Simulation Video
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Mean First Passage Time for a small periodic moving
trap inside a reflecting disk
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
The University of British Columbia, Vancouver.
June 12, 2017
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
2 / 30
Overview
Introduction to Mean first passage time (MFPT)
MFPT problems
Stationary trap
Moving trap
Some published results
Current work
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Introduction to Mean first passage
time
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Introduction to Mean first passage time (MFPT)
First passage time is the time it a particle to get to a specific point or
exit a domain/region starting from a specific location.
Mean first passage time is the average of the first passage time
distribution
Application: Immune cell activation, predator-prey interaction, e.t.c.
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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MFPT for stationary trap
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Derivation Mean first passage time (MFPT)
Consider a random walk on a one-dimensional domain Ω = [0, L] with
reflecting boundaries at x = 0 and x = L, and an absorbing stationary
trap at x0 ∈ Ω
Let u(x) be the MFPT for a particle starting at position x ∈ Ω to get
absorbed in the trap
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Derivation Mean first passage time (MFPT)
We can write the MFPT of the particle starting at point x in terms of
the MFPT starting at the neighboring points of x, that is,
1
1
u(x) = u(x − ∆x) + u(x + ∆x) + ∆t
2
2
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Derivation Mean first passage time (MFPT)
Taylor expanding and simplifying
−1 =
(∆x)2 00
u (x).
2 ∆t
Taking the limit ∆x −→ 0 and ∆t −→ 0 such that D ≡
have
−1 = D u 00 (x)
(∆x)2
2 ∆t ,
we
with the following boundary condition u 0 (0) = 0 and u 0 (L) = 0
At x0 , u(x0 ) = 0
Expect/average MFPT over the domain is
Z
1
u=
u(x) dx.
|Ω| Ω
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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MFPT on a disk
Consider a Brownian particle in a reflecting disk-shaped region Ω of
radius R with a circular trap Ω0 ⊂ Ω of radius D ∆u = −1,
∂n u = 0, on ∂Ω
x ∈ Ω \ Ω0 ,
and u = 0 on ∂Ω0 .
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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MFPT on a disk
Parameters: D = 1, R = 1, and = 0.1.
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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MFPT for oscillating trap
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Derivation Mean first passage time (MFPT)
Consider a random walk on a one-dimensional domain Ω = [0, 1] with
reflecting boundaries at x = 0 and x = 1, and an absorbing trap that
oscillates around at x = 1/2 (with small amplitude)
MFPT depends on:
location of particle
location of trap
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Derivation Mean first passage time (MFPT)
Suppose the position of the trap is determined by some function. For
example, η(t) = 1/2 + sin(ωt)
Figure ref: Tzou, Justin C., Shuangquan Xie, and Theodore Kolokolnikov.
”Drunken robber, tipsy cop: First passage times, mobile traps, and Hopf
bifurcations.” arXiv preprint arXiv:1410.1391 (2014).
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Derivation Mean first passage time (MFPT)
Let u(x, t) be the stationary MFPT for a particle at location x at
time t
1
1
u(x, t) = u(x − ∆x, t + ∆t) + u(x + ∆x, t + ∆t) + ∆t.
2
2
Taylor expanding, simplifying, and taking the limit ∆x −→ 0 and
2
∆t −→ 0 such that D ≡ (∆x)
2 ∆t ,
∂u
∂2u
= −D 2 − 1.
∂t
∂x
backward-time diffusion equation
Expect/average MFPT over a period
Z Z
1 1
u=
u(x, t) dΩ dt.
|Ω| T T Ω
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Simulation result
Parameters: D = 1, = 0.2, ω = 80, and x = 0.5.
Figure: Surface plot of MFPT.
Ref: Tzou, Justin C., Shuangquan Xie, and Theodore Kolokolnikov. ”Drunken robber,
tipsy cop: First passage times, mobile traps, and Hopf bifurcations.”
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Comparison of simulation result
Figure: Comparison of MFPT obtained from PDE solution and Monte Carlo
simulation.
Ref: Tzou, Justin C., Shuangquan Xie, and Theodore Kolokolnikov. ”Drunken robber,
tipsy cop: First passage times, mobile traps, and Hopf bifurcations.”
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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MFPT on a disk with a periodic
moving trap
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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MFPT on a disk with an oscillating trap
Consider a Brownian particle in a disk-shaped region with a reflecting
boundary and a trap that rotates about the center of the trap
MFPT problem is (τ = −t)
∂u
= D∆u + 1, x ∈ Ω \ Ω0 , τ > 0,
∂τ
∂n u = 0 on ∂Ω, u = 0 on Ω0 (τ ),
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
2π
u(x, 0) = u x,
ω
June 12, 2017
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Simulation video
Simulation Video
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Some screen shots from the simulation (t=0, 0.0265,
0.1909, 0.5655)
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Comparison of simulated result (Monte Carlo, Flex PDE,
and Finite Difference & Closest point method)
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Some important questions to ask about the method.
Original Problem: ’Boundary Value Problem’
∂u
= D∆ u + 1,
∂τ
x ∈ Ω \ Ω0 , τ > 0,
∂n u = 0 on ∂Ω, u = 0 on Ω0 (τ ),
2π
u(x, 0) = u x,
ω
Problem considered: Initial Value Problem
∂u
= D∆ u + 1, x ∈ Ω \ Ω0 , τ > 0,
∂τ
∂n u = 0 on ∂Ω, u = 0 on Ω0 (τ ),
u(x, 0) = static solution ???
Question: Does IVP solved after n periods converge to BVP?
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Some published results ...
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
24 / 30
Some published results
Optimizing the radius of rotation of the trap that minimizes average
MFPT
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Problem description
For a given angular frequency ω,
find the optimal radius of rotation, r0opt of the trap that minimizes
average MFPT
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Result: Plot of r0opt vs ω
bifurcation at ωc ≈ 3.026 such that for ω < ωc , r0opt = 0
result is not monotonic
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Conclusion
Follow up question:
Suppose the trap does not oscillate about the center of the disk
will the result be the similar?
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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References
Tzou JC, Kolokolnikov T. Mean first passage time for a small rotating trap inside a
reflective disk. Multiscale Modeling & Simulation. 2015 Jan 22;13(1):231-55.
Tzou JC, Xie S, Kolokolnikov T. Drunken robber, tipsy cop: First passage times,
mobile traps, and Hopf bifurcations. arXiv preprint arXiv:1410.1391. 2014 Oct 3.
Tzou JC, Xie S, Kolokolnikov T. First-passage times, mobile traps, and Hopf
bifurcations. Physical Review E. 2014 Dec 29;90(6):062138.
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
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Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald
MFPT for moving
(UBC) trap
June 12, 2017
30 / 30