随机环境下种群增长模型的数值模拟与最优解  

NUMERICAL SIMULATION AND OPTIMAL SOLUTION FOR THE MODEL OF POPULATION GROWTH IN A STOCHASTIC ENVIRONMENT

作　　者：余味 乔韵璇[1] YU Wei QIAO Yunxuan(Center for Applied Statistics, School of Statistics, Renmin University of China, Zhongguancun Street 59, Haidian District, Beijing 100872)

机构地区：[1]中国人民大学统计学院,北京100872

出　　处：《系统科学与数学》2016年第11期2087-2098,共12页Journal of Systems Science and Mathematical Sciences

基　　金：国家自然科学基金(11471335)资助课题

摘　　要：由于很多随机微分方程没有显式解,其数值算法是近年研究的一个热点问题.文章用Milstein高阶法对随机Logistic模型进行数值求解.随机Logistic模型以及由它衍生出的一系列模型是生物经济中常用的模型,对于研究生物资源的适度开发和控制捕获以获得最大的经济效益具有重要的意义.文章首先用数值模拟得到此模型在不同参数下的样本轨道,并且与直接用其强解的表达式得到的轨道相对比.然后考虑三种不同的捕获模式-固定捕获努力量,两点分布的捕获努力量和与种群数量变化比例相同的捕获努力量,其中后两种捕获努力量是与种群数量有关的随机过程.用模拟的方法得出这三种捕获模式的长期平均产出的变化规律,并且比较这三种捕获模式的优劣.发现设计的后两种捕获模式相比于固定捕获努力量各有其优点.最后用随机Logistic模型分析一个北冰洋鳕鱼的数据.Since many stochastic differential equations have no closed-form solution, many researchers focus on developing numerical algorithms. In this paper, we use the Milstein high order algorithm to study the stochastic Logistic model. The stochastic Logistic model is frequently used in bio-economic analysis, along with various models derived from it. They are useful and meaningful for the studies about gain of economic benefits through moderate exploitation and capture monitoring of biological resources. In our paper, we will first simulate the sample path of the stochastic Logistic model by the Milstein high order algorithm under different model parameters and compare with the path generated out of the strong solution directly. Then we consider 3 different fishing patterns-- A fixed fishing effort, a binary fishing effort and a fishing effort changing proportional to the population. We use simulations to compare the long- time average yield of the 3 patterns and find that under pattern 2 we can invest large fishing effort when the population grows and do nothing otherwise, such that some organization cost can be saved. And pattern 3 can provide larger long-time yield at some cases. Furthermore, under pattern 3, the time before the population becomes stable are much smaller than the other two. At last we use the model to fit a data set about north east Arctic cod stock.

关 键 词：种群增长模型 随机微分方程 蒙特卡洛模拟 捕获努力量 最优解 

分 类 号：O211.63[理学—概率论与数理统计]