Stationary distribution and pathwise estimation of n-species mutualism system with stochastic perturbation
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1 Available online at J. Nonlinear Sci. Appl. 9 6), Research Article Stationary distribution and pathwise estimation of n-species mutualism system with stochastic perturbation Weiwei Fang a, Qixing Han b, Xiangdan Wen c,, Qiuyue Li d a School of Mathematical Sciences, Harbin Normal University, Harbin, 55, P. R. China. b School of Mathematics, Changchun Normal University, Changchun 33, P. R. China. c Department of Mathematics, Yanbian University, Yanji 33, P. R. China. d Department of Foundation Courses, Aviation University of Airforce, Changchun 3, P. R. China. Communicated by R. Saadati Abstract In this paper, we develop a new stochastic mutualism population model d t) = t) a ij x j t) dt + σ i t)db i t), i =,,, n. By constructing suitable Lyapunov functions, we show the system has a stationary distribution. We also discuss the pathwise behaviour of the solution. The conclusions of this paper is very powerful since they are independent both of the system parameters and of the initial value. It is also independent of the noise intensity as long as the noise intensity σi >. Computer simulations are used to illustrated our results. c 6 All rights reserved. Keywords: Mutualism model, Itô s formula, stationary distribution, ergodicity, pathwise estimation. MSC: 65C3, 3k5, 9D5.. Introduction Consider a n-species Lotka-Volterra mutualism model Corresponding author addresses: fangww@63.com Weiwei Fang), hanqixing3@63.com Qixing Han), xdwen5@yeah.net Xiangdan Wen), qiuyueli:8836@qq.com Qiuyue Li) Received 5--3
2 W. Fang, Q. Han, X. Wen, Q. Li, J. Nonlinear Sci. Appl. 9 6), d t) = t) a ij x j t) dt, i =,,, n,.) where t) is the ith species population density at time t, r i is the intrinsic growth rate of species, a ii represents the population decay rate in the competition among the ith species and a ij represents the ith species population increase rate in the mutualism among the other species x j i, j =,,, n, i j). In particular, Chen and Song [] have studied the sufficient conditions for the global stability of positive equilibrium of model.), the sufficient conditions are as follows: i) There is a matrix G = G ij ) n n, such that a ii G ii, a ij G ij i j) hold for all i, j =,,, n. ii) All of the principal minors of G are positive. There are many other researchers who have studied the dynamics of mutualism model see [,, 6] and references therein). In fact, mutualism population dynamics is inevitably affected by environmental white noise, which is always present [6, 7, 9,,,, 5]. In practice, we usually estimate parameters by an average value plus errors. We may assume that the errors follow normal distributions, but the standard deviations of the errors, known as the noise intensities, may depend on the population sizes [, ]. Therefore, we replace parameter r i in model.) by r i r i + σ i t)ḃt), i =,,, n). Then we get the following new stochastic system d t) = t) a ij x j t) dt + σ i t)db i t), i =,,, n,.) where B i t) is standard one-dimensional independent Wiener processes, σi are the intensity of the white noise and σi >. The organization of this paper is as follows. In next section, we will investigate the pathwise behaviour of the solution of system.). In Section 3, we show that the system has a stationary distribution with no parametric restriction if σi >. We illustrate our results through an example in Section. Finally, the conclusion is presented in Section 5. Throughout this paper, let Ω, F, P ) be a complete probability space with a filtration {F t } t satisfying the usual conditions i.e. it is increasing and right continuous while F contains all P-null sets). We denote by R+ n the positive cone in R n, that is R+ n = {x R n : for all i n}. In general, we consider a d-dimensional stochastic differential equation [3] dxt) = fxt), t)dt + gxt), t)dbt) on t t.3) with initial value xt ) = x R d, where Bt) denotes d-dimensional standard Brownian motion. Define the differential operator L associated with Equation.3) by L = t + d k= f k x, t) x k + If L acts on a function V C, S h R + ; R + ), then d [g T x, t)gx, t)] kj. x k x j k, LV x, t) = V t x, t) + V x x, t)fx, t) + trace[gt x, t)v xx gx, t)], where C, S h R + ; R + ) is the family of all nonnegative functions V x, t) defined on S h R + such that they are continuously twice differentiable in x and once in t, and V t = V t, V x = V, V,, V V ), V xx = ) d d. x x x d x k x j
3 W. Fang, Q. Han, X. Wen, Q. Li, J. Nonlinear Sci. Appl. 9 6), Asymptotic pathwise estimation In this section, we will investigate pathwise behaviour of the solution of system.), which is one of desired population dynamical properties of population system. Theorem.. For any given initial value x R n +, the solution of system.) has the following property: lim sup t log xt) Proof. Define a Lyapunov function for any p, ) a.s. V x) = x p i. Then Let dv x) = p By Itô s formula, we get and p a ij x j p σ i dt + p p n σ i x p+) i zx) = V x), Kx) = p d log V x) = de t log V x) = p σ i x p+) i db i t). a ij x j p σ i. ) Kx) V x) z x) dt + zx)dbt) log V x) + Kx) ) V x) z x) dt + zx)dbt). Integrating both sides from to t yields e t log V xt)) = log V x)) + e s log V xs)) + Kxs)) ) V xs)) z xs)) ds + Using the Exponential Martingale Inequality, for any α, β, T >, we have { [ P ω : sup e s zxs))dbs) α ] } e s z xs))ds β t T e αβ. Choose T = Kδ, α = e Kδ, β = +δ)ekδ logkδ), where < δ <, < <. Note that K= <, K) +δ e s zxs))dbs). an application of the Borel-Cantelli lemma yields that for almost all ω Ω, there is a random integer n = n ω) > such that e s zxs))dbs) + δ)ekδ logkδ) + δ)ekδ logkδ) + e Kδ + e s z xs))ds e s z xs))ds, t Kδ, n n..)
