Assuming that the control constraints are not binding (i.e., the optimal solution does not occur at 𝜏=𝜏_min or 𝜏=𝜏_max). we have a singular control given by 𝜕𝐻/𝜕𝜏=0. Hence, we can obtain 𝜆₃=0.

The adjoint equations are 𝜕𝐻𝜕𝜆₁=−𝜕𝐻𝜕𝑆=𝜆₁𝐾𝐷+^𝑆𝜇𝑥𝛿𝐾_𝑆+𝑆²−𝜆₂𝐾_𝑆𝜇𝑥𝐾_𝑆+𝑆²,𝜕𝐻𝜕𝜆₂=−𝜕𝐻𝜕𝑥=−𝑝𝑒^−𝛿𝑡𝜆𝐸+¹𝜇𝑆𝛿𝐾_𝑆+𝑆−𝜆₂𝜇𝑆𝐾_𝑆,+𝑆−𝐷−𝐸𝜕𝐻𝜕𝜆₃=−𝜕𝐻𝜕𝐸=−𝑒^𝛿𝑡𝑝𝑥−𝑐+𝜆₂𝑥. (3.4)(3.5)(3.6) We obtain from (3.6) that 𝜆₂=𝑒^−𝛿𝑡𝑐𝑝−𝑥. (3.7) In order to obtain an optimal equilibrium solution, by considering the interior equilibrium 𝑃₂, (3.4) can be rewritten as 𝜕𝐻𝜕𝜆₁=𝜆₁𝐾𝐷+^𝑆𝜇𝑥^∗𝛿𝐾_𝑆+𝑆^∗2−𝑒^−𝛿𝑡𝑐𝑝−𝑥^∗𝐾_𝑆𝜇𝑥^∗𝐾_𝑆+𝑆^∗2. (3.8) we can obtain 𝜆₁(𝑡)=𝑝−𝑐/𝑥𝐾_𝑆𝜇𝑥^∗/𝐾_𝑆+𝑆^∗2𝐴𝑒^−𝛿𝑡, (3.9) where 𝐴=𝐷+𝐾_𝑆𝜇𝑥^∗/(𝛿(𝐾_𝑆+𝑆^∗)²). Substituting (3.7) and (3.9) into (3.5), we have 𝛿𝑐𝑝−𝑥^∗=𝑝𝐸^∗−𝑝−𝑐/𝑥^∗𝐾_𝑆𝜇²𝑥^∗𝑆^∗𝛿𝐴𝐾_𝑆+𝑆^∗3+𝑐𝑝−𝑥^∗𝜇𝑆^∗𝐾_𝑆+𝑆^∗−𝐷−𝐸^∗, (3.10) which provides an equation to the singular path and gives the optimal equilibrium levels of microorganism 𝑆^∗=𝑆^∗_𝛿, 𝑥^∗=𝑥^∗_𝛿. Then the optimal equilibrium levels of the harvest effort and tax can be obtained as follows: 𝐸^∗_𝛿=𝜇𝑆^∗_𝛿𝐾_𝑆+𝑆^∗_𝛿−𝐷,𝜏_𝛿𝑐=𝑝−𝑥^∗_𝛿. (3.11) Next, with the help of MATLAB, a simulation work with a hypothetical set of parameters is performed to understand the theoretical results which have been established.

Let the parameters be 𝑆⁰=5, 𝐾_𝑆=0.2, 𝜇=1, 𝑝=3, 𝐷=0.6, 𝛿=0.6, and 𝑐=2. For the system (2.2), the range of the taxation can be obtained 𝜏∈[0,2.267399] in view of the positive equilibrium 𝑃₂(𝑆^∗,𝑥^∗,𝐸^∗). According to the parameters given above, (3.10) can be numerically computed as follows: 3−2.778/(3−𝜏)+𝒜()−𝜏(())2.65−2.77/3−𝜏+𝒜−3.6−2.77/3−𝜏+2.35+𝒜(3−𝜏)(0.6+1/((3−𝜏)(2.65−2.77/(3−𝜏)+𝒜)))(2.65−2.77/(3−𝜏)+𝒜)=0, (3.12) where 𝒜 denotes (1/2)(5.55/(3−𝜏)−4.7)²+6. By solving the above equation, two real roots can be obtained, 𝜏=0.2443807704 and 𝜏=5.204521452. It is obvious that only 𝜏=0.2443807704 satisfies the range 𝜏∈[0,2.267399]. Consequently, the optimal tax is 𝜏=0.2443807704, then the optimal equilibrium levels of the population and harvest effort can be also obtained (𝑆^∗_𝛿,𝑥^∗_𝛿,𝐸^∗_𝛿)=(3.158783855, 0.7257896803, 0.3132643112) (see Figure 1).

(a)

(b)

(c)

(d)

Figure 1 

The positive equilibrium of the optimal purification. (a) Time series of the substrate concentration. (b) Time series of the microorganism concentration. (c) Time series of the purification effort. (d) Phase space trajectories corresponding to the optimal 𝜏=0.2443807704.

4. Discussion

In this paper, a bioeconomic model is established to investigate the effects of the harvest effort on the dynamic behavior of the chemostat. In Theorem 2.2, we obtain the microorganism-free equilibrium 𝑃₀(𝑆⁰,0,0) is stable if 𝜇𝑆⁰/(𝐾_𝑆+𝑆⁰)<𝐷 holds, which is simulated in Figure 2. Theorem 2.3 shows that the equilibrium 𝑃₁ is globally asymptotically stable, if (𝑝−𝜏)𝛿𝑆⁰<((𝑝−𝜏)𝛿𝐾_𝑆𝐷)/(𝜇−𝐷)+𝑐 holds (see Figure 3). The existence and global stability of the positive equilibrium is proven in Theorem 2.4.

(a)

(b)

(c)

(d)

Figure 2 

The dynamic behavior of the microorganism-free equilibrium with the parameters 𝐷=0.4, 𝑆0=0.3, 𝛿=1, 𝐾𝑠=0.9, 𝑐=1, 𝑘=0.1, and 𝑝=0.3. (a) Time series of the substrate concentration. (b) Time series of the microorganism extinction. (c) Time series of the effort loss. (d) Phase space trajectories of the microorganism-free equilibrium.

(a)

(b)

(c)

(d)

Figure 3 

The dynamic behavior of the effort-free equilibrium with the parameters 𝐷=0.4, 𝑆0=2, 𝛿=1, 𝐾𝑠=0.5, 𝑐=3, 𝑘=0.3, and 𝑝=3. (a) Time series of the substrate concentration. (b) Time series of the microorganism concentration. (c) Time series of the effort loss. (d) Phase space trajectories of the effort-free equilibrium.

Nowadays, the biological resources in the chemostat model are mostly harvested with the aim of achieving economic interest and the taxation is used as an economic control instrument to protect the resources from overexploitation, which motivates the introduction of the harvest effort and tax into the proposed model. The application of the control theory enabled us to show the existence of a unique optimal equilibrium point which is stable. These results can be used as a microorganism culture such as ethanol fermentation and lactic acid fermentation to obtain a more economic profit.

Acknowledgment

This work is supported by the innovation fund for technology based firms (12C26213313100) and innovation fund for Ningbo based firms (2011B710010).