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首页> 外文期刊>Journal of Mathematical Biology >Harvesting and seeding of stochastic populations: analysis and numerical approximation
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Harvesting and seeding of stochastic populations: analysis and numerical approximation

机译：随机群体的收获和播种：分析和数值逼近

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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 thresholds such that: (1) if the population size is 'low', so that it lies in (0,L1] there is seeding using the maximal seeding rate; (2) if the population size 'moderate', so that it lies in (L1,L2) there is no harvesting or seeding; (3) if the population size is 'high', so that it lies in the interval [L2,infinity) 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 thresholds L1(x2)
机译：我们研究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，我们都可以收获或种子（重新填充）物种。收获带来了经济增益，同时播种了费用。问题是找到最佳的收获播种策略，最大化预期的总收入减去花费的费用必须支付各种物种的种子。在hpenet等。 （J数学BIOL 79（2）：533-570,2019B）当一个人对人口绝对控制时，我们考虑了这个问题（无限收获和播种率也是可能的）。在许多情况下，这些近似不会使生物学意义，并且必须考虑当播种和收获率的一个或两者的界限时会发生什么。本文的重点是对这三种新颖设置的分析：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。即使一个维度收获问题也可能很难解决。一旦看着一个以上物种的生态系统，分析结果通常会变得棘手。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近似方法。这是通过Markov链接的连续时间动态来完成的，然后显示近似收敛到正确的最佳策略，因为网格尺寸变为零。通过实施这些数值近似，我们能够在特定关键示例中获得有关如何最好的收获和种子种类的定性信息。我们能够通过数值实验表明，在单一物种中设置最佳播种机策略始终是阈值类型。这意味着有阈值，使得：（1）如果人口大小为“低”，则它位于（0，l1]中使用最大播种率;（2）如果人口大小“中等”，因此，它位于（L1，L2）中没有收获或播种;（3）如果人口大小为“高”，则它位于间隔[L2，Infinity）使用最大收集率是否有收获。一旦我们有一个具有至少两个物种的系统，数值实验表明，恒定的阈值策略不再最佳。假设有两个竞争物种，我们只允许收获或种子物种1.接种和收获的最佳策略涉及依赖于密度x（2）的下阈值和上阈值L1（x2）

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