Quasi-steady-state analysis of two-dimensional random intermittent search processes

We use perturbation methods to analyze a two-dimensional random intermittent search process, in which a searcher alternates between a diffusive search phase and a ballistic movement phase whose velocity direction is random. A hidden target is introduced within a rectangular domain with reflecting boundaries. If the searcher moves within range of the target and is in the search phase, it has a chance of detecting the target. A quasi-steady-state analysis is applied to the corresponding Chapman-Kolmogorov equation. This generates a reduced Fokker-Planck description of the search process involving a nonzero drift term and an anisotropic diffusion tensor. In the case of a uniform direction distribution, for which there is zero drift, and isotropic diffusion, we use the method of matched asymptotics to compute the mean first passage time (MFPT) to the target, under the assumption that the detection range of the target is much smaller than the size of the domain. We show that an optimal search strategy exists, consistent with previous studies of intermittent search in a radially symmetric domain that were based on a decoupling or moment closure approximation. We also show how the decoupling approximation can break down in the case of biased search processes. Finally, we analyze the MFPT in the case of anisotropic diffusion and find that anisotropy can be useful when the searcher starts from a fixed location.

Figure 1

Stochastic model of random intermittent search.

Figure 2

The MFPT $T_{\mathrm{av}}$ of an unbiassed random intermittent search process on a square domain of length $L$. $T$ is plotted as a function of (a) the transition rate $\alpha$ and (b) the transition rate $\beta$. The target is located at the center of the domain and the target radius is $\rho =0.05L$. In this and subsequent figures we reabsorb the scaling factor $\epsilon$ into $\alpha ,\beta$ and express parameters in physical units. For the sake of illustration, we take $L=10$ $\mu$m, $v=1\hspace{0.5em}\mu {\mathrm{m}}^{\mathrm{\hspace{0.5em}}}$, and $k=0.5$ ${\mathrm{s}}^{-1}$ and time is measured in seconds.

Figure 3

Averaged MFPT, $T_{\mathrm{av}}$, in the $(\alpha ,\beta )$ plane. (a) Analytical approximation (3.30) with effective diffusivity and detection rate given by (3.2). The minimum is marked with an “O.” (b) Decoupling approximation, with the minimum at $({\alpha }_{\mathrm{opt}},{\beta }_{\mathrm{opt}})$ marked with an asterisk. (c) The amount of ${10}^5$ averaged Monte Carlo simulations for each value of $(\alpha ,\beta )$ on a 20$\times$20 grid. The minimum is marked by with “X,” which is also shown in (a) and (b) for comparison. Parameter values are $L=10$ $\mu$m, $\rho =0.35$ $\mu$m, $v=1$ $\mu$m ${\mathrm{s}}^{-1}$, and $k=1$ ${\mathrm{s}}^{-1}$.

Figure 4

Relative error for both approximations in Fig. 3 compared to Monte Carlo simulations. The magnitude of the relative error is shown on the color bar (gray) scale [dark red (gray) shows error greater than or equal to 0.1]. Contours show the averaged MFPT from Monte Carlo simulations in Fig. 3c. (a) The QSS reduction. The cone of accuracy reflects the asymptotic nature of the QSS approximation. (b) The decoupling approximation. The more uniform accuracy of the decoupling approximation give a slightly better estimate of the optimal transition rates, but the QSS approximation gives a slightly better estimate of the minimum MFPT.

Figure 5

Anisotropy in the diffusion tensor effectively alters the search domain. With no anisotropy (i.e., $q=1/2$) the two domains are both the unit square. As we increase $q$, diffusion in the $y$ direction is inhibited, which effectively decreases the distance to the target, as seen in the rescaled domain.

Figure 6

Effect of anisotropy, $q$, on the MFPT as the target location ${\mathbf{r}}_0$ is varied while the initial position $\mathbf{r}$ of the searcher is fixed. (a) Plot of MFPT against $q$ for various target locations. (b) Plot of MFPT against target location for various $q$. Domain parameters $L$ and $\rho$ are the same as in Fig. 2. The target location is ${\mathbf{r}}_0=(L-3\rho ,y_0)$ with $\rho ⩽y_0⩽L/2$ and the initial position of the searcher is $\mathbf{r}=(0,L/2)$. Here $L=10$ $\mu$m and $\rho =0.5$ $\mu$m. Transition rates are $\alpha =4$ ${\mathrm{s}}^{-1}$ and $\beta =6$ ${\mathrm{s}}^{-1}$ while the detection rate is $k=0.5$ ${\mathrm{s}}^{-1}$.