On the exploitation of a population given by a system of linear equations with random parameters

M. S. Woldeab, L. I. Rodina
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Abstract

We consider a population whose dynamics in the absence of exploitation is given by a system of linear homogeneous differential equations, and some random shares of the resource of each species at fixed times, are extracted from this population. We assume that the harvesting process can be controlled in such a way as to limit the amount of the extracted resource in order to increase the size of the next harvesting. A method for harvesting a resource is described, in which the largest value of the average time benefit is reached with a probability of one, provided that the initial amount of the population is constantly maintained or periodically restored. The harvesting modes are also considered in which the average time benefit is infinite. To prove the main assertions, we use the corollary of the law of large numbers proved by A.N. Kolmogorov. The results on the optimal resource extraction for systems of linear difference equations, a particular case of which are Leslie and Lefkovich population dynamics models, are given.
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一类具有随机参数的线性方程组给出的总体的开发
我们考虑一个种群,在没有开发的情况下,其动态由线性齐次微分方程系统给出,并且从该种群中抽取每个物种在固定时间的一些随机资源份额。我们假设采集过程可以通过这样一种方式进行控制,即限制提取资源的数量,以便增加下一次采集的规模。本文描述了一种获取资源的方法,只要不断保持或定期恢复种群的初始数量,以1的概率达到平均时间效益的最大值。同时考虑了平均时间效益无限的收获模式。为了证明主要的断言,我们使用柯尔莫哥洛夫证明的大数定律的推论。以Leslie和Lefkovich种群动力学模型为例,给出了线性差分方程系统的最优资源提取的结果。
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