Abstract

This paper considers environmental–economic models of optimal harvesting, given by impulsive differential equations, which depend on random parameters. We assume that lengths of intervals \({{\theta }_{k}}\) between the moments of impulses \({{\tau }_{k}}\) are random variables and the sizes of impulse influence depend on random parameters \({{{v}}_{k}},\)\(k = 1,2, \ldots \) One example of such objects is an impulsive equation that simulates dynamics of the population subject to harvesting. In the absence of harvesting, the population development is described by the differential equation \(\dot {x} = g(x),\) and in time points \({{\tau }_{k}}\) some random share of resource \({{{v}}_{k}},\)\(k = 1,2, \ldots \) is taken from the population. We can control the gathering process so that to stop harvesting when its share will appear big enough to save the biggest possible remaining resource to increase the size of the next gathering. Let the equation \(\dot {x} = g(x)\) have an asymptotic stable solution \(\varphi (t) \equiv K,\) whose attraction region is the interval \(({{K}_{1}},{{K}_{2}}),\) where \(0 \leqslant {{K}_{1}} < K < {{K}_{2}}.\) We construct the control \(u = ({{u}_{1}}, \ldots ,{{u}_{k}}, \ldots ),\) which limits a share of the harvested resource at each moment of time \({{\tau }_{k}}\) so that the quantity of the remaining resource, since some moment \({{\tau }_{{{{k}_{0}}}}},\) would be not less than the given value \(x \in ({{K}_{1}},K).\) For any \(x \in ({{K}_{1}},K)\) the estimates of the average temporal benefit, valid with the probability of one, are obtained. It is shown that there is a unique \(x* \in ({{K}_{1}},K),\) at which the lower estimate reaches the greatest value. Thus, we described the way of population control at which the value of the average time profit can be lower estimated with the probability of one by the greatest number whenever possible.

中文翻译：

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在概率环境经济模型中估算平均时间收益

摘要

本文考虑了由随机参数决定的脉冲微分方程给出的最佳收获的环境经济模型。我们假设脉冲\（{{\ tau} _ {k}} \）之间的间隔\（{{\ theta __ {k}} \）的长度是随机变量，并且脉冲影响的大小取决于随机参数\（{{{v}} _ {k}}，\）\（k = 1,2，\ ldots \）这样的对象的一个​​例子是一个脉冲方程，它模拟了受收获人口的动态。在没有收获的，人口发展由差分方程来描述\（\点{X} = G（X），\）和在时间点\（{{\ tau蛋白} _ {K}} \）一些随机资源份额\（{{{v}} _ {k}}，\）\（k = 1,2，\ ldots \）来自总体。我们可以控制收集过程，以便在其份额显示足够大时停止收集，以节省最大可能剩余的资源，从而增加下一次收集的规模。令方程\（\ dot {x} = g（x）\）具有一个渐近稳定解\（\ varphi（t）\ equiv K，\），其吸引区域为区间\（（{{K} _ { 1}}，{{K} _ {2}}），\）其中\（0 \ leqslant {{K} _ {1}} <K <{{K} _ {2}}。\）我们构造了控制\（u =（{{u} _ {1}}，\ ldots，{{u} _ {k}}，\ ldots）\），它限制了每个时间\（ {{\ tau} _ {k}} \）使得剩余资源的数量，因为一些力矩\（{{\ tau蛋白} _ {{{{K} _ {0}}}}}，\）将是不小于给定值\（X \在（{{K} _ {1}}，K）。\）对于任何\（x \ in（{{K} _ {1}}，K）\），平均时间收益的估计值与概率一起有效之一。结果表明，存在一个唯一的\（x * \ in（{{K} _ {1}}，K）\），较低的估计值会达到最大值。因此，我们描述了一种人口控制方式，在这种方式下，只要有可能，平均时间利润的值就可以以最大概率乘以一的概率降低。