A Note on Stability of Stochastic Logistic Model by Incorporating the Ornstein-Uhlenbeck Process

In this research, we first prove that the stochastic logistic model (10) has a positive global solution. Subsequently, we introduce the sufficient conditions for the stochastically stability of the general form of stochastic differential equations (SDEs) in terms of equation (1), for zero solution by using the Lyapunov function. This result is verified via several examples in Appendix A. Besides; we prove that the stochastic logistic model, by incorporating the Ornstein-Uhlenbeck process is stable in zero solution. Furthermore, the simulated results are displayed via the 4-stage stochastic Runge-Kutta (SRK4) numerical method.


Introduction
Stochastic differential equations (SDEs) have been intensively used to model the natural phenomena in last decades and these equations play a prominent applied role in various fields such as: Finance (Delong, 2013), Chemical (Coffey & Kalmykov, 2012), Biology (Wilkinson, 2011), Neural network (Zhang, et al., 2018), Prostate cancer (Assia & Wendi 2019) and so on. Most SDEs do not have explicit solutions. Nevertheless, these equations can be solved numerically (Ayoubi, 2019;Bahar and Mao, 2004;Burrage and Burrage, 1996;Xiao & Tang, 2016). We use SRK4 method to approximate the solutions numerically. Whereas, Rüemelin (1982) proposed the S-stage stochastic Runge-Kutta explicit method which was based on Brownian motion. Likewise, Xiao and Tang (2016) introduced the High strong order stochastic Runge-Kutta methods for stochastic differential equations in case of Stratonovich hence our model is in Stratonovich sense. The classical Brownian motion introduced by Scottish Botanist Robert Brown in (1827), describes this motion based on random movements of pollen grains in liquid or gas. However, Brown did not solve the problem by himself (Mishura & Mishura, 2008). Norbert Wiener (1923) explained full mathematical theory of Brownian motion which existed as a rigorously defined mathematical objective to recognize his contribution. Brownian motion is a simple continuous stochastic process which is extensively used to model phenomena in various fields such as industry, dynamic process, physics, finance and fermentation process, (Arifah, 2005;Ayoubi et al., 2015;Ayoubi, 2015;Bahar & Mao 2004;Bazli, 2010;Mazo, 2002). This Brownian motion is the cause of instability (turbulence) in dynamic process Ayoubi, 2015;Liu & Wang, 2013). Therefore, the deterministic models are inadequate to describe the dynamic process which containts random fluctuations. Bahar and Mao (2004) introduced the stochastic logistic model to illustrate the population growth, which is affected by environmental noise (Liu & Wang, 2013). The environmental noise destroys the stability in dynamic process.
Aleksandr Mikhailovich Lyapunov in (1892), proposed the sense of stability for nonlinear dynamic system. He introduced an approach to determine the stability of the system without solving the system. Likewise, the stability theory for SDE introduced by Khasminskii (2011) and Mao (1991), Mao (1994) was explained some basic principles of different types of SDE. However, there is no specific research on stability stochastic logistic model with Ornsttein-Uhlenbeck process. Nevertheless, there are some previous research works which have been done on stability of logistic equation with white noise which are (Golec & Sathananthan, 2003;Jiang, et al., 2008;Liu & Wang, 2013;Sung & Wang, 2008). None of these researches investigated the stability of stochastic logistic model with Ornsttein-Uhlenbeck process. This research establishes the sufficient conditions for SDEs and stochastic logistic equation for zero solution by using Lyapunov function. In addition, we showed that noise is unfavorable for stability of population growth. Moreover, we apply the SRK4 method to evaluate the numerical solution.
Numerical simulation and Conclusion.

Preliminaries and Model Description
T denotes the terminal time, t is time and 0 t is the initial time, () xt corresponds to the highest data size, 0 x is initial data size, max  denotes the maximum specific growth rate while max x illustrates a carrying capacity and  indicates the random fluctuation.
Definition 2.1 (Arifah, 2005) is related to an m  vector valued function and is an m  vector stochastic process. System (1) has global unique solution which is x(t; There is a zero solution for system (1), at 0 0 This solution is devoted to the origin point or zero solution. This paper investigates the stability of SDEs and stochastic logistic model for zero solution by using Lyapunov function.
Theorem 2.1 (Mao, 1991) , then, the trivial solution is stable. If exist a positive-definite decrescent function  (1) is stochastically asymptotically stable, if there is exist a decrescent function,

the trivial solution of equation
is said to be negative definite. (1)

the trivial solution of equation
is said to be negative definite.

