随机扩散种群模型动力学行为的研究
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摘要
种群生态学起源于人口统计学,是研究生物种群发展规律的科学,其研究方法主要通过数学模型理解、解释和预测自然界中各物种的发展变化规律,从而达到更好地管理和保护生态环境中生物种群的目的.早期,学者们通过建立确定性模型来研究种群生态系统.Lotka-Volterra模型是理论生态学的一个里程碑,此后各种生态模型被相继提出,并有很多学者系统的研究了确定性种群体系的动力学性质.扩散现象在各物种的历史发展轨迹中都起到重要作用,它在几乎所有的物种间普遍存在,并影响着各物种的持久与发展.很多学者将扩散现象引入到确定性模型中加以研究.然而,由于自然界中处处存在的不确定性和随机现象使生态系统中的各个物种会受到不同形式的随机干扰.因此用随机微分方程模型来描述种群动力学在某种程度上能更精确地反映实际现象.
本文主要考虑带有扩散的单种群模型和捕食-被捕食模型中的内禀增长率受到环境白噪声影响时的动力学行为,并研究了扩散单种群模型在环境白噪声和彩色噪声的共同影响下的渐进性质.首先通过Lyapunov泛函方法给出随机系统全局正解的存在唯一性,这是研究系统动力学行为的基础.然后研究系统的p阶矩有界性,以此为基础研究种群系统是否具有随机持久性和均值意义下的持久性.在确定性扩散单种群系统和捕食-被捕食系统中,若满足一定的条件系统存在平衡点,但引入随机扰动后,随机系统不存在平衡点.但是当白噪声强度较小时,随机系统的解表现为在对应确定性系统的某邻域内波动,即随机系统存在平稳分布,具有遍历性.而强度较大的白噪声会导致系统的灭绝.
总之,研究表明,当白噪声小的时候,随机系统具有类似相应的确定系统的性质；当白噪声大的时候,随机系统会出现完全不同于确定性系统的性质,如非持久性和灭绝性.在现实世界中,这种强的白噪声可以理解为突发的恶劣天气,环境急剧变化等.
Population ecology originated from the population statistics, it is a science to study the development law of the species, and its research methods is using math-ematical models to understand, explain and predict the change of each species, so as to manage and protect the species more well. In the beginning, scholars built up deterministic mathematical models of ecological system and studied their dynamic behaviors. Lotka-Volterra model is a milestone in theoretical ecology. Various ecological model have been proposed later, and many scholars systematic studied their dynamic properties. Dispersal is a life history trait that has profound effects on both species persistence and evolution and it is prevalent in almost all species. Many scholars introduce diffusion phenomena to the deterministic model. However, there always exists white noise in the environment, which will lead to var-ious species in the ecosystem are subject to various forms of random interference. Therefore, stochastic differential equations can reflect the reality more accurately. In this paper, we consider the dynamic behaviors when the intrinsic rate is stochas-tic perturbed in single population models and predator-prey model with diffusion, respectively.
In this paper, we consider the dynamical behavior of diffusion single-species models and diffusion predator-prey model when the intrinsic rate of increase is disturbed by environmental white noise, and study the asymptotical properties of the diffusion single-species models under the combined effect of environmental white noise and color noise. First, we show there exists a unique positive solution of the stochastic systems by Lyapunov analysis method, which is the base to study the dynamics of the systems. Then we study the p-th moment boundedness, and based on it, we study that whether the system has random persistence and the mean time persistence. In the diffusion deterministic single population models and predator-prey model, they have positive equilibrium points under the certain conditions, but the introduction of random perturbations cause the random systems have no equilibrium points. However, the solutions of the random system is in a neighborhood of the deterministic system when the intensity of the white noise is small, The performance of solutions in a fluctuations, that is the random system has a stationary distribution and has ergodicity. Specially, the large white noise may bring the extinction of the species.
All in all, in this paper, we point out that the stochastic systems imiate the corresponding deterministic systems if the white noise is small; while if the white noise is large, the stochastic systems have more different properties, such as unper-sistence and extinction. In the reality, the large white noise can be considered as the bad weather and rapidly changing environment etc.
