Asymptotic Analysis of First Passage Time Problems Inspired by Ecology

Abstract

A hybrid asymptotic–numerical method is formulated and implemented to accurately calculate the mean first passage time (MFPT) for the expected time needed for a predator to locate small patches of prey in a 2-D landscape. In our analysis, the movement of the predator can have both a random and a directed component, where the diffusivity of the predator is isotropic but possibly spatially heterogeneous. Our singular perturbation methodology, which is based on the assumption that the ratio $\varepsilon$ of the radius of a typical prey patch to that of the overall landscape is asymptotically small, leads to the derivation of an algebraic system that determines the MFPT in terms of parameters characterizing the shapes of the small prey patches together with a certain Green’s function, which in general must be computed numerically. The expected error in approximating the MFPT by our semi-analytical procedure is smaller than any power of ${-1/\log \varepsilon }$, so that our approximation of the MFPT is still rather accurate at only moderately small prey patch radii. Overall, our hybrid approach has the advantage of eliminating the difficulty with resolving small spatial scales in a full numerical treatment of the partial differential equation (PDE). Similar semi-analytical methods are also developed and implemented to accurately calculate related quantities such as the variance of the mean first passage time (VMFPT) and the splitting probability. Results for the MFPT, the VMFPT, and splitting probability obtained from our hybrid methodology are validated with corresponding results computed from full numerical simulations of the underlying PDEs.

Appendix: The Logarithmic Capacitance of a Two-Disk Cluster

We first derive the result (50) for the logarithmic capacitance $d_{1c}$ of two disjoint circular patches $\varOmega _{1,0}$ and $\varOmega _{1,1}$ of radii $a_0$ and $a_1$, respectively. Since $d_{1c}$ is invariant under coordinate rotations, we can without loss of generality choose the centers of the circles to lie along the horizontal $y_2=0$ axis at some $y_1=b_0>0$ and $y_1=-b_1<0$, respectively.

For this special two-trap cluster, the inner problem (49) with $\mathbf y =(y_1,y_2)$, is

$$\begin{aligned} \varDelta _\mathbf{y } q_c&= 0 , \quad \mathbf y \notin \varOmega _{1,j} , \quad j=0,1 \,; \quad q=0 , \quad \mathbf y \in \partial \varOmega _{1,j} , \quad j=0,1, \nonumber \\ q&\sim \log |\mathbf y | , \quad \hbox {as} \quad |\mathbf y |= (y_1^2+y_2^2)^{1/2} \rightarrow \infty , \end{aligned}$$

(75)

where $\varOmega _{1,1}$ and $\varOmega _{1,0}$ are the circles $(y_1+b_1)^2+y_2^2=a_1^2$ and $(y_1-b_0)^2+y_2^2=a_0^2$, respectively. The logarithmic capacitance $d_{1c}$ of the two-circle cluster is defined in terms of the solution to (75) by the far-field condition

$$\begin{aligned} q_c - \log |\mathbf y | = -\log {d} + o(1) , \quad \hbox {as} \quad |\mathbf y |=(y_1^2+y_2^2)^{1/2} \rightarrow \infty . \end{aligned}$$

(76)

To solve (75), we introduce bipolar coordinates $\xi$ and $\eta$ defined by

$$\begin{aligned} y_1 = \frac{c \sinh \xi }{\cosh \xi -\cos \eta } , \quad y_2 = \frac{c \sin \eta }{\cosh \xi -\cos \eta } . \end{aligned}$$

(77)

Then, $|\mathbf y |\rightarrow \infty$ corresponds to $\rho \equiv (\xi ^2+\eta ^2)^{1/2}\rightarrow 0$. From (77), we obtain $|\mathbf y |\sim {2c/\rho }$ as $|\mathbf y |\rightarrow \infty$. Therefore, the far-field behavior in (75) is equivalent to $q_c\sim -\log \rho$ as $\rho =(\xi ^2+\eta ^2)^{1/2}\rightarrow 0$.

