Filling of a Poisson trap by a population of random intermittent searchers

We extend the continuum theory of random intermittent search processes to the case of $N$ independent searchers looking to deliver cargo to a single hidden target located somewhere on a semi-infinite track. Each searcher randomly switches between a stationary state and either a leftward or rightward constant velocity state. We assume that all of the particles start at one end of the track and realize sample trajectories independently generated from the same underlying stochastic process. The hidden target is treated as a partially absorbing trap in which a particle can only detect the target and deliver its cargo if it is stationary and within range of the target; the particle is removed from the system after delivering its cargo. As a further generalization of previous models, we assume that up to $n$ successive particles can find the target and deliver its cargo. Assuming that the rate of target detection scales as $1/N$, we show that there exists a well-defined mean-field limit $N\to \infty$, in which the stochastic model reduces to a deterministic system of linear reaction-hyperbolic equations for the concentrations of particles in each of the internal states. These equations decouple from the stochastic process associated with filling the target with cargo. The latter can be modeled as a Poisson process in which the time-dependent rate of filling $\lambda \left(t\right)$ depends on the concentration of stationary particles within the target domain. Hence, we refer to the target as a Poisson trap. We analyze the efficiency of filling the Poisson trap with $n$ particles in terms of the waiting time density $f_n\left(t\right)$. The latter is determined by the integrated Poisson rate $\mu \left(t\right)={\int }_0^t\lambda \left(s\right)ds$, which in turn depends on the solution to the reaction-hyperbolic equations. We obtain an approximate solution for the particle concentrations by reducing the system of reaction-hyperbolic equations to a scalar advection-diffusion equation using a quasisteady-state analysis. We compare our analytical results for the mean-field model with Monte Carlo simulations for finite $N$. We thus determine how the mean first passage time (MFPT) for filling the target depends on $N$ and $n$.

Figure 1

Schematic diagram illustrating a population model of motor-driven particles moving along a one-dimensional track. The particles can transition from a moving state with velocity $\pm v$ at a rate ${\beta }_{\pm }$ and from a stationary searching state at a rate $\alpha$. A partially absorbing trap is located at $x=X$ that fills up according to a Poisson process.

Figure 2

Unbiased $({\beta }_{+}={\beta }_{-}=\beta$) random intermittent search in the mean-field population model. The MFPT vs the (a) average search time and (b) the average duration of the moving (forward and backward) state for different values of the Poisson trap size $n$. Each curve has a minimum MFPT for a given value of $\alpha$ and $\beta$. Parameter values used are $\epsilon =0.1$, $k=5/N$, $X=50$, and $l=0.25$. For (a) we use $\beta =1/\epsilon$, and for (b) we use $\alpha =2/\epsilon$.

Figure 3

First passage time density for an unbiased search and a single-capacity trap ($n=1$). The solid curve shows the analytical density function in the mean-field limit, and the remaining curves are histograms obtained from ${10}^4$ Monte Carlo simulations for different numbers of searchers $N$. Parameter values are the same as in Fig. 2.

Figure 4

First passage time density for a biased search and a single-capacity trap ($n=1$). The black curve shows the analytical density function in the mean-field limit, and the remaining curves are histograms obtained from $5\times {10}^4$ Monte Carlo simulations. Parameter values used are $\alpha =1/\epsilon$, ${\beta }_{+}=1/\epsilon$, ${\beta }_{-}=2/\epsilon$, $\epsilon =0.1$, $k=5/N$, $X=50$, and $l=0.25$.

Figure 5

The MFPT vs hitting probability for the single-capacity trap ($n=1$). Each curve is parameterized by $0⩽{\beta }_{+}⩽{\beta }_{-}$. Analytical results are shown as lines, with the single searcher in gray and the mean-field limit ($N=\infty$) in black. Averaged Monte Carlo simulations (${10}^4$ simulation each) are shown as circles, with sets ranging from light gray to black, with a different value of $N$ used in each set. From light gray to black there are six sets of simulations with $N=1,2,3,4,5,25$, respectively. Parameter values are the same as in Fig. 4.

Figure 6

The first passage time density for different values of $n=1,2,3,4,5$ with $n$ increasing from left to right. Solid curves show the mean-field limit, and histograms are generated from ${10}^4$ Monte Carlo simulations with $N=50$ searchers. (a) The unbiased case $V=0$. (b) The biased case $V>0$. Parameter values used are the same as in Figs. 2 and 4 for (a) and (b), respectively.

Figure 7

The MFPT vs hitting probability for different trap capacities $n$. Each curve is parameterized by $0⩽{\beta }_{+}⩽{\beta }_{-}$. The mean-field limit is shown as solid curves, and ${10}^3$ averaged Monte Carlo simulations with $N=50$ are shown as circles. Parameter values are the same as in Fig. 4.