本文主要考虑带有扩散的单种群模型和捕食-被捕食模型中的内禀增长率受到环境白噪声影响时的动力学行为,并研究了扩散单种群模型在环境白噪声和彩色噪声的共同影响下的渐进性质.首先通过Lyapunov泛函方法给出随机系统全局正解的存在唯一性,这是研究系统动力学行为的基础.然后研究系统的p阶矩有界性,以此为基础研究种群系统是否具有随机持久性和均值意义下的持久性.在确定性扩散单种群系统和捕食-被捕食系统中,若满足一定的条件系统存在平衡点,但引入随机扰动后,随机系统不存在平衡点.但是当白噪声强度较小时,随机系统的解表现为在对应确定性系统的某邻域内波动,即随机系统存在平稳分布,具有遍历性.而强度较大的白噪声会导致系统的灭绝.
总之,研究表明,当白噪声小的时候,随机系统具有类似相应的确定系统的性质；当白噪声大的时候,随机系统会出现完全不同于确定性系统的性质,如非持久性和灭绝性.在现实世界中,这种强的白噪声可以理解为突发的恶劣天气,环境急剧变化等.
Population ecology originated from the population statistics, it is a science to study the development law of the species, and its research methods is using math-ematical models to understand, explain and predict the change of each species, so as to manage and protect the species more well. In the beginning, scholars built up deterministic mathematical models of ecological system and studied their dynamic behaviors. Lotka-Volterra model is a milestone in theoretical ecology. Various ecological model have been proposed later, and many scholars systematic studied their dynamic properties. Dispersal is a life history trait that has profound effects on both species persistence and evolution and it is prevalent in almost all species. Many scholars introduce diffusion phenomena to the deterministic model. However, there always exists white noise in the environment, which will lead to var-ious species in the ecosystem are subject to various forms of random interference. Therefore, stochastic differential equations can reflect the reality more accurately. In this paper, we consider the dynamic behaviors when the intrinsic rate is stochas-tic perturbed in single population models and predator-prey model with diffusion, respectively.
In this paper, we consider the dynamical behavior of diffusion single-species models and diffusion predator-prey model when the intrinsic rate of increase is disturbed by environmental white noise, and study the asymptotical properties of the diffusion single-species models under the combined effect of environmental white noise and color noise. First, we show there exists a unique positive solution of the stochastic systems by Lyapunov analysis method, which is the base to study the dynamics of the systems. Then we study the p-th moment boundedness, and based on it, we study that whether the system has random persistence and the mean time persistence. In the diffusion deterministic single population models and predator-prey model, they have positive equilibrium points under the certain conditions, but the introduction of random perturbations cause the random systems have no equilibrium points. However, the solutions of the random system is in a neighborhood of the deterministic system when the intensity of the white noise is small, The performance of solutions in a fluctuations, that is the random system has a stationary distribution and has ergodicity. Specially, the large white noise may bring the extinction of the species.
All in all, in this paper, we point out that the stochastic systems imiate the corresponding deterministic systems if the white noise is small; while if the white noise is large, the stochastic systems have more different properties, such as unper-sistence and extinction. In the reality, the large white noise can be considered as the bad weather and rapidly changing environment etc.
引文
[1]Mao X. Stochastic Differential Equations and Applications[M]. New York: Horwood,1997.
[2]Friedman A. Stochastic Differential Equations and their Applications[M]. New York:Academic Press,1976.
[3]Arnold L. Stochastic Differential Equations:Theory and Applications[M]. New York:Wiley,1972.
[4](?)ksendal B. Stochastic Differential Equations[M]. Singapore:World Scien-tific,2003.
[5]Okubo A. Diffusion and ecological problems:mathematical models[M]. In Biomathematics,1980.
[6]Allen L. Persistence, extinction, and critical patch number for island popu-lations[J]. Bull Math Biol,1987,65:1-12.
[7]Li M, Shuai Z. Global-stability problem for coupled systems of differential equations on networks[J]. J Differential Equations,2010,248:1-20.
[8]闫理坦,鲁立刚,许志强.随机积分与不等式[M].北京：科学出版社,2005.
[9]Mahbuba R, Chen L. On the non-autonomous Lotka-Volterra competition system with diffusion [J]. Differential Equations and Dynamical Systems, 1994,2:243-253.
[10]Wang W, Chen L. Global stability of a population dispersal in a two-patch environment [J]. Dynamic Systems and Applications,1997,6:207-216.
[11]Zhang J, Chen L. Periodic solutions of single-species non-autonomous diffu-sion models with continuous time delays[J]. Math Comput Modelling,1996, 23:17-28.
[12]Cui J, Takeuchi Y, Lin L. Permanence and extinction for dispersal population systems[J]. J Math Anal Appl,2004,298:73-93.
