_{1}

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Current research is concerned with the stability of stochastic logistic equation with Ornstein-Uhlenbeck process. First, this research proves that the stochastic logistic model with Ornstein-Uhlenbeck process has a positive solution. After that, it also introduces the sufficient conditions for stochastically stability of stochastic logistic model for cell growth of microorganism in fermentation process for positive equilibrium point by using Lyapunov function. In addition, this research establishes the sufficient conditions for zero solution as mentioned in
Appendix A due to the cell growth of microorganism
*μ*
_{max}, which cannot be negative in fermentation process. Furthermore, for numerical simulation, current research uses the 4-stage stochastic Runge-Kutta (SRK4) method to show the reality of the results.

Stochastic differential equations (SDEs) have been intensively used to model the natural phenomena in the last decades and these equations play a prominent applied role in various fields [

This research deals with the stability of stochastic logistic model with Ornstein-Uhlenbeck process. This process is a batter model of Brownian motion ( [

The classical Brownian motion was introduced by Scottish Botanist Robert Brown in (1827). He described this motion based on random movements of pollen grains in liquid or gas [

Fermentation process converts sugar into alcohol with the help of yeast [

This paper is organized in five main sections; Introduction, Preliminaries and Models Description, Main Results, Numerical Simulation and Conclusion.

Throughout this paper; the notation E [ x ( t ) ] is the expectation of ( x ( t ) ) , T shows the terminal time, t is time, t 0 illustrates the initial time, x ( t ) corresponds to the highest cell size, x 0 is initial cell size, μ max denotes the maximum specific growth rate, x max illustrates a carrying capacity, σ indicates the random fluctuation, W ( t ) shows Brownian motion and b is a constant.

The simplest mathematical model to illustrate the exponential phase for cell growth in fermentation process is:

d x ( t ) d t = μ max × x ( t ) . (1)

The solution of Equation (1) is:

x ( t ) = x 0 e t × μ max , (2)

where t is time, x 0 corresponds to the initial cell size and μ max denotes the maximum specific growth rate. If μ max > 0 , Equation (1) is strongly ascending, and if, μ max < 0 , strongly descending. Hence, Equation (1) is not adequate to model stationary phase in fermentation. Therefore, Pierre Francois Verhulst (1838) introduced new model containing stationary phase in fermentation process. Thus, the exponential growth model (1) is augmented by the inclusion of multiplicative

factor of 1 − x ( t ) x max . Hence, the logistic ordinary differential equation is:

d x ( t ) = μ max ( 1 − x ( t ) x max ) x ( t ) d t , (3)

where x max is a carrying capacity of a microbial species and Equation (3) can be solved analytically by determining the solution is:

x ( t ) = x max x 0 e μ max t x max + x 0 ( − 1 + e μ max t ) , (4)

where x ( t ) is the highest cell size. Madihah, in [

ρ → ρ + σ d W ( t ) d t , (5)

where ρ = μ max x max is diffusion coefficient. Model (3) with perturbation (5) is:

d x ( t ) = μ max ( 1 − x ( t ) x max ) x ( t ) d t + σ x 2 ( t ) d W ( t ) , t ∈ [ 0 , T ] , (6)

where W ( t ) is m-dimensional Weiner process and σ is corresponding to the random fluctuation. Bazli, in [

Remarks 2.1: Liu et al., in [

y ″ ( t ) = − b y ′ ( t ) + d W ( t ) (7)

where y ( t ) stands for positive Brownian motion at time t, b > 0 shows the coefficient friction, σ indicates the diffusion coefficient and W ( t ) is white noise.

By substituting

y ′ ( t ) = W ( t ) , (8)

into Equation (7) hence, the Ornstein-Uhlenbeck process becomes;

d W ( t ) = − b W ( t ) d t + σ d W ( t ) . (9)

Substituting Equation (9) into Equation (6) yields

d x ( t ) = μ max ( 1 − x ( t ) x max ) x ( t ) d t − σ x 2 ( t ) b W ( t ) d t + x 2 ( t ) σ 2 d W ( t ) . (10)

Equation (10) is stochastic logistic model with Ornstein-Uhlenbeck process. □

First, it is needed to prove that Equation (10) has a unique positive solution and later on we will pay attention on stability.