4 W. Fang, Q. Han, X. Wen, Q. Li, J. Nonlinear Sci. Appl. 9 6), Hence log V xt)) e t log V x)) + + δ)ekδ t logkδ) + e s t log V xs)) + Kxs)) V xs)) )z xs)) ) ds. Applying inequalities log x x and n p ) x p n x p i n p ) x p, one see that log V xt)) + Kxt)) V xt)) )z xt)) V xt)) + Kxt)) V xt)) x p i + př + pǎ ij C, n x where ř = max {r, r,, r n }, ǎ ij = max {a ij }, i, j =,, n and C is a constant. Therefore log V xt)) e t log V x)) + + δ)ekδ t logkδ) + C e t ). It then follows that for almost all ω Ω, if n n, K )δ t Kδ, log xt) p log V xt)) logk )δ [e t log V x)) + C] + + δ)eδ logkδ). logk )δ This implies Letting δ, gives Letting p we obtain This completes the proof. lim sup t log xt) p lim sup t lim sup t log xt) log xt) + δ)eδ p a.s. a.s. a.s. 3. Existence of stationary distribution In this section, we prove the existence of stationary distribution of system.). The following theorem gives a criterion for the existence of stationary distribution in terms of Lyapunov function see [3], Chapter 3, p.3, [8], [7], p.63). Let Xt) be a homogeneous Markov Process in E l E l denotes l dimensional Euclidean space) and is described by the following stochastic equation, The diffusion matris defined as follows, dxt) = bx)dt + k g r X)dB r t). 3.) r= Λx) = λ ij x)), λ ij x) = k grx)g i rx). j r=
5 W. Fang, Q. Han, X. Wen, Q. Li, J. Nonlinear Sci. Appl. 9 6), Theorem 3.. The Markov process Xt) has a unique ergodic stationary distribution µ ) if there exists a bounded domain U E l with regular boundary Γ and A : there is a positive number M such that l i, λ ij x)ξ i ξ j M ξ, x U, ξ R l, A : there exist a nonnegative C function V such that LV is negative for any E l \ U. Then { P x lim T T T } fxt))dt = fx)µdx) = E l for all x E l, where f ) is a function integrable with respect to the measure µ. Theorem 3.. Assume that r i > σ i, i =,,, n, then for any initial value x R n +, there is a stationary distribution µ.) for system.) and it has ergodic property. Proof. For any p, ), θ, ), define a nonnegative C function V by V x, x,, x n ) = x p i + ). x θ i Denote Applying Itô s formula, we get LV = p p p p p p r i x p i + r i x p i + M p p) V = x p i, V = θ. a ij x j p σ i ) a ij x j p p) σ +p i a ij p + x j ) p p) σ +p i a ij p a ji + p p) σ +p i σi x +p i, where M = sup {p n r i x p i + n a ij x R+ n p + n a ji LV = ) p p) σi +p } <. a ij x j θ + σ i r i + a ii θ + ) σ i. 3.) 3.3)
6 W. Fang, Q. Han, X. Wen, Q. Li, J. Nonlinear Sci. Appl. 9 6), Thus LV = LV + LV M p p) σi x +p i θ r i + a ii θ + ) σ i. 3.) Define U = {x, x,,, x n ) R n +, }, where is sufficiently small number. Now, for any fixed m m n), if < x m <, we have LV M Noting that M p p) p p) σi x +p i θr m x m θ σi +p θ n a ii θ + ) σ i. a ii θ+ σ i ) is bounded, we obtain LV M θr m. 3.5) If x m >, we get from 3.) that LV M M p p) 8 p p) 8 σi x +p i θ σ m +p, r i + a ii θ + ) σ i where M = sup {M p p) x R+ n 8 σi +p θ n x i r i + a ii θ+ σ i x ) i } <. Now, we can choose sufficiently small such that p p) σ +p 8 mx m 3.6) M θr m < and p p) M 8 which together with 3.5) and 3.6) yields that σ m LV < <, +p for any x R n +\U. Hence condition A in Theorem 3. is satisfied. Besides, the diffusion matrix of Equation.) is Λ = diag{σ x, σ x,, σ nx n }. Choosing M = min {σ x, σ x,, σ nx n, x, x,, x n ) U}, we get λ ij x)ξ i ξ j = i, σi x i ξ i M ξ. Hence condition A is satisfied. According to Theorem 3., the desired results can be obtained.