[13]Lu Z, Takeuchi Y. Global asymptotic behavior in single-species discrete dif-fusion systems[J]. J Math Biol,1993,32:67-77.
[14]Levin S. Dispersion and population interactions[J]. Am Nat,1974,108:207-228.
[15]Allen L. Persistence and extinction in single-species reaction-diffusion mod-els[J]. Bull Math Biol,1983,45:209-227.
[16]Mao X, Yuan C, Zou J. Stochastic differential delay equations of population dynamics[J]. J Math Anal Appl,2005,304:296-320.
[17]Mao X, Yuan C. Stochastic Differential Equations with Markovian Switch-ing [M]. Imperial College Press,2006.
[18]Ji C, Jiang D, Liu H. Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation[J]. Math Prob Engi,2010, 10:1155-1172.
[19]Mao X. Stochastic differential equations and applications[M]. Horwood, New York,1997.
[20]Li X, Jiang D, Mao X. Population dynamical behavior of Lotka-Volterra system under regime switching[J]. J Comput Appl Math,2009,232:427-448.
[21]Li X, Gray A, Jiang D, Mao X. Sufficient and necessary conditions of stochas-tic permanence and extinction for stochastic logistic populations under regime switching[J]. J Math Anal Appl,2011,376:11-28.
[22]L. Chen, J. Chen. Nonlinear Biological Dynamical System, Science Press[M]. Beijing,1993.
[23]Ji C, Jiang D, Shi N. Analysis of a predator-prey model with modified Leslie-Gower and Holling-type 11 schemes with stochastic perturbationc[J]. J Math Anal Appl,2009,359:482-498.
[24]Ikeda N, Watanabe S. "Stochastic Differential Equations and Diffusion Pro-cesses," 2nd edition, North-Holland Mathematical Library,24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo,1989.
[25]Lotka A J. Elements of Physical Biology[M]. Galtimore:Williams and Wilkins,1925.
[26]Zhu C, Yin G. Asymptotic properties of hybrid diffusion systems[J]. SIAM J Control Optim,2007,46:1155-1179.
[27]Volterra V. Variazioni e fluttuazioni del numero d'individui in specie animali conviventi[J]. Mem R Accad Naz dei Lincei,1926,2:31-113.
[28]Gopalsamy K. Stability and Oscillation in Delay Differential Equations of Population Dynamics [M]. Netherlands:Kluwer Academic Publishers Group, 1992.
[29]Kuang Y. Delay Differential Equations with Applications in Population Dy-namies[M]. Boston:Academic Press,1993.
[30]May R M. Stability and Complexity in Model Ecosystems[M]. New Jersey: Princeton University Press,2001.
[31]Du N, Kon R, Sato K, Takeuchi Y. Dynamical behavior of Lotka-Volterra competition systems:Non-autonomous bistable case and the effect of tele-graph noise[J]. J Comput Appl Math,2004,170:399-422.
[32]Stakin M. The dynamics of a population in a Markovian environment[J]. Ecology,1987,59:249-256.
[33]Luo Q, Mao X. Stochastic population dynamics under regime switching[J]. J Math Anal Appl,2007,334:69-84.
[34]Kuang Y, Takeuchi Y. Predator-Prey dynamics in models of prey dispersal in two-patch environments[J]. J Math Biol 1994,120:77-98.
[35]Li M, Shuai Z. Global-stability problem for coupled systems of differential equations on networks[J]. J Differential Equations,2010,248:1-20.
[36]Cai G, Lin Y. Stochastic analysis of predator-prey type ecosystems[J]. Ecol Complex,2007,4:242-249.
[37]Zhang L, Teng Z. Boundedness and permanence in a class of periodic time-dependent predator-prey system with prey dispersal and predator density-independence[J]. Chaos Soli Frac,2008,36:729-739.
[38]Ikeda N, Wantanabe S. Stochastic Differential Equations and Diffusion Pro-cesses[M]. North-Holland:Amsterdam,1981.
[39]Hasminskii R. Stochastic Stability of Differential Equations. The Nether-lands:Sijthoff Noordhoff, Alphen aan den Rijn,1980.
[40]Gard T C. Introduction to Stochastic Differential Equations[M]. New York: Dekker,1988.
[41]胡适耕,黄乘明,吴付科.随机微分方程[M].北京：科学出版社,2008.
[42]Zhu C, Yin G. Asymptotic properties of hybrid diffusion systems [J]. SI AM J Control Optim,2007,46:1155-1179.