Theorem 3.1: For all t ≥ 0 and for any 0 < x ( 0 ) = x 0 Equation (10) has a unique positive solution.

Proof: For any given initial value x 0 ∈ R + n , the coefficients of Equation (10) are locally Lipschitz continuous. Hence, there is a unique locally solution x ( t ) , t ∈ [ 0, i e ) Where i e shows the explosion time Arnold in [

really necessary to determine that i e → ∞ . Let k 0 ∈ [ 1 k 0 , k 0 ] , k 0 > 0 and k > k 0

is sufficiently large for each component of x 0 , the stopping time for every integer k > k 0 is:

t k = inf { t ∈ [ 0 , i e ) : x ( t ) is not in ( 1 k , k ) } .

k 0 is sufficiently large means that the t k is increasing as k → ∞ . Hence, t ∞ = lim k → ∞ t k , whereas t ∞ ≤ i e .

If we can show that i ∞ = ∞ a.s., then i ∞ = ∞ a.s. and x ( t ) ∈ R + n a.s. for all t ≥ 0 . In other words, to complete the proof all we need to show is that i ∞ = ∞ a.s. For if this statement is false, then there is a pair of constants T > 0 and ϵ ∈ ( 0,1 ) such that

P { i ∞ ≤ T } ≥ ϵ .

Therefore, there exist an integer k 1 ≥ k 0

P { i ∞ ≤ T } ≥ ϵ , ∀ k ≥ k 1 (11)

Define V ( x ( t ) ) = ( x ( t ) − 1 − 1 2 ln x ( t ) ) for all x ( t ) ∈ [ 0, i e ) and the nonnegativity of this function can be seen from

u − 1 − 1 2 ln u

Itô ( [

d ( V ( x ( t ) ) ) = 1 2 x ( t ) − 1 2 x ( t ) [ μ max ( 1 − x ( t ) x max ) x ( t ) d t ] ︸ A − [ σ x 2 ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ] ︸ A + 0.5 [ 0.25 x 3 ( t ) − 1 2 x 2 ( t ) ( μ max ( 1 − x ( t ) x max ) x ( t ) d t ) ︸ B − σ x 2 ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ] 2 ︸ B (12)

To simplify A and B separately yields

A = 1 2 x ( t ) μ max ( 1 − x ( t ) x max ) x ( t ) d t − 1 2 x ( t ) σ x 2 ( t ) b W ( t ) d t + 1 2 x ( t ) σ 2 x 2 ( t ) d W ( t ) − 1 2 x ( t ) μ max ( 1 − x ( t ) x max ) x ( t ) d t + 1 2 x ( t ) σ x 2 ( t ) b W ( t ) d t − 1 2 x ( t ) σ 2 x 2 ( t ) d W (t)

B = 0.25 x 3 ( t ) − 1 2 x 2 ( t ) [ ( μ max ( 1 − x ( t ) x max ) x ( t ) d t ) 2 + 2 ( μ max ( 1 − x ( t ) x max ) x ( t ) d t ) ( − σ x 2 ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ) + ( − σ x 2 ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ) 2 ]

B = 0.25 x 3 ( t ) − 1 2 x 2 ( t ) [ ( μ max 2 x 2 ( t ) d t × d t − 2 μ max 2 x max x 2 ( t ) d t × d t + ( μ max x max x ( t ) d t ) 2 ) + 2 ( μ max x ( t ) d t − μ max x max x 2 ( t ) d t ) × ( − σ x 2 ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ) + ( − σ x 2 ( t ) b W ( t ) d t ) 2 + 2 ( − σ x 2 ( t ) b W ( t ) d t ) ( σ 2 x 2 ( t ) d W ( t ) ) + ( σ 2 x 2 ( t ) d W ( t ) ) 2 ]

An application of these facts d t × d t = 0 , d t × d w ( t ) = 0 and d W ( t ) × d W ( t ) = d t ( [

