J. Appl. Math. Comput. DOI 10.1007/s12190-014-0799-9 ORIGINAL RESEARCH

Nonlinear modelling of ethanol inhibition with the state feedback control Zhong Zhao · Junyi Zhang · Liuyong Pang · Ying Chen

Received: 3 April 2014 © Korean Society for Computational and Applied Mathematics 2014

Abstract Considering the ethanol fermentation is a typical product-inhibiting process, a mathematical model of the ethanol fermentation with the state feedback control is proposed in this paper. The sufficient conditions for existence of the positive period-1 solution and period-2 solution are obtained based on the theory of the impulsive semi-dynamical system and the qualitative properties of the corresponding continuous system. We prove that ethanol fermentation with impulsive state feedback control tends to an order-1 periodic solution or order-2 periodic solution if the control measures are achieved during the fermentation. Furthermore, mathematical results are justified by some numerical simulations. Keywords Feedback control · Ethanol inhibition · Order-1 periodic solution · Order-2 periodic solution Mathematical Subject Classification

34C05 · 92D25

1 Introduction Ethanol as an alternative source of fuel is an important industrial chemical due to high prices and environmental problems caused by fossil fuels [1]. It is a relatively clean burning fuel which can contribute to improving the nation’s air quality and is becoming more and more popular and important for mitigating the current depletion of crude oil This work is supported by the National Natural Science Foundation of China (No. 11371164), NSFC-Talent Training Fund of Henan(U1304104) and the young backbone teachers of Henan (No. 2013GGJS-214). Z. Zhao (B) · J. Zhang · L. Pang · Y. Chen Department of Mathematics, Huanghuai University, Zhumadian 463000, Henan, People’s Republic of China e-mail: [email protected]

123

Z. Zhao et al.

and environmental deterioration [2]. Generally speaking, ethanol production depends mostly on the fermentation of sucrose from sugarcane and molasses or glucose derived from starch-based crops such as corn, wheat, and cassava [3]. Ethanol production from the microbial fermentation has an immense promise for its low cost, ease of operation and no pollution. However, it has been experimentally shown the concentration of extracellular product strongly affects the dynamic behavior of cells, which leads to the incidence of the sustained oscillation and multiplicity under certain conditions [4, 5]. Ethanol productivity decreases during the oscillations, leading to higher level of residue substrate which is subsequently lost from production [6]. In order to achieve the continuous production and reduce the inhibition effect to a reasonable range, it is necessary to keep the ethanol concentration lower than a predetermined threshold. Recently,impulsive state feedback control strategy has widely been introduced into population dynamics and chemostat model by many researches [7–12]. Authors [11] investigate a prey-predator model with Allee effect and state-dependent impulsive harvesting. In [12], chemostat model with impulsive state feedback control is proposed and the conditions for the existence of the order-1 periodic solution are obtained by using the existence criteria of periodic solution of a general planar impulsive autonomous system. However, few papers have discussed the mathematical model of the ethanol fermentation with the impulsive state feedback control. An outline of this paper is as follows: an autonomous system with the impulsive state feedback strategy is introduced into the bioreactor in Sect. 2. In addition, some definitions and existence criteria of the periodic solution for a general planar impulsive autonomous system are also given in Sect. 2. In Sect. 3, the qualitative analysis is given and the existence of order-1 periodic solution is investigated. Finally, we give some numerical simulations and a brief discussion. 2 Model description and preliminaries Many ethanol fermentation processes are the ones inhibited by the end-product, that is, microorganism growth is inhibited by the metabolic product when the concentration of the metabolic product reaches some critical threshold. Incorporating the repression caused by the ethanol concentration during the fermentation process, Li [13] proposed the following model for ethanol fermentation in a chemostat. ⎧ dS μSx 0 ⎪ dt = D(S − S) − δ1 (K s +S)(K +P) , ⎪ ⎪ ⎨ μSx dx dt = (K s +S)(K +P) − Dx, ⎪ ⎪ δ2 μSx ⎪ ⎩ dP = − DP, dt

(2.1)

δ1 (K s +S)(K +P)

where S(t) denotes the concentration of the substrate (or glucose) at time t, x(t) shows the concentration of Z. mobilis at time t, P(t) stands for the ethanol concentration at time t. S 0 is the concentration of the input substrate, D is the washout rate, δ1 and δ2 are the yield coefficients, μ is the maximal specific growth rate of biomass, K s and K are the half-saturation constants.

