Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system

Applied Mathematics Letters 25 (2012) 1980–1985

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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml

Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system Meng Liu ∗ , Ke Wang Department of Mathematics, Harbin Institute of Technology, Weihai 264209, PR China

article

abstract

info

Article history: Received 31 March 2011 Accepted 12 March 2012

This paper is concerned with a stochastic generalized logistic equation dx = x[r − axθ ]dt +

n  i=1

Keywords: Generalized logistic equation Stochastic perturbations Stationary distribution Ergodic Extinction

αi xdBi (t ) +

n 

βi x1+θ dBi (t ),

i=1

where Bi (t )(i = 1, . . . , n) are independent Brownian motions. We show that if the intensities of the white noises are sufficiently small, then there is a stationary distribution to this equation and it has an ergodic property. If the intensities of the white noises are sufficiently large, then the equation is extinctive. Some numerical simulations are introduced to support the main results at the end. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction The study of the logistic system has long been and will continue to be one of the dominant themes in mathematical ecology due to its universal existence and importance. The famous deterministic generalized logistic model (Gilpin–Ayala model) takes the form dx dt

= x[r − axθ ]

(1)

where a > 0, θ > 0. It is well-known that Eq. (1) has a positive equilibrium x∗ = (r /a)1/θ and x∗ is globally asymptotically stable provided r > 0. However, population systems are often subject to environmental noise. In reality, due to environmental noise, coefficients in the system are not constants; they always fluctuate around some average values. May [1] has claimed that due to environmental fluctuation, the growth rates, competition coefficients and all other parameters in the system exhibit stochastic fluctuation, and as a result the solution of the model never attains a steady point, but fluctuates around some average values. Thus many authors have studied the stochastic population systems n(see e.g. [2–7]). Suppose that the growth rate r is subject to stochastic noises with r → r + i=1 αi B˙ i (t ) and a is subject to stochastic n noises with −a → −a + i=1 βi B˙ i (t ), then we obtain the stochastic equation dx = x[r − axθ ]dt +

n 

αi xdBi (t ) +

i =1

n 

βi x1+θ dBi (t ),

(2)

i=1

where B(t ) = (B1 (t ), . . . , Bn (t ))T is an n-dimensional Brownian motion and αi2 and βi2 stand for the intensities of the white noises. The reason why we use an n-dimensional Brownian motion B(t ) to model the stochastic noises is that the noise

∗

Corresponding author. E-mail address: [email protected] (M. Liu).

0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.03.015

M. Liu, K. Wang / Applied Mathematics Letters 25 (2012) 1980–1985

1981

terms on r and a may or may not correlate to each other. If the noise terms on r and a are independent, we may choose α1 ̸= 0, α2 = · · · = αn = 0 and β2 ̸= 0, β1 = β3 · · · = βn = 0. If we choose α1 ̸= 0, α2 ̸= 0, α3 = · · · = αn = 0 and β1 ̸= 0, β2 = · · · = βn = 0, then the noise terms on r and a are correlate. As pointed out above, the positive equilibrium x∗ of (1) is globally asymptotically stable provided r > 0, which indicates that if the deterministic perturbation is small, the properties of the solution will not be changed. When it is subject to stochastic noise, it is interesting to study whether there also exists some stabilities. However, in this case there is no positive equilibrium. Therefore, the solution of Eq. (2) will not tend n  to a fixed positive point. In this paper, we first show that if 2 1/θ 0 < θ ≤ 1, a > 0.5(r /a)1/θ > 0.5 ni=1 αi2 , then there is a stationary distribution to system (2) and i=1 βi and a(r /a) it is ergodic. Ergodic property is one of the most important properties of Markov processes, which has been widely applied n in statistics theory, probability theory, Lie theory and harmonic analysis (see e.g. [8,9]). Then we show that if r < 0.5 i=1 αi2 , then the solution of (2) is extinctive. 2. Main results To begin with, let us prepare a lemma (see [9]). Let X (t ) be a homogeneous Markov process in E l (E l denotes euclidean l-space) described by the following stochastic differential equation: dX (t ) = b(X )dt +

k 

σm (X )dBm (t ).

m=1

The diffusion matrix is A(x) = (aij (x)),

aij =

k 

σm(i) (x)σm(j) (x).

m=1

Assumption 1. There exists a bounded domain U ⊂ E l with regular boundary Γ , having the properties that (A1) In the domain U and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix A(x) is bounded away from zero. (A2) If x ∈ E l \ U, the mean time τ at which a path issuing from x reaches the set U is finite, and supx∈K Ex τ < +∞ for every compact subset K ∈ E l . Lemma 1 (Hasminskii [9]). If Assumption 1 holds, then the Markov process X (t ) has a stationary distribution µ(·). Let f (·) be a function integrable with respect to the measure µ. Then

 lim

P

T →+∞

1

T



T

f (X (s))ds = 0



 El

f (x)µ(dx)

= 1.

