【作者】 祖力；

【导师】 蒋达清；

【作者基本信息】 东北师范大学 ， 应用数学， 2013， 博士

【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.更多还原

【Abstract】 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.更多还原