Target competition for resources under multiple search-and-capture events with stochastic resetting

Abstract

We develop a general framework for analysing the distribution of resources in a population of targets under multiple independent search-and-capture events. Each event involves a single particle executing a stochastic search that resets to a fixed location x_r at a random sequence of times. Whenever the particle is captured by a target, it delivers a packet of resources and then returns to x_r, where it is reloaded with cargo and a new round of search and capture begins. Using renewal theory, we determine the mean number of resources in each target as a function of the splitting probabilities and unconditional mean first passage times of the corresponding search process without resetting. We then use asymptotic PDE methods to determine the effects of resetting on the distribution of resources generated by diffusive search in a bounded two-dimensional domain with N small interior targets. We show that slow resetting increases the total number of resources M_tot across all targets provided that $\sum _{j=1}^{N}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_j)<0$, where G is the Neumann Green’s function and x_j is the location of the j-th target. This implies that M_tot can be optimized by varying r. We also show that the k-th target has a competitive advantage if $\sum _{j=1}^{N}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_j)>NG({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_k)$.

1. Introduction

Random search strategies are used throughout nature as a means of efficiently searching for one or more targets of unknown location. Examples include animals foraging for food or shelter [1–3], proteins searching for particular sites on DNA [4–7] and biochemical reaction kinetics [8,9]. One major example of a search protocol is random intermittent search, in which a particle randomly switches between a slow search phase and a faster non-search phase [10–13].

At the macroscopic scale, intermittent motion has been observed in a variety of animal species during exploratory behaviour. Intermittent motion also occurs at the microscopic level of reaction kinetics and molecular transport within biological cells. For example, diffusing molecules within the cellular environment intermittently bind and unbind to molecular motors that execute ballistic transport along cytoskeletal filaments powered by ATP hydrolysis; reaction with a target molecule can only occur during the free diffusive phase [14]. Another example of an intermittent search strategy occurs in computational search algorithms, such as simulated annealing, which try to locate the global minimum of a complex (free) energy landscape. However, at low ‘temperatures’, the system may become trapped in a metastable, local minimum for a long time. One way to speed up the search process is to restart the algorithm from the initial configuration at a random sequence of times, which can be modelled probabilistically as a random walk with restart [15–17].

There has recently been an explosion of interest in search processes with stochastic resetting or restart, since they provide a general theoretical framework for understanding a wide range of naturally and artificially occurring phenomena [18]. The fundamental idea is that the position of a particle performing a stochastic search for some target is reset to a fixed location at a random sequence of times, which is typically (but not necessarily) generated by a Poisson process. In many cases, there exists an optimal resetting rate for minimizing the mean first passage time (MFPT) to reach a target. This was originally established for Brownian motion in an unbounded domain [19–21], but has subsequently been observed in a wide range of stochastic processes with resetting, including diffusion in an interval [22], anomalous diffusion processes such as Levy flights [23], velocity jump processes [24,25] and resetting with finite return times [26–28] or refractory periods [29,30]. There is also growing evidence that stochastic search processes with optimal resetting display some form of universality [27,31–35].

Almost all studies of search processes have focused on the first passage time (FPT) problem to be captured by a single target, after which the search process ends; we will refer to this as a single search-and-capture event. However, there are a number of examples in cell biology where the arrival of a particle results in the delivery of some resource to a target, after which the particle escapes and returns to its initial position. The particle may then be resupplied with cargo and initiate a new search process that delivers cargo to a different target. Over multiple search-and-capture events, a distribution of resources builds up across the population of competing targets. One example is the motor-driven intracellular transport of vesicles to synaptic targets in the axons and dendrites of neurons [36–38]. The particle (searcher) represents a molecular motor complex moving along a polymerized filament such as a microtubule. After delivering its cargo to a synapse, the complex returns to the cell body, where it can be reloaded with vesicles and start a new search-and-capture process. Another example is the transport of morphogen at the tips of growing actin-rich filaments (cytonemes) to target cells during embryonic development in vertebrates [39–41]. Now the particle represents the tip of a cytoneme growing from a source cell. When the tip makes contact with a target cell, it delivers its cargo and then rapidly retracts back to the source cell. A new cytoneme can then nucleate from the source cell and initiate a new round of search and capture. In both examples, the particle may return to its initial position without delivering its cargo, which corresponds to a resetting or restart event with a finite return time and a refractory period.

In the particular case of cytoneme-based morphogenesis, we have recently shown [42,43] how multiple search-and-capture events can be formulated in terms of a G/M/∞ queue [44], under the additional assumption that the build-up of resources within a set of target cells is counterbalanced by degradation, so that there is a steady-state probability distribution for the number of packets in the long-time limit. Queuing theory concerns the mathematical analysis of waiting lines formed by customers randomly arriving at some service station, and staying in the system until they receive service from a group of servers. A sequence of search-and-capture events can be mapped onto a queuing process as follows: individual resource packets are analogous to customers, the delivery of a packet corresponds to a customer arriving at the service station, and the degradation of a resource packet is the analogue of a customer exiting the system after being serviced. Finally, assuming that the packets are degraded independently of each other, the effective number of servers in the corresponding queuing model is infinite, that is, the presence of other customers does not affect the service time of an individual customer. One of the advantages of formulating the search problem as a queuing process is that one can use renewal theory to calculate moments of the distribution of resources in steady state. Within the context of cytonemes, the mean distribution corresponds to a morphogen gradient [42].

In this paper, we consider the general problem of how resources are distributed across a population of competing targets following multiple search-and-capture events with non-instantaneous stochastic resetting. We proceed by extending our previous work on cytoneme-based morphogenesis [42,43], which focused on one-dimensional search processes with partially absorbing target domains, to higher-dimensional search processes with totally absorbing target boundaries. We begin in §2, by considering a general stochastic search process without resetting in ${\mathbb{R}}^d$, and use queuing theory to derive an expression for the steady-state mean number of resource packets in each of N targets. (One can also derive higher-order moments such as the variance.) In particular, we show that the mean number of resources in the k-th target is given by 〈M_k〉 = π_k/(γ(T + τ_cap)), where π_k and T are the splitting probabilities and unconditional MFPT of the underlying search process, γ is the degradation rate and τ_cap is the mean time to load and unload cargo. In §3, we calculate the corresponding mean number of resources in the presence of resetting with delays. We exploit the fact that, following each resetting event, the particle loses all memory of previous search phases, which means that we can condition on whether or not the particle resets at least once, even though a reset event occurs at random times. (In other words, the stochastic process satisfies the strong Markov property.) This allows us to express the splitting probabilities and unconditional FPTs with resetting in terms of the splitting probabilities and conditional FPT densities of the search process without resetting. In general, explicit expressions for the latter cannot be obtained in the case of multiple targets and two-dimensional (2D) and three-dimensional (3D) search processes. On the other hand, progress can be made when the targets are sufficiently small and the search process is diffusion-based, since asymptotic PDE methods can be used to approximate the splitting probabilities and moments of the conditional FPT densities [45–50]. Therefore, in §4, we combine our stochastic analysis with asymptotic PDE methods to determine the effects of resetting on the distribution of resources generated by a diffusive search in a bounded 2D domain with small targets. In particular, we show that when the mean resetting rate r is sufficiently small, the number of resources in the k-th target increases if and only if $N^{-1}\sum _{j=1}^{N}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_j)>G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_k)$, where G is the Neumann Green’s function for the diffusion process, x_r is the reset location and x_j is the location of the j-th target. We also show that the total number of resources M_tot across all targets increases with r provided that $\sum _{j=1}^{N}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_j)<0$. Since M_tot → 0 as r → ∞, this suggests that M_tot can then be optimized by varying r. Analogous results hold in three dimensions, provided that the sums over Green’s functions are weighted by the shape capacitances of the targets.

2. Queuing theory of target resource allocation

Consider a particle (searcher) subject to stochastic motion in $\mathcal{U}\subset {\mathbb{R}}^d$. Suppose that there exist N fixed targets ${\mathcal{U}}_i\in \mathcal{U}$, i = 1, …, N, as shown in figure 1. The probability density p(x, t|x₀) for the particle to be at position x at time t, having started at x₀, evolves according to the master equation

$$\frac{\mathrm{\partial }p(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0)}{\mathrm{\partial }t}=\mathbb{L}p(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0)=-\mathrm{\nabla }\cdot \mathbf{\text{J}}(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0),$$ 2.1

where $\mathbb{L}$ is the infinitesimal generator of the stochastic process and J(x, t|x₀) denotes the probability flux. This is supplemented by the boundary conditions

$${\mathrm{\partial }}_{n}P(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0)=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }\mathcal{U};p(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0)=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }{\mathcal{U}}_a=\bigcup _{j=1}^N\mathrm{\partial }{\mathcal{U}}_j,$$ 2.2

and the initial condition p(x, 0|x₀) = δ(x − x₀). That is, the exterior boundary $\mathrm{\partial }\mathcal{U}$ is reflecting and the interior boundaries $\mathrm{\partial }{\mathcal{U}}_j$ are absorbing.

Figure 1.

Particle searching for one of N targets ${\mathcal{U}}_i\subset \mathcal{U}\subset {\mathbb{R}}^d$, i = 1, …, N. Each time the particle finds a target, it delivers a packet of resources and returns to its initial position x₀. (Online version in colour.)