Logistic Models
The simplest mathematical model for exponential growth is The solution of equation (2) is If max 0   equation (3) is strongly ascending, and max 0   strongly descending. Thus, the exponential growth model of (2) is augmented by the inclusion of a multiplicative factor of max () 1, xt x  and the ordinary logistic equation is equation (4) can be solved analytically and the solution is that equation (4) equation (4) particularly is used in [8].

Mean Results
Remarks 2.1: In (Golec & Sathananthan, 2003;Jiang, & Li, 2008;Liu & Wang, 2013;Sun & Wang, 2008) under different conditions, showed that equation (6) is stable. All of them considered the white noise. Nevertheless, this research investigates the stability of stochastic logistic model with Ornsttein-Uhlenbeck process in zero solution via Lyapunove function. The Ornstein-Uhlenbeck process is: where   yt indicates the Brownian motion at time t ,   Wt is white noise, 0 b  shows the coefficient friction and  is the diffusion coefficient.
into the equation (7) hence, the Ornstein-Uhlenbeck process becomes: The expectation and variance of Ornstein-Uhlenbeck process is: Whereas, the normal distribution of white noise is a normal Gaussian. By substituting equation (9) into equation (6) the new logistic model is given as: Equation (10) is a stochastic logistic model with Ornstein-Uhlenbeck process.
First, it is necessary to prove that equation (10) has a unique positive solution then, we focus on stability.  (Arnold, 1972;Friedman, 1976).
We define the Lyapunov function  (Mao, 2007) for Lyapunov function and taking into account the equation (10) then we have: computing A and B separately we get: Somewhat lengthy calculation and application of the facts that It is worth mentioning, , it follows from the boundedness of  . Hence, there is nonnegative number 1 Q which is independent of x and t . If , obviously there is a positive number 2 Q which is independent of x and t , and it follows from the boundedness  , such that As results, we determine that there exist a positive number Q that is independent of x and t such that Substituting of the inequality (15) into (14) we get: It is obvious that the expectation of Brownian motion is zero (Mao, 2007) Consequently, we must have    and the proof is completed. Theorem 3.2: The differential operator   () LV x t which is associated with equation (1): TT T This result is achieved based on Lyapunov quadratic function and by taking into account equation (1). Suppose that the Lyapunov quadratic function is given by Equation (19) in some neighborhood of   0 0, x t t t    , with respect to the equation (1) is negative definite and it is stochastically asymptotically stable, in zero solution (origin point). Proof: To achieve the goal, we use the basic concept of derivative and Lyapunov function, to yield:

t Mx t x t Mf t x t dt x t Mg t x t dW t Mx t f t x t dt f t x t dtMf t x t dt Mf t x t dt
By using the facts that (Gardiner, 2004 pp87) and with somewhat lengthy calculation we have: By applying expectation in the above system, we get:

T T T E dV x t E x t Mf t x t dt Mx t f t x t dt
Mg t x t g t x t dt E LV x t dt (Mao, 2007 p.108 and Consequently, based on  we see that the equation (1) is stable, asymptotically stable or asymptotically stable in large and the above proof is completed.  (10) and hence, function   () V x t is negative definite. Therefore, the zero solution is stable and we reach to desired goal. To support our theory, this research considers numerical simulation which is presented in below section.

Numerical Simulation
In this section we consider a strong and accurate numerical method SRK4 to elaborate the analytical results Ayoubi, 2015). Bear in mind, we cannot use the SRK to approximate the numerical solution of equation (10). Since, it is in Itô sense (Rosli et al., 2010). Thus, equation (10) can be converted into Stratonovich sense by using the below formula where () dW t denotes the Stratonovich form. Equations (10) and (26), represent some solution under different approach. We use SRK4 method to approximate the numerical solution. SRK4 was based on the increment of Wiener process (Rosli et al., 2010).

Conclusion and Recommendations
This research was assigned for the stability of equation (1) and (10). In addition, the research presented a general theory to state the stability of SDEs for zero solution via Lyapunov function (see Theorem 3.2) which is verified by the stochastic logistic model (10)