Asymptotic analysis of the 2D narrow-capture problem for partially accessible targets
File(s)
Accepted version
Author(s)
Bressloff, Paul C
Type
Journal Article
Abstract
In this paper we use singular perturbation theory to solve the 2D narrow capture problem for a set of partially accessible targets U𝑘, 𝑘=1,…,𝑁, in a bounded domain Ω⊂ℝ2. In contrast to previous models of narrow capture, we assume that when a searcher finds a target by attaching to the partially adsorbing surface 𝜕U𝑘 it does not have immediate access to the resources within the target interior. Instead, the searcher remains attached to the surface for a random waiting time 𝜏, after which it either gains access to the resources within (surface absorption) or detaches and continues its search process (surface desorption). We also consider two distinct desorption scenarios – either the particle continues its search from the point of desorption or rapidly returns to its initial search position. In applications to animal foraging, the latter would correspond to an animal returning to its home base whereas the resources within a target could represent food or shelter. We formulate the narrow capture problem in terms of a set of renewal equations that relate the probability density and target flux densities for absorption to the corresponding quantities for irreversible adsorption. The renewal equations, which effectively sew together successive rounds of adsorption and desorption prior to the final absorption event, provide a general probabilistic framework for incorporating non-Markovian models of desorption/absorption and different search scenarios following desorption. We solve the general renewal equations in two stages. First, we calculate the Laplace transformed target fluxes for irreversible adsorption by solving a Robin boundary value problem (BVP) in the small-target limit using matched asymptotic analysis. We then use the inner solution of the BVP to solve the corresponding Laplace transformed renewal equations for non-Markovian desorption/absorption, which leads to explicit Neumann series expansions of the corresponding target fluxes. Finally, the latter are used to determine the corresponding splitting probabilities and conditional mean FPTs for absorption.
Date Issued
2025-10-01
Date Acceptance
2025-07-09
Citation
SIAM Journal on Applied Mathematics, 2025, 85 (5), pp.2122-2144
ISSN
0036-1399
Publisher
Society for Industrial and Applied Mathematics
Start Page
2122
End Page
2144
Journal / Book Title
SIAM Journal on Applied Mathematics
Volume
85
Issue
5
Copyright Statement
© 2025 Society for Industrial and Applied Mathematics. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Publication Status
Published
Date Publish Online
2025-09-23