B = 0.125 x 3 ( t ) − 0.25 x 2 ( t ) ( σ 4 x 4 ( t ) d t )

⇒ B = 0.125 σ 4 x 4 ( t ) d t x 3 ( t ) − 0.25 σ 4 x 4 ( t ) d t x 2 (t)

By substituting the value of A and B into Equation (12) yields

d ( V ( x ( t ) ) ) = [ 1 2 x ( t ) μ max ( 1 − x ( t ) x max ) x ( t ) d t − 1 2 x ( t ) σ x 2 ( t ) b W ( t ) d t + 1 2 x ( t ) σ 2 x 2 ( t ) d W ( t ) − 1 2 x ( t ) μ max ( 1 − x ( t ) x max ) d t + 0.5 σ x ( t ) b W ( t ) d t − 0.5 σ 2 x ( t ) d W ( t ) ] + 0.125 σ 4 x 4 ( t ) d t x 3 ( t ) − 0.25 σ 4 x 4 ( t ) d t x 2 ( t ) (13)

d ( V ( x ( t ) ) ) = [ 0.5 x 0.5 ( t ) μ max ( 1 − x ( t ) x max ) − 0.5 x − 0.5 ( t ) μ max ( 1 − x ( t ) x max ) − 0.5 σ x ( t ) b W ( t ) + 0.5 σ x ( t ) b W ( t ) + 0.125 σ 4 x 2.5 ( t ) − 0.125 σ 4 x 2 ( t ) ] d t − 0.5 σ 2 x 2 ( t ) d W ( t ) d t − 0.5 σ 2 x ( t ) d W ( t ) (14)

d ( V ( x ( t ) ) ) = [ 0.5 x 0.5 ( t ) μ max ( 1 − x ( t ) x max ) − 0.5 x − 0.5 ( t ) μ max ( 1 − x ( t ) x max ) + 0.5 σ x ( t ) b W ( t ) ( 1 − x 0.5 ( t ) ( t ) ) + 0.25 σ 4 x 2 ( 0.5 x ( t ) − 1 ) ] d t + 0.5 σ 2 x ( t ) d W ( t ) ( x 0.5 ( t ) − 1 ) (15)

Worth mentioning,

0.5 x 0.5 ( t ) μ max ( 1 − x ( t ) x max ) − 0.5 x − 0.5 ( t ) μ max ( 1 − x ( t ) x max ) + 0.25 σ 4 x 2 ( 0.5 x ( t ) − 1 ) (16)

≤ 0.5 x 0.5 ( t ) μ max ∨ ( 1 − x ( t ) x max ) − 0.5 x − 0.5 ( t ) μ max ∨ ( 1 − x ( t ) x max ) + 0.25 sup t ∈ R + | σ 4 | x 2 ( t ) ( 0.5 x ( t ) − 1 ) . (17)

If x ( t ) ≤ x max then ( 1 − x ( t ) x max ) ≥ 0 , it follows the boundedness σ . Therefore, there is nonnegative number Q 1 which independent of x and t.

If 0 < x ( t ) < x max , obviously there is a positive number Q 2 which is independent of x and t, and it follow bounded σ , such that

0.5 x 0.5 ( t ) μ max ( 1 − x ( t ) x max ) − 0.5 x − 0.5 ( t ) μ max ( 1 − x ( t ) x max ) + 0.25 σ 4 x 2 ( 0.5 x ( t ) − 1 ) ≤ 0.5 x 0.5 ( t ) μ ^ max ( 1 − x ( t ) x max ) − 0.5 x − 0.5 ( t ) μ ^ max ( 1 − x ( t ) x max ) + 0.25 sup t ∈ R + | σ 4 | x 2 ( 0.5 x ( t ) − 1 ) ≤ Q 2 . (18)

On the other hand, we show that there is a nonnegative number Q which independent of x and t such that

0.5 x 0.5 ( t ) μ max ( 1 − x ( t ) x max ) − 0.5 x − 0.5 ( t ) − μ max ( 1 − x ( t ) x max ) + 0.25 σ 4 x 2 ( 0.5 x ( t ) − 1 ) ≤ Q . (19)