Let ${\mathcal{T}}_k({\mathbf{\text{x}}}_0)$ denote the FPT that the particle is captured by the k-th target, with ${\mathcal{T}}_k({\mathbf{\text{x}}}_0)=\mathrm{\infty }$ indicating that it is not captured. Define Π_k(x₀, t) to be the probability that the particle is captured by the k-th target after time t, given that it started at x₀

$${\mathrm{\Pi }}_k({\mathbf{\text{x}}}_0,t)=\mathbb{P}[t<{\mathcal{T}}_k({\mathbf{\text{x}}}_0)<\mathrm{\infty }]={\int }_t^{\mathrm{\infty }}{J}_k({\mathbf{\text{x}}}_0,t^{\prime })\hspace{0.17em}\text{d}{t}^{\prime },$$ 2.3

where

$$J_k({\mathbf{\text{x}}}_0,t)=-{\int }_{\mathrm{\partial }{\mathcal{U}}_k}\mathbf{\text{J}}(\mathbf{\text{y}},t|{\mathbf{\text{x}}}_0)\cdot \hspace{0.17em}\text{d}\mathbf{\text{y}}.$$ 2.4

The negative sign indicates that the flux is into the domain ${\mathcal{U}}_k$. The splitting probability π_k(x₀) and conditional MFPT T_k(x₀) for the particle to be captured by the k-th target are then

$${\pi }_k({\mathbf{\text{x}}}_0)={\mathrm{\Pi }}_k({\mathbf{\text{x}}}_0,0)={\int }_0^{\mathrm{\infty }}{J}_k({\mathbf{\text{x}}}_0,t)\hspace{0.17em}\text{d}t$$ 2.5

and

$$T_k({\mathbf{\text{x}}}_0)=\mathbb{E}[{\mathcal{T}}_k|{\mathcal{T}}_k<\mathrm{\infty }]=\frac{1}{{\pi }_k({\mathbf{\text{x}}}_0)}{\int }_0^{\mathrm{\infty }}{\mathrm{\Pi }}_k({\mathbf{\text{x}}}_0,t)\hspace{0.17em}\text{d}t.$$ 2.6

We will assume that $\sum _k{\pi }_k({\mathbf{\text{x}}}_0)=1$, which implies that the particle is eventually captured by a target with probability 1. Finally, note that integrating equation (2.1) with respect to x and t implies that the survival probability up to time t is

$$Q({\mathbf{\text{x}}}_0,t)={\int }_{\mathcal{U}\mathrm{\setminus }{\mathcal{U}}_a}p(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0)\hspace{0.17em}\text{d}\mathbf{\text{x}}=\sum _{k=1}^N{\mathrm{\Pi }}_k({\mathbf{\text{x}}}_0,t).$$ 2.7

(Note that in [42,43] the search domain is one-dimensional and ${\mathcal{U}}_j$ is a partially absorbing domain: if $\mathbf{\text{x}}\in {\mathcal{U}}_j$ then the particle is absorbed by the corresponding target at a rate κ so that the flux is $J_k({\mathbf{\text{x}}}_0,t)=\kappa {\int }_{{\mathcal{U}}_k}p(\mathbf{\text{y}},t|{\mathbf{\text{x}}}_0)\hspace{0.17em}\text{d}\mathbf{\text{y}}$.)

Now suppose that, rather than being permanently absorbed or captured by a target, the particle delivers a discrete packet of some resource to the target and then returns to x₀, initiating another round of search and capture. (The return to x₀ is distinct from resetting, which can occur at any time; see §3.) We will refer to the delivery of a single packet as a burst event. The sequence of bursts resulting from multiple rounds of search and capture leads to an accumulation of packets within the targets, which we assume is counteracted by degradation at some rate γ. This is illustrated in figure 2 for two targets in a rectangular domain. We will assume that the total time for the particle to unload its cargo, return to x₀ and start a new search process is given by the random variable $\hat{\tau }$, which for simplicity is taken to be independent of the location of the targets. (This is reasonable if the sum of the mean loading and unloading times is much larger than a typical return time.) Let n ≥ 1 label the n-th burst event and denote the target that receives the n-th packet by j_n. If ${\mathbb{T}}_n$ is the time of the n-th burst, then the inter-arrival times are

$${\mathrm{\Delta }}_n:={\mathbb{T}}_n-{\mathbb{T}}_{n-1}={\hat{\tau }}_n+{\mathcal{T}}_{j_n},n\ge 1,$$ 2.8

with $\mathbb{E}[{\mathcal{T}}_j]={\pi }_j{T}_j$. Finally, given an inter-arrival time Δ, we denote the identity of the target that captures the particle by $\mathcal{K}(\mathrm{\Delta })$. We can then write, for each target j,

$$F_j(t)=\mathbb{P}[\mathrm{\Delta }<t,\mathcal{K}(\mathrm{\Delta })=j]=\mathbb{P}[\mathrm{\Delta }<t,|\mathcal{K}(\mathrm{\Delta })=j]\mathbb{P}[\mathcal{K}(\mathrm{\Delta })=j]={\pi }_j{\int }_0^t{\mathcal{F}}_j(\mathrm{\Delta })\hspace{0.17em}\text{d}\mathrm{\Delta },$$ 2.9

where ${\mathcal{F}}_j(\mathrm{\Delta })$ is the conditional inter-arrival time density for the j-th target. Let $\rho (\hat{\tau })$ denote the waiting time density of the delays ${\hat{\tau }}_n$. Then

$${\mathcal{F}}_j(\mathrm{\Delta })={\int }_0^{\mathrm{\Delta }}\hspace{0.17em}\text{d}t{\int }_0^{\mathrm{\Delta }}d\hat{\tau }\delta (\mathrm{\Delta }-t-\hat{\tau })f_j(t)\rho (\hat{\tau })={\int }_0^{\mathrm{\Delta }}{f}_j(t)\rho (\mathrm{\Delta }-t)\hspace{0.17em}\text{d}t,$$

where f_j(t) = J_j(t)/π_j is the conditional FPT density for a single search-and-capture event that delivers a packet to the j-th target. (For notational simplicity, we drop the explicit dependence on the initial position x₀.) In particular,

$$T_j={\int }_0^{\mathrm{\infty }}t{f}_j(t)\hspace{0.17em}\text{d}t=-\frac{1}{{\pi }_j}{\int }_0^{\mathrm{\infty }}t\frac{\text{d}{\mathrm{\Pi }}_j(t)}{\text{d}t}\hspace{0.17em}\text{d}t=\frac{{\tilde{\mathrm{\Pi }}}_j(0)}{{\pi }_j}.$$ 2.10

Laplace transforming the convolution equation then yields

$${\tilde{\mathcal{F}}}_j(s)={\tilde{f}}_j(s)\tilde{\rho }(s).$$ 2.11

Figure 2.

Multiple search-and-capture events. (a) Particle searching in a rectangular domain with two targets. Each time the particle reaches a target it delivers a discrete packet of resources (burst event) and then returns to x₀, where it is loaded with another packet and the process repeats. (b) Sample trajectory projected onto the z-coordinate showing a sequence of burst events. The delay time $\hat{\tau }$ between a burst event and initiation of a new search is generated from a waiting time density $\rho (\hat{\tau })$. (The lines with arrows do not represent actual trajectories.) (c) The burst sequence results in an accumulation of resource packets within each target, which is counteracted by degradation at some rate γ. (Online version in colour.)

We wish to determine the steady-state amount of resources accumulated by each of the targets. As we have previously shown within the context of cytoneme-based morphogenesis [42,43], this can be achieved by reformulating a multiple search-and-capture model as a G/M/∞ queuing process [44], following along similar lines to studies of gene transcription [51]. Here, we briefly sketch the basic steps; further details can be found in the cited references. Consider a particular target k. Let M_k(t) be the number of resource packets in the target that have not yet degraded at time t. Introduce the generating function G_k(z, t) and corresponding binomial moments according to

$$G_k(z,t)=\sum _{l=0}^{\mathrm{\infty }}{z}^l\mathbb{P}[M_k(t)=l],B_{m,k}(t)=\sum _{l=m}^{\mathrm{\infty }}\frac{l!}{(l-m)!m!}\mathbb{P}[M_k(t)=l],\hspace{0.17em}m\ge 1.$$ 2.12

Suppose that the targets have no resources at time t = 0. Using the theory of conditional expectations, it can be shown that the generating function satisfies the integral equation

$$\begin{array}{rl}{G}_k(z,t)& =\sum _{j=1}^N{\int }_t^{\mathrm{\infty }}\hspace{0.17em}\text{d}{F}_j(y)+\sum _{j\ne k}{\int }_0^t{G}_k(z,t-y)\hspace{0.17em}\text{d}{F}_j(y)\\ & +{\int }_0^t[z+(1-z)\mathrm{\Phi }(t-y)]G_k(z,t-y)\hspace{0.17em}\text{d}{F}_k(y).\end{array}$$ 2.13

Here, Φ(t) = 1 − e^−γt is the probability that a packet has not degraded over a time interval t. Differentiating equation (2.13) with respect to z and using

$$B_{m,k}(t)=\frac{1}{m!}{\left.\frac{{\text{d}}^m{G}_k(z,t)}{\text{d}{z}^m}\right|}_{z=1}$$

yields an iterative integral equation for the binomial moments

$$B_{m,k}(t)=\sum _{j=1}^N{\int }_0^t{B}_{m,k}(t-y)\hspace{0.17em}\text{d}{F}_j(y)+{\int }_0^t{B}_{m-1,k}(t-y)[1-\mathrm{\Phi }(t-y)]\hspace{0.17em}\text{d}{F}_k(y).$$ 2.14

Laplace transforming equation (2.14) after making the substitutions $\text{d}{F}_j(y)={\pi }_j{\mathcal{F}}_j(y)\hspace{0.17em}\text{d}y$ and 1 − Φ(t) = e^−γt, we find that

$${\tilde{B}}_{m,k}(s)=\left[\frac{{\pi }_k{\tilde{\mathcal{F}}}_k(s)}{1-\sum _{j=1}^N{\pi }_j{\tilde{\mathcal{F}}}_j(s)}\right]{\tilde{B}}_{m-1,k}(s+\gamma ).$$ 2.15

Multiplying both sides by s and taking the limit s → 0⁺ yields

$$B_{m,k}^{\ast }=\underset{t\to \mathrm{\infty }}{lim}{B}_{m,k}(t)=\underset{s\to 0^{+}}{lim}s{\tilde{B}}_{m,k}(s)={\lambda }_k{\tilde{B}}_{m-1,k}(\gamma ),$$

where

$${\lambda }_k:=\underset{s\to 0^{+}}{lim}\frac{s{\pi }_k{\tilde{\mathcal{F}}}_k(s)}{1-\sum _{j=1}^N{\pi }_j{\tilde{\mathcal{F}}}_j(s)}=\frac{{\pi }_k}{\sum _{j=1}^N{\pi }_j(T_j+⟨\hat{\tau }⟩)}.$$ 2.16

We have used L’Hôpital’s rule and equation (2.11), together with the following properties: ${\tilde{f}}_j(0)=1=\tilde{\rho }(0)$, ${\tilde{f}}_j^{\prime }(0)=-T_j$ and ${\tilde{\rho }}^{\prime }(0)=-{\tau }_{\text{cap}}.$ Here, τ_cap is the mean time for loading and unloading a packet. Equations (2.15) and (2.16) completely determine the steady-state binomial moments. In particular, since B_0,k(t) = 1 and ${\tilde{B}}_{0,k}(s)=1/s$, the mean number of packets in the k-th target is

$$B_{1,k}^{\ast }\equiv ⟨M_k⟩=\frac{{\lambda }_k}{\gamma }=\frac{{\pi }_k}{\gamma \sum _{j=1}^N{\pi }_j(T_j+{\tau }_{\text{cap}})}=\frac{{\pi }_k}{\gamma (T+{\tau }_{\text{cap}})},$$ 2.17

where $T=\sum _j{\pi }_j{T}_j$ is the unconditional MFPT. We can thus interpret λ_k as the mean rate at which a packet is delivered to the k-th target. This is consistent with equation (2.16), since T + τ_cap is the mean time for one successful delivery of a packet to any one of the targets and initiation of a new round of search and capture. Hence, its inverse is the mean rate of resource bursts and π_k are the fractions that are delivered to the k-th target (over many trials). The dependence of the mean 〈M_k〉 on the target label k specifies the steady-state allocation of resources across the set of targets. It will depend on the details of the particular search process (2.1), the positions and shapes of the targets within the domain $\mathcal{U}$, the initial position x₀ and the rate of degradation γ.