With bipolar coordinates, lines of constant $\xi$ map to disks of the form $(y_1-y_{c})^2+y_2^2=a^2$, where $y_c={c/\tanh \xi }$ and $a={c/|\sinh \xi |}$. As such, the right circle $\varOmega _{1,0}$ with center $\mathbf y =(b_0,0)$ and radius $a_0$ corresponds to $\xi =\xi _0>0$, where

$$\begin{aligned} a_0 = {c/\sinh \xi _0} , \quad b_0 = {c/\tanh \xi _0} . \end{aligned}$$

(78)

In contrast, the left circle $\varOmega _{1,1}$ with center $\mathbf y =(-b_1,0)$ and radius $a_1$ corresponds to $\xi =-\xi _1<0$, so that $\xi _1>0$, where

$$\begin{aligned} a_1 = {c/\sinh \xi _1} , \quad b_1 = {c/\tanh \xi _1} . \end{aligned}$$

(79)

We label the center-to-center distance between the two disks as $l$, so that $l=b_0+b_1$.

From (78) and (79), we obtain that $\xi _0>0$, $\xi _1>0$ and $c$ are determined in terms of the disk radii $a_0$ and $a_1$, and the center-to-center distance $l$, by the three equations

$$\begin{aligned} a_0={c/\sinh \xi _0} , \quad a_1={c/\sinh \xi _1} , \quad l = {c/\tanh \xi _0} + {c/\tanh \xi _1} . \end{aligned}$$

(80)

From this system, we readily derive that $l = \sqrt{c^2 + a_0^2} + \sqrt{c^2 + a_1^2}$. By squaring this relation, we can solve for $c$ in terms of $l$ to obtain, after some algebra, that $c$ is given by (51). With $c$ known, $\xi _0$ and $\xi _1$ are obtained as in (52). We remark that, with $c$ as given in (51), the centers of the two disks are at $\mathbf y =(\sqrt{c^2+a_0^2},0)$ and $\mathbf y =(-\sqrt{c^2+a_1^2},0)$.

Upon transforming (75) to bipolar coordinates, we obtain that $Q(\xi ,\eta )\equiv q_c\left[ y_1(\xi ,\eta ),y_2(\xi ,\eta )\right]$ satisfies

$$\begin{aligned} Q_{\xi \xi } + Q_{\eta \eta }&= 0 , \quad -\xi _1\le \xi \le \xi _0 , \quad |\eta |\le \pi , \nonumber \\&\quad Q=0 \quad \hbox {on}\quad \xi =-\xi _1,\,\,\,\xi =\xi _0\,; \quad Q, Q_\eta \quad 2\pi \,\,\hbox {periodic in } \eta , \nonumber \\&\quad Q \sim -\frac{1}{2}\log \left( \xi ^2+\eta ^2\right) + {\mathcal O}(1) , \quad \hbox {as} \quad \xi ^2+\eta ^2 \rightarrow 0 . \end{aligned}$$

(81)

To solve (81), we first observe that a special solution to $Q_{\xi \xi }+Q_{\eta \eta }=0$ with the singularity behavior in (81) is

$$\begin{aligned} Q_{f}(\xi ,\eta )\equiv -\frac{1}{2}\log \left( \cosh \xi -\cos \eta \right) = -\frac{|\xi |}{2} + \frac{\log {2}}{2} + \sum _{m=1}^{\infty } \frac{e^{-m|\xi |}}{m} \cos (m\eta ) . \quad \end{aligned}$$

(82)

We then decompose $Q=Q_f+ Q_{p}$, so that $Q_p$ satisfies $\varDelta _\mathbf{y } Q_p=0$, is $2\pi$ periodic in $\eta$, and satisfies the boundary conditions

$$\begin{aligned} Q_{p}(\xi _0,\eta )&= \frac{\xi _0}{2} -\frac{\log {2}}{2} - \sum _{m=1}^{\infty } \frac{e^{-m\xi _0}}{m}\cos (m\eta ) , \nonumber \\ Q_{p}(-\xi _1,\eta )&= \frac{\xi _1}{2} -\frac{\log {2}}{2} - \sum _{m=1}^{\infty } \frac{e^{-m\xi _1}}{m}\cos (m\eta ) . \end{aligned}$$