Remark 1. To verify (A1), it is sufficient to show that G is uniformly elliptical in U, where Gu = b(x)ux + 0.5trace(A(x)uxx ), that is, there is a positive number N such that k 

aij (x)ηi ηj > N |η|2 ,

x ∈ U , η ∈ Rk

i,j=1

(see [10, p. 103] and Rayleigh’s principle in [11, p. 349]). To validate (A2), it is sufficient to prove that there is a neighborhood U and a non-negative C 2 -function such that for any x ∈ E l \ U , LV (x) is negative (see [12, p. 1163]). Remark 2. The diffusion matrix of Eq. (2) is A(x) =

n

i =1

[αi x + βi x2 ]2 .

Lemma 2. For any given initial value x(0) = x0 ∈ R+ = {x : x > 0}, there is a unique solution x(t ) to (2) on t ≥ 0 and the solution will remain in R+ almost surely (a.s., i.e., with probability one). Proof. The proof is similar to Liu and Wang [5] and hence is omitted.



Now we are in the position to give our main results. Theorem 3. Suppose that 0 < θ ≤ 1. If a > 0.5(r /a)1/θ distribution µ(·) for system (2) and it has ergodic property:

 P

lim

t →+∞

1 t

t



x(s)ds = 0

 R+

 z µ(dz )

= 1.

n

i =1

βi2 and a(r /a)1/θ > 0.5

n

i=1

αi2 then there is a stationary

1982

M. Liu, K. Wang / Applied Mathematics Letters 25 (2012) 1980–1985

Proof. Applying the Itô formula leads to d(et x) = et xdt + et dx t





θ

= e x + x[r − ax ] dt + et

n 

αi xdBi (t ) + et

n 

i=1 n 

≤ K1 et dt + et

αi xdBi (t ) + et

βi x1+θ dBi (t )

i=1 n 

i =1

βi x1+θ dBi (t ),

i=1

where K1 is a positive number. Then we have E [x(t )] ≤ e−t x(0) + K1 (1 − e−t ). In other words, we have already shown that lim supt →+∞ E [x(t )] ≤ K1 . Then there is a T > 0 such that E [x(t )] ≤ 2K1 for t ≥ T . At the same time note that E [x(t )] is continuous, then there is a positive constant K2 such that E [x(t )] < K2 for 0 ≤ t < T . Define K = max{2K1 , K2 }. Then E [x(t )] ≤ K ;

t ≥ 0.

(3) xθ x∗

Define V (x) = xθ − x∗ − x∗ ln dV (x) = θ

 x

θ−1

−

x∗



x

. By the famous Itô formula

 dx + 0.5θ (θ − 1)x

θ−2

+

x∗



x2

= θ (xθ − x∗ )[r − axθ ]dt + 0.5θ [(θ − 1)xθ + x∗ ]

(dx)2 x2

n 

[αi + βi xθ ]2 dt

i=1

  n n   θ + θ (x − x ) αi dBi (t ) + βi x dBi (t ) θ

∗

i =1

i=1

  n n n    θ 2 θ ∗ θ ≤ −aθ (x − x ) dt + 0.5θ x [αi + βi x ] dt + θ (x − x ) αi dBi (t ) + βi x dBi (t ) θ

∗ 2

∗

i =1

 = −θ a − 0.5x∗  + θ (xθ − x∗ )

n 



i =1



βi2 x2θ + θ 2ax∗ + x∗

n 

i =1

i=1

n 



αi dBi (t ) +

i=1

n 

 αi βi xθ + 0.5θ x∗

i=1 n 

αi2 − aθ (x∗ )2

i=1

βi xθ dBi (t )

i =1

 2   2   n n    ∗ ∗ ∗ ∗     2ax + x α β 2ax + x α β i i i i n     θ i=1 i=1  +     = θ − a − 0.5x∗ βi2  x −   n n     i =1  βi2 βi2 2 a − 0.5x∗ 4 a − 0.5x∗  i=1

+ 0.5x∗

n 

αi2 − a(x∗ )2

        

i=1

  n n   θ + θ (x − x ) αi dBi (t ) + βi x dBi (t ) θ

∗

i =1

i=1

  n n   θ ∗ θ = LV (x) + θ (x − x ) αi dBi (t ) + βi x dBi (t ) , i=1

i=1

i =1

M. Liu, K. Wang / Applied Mathematics Letters 25 (2012) 1980–1985

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where

 2 2   n n    ∗ ∗ ∗ ∗     2ax + x αi βi [2ax + x αi βi ] n     θ i =1 i=1  +     LV (x) = θ − a − 0.5x∗ βi2  x −   n n     2 2 ∗ ∗ i =1  2 a − 0 . 5x β 4 a − 0 . 5x β  i i i=1