The above analysis can also be used to determine the higher-order statistics of the resource distribution. For example,

$$B_{2,k}^{\ast }\equiv \frac{1}{2}(⟨M_k^2⟩-⟨M_k⟩)={\lambda }_k{\tilde{B}}_1(\gamma )=\frac{{\lambda }_k}{2\gamma }\frac{{\pi }_k{\tilde{\mathcal{F}}}_k(\gamma )}{1-\sum _{j=1}^N{\pi }_j{\tilde{\mathcal{F}}}_j(\gamma )}.$$ 2.18

Hence, the variance of the number of resource packets is

$$\text{Var}[M_k]=2B_{2,k}^{\ast }+B_{1,k}^{\ast }(1-B_{1,k}^{\ast })=\frac{{\lambda }_k}{\gamma }\left[\frac{{\pi }_k{\tilde{\mathcal{F}}}_k(\gamma )}{1-\sum _{j=1}^N{\pi }_j{\tilde{\mathcal{F}}}_j(\gamma )}+1-\frac{{\lambda }_k}{\gamma }\right].$$ 2.19

Although we focus on a single searcher in this paper, it is straightforward to extend the results to the case of $\mathcal{N}$ independent searchers. In that case both the mean and variance scale as $\mathcal{N}$. (A related problem of multiple searchers seeking a single hidden target and subject to finite random lifetimes has been considered elsewhere [52,53].

3. Search and capture with stochastic resetting

Now suppose that, prior to being absorbed by one of the targets, the particle can reset to a fixed location x_r at a random sequence of times generated by an exponential probability density ψ(τ) = re^−rτ, where r is the resetting rate. The probability that no resetting has occurred up to time τ is then $\mathit{\Psi }(\tau )=1-{\int }_0^{\tau }\psi (s)\hspace{0.17em}\text{d}s={\mathbf{\text{e}}}^{-r\tau }$. In the following, we identify x_r with the initial position by setting x₀ = x_r. The particle returns to x_r at a constant speed V following a resetting event; during the return phase the particle cannot be captured by a target. This means that the return time depends on the location x of the particle at reset according to |x − x_r|/V. (This contribution to the delay $\hat{\tau }$ following target capture was neglected in §2. Another motivation for such an approximation is that the speed of a particle after unloading its cargo could be significantly faster.) In addition, whenever the particle returns to x_r, it is subject to a refractory period before the search begins again; this is distinct from the reloading time following a capture event. The refractory period is itself a random variable with a corresponding waiting time density Φ_ref, which is taken to have a finite mean τ_ref. The various stages of a single search-and-capture event are illustrated in figure 3.

Figure 3.

Particle searching in one dimension for a target at x = 0, which is found after two resettings. After each resetting event, the particle returns to the point x_r at a constant speed V, after which it remains at x_r for a refractory period before re-entering the search phase. Also shown is the decomposition of the FPT, $\mathcal{T}=\mathcal{S}+\mathcal{N}+\mathcal{R}$. (Online version in colour.)

A number of authors have recently calculated the splitting probabilities and conditional MFPTs of a single search-and-capture event in the presence of stochastic resetting and a single target or a pair of targets [22,27,28,32–34]. (Several of these studies also allow for non-exponential resetting, which we do not consider in this paper.) The basic idea is to exploit the fact that once the particle has returned to x_r it has lost all memory of previous search phases, which means that one can condition on whether or not the particle resets at least once, even though a reset event occurs at random times. (In other words, the stochastic process satisfies the strong Markov property.) Renewal theory can then be used to express statistical quantities with resetting in terms of statistical quantities without resetting. In order to distinguish between the two cases, we will add a subscript r to the splitting probabilities etc. of the former. Here, we extend the particular formulation of [43] to calculate the Laplace transform of the conditional FPT densities for multiple targets under resetting; the previous analysis focused on a one-dimensional search, partially absorbing target domains and calculated low-order moments of the conditional FPT density, rather than the density itself.

(a). Splitting probabilities and conditional first passage time densities

Let I(t) denote the number of resettings in the time interval [0, t], and consider the following set of FPTs (see figure 3):

$$\begin{array}{rl}& {\mathcal{T}}_k=inf\{t\ge 0;\mathbf{\text{X}}(t)\in \mathrm{\partial }{\mathcal{U}}_k,\hspace{0.17em}I(t)\ge 0\},\\ & \mathcal{S}=inf\{t\ge 0;\mathbf{\text{X}}(t)={\mathbf{\text{x}}}_r,\hspace{0.17em}I(t)=1\},\\ & {\mathcal{R}}_k=inf\{t\ge 0;\mathbf{\text{X}}(t+\mathcal{S}+\mathcal{N})\in \mathrm{\partial }{\mathcal{U}}_k,\hspace{0.17em}I(t+\mathcal{S}+\mathcal{N})\ge 1\}.\end{array}$$

Here, ${\mathcal{T}}_k$ is the FPT for finding the k-th target irrespective of the number of resettings, $\mathcal{S}$ is the FPT for the first resetting and return to x_r without being captured by any target, $\mathcal{N}$ is the first refractory time and ${\mathcal{R}}_k$ is the FPT for finding the k-th target given that at least one resetting has occurred. Next, we define the sets ${\mathit{\Omega }}_k=\{{\mathcal{T}}_k<\mathrm{\infty }\}$ and ${\mathit{\Gamma }}_k=\{\mathcal{S}<{\mathcal{T}}_k<\mathrm{\infty }\}\subset {\mathit{\Omega }}_k$, where Ω_k is the set of all events for which the particle is eventually absorbed by the k-th target without being absorbed by any other target and Γ_k is the subset of events in Ω_k for which the particle resets at least once. It immediately follows that ${\mathit{\Omega }}_k\mathrm{\setminus }{\mathit{\Gamma }}_k=\{{\mathcal{T}}_k<\mathcal{S}=\mathrm{\infty }\}$, where ${\mathit{\Omega }}_k\mathrm{\setminus }{\mathit{\Gamma }}_k$ is the set of all events for which the particle is captured by the k-th target without any resetting.

The splitting probability π_k can be decomposed as

$${\pi }_{r,k}({\mathbf{\text{x}}}_r):=\mathbb{P}[{\mathit{\Omega }}_k]=\mathbb{P}[{\mathit{\Omega }}_k\mathrm{\setminus }{\mathit{\Gamma }}_k]+\mathbb{P}[{\mathit{\Gamma }}_k].$$ 3.1

The probability that the particle is captured by the k-th target in the interval [τ, τ + dτ] without any returns to x_r is Ψ(τ)J_k(x_r, τ)dτ with J_k(x_r, τ) given by equation (2.4). Hence,

$$\mathbb{P}[{\mathit{\Omega }}_k\mathrm{\setminus }{\mathit{\Gamma }}_k]={\int }_0^{\mathrm{\infty }}\mathrm{\Psi }(\tau )J_k({\mathbf{\text{x}}}_r,\tau )\hspace{0.17em}\text{d}\tau =-{\int }_0^{\mathrm{\infty }}{\mathbf{\text{e}}}^{-r\tau }\frac{\text{d}{\mathrm{\Pi }}_k({\mathbf{\text{x}}}_r,\tau )}{\text{d}\tau }\hspace{0.17em}\text{d}\tau ={\pi }_k({\mathbf{\text{x}}}_0)-r{\tilde{\mathrm{\Pi }}}_k({\mathbf{\text{x}}}_0,r),$$ 3.2

after integrating by parts. Next, from the definitions of the FPTs, we have

$$\mathbb{P}[{\mathit{\Gamma }}_k]=\mathbb{P}[\mathcal{S}<\mathrm{\infty }]\mathbb{P}[{\mathcal{R}}_k<\mathrm{\infty }],$$ 3.3

and memoryless return to x_r implies that $\mathbb{P}[{\mathcal{R}}_k<\mathrm{\infty }]={\pi }_{r,k}({\mathbf{\text{x}}}_r)$. In addition

$$\mathbb{P}[\mathcal{S}<\mathrm{\infty }]=r{\int }_0^{\mathrm{\infty }}{\mathbf{\text{e}}}^{-r\tau }Q({\mathbf{\text{x}}}_r,\tau )d\tau =r\tilde{Q}({\mathbf{\text{x}}}_r,r)=r\sum _{k=1}^N{\tilde{\mathrm{\Pi }}}_k({\mathbf{\text{x}}}_r,r).$$ 3.4

We have used equation (2.7) and the fact that the probability of resetting in the time interval [τ, τ + dτ] is equal to the product of the reset probability ψ(τ) dτ and the survival probability Q(x_r, t). Hence, equation (3.3) becomes

$$\mathbb{P}[{\mathit{\Gamma }}_k]={\pi }_{r,k}({\mathbf{\text{x}}}_r)r\tilde{Q}({\mathbf{\text{x}}}_r,r).$$ 3.5

