Simulation Video Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 1 / 30
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 3 / 30
Introduction to Mean first passage time Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 4 / 30
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 5 / 30
MFPT for stationary trap Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 6 / 30
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 x 0 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 7 / 30
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, u(x) = 1 2 u(x x) + 1 u(x + x) + t 2 Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 8 / 30
Derivation Mean first passage time (MFPT) Taylor expanding and simplifying 1 = ( x)2 2 t u (x). Taking the limit x 0 and t 0 such that D ( x)2 2 t, we have 1 = D u (x) with the following boundary condition u (0) = 0 and u (L) = 0 At x 0, u(x 0 ) = 0 Expect/average MFPT over the domain is u = 1 u(x) dx. Ω Ω Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 9 / 30
MFPT on a disk Consider a Brownian particle in a reflecting disk-shaped region Ω of radius R with a circular trap Ω 0 Ω of radius ɛ D u = 1, x Ω \ Ω 0, n u = 0, on Ω and u = 0 on Ω 0. Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 10 / 30
MFPT on a disk Parameters: D = 1, R = 1, and ɛ = 0.1. Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 11 / 30
MFPT for oscillating trap Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 12 / 30
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 13 / 30
Derivation Mean first passage time (MFPT) Suppose the position of the trap is determined by some function. For example, η(t) = 1/2 + ɛ 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 14 / 30
Derivation Mean first passage time (MFPT) Let u(x, t) be the stationary MFPT for a particle at location x at time t u(x, t) = 1 2 u(x x, t + t) + 1 u(x + x, t + t) + t. 2 Taylor expanding, simplifying, and taking the limit x 0 and t 0 such that D ( x)2 2 t, u t = D 2 u x 2 1. backward-time diffusion equation Expect/average MFPT over a period u = 1 1 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 15 / 30
Simulation result Parameters: D = 1, ɛ = 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 16 / 30
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 17 / 30
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 18 / 30
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 τ = D u + 1, x Ω \ Ω 0, τ > 0, n u = 0 on Ω, u = 0 on Ω 0 (τ), u(x, 0) = u ( x, 2π ) ω Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 19 / 30
Simulation video Simulation Video Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 20 / 30
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 21 / 30
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 22 / 30
Some important questions to ask about the method. Original Problem: Boundary Value Problem u τ = D u + 1, x Ω \ Ω 0, τ > 0, n u = 0 on Ω, u = 0 on Ω 0 (τ), u(x, 0) = u ( x, 2π ) ω Problem considered: Initial Value Problem u τ = D u + 1, x Ω \ Ω 0, τ > 0, n u = 0 on Ω, u = 0 on Ω 0 (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 23 / 30
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 25 / 30
Problem description For a given angular frequency ω, find the optimal radius of rotation, r opt 0 of the trap that minimizes average MFPT Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 26 / 30
Result: Plot of r opt 0 vs ω bifurcation at ω c 3.026 such that for ω < ω c, r opt 0 = 0 result is not monotonic Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 27 / 30
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 28 / 30
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 29 / 30
Sarafa Iyaniwura, Kelly Huang, Justin Tzou, and Colin Macdonald MFPT for moving (UBC) trap June 12, 2017 30 / 30