Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo, 152-8551, Japan.
Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA.
J Math Biol. 2021 Jan 19;82(1-2):2. doi: 10.1007/s00285-021-01565-7.
This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the population of a single species with logistic growth residing in a patchy environment and study the effects of dispersal and spatial heterogeneity of patches on the total population at equilibrium. Our objective is to maximize the total population by redistributing the resources among the patches under the constraint that the total amount of resources is limited. It is shown that the global maximizer can be characterized for any number of patches when the diffusion rate is either sufficiently small or large. To show this, we compute the first variation of the total population with respect to resources in the two patches case. In the case of three or more patches, we compute the asymptotic expansion of all patches by using the Taylor expansion with respect to the diffusion rate. To characterize the shape of the global maximizer, we use a recurrence relation to determine all coefficients of all patches.
本文研究了一个自然出现在种群生物学中的非线性优化问题。我们考虑了一个在斑块环境中具有逻辑增长的单一物种的种群，并研究了扩散和斑块空间异质性对平衡时总种群的影响。我们的目标是通过在资源总量有限的约束下在斑块之间重新分配资源来最大化总种群。结果表明，当扩散率足够小时或足够大时，可以为任意数量的斑块特征化全局最大值器。为了证明这一点，我们在两个斑块的情况下计算了总种群相对于资源的一阶变分。在三个或更多斑块的情况下，我们通过使用相对于扩散率的泰勒展开式来计算所有斑块的渐近展开式。为了描述全局最大值器的形状，我们使用递归关系来确定所有斑块的所有系数。
