SIAM Journal on Applied Mathematics, Volume 81, Issue 1, Page 153-172, January 2021.
Following several recent works devoted to the analysis of spatial heterogeneity in reaction-diffusion equations, we investigate the problem of optimizing the total population size for logistic diffusive models with respect to resources distributions. Using the spatially heterogeneous Fisher-KPP equation, we obtain a surprising fragmentation phenomenon: depending on the scale of diffusivity (i.e., the dispersal rate), it is better to either concentrate or fragment resources. Our main result is that, the smaller the dispersal rate of the species in the domain, the more optimal resources distributions tend to oscillate. This is in sharp contrast with other criteria in population dynamics, such as the classical problem of optimizing the survival ability of a species, where concentrating resources is always favorable, regardless of the diffusivity. Our study is completed by numerous numerical simulations that confirm our results.

中文翻译：

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非能量最优控制问题的碎片现象：逻辑扩散模型中总人口规模的优化

SIAM应用数学杂志，第81卷，第1期，第153-172页，2021年1月。
继最近几篇致力于分析反应扩散方程中空间异质性的工作之后，我们研究了针对资源分配的逻辑扩散模型优化总人口规模的问题。使用空间异构的Fisher-KPP方程，我们得到一个令人惊讶的碎片现象：根据扩散的规模（即分散率），最好集中或碎片化资源。我们的主要结果是，物种在该域中的扩散率越小，越倾向于优化资源分配。这与种群动态中的其他标准形成鲜明对比，例如优化物种生存能力的经典问题，即无论扩散程度如何，资源的集中总是有利的。