123

Nonlinear modelling of ethanol inhibition Fig. 1 A Fresh water, B substrate (Glucose), C and F switch, D Biomass indicator, E light source

Since control measures are taken only when the ethanol concentration reaches a predetermined threshold, we will adopt the impulsive state feedback control to model the fermentation process.Based on a continuous culture system of (2.1), the sketch map of the apparatus can be seen in Fig. 1. The apparatus includes an optical sensing device which continuously monitors the ethanol concentration in the bioreactor and two control switches(C and D) connected with a computer. When the ethanol concentration is lower than a predetermined threshold, the switch C is closed and the switch F keeps the flow unchanged. In this case, the ethanol concentration is increasing by consuming the substrate in the bioreactor. Once the ethanol concentration reaches the predetermined threshold, then the switch C is opened and fresh water is flowed into the bioreactor. At the same time, the switch F is enlarged to expand the flow, which can make the ethanol concentration lower than the predetermined threshold. Based on Fig. 1 and system (2.1), we formulate the following ethanol fermentation model with the state feedback control. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

dS dt

= D(S 0 − S) −

dx dt

=

μSx δ1 (K s +S)(K +P) ,

μSx (K s +S)(K +P) − Dx, δ2 μSx δ1 (K s +S)(K ⎫+P) − DP,

dP dt = ⎪ S = −q S, ⎬ ⎪ ⎪ ⎪ ⎪ x = −q x, ⎪ ⎪ ⎭ ⎪ ⎪ P = −q P, ⎪ ⎩ S(0) = S0 , x(0) = x0 , P(0) = P0 ,

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

P < h, (2.2) P = h,

123

Z. Zhao et al.

where q(0 < q < 1) is the dilution rate of the biomass concentration due to the feedback control when the ethanol concentration P reaches the predetermined threshold h (see Fig. 1). Other parameters are the same as ones of system (2.1). We will show the initial value region of system (2.2) can be divided into two parts; one part is the one from which the ethanol concentration of any solution cannot reach h and the trajectory is free from the impulsive effect, while the other is the one from which the ethanol concentration of any solution can reach h, and thus impulsive actions must be implemented so that the ethanol concentration is kept no larger than h. For convenience, we give some definitions and lemmas. Definition 2.1 ([14]) An triple (X, π, R+ ) is said to a semi-dynamical system if X is a metric space, R+ is the set of all non-negative reals and π : X × R+ → X is a continuous function such that (i) π(x, 0) = x for all x ∈ X ; (ii) π(π(π, s), t) = π(x, t + s) for all x ∈ X and t, s ∈ R+ . We denote a semi-dynamical system (X, π, R+ ) by (X, π ). For any x ∈ X , the function πx : R+ → X defined as πx (t) = π(x, t) is continuous and we call πx the trajectory of x. The set C + (x) = {π(x, t)|t ∈ R+ } is called the positive orbit of x. For any subset M of X , we let M + (x) = C + (x)∩ M −{x} and M − (x) = G(x)∩ M −{x}, where G(x) = ∪{G(x, t)|t ∈ R+ } and G(x, t) = {y|π(y, t) = x} is the attainable set of x at t ∈ R+ . Finally, we set M(x) = M + (x) ∪ M − (x). Definition 2.2 ([14]) An impulsive semi-dynamical system (X, π, M, I ) consists of a semi-dynamical system (X, π ) together with a nonempty closed subset M of X and a continuous function I : M → X such that the following properties: (i) No point x ∈ X is a limit point of M(x). (ii) {t|G(x, t) ∩ M = ∅} is a closed subset of R+ . According to the denotations [14], we write N = I (M) = {y ∈ X |y = I (x), x ∈ M and f or any x ∈ X, I (x) = x + }. M is called the impulsive set and the function I is called the impulsive function. Defining a function  : X → R+ ∪ {∞} as follows: (x) =

∞ i f M + (x) = ∅, s i f π(x, t) ∈ M for 0 < t < s and π(x, s) ∈ M,

(2.3)

where s is called the time without impulse of x. i.e. s is the first time when π(x, 0) hits M. Definition 2.3 ([14]) Let (X, π, M, I ) be an impulsive semi-dynamical system and x ∈ X and x ∈ M. The trajectory of x is a function

πx defined on subset [0, s) of R+ (s may be ∞) to X inductively as follows: π (

xn−1 , t) , τn−1 < t < τn ,

πx =

+ + where xn is the sequence of impulsive point of x, which satisfies π(xn−1 , (xn−1 )) = n−1 + xn . τn is the sequence of impulsive time relative to {xn }, τn = k=0 (xk ).