[43]Guo H B, Li M Y, Shuai Z S. Global stability of the endemic equilibrium of multigroup SIR epidemic models[J]. Can Appl Math Q,2006,14:259-284.
[44]Li M Y, Shuai Z S, Wang C C. Global stability of multi-group epidemic models with distributed delays[J]. J Math Anal Appl,2010,361:38-47.
[45]Mao X R, Marion G, Renshaw E. Environmental Brownian noise suppresses explosions in population dynamics[J]. Stochastic Process Appl,2002:95-110.
[46]Mao X R, Sabanis S, Renshaw E. Asymptotic behaviour of the stochastic Lotka-Volterra model[J]. J Math Anal Appl,2003,287:141-156.
[47]Brauer F, Castillo-Chavez C. Mathematical Models in Population Biology and Epidemiology[M]. New York:Springer-Verlag,2000.
[48]Hastings A. Population Biology Concepts and Models[M]. New York: Springer-Verlag,1997.
[49]Berman A, Plemmons R J. Nonnegative Matrices in the Mathematical Sci-ences[J]. New York:Academic Press,1979.
[50]Li X Y, Mao X R, Shen Y. Approximate solutions of stochastic differential delay equations with Markovian switching[J]. J Difference Equ Appl,2010, 16:195-207.
[51]Kliemann W. Recurrence and invariant measures for degenerate diffusions [J]. Ann Probab,1987,15:690-707.
[52]王克.随机生物数学模型[M].北京：科学出版社,2010.
[53]Strang G. Linear Algebra and its Applications [M]. Singapore:Thomson Learning,1988.
[54]Freedman H, Takeuchi Y. Global stability and predator dynamics in a model of prey dispersal in a patchy environment [J]. Nonlinear Anal Theory Methods Appl,1989,13:993-1002.
[55]Highm D. An algorithmic introduction to numerical simulation of stochastic differential equations[J]. SIAM Rev,2001,43:525-546.
[56]Atar R, Budhiraja A, Dupuis P. On positive recurrence of constrained diffu-sion processes [J]. Ann Probab,2001,29:979-1000.
[57]Bahar A, Mao X R. Stochastic delay Lotka-Volterra model[J]. J Math Anal Appl,2004,292:364-380.
[58]Arnold L, Horsthemke W, Stucki J W. The influence of external real and white noise on the Lotka-Volterra model[J]. Biometrical J,1979,21:451-471.
[59]Bandyopadhyay M, Chattopadhyay J. Ratio-dependent predator-prey model: effect of environmental fluctuation and stability[J]. Nonlinearity,2005,18: 913-936.
[60]Jiang D Q, Shi N Z, Zhao Y N. Existence, uniqueness, and global stabil-ity of positive solutions to the food-limited population model with random perturbation[J]. Math Comput Model,2005,42:651-658.
[61]Jiang D Q, Shi N Z. A note on nonautonomous logistic equation with random perturbation[J]. J Math Anal Appl,2005,303:164-172.
[62]Du N H, Sam V H. Dynamics of a stochastic Lotka-Volterra model perturbed by white noise[J]. J Math Anal Appl 2006,324:82-97.
[63]Jiang D Q, Zhang B X, Wang D H, Shi N Z. Existence, uniqueness and global attractivity of positive solutions and MLE of the parameters to the Logistic equation with random perturbation[J]. Sci China (Ser. A),2007,50:977-986.
[64]Mao X R, Lam J, Huang L R. Stabilisation of hybrid stochastic differential equations by delay feedback control [J]. Systems Control Lett,2008,57:927-935.
[65]Jiang D Q, Shi N Z, Li X Y. Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation[J]. J Math Anal Appl,2008,340:588-597.
[66]Saha T, Bandyopadhyay M. Dynamical analysis of a delayed ratio-dependent prey-predator model within fluctuating environment [J]. Appl Math Comput, 2008,196:458-478.
[67]Xu R, Chen L. Persistence and stability of a two species ratio-dependent predator-prey system with time delay in a two-patch environment[J]. J Com-put Math Appl,2000,40:577-588.
[68]Cui J. The effect of dispersal on permanence in a predator-prey population growth model[J]. J Comput Math Appl,2002,44:1085-1097.
[69]Z. Teng, L. Chen, Permanence and extinction of periodic predator-prey sys-tems in patchy environment with delay[J]. Nonlinear Anal:RWA,2003,4: 335-364.
