【作者】 程亚焕；

【导师】 李冬梅；

【作者基本信息】 哈尔滨理工大学 ， 基础数学， 2005， 硕士

【摘要】 随着环境污染的日益严重,人们对具有毒素影响的种群生存状态日渐关注。研究污染环境中种群的生存状况极其变化规律也随之成为数学生态学领域中的热点问题。污染和捕获是影响种群生存和发展的两个重要因素,而且种群的生存空间日益狭小,捕获量也往往随季节、种群数量等因素的改变而变化。因此,在污染环境容量较小且捕获量为变量的条件下,讨论种群的生存状态具有一定实际意义。目前,针对单种群模型在多种生存条件下的种群发展趋势,及同时讨论其多种生存状况的系统分析研究结果较少。对两种群的研究多局限于二维Volterra 捕食-食饵系统和竞争系统,且大多是在污染环境容量较大,忽略毒素通过食物链对种群产生影响的条件下进行讨论,对互惠系统的研究则很少见。本文主要采用积分均值法、比较原理及微分不等式等微积分方法,研究在容量较大的环境中,一类改进的广义Logistic 单种群受污染和捕获影响时的生存状况,给出种群β-持续生存、β-绝灭及在有限时间内零绝灭的充分条件,并得到保证种群β-持续生存的最大收获量。当环境容量较小且捕获量是关于种群数量的线性函数时,还得到种群弱持续生存和绝灭的充分条件,使单种群领域的研究结果更为深入和全面。利用上述方法,本文又进一步研究了污染环境容量较小且毒素通过食物链对种群产生影响的二维Volterra 互惠系统,得到种群弱平均持续生存、绝灭、β-持续生存和β-绝灭的充分条件,并给出了系统参数波动与解的位置的关系式,确定了相应条件下解的最终变化范围,为实际应用提供了理论依据。更多还原

【Abstract】 With environment pollution being more serious, people pay close attention to existence of population with toxin gradually. It becomes hot spot problem to study the existence of population and their change law in a polluted environment. Pollution and capture are two important factors which affect the existence and development of population, and capture changes with the transformation of season and population when existence space is smaller and smaller. Therefore,it has practical sense to study the existence of population under conditions that capacity is small and capture is variable. Now, the study result about single community under several existence conditions is few, and people seldom discuss single community’s existence states at the same time. The research for two community only confines to two-dimensional Volterra predator-prey system and competing system. Furthermore, the discussion progresses under the conditions that capacity is large and the effect that toxin affects population by food chain is neglectful. This paper studies the existence of a kind of general Logistic population which live in a large capacity and polluted environment with capture by methods of integral in mean,comparability principle and differential inequality principle.It gives the sufficient conditions for β-persistence, β-extinction and extinction in finite time,and obtains the maximal harvesting rate when the population is β-persistence.When the capacity is small and the capture is a linear function for population,it gets the sufficient conditions for weak persistence and extinction.This makes the research in region for single community even more deeper and fuller. Using the same methods,this paper also studies the existence of two-dimensional Volterra cooperative system in a small capacity and polluted environment when toxin affects population by food chain.It obtains the sufficient conditions for weak persistence in the mean,extinction, β-persistence and β-extinction.It also gives the relational expression about parameter fluctuation and solution position,and defines the final changing range of solutions under corresponding conditions.This offers theoretical foundation for practical applications.更多还原