123

Nonlinear modelling of ethanol inhibition

Definition 2.4 ([14]) A trajectory

πx is said to be periodic of period τ and order k if there exist positive integers m ≥ 1 and k ≥ 1 such that k is the smallest integer for m+k−1 + and τ = i=m (xi+ ). xm+ = xm+k Consider the following general autonomous impulsive differential equations: ⎧ ⎪ ⎪ ⎪ ⎨

dx dt dy dt

= P(x, y),



(x, y) ∈ M, = Q(x, y), ⎪ ⎪ x = I1 (x, y), ⎪ ⎩ (x, y) ∈ M, y = I2 (x, y),

(2.4)

where (x, y) ∈ R 2 . P, Q, I1 and I2 are all functions mapping R 2 into R, M ⊂ R 2 is the impulsive set, and we assume: (H2.1) P(x, y) and Q(x, y) are all continuous with respect to x, y ∈ R 2 . (H2.2) M ⊂ R 2 is a set consisting of a line, I1 (x, y) and I2 (x, y) are linear functions of x and y. For each point S(x, y) ∈ M, we define I : R 2 → R 2 : I (S) = (x + , y + ) ∈ R 2 , x + = x + I1 (x, y), y + = y + I2 (x, y). Obviously, N = I (M) is also a linear function of R 2 or a subset of a line and we assume M ∩ N = ∅. From Definition 2.2, we know system (2.4) is an impulsive semidynamical system. The following lemma gives the conditions under which system (2.4) has an order-1 periodic solution by Definition 2.4. Lemma 2.5 ([15] If system (2.4) satisfies assumptions (H2.1) and (H2.2), and there exists a boundedly closed and simply connected region D which has the following properties: (i) There is no singularity in it and the boundary ∂ D satisfies (D − ∂ D) ∩ M = ∅, (ii) L 1 = D ∩ M cannot be tangent with trajectories of (2.4) except at end-points and I (L 1 ) ⊂ D, (iii) Trajectories with initial point in ∂ D − L 1 will enter into the interior of D, then there must exist an order-1 periodic solution in region D. 3 Qualitative analysis for system (2.1) Before discussing the periodic solution of system (2.2), we should consider the qualitative property of (2.2) without the impulsive effect. Lemma 3.1 Suppose ω(t) = (S(t), x(t), P(t)) is a solution of (2.1) subject to ω(0+ ) ≥ 0, then ω(t) ≥ 0 for all t ≥ 0,and further ω(t) > 0, t ≥ 0 if ω(0+ ) > 0. A dynamical system is said to be dissipative if all positive trajectories eventually lie in a bounded set [16].

123

Z. Zhao et al.

Theorem 3.2 The system (2.1) is dissipative. Proof Define a function V1 (t) = δ1 S(t)+ x(t). We calculate the upper right derivative of V1 (t) along a solution of the first and second equations of system (2.1) and get the following differential equation: d V1 = δ1 DS 0 − DV1 (t), dt we have δ1 S(t) + x(t) → δ1 S 0 , t → ∞.

(3.1)

According to the positivity of S(t) and x(t), we obtain S(t) ≤ S 0 , x(t) ≤ δ1 S 0 . Again from third equation, we get 2

δ2 μS 0 dP ≤ − DP. dt δ1 (K s + S 0 )K Thus, P ≤

2  δ2 μS 0 = δ1 (K s +S 0 )D K

M . The proof is completed.

Next, we explore the asymptotical behavior of the system (2.1). From Eq. (3.1), that is, δ1 S(t) + x(t) → δ1 S 0 , t → ∞, we consider the limiting system of system (2.1) as follows: ⎧ 0 ⎨ d S = D(S 0 − S) − μS(S −S) , dt (K s +S)(K +P) (3.2) ⎩ d P = δ2 μS(S 0 −S) − DP, dt (K s +S)(K +P) An equilibrium point of system (3.2) satisfies the system ⎧ ⎨ D(S 0 − S) − ⎩

δ2 μS(S 0 −S) (K s +S)(K +P)

μS(S 0 −S) (K s +S)(K +P)

= 0,

− D P = 0.

(3.3)

It can be seen that system (3.3) has an ethanol-free equilibrium of the form E 0 = (S 0 , 0). We start by analyzing the behavior of the system (3.3) near E 0 . The characteristic equation of the linearization of (3.2) near E 0 is ⎛ det ⎝

μS 0 K (K s +S 0 ) δ2 μS 0 K (K s +S 0 )

− D−λ

0 −D − λ

⎞ ⎠ = 0.

Two eigenvalues are λ1 = −D, λ2 = K (KμS+S 0 ) − D, respectively. We obtain the s ethanol-free equilibrium of system (3.2) is unique and locally asymptotically stable if 0 the condition K (KμS+S 0 ) < D holds. 0

s

123

Nonlinear modelling of ethanol inhibition

Define R1 =

μS 0 . D K (K s + S 0 )

Theorem 3.3 The ethanol-free equilibrium E 0 is asymptotically stable if R1 < 1. E 0 is unstable if R1 > 1. From (3.3), it follows that when the trivial equilibrium E 0 of system (3.2) is asymptotically stable, then positive equilibrium does not exist. When R1 > 1, system (3.2) has a unique positive equilibrium E ∗ (S ∗ , P ∗ ), where ∗

