Harvesting and seeding of stochastic populations: analysis and numerical approximation
Harvesting and seeding of stochastic populations: analysis and numerical approximation
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随机种群的收获和播种：分析和数值近似
DOI：
10.1007/s00285-020-01502-0
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发表时间：
2020
影响因子：
1.9
通讯作者：
Tran, Ky Quan
中科院分区：
文献类型：
--
作者：
Hening, Alexandru;Tran, Ky Quan
我们研究的是受随机环境波动影响的相互作用物种的生态系统。在任何时候，我们都可以收获或播种（重新繁殖）物种。收获带来经济收益，而播种则产生成本。问题是要找到最佳的收获播种策略，最大限度地提高预期的总收入从收获减去成本必须支付的各种物种的播种。在Hening等人（J Math Biol 79（2）：533-570，2019 b）中，我们考虑了当一个人对种群具有绝对控制时的这个问题（无限的收获和播种率是可能的）。在许多情况下，这些近似值在生物学上是没有意义的，人们必须考虑当播种率和收获率之一或两者都有限时会发生什么。本文的重点是分析这三个新的设置：有界播种和无限收获，有界播种和有界收获，无限播种和有界收获。即使是一维的收获问题也很难解决。一旦观察一个有不止一个物种的生态系统，分析结果通常变得难以处理。为了获得关于系统的定性行为的信息，我们开发了严格的数值逼近方法。这是通过近似的连续时间动态马尔可夫链，然后表明，近似收敛到正确的最佳策略，网格大小为零。通过实施这些数值近似，我们能够获得有关如何在特定关键示例中最佳收获和播种物种的定性信息。通过数值实验，我们能够证明，在单物种设置的最优播种收获策略总是阈值型。这意味着存在这样的阈值：（1）如果种群大小为“低”，那么它位于，使用最大播种率播种;（2）如果种群大小为“中等”，那么它位于，没有收获或播种;（3）如果种群大小为“高”，那么它位于区间内，使用最大收获率收获。一旦我们有一个至少有两个物种的系统，数值实验表明，恒定阈值策略不再是最优的。假设有两个竞争物种，我们只允许收获或播种物种1。播种和收获的最佳策略将涉及较低和较高的阈值，这取决于物种的密度2.
We study an ecosystem of interacting species that are influenced by random environmental fluctuations. At any point in time, we can either harvest or seed (repopulate) species. Harvesting brings an economic gain while seeding incurs a cost. The problem is to find the optimal harvesting-seeding strategy that maximizes the expected total income from harvesting minus the cost one has to pay for the seeding of various species. In Hening et al. (J Math Biol 79(2):533–570, 2019b) we considered this problem when one has absolute control of the population (infinite harvesting and seeding rates are possible). In many cases, these approximations do not make biological sense and one must consider what happens when one, or both, of the seeding and harvesting rates are bounded. The focus of this paper is the analysis of these three novel settings: bounded seeding and infinite harvesting, bounded seeding and bounded harvesting, and infinite seeding and bounded harvesting. Even one dimensional harvesting problems can be hard to tackle. Once one looks at an ecosystem with more than one species analytical results usually become intractable. In order to gain information regarding the qualitative behavior of the system we develop rigorous numerical approximation methods. This is done by approximating the continuous time dynamics by Markov chains and then showing that the approximations converge to the correct optimal strategy as the mesh size goes to zero. By implementing these numerical approximations, we are able to gain qualitative information about how to best harvest and seed species in specific key examples. We are able to show through numerical experiments that in the single species setting the optimal seeding-harvesting strategy is always of threshold type. This means there are thresholdssuch that: (1) if the population size is ‘low’, so that it lies in, there is seeding using the maximal seeding rate; (2) if the population size ‘moderate’, so that it lies in, there is no harvesting or seeding; (3) if the population size is ‘high’, so that it lies in the interval, there is harvesting using the maximal harvesting rate. Once we have a system with at least two species, numerical experiments show that constant threshold strategies are not optimal anymore. Suppose there are two competing species and we are only allowed to harvest or seed species 1. The optimal strategy of seeding and harvesting will involve lower and upper thresholdswhich depend on the densityof species 2.
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