By substituting the inequality (19) into Equation (13) yields

d ( V ( x ( t ) ) ) ≤ Q d t + 0.5 σ x ( t ) b W ( t ) ( 1 − x 0.5 ( t ) ( t ) ) d t + 0.5 σ 2 x ( t ) d W ( t ) ( x 0.5 ( t ) − 1 ) . (20)

where W ( t ) is Brownian motion, and taking integral and expectation

E ( V ( x ( i n ∧ T ) ) ) ≤ V ( x ( 0 ) ) + Q E ( i n ∧ T ) ≤ V ( x ( 0 ) ) + Q T . (21)

By taking into account, the inequality (11) then we have

P ( i n ≤ T ) ≥ ϵ .

For every ϵ ∈ ( i n ≤ T ) , x ( i n , ω ) = n or ( i , ω ) = 1 n . Therefore, V ( x ( t ) ) is no less than either

min { k − 1 − 1 2 ln ( k ) , 1 k − 1 + 1 2 ln ( k ) } .

Based on (20) we have

V ( x ( 0 ) ) + Q T ≥ E [ { i n ≤ T } ( ω ) V ( x ( i n ) ) ] ≥ ϵ min { k − 1 − 1 2 ln ( k ) , 1 k − 1 + 1 2 ln ( k ) } .

where { i n ≤ T } ( ω ) illustrates the function { i n ≤ T } . Letting n tends to infinity leads to dissidence

∞ > V ( x ( 0 ) ) + Q T = ∞

Therefore, we need to have i ∞ = ∞ . Eventually, our claim is proved truly. □

In the next section we prove that Equation (10) with Ornstein-Uhlenbeck process is globally asymptotically stable in positive solution which illustrates the highly randomness in stationary phase.

Theorem 3.2: Equation (10) is globally asymptotically stable in positive equilibrium points x max , under following assumptions a.s.

H1: For

1) 0 ≤ t 0 , 0 < x 0 < σ < μ max < b < x max

2) 0 ≤ t 0 , 0 < x 0 < σ < b < x max < μ max

and the

lim sup t → ∞ ∫ 0 t x ( s ) d s t = b ∨ and lim inf t → ∞ ∫ 0 t x ( s ) d s t = b ^ .

where, b ∨ and b ^ are positive constant then the solution x ( t ) satisfies the lim t → ∞ x ( t ) = x max .

Proof: An application Itô’s formula ( [

x ( t ) = x max 1 + [ x 0 exp ( σ b ∫ 0 t x ( s ) W ( s ) d s + σ 2 ∫ 0 t x ( s ) d W ( s ) ) × 1 exp ( − μ max x max ∫ 0 t x ( s ) d s − σ 4 2 ∫ 0 t x 2 ( s ) d s − μ max ∫ 0 t d s ) ( x max x 0 − 1 ) ] (22)

For the convenience we set the denominator of Equation (22) to Q and according to the ( [

Q = 1 + e ( − ∫ 0 t μ max d s − μ max x max ∫ 0 t x ( s ) d s − ∫ 0 t σ b W ( s ) d W ( s ) × ∫ 0 t x ( s ) d W ( s ) ) × e ( σ 2 ∫ 0 t x ( s ) d W ( s ) − 1 2 σ 4 ∫ 0 t x 2 ( s ) d s ) .

