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摘要人口发展过程中的老龄化进程加快，出生比例持续提高，可以说人口发展的预测已经是我们不得不正视的一个问题。人口的预测模型并不是今天才开始研究的，在漫长的模型发展史中有精确的模型，也有在人口发展过程中逐渐发现有误的，其中最有名的模型莫过于Malthus模型和Logistic模型。本文是在这两种模型的基础上引进了第三种模型即Volterra 模型。Volterra 模型是在Logistic模型的基础加入积分表达式，以达到分析毒素在种群内部的增长作用。在引进了Volterra 模型后，通过比较以前对于Volterra 模型的求解较于繁琐，本文提出了一种新式解法，即使用Taylor 级数展开去近似求解Volterra 模型的解，但是Taylor级数展开所求出的近似解通过与之前的解法比较还是出现了较大的误差，故而本人运用Pade逼近去提高解的精度，得到了精度较高的解，并运用Mathmatica软件使计算简单。通过一系列的比较得出各个参数在Volterra 模型的微分方程中有什么做作用与影响。23853
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毕业论文关键词：Volterra 人口模型，Taylor级数，Pade逼近，
Abstract
With the accelerating pace of olding population during the process of people development, constantly ascending birth rate, we have to face a problem regarding forecast of people development. The model of forecasting population doesn’t start from today. Some models are accurate over the long-term development of population forecast model, many of them are gradually proved to be false instead. The most famous models are Malthus Model and Logistic Model among these. Volterra Model as the third one is introduced based on both two previous models, which will be analyzed in this paper. Specifically, the formula of Volterra Model is evolved from the combination of Logistic Model and integration formula, in which the integration formula represents the effect of toxin accumulation on the species.A novel solution, using Taylor-series expansion to get approximately answer of Volterra Model, will be worked out after Volterra Model is introduced since the solution process, through comparing past solutions to Volterra with ones currently is proved to be redundant. However, the approximate solution to Taylor-series expansion comparing with previous solutions still causes a large number of margins of error. Thus, Pade approximation is taken to improve the accuracy of solution and also, the utilization of Mathmatica software makes calculation simple, for purpose of researching how influence of different parameter will produce on Volterra Model.
Key words：Volterra population Model, Taylor-series , Pade approximation
目录
1 绪论 4
2 人口模型发展过程 5
2.1 Malthus模型 5
2.2 Logistic模型 6
3 Volterra 人口模型建立 9
3.1文多•沃尔泰拉(Vito Volterra)介绍 9
3.2模型建立 9
4 模型求解 11
4.1 小参数的奇异摄动法 11
4.2用数值系统解问题 12
4.3Taylor 展开 13
4.4 Pade逼近的定义及其结构特征 14
4.5求解结果 15
5 结论 19
答谢 20
参考文献 21
附录 22
1 绪论
严格地讲，讨论人口问题所建立的模型应属于离散性模型。但在人口基数很大的情况下，突然增加或减少的只是单一的个体或少数几个个体，相对于全体数量而言，这种改变量是极其微小的，甚至可以说是忽略不计的。因此我们可以近似地假设人口随时间连续变化甚至是可微的。这样我们就可以采用微分方程的工具来研究这一问题。