【作者】 汪洋；

【导师】 王克；

【作者基本信息】 哈尔滨工业大学 ， 计算数学， 2011， 硕士

【摘要】 生物数学的领域里有许多关于捕食者-食模型的研究。依赖于被捕食者的捕食者-食模型功能性反应的文章很难描述捕食者系统对模型的干扰。一些生物学家认为,当捕食者相互之间竞争猎取食物时,功能性反应既依赖于被捕食者的密度,也依赖于捕食者的密度。这个观点得到了一些来自实际观测资料的支持。Skalski和Gilliam收集了19种捕食者-食系统观测统计资料。他们指出3种既依赖于被捕食者密度,也依赖于捕食者密度的功能性反应,分别为Hassell-Varley , Beddington-DeAngelis以及Crowley-Martin .它们可以与实际观测资料符合的很好,在某些情况下, Beddington-DeAngelis型功能性反应的效果更加令人满意。另一方面,现实世界中的随机干扰无处不在,并且捕食者种群种内密度制约因素也应考虑进来。然而这些因素在现有的研究中几乎都被忽略了。这篇文章的创新点在于把随机因素和捕食者种内密度制约因素考虑进来,提出一个Beddington-DeAngelis型功能性反应的随机捕食者-食模型。并且这个模型的种群动力学系统是非自治的,因此模型的研究更加复杂和困难,比如以前确定性模型中的有界性在随机模型(SBD)中就被随机干扰所破坏。然后作者克服了这些困难,依然给出模型的种群动力学性质。这篇文章首先介绍具有Beddington-DeAngelis型功能性反应的随机捕食者-食模型的研究背景,所做的工作以及意义。由于模型表示的是生物种群,那么就需要证明其存在正的全局解。证明过程主要利用伊藤公式和基本的变换技巧,给出其解的存在唯一性。紧接着利用伊藤公式并通过构造合适的Lyapunov函数,证明了解的矩有界性以及随机上有界性。然后讨论生物的生存和灭绝性,依然利用伊藤公式以及Lyapunov函数给出了各个物种生存和灭绝的充分条件。然后,作者利用伊藤公式以及多个Lyapunov函数给出了系统解的全局吸引性的充分条件。最后,作者继续利用Milstein方法进行数值模拟,并给出了数值模拟的图像。通过上述的工作,给出主要结论。更多还原

【Abstract】 There are many different kinds of predator-prey systems in the mathematical ecologyliterature. It is well known that the traditional predator-prey systems with prey-dependentfunctional response fail to model the interference among predators. Some biologists haveargued that in many cases, especially when predators have to search for food and there-fore, have to share or compete for food, the functional response in a prey-predator mod-el should be predator-dependent. There are many significant evidences to suggest thatpredator dependence in the functional response occurs quite frequently in natural system-s and laboratory . Especially, by comparing the statistical evidence from 19 predator-prey systems with the three classical predator-dependent functional responses (Hassell-Varley , Beddington-DeAngelis and Crowley-Martin ), Skalski and Gilliam claimed thatthe predator-dependent can provide better descriptions of predator feeding over a range ofpredator-prey abundances, and in some cases the Beddington-DeAngelis type functionalresponse preformed even better. On the one hand, in the real world, population systemsare inevitably affected by stochastic noises. In addition, the density-dependence of thepredator population should be taken into account. However, both the stochastic factor andthe density-dependence of the predator were neglected in almost all existing studies.In this paper author shall propose a stochastic non-autonomous predator-prey systemwith Beddington-DeAngelis functional response by taking into account the stochastic fac-tor and the density-dependence of the predator. The population system is non-autonomousand therefore more complicated and the mathematics presented is more diffcult, for ex-ample, the boundedness of x(t) and y(t) in deterministic system is destroyed by stochasticnoises in (SBD). Author overcomes these problems, and studies the population dynamicalproperties. They are the innovations of this paper.Firstly , author introduces the research background of this paper. Since the sys-tem denotes a population system, then author need to show that the model has a positiveand global solution. Author uses It(?) formula and substitution technique to show theexistence, the uniqueness and the positivity of the solution. Then by using It(?) formulaand constructing some appropriate Lyapunov functions, author studys the boundednessof moments and the upper-growth rate of the solution. After that author discusses thepersistence and extinction of the model. Author establishes the su?cient conditions for persistence and extinction of each population by using It(?) formula and Lyapunov func-tions. Then author uses It(?) formula and Lyapunov functions to investigate the globalattractivity of the system. At the end , author introduces some numerical simulationsto confirm the results by using Milstein method. Finally, author closes the paper withconclusions and discussions.更多还原