Taking limit from both sides of above equation yields:

lim t → ∞ Q = 1 + lim t → ∞ e ( − t ) ( ∫ 0 t μ max d s t + μ max x max ∫ 0 t x ( s ) d s t + ∫ 0 t σ b W ( s ) d W ( s ) t ∫ 0 t x ( s ) d W ( s ) t × lim t → ∞ e ( − t ) − σ 2 ∫ 0 t x ( s ) d W ( s ) t + 1 2 σ 4 ∫ 0 t x 2 ( s ) d s t ) . (23)

where,

∫ 0 t x ( s ) d W ( s ) = M ( t ) (24)

is a Martingale with quadratic variation

〈 M ( t ) , M ( t ) 〉 = ∫ 0 t x 2 ( s ) d ( s ) (25)

Based on ( [

lim t → ∞ sup M ( t ) t = 0 a . s . (26)

and

lim t → ∞ sup 〈 M ( t ) , M ( t ) 〉 t = C a . s . (27)

where C ∈ R + then Equation (23) becomes

lim t → ∞ Q = 1 + lim t → ∞ e ( − t ) ( ∫ 0 t μ max d s t + μ max x max ∫ 0 t x ( s ) d s t + 1 2 σ 4 C ) (28)

lim t → ∞ Q = 1 + lim t → ∞ e ( − t ) ( ∫ 0 t μ max d s t + μ max x max ∫ 0 t x ( s ) d s t + 1 2 σ 4 C ) = 0 a . s . (29)

By substituting Equation (29) into Equation (22) we have:

lim t → ∞ x ( t ) = lim t → ∞ x max

⇒ lim t → ∞ x ( t ) = x max .

Ultimately, prove is completed. □

Remark 3.1: Liu et al., in [

In this section we consider a strong and accurate numerical method (SRK4) to elaborate the analytical results ( [

f ¯ ( t , x ( t ) ) = f ( t , x ( t ) ) − 1 2 g ( t , x ( t ) ) ∂ g ∂ x ( t , x ( t ) ) (30)

Hence,

d x ( t ) = [ μ max ( 1 − x ( t ) x max ) x ( t ) − σ x 2 ( t ) b W ( t ) ] d t − σ 4 x 3 ( t ) ∘ d W ( t ) (31)

where ∘ is used to denote the Stratonovich form of SDE (i.e. ∘ d W ( t ) . Equation (10) and Equation (31) present some solution under different approach. We use the SRK4 method for numerical approximation. This method was introduced by Rümellin [

process, ∫ t n t n + 1 ∘ d W ( t ) . Furthermore, Xiao in [

stochastic Runge-Kutta methods for stochastic differential equations in case of Stratonovich with scalar noise. Since our model is in Stratonovich sense.

The blue line indicates the simple path of Equation (10) and the red line shows the simple path of Equation (3) respectively. In

In both

Remark 4.1: The red line shows the simple path of Equation (3) and the blue line indicates the simple path of Equation (10) respectively. If σ → ± ∞ , there is no phases for cell growth which means this random fluctuations will destroy the phases in fermentation process on the other hands the process will tend to infinity.

Remark 4.2: Figures 1-4 show the stability of Equation (3) and Equation (10) in different equilibrium points.

This research is conducted on stability of stochastic logistic model with Ornstein-Uhlenbeck process for cell growth of microorganism in fermentation.

This research proved that Equation (10) has a unique positive solution (see Theorem 3.1). Moreover, we proved that Equation (10) was stochastically stable in zero solution and positive equilibrium point (see Theorem 3.2 and Appendix A Theorem 1) and viewed in

This work is jointly supported by the National Natural Science Foundation of China under Grant Nos. 61573291, the Fundamental Research Funds for Central Universities XDJK2016B036.

The author declares no conflicts of interest regarding the publication of this paper.

Ayoubi, T. (2019) Stability of Stochastic Logistic Model with Ornstein-Uhlenbeck Process for Cell Growth of Microorganism in Fermentation Process. Applied Mathematics, 10, 659-675. https://doi.org/10.4236/am.2019.108047

Theorem 1: Equation (10) under the following assumption is globally asymptotically stable in zero solution a.s (almost surely).

(H2): For 0 ≤ t 0 , μ max < 0 < x 0 < σ < x max , then the lim t → ∞ x ( t ) = 0 .

Proof: An application of Itô’s formula for Equation (10) ( [

d ( ln | x ( t ) | ) = 1 x ( t ) [ μ max ( 1 − x ( t ) x max ) x ( t ) d t − σ x 2 ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ] ︸ A + − 1 2 x 2 ( t ) [ μ max ( 1 − x ( t ) x max ) x ( t ) d t − σ x 2 ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ] 2 ︸ B (32)

Equation (32) has two parts A and B.