Finally, combining equations (3.2) and (3.5) and rearranging leads to the following result:

$${\pi }_{r,k}({\mathbf{\text{x}}}_r)=\frac{{\pi }_k({\mathbf{\text{x}}}_r)-r{\tilde{\mathrm{\Pi }}}_k({\mathbf{\text{x}}}_r,r)}{1-r\tilde{Q}({\mathbf{\text{x}}}_r,r)}.$$ 3.6

The analysis of the conditional FPT densities is a little more involved. First note that

$${\pi }_{r,k}({\mathbf{\text{x}}}_r){\tilde{f}}_{r,k}({\mathbf{\text{x}}}_r,s)=\mathbb{E}[{\mathbf{\text{e}}}^{-s{\mathcal{T}}_k}{1}_{{\mathit{\Omega }}_k}].$$ 3.7

Proceeding along similar lines to the splitting probability, we introduce the decomposition

$$\mathbb{E}[{\mathbf{\text{e}}}^{-s{\mathcal{T}}_k}{1}_{{\mathit{\Omega }}_k}]=\mathbb{E}[{\mathbf{\text{e}}}^{-s{\mathcal{T}}_k}{1}_{{\mathit{\Omega }}_k\mathrm{\setminus }{\mathit{\Gamma }}_k}]+\mathbb{E}[{\mathbf{\text{e}}}^{-s{\mathcal{T}}_k}{1}_{{\mathit{\Gamma }}_k}].$$ 3.8

The first expectation can be evaluated by noting that it is the FPT density for capture by the k-th target without any resetting, and the probability density for such an event is Ψ(τ)J_k(x_r, τ) dτ

$$\begin{array}{rl}\mathbb{E}[{\mathbf{\text{e}}}^{-s{\mathcal{T}}_k}{1}_{{\mathit{\Omega }}_k\mathrm{\setminus }{\mathit{\Gamma }}_k}]& ={\int }_0^{\mathrm{\infty }}{\mathbf{\text{e}}}^{-(s+r)\tau }{J}_k({\mathbf{\text{x}}}_r,\tau )\hspace{0.17em}\text{d}\tau =-{\int }_0^{\mathrm{\infty }}{\mathbf{\text{e}}}^{-(s+r)\tau }\frac{\text{d}{\mathrm{\Pi }}_k({\mathbf{\text{x}}}_r,\tau )}{\text{d}\tau }\hspace{0.17em}\text{d}\tau \\ & ={\pi }_k({\mathbf{\text{x}}}_r)-(r+s){\tilde{\mathrm{\Pi }}}_k({\mathbf{\text{x}}}_0,r+s).\end{array}$$ 3.9

The second expectation can be written as

$$\begin{array}{rl}\mathbb{E}[{\mathbf{\text{e}}}^{-s{\mathcal{T}}_k}{1}_{{\mathit{\Gamma }}_k}]& =\mathbb{E}[{\mathbf{\text{e}}}^{-s[\mathcal{S}+\mathcal{N}+{\mathcal{R}}_k]}{1}_{{\mathit{\Gamma }}_k}]\\ & ={\int }_0^{\mathrm{\infty }}r{\mathbf{\text{e}}}^{-r{\tau }_1}\left[{\int }_{\mathcal{U}\mathrm{\setminus }{\mathcal{U}}_a}{\mathbf{\text{e}}}^{-s({\tau }_1+|\mathbf{\text{x}}-{\mathbf{\text{x}}}_r|/V)}p(\mathbf{\text{x}},{\tau }_1|{\mathbf{\text{x}}}_r)\hspace{0.17em}\text{d}\mathbf{\text{x}}\right]\text{d}{\tau }_1\\ & \times \left[{\int }_0^{\mathrm{\infty }}{\mathbf{\text{e}}}^{-s{\tau }_2}{\varphi }_{\text{ref}}({\tau }_2)d{\tau }_2\right]{\pi }_{r,k}({\mathbf{\text{x}}}_r){\tilde{f}}_{r,k}({\mathbf{\text{x}}}_r,s).\end{array}$$ 3.10

We have used the fact that the probability that the first return is initiated in the interval [τ₁, τ₁ + dτ₁], given that X(τ₁) = x and the particle has not been captured by a target, is ψ(τ)p₀(x, τ|x_r)dτ₁. The particle then takes an additional time |x − x_r|/V to return to x_r, after which it spends a time τ₂ in the refractory state with waiting time density ϕ_ref(τ₂). The remaining time to find the k-th target has the same conditional FPT density as ${\mathcal{T}}_k$. Combining equations (3.8)–(3.10) and expressing in terms of Laplace transforms,

$$\begin{array}{rl}{\pi }_{r,k}({\mathbf{\text{x}}}_r){\tilde{f}}_{r,k}({\mathbf{\text{x}}}_r,s)& ={\pi }_k({\mathbf{\text{x}}}_0)-(r+s){\tilde{\mathrm{\Pi }}}_k({\mathbf{\text{x}}}_0,r+s)\\ & +r\left[{\int }_{\mathcal{U}\mathrm{\setminus }{\mathcal{U}}_a}{\mathbf{\text{e}}}^{-s|\mathbf{\text{x}}-{\mathbf{\text{x}}}_r|/V}\tilde{p}(\mathbf{\text{x}},s+r|{\mathbf{\text{x}}}_r)\hspace{0.17em}\text{d}\mathbf{\text{x}}\right]{\tilde{\varphi }}_{\text{ref}}(s){\pi }_{r,k}({\mathbf{\text{x}}}_r)\hspace{0.17em}{\tilde{f}}_{r,k}({\mathbf{\text{x}}}_r,s).\end{array}$$

Rearranging then yields the conditional FPT density

$${\pi }_{r,k}({\mathbf{\text{x}}}_r){\tilde{f}}_{r,k}({\mathbf{\text{x}}}_r,s)=\frac{{\pi }_k({\mathbf{\text{x}}}_r)-(r+s){\tilde{\mathrm{\Pi }}}_k({\mathbf{\text{x}}}_r,r+s)}{1-r{\tilde{\varphi }}_{\text{ref}}(s){\int }_{\mathcal{U}\mathrm{\setminus }{\mathcal{U}}_a}{\mathbf{\text{e}}}^{-s|\mathbf{\text{x}}-{\mathbf{\text{x}}}_r|/V}\tilde{p}(\mathbf{\text{x}},s+r|{\mathbf{\text{x}}}_r)\hspace{0.17em}\text{d}\mathbf{\text{x}}}.$$ 3.11

The Laplace transform of the FPT density is the moment generator of the conditional FPT ${\mathcal{T}}_k$

$$T_{r,k}^{(n)}=\mathbb{E}[{\mathcal{T}}_k^n{1}_{{\mathit{\Omega }}_k}]={\left.{\left(-\frac{\text{d}}{\text{d}s}\right)}^n\mathbb{E}[{\mathbf{\text{e}}}^{-s{\mathcal{T}}_k}{1}_{{\mathit{\Omega }}_k}]\right|}_{s=0}.$$ 3.12

For example, the conditional MFPT $T_{r,k}=T_{r,k}^{(1)}$ is

$$\begin{array}{rl}{\pi }_{r,k}({\mathbf{\text{x}}}_r)T_{r,k}({\mathbf{\text{x}}}_r)& =\frac{{\tilde{\mathrm{\Pi }}}_k({\mathbf{\text{x}}}_r,r)+r{\tilde{\mathrm{\Pi }}}_k^{\prime }({\mathbf{\text{x}}}_r,r)}{1-r\tilde{Q}({\mathbf{\text{x}}}_r,r)}+\left[\frac{{\pi }_k({\mathbf{\text{x}}}_r)-r{\tilde{\mathrm{\Pi }}}_k({\mathbf{\text{x}}}_r,r)}{1-r\tilde{Q}({\mathbf{\text{x}}}_r,r)}\right]\\ & \times \left[\frac{-r{\tilde{\varphi }}_{\text{ref}}^{\prime }(0)\tilde{Q}({\mathbf{\text{x}}}_r,r)-r\sum _k{\tilde{\mathrm{\Pi }}}_k^{\prime }({\mathbf{\text{x}}}_r,r)+r\tilde{F}({\mathbf{\text{x}}}_r,r)/V}{1-r\tilde{Q}({\mathbf{\text{x}}}_r,r)}\right],\end{array}$$ 3.13

where ′ denotes differentiation with respect to r, and $\tilde{F}({\mathbf{\text{x}}}_r,r)$ is the Laplace transform of

$$F({\mathbf{\text{x}}}_r,t)={\int }_{\mathcal{U}\mathrm{\setminus }{\mathcal{U}}_a}|\mathbf{\text{x}}-{\mathbf{\text{x}}}_r|p(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_r)\hspace{0.17em}\text{d}\mathbf{\text{x}}.$$ 3.14

Using equation (3.6) and $-{\tilde{\varphi }}_{\text{ref}}^{\prime }(0)={\tau }_{\text{ref}}$, we obtain the following expression for the conditional MFPTs:

$$\begin{array}{rl}{\pi }_{r,k}({\mathbf{\text{x}}}_r)T_{r,k}({\mathbf{\text{x}}}_r)& =\frac{{\tilde{\mathrm{\Pi }}}_k({\mathbf{\text{x}}}_r,r)+r{\tilde{\mathrm{\Pi }}}_k^{\prime }({\mathbf{\text{x}}}_r,r)}{1-r\tilde{Q}({\mathbf{\text{x}}}_r,r)}\\ & +{\pi }_{r,k}({\mathbf{\text{x}}}_r)\left[\frac{r{\tau }_{\text{ref}}\tilde{Q}({\mathbf{\text{x}}}_r,r)-r\sum _k{\tilde{\mathrm{\Pi }}}_k^{\prime }({\mathbf{\text{x}}}_r,r)+r\tilde{F}({\mathbf{\text{x}}}_r,r)/V}{1-r\tilde{Q}({\mathbf{\text{x}}}_r,r)}\right].\end{array}$$ 3.15

Finally, summing over k yields the unconditional MFPT

$$T_r({\mathbf{\text{x}}}_r):=\sum _{k=1}^N{\pi }_{r,k}({\mathbf{\text{x}}}_r)T_{r,k}({\mathbf{\text{x}}}_r)=\frac{\tilde{Q}({\mathbf{\text{x}}}_r,r)}{1-r\tilde{Q}({\mathbf{\text{x}}}_r,r)}+\left[\frac{r{\tau }_{\text{ref}}\tilde{Q}({\mathbf{\text{x}}}_r,r)+r\tilde{F}({\mathbf{\text{x}}}_r,r)/V}{1-r\tilde{Q}({\mathbf{\text{x}}}_r,r)}\right].$$ 3.16