S =

K + δ2 S 0 −

μ D

− K s δ2 + 2δ2

√ 

,

P ∗ = δ2 (S 0 − S ∗ ), μ where  = ( D + K s δ2 − K −δ2 S 0 )2 +4δ2 K s (K +δ2 S 0 ). We can easily obtain S ∗ > 0 ∗ and P > 0 for R1 > 1. Next, we analyze the stability of the positive equilibrium E ∗ (S ∗ , x ∗ ). The characteristic equation at E ∗ (S ∗ , P ∗ ) is

⎛ det ⎝

λ+ D+

μ(K s S 0 −2K s S ∗ −S ∗2 ) (K s +S ∗ )2 (K +P ∗ )

0 ∗ ∗2 s S −2K s S −S ) − δ2 μ(K (K s +S ∗ )2 (K +P ∗ )

∗

0

s

λ+ D+

δ2 μ(S 0 −S ∗ )S ∗ (K s +S ∗ )(K +P ∗ )2

After a few computations, we know λ1 = −D, λ2 = −D − −

δ2 μ(S 0 −S ∗ )S ∗ (K s +S ∗ )(K +P ∗ )2

⎞

∗

(S −S ) − (K μS +S ∗ )(K +P ∗ )2

⎠ = 0.

μ(K s S 0 −2K s S ∗ −S ∗2 ) (K s +S ∗ )2 (K +P ∗ )

< 0 if and only if R1 > 1. Therefore, we have:

Theorem 3.4 As long as R1 > 1 holds, the positive equilibrium of system (3.2) is locally asymptotically stable. Now, let us discuss the global stability of system (3.2). Theorem 3.5 If R1 > 1 holds, then the positive equilibrium (S ∗ , P ∗ ) of system (3.2) is globally asymptotically stable. ) δ2 μS(S −S) Proof Let F1 (S, P) = D(S 0 − S) − (KμS(S−S +P)(K s +S) , F2 (S, P) = (K +P)(K s +S) − D P. From Theorem 3.2, we obtain that ν = {(S, P) ∈ R 2 : 0 < S ≤ S 0 , 0 < P ≤ M } is a positive invariant for system (3.2). Choose a Dulac function B(S, P) = (S 0 − S)−1 . Then B(S, P), F1 (S, P) and F2 (S, P) are continuously differentiable functions on the region υ, and 0

0

μK s δ2 μS ∂(B F1 ) ∂(B F2 ) D + =− − < 0. − 0 ∂S ∂P (K s + S)2 (K + P) (K s + S)(K + P)2 S −S According to the Bendixson–Dulac theorem [17], there is no closed orbit in the region υ. Therefore, the equilibrium E ∗ (S ∗ , P ∗ ) is globally asymptotically stable.

123

Z. Zhao et al.

3.1 Existence of the order-1 periodic solution From system (2.2) and (3.1), we obtain the dynamical behavior of system (2.2) can be determined by the following system: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

dS dt

= D(S 0 − S) −

−S) = (KμS(S − D P, s +S)(K

+P) ⎪ S = −q S, ⎪ ⎪ ⎪ ⎪ x = −q P, ⎪ ⎩ S(0) = S0 , P(0) = P0 , 0

dP dt

⎫

μS(S 0 −S) ⎬ δ1 (K s +S)(K +P) ,

⎭

P < h, (3.4) P = h,

For the initial point which satisfies P(0) < P ∗ (P ∗ is defined in Sect. 3) and if h > P ∗ , then all the solutions of system (3.4) tend to the equilibrium (S ∗ , P ∗ ) and no impulse occurs, which implies the ethanol concentration does not affect the ethanol product. Therefore, control measures should not be taken under the circumstances. In fact, endproduct inhibition was observed in the microorganism as described as in the previous report [18]. So we mainly focus our attention on the case h < P ∗ , P(0) < P ∗ and S(0) ≤ S 0 . The line P = h intersects the isoclinal line ddtP = 0 at the point (SC , h), where SC

−S) 0 0 is satisfied (Kδ2sμS(S +S)(K +h) = Dh and interacts the line S = S at the point B(S , h). The impulsive set M lies on the segment C B. The impulsive set M = C B, C B = {(S, x)|x = h, S A ≤ S ≤ S 0 }. The impulsive functions I1 and I2 map the impulsive set M as N = I (M) = D E, D E = {(S, P) : P = (1 − q)h, (1 − q)S A ≤ S ≤ (1 − q)S 0 }, where D = (S D , (1 − q)h), S D = (1 − q)S A , S E = (1 − q)S 0 . From the third equation of (3.4), we know that S + = (1 − q)S for P = h and furthermore S D = (1 − q)S A . According to the value of S E , we mainly discuss the following cases: 0