+ 0.5x∗

n 

αi2 − a(x∗ )2

   

i =1





=: θ −C1 x − Note that if a > 0.5x∗

i=1

    

2

C2

+

2C1

C22 4C1

+ 0.5x

∗

n 

 α − a(x )

∗ 2

2 i

.

i =1

 βi2 and ax∗ > 0.5 ni=1 αi2 , then LV (x) < 0 for      2 n C  C2 2 2  ∗ ∗ 2 C1 + x ∈ R+ \ U1 := R+ \ − + 0.5x αi − a(x ) , 2 n

i=1

4C1

2C1

i =1

     2 n C  C 2  2 + 0.5x∗ . αi2 − a(x∗ )2 C1 + 2 4C1

2C1

i=1

Let U be a neighborhood of U1 such that 0 ̸∈ U, then for xn ∈ R+ \ U, we obtain LV (x) < 0. In other words, Assumption (A2) holds. On the other hand, there is N > 0 such that i=1 [αi x + βi x1+θ ]2 η2 ≥ N η2 for x ∈ U¯ and η ∈ R, where U¯ is the closure of U. That is to say, Assumption (A1) is satisfied. Consequently, Eq. (2) has a stable stationary distribution µ(·) and it is ergodic. By the ergodic property, for H > 0, we get lim

t →+∞

1

t



t

[x(s) ∧ H ]ds =



0

[z ∧ H ]µ(dz ) a.s.

(4)

R+

Making use of the famous dominated convergence theorem and (3), one can see that

 E

1

lim



t

t →+∞



t

[x(s) ∧ H ]ds = lim

t →+∞

0

1 t



t





E x(s) ∧ H ds ≤ K . 0

This, together with (4), means R [z ∧ H ]µ(dz ) ≤ K . Letting H → +∞ results in R z µ(dz ) ≤ K . Thus the function f (x) = x + + is integrable with respect to the measure µ(·). Then the desired assertion follows from Lemma 1 immediately. 



Theorem 4. If b := r − 0.5



n

i=1

αi2 < 0, then the solution x(t ) of (2) obeys limt →+∞ x(t ) = 0 a.s.

Proof. Applying Itô’s formula to (2), we can observe that d ln x =

dx x

−



(dx)2

θ

= b − ax − 0.5

2x2

n 

 2 2θ i x

β

dt +

i=1

n 

αi dBi (t ) +

i =1

n 

βi xθ dBi (t ).

i=1

Then we get ln x(t ) − ln x0 = bt − a

t

 0

where Mi (t ) =

t 0

n 

P

t



x2θ (s)ds +

0

i =1

a.s.

The quadratic variation of Mi (t ) is ⟨Mi (t ), Mi (t )⟩ = see that



βi2

0≤t ≤k

i=1

αi Bi (t ) +

n 

Mi (t ),

(5)

i =1

(6)

t 0

βi2 x2θ (s)ds. In view of the exponential martingale inequality, we can

  sup Mi (t ) − ⟨Mi (t ), Mi (t )⟩ > 2 ln k ≤ 1/k2 . 

n 

βi xθ (s)dBi (s). Note that for all 1 ≤ i ≤ n,

lim Bi (t )/t = 0

t →+∞

xθ (s)ds − 0.5

1

2

1984

M. Liu, K. Wang / Applied Mathematics Letters 25 (2012) 1980–1985

a

b

c

Fig. 1. Solutions of system (2) for r = 0.5, a = θ = 1, n = 2, α1 = 0.1, β1 = β2 = 0.1, x(0) = 0.3. (a) and (b) are with α2 = 0.08. (a) is the stationary distribution and (b) is the sample path of (2); (c) is with α2 = 1.1.

Making use of Borel–Cantelli lemma yields that for almost all ω ∈ Ω , there is a random integer k0 = k0 (ω) such that for k ≥ k0 , sup0≤t ≤k [Mi (t ) − 21 ⟨Mi (t ), Mi (t )⟩] ≤ 2 ln k. That is to say Mi (t ) ≤ 2 ln k +