(b). Small r-expansion

We incorporate the above analysis into the queuing theory of §2 in order to investigate the distribution of resources across a set of targets when the search process is subject to stochastic resetting with x₀ = x_r. From equation (2.17), the steady-state mean becomes

$$⟨M_{r,k}({\mathbf{\text{x}}}_r)⟩=\frac{{\pi }_{r,k}({\mathbf{\text{x}}}_r)}{\gamma \sum _{j=1}^N{\pi }_{r,j}({\mathbf{\text{x}}}_r)(T_{r,j}({\mathbf{\text{x}}}_r)+{\tau }_{\text{cap}})}=\frac{{\pi }_{r,k}({\mathbf{\text{x}}}_r)}{\gamma (T_r({\mathbf{\text{x}}}_r)+{\tau }_{\text{cap}})},$$ 3.17

with π_r,k(x_r) and T_r(x_r) given by equations (3.6) and (3.16), respectively. Finding exact expressions for 〈M_r,k(x_r)〉 is not possible except in a few limited cases. However, insights into the effects of resetting on the distribution of resources can be obtained by carrying out a small-r expansion. (Here, small means relative to other fundamental time scales of the search process such as refractory periods, the degradation rate and the MFPT in the absence of resetting.) For example, establishing that the MFPT of a search process with stochastic resetting initially decreases with r indicates that the MFPT can be optimized by varying r, since the MFPT becomes infinite in the limit r → ∞ [27,33,34]. We consider the effects of resetting on two distinct properties of target resource allocation: (i) the relative distribution of resources as specified by the splitting probabilities π_r,k(x_r); (ii) the mean number of resources per target,

$${\mathcal{M}}_r({\mathbf{\text{x}}}_r)=\frac{1}{N}\sum _{k=1}^N⟨M_{r,k}({\mathbf{\text{x}}}_r)⟩=\frac{1}{\gamma N(T_r({\mathbf{\text{x}}}_r)+{\tau }_{\text{cap}})}.$$ 3.18

First note that, from the definition of Π_k(t), we have

$$r{\tilde{\mathrm{\Pi }}}_k(r)={\pi }_k-{\tilde{J}}_k(r).$$ 3.19

(For notational simplicity, we drop the explicit dependence on the reset position x_r.) Since the probability flux J_k is the moment generator of the FPT density into the k-th target, it has the following Taylor series expansion with respect to r:

$${\tilde{J}}_k(r)={\tilde{J}}_k(0)+r{\tilde{J}}_k^{\prime }(0)+\frac{r^2}{2}{\tilde{J}}_k^{″}(0)+o(r^2)={\pi }_k-r{\pi }_k{T}_k+\frac{r^2}{2}{\pi }_k{T}_k^{(2)}+o(r^2),$$

where $T_k^{(2)}={\int }_0^{\mathrm{\infty }}{t}^2{J}_k(t)\hspace{0.17em}\text{d}t$. Substituting into equation (3.19) implies that

$${\tilde{\mathrm{\Pi }}}_k(r)={\pi }_k{T}_k-\frac{r}{2}{\pi }_k{T}_k^{(2)}+o(r).$$ 3.20

In addition, summing over k shows that

$$\tilde{Q}(r)=T-\frac{r}{2}\sum _{k=1}^s{\pi }_k{T}_k^{(2)}+o(r)=T-\frac{r}{2}({\sigma }_T^2+T^2)+o(r),$$ 3.21

where $T=\sum _{k=1}^N{\pi }_k{T}_k$ and ${\sigma }_T^2=\sum _{k=1}^N{\pi }_k{T}_k^{(2)}-T^2$ is the unconditional variance without resetting.

Using this expansion, we can obtain small-r approximations of the splitting probabilities and conditional MFPTs. First, from equations (3.6), we have

$${\pi }_{r,k}=\frac{{\pi }_k-r{\tilde{\mathrm{\Pi }}}_k(r)}{1-r\sum _{l=1}^N{\tilde{\mathrm{\Pi }}}_l(r)}=\frac{{\pi }_k-r{\pi }_k{T}_k+o(r)}{1-r\sum _{l=1}^N{\pi }_l{T}_l+o(r)}={\pi }_{r,k}={\pi }_k+r{\pi }_k\left(\sum _{l=1}^N{\pi }_l{T}_l-T_k\right)+o(r).$$ 3.22

This implies that a small resetting rate will increase the probability of finding the k-th target, and thus the relative amount of resources allocated to the target in steady state, provided that

$$T_k<\sum _{l=1}^N{\pi }_l{T}_l.$$ 3.23

This is the multi-target extension of the result previously obtained for Bernoulli trials in [34]. Next, substituting (3.21) into (3.16) gives

$$T_r=T+\frac{r}{2}\left(T^2-{\sigma }_T^2+\frac{2\tilde{F}(0)}{V}+2{\tau }_{\text{ref}}T\right)+o(r).$$ 3.24

It follows that adding a small rate of resetting reduces the unconditional MFPT if and only if

$$T^2-{\sigma }_T^2+\frac{2\tilde{F}(0)}{V}+2{\tau }_{\text{ref}}T<0.$$ 3.25

This is a generalization of the condition for the coefficient of variation (CV) obtained in the case of instantaneous resetting [33], which is recovered by taking τ_ref = 0 and V → ∞

$$\text{CV}:=\frac{{\sigma }_T}{T}>1.$$ 3.26

If we only include the effects of refractory periods, then we have a quadratic in T and the condition becomes $T<\sqrt{{\tau }_{\text{ref}}^2+{\sigma }_T^2}$. Within the context of resource allocation, the reduction of the unconditional MFPT for small r increases the mean number of resources in steady state, ${\mathcal{M}}_r$; see equation (3.18).

4. Diffusion in a bounded two-dimensional domain with small interior targets

We have reduced the problem of analysing the effects of stochastic resetting in the small-r regime to that of evaluating the splitting probabilities and moments of the conditional FPT densities in the absence of resetting. However, the latter is itself a non-trivial problem in the case of multiple competing targets in 2D or 3D domains. Nevertheless, one can proceed further in the case of small targets by using asymptotic PDE methods. In order to illustrate this, consider a Brownian particle diffusing in a bounded 2D domain $\mathcal{U}$ with N small targets (or traps) in the interior of the domain. In the absence of resetting, the probability density evolves according to the diffusion equation

$$\frac{\mathrm{\partial }p(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0)}{\mathrm{\partial }t}=D{\mathrm{\nabla }}^{2}p(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0)=-D\mathrm{\nabla }\cdot \mathbf{\text{J}}(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0)$$ 4.1a

and

$${\mathrm{\partial }}_{n}p(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0)=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }\mathcal{U};p(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0)=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }{\mathcal{U}}_a=\bigcup _{j=1}^N\mathrm{\partial }{\mathcal{U}}_j.$$ 4.1b

Each target is assumed to have a size $|{\mathcal{U}}_j|={ϵ}^2|\mathcal{U}|$ with ${\mathcal{U}}_j\to {\mathbf{\text{x}}}_j\in \mathcal{U}$ uniformly as ϵ → 0, j = 1, …, N. The traps are also taken to be well separated in the sense that |x_i − x_j| = O(1), j ≠ i, and $\text{dist}\hspace{0.17em}(x_j,\mathrm{\partial }\mathcal{U})=O(1)$. It is well known that the FPT problem in the case of small targets in 2D (and 3D) domains can be analysed using matched asymptotic expansions and Green’s function methods [45–50]. More specifically, the splitting probabilities π_k, conditional MFPTs T_k and higher-order moments such as $T_k^{(2)}$ satisfy time-independent boundary value problems (BVPs) that can be solved by constructing an inner or local solution valid in an O(ϵ) neighbourhood of each target, and then matching to an outer or global solution that is valid away from each neighbourhood. We focus on the 2D case, but indicate the analogous results in three dimensions.

(a). Asymptotic analysis of first passage time problem in the absence of resetting

We now calculate π_k, T_k and $T_k^{(2)}$ in two dimensions by extending the asymptotic analysis of [49] to the case of conditional FPTs. (Conditional FPTs have been explicitly calculated in three dimensions [48].) The splitting probability π_k(x) satisfies the BVP

$${\mathrm{\nabla }}^2{\pi }_k(\mathbf{\text{x}})=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathcal{U}\mathrm{\setminus }{\mathcal{U}}_a\text{and}{\mathrm{\partial }}_n{\pi }_k(\mathbf{\text{x}})=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }\mathcal{U},$$ 4.2a

with

$${\pi }_k(\mathbf{\text{x}})=1,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }{\mathcal{U}}_k;{\pi }_k(\mathbf{\text{x}})=0,\hspace{0.17em}\mathbf{\text{x}}\in \bigcup _{j\ne k}\mathrm{\partial }{\mathcal{U}}_j.$$ 4.2b

In the inner region around the j-th target, we introduce the stretched coordinates y = ϵ⁻¹(x − x_j) and set u_k,j(y) = π_k(x_j + ϵy). Expand the inner solution as $u_{k,j}={\delta }_{j,k}+\nu A_{k,j}(\nu )u_{k,j}^{(0)}+\dots$, where ν = −1/lnϵ and A_j is an unknown constant. Higher-order terms are assumed to be beyond-all-orders with respect to the logarithmic terms. It follows that $u_{k,j}^{(0)}(\mathbf{\text{y}})$ satisfies the BVP,

$${\mathrm{\nabla }}_{\mathbf{\text{y}}}^2{u}_{k,j}^{(0)}(\mathbf{\text{y}})=0,\hspace{0.17em}\mathbf{\text{y}}\in {\mathbb{R}}^2\mathrm{\setminus }{\mathcal{U}}_j;u_{k,j}^{(0)}(\mathbf{\text{y}})=0,\hspace{0.17em}\mathbf{\text{y}}\in \mathrm{\partial }{\mathcal{U}}_j,$$ 4.3

together with the far-field condition

$$u_{k,j}^{(0)}(\mathbf{\text{y}})\sim \ln |\mathbf{\text{y}}|-\ln d_j+O(|\mathbf{\text{y}}{|}^{-1}),$$ 4.4

where d_i is known as the logarithmic capacitance of ${\mathcal{U}}_j$, and depends on the shape but not the orientation of the target. The unspecified O(|y|⁻¹) term is the corresponding dipole term.

