Applied Mathematics Letters. Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system

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1 Applied Mathematics Letters 5 (1) Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system Meng Liu, Ke Wang Department of Mathematics, Harbin Institute of Technology, Weihai 649, PR China a r t i c l e i n f o a b s t r a c t Article history: Received 31 March 11 Accepted 1 March 1 Keywords: Generalized logistic equation Stochastic perturbations Stationary distribution Ergodic Extinction This paper is concerned with a stochastic generalized logistic equation n n dx = x[r ax θ ]dt + α i xdb i (t) + β i x 1+θ db i (t), where B i (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. 1 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 ] where a >, θ >. It is well-known that Eq. (1) has a positive equilibrium x = (r/a) 1/θ and x is globally asymptotically stable provided r >. 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 (see e.g. [ 7]). Suppose that the growth rate r is subject to stochastic noises with r r + n α iḃi(t) and a is subject to stochastic noises with a a + n β iḃi(t), then we obtain the stochastic equation n n dx = x[r ax θ ]dt + α i xdb i (t) + β i x 1+θ db i (t), () where B(t) = (B 1 (t),..., B n (t)) T is an n-dimensional Brownian motion and α i and 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 (1) Corresponding author. address: liumeng557@sina.com (M. Liu) /$ see front matter 1 Elsevier Ltd. All rights reserved. doi:1.116/j.aml

2 M. Liu, K. Wang / Applied Mathematics Letters 5 (1) 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 α 1, α = = α n = and β, β 1 = β 3 = β n =. If we choose α 1, α, α 3 = = α n = and β 1, β = = β n =, 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 >, 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. () will not tend to a fixed positive point. In this paper, we first show that if < θ 1, a > (r/a) 1/θ n β i and a(r/a) 1/θ > n α i, then there is a stationary distribution to system () and it is ergodic. Ergodic property is one of the most important properties of Markov processes, which has been widely applied in statistics theory, probability theory, Lie theory and harmonic analysis (see e.g. [8,9]). Then we show that if r < n α i, then the solution of () is extinctive.. 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: k dx(t) = b(x)dt + σ m (X)dB m (t). The diffusion matrix is m=1 A(x) = (a ij (x)), a ij = k m=1 σ (i) (j) m (x)σ (x). m 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. (A) If x E l \ U, the mean time τ at which a path issuing from x reaches the set U is finite, and sup x K E x 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 1 T P lim f (X(s))ds = f (x)µ(dx) = 1. T + T E l Remark 1. To verify (A1), it is sufficient to show that G is uniformly elliptical in U, where Gu = b(x)u x + trace(a(x)u xx ), that is, there is a positive number N such that k a ij (x)η i η j > N η, i,j=1 x U, η R k (see [1, p. 13] and Rayleigh s principle in [11, p. 349]). To validate (A), it is sufficient to prove that there is a neighborhood U and a non-negative C -function such that for any x E l \ U, LV(x) is negative (see [1, p. 1163]). Remark. The diffusion matrix of Eq. () is A(x) = n [α ix + β i x ]. Lemma. For any given initial value x() = x R + = {x : x > }, there is a unique solution x(t) to () on t 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 < θ 1. If a > (r/a) 1/θ n β i and a(r/a) 1/θ > n α i then there is a stationary distribution µ( ) for system () and it has ergodic property: P lim t + 1 t t x(s)ds = R+ zµ(dz) = 1.

3 198 M. Liu, K. Wang / Applied Mathematics Letters 5 (1) Proof. Applying the Itô formula leads to d(e t x) = e t xdt + e t dx = e x t + x[r ax θ ] dt + e t K 1 e t dt + e t n α i xdb i (t) + e t n α i xdb i (t) + e t n β i x 1+θ db i (t) n β i x 1+θ db i (t), where K 1 is a positive number. Then we have E[x(t)] e t x() + K 1 (1 e t ). In other words, we have already shown that lim sup t + E[x(t)] K 1. Then there is a T > such that E[x(t)] K 1 for t T. At the same time note that E[x(t)] is continuous, then there is a positive constant K such that E[x(t)] < K for t < T. Define K = max{k 1, K }. Then E[x(t)] K; t. (3) Define V(x) = x θ x x ln xθ x. By the famous Itô formula dv(x) = θ x θ 1 x dx + θ (θ 1)x θ + x (dx) x x x = θ(x θ x )[r ax θ ]dt + θ[(θ 1)x θ + x ] n + θ(x θ x ) α i db i (t) + aθ(x θ x ) dt + θx = θ a x n n n [α i + β i x θ ] dt n n [α i + β i x θ ] dt + θ(x θ x ) α i db i (t) + x θ + θ n + θ(x θ x ) α i db i (t) + = θ a x n ax + x n α i β i x θ + θx n xθ n + x α i a(x ) n + θ(x θ x ) α i db i (t) + n n n α i aθ(x ) ax + x n α i β i ax + x n α i β i + a x 4 a x n n = LV(x) + θ(x θ x ) α i db i (t) + n, n