√

+(1−q)h)− 1 , Case (1) S D < S E < SG (see Fig. 2a), where SG = μS δ2 −D(1−q)D(K 2μδ2 1 = (D(1 − q)h(K + (1 − q)h) − μδ2 S 0 )2 − 4μδ2 D(1 − q)h(K + (1 − q)h)K s > 0, μS 0 δ2 > D(1 − q)D(K + (1 − q)h. 0

Case (2) SG ≤ S E ≤ SG 1 (see Fig. 2b), where −

μSG 1 + D = 0, (K s + SG 1 )(K + (1 − q)h) D K s (K + (1 − q)h) SG 1 = . μ − D(K + (1 − q)h) √

+(1−q)h)+ 1 Case (3) SG 1 < S E ≤ SG 2 (see Fig. 3a), where SG 2 = μS δ2 −D(1−q)D(K , 2μδ2 0 2 1 = (D(1 − q)h(K + (1 − q)h) − μδ2 S ) − 4μδ2 D(1 − q)h(K + (1 − q)h)K s > 0, μS 0 δ2 > D(1 − q)D(K + (1 − q)h. 0

Case (4) SG 2 < S E < S 0 (see Fig. 3b).

123

Nonlinear modelling of ethanol inhibition

Fig. 2 Existence of the order-1 periodic solution. a S D < S E < SG , b SG < S E < SG 1

Fig. 3 Existence of the order-1 periodic solution. a SG 1 < S E < SG 2 , b SG 2 < S E < S 0

Firstly, we discuss Case 1, S D < S E < SG , the illustration can be seen in Fig. 2a. From the qualitative characteristic of (3.2), it is easily known that all the trajectories of (3.4) starting from the region {P(0) > h, ddtS |(S(0),P(0)) ≤ 0} do not interact the line P = h and the trajectories starting from the region {P(0) < h, ddtS |(S(0),x(0)) ≤ 0} must interact with the segment C B. Theorem 3.6 Suppose h < P ∗ , P(0) < h and R1 > 1, then system (3.4) has an 0 . order-1 periodic solution, where R1 = (K μS +S 0 )K s

Proof We will construct a closed region (see Fig. 2a) in order to use Lemma 2.5. By a simple computation, we obtain that S D = (1 − q)SC < SG , then the straight line l passing through the points S E ((1 − q)S0 , (1 − q)h) and SC (SC , h) satisfies 

l(S, x) = P −

qh (S − SC ) − h. SC − (1 − q)S 0

From Case (1) and Fig. 2a, we have SC > (1 − q)S 0 . The derivative of l(S, x) along the trajectories of (2.3) is qh dS dP dl |(2.3) = − < 0, dt dt SC − (1 − q)S 0 dt

123

Z. Zhao et al.

since ddtS > 0, ddtP < 0 for the segment EC. We know that ddtS < 0, ddtP < 0 for S = S 0 and ddtS = 0 for P = 0. Furthermore, the perpendicular line S = S D = (1 − p)SC interacts the axis P = 0 at the point I and interacts the line ddtP = 0 at the point H . According to the qualitative property of system (3.2), we obtain ddtS | D H > 0, ddtP | D H < 0 and ddtS | H I > 0, ddtP | H I > 0. According to Lemma 2.5, we obtain that system (3.4) has an order-1 periodic solution. Case (2) When SG ≤ S E ≤ SG 1 , (see Fig. 2b), We will construct a closed region   G. C G is a part of the (see Fig. 2b), which is composed of G D, G I , I A, AB, BC, C dP isoclinal line dt = 0. According to the qualitative property of system (3.2), we have dS dP   G > 0, dt |C G < 0, and other segment is the same as the case 1. According to dt |C Lemma 2.5, we can obtain that system (3.4) has an order-1 periodic solution. The proofs of the case (3) and case (4) are similar to case (2), therefore, we omit it.

3.2 The existence of the order-2 periodic solution To discuss the dynamics of system (3.4), we choose two sections X 0 = {(S, P) |S ≥ 0, P = (1 − q)h} and X 1 = {(S, P)|S ≥ 0, P = h} to establish a Poincaré map. Suppose the point Bk (Sk , h) is on the Poincaré section X 1 . Then Bk+ ((1 − q)Sk , (1 − q)h) is on X 0 due to impulsive effect, and the trajectory with the initial point Bk+ intersects the Poincaré section X 1 at the point Bk+1 = (Sk+1 , h), where Sk+1 is decided by Sk and the parameter q. We get the following Poincaré map Sk+1 = F(q, Sk ).

(3.5)

From the dependence of the solutions on the initial conditions, the function is continuous on q and Sk . For each fixed point of the Poincaré map, there is an associated periodic solution of system (3.4), and vice versa. From Theorem 3.6, we know that system (3.4) has an order-1 periodic solution. In this section, we will discuss the existence of the order-2 periodic solution.