Let U_k(x) denote the leading-order contribution to the outer solution, which satisfies the BVP

$${\mathrm{\nabla }}^2{U}_k(\mathbf{\text{x}})=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathcal{U}\mathrm{\setminus }\{{\mathbf{\text{x}}}_1,\dots ,{\mathbf{\text{x}}}_N\},\hspace{0.17em}{\mathrm{\partial }}_n{U}_k(\mathbf{\text{x}})=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }\mathcal{U},$$ 4.5

together with singularity conditions as x → x_j, j = 1, …, N. The latter are determined by matching to the inner solution. That is,

$$U_k(\mathbf{\text{x}})\sim {\delta }_{j,k}+\nu A_{k,j}(\nu )\ln |\mathbf{\text{x}}-{\mathbf{\text{x}}}_j|/d_jϵ,\text{as}\hspace{0.17em}\mathbf{\text{x}}\to {\mathbf{\text{x}}}_j.$$ 4.6

In other words, U_k satisfies the inhomogeneous equation

$${\mathrm{\nabla }}^2{U}_k(\mathbf{\text{x}})=2\pi \sum _{j=1}^N\nu A_{k,j}(\nu )\delta (\mathbf{\text{x}}-{\mathbf{\text{x}}}_j),\hspace{0.17em}\mathbf{\text{x}}\in \mathcal{U};{\mathrm{\partial }}_n{U}_k(\mathbf{\text{x}})=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }\mathcal{U}.$$ 4.7

Applying the divergence theorem to equation (4.7) leads to the constraint

$$\sum _{j=1}^N{A}_{k,j}(\nu )=0.$$ 4.8

Equation (4.7) can be solved in terms of the Neumann Green’s function

$${\mathrm{\nabla }}^{2}G(\mathbf{\text{x}};{\mathbf{\text{x}}}^{\prime })=-\frac{1}{|\mathcal{U}|}+\delta (\mathbf{\text{x}}-{\mathbf{\text{x}}}^{\prime }),\hspace{0.17em}\mathbf{\text{x}}\in \mathcal{U};\hspace{0.17em}{\mathrm{\partial }}_{n}G=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }\mathcal{U}$$ 4.9a

and

$$G(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })=\frac{1}{2\pi }\ln |\mathbf{\text{x}}-{\mathbf{\text{x}}}^{\prime }|+R(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime }),\hspace{0.17em}{\int }_{\mathcal{U}}G(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })\hspace{0.17em}\text{d}\mathbf{\text{x}}=0,$$ 4.9b

with R(x, x′) corresponding to the regular part of Green’s function. Given G, the solution can be written as

$$U_k(\mathbf{\text{x}})=2\pi \sum _{j=1}^N\nu A_{k,j}(\nu )G(\mathbf{\text{x}},{\mathbf{\text{x}}}_j)+{\chi }_k,$$ 4.10

with unknown constant ${\chi }_k={|\mathcal{U}|}^{-1}{\int }_{\mathcal{U}}{U}_k(\mathbf{\text{x}})\hspace{0.17em}\text{d}\mathbf{\text{x}}$. Note that $\sum _{k=1}^N{\chi }_k=1$, since $\sum _{k=1}^N{U}_k(\mathbf{\text{x}})=1$.

For a fixed target k, we have N + 1 constants A_k,j(ν), j = 1, …, N, and χ_k, which can be determined by imposing the constraint (4.8) and N matching conditions. The latter are obtained by matching the far-field behaviour of the inner solution u_k,j with the near-field behaviour of the outer solution U_k in a neighbourhood of ${\mathcal{U}}_j$ for j = 1, …, N,

$$2\pi \sum _{i\ne j}\nu A_{k,i}(\nu )G({\mathbf{\text{x}}}_i,{\mathbf{\text{x}}}_j)-A_{k,j}(\nu )+\nu A_{k,j}(\nu )[\ln d_j+2\pi R({\mathbf{\text{x}}}_j,{\mathbf{\text{x}}}_j)]+{\chi }_k={\delta }_{j,k}.$$ 4.11

This is a matrix equation that has the formal solution

$$A_{k,i}(\nu )=\sum _{j=1}^N{[{\delta }_{i,j}-2\pi \nu {\mathcal{G}}_{ij}]}^{-1}({\chi }_k-{\delta }_{j,k}),$$ 4.12

where ${\mathcal{G}}_{ij}=G({\mathbf{\text{x}}}_i,{\mathbf{\text{x}}}_j)$ for i ≠ j and ${\mathcal{G}}_{ii}=R({\mathbf{\text{x}}}_i,{\mathbf{\text{x}}}_i)+\ln d_i/2\pi .$ The presence of the small parameter ν rather than ϵ in the matched asymptotic expansions is a common feature of strongly localized perturbations in 2D domains [50]. It is well known that ν → 0 more slowly than ϵ → 0. Hence, if one is interested in obtaining o(ϵ) accuracy, then it is necessary to sum over the logarithmic terms non-perturbatively along the above lines by matching the inner and outer solutions using the Neumann Green’s function. (It is also possible to include O(ϵ) and higher-order terms as illustrated in [54].) This is equivalent to calculating the asymptotic solution for all terms of O(ν^k) for any k. We could now proceed by numerically solving the system of linear equations (4.11) along the lines of [49]. However, within the context of investigating the effects of stochastic resetting on the accumulation of resources, it is convenient to use a regular Taylor expansion of the coefficients rather than the full non-perturbative solution. Thus, solving for A_k,j(ν) to o(ν) and imposing equation (4.8) gives

$$A_{k,j}(\nu )\sim {\chi }_k-{\delta }_{j,k}+2\pi \nu \sum _{i=1}^N({\chi }_k-{\delta }_{i,k}){\mathcal{G}}_{ij}+o(\nu ),\hspace{0.17em}{\chi }_k\sim \frac{1}{N}-\frac{2\pi \nu }{N}\sum _{i,j=1}^N(N^{-1}-{\delta }_{i,k}){\mathcal{G}}_{ij}+o(\nu ).$$

The outer solution then yields the following approximation for the splitting probabilities:

$${\pi }_k(\mathbf{\text{x}})\sim \frac{1}{N}+2\pi \nu \left[\frac{1}{N}\sum _{j=1}^{N}G(\mathbf{\text{x}},{\mathbf{\text{x}}}_j)-G(\mathbf{\text{x}},{\mathbf{\text{x}}}_k)\right]-\frac{2\pi \nu }{N}\sum _{i,j=1}^N(N^{-1}-{\delta }_{i,k}){\mathcal{G}}_{ij}+o(\nu ).$$ 4.13

An analogous asymptotic analysis can be used to determine the conditional MFPTs T_k and second moments $T_k^{(2)}$. First, T_k satisfies a BVP of the form

$${\mathrm{\nabla }}^2({\pi }_k(\mathbf{\text{x}})T_k(\mathbf{\text{x}}))=-\frac{{\pi }_k(\mathbf{\text{x}})}{D},\hspace{0.17em}\mathbf{\text{x}}\in \mathcal{U}\mathrm{\setminus }{\mathcal{U}}_a;{\mathrm{\partial }}_n({\pi }_k(\mathbf{\text{x}})T_k(\mathbf{\text{x}}))=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }\mathcal{U}$$ 4.14a

and

$${\pi }_k(\mathbf{\text{x}})T_k(\mathbf{\text{x}})=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }{\mathcal{U}}_k;{\pi }_k(\mathbf{\text{x}})T_k(\mathbf{\text{x}})\to 0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }{\mathcal{U}}_j,\hspace{0.17em}j\ne k.$$ 4.14b

Let w_k,j(y) denote the inner solutions around the j-th target and let W_k(x) denote the outer solution. The analysis of the inner solution is identical to that of the splitting probability, so that

$$w_{k,j}={\delta }_{j,k}+\nu B_{k,j}(\nu )w_{k,j}^{(0)}+\dots ,$$ 4.15

with B_j to be determined. We thus have to solve the outer BVP

$${\mathrm{\nabla }}^2{W}_k(\mathbf{\text{x}})=-\frac{U_k(\mathbf{\text{x}})}{D},\hspace{0.17em}\mathbf{\text{x}}\in \mathcal{U}\mathrm{\setminus }{\mathcal{U}}_a;{\mathrm{\partial }}_n{W}_k(\mathbf{\text{x}})=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }\mathcal{U}$$ 4.16

and

$$W_k(\mathbf{\text{x}})=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }{\mathcal{U}}_k;W_k(\mathbf{\text{x}})\to 0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }{\mathcal{U}}_j,\hspace{0.17em}j\ne k,$$ 4.17

supplemented by the singularity conditions

$$W_k(\mathbf{\text{x}})\sim {\delta }_{j,k}+\nu B_{k,j}(\nu )\ln |\mathbf{\text{x}}-{\mathbf{\text{x}}}_j|/d_jϵ,\text{as}\hspace{0.17em}\mathbf{\text{x}}\to {\mathbf{\text{x}}}_j.$$ 4.18

The solution of the inhomogeneous outer equation can be decomposed into a particular and complementary solution

$$W_k(\mathbf{\text{x}})=-D^{-1}{\int }_{\mathcal{U}}G(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })U_k({\mathbf{\text{x}}}^{\prime })\hspace{0.17em}\text{d}{\mathbf{\text{x}}}^{\prime }+2\pi \sum _{j=1}^N\nu B_{k,j}(\nu )G(\mathbf{\text{x}},{\mathbf{\text{x}}}_j)+{\hat{\chi }}_k,$$ 4.19

with unknown constant ${\hat{\chi }}_k={|\mathcal{U}|}^{-1}{\int }_{\mathcal{U}}{W}_k(\mathbf{\text{x}})\hspace{0.17em}\text{d}\mathbf{\text{x}}$ and the constraint

$$\sum _{j=1}^N{B}_{k,j}(\nu )=\frac{{\chi }_k|\mathcal{U}|}{2\pi \nu D}.$$ 4.20

Matching the inner and outer solutions generates the N additional conditions

$$\begin{array}{r}2\pi \sum _{i\ne j}\nu B_{k,i}(\nu )G({\mathbf{\text{x}}}_i,{\mathbf{\text{x}}}_j)-B_{k,j}(\nu )+\nu B_{k,j}(\nu )[\ln d_j+2\pi R({\mathbf{\text{x}}}_j,{\mathbf{\text{x}}}_j)]+{\hat{\chi }}_k=D^{-1}{\int }_{\mathcal{U}}G({\mathbf{\text{x}}}_j,{\mathbf{\text{x}}}^{\prime })U_k({\mathbf{\text{x}}}^{\prime })d{\mathbf{\text{x}}}^{\prime }.\end{array}$$ 4.21

