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.

本文关注的是种群生物学中自然出现的非线性优化问题。我们考虑居住在斑驳环境中的具有逻辑增长的单个物种的种群，并研究斑块的分散和空间异质性对平衡总种群的影响。我们的目标是在资源总量有限的约束下，通过在补丁之间重新分配资源来最大化总人口。结果表明，当扩散率足够小或足够大时，全局最大化器可以针对任意数量的补丁进行表征。为了表明这一点，我们计算了两个补丁案例中总人口相对于资源的第一个变化。在三个或更多补丁的情况下，我们通过使用关于扩散率的泰勒展开来计算所有补丁的渐近展开。为了表征全局最大化器的形状，我们使用递归关系来确定所有补丁的所有系数。