Maximization of the total population in a reaction–diffusion model with logistic growth
Maximization of the total population in a reaction–diffusion model with logistic growth
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DOI：
10.1007/s00526-018-1353-7
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发表时间：
2018-04
影响因子：
2.1
通讯作者：
Kentaro Nagahara;E. Yanagida
中科院分区：
文献类型：
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作者：
Kentaro Nagahara;E. Yanagida
本文研究了种群生物学中自然出现的一个非线性优化问题。我们考虑的空间异质性的影响，在一个生物物种的总人口在一个稳定的状态，使用反应扩散逻辑模型。我们的目标是在资源总量有限的情况下，当资源在栖息地中分布时，使总种群数量最大化，以控制种群的内在增长率。证明了在一定条件下，局部最大化器必须是“bang-bang”型的，这部分回答了Ding等人（Nonlinear Anal真实的World Appl 11（2）：688-704，2010）提出的猜想.为此，我们计算总人口的第一和第二变化。当增长率不是bang-bang型时，在某些情况下，第一个变差变为非零，因此资源分配不是局部最大化。当第一个变量为零时，我们证明第二个变量为正。这些结果表明，bang-bang性质是必不可少的总人口最大化。
This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the effect of spatial heterogeneity on the total population of a biological species at a steady state, using a reaction–diffusion logistic model. Our objective is to maximize the total population when resources are distributed in the habitat to control the intrinsic growth rate, but the total amount of resources is limited. It is shown that under some conditions, any local maximizer must be of “bang–bang” type, which gives a partial answer to the conjecture addressed by Ding et al. (Nonlinear Anal Real World Appl 11(2):688–704, 2010). To this purpose, we compute the first and second variations of the total population. When the growth rate is not of bang–bang type, it is shown in some cases that the first variation becomes nonzero and hence the resource distribution is not a local maximizer. When the first variation becomes zero, we prove that the second variation is positive. These results implies that the bang–bang property is essential for the maximization of total population.