摘 要：人口增长预测是社会经济规划和资源分配的重要基础，而差分方程作为一种数学工具，在离散时间序列分析中具有独特优势。本研究以差分方程为核心方法，构建了适用于不同场景的人口增长模型，旨在提高人口预测的精确性和适应性。通过引入非线性项和动态参数调整机制，模型能够更好地反映实际人口增长中的复杂因素，如迁移、政策干预和年龄结构变化等。研究采用历史人口数据进行模型校准与验证，并对比传统模型（如指数增长模型和Logistic模型）的结果，发现所提方法在长期预测中表现出更高的稳定性和准确性。实验结果表明，改进后的差分方程模型能够有效捕捉人口增长的非线性特征，尤其在处理波动性较强的数据时优势明显。本研究的主要贡献在于提出了一种灵活且可扩展的建模框架，为人口预测及相关决策提供了新的理论支持和技术手段，同时为差分方程在其他领域的时间序列预测应用提供了参考。
关键词：人口增长预测；差分方程模型；非线性特征；动态参数调整；模型校准与验证
Predictive Analysis of Differential Equations in Population Growth Models
英文人名
Directive teacher:×××
Abstract：Population growth forecasting serves as a crucial foundation for socioeconomic planning and resource allocation, while difference equations, as a mathematical tool, exhibit unique advantages in the analysis of discrete time series. This study focuses on difference equations to construct population growth models adaptable to various scenarios, aiming to enhance the accuracy and adaptability of population predictions. By incorporating nonlinear terms and a dynamic parameter adjustment mechanism, the model can better reflect complex factors in actual population growth, such as migration, policy intervention, and changes in age structure. Historical population data were utilized for model calibration and validation, with comparisons made to traditional models, including exponential growth and Logistic models. The proposed method demonstrated superior stability and accuracy in long-term forecasts. Experimental results indicate that the improved difference equation model effectively captures the nonlinear characteristics of population growth, showing significant advantages when dealing with highly volatile data. The primary contribution of this study lies in the proposal of a flexible and scalable modeling fr amework, providing new theoretical support and technical approaches for population forecasting and related decision-making, as well as offering reference for the application of difference equations in time series prediction across other domains.
Keywords: Population Growth Forecast；Difference Equation Model；Nonlinear Characteristics；Dynamic Parameter Adjustment；Model Calibration And Validation
目 录
引言 1
一、差分方程的基本理论 1
(一)差分方程的定义与分类 1
(二)差分方程的稳定性分析 2
(三)差分方程在动态系统中的应用 2
二、人口增长模型的基础构建 3
(一)传统人口增长模型概述 3
(二)差分方程在人口模型中的引入 3
(三)模型参数的选择与意义 4
三、差分方程在人口预测中的实现 4
(一)时间序列数据的处理方法 4
(二)预测模型的构建与求解 5
(三)模型结果的验证与误差分析 5
四、差分方程预测的实际应用案例 6
(一)典型地区的人口增长分析 6
(二)不同政策对人口预测的影响 6
(三)长期趋势与短期波动的结合研究 7
结论 7
参考文献 9
致谢 9