Solving for B_k,j(ν) to o(1) and imposing equation (4.20) gives

$$B_{k,j}(\nu )\sim {\hat{\chi }}_k\left[1+2\pi \nu \sum _{i=1}^N{\mathcal{G}}_{ij}\right]+o(\nu ),{\hat{\chi }}_k\sim \frac{{\chi }_k|\mathcal{U}|}{2\pi \nu DN}\left[1-\frac{2\pi \nu }{N}\sum _{i,j=1}^N{\mathcal{G}}_{ij}\right]+o(1).$$ 4.22

Hence,

$$\begin{array}{r}{\pi }_k(\mathbf{\text{x}})T_k(\mathbf{\text{x}})\sim \frac{|\mathcal{U}|}{D{N}^2}\sum _{j=1}^{N}G(\mathbf{\text{x}},{\mathbf{\text{x}}}_j)+\frac{|\mathcal{U}|}{2\pi \nu D{N}^2}\left[1-\frac{2\pi \nu }{N}\sum _{i,j=1}^N{\mathcal{G}}_{ij}\right]-\frac{|\mathcal{U}|}{D{N}^2}\sum _{i,j=1}^N(N^{-1}-{\delta }_{i,k}){\mathcal{G}}_{ij}+o(1).\end{array}$$ 4.23

Finally, $T_k^{(2)}$ satisfies the BVP

$${\mathrm{\nabla }}^2({\pi }_k(\mathbf{\text{x}})T_k^{(2)}(\mathbf{\text{x}}))=-\frac{2{\pi }_k(\mathbf{\text{x}})T_k(\mathbf{\text{x}})}{D},\hspace{0.17em}\mathbf{\text{x}}\in \mathcal{U}\mathrm{\setminus }{\mathcal{U}}_a;{\mathrm{\partial }}_n({\pi }_k(\mathbf{\text{x}})T_k^{(2)}(\mathbf{\text{x}}))=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }\mathcal{U}$$ 4.24a

and

$${\pi }_k(\mathbf{\text{x}})T_k^{(2)}(\mathbf{\text{x}})=0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }{\mathcal{U}}_k;{\pi }_k(\mathbf{\text{x}})T_k^{(2)}(\mathbf{\text{x}})\to 0,\hspace{0.17em}\mathbf{\text{x}}\in \mathrm{\partial }{\mathcal{U}}_j,\hspace{0.17em}j\ne k.$$ 4.24b

Denoting the outer solution by $W_k^{(2)}(\mathbf{\text{x}}),$ we have

$$W_k^{(2)}(\mathbf{\text{x}})=-\frac{2}{D}{\int }_{\mathcal{U}}G(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })W_k({\mathbf{\text{x}}}^{\prime })\hspace{0.17em}\text{d}{\mathbf{\text{x}}}^{\prime }+2\pi \sum _{j=1}^N\nu B_{k,j}^{(2)}(\nu )G(\mathbf{\text{x}},{\mathbf{\text{x}}}_j)+{\hat{\chi }}_k^{(2)},$$ 4.25

with unknown constant ${\hat{\chi }}_k^{(2)}={|\mathcal{U}|}^{-1}{\int }_{\mathcal{U}}{W}_k^{(2)}(\mathbf{\text{x}})\hspace{0.17em}\text{d}\mathbf{\text{x}}$ and the constraint

$$\sum _{j=1}^N{B}_{k,j}^{(2)}(\nu )=\frac{{\hat{\chi }}_k|\mathcal{U}|}{\pi \nu D}.$$ 4.26

Matching the inner and outer solutions generates the N additional conditions

$$\begin{array}{r}2\pi \sum _{i\ne j}\nu B_{k,i}^{(2)}(\nu )G({\mathbf{\text{x}}}_i,{\mathbf{\text{x}}}_j)-B_{k,j}^{(2)}(\nu )+\nu B_{k,j}^{(2)}(\nu )[\ln d_j+2\pi R({\mathbf{\text{x}}}_j,{\mathbf{\text{x}}}_j)]+{\hat{\chi }}_k^{(2)}=\frac{2}{D}{\int }_{\mathcal{U}}G({\mathbf{\text{x}}}_j,{\mathbf{\text{x}}}^{\prime })W_k({\mathbf{\text{x}}}^{\prime })\hspace{0.17em}\text{d}{\mathbf{\text{x}}}^{\prime }.\end{array}$$ 4.27

Solving for $B_{k,j}^{(2)}(\nu )$ to o(1/ν) and imposing equation (4.26) gives

$$B_{k,j}^{(2)}(\nu )\sim {\hat{\chi }}_k^{(2)}\left[1+2\pi \nu \sum _{i=1}^N{\mathcal{G}}_{ij}\right]+o(1/\nu ),{\hat{\chi }}_k^{(2)}\sim \frac{{\hat{\chi }}_k|\mathcal{U}|}{\pi \nu DN}\left[1-\frac{2\pi \nu }{N}\sum _{i,j=1}^N{\mathcal{G}}_{ij}\right]+o(1/\nu ).$$ 4.28

Hence,

$$\begin{array}{rl}{\pi }_k(\mathbf{\text{x}})T_k^{(2)}(\mathbf{\text{x}})& \sim \frac{|\mathcal{U}{|}^2}{\pi \nu D^2{N}^3}\sum _{j=1}^{N}G(\mathbf{\text{x}},{\mathbf{\text{x}}}_j)+\frac{|\mathcal{U}{|}^2}{2{\pi }^2{\nu }^2{D}^2{N}^3}\left[1-\frac{4\pi \nu }{N}\sum _{i,j=1}^N{\mathcal{G}}_{ij}\right]\\ & -\frac{|\mathcal{U}{|}^2}{\pi \nu D^2{N}^3}\sum _{i,j=1}^N(N^{-1}-{\delta }_{i,k}){\mathcal{G}}_{ij}+o\left(\frac{1}{\nu }\right).\end{array}$$ 4.29

(b). Effects of stochastic resetting in the small-r regime

We can now analyse how resetting affects the search process in the small-r regime by taking x = x_r and using the results of §3b. From equations (4.13), (4.23) and (4.29), we have the leading-order approximation

$$T\sim \frac{|\mathcal{U}|}{DN}\sum _{j=1}^{N}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_j)+\frac{|\mathcal{U}|}{2\pi \nu DN}\left[1-\frac{2\pi \nu }{N}\sum _{i,j=1}^N{\mathcal{G}}_{ij}\right].$$

Recall from equation (3.23) that resetting increases the splitting probability π_r,k to the k-th target provided that $T_k<T=\sum _{k=1}^N{\pi }_l{T}_l$. Equations (4.13) and (4.23) then imply that π_r,k > π_k for small r and μ if and only if

$$\frac{1}{N}\sum _{j=1}^{N}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_j)>G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_k).$$ 4.30

As might be expected, this condition is clearly satisfied when x_r ≈ x_k, since G(x_r, x_k) ≈ ln|x_r − x_k|/2π ≪ 0. That is, a target close to x_r benefits from resetting. Next recall from equation (3.26) that, in the absence of delays, resetting will decrease the unconditional MFPT provided that ${\sigma }_T^2>T^2$. This reduces to the condition

$$\sum _{k=1}^N{\pi }_k({\mathbf{\text{x}}}_r)T_k^{(2)}({\mathbf{\text{x}}}_r)-2T^2\sim -\frac{|\mathcal{U}{|}^2}{\pi \nu D^2{N}^2}\sum _{j=1}^{N}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_j)>0.$$

That is, the total amount of resources M_tot(x_r) will increase provided that

$$\sum _{j=1}^{N}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_j)<0.$$ 4.31

A significant feature of these conditions is that the geometry of the search domain is succinctly taken into account by Green’s functions.

For the sake of illustration, suppose that the targets are located in a circular domain $\mathcal{U}\subset {\mathbb{R}}^2$ of unit radius centred at the origin. The 2D Neumann Green’s function is given by [49]

$$G(\mathbf{\text{x}},{\mathbf{\text{x}}}_r)=\frac{1}{2\pi }\left[\ln (|\mathbf{\text{x}}-{\mathbf{\text{x}}}_r|)+\ln \left(\left|\mathbf{\text{x}}|{\mathbf{\text{x}}}_r|-\frac{{\mathbf{\text{x}}}_r}{|{\mathbf{\text{x}}}_r|}\right|\right)-\frac{1}{2}(|\mathbf{\text{x}}{|}^2+|{\mathbf{\text{x}}}_r{|}^2)+\frac{3}{4}\right].$$ 4.32

Suppose that the reset point is at the origin, x_r = 0, so G(x, x_r) = G(ρ) with ρ = |x| and

$$G(\rho )=\frac{1}{2\pi }\left[\ln \rho -\frac{{\rho }^2}{2}+\frac{3}{4}\right].$$ 4.33

Substituting into equation (4.30) and exponentiating gives

$${\rho }_k{\mathbf{\text{e}}}^{-{\rho }_k^2/2}<{\left(\prod _{j=1}^k{\rho }_j{\mathbf{\text{e}}}^{-{\rho }_j^2/2}\right)}^{1/N},$$ 4.34

where ρ_j is the radial distance of the j-th target from the origin. That is, introducing the function $f(\rho )=\rho {\mathbf{\text{e}}}^{-{\rho }^2/2}$, we see that resetting will increase the amount of resources in the k-th target (at least for small r) if f(ρ_k) is less than the geometric mean of f(ρ) over all targets. In the case of two targets (N = 2), the condition reduces to f(ρ₁) < f(ρ₂) for k = 1. Since f′(ρ) > 0 for all ρ ≤ 1, it follows that r₁ < r₂. Given the symmetry of the geometric configuration, this is consistent with the intuitive result that the target closer to the reset point has a competitive advantage. Turning to the second condition (4.31), we explore its implications for a single target at a radial distance ρ. The total number of resources M_tot will initially be increased by resetting provided that G(ρ) < 0, that is, f(ρ) < e^−3/4. This yields a critical radius ρ_c ≈ 0.55 such that resetting initially increases M_tot if and only if ρ < ρ_c. On the other hand, M_tot → 0 as r → ∞, since the particle resets so often that it never has a chance to reach the target. This suggests that M_tot can be optimized by varying r when ρ < ρ_c.