Suppose that (

S(t), P(t)) is a periodic solution of system (3.4), then (

S0 , (1 − p)h) ∈ N ⊆ D E and (

S0 , h) ∈ M ⊆ C B. It is easily obtained that

S0 <

S0 due to the impulsive effect. Let (S0 , P0 ) ∈ N ⊆ D E and (S0 , P0 ) ∈ M ⊆ C B. It is also easily obtained that S0 < S0 since S(t + ) = (1 − p)S(t) by third equation of system (3.4). Here, we denote an arbitrary solution of system (3.4) by (S(t), P(t)). The second interaction point of the trajectory and set M(P = h) is denoted as the point (S1 , h). After a series of impulse, the corresponding interaction points of trajectory and set M are (Si , h) (i = 3, 4, . . .). From the Poincaré map (3.5), we have S1 = F(q, S0 ), S2 = F(q, S1 ) and Sn+1 = F(q, Sn )(n = 3, 4, . . .). By the qualitative analysis of system (3.2), we know that ddtP < 0 for the region (S, P) ∈ C J OC and ddtP > 0 for the region (S, P) ∈ C O ABC (see Fig. 2b). So we consider the following cases:

123

Nonlinear modelling of ethanol inhibition

Case 1: If the periodic solution is in the region COABC. The trajectory starting from the point (S0 , (1 − p)h) will interact the segment C B at the point (S0 , h) and then jumps to (S0+ , (1 − p)h). The trajectory starting from the point (S0+ , (1 − p)h) will interact the segment C B at the point (S1 , h). The trajectory starting from the point (S1+ , (1 − p)h) will again interact the segment at the point (S2 , h) and so on. Without S0 . By the qualitative property of the system loss of generality, we suppose that S0 <

(3.2) in the region COABC, one and only one of the following sequences holds: (A) S0 ≤ S1 ≤ S2 ≤ S3 ≤ · · · ≤

S0 , (B)

S0 ≥ S0 ≥ S1 ≥ S2 ≥ S3 ≥ · · · . It is known that the sequences tend to be periodic since the sequences are monotone and ultimately bounded. It follows by Definition 2.4 that order-2 periodic solution does not exist in this case. Case 2: If the periodic solution is in the region CJOC (see Fig. 2b), then we have dP dt < 0. For any two points Bm (Sm , h) and B j (S j , h) in the region CJOC, where Sm < S j . We have Sm+ = (1 − q)Sm and S + j = (1 − q)S j due to impulsive effect. Then it follows from the vector field of system (3.2) that 0 < S j+1 < Sm+1 < 1, that is 0 < S j+1 < Sm+1 < 1 f or 0 < Sm < S j < 1.

(3.6)

From the poincaré map (3.5), we have S1 = F(q, S0 ), S2 = F(q, S1 ), and Sn+1 = F(q, Sn )(n = 3, 4 . . .). 1. If S0 = S1 , then system (3.4) has a positive period-1 solution; 2. If S0 = S1 , without loss of generality, suppose that S1 < S0 . It follows from (3.6) that S2 > S1 . Furthermore, if S2 = S0 , then system (3.4) has a positive period-2 solution. 3. If S0 = S1 = S2 = S3 = · · · Sk−1 (k ≥ 3) and S0 = Sk , then system (3.4) has a positive period-k solution. In fact, it is impossible. If S0 < S1 then from (3.6), we have S1 > S2 and then S2 < S0 < S1 or S0 < S2 < S1 . If S0 > S1 , then from (3.6), we have S1 < S2 and then S1 < S2 < S0 or S1 < S0 < S2 . Thus, the relations of S0 , S1 and S2 is one in the following: S2 < S0 < S1 , S0 < S2 < S1 , S1 < S2 < S0 , S1 < S0 < S2 . (i) S2 < S0 < S1 If S2 < S0 < S1 holds, then from (3.6), we have S3 > S1 > S2 . It is also true that S3 > S1 > S0 > S2 . We again obtain S4 < S2 < S1 < S3 and then S4 < S2 < S0 < S1 < S3 . By means of induction, we have 0 < · · · < S2k < · · · S4 < S2 < S0 < S1 < S3 < S5 < · · · < · · · < S2k+1 < · · · < 1.

(3.7)

Similar to (i), we have

123

Z. Zhao et al.

(ii) S0 < S2 < S1 0 < S0 < S2 < S4 < · · · < S2k < · · · < S2k+1 < · · · < S5 < S3 < S1 < 1.

(3.8)

(iii) S1 < S2 < S0 0 < S1 < S3 < S5 < · · · < S2k+1 < · · · < S2k < · · · < S4 < S2 < S0 < 1.

