【作者】 陈波；

【导师】 苏晓明；

【作者基本信息】 沈阳工业大学 ， 数学， 2023， 硕士

【摘要】 种群动力学模型是一种用于描述生物种群动态行为,预测与控制生物种群数量变化趋势的生物数学模型,捕食者-猎物模型就是最基本的模型之一。由于自然界中的随机干扰客观存在且无法避免,因此采用随机生物数学模型对种群的动态行为进行模拟比确定性模型更加符合实际。本文利用相关的数学理论与随机微分理论研究了两类随机生物数学模型,主要内容概括如下:针对具有Crowley-Martin型功能反应和气候效应项的随机非自治模型,研究了模型的动力学行为。首先通过伊藤公式,微分不等式,证明了模型全局正解的存在唯一性;然后利用伊藤公式、强大数定律、切比雪夫不等式以及构造辅助函数等方法,得到了模型随机最终有界、均值持续生存及灭绝、随机持久生存的充分条件;最后通过数值模拟,验证了所得结果的有效性。针对具有避难所和Ornstein-Uhlenbeck过程的随机三种群捕食模型,研究了模型的持久与灭绝问题。首先利用伊藤公式、随机比较定理,证明了模型存在正解且唯一;然后利用伊藤公式、微分不等式,得到了模型平均持久与灭绝的充分条件;最后通过数值模拟,验证了所得结果的有效性。更多还原

【Abstract】 The population dynamics model is a kind of biological mathematical model used to describe the dynamic behavior of biological population,predict and control the changing trend of biological population.The predator-prey model is one of the most basic models.Because the random interference in nature exists objectively and cannot be avoided,it is more practical to use stochastic biological mathematical model to simulate the dynamic behavior of the population than the deterministic model.In this paper,two kinds of stochastic biological mathematical models are studied by using relevant mathematical theory and stochastic differential theory.The main contents are summarized as follows:The dynamical behavior of the model is investigated for stochastic non-autonomous models with Crowley-Martin type functional response and climate effect terms.Firstly,the existence and uniqueness of global positive solutions are proved by It(?)’s formula and differential inequality;Then,by using It(?)’s formula,strong law of numbers,Chebyshev’s inequality and constructing auxiliary functions,the sufficient conditions of stochastic final bounded,mean persistent survival and extinction,and stochastic persistent survival are obtained;Finally,the validity of the results is verified by numerical simulation.A stochastic three-population predation model with shelters and Ornstein-Uhlenbeck processes is studied for its persistence and extinction.Firstly,It(?)’s formula and random comparison theorem are used to prove that the model has a positive and unique solution;Then,by using It(?)’s formula and differential inequality,the sufficient conditions of average durability and extinction of the model are obtained;Finally,the validity of the results is verified by numerical simulation.更多还原