Volterra人口模型求解Taylor展开和Pade逼近
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。