The spatial distribution of resources for diffusive populations can have a strong effect on population abundance. We investigate the optimal allocation of resources for a diffusive population. Population dynamics are represented by a parabolic partial differential equation with density-dependent growth and resources are represented through their space- and time-varying influence on the growth function. We consider both local and integral constraints on resource allocation. The goal is to maximize the abundance of the population while minimizing the cost of resource allocation. After characterizing the optimal control in terms of the population solution and the adjoint functions, we illustrate several scenarios numerically. The effects of initial and boundary conditions are important for the optimal allocation of resources.

中文翻译：

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扩散人口模型的最佳资源分配

分散种群资源的空间分布对种群丰度有很大的影响。我们研究了分散人口的最佳资源分配。人口动态由具有密度依赖性增长的抛物线偏微分方程表示，资源通过其对增长函数的空间和时间变化影响来表示。我们考虑资源分配的局部约束和整体约束。目标是最大化人口的丰度，同时最小化资源分配的成本。在根据总体解和伴随函数表征最优控制之后，我们用数值说明了几种情况。初始条件和边界条件的影响对于资源的优化配置很重要。