[2]Friedman A. Stochastic Differential Equations and their Applications[M]. New York:Academic Press,1976.
[3]Arnold L. Stochastic Differential Equations:Theory and Applications[M]. New York:Wiley,1972.
[4](?)ksendal B. Stochastic Differential Equations[M]. Singapore:World Scien-tific,2003.
[5]Okubo A. Diffusion and ecological problems:mathematical models[M]. In Biomathematics,1980.
[6]Allen L. Persistence, extinction, and critical patch number for island popu-lations[J]. Bull Math Biol,1987,65:1-12.
[7]Li M, Shuai Z. Global-stability problem for coupled systems of differential equations on networks[J]. J Differential Equations,2010,248:1-20.
[8]闫理坦,鲁立刚,许志强.随机积分与不等式[M].北京：科学出版社,2005.
[9]Mahbuba R, Chen L. On the non-autonomous Lotka-Volterra competition system with diffusion [J]. Differential Equations and Dynamical Systems, 1994,2:243-253.
[10]Wang W, Chen L. Global stability of a population dispersal in a two-patch environment [J]. Dynamic Systems and Applications,1997,6:207-216.
[11]Zhang J, Chen L. Periodic solutions of single-species non-autonomous diffu-sion models with continuous time delays[J]. Math Comput Modelling,1996, 23:17-28.
[12]Cui J, Takeuchi Y, Lin L. Permanence and extinction for dispersal population systems[J]. J Math Anal Appl,2004,298:73-93.
[13]Lu Z, Takeuchi Y. Global asymptotic behavior in single-species discrete dif-fusion systems[J]. J Math Biol,1993,32:67-77.
[14]Levin S. Dispersion and population interactions[J]. Am Nat,1974,108:207-228.
[15]Allen L. Persistence and extinction in single-species reaction-diffusion mod-els[J]. Bull Math Biol,1983,45:209-227.
[16]Mao X, Yuan C, Zou J. Stochastic differential delay equations of population dynamics[J]. J Math Anal Appl,2005,304:296-320.
[17]Mao X, Yuan C. Stochastic Differential Equations with Markovian Switch-ing [M]. Imperial College Press,2006.
[18]Ji C, Jiang D, Liu H. Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation[J]. Math Prob Engi,2010, 10:1155-1172.
[19]Mao X. Stochastic differential equations and applications[M]. Horwood, New York,1997.
[20]Li X, Jiang D, Mao X. Population dynamical behavior of Lotka-Volterra system under regime switching[J]. J Comput Appl Math,2009,232:427-448.
[21]Li X, Gray A, Jiang D, Mao X. Sufficient and necessary conditions of stochas-tic permanence and extinction for stochastic logistic populations under regime switching[J]. J Math Anal Appl,2011,376:11-28.
[22]L. Chen, J. Chen. Nonlinear Biological Dynamical System, Science Press[M]. Beijing,1993.
[23]Ji C, Jiang D, Shi N. Analysis of a predator-prey model with modified Leslie-Gower and Holling-type 11 schemes with stochastic perturbationc[J]. J Math Anal Appl,2009,359:482-498.
[24]Ikeda N, Watanabe S. "Stochastic Differential Equations and Diffusion Pro-cesses," 2nd edition, North-Holland Mathematical Library,24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo,1989.
[25]Lotka A J. Elements of Physical Biology[M]. Galtimore:Williams and Wilkins,1925.
[26]Zhu C, Yin G. Asymptotic properties of hybrid diffusion systems[J]. SIAM J Control Optim,2007,46:1155-1179.
[27]Volterra V. Variazioni e fluttuazioni del numero d'individui in specie animali conviventi[J]. Mem R Accad Naz dei Lincei,1926,2:31-113.
[28]Gopalsamy K. Stability and Oscillation in Delay Differential Equations of Population Dynamics [M]. Netherlands:Kluwer Academic Publishers Group, 1992.
[29]Kuang Y. Delay Differential Equations with Applications in Population Dy-namies[M]. Boston:Academic Press,1993.
[30]May R M. Stability and Complexity in Model Ecosystems[M]. New Jersey: Princeton University Press,2001.
[31]Du N, Kon R, Sato K, Takeuchi Y. Dynamical behavior of Lotka-Volterra competition systems:Non-autonomous bistable case and the effect of tele-graph noise[J]. J Comput Appl Math,2004,170:399-422.
[32]Stakin M. The dynamics of a population in a Markovian environment[J]. Ecology,1987,59:249-256.
