Mean First Passage Times for Transport Equations

TL;DR

The paper develops a comprehensive MFPT theory for velocity-jump (kinetic) transport, deriving forward and backward equations and establishing MFPT equations for isotropic diffusive and anisotropic regimes. A key contribution is the parabolic scaling that yields a macroscopic, velocity-averaged MFPT equation with a diffusion tensor $\mathbb{D}(x) = \frac{1}{\mu} \int_V v\otimes v\, q(x,v) \, dv$, linking microscopic turning dynamics to a diffusion-like MFPT for $T(x)$. The authors provide explicit, solvable results in circular domains and annuli under radial or tangential biases, and demonstrate applications to ecological landscapes with linear features such as roads or seismic lines. Together, the framework bridges microscopic transport mechanisms and macroscopic first-arrival times, offering analytical and numerical tools for modeling MFPT in biological and ecological contexts.

Abstract

Many transport processes in ecology, physics and biochemistry can be described by the average time to first find a site or exit a region, starting from an initial position. Typical mathematical treatments are based on formulations that allow for various diffusive forms and geometries but where only initial and final positions are taken into account. Here, we develop a general theory for the mean first passage time (MFPT) for velocity jump processes. For random walkers, both position and velocity are tracked and the resulting Fokker-Planck equation takes the form of a kinetic transport equation. Starting from the forward and backward formulations we derive a general elliptic integro-PDE for the MFPT of a random walker starting at a given location with a given velocity. We focus on two scenarios that are relevant to biological modelling; the diffusive case and the anisotropic case. For the anisotropic case we also perform a parabolic scaling, leading to a well known anisotropic MFPT equation. To illustrate the results we consider a two-dimensional circular domain under radial symmetry, where the MFPT equations can be solved explicitly. Furthermore, we consider the MFPT of a random walker in an ecological habitat that is perturbed by linear features, such as wolf movement in a forest habitat that is crossed by seismic lines.

Figures (4)

• Figure 1: Schematic of anisotropic transport in a bounded planar region $\Omega\in\mathbb{R}^2$ in the presence of one-dimensional substructures. Diffusion is one-dimensional along the linear features and two-dimensional away from them. This schematic can represent, for example, organelle transport within a two-dimensional cell, grown on a flat surface, whereby particles transition from performing linear random walks along microtubules and/or other filaments emanating from the nucleus, to undergoing two-dimensional planar diffusion in the cytoplasm. Similarly, it can represent animal motion in a given environment with roads or other one-dimensional features that direct the animal's movement. Due to the stochastic nature of the transport process, each individual trajectory starting at a given position $x \in \Omega$ and with a given velocity $v$ will reach the target area, shown in red, at a specific time. Our goal is to determine the mean first arrival time to the target. We can model this transport scenario using a kinetic transport equation with turning kernel given by the bimodal von-Mises distribution in a circular geometry and assuming that motion is biased along the radial direction.
• Figure 2: The MFPT \ref{['eqn:MFPT_radial']} for $D=0.5$ on an annulus with domain $\rho<r<R_0$, inner circle radius $\rho = 0.5$, outer circle radius $R_0 = 3$ and several values of $\alpha \in (-1,1)$. Panel (a): Particles exit the annulus only through the inner circle; the corresponding boundary conditions under which to solve \ref{['disk2']} are $T_\alpha(\rho) = T'_\alpha(R_0)=0$. Note that $T_\alpha(r)$ increases with $\alpha$ for given $r > \rho$ and $T_\alpha(r > \rho)\to\infty$ as $\alpha \to -1^{+}$ as shown in \ref{['circinner']}. Under purely circular motion particles move tangentially and cannot decrease their distance from the origin, hence they will never exit the annulus. Panel (b): Particles may exit the annulus through both the inner and outer circles; the corresponding boundary conditions under which to solve \ref{['disk2']} are $T_\alpha(\rho) = T_\alpha(R_0)=0$. In the limit $\alpha\to-1^+$ the solution $T_\alpha(r)$ displays a discontinuity at $r=\rho$ as particles located at $r \to \rho^{+}$ will be able to exit the annulus only through the outer circle at $r= R_0$. Particle simulations (black squares) based on $N=10^4$ independent stochastic simulations of \ref{['kinetic1']} with $\mu = 10^4$, $\sigma = \sqrt{\mu}$ agree with theoretical predictions. Specialized numerical algorithms are needed to simulate the exit from the inner circle as $\alpha \to {-1}^{+}$ since $T_{\alpha}\gg1$. Indeed, we note the difference in the scale of the vertical axes between the two panels. All quantities are in arbitrary units.
• Figure 3: Top row: The MFPT to the boundary of the domain $\Omega = [-1,1]^2$ in the presence of one or multiple linear segments orienting particle motion. Results are obtained by numerically solving \ref{['anisotropicMFPT3']} for $T(x)$ in the domain $\Omega = [-1,1]^2$ under the bimodal von-Mises distribution \ref{['eq:vonMises']} modulated by the linear structures shown in the inset as described in \ref{['eq:linear_eqn']}. The cutoff distance $d_0$ for particles to move along the linear structures is set at $d_0 = 0.02$ and the corresponding measure of concentration is $k_0=25$. The anisotropy profiles from left to right are a vertical line, a slant line, 3 parallel vertical lines and 10 randomly intersecting lines. Bottom row: The difference between the MFPT under anisotropic transport and the MFPT for a random walker in the same two-dimensional domain. The scenario of 10 randomly intersecting segments in the right-most panels yields the smallest difference between the two MFPTs. All quantities are in arbitrary units. \newlabelfig:MFPT_linear0
• Figure 4: Calculation of the MFPT on an ecological landscape. Panel (a): Aerial snapshot of the boreal forest in Western Canada. Panel (b): A digitized and thresholded version of the same image highlighting roads and seismic lines with bidirectional information. Panel (c): Contour plot of the MFPT as derived from \ref{['anisotropicMFPT3']}, \ref{['eq:DiffVonMises']} and \ref{['eq:linear_eqn']} using the method described in Section \ref{['sec:linear']}. Also shown are three single trajectories from particle simulations of \ref{['kinetic1']}, each initialized at the origin with parameters $\mu = 10^4$, $\sigma = 10^4$ and bimodal von-Mises turning kernel \ref{['eq:vonMises']}. Particle trajectories remain closely anchored to the linear segments before eventual absorption at different edges of $\partial\Omega$. All quantities are in arbitrary units.

Theorems & Definitions (4)

• Lemma 1
• Proof 1
• Lemma 2
• Proof 2