(3.9)

(iv) S1 < S0 < S2 0 < · · · < S2k+1 < · · · < S5 < S3 < S1 < S0 < S2 < S4 < · · · < S2k < · · · < 1.

(3.10)

If there exists a period-k solution (k ≥ 3) in system (3.4), then we have S0 = S1 = S2 = S3 = · · · Sk−1 , Sk = S0 , which is a contradiction to (3.7)–(3.10). So there exists no period-k solution (k ≥ 3) in system (3.4). In fact, there exists stable period-1 or period-2 solution in this case. It follows from (3.7) that limn→∞ S2k = S0∗ and limn→∞ S2k+1 = S1∗ , where 0 < S0∗ < S1∗ < 1. Thus S1∗ = F(q, S0∗ ) and S0∗ = F(q, S1∗ ). So system (3.4) has a stable period-2 solution in the case (i). Similarly, we obtain that system (3.4) has a stable period-1 solution in the case (ii) and (iii) and has a period-2 solution in the case (iv) 4 Numerical simulations and discussion End-product inhibition during the alcohol fermentation is widely acknowledged in all enzymatic reactions. Therefore, there is a strong economic incentive to develop efficient control strategies that would enable rapid startup and stabilization of steady states in the bioreactors subject to the inhibition of the product concentration. In this paper, we have formulated the mathematical model of ethanol fermentation with the impulsive state feedback control and investigated the existences of order-1 periodic solution and order-2 periodic solution of the ethanol fermentation in a bioreactor. By the existence criteria of periodic solution of a general planar impulsive autonomous system, the conditions for the existence of order-1 and order-2 periodic solutions of the system (3.4) are obtained. Now we will give an numerical analysis on the ethanol inhibition. Firstly, system (3.4) without the state feedback control has two equilibria (S 0 , 0) and (S ∗ , P ∗ ). By the qualitative analysis, we obtain that the freeethanol equilibrium (S 0 , 0) is asymptotically stable if R1 < 1 (R1 is defined above). μS 0 R1 = (K +S 0 )D K < 1 indicates that the ethanol concentration in the bio-reactor is less s

than the wash-out rate, that is, the fermentation ends eventually. In Fig. 4, let S 0 =

123

Nonlinear modelling of ethanol inhibition

Fig. 4 Time-series of the system (3.2) with continuous input and S 0 = 9, μ = 0.08, K s = 0.1, K = 0.1, D = 0.8, δ2 = 0.06, S(0) = 0.01, P(0) = 0.01, R1 = 0.989 < 1

Fig. 5 Time-series of the system (3.2) with continuous input. S 0 = 9, μ = 0.5, K s = 0.1, K = 0.1, D = 0.8, δ2 = 0.06, S(0) = 0.01, P(0) = 0.01, R1 = 6.18 > 1, (S ∗ , P ∗ ) = (1.1105, 0.4733)

9, μ = 0.08, K s = 0.1, K = 0.1, D = 0.8, δ2 = 0.06, S(0) = 0.01, P(0) = 0.01, R1 = 0.989 < 1, we illustrate the ethanol concentration will decrease to zero and the substrate concentration (glucose) will keep constant. From Theorem 3.5, the positive equilibrium (S ∗ , P ∗ ) is globally asymptotically stable if R1 > 1. The globally asymptotical stability of the positive equilibrium (S ∗ , P ∗ ) shows the ethanol concentration will ultimately reach the predetermined threshold and will not again increase even if the substrate concentration is added into the culture vessel. It implies the product concentration has a great inhibition on the product, which is simulated in Fig. 5 with the parameters S 0 = 9, μ = 0.5, K s = 0.1, K = 0.1, D = 0.8, δ2 = 0.06, S(0) = 0.01, P(0) = 0.01, R1 = 6.18 > 1, (S ∗ , P ∗ ) = (1.1105, 0.4733). During the practical industry product, manufacturers always consider how to keep a sustainable and steady output of ethanol by a lower cost. Therefore, from Fig. 5, we should control the ethanol concentration under the predetermined threshold of P ∗ = 0.4733 in order to enhance the ethanol output and decrease the product cost. Secondly, in Theorem 3.6, we prove the existence of the order-1 periodic solution which is also shown in Fig. 6 with the parameters q = 0.6, h = 0.4733, and other parameters are the same as in Fig. 5. In Fig. 6, when the ethanol concentration reaches some predetermined threshold h = 0.4733, fresh water and substrate are added to the culture and an equal volume is removed so that the ethanol concentration is under the predetermined threshold. Therefore we obtain an order-1 periodic solution, which shows that we can control the ethanol concentration so as to achieve a sustainable output and decrease the product cost. In Fig. 7, we shows system (2.2) exists an order-1 periodic solution with the parameters D = 0.1, S 0 = 9.8, δ1 = 3, K = 0.1, δ2 = 0.06, q = 0.6, h = 0.4733, S(0) = 0.1, x(0) = 0.11, P(0) = 0.01.