7 W. Fang, Q. Han, X. Wen, Q. Li, J. Nonlinear Sci. Appl. 9 6), Simulations In this section, we will consider -species mutualism system with stochastic perturbation which can be represented by { dx t) = x t)[r a x t) + a x t))dt + σ x t)db t)],.) dx t) = x t)[r + a x t) a x t))dt + σ x t)db t)]. Using the Milstein method mentioned in [5], we get the corresponding discretization equation: { x,k+ = x,k + x,k [r a x,k + a x,k ) t + σ x,k,k t + σ x,k,k t t)], x,k+ = x,k + x,k [r + a x,k a x,k ) t + σ x,k,k t + σ x,k,k t t)]. Let r =.7, r =.7, a =.6, a =., a t) =.3, a t) =.8, x ) =., x ) =.5. xt) t 3 xt) 6 5 t Figure : Solution x t), x t)) for system.) compared to the deterministic system with σ =., σ =. and its corresponding histogram. xt) xt) 6 5 t 6 5 t Figure : Solution x t), x t)) for system.) compared to the deterministic system with σ =.85, σ =.9 and its corresponding histogram.
8 W. Fang, Q. Han, X. Wen, Q. Li, J. Nonlinear Sci. Appl. 9 6), In Figure, we choose σ =., σ =., According to Theorem 3., we can conclude that there is a stationary distribution for system.). Figure shows the histogram of the approximate stationary distribution of system.). In Figure, we choose the same parameters as in Figure, but increase the intensities of the white noise σ =.85, σ =.9), Theorem 3. is still satisfied. We can see that with the increasing of the white noise, the zone which the solution is fluctuating in is getting large compared to the earlier case. However, there is still a stationary distribution for system.) see the histogram on the right in Figure ). 5. Conclusion In this paper, we have considered a stochastic mutualism system. By constructing suitable Lyapunov functions, we have shown that the system has a stationary distribution with no parametric restriction. The result is very interesting, since the existence of a stationary distribution is independent both of the system parameters and of the initial value. It is also independent of the noise intensity as long as the noise intensity σi >. Usually, for the stochastic system, the strong white noise may make the system to be extinct, however, the new stochastic system does not. On the other hand, the stability of positive equilibrium of the corresponding deterministic system.) need some restrictions on the parameters see []), but the stochastic system.) has a stationary distribution with no parametric restriction if σi >. The existence of a stationary distribution means stochastic stability to some extent, namely that noise is helpful for the stability of the population system. Acknowledgements The work was supported by NSFC of ChinaNo: 3785), the Ph.D. Programs Foundation of Ministry of China No. 98), Scientific Research Fund of Jilin Provincial Education Department 5 No:, 5 No:356) and Natural Science Foundation of Changchun Normal University. References [] A. Bahar, X. Mao, Stochastic delay LotkaVolterra model, J. Math. Anal. Appl., 9 ), [] L. Chen, X. Song, Z. Lu, Mathematical Models and Methods in Ecology, Sichuan Science and Techology Press, Sichuan, 3)., 5 [3] T. C. Gard, Introduction to Stochastic Differential Equations, Marcel Dekker, New York, 988). 3 [] B. S. Goh, Stability in models of mutualism, Amer. Natur., 3 979), [5] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 3 ), [6] C. Ji, D. Jiang, Persistence and non-persistence of a mutualism system with stochastic perturbation, Discrete Contin. Dyn. Syst., 3 ), [7] C. Ji, D. Jiang, H. Liu, Q. Yang, Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation, Math. Prob. Eng., ), 8 pages. [8] R. Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Alphen aan den Rijn, Germantown, Netherland, 98). 3 [9] R. Z. Khasminskii, F. C. Klebaner, Long term behavior of solution of the Lotka-Volterra system under small random perturbations, Ann. Appl. Probab., ), [] Y. Li, H. Zhang, Existence of periodic solutions for a periodic mutualism model on time scales, J. Math. Anal. Appl., 33 8), [] M. Liu, K. Wang, Analysis of a stochastic autonomous mutualism model, J. Math. Anal. Appl., 3), [] Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 33 7), [3] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 997). [] X. Mao, Delay population dynamics and environmental noise, Stoch. Dyn., 5 5) 9 6. [5] X. Mao, G. Marion, E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97 ), 95. [6] C. Y. Wang, S. Wang, F. P. Yang, L. R. Li, Global asymptotic stability of positive equilibrium of three-species Lotka-Volterra mutualism models with diffusion and delay effects, Appl. Math. Model., 3 ), [7] C. Zhu, G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 6 7),
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