Permanence and Extinction of Stochastic Logistic System with Feedback Control under Regime Switching

We study a stochastic logistic system with feedback control under regime switching. Sufficient conditions for extinction, nonpersistence in themean, weak persistence, and persistence in themean are established. A very important fact is found in our results; that is, the feedback control is harmless to the permanence of species even under the regime switching and stochastic perturbation environments. Finally, some examples are introduced to illustrate the main results.


Introduction
The classical logistic equation with feedback control is ẋ () =  () ( () −  ()  () −  ()  ()) , u () = − ()  () +  ()  () , where () denotes the population size at time  and () is an "indirect control" variable (see [1]).It has been studied extensively and many important results on the global dynamics of solutions have been founded (see [2][3][4][5] and references therein).On the other hand, population systems in the real world are often affected by environmental noise and there are various types of environmental noise, for example, white or color noise (see [6][7][8][9][10][11][12][13][14] and references therein) and it has been shown that the presence of such noise affects population systems significantly.But the white and color noise are unobservable, and we can only observe the species ().Hence, we can use the same feedback control with system (1) to regulate the species which is affected by the environmental noises.To the best of the authors' knowledge, few scholars still consider the stochastic perturbation logistic system with feedback controls under regime switching.And we have known very little about how the feedback control affects the survival of species which is under the random factors and switching environment.
In this work, our purpose is to design feedback controls such that the system becomes permanent or extinct.We will establish the sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and persistence in the mean of system (2).
In system (2), () is the size of the species and () is the regulator; thus, we are only interested in the positive solutions.Moreover, in order for a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (cf.Mao [15]).However, the coefficients of system (2) do not satisfy the linear growth condition, though they are locally Lipschitz continuous.In this section, using the comparison theorem of stochastic equations (see [16]), we will show there is a unique positive solution with positive initial value of system (2).
Proof.Since the coefficients of the equation are locally Lipschitz continuous, it is known that for any given initial value ( 0 ,  0 ) ∈  2  + there is a unique maximal local solution ((), ()) for all  ∈ [0,   ), where   is the explosion time.Furthermore, by Theorem 2.1 in [17], we have where () = () − 0.5 2 () for each  ∈ S. Hence, to show this solution is global positive, we only show that   = ∞ a.s.By the first equation of ( 2 From Theorem 2.1 in [17], we known that there exists a unique continuous positive solution () of system (8) for any positive initial value  0 , which will remain in  + with probability one.Consequently, by the comparison theorem of stochastic differential equation, we have Therefore, () < ∞ for all  > 0 a.s.By the second equation of (2), we can represent () by From this, we can find that if () is global, then () also is a global solution; that is,   = ∞ a.s.This completes the proof of the theorem.Now, we will discuss extinction and persistence of system (2).For convenience and simplicity in the following discussion, we denote () = () − 0.5 2 () and b = ∑  =1   () for  ∈ S and write ((), ()) = ((, 0,  0 ,  0 ), (, 0,  0 ,  0 )) simply for any ( 0 ,  0 ) ∈ where () = ∫  0 (())d().By the second equation of system (2), we have Note that () is a local martingale.Making use of the strong law of large numbers for local martingales (see Mao [15]), we have lim We denote Ω 0 = {lim  → ∞ ()/ = 0}; obviously, P(Ω 0 ) = 1.

Remarks
In Theorem 4, we used a new method to study the weakly persistent in the mean of species .If () ≡ 0 for all  ∈ S, system (2) becomes a stochastic logistic system without feedback control under the regime switching which has been studied by [10].By Theorem 4, species  is weakly persistent in the mean if b > 0, and the condition   > 0 is not necessary.Hence, Theorem 3 in [10] is improved by Theorem 4 in this paper.
In [10], the authors studied system (2) without the feedback control and obtained the critical value between weak persistence and extinction; that is, species  is weakly persistent if b > 0 and goes to extinction if b < 0. Hence, from Theorems 2-5, we can find that if   and   are positive in system (2), species  has the same extinction and permanence property as the system without feedback control under the same conditions.Therefore, the feedback control is harmless to the permanence of species even under the white and color noise perturbation environments.
In a similar discussion with Theorem 3, we can obtain ⟨⟩ * ≥ ]  /(    +     ).Since  is arbitrary, we obtain ⟨⟩ * ≥ b  /(    +     ) :=  for all  ∈ Ω 0 .Now, we will prove ⟨⟩ * also has a lower bound.From the above proof, we can imply for any  > 0 and  ∈ Ω 0 that there is a positive constant  such that ∫