Asymptotic analysis of the 2D narrow-capture problem for partially accessible targets

TL;DR

The paper addresses a 2D narrow-capture problem with multiple partially accessible targets, distinguishing adsorption on target surfaces from internal absorption and introducing a renewal framework to couple adsorption and absorption in both Markovian and non-Markovian settings. It develops a two-stage approach: first solving the Laplace-transformed flux for irreversible adsorption via a Robin boundary-value problem in the small-target limit using matched asymptotics, and then using the inner (local) solution to solve renewal equations for non-Markovian desorption/absorption, yielding explicit Neumann-series expressions that can be summed non-perturbatively in the logarithmic small-target parameter $\nu=-1/\ln\epsilon$. The method also accommodates different post-desorption behaviours, including returning to the home base, and provides practical formulas for splitting probabilities $\pi_k(\mathbf{x}_0)$ and conditional MFPTs $\mathfrak{T}_k(\mathbf{x}_0)$, with concrete illustration via a pair of targets in the unit disc. The results advance understanding of how adsorption, desorption, and interior-access constraints shape search efficiency, and they open avenues for extensions to non-circular geometries, 3D settings, and encounter-based non-Markovian mechanisms in biological and ecological contexts.

Abstract

In this paper we use singular perturbation theory to solve the 2D narrow capture problem for a set of partially accessible targets $\calU_k$, $k=1,\ldots,N$, in a bounded domain $Ω\subset \R^2$. In contrast to previous models of narrow capture, we assume that when a searcher finds a target by attaching to the partially adsorbing surface $\partial \calU_k$ 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 ({\em surface absorption}) or detaches and continues its search process ({\em 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. 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 Robbin 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.

Figures (5)

• Figure 1.1: Diffusion of a searcher (particle) in a bounded domain $\Omega$ with $N$ partially accessible targets ${\mathcal{U}}_j$, $j=1,\ldots,N$. Each boundary surface $\partial {\mathcal{U}}_j$ is taken to be partially adsorbing. However, adsorption of the particle at a point on $\partial {\mathcal{U}}_j$ does not give it immediate access to the resources within ${\mathcal{U}}_j$. After some random waiting time attached to the surface, the particle either succeeds in entering the interior ${\mathcal{U}}_j$ (absorption) or detaches from the target to continue its search (desorption).
• Figure 3.1: Formulation of the multi-target search process with irreversible adsorption as a singularly perturbed narrow capture problem. (a) Original unscaled domain. (b) Construction of the inner solution in terms of stretched coordinates $\mathbf{z}=\epsilon^{-1}(\mathbf{x}-{\mathbf{x}}_j)$, where ${\mathbf{x}}_j$ is the centre of the $j$-th target. The rescaled radius is $\ell_j$ and the region outside the compartment is taken to be ${\mathbb R}^2$ rather than the bounded domain $\Omega$. (c) Construction of the outer solution. The $k$th target is shrunk to a single point $\mathbf{x}_k$. The outer solution is expressed in terms of the corresponding modified Neumann Green's function and then matched with the inner solution around each target.
• Figure 4.1: A pair of identical targets of radii $\epsilon$ are placed at the positions $\mathbf{x}_1=(a,0)$ and $\mathbf{x}_2=(-a,0)$ in the unit disc with $a=0.5$. The contour plot shows the splitting probability $\overline{\pi}_1(\mathbf{x}_0)$ for irreversible adsorption, see equation (\ref{['2targ']}a), as a function of $\mathbf{x}_0=(r\cos \theta,r\sin \theta)$ with $0<r<1$ and $0\leq \theta \leq 2\pi$. Other parameters are $\nu=0.1$, $D=1$ and $\kappa_0\rightarrow \infty$.
• Figure 4.2: Contour plots of the splitting probability $\pi_1(\mathbf{x}_0)$ for absorption, equation (\ref{['resbpi']}), as a function of $\mathbf{x}_0=(r\cos \theta,r\sin \theta)$: (a) $\sigma_1=0.25,\sigma_2 =0.75$, (b) $\sigma_1=\sigma_2=0.5$, (c) $\sigma_1 =0.75,\sigma_2=0.25$. Same target configuration and other parameters as Fig. \ref{['fig4']}.
• Figure 4.3: Contour plots of the unconditional MFPT ${\mathfrak T}(\mathbf{x}_0)$ for absorption, equation (\ref{['calT']}), as a function of $\mathbf{x}_0=(r\cos \theta,r\sin \theta)$: (a) $\sigma_1=0.25,\sigma_2 =0.75$, (b) $\sigma_1=\sigma_2=0.5$, (c) $\sigma_1 =0.75,\sigma_2=0.25$. We also set $\langle \tau\rangle =0$ for both targets.Same target configuration and other parameters as Fig. \ref{['fig4']}.