hwk11 - a the speed of the front connecting the rest state...

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Phys. 7268 Assignment 11 Due: 4/21/11 Problem 1 Piecewise Linear Model. Consider the reaction-diffusion system t u = η - 1 f ( u,v ) + 2 u, t v = g ( u,v ) , (1) with a “piecewise linear” form of the reaction kinetics: f ( u,v ) = θ ( u - a ) - u - v, g ( u,v ) = u - bv, (2) where θ ( · ) is a Heaviside step function and η ± 1. (a) Sketch the u and v nullclines for a = 0 . 25 and (i) b = 0 . 2 and (ii) b = 1. For what ranges of b is the reaction kinetics monostable and bistable for this value of a . (b) Consider the front connecting, for a = 0 . 3, the rest state u = 0, v = 0 with the excited state at this value of v , i.e. u = 1, v = 0. Plot the effective potential Φ( u ). (c) What is the value of v for which the front connecting the small u and large u portions of the u -nullcline is stationary for a = 0 . 25 (i.e. v = v * giving the stall solution)? (d) Calculate as a function of the parameter
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Unformatted text preview: a the speed of the front connecting the rest state u = 0, v = 0 with the excited state at this value of v , i.e. u = 1, v = 0. Verify that in the approximation we used this front speed is zero for a = 0 . 5. (e) Find the propagation speed c and calculate and plot u ( x-ct ) and v ( x-ct ) for the excitation pulse propagating in the rest state of this system with a = 0 . 25, b = 0 . 2. (f) Calculate and plot expressions for the dispersion relationship C ( T ) for the scaled speed C = 1 / 2 c as a function of the temporal period T for propagating wave trains for a = 0 . 25, b = 0 . 2. Do not worry about the breakdown of the scaling that occurs for small C ....
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