Optimizing Resource Distribution in a One-Dimensional Logistic Diffusion Model

TL;DR

The paper tackles maximizing the equilibrium population in a bounded 1D logistic diffusion model by optimally distributing a resource $m$ under a mean constraint. It develops a block-decomposition framework to handle potential fragmentation of optimal resources and introduces the advantage function $H(l,b)$ to quantify the gain from allocating resources on subintervals, recasting the optimization into a convexity analysis of $H$. A key result is the superlinear (convex) behavior of $H$ for small total resource, which combined with block refinement yields an explicit structure for the optimal control. The authors also derive an analytic expansion for the total population $F_\mu(m)$ in the small-resource regime and obtain first-order derivative information, enabling a precise determination of the optimal resource pattern: for sufficiently small $m_0$, the optimum concentrates on a single end-interval $m_\mu^* = \chi_{(0,b)}$ (up to symmetry). This work provides structural insights and explicit optimal configurations in a 1D setting, with potential implications for resource allocation in diffusion-dominated ecological systems.

Abstract

In this article, we study the optimization of resource distributions in a one-dimensional logistic diffusive model. The goal is to determine a distribution on a bounded one-dimensional domain that maximizes the total population at equilibrium. Previous works have shown that optimal resources are bang-bang, and in one dimension, a sufficiently large dispersal rate forces the optimal resource to be concentrated. For general dispersal rates, however, the analysis becomes more difficult because the equilibrium population may behave irregularly, and the optimal resource may be fragmented. To address this, we introduce a block decomposition that reduces fragmented resources to a collection of concentrated blocks. We then define an advantage function, which measures the gain in the equilibrium population obtained by allocating resources on a fixed interval and is used to analyze the contribution of each block to the total population. This function also allows us to reformulate the optimization problem as a convexity analysis of the advantage function. We prove the superlinearity of this function when the total resource is small enough, and this property leads to an explicit characterization of the optimal control with sufficiently small total resource.

Figures (5)

• Figure 1: Block Decomposition of $\theta_{m, \mu}$
• Figure 2: Advantage Function $H(l, b)$
• Figure 3: The Phase Diagram of $\theta$
• Figure 4: The Phase Diagram of $\theta_{\tilde{m}_n, \mu}$
• Figure 5: The Phase Diagram of $\theta_{m_n, \mu}$

Theorems & Definitions (20)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem 1: Block decomposition
• Theorem 2: Optimal control with small resource
• Lemma 1
• proof
• Lemma 2