4 M. Liu, K. Wang / Applied Mathematics Letters 5 (1) where LV(x) = θ a x =: θ n xθ n + x α i a(x ) C1 x C + C + x C 1 4C 1 4C 1 [ax + x n α i β i ] ax + x n α i β i + a x 4 a x n n α i a(x ). Note that if a > x n β i and ax > n α i, then LV(x) < for x R + \ U 1 := R + \ C n + x α i a(x ) C 1 + C, C 1 C n + x α 4C i a(x ) C 1 + C. C 1 1 n Let U be a neighborhood of U 1 such that U, then for x R + \ U, we obtain LV(x) <. In other words, Assumption (A) holds. On the other hand, there is N > n such that [α ix + β i x 1+θ ] η Nη for x Ū and η R, where Ū is the closure of U. That is to say, Assumption (A1) is satisfied. Consequently, Eq. () has a stable stationary distribution µ( ) and it is ergodic. By the ergodic property, for H >, we get 1 t lim [x(s) H]ds = [z H]µ(dz) a.s. (4) t + t R+ Making use of the famous dominated convergence theorem and (3), one can see that 1 t 1 t E lim [x(s) H]ds = lim E x(s) H ds K. t + t t + t This, together with (4), means [z H]µ(dz) K. Letting H + results in zµ(dz) K. Thus the function f (x) = x R+ R+ is integrable with respect to the measure µ( ). Then the desired assertion follows from Lemma 1 immediately. Theorem 4. If b := r n α i <, then the solution x(t) of () obeys lim t + x(t) = a.s. Proof. Applying Itô s formula to (), we can observe that d ln x = dx n x (dx) = b ax θ β x i xθ dt + Then we get ln x(t) ln x = bt a t x θ (s)ds n t where M i (t) = t β ix θ (s)db i (s). Note that for all 1 i n, lim B i(t)/t = t + a.s. n α i db i (t) + x θ (s)ds + n. n α i B i (t) + n M i (t), (5) The quadratic variation of M i (t) is M i (t), M i (t) = t β i xθ (s)ds. In view of the exponential martingale inequality, we can see that P sup t k M i (t) 1 M i(t), M i (t) > ln k 1/k. (6)

5 1984 M. Liu, K. Wang / Applied Mathematics Letters 5 (1) a b c Fig. 1. Solutions of system () for r =, a = θ = 1, n =, α 1 =.1, β 1 = β =.1, x() =.3. (a) and (b) are with α =.8. (a) is the stationary distribution and (b) is the sample path of (); (c) is with α = 1.1. Making use of Borel Cantelli lemma yields that for almost all ω Ω, there is a random integer k = k (ω) such that for k k, sup t k [M i (t) 1 M i(t), M i (t) ] ln k. That is to say M i (t) ln k + 1 M i(t), M i (t) = ln k + t xθ (s)ds for all t k, k k almost surely. Substituting this inequality into (5), we can obtain that ln x(t) ln x bt a t x θ (s)ds + n ln k + n α i B i (t) bt + n ln k + n α i B i (t) for all t k, k k almost surely. In other words, we have already shown that for < k 1 t k, k k, we have t 1 {ln x(t) ln x } b + n(k 1) 1 ln k + n α ib i (t)/t. Making use of (6) gives lim sup t + t 1 ln x(t) b. That is to say, if b <, then lim t + x(t) =. Remark 3. Consider the stochastic logistic equation dx = x[r ax]dt + n α i xdb i (t) + It then follows from Theorems 3 and 4 that if a /r > n β i and r > n n β i x db i (t). (7) µ( ) for system (7) and it has ergodic property: P α i lim t + t 1 t x(s)ds = R+ zµ(dz), then there is a stationary distribution = 1; If r < n α i, then

6 M. Liu, K. Wang / Applied Mathematics Letters 5 (1) the solution x(t) of (7) satisfies lim t + x(t) =. It is easy to see that under the condition a /r > n β i, we give the sufficient and necessary conditions of existence of ergodic stationary distribution and extinction for system (7). 3. Numerical simulations Let us use the Monte Carlo simulation method to illustrate our results. In Fig. 1, we choose r =, a = θ = 1, n =, α 1 =.1, β 1 = β =.1, then a /r > n β i. The only difference between condition of Fig. 1(a) (c) is that the value of α is different. In Fig. 1(a) and (b), we choose α =.8, then r > α i. In view of Theorem 3, there is a stationary distribution µ( ) for Eq. () and it has ergodic property. Fig. 1(a) is the stationary distribution and Fig. 1(b) is the sample path of (). In Fig. 1(c), we choose α = 1.1, then r < α i. By Theorem 4, the solution of () is extinctive. Fig. 1(c) confirms this. 4. Conclusions and future directions A stochastic generalize logistic equation is studied. We have shown that if < θ 1 and the intensities of the white noises are sufficiently small in the sense that a > (r/a) 1/θ n β i and a(r/a) 1/θ > n α i, then there is a stationary distribution to this equation and it has ergodic property. If r < n α i, 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 , , , 1113 and ), the Postdoctoral Science Foundation of China (Grant No ), Shandong Provincial Natural Science Foundation of China (Grant No. ZR11AM4), and the NSFC of Shandong Province (No. ZR1AQ1). References [1] R.M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press, [] D. Jiang, N. Shi, X. Li, Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 34 (6) [3] M. Liu, K. Wang, Extinction and permanence in a stochastic nonautonomous population system, Appl. Math. Lett. 3 (1) [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 (11) [5] M. Liu, K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl. 375 (11) [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 (11) [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 (11) [8] R. Atar, A. Budhiraja, P. Dupuis, On positive recurrence of constrained diffusion processes, Ann. Probab. 9 (1) [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, 198. [1] T.C. Gard, Introduction to Stochastic Differential Equations, New York, [11] G. Strang, Linear Algebra and its Applications, Thomson Learning, Inc., [1] C. Zhu, G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim. 46 (7)