(83)

To determine $Q_p$, we first represent $Q_p$ as a Fourier cosine series in $\cos (m\eta )$ and then use the boundary conditions (83) to identify the coefficients in the Fourier series. In this way, we obtain that

$$\begin{aligned} Q_p(\xi ,\eta ) =C_0 + D_0 \xi + \sum _{m=1}^{\infty } \left[ C_m \cosh (m\xi ) + D_m \sinh (m\xi )\right] \cos (m\eta ) , \end{aligned}$$

(84a)

where

$$\begin{aligned} C_0&= -\frac{\log {2}}{2} + \frac{\xi _0\xi _1}{\xi _0+\xi _1}\nonumber \\ C_m&= -\frac{ \left[ e^{-m\xi _0}\sinh (m\xi _1)+e^{-m\xi _1}\sinh (m\xi _0)\right] }{m \sinh \left[ m(\xi _0+\xi _1)\right] } , \quad \hbox {for} \quad m\ge 1, \end{aligned}$$

(84b)

with similar formulae for the coefficients $D_m$ for $m\ge 0$.

Finally, to identify the logarithmic capacitance $d_{1c}$ of the cluster, we expand $Q=Q_f+Q_p$ as $(\xi ,\eta )\rightarrow 0$ to obtain

$$\begin{aligned} Q \sim -\frac{1}{2}\log \left( \frac{\xi ^2+\eta ^2}{2}\right) + Q_{p}(0,0) . \end{aligned}$$

(85)

Since $\xi ^2+\eta ^2\sim {4c^2/|\mathbf y |^2}$ from (77), we obtain from (85) that

$$\begin{aligned} q_c \sim \log |\mathbf y | - \log \left( 2\sqrt{c}\right) + Q_p(0,0) , \quad \hbox {as} \quad |\mathbf y |\rightarrow \infty . \end{aligned}$$

(86)

From this relation, together with $Q_{p}(0,0)=\sum _{m=0}^{\infty } C_m$ from (83), we identify $d_{1c}$ in (76) as

$$\begin{aligned} \log {d_{1c}} = \log \left( 2 c\right) - \frac{\xi _0\xi _1}{\xi _0+\xi _1} - \sum _{m=1}^{\infty } C_m, \end{aligned}$$

(87)

where $C_m$ for $m\ge 1$ is given in (84b). This completes the derivation of (50).

Next, we derive (53) for $q^{\star }_{\infty }$ by first transforming (48) to bipolar coordinates. In this way, we obtain that $Q^{\star }(\xi ,\eta )\equiv q^{\star }\left[ y_1(\xi ,\eta ),y_2(\xi ,\eta )\right]$ is the smooth function that satisfies

$$\begin{aligned} Q^{\star }_{\xi \xi } + Q^{\star }_{\eta \eta }&= 0 , \quad -\xi _1\le \xi \le \xi _0 , \quad |\eta |\le \pi , \nonumber \\ Q^{\star }=0 \quad \hbox {on}\quad \xi&= -\xi _1\,; \quad Q^{\star }=1 \quad \hbox {on}\quad \xi =\xi _0\,; \quad Q^{\star }, Q^{\star }_\eta \quad 2\pi \,\, \hbox {periodic in } \eta .\nonumber \\ \end{aligned}$$

(88)

The solution to this problem is simply $Q^{\star }={(\xi +\xi _1)/(\xi _0+\xi _1)}$. Since $r\rightarrow \infty$ as $(\xi ,\eta )\rightarrow 0$, we readily identify the far-field behavior in (48) as $q^{\star }_{\infty }=Q^{\star }(0,0)={\xi _1/(\xi _0+\xi _1)}$, which confirms (53).