[33]Luo Q, Mao X. Stochastic population dynamics under regime switching[J]. J Math Anal Appl,2007,334:69-84.
[34]Kuang Y, Takeuchi Y. Predator-Prey dynamics in models of prey dispersal in two-patch environments[J]. J Math Biol 1994,120:77-98.
[35]Li M, Shuai Z. Global-stability problem for coupled systems of differential equations on networks[J]. J Differential Equations,2010,248:1-20.
[36]Cai G, Lin Y. Stochastic analysis of predator-prey type ecosystems[J]. Ecol Complex,2007,4:242-249.
[37]Zhang L, Teng Z. Boundedness and permanence in a class of periodic time-dependent predator-prey system with prey dispersal and predator density-independence[J]. Chaos Soli Frac,2008,36:729-739.
[38]Ikeda N, Wantanabe S. Stochastic Differential Equations and Diffusion Pro-cesses[M]. North-Holland:Amsterdam,1981.
[39]Hasminskii R. Stochastic Stability of Differential Equations. The Nether-lands:Sijthoff Noordhoff, Alphen aan den Rijn,1980.
[40]Gard T C. Introduction to Stochastic Differential Equations[M]. New York: Dekker,1988.
[41]胡适耕,黄乘明,吴付科.随机微分方程[M].北京：科学出版社,2008.
[42]Zhu C, Yin G. Asymptotic properties of hybrid diffusion systems [J]. SI AM J Control Optim,2007,46:1155-1179.
[43]Guo H B, Li M Y, Shuai Z S. Global stability of the endemic equilibrium of multigroup SIR epidemic models[J]. Can Appl Math Q,2006,14:259-284.
[44]Li M Y, Shuai Z S, Wang C C. Global stability of multi-group epidemic models with distributed delays[J]. J Math Anal Appl,2010,361:38-47.
[45]Mao X R, Marion G, Renshaw E. Environmental Brownian noise suppresses explosions in population dynamics[J]. Stochastic Process Appl,2002:95-110.
[46]Mao X R, Sabanis S, Renshaw E. Asymptotic behaviour of the stochastic Lotka-Volterra model[J]. J Math Anal Appl,2003,287:141-156.
[47]Brauer F, Castillo-Chavez C. Mathematical Models in Population Biology and Epidemiology[M]. New York:Springer-Verlag,2000.
[48]Hastings A. Population Biology Concepts and Models[M]. New York: Springer-Verlag,1997.
[49]Berman A, Plemmons R J. Nonnegative Matrices in the Mathematical Sci-ences[J]. New York:Academic Press,1979.
[50]Li X Y, Mao X R, Shen Y. Approximate solutions of stochastic differential delay equations with Markovian switching[J]. J Difference Equ Appl,2010, 16:195-207.
[51]Kliemann W. Recurrence and invariant measures for degenerate diffusions [J]. Ann Probab,1987,15:690-707.
[52]王克.随机生物数学模型[M].北京：科学出版社,2010.
[53]Strang G. Linear Algebra and its Applications [M]. Singapore:Thomson Learning,1988.
[54]Freedman H, Takeuchi Y. Global stability and predator dynamics in a model of prey dispersal in a patchy environment [J]. Nonlinear Anal Theory Methods Appl,1989,13:993-1002.
[55]Highm D. An algorithmic introduction to numerical simulation of stochastic differential equations[J]. SIAM Rev,2001,43:525-546.
[56]Atar R, Budhiraja A, Dupuis P. On positive recurrence of constrained diffu-sion processes [J]. Ann Probab,2001,29:979-1000.
[57]Bahar A, Mao X R. Stochastic delay Lotka-Volterra model[J]. J Math Anal Appl,2004,292:364-380.
[58]Arnold L, Horsthemke W, Stucki J W. The influence of external real and white noise on the Lotka-Volterra model[J]. Biometrical J,1979,21:451-471.
[59]Bandyopadhyay M, Chattopadhyay J. Ratio-dependent predator-prey model: effect of environmental fluctuation and stability[J]. Nonlinearity,2005,18: 913-936.
[60]Jiang D Q, Shi N Z, Zhao Y N. Existence, uniqueness, and global stabil-ity of positive solutions to the food-limited population model with random perturbation[J]. Math Comput Model,2005,42:651-658.
[61]Jiang D Q, Shi N Z. A note on nonautonomous logistic equation with random perturbation[J]. J Math Anal Appl,2005,303:164-172.
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