123

Z. Zhao et al.

Fig. 6 Existence of the order-1 periodic solution with the parameters q = 0.6, h = 0.4733, and other parameters are the same as ones in Fig. 5 4.5

25

4 20

3.5

15

2.5

x

S

3

2

10

1.5 1

5

0.5 0

0

100

200

300

400

0

500

0

100

200

300

t

t

(a)

(b)

400

500

0.5

0.4

0.5

0.35

0.4

0.3

0.3

P

P

0.45

0.25

0.2

0.2

0.1

0.15

0 30

0.1

20

0.05 0

S 0

100

200

300

t

(c)

400

500

10 0

0

1

2

4

3

5

x

(d)

Fig. 7 Existence of the order-1 periodic solution of system (2.2). D = 0.1, S 0 = 9.8, δ1 = 3, K = 0.1, δ2 = 0.06, q = 0.6, h = 0.4733, S(0) = 0.1, x(0) = 0.11, P(0) = 0.01

References 1. Khaw, T.S., Katakura, Y., Ninomiya, K., Moukamnerd, C., Kondo, A., Ueda, M., Shioya, S.: Enhancement of ethanol production by promoting surface contact between starch granules and arming yeast in direct ethanol fermentation. J. Biosci. Bioeng. 103, 95–97 (2007) 2. Honda, H., Taya, M., Kobayashi, T.: Ethanol fermentation associated with solvent extraction using immobilized growing cells of Saccharomyces cerevisiae and its lactose fermentable fusant. J. Chem. Eng. Jpn. 19, 268–273 (1986) 3. Hu, Z., Wen, Z.: Effects of carbon dioxide feeding rate and light intensity on the fed-batch pulse-feeding cultivation of Spirulina platensis in helical photobioreactor. Biochem. Eng. J. 38, 369–378 (2008)

123

Nonlinear modelling of ethanol inhibition 4. Ricci, M., Martini, S., Bonechi, C., Trabalzini, L., Santucci, A., Rossi, C.: Inhibition effects of ethanol on the kinetics of glucose metabolism by S. cerevisiae: NMR and modelling study. Chem. Phys. Lett. 387, 377–382 (2004) 5. Lian, J.G., Zhang, H.K.: Stability of T-periodic solution on the extended simplified brusselator model. Int. J. Biomath. 1, 19–27 (2008) 6. McLellan, P.J., Daugulis, A.J., Li, J.H.: The incidence of oscillatory behavior in the continuous fermentation of Zymomonas mobilis. Biotechnol. Prog. 15, 667–680 (1997) 7. Tang, S.Y., Chen, L.S.: Modelling and analysis of integrated pest management strategy. Discrete Contin. Dyn. Syst. Ser. B 4, 759–768 (2004) 8. Zhu, G.H., Meng, X.Z., Chen, L.S.: The dynamics of a mutual interference age structured predator-prey model with time delay and impulsive perturbations on predators. Appl. Math. Comput. 216, 308–316 (2010) 9. Jiang, G.R., Lu, Q.S., Qian, L.N.: Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos Solitons Fractals 31(2), 448–461 (2007) 10. Zhao, Z., Yang, L., Chen, L.S.: Impulsive state feedback control of the microorganism culture in a turbidostat. J. Math. Chem. 47, 1224–1239 (2010) 11. Wei, C.J., Chen, L.S.: Periodic solution and heteroclinic bifurcation in a predator-prey system with Allee effect and impulsive harvesting. Nonlinear Dyn. 76, 1109–1117 (2014) 12. Guo, H.J., Chen, L.S.: Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control. J. Appl. Math. Comput. 33, 193–208 (2010) 13. Li, C.C.: Mathematical models of ethanol inhibition effects during alcohol fermentation. Nonlinear Anal. Real World Appl. 71, 1608–1619 (2009) 14. Lakshmikantham, V., Bainov, D.D., Simeonov, P.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) 15. Zeng, G.Z.: Existence of periodic solution of order one of state-depended impulsive differential equations and its application in pest control. J. Biomath. 22, 652–660 (2007). (in China) 16. Ball, J.M.: Global attractors for damped semilinear wave equations. Discrete Contin. Dyn. Syst. 10, 31–52 (2014) 17. Zhang, Z.F., Ding, T.R., Huang, Z.W.: Qualitative Theory of Differential Equations, Translations of Mathematical Monographs 101. Amer. Math. Soc, Providence, RI (1992) 18. Luong, J.H.T.: Kinetics of ethanol inhibition in alcohol fermentation. Biotechnol. Bioeng. 27, 180–285 (1985)

123