If one is interested in more accurate approximations of the conditions (4.30) and (4.31), which take into account the shape and size of each target, then it is necessary to numerically solve the linear systems (4.11), (4.21) and (4.27) along the lines of [49]. For the sake of illustration, consider one of the target configurations studied in [49]. This consists of a unit disc containing four circular targets of radius ϵ = 0.025 equally spaced on a ring of radius ρ, 0 < ρ < 1; see figure 4a. The time scale is fixed by setting D = 1. In figure 4b, we sketch plots of the unconditional MFPT T and standard deviation σ_T as a function of ρ that were found in [49]. The two curves meet at a critical radius ρ_c ≈ 0.4 that marks the transition between increasing and decreasing M_tot under resetting.

Figure 4.

(a) Target configuration consisting of four circular targets with radius ϵ = 0.025, which are equally spaced on a ring of radius ρ inside the unit disc. (b) Sketches of the unconditional MFPT T and standard deviation σ_T as a function of the radius ρ. The critical radius ρ_c marks the transition between increasing and decreasing M_tot under resetting. (Online version in colour.)

(c). Comparison with a three-dimensional search process

Asymptotic methods can also be used to calculate the splitting probabilities and moments of the conditional FPT densities in the case of diffusion to small targets in 3D domains [46,48]. The major difference is that the far-field behaviour of an inner solution such as (4.4) becomes

$$u_{k,j}(\mathbf{\text{y}})\sim \frac{C_j}{|\mathbf{\text{y}}|}+\frac{{\mathbf{\text{P}}}_j\cdot \mathbf{\text{y}}}{|\mathbf{\text{y}}{|}^3}+\dots \text{as }|\mathbf{\text{y}}|\to \mathrm{\infty },$$ 4.35

where C_j is the capacitance and P_j is the dipole vector of an equivalent charged conductor with the shape ${\mathcal{U}}_j$. (Here, C_j has the units of length.) One now matches inner and outer solutions using regular perturbation expansions to second order in ϵ. Using the resulting asymptotic expansions calculated in [48], we find that

$$\begin{array}{rl}{T}_k(\mathbf{\text{x}})& \sim \frac{|\mathcal{U}|}{4\pi ϵDN\overline{C}}+\frac{|\mathcal{U}|}{D}G(\mathbf{\text{x}},{\mathbf{\text{x}}}_k)-\frac{|\mathcal{U}|}{D{(N\overline{C})}^2}\sum _{i,j=1}^N{C}_i{\mathcal{G}}_{ij}{C}_j+o(1),\end{array}$$ 4.36a

$$\begin{array}{rl}T& \sim \frac{|\mathcal{U}|}{4\pi ϵND\overline{C}}\left[1+4\pi ϵ\sum _{j=1}^N{C}_{j}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_j)-\frac{4\pi ϵ}{N\overline{C}}\sum _{i,j=1}^N{C}_i{\mathcal{G}}_{ij}{C}_j\right]\end{array}$$ 4.36b

$$\begin{array}{rl}\text{and}{\sigma }_T^2& \sim \frac{|\mathcal{U}{|}^2}{{(4\pi ϵND\overline{C})}^2}-\frac{|\mathcal{U}{|}^2}{2\pi ϵD^2{(N\overline{C})}^3}\sum _{i,j=1}^N{C}_i{\mathcal{G}}_{ij}{C}_j.\end{array}$$ 4.36c

Here, ${\mathcal{G}}_{ij}=G({\mathbf{\text{x}}}_i,{\mathbf{\text{x}}}_j),\hspace{0.17em}\text{for }i\ne j$ and ${\mathcal{G}}_{ii}=R({\mathbf{\text{x}}}_i,{\mathbf{\text{x}}}_i)$, with G the 3D Neumann Green’s function

$$G(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })=-\frac{1}{4\pi |\mathbf{\text{x}}-{\mathbf{\text{x}}}^{\prime }|}+R(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime }).$$ 4.37

The 3D analogues of the conditions (4.30) and (4.31) can then be written down. First, equations (3.23) and (4.36a) imply that resetting will initially increase the splitting probability to the k-th target provided that

$$\frac{1}{N}\sum _{j=1}^N{C}_{j}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_j)>\overline{C}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_k).$$ 4.38

Second, equations (3.26) and (4.36b), (4.36c) show that instantaneous resetting will decrease the unconditional MFPT provided that

$${\sigma }_T^2-T^2\sim \frac{|\mathcal{U}{|}^2}{2\pi ϵ{(DN\overline{C})}^2}\sum _{j=1}^N{C}_{j}G({\mathbf{\text{x}}}_r,{\mathbf{\text{x}}}_j)<0.$$ 4.39

Note that the results for two dimensions and three dimensions are almost identical, except that in the latter case the sums over Green’s functions are weighted by the target capacitances. Again we illustrate our results by considering a simple geometric configuration and take all targets to have identical shapes so that C_j = C for all j. Let $\mathcal{U}\subset {\mathbb{R}}^3$ be the unit sphere centred about the origin. The 3D Neumann Green’s function takes the form [46]

$$G^{(3)}(\mathbf{\text{x}},{\mathbf{\text{x}}}_r)=-\frac{1}{4\pi }\left[\frac{1}{|\mathbf{\text{x}}-{\mathbf{\text{x}}}_r|}+\frac{1}{|\mathbf{\text{x}}|r^{\prime }}+\ln \left(\frac{2}{1-|\mathbf{\text{x}}||{\mathbf{\text{x}}}_r|\cos \theta +|\mathbf{\text{x}}|r^{\prime }}\right)+\frac{1}{2}(|\mathbf{\text{x}}{|}^2+|{\mathbf{\text{x}}}_r{|}^2)-2.8\right],$$ 4.40

where

$$\cos \theta =\frac{\mathbf{\text{x}}\cdot {\mathbf{\text{x}}}_r}{|\mathbf{\text{x}}||{\mathbf{\text{x}}}_r|},{\mathbf{\text{x}}}^{\prime }=\frac{\mathbf{\text{x}}}{|\mathbf{\text{x}}{|}^2}\text{and}{r}^{\prime }=|{\mathbf{\text{x}}}^{\prime }-{\mathbf{\text{x}}}_r|.$$

If x_r = 0 then G(x, x_r) = G(ρ) with

$$G(\rho )=-\frac{1}{4\pi }\left[\frac{1}{\rho }+\frac{{\rho }^2}{2}-1.8\right].$$ 4.41

As in the case of the disc, targets that are closer to the origin have a competitive advantage. Moreover, there exists a critical radius ρ_c for a single target to benefit from the inclusion of resetting, which satisfies the condition ${\rho }_c^{-1}+{\rho }_c^2/2=1.8$, that is, ρ_c ≈ 0.61.

5. Conclusion

In this paper, we combined renewal theory with classical asymptotic PDE methods to investigate the distribution of resources across a set of targets in a bounded 2D domain. The targets are supplied by a particle (or set of independent particles) undergoing multiple rounds of search and capture with stochastic resetting and delays. The delays arise from finite return times and refractory periods. Our analysis was motivated by certain problems in cell biology; namely, motor-driven transport and delivery of proteins to synapses, and cytoneme-based transport and delivery of morphogen to target cells. In both cases the search process is subject to non-instantaneous resetting: return of a motor complex to the cell body and reinjection into the axon; retraction of a cytoneme back to a source cell and subsequent renucleation. However, the mathematical framework developed here is applicable to any type of search problem in which the goal of the search process is not simply target detection, but delivery of some resource to the target. Hence, the modelling framework is target-centric rather than searcher-centric. This contrasts with search processes in animal foraging, for example, where the goal is for the forager (particle) to find a target in order to obtain some resource such as food. In this case, multiple search-and-capture events lead to the accumulation of resources within the searcher, and competition is between multiple searchers rather than multiple targets.

From a mathematical perspective, there are various possible extensions of our work. First, although we restricted our asymptotic analysis of target competition in §4 to the small-r regime, it should be possible to consider larger values of r. For example, similar asymptotic methods could be used to determine the Laplace transform of the survival probability without resetting, $\tilde{Q}({\mathbf{\text{x}}}_r,r)$, for arbitrary values of r; see equation (2.7). The latter completely determines the unconditional MFPT T_r(x_r) in the absence of finite return times; see equation (3.16). A more challenging problem, which we hope to address elsewhere, is to use asymptotic methods to determine the Laplace transform ${\tilde{\mathrm{\Pi }}}_k({\mathbf{\text{x}}}_0,s)$ of the integrated flux into the k-th target as defined by equation (2.3). This is necessary in order to calculate the splitting probability π_r,k(x₀) outside the small-r regime; see equation (3.6). In addition, ${\tilde{\mathrm{\Pi }}}_k({\mathbf{\text{x}}}_0,s)$ appears in the expression for the conditional FPT densities, equation (3.13), which are needed in order to calculate higher-order moments of the steady-state resource distribution such as the variance; see equation (2.19). A second extension would be to consider a higher-dimensional version of the search-and-capture processes considered in [43]. This would involve treating the sets ${\mathcal{U}}_k$ in figure 1 as partially absorbing domains so that the flux into the k-th target is of the form $J_k({\mathbf{\text{x}}}_0,t)=\kappa {\int }_{{\mathcal{U}}_k}p(\mathbf{\text{x}},t|{\mathbf{\text{x}}}_0)\hspace{0.17em}\text{d}\mathbf{\text{x}}$. Another variation on a theme would be to consider problems in which a single search-and-capture event without resetting does not have finite moments of the FPT density. This would occur when $\mathcal{U}$ is unbounded, for example. Resetting then plays a critical role in rendering these moments finite. Finally, it would be interesting to explore the effects of interactions between multiple searchers, rather than treating them as independent. One source of interactions in the case of one-dimensional search processes such as motor-driven transport is exclusion effects.

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Competing interests

I declare I have no competing interest.

Funding

This work was supported by the National Science Foundation (DMS 1613048).