1 2

⟨Mi (t ), Mi (t )⟩ = 2 ln k + 0.5



t

βi2 x2θ (s)ds

0

for all 0 ≤ t ≤ k, k ≥ k0 almost surely. Substituting this inequality into (5), we can obtain that ln x(t ) − ln x0 ≤ bt − a

t



xθ (s)ds + 2n ln k +

0

n 

αi Bi (t ) ≤ bt + 2n ln k +

i =1

n 

αi Bi (t )

i=1

for all 0 ≤ t ≤ k, k ≥ k0 almost surely. In other words, we have already shown that for 0 < k − 1 ≤ t ≤ k, k ≥ k0 , we n have t −1 {ln x(t ) − ln x0 } ≤ b + 2n(k − 1)−1 ln k + i=1 αi Bi (t )/t. Making use of (6) gives lim supt →+∞ t −1 ln x(t ) ≤ b. That is to say, if b < 0, then limt →+∞ x(t ) = 0.  Remark 3. Consider the stochastic logistic equation dx = x[r − ax]dt +

n  i=1

αi xdBi (t ) +

n 

βi x2 dBi (t ).

(7)

i =1

It then follows from Theorems 3 and 4 that if a2 /r > 0.5

 βi2 and r > 0.5 ni=1 αi2 , then there is a stationary distribution    n −1 t µ(·) for system (7) and it has ergodic property: P limt →+∞ t x(s)ds = R z µ(dz ) = 1; If r < 0.5 i=1 αi2 , then 0 + 

n

i =1

M. Liu, K. Wang / Applied Mathematics Letters 25 (2012) 1980–1985

1985

the solution x(t ) of (7) satisfies limt →+∞ x(t ) = 0. It is easy to see that under the condition a2 /r > 0.5 i=1 βi2 , we give the sufficient and necessary conditions of existence of ergodic stationary distribution and extinction for system (7).

n

3. Numerical simulations Let us use the Monte Carlo simulation method nto illustrate our results. In Fig. 1, we choose r = 0.5, a = θ = 1, n = 2, α1 = 0.1, β1 = β2 = 0.1, then a2 /r > 0.5 i=1 βi2 . The only difference between condition of Fig. 1(a)–(c) is that the

value of α2 is different. In Fig. 1(a) and (b), we choose α2 = 0.08, then r > 0.5 i=1 αi2 . In view of Theorem 3, there is a stationary distribution µ(·) for Eq. (2) and it has ergodic property. Fig. 1(a) is the stationary distribution and Fig. 1(b) is the 2 sample path of (2). In Fig. 1(c), we choose α2 = 1.1, then r < 0.5 i=1 αi2 . By Theorem 4, the solution of (2) is extinctive. Fig. 1(c) confirms this.

2

4. Conclusions and future directions A stochastic generalize logistic equation is studied. We have shown if 0 < θ ≤ 1 and theintensities of the white n that 2 noises are sufficiently small in the sense that a > 0.5(r /a)1/θ /a)1/θ > 0.5 ni=1 αi2 , then there is a i=1 βi and a(r n stationary distribution to this equation and it has ergodic property. If r < 0.5 i=1 αi2 , then the system is extinctive. Particularly, for the classical stochastic logistic equation (7), we obtained the sufficient and necessary conditions of existence of ergodic stationary distribution and extinction under a simple condition. Some interesting topics deserve further investigation. It is interesting to study the density function of the stationary distribution µ(·). It is also interesting to investigate the higher-dimensional stochastic systems, for example, stochastic competitive system. We leave these investigations for future work. Acknowledgments We thank G. Hu for valuable program files of the figures. We also thank the NSFC of PR China (Nos. 11126219, 11171081, 11171056, 11001032 and 11101183), the Postdoctoral Science Foundation of China (Grant No. 20100481339), Shandong Provincial Natural Science Foundation of China (Grant No. ZR2011AM004), and the NSFC of Shandong Province (No. ZR2010AQ021). References [1] R.M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press, 1973. [2] D. Jiang, N. Shi, X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 340 (2006) 588–597. [3] M. Liu, K. Wang, Extinction and permanence in a stochastic nonautonomous population system, Appl. Math. Lett. 23 (2010) 1464–1467. [4] M. Liu, K. Wang, Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol. 73 (2011) 1969–2012. [5] M. Liu, K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl. 375 (2011) 443–457. [6] X. Li, A. Gray, D. Jiang, X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl. 376 (2011) 11–28. [7] C. Ji, D. JIang, N. Shi, A note on a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl. 377 (2011) 435–440. [8] R. Atar, A. Budhiraja, P. Dupuis, On positive recurrence of constrained diffusion processes, Ann. Probab. 29 (2001) 979–1000. [9] R.Z. Hasminskii, Stochastic Stability of Differential Equations, in: Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 7, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. [10] T.C. Gard, Introduction to Stochastic Differential Equations, New York, 1988. [11] G. Strang, Linear Algebra and its Applications, Thomson Learning, Inc., 1988. [12] C. Zhu, G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim. 46 (2007) 1155–1179.