So,

A = μ max ( 1 − x ( t ) x max ) x ( t ) d t − σ x 2 ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W (t)

and

B = [ μ max ( 1 − x ( t ) x max ) x ( t ) d t − σ x 2 ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ] 2 .

To simplify the B we have

B = [ μ max ( 1 − x ( t ) x max ) d t − σ x ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ] 2

B = ( μ max ( 1 − x ( t ) x max ) d t ) 2 + 2 ( μ max ( 1 − x ( t ) x max ) d t ) × ( − σ x ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ) + ( − σ x ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ) 2

B = ( μ max 2 d t × d t − 2 μ max x max x 2 ( t ) d t × d t + ( x ( t ) x max d t ) 2 ) + 2 ( μ max d t − μ max x max x 2 ( t ) d t ) × ( − σ x ( t ) b W ( t ) d t + σ 2 x ( t ) d W ( t ) ) + ( − σ x ( t ) b W ( t ) d t + σ 2 x ( t ) d W ( t ) ) 2 + ( − σ x ( t ) b W ( t ) d t ) 2 + 2 ( − σ x ( t ) b W ( t ) d t ) ( σ 2 x ( t ) d W ( t ) ) + ( σ 2 x 2 ( t ) d W ( t ) ) 2

Using these facts d t × d t = 0 , d t × d w ( t ) = 0 and d W ( t ) × d W ( t ) = d t ( [

B = σ 4 x 4 ( t ) d t .

By substituting the value of A and B into Equation (32) yields:

d ( ln | x ( t ) | ) = 1 x ( t ) [ μ max ( 1 − x ( t ) x max ) x ( t ) d t − σ x 2 ( t ) b W ( t ) d t + σ 2 x 2 ( t ) d W ( t ) ] + − 1 2 x ( t ) 2 ( σ 4 x 4 ( t ) d t ) (33)

d ( ln | x ( t ) | ) = [ μ max ( 1 − x ( t ) x max ) d t − σ x ( t ) b W ( t ) d t + σ 2 x ( t ) d W ( t ) ] − 1 2 ( σ 4 x 2 ( t ) d t ) (34)

x ( t ) = e μ max ∫ 0 t ( 1 − x ( s ) x max ) d s − σ b ∫ 0 t x ( s ) W ( s ) d s + σ 2 ∫ 0 t x ( s ) d W ( s ) − 1 2 σ 4 ∫ 0 t x 2 ( s ) d s (35)

where W ( t ) is Brownian motion and taking expectation from both sides of above equation. We know the expectation of Brownian motion is zero ( [

E [ x ( t ) ] = E [ exp ( − μ max x max ∫ 0 t x ( s ) d s − σ 4 2 ∫ 0 t x 2 ( s ) d s + μ max ∫ 0 t d s ) ] (36)

Takes limit

lim t → ∞ E [ x ( t ) ] = lim t → ∞ E [ exp ( − μ max x max ∫ 0 t x ( s ) d s − σ 4 2 ∫ 0 t x 2 ( s ) d s + μ max ∫ 0 t d s ) ] (37)

lim t → ∞ E [ x ( t ) ] = lim t → ∞ E [ exp ( − μ max x max ∫ 0 t x ( s ) d s t − σ 4 2 ∫ 0 t x 2 ( s ) d s t + μ max ∫ 0 t d s t ) ] (38)

For 0 ≤ t ≤ T and based on ( [

lim t → ∞ E [ e t ( μ max ∫ 0 t d s t ) ] . (39)

Under H2 yields

lim t → ∞ E [ e t ( μ max ∫ 0 t d s t ) ] = 0. (40)

Therefore, Equation (38) becomes

lim t → ∞ x ( t ) = 0.

Ultimately, our claim is proved. □

For plotting

Note: The thick red line shows the stability of Equation (10) without Ornstein-Uhlenbeck process. It is thick due to the growth rate μ max , is increased, compare the value of μ max with