traveling waves in a bistable reaction-diffusion...
TRANSCRIPT
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Traveling Waves in a bistable Reaction-Diffusion System
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Wave phenomena in RD systems
€
∂u∂t
= f (u) + D∂2u∂x 2
, ∂u∂x ±∞
= 0
Keener & Sneyd (1998) Mathematical Physiology; p 270-275
€
f (u) = au(u −1)(α − u) 0 <α <1
f (u) = −u + H(u −α) 0 <α <1
With f cubic:
or
Piecewise linear:
1 α
McKean, 1970
f(u)
f(u)
U is the shape of the wave, z is the position along the wave front
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Bistable well-mixed system
1 α Steady states:
u = 0, α, 1
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With diffusion
1 α
u=1
u=0 x
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Moving Wave
u=1
u=0 x
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Wave profile
z
Traveling wave coordinates: z=x-ct, U(z)=u(x,t)
u=1
u=0
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What is the wave profile? Traveling wave coordinates: z=x-ct, U(z)=u(x,t) Scaling and transformation of coordinates:
€
∂∂t→−c d
dz∂∂x
→ddz
∂u∂t
= f (u) +∂ 2u∂x 2
→ − c dUdz
= f (U) +d2Udz2PDE ODE
€
Uzz − cUz + f (U) = 0
Uz =WWz = cW − f (U)
2nd order (nonlin) ODE
Equivalent ODE system: We can study this qualitatively in the UW phase plane to get insight into the shape of the wave.
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Example: f(u)=u(1-u)(u- α)
€
∂u∂t
= f (u) +∂ 2u∂x 2
W
U U
W
W’=0
U’=0
€
Uzz − cUz + f (U) = 0
Uz =WWz = cW − f (U)
(Rescaled)
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Xpp file # bistable.ode # classic example of a wave joining 0 and 1 # u_t = u_xx + u(1-u)(u-a) # # -cu'=u''+u(1-u)(u-a) f(u)=u*(1-u)*(u-a) par c=0,a=.25 u'=up up'=-c*up-f(u) init u=1 @ xp=u,yp=up,xlo=-.5,xhi=1.5,ylo=-.5,yhi=.5 done
Bard Ermentrout’s XPP file
Bistable.ode
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Interpretation: Each trajectory describes how U (and W) vary as z increases over range -∞< z < ∞
In bio applications: u= density>0, so we want: a positive bounded wave: Look for a positive bounded trajectory connecting two steady states.
W
U U
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Here is a heteroclinic trajectory:
W
U
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Here it is on its own:
U
W
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And here is the shape of the wave it represents:
U
z
![Page 14: Traveling Waves in a bistable Reaction-Diffusion Systemkeshet/MCB2012/SlidesLEK/TravWaves_ppt.pdf · 2012. 4. 12. · Bistable.ode! Interpretation:! Each trajectory describes how](https://reader035.vdocuments.mx/reader035/viewer/2022070213/610af709f83f56203f759ba4/html5/thumbnails/14.jpg)
What is the speed of the wave? • A reaction-diffusion equation with “bistable” kinetics will admit a traveling wave that looks like a moving “front”, with high level at back, and low level ahead.
• But how fast does the wave move and does it ever stop?
There is a cute trick that allows us to answer this question for arbitrary function f(u).
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Determining wave speed
€
∂u∂t
= f (u) +∂ 2u∂x 2
, z = x − ct
−c dUdz
− f (U) =∂ 2Udz2
−c dUdz
⎛
⎝ ⎜
⎞
⎠ ⎟ 2
− f (U) dUdz
⎛
⎝ ⎜
⎞
⎠ ⎟ =12ddz
dUdz
⎛
⎝ ⎜
⎞
⎠ ⎟ 2
−c dUdz
⎛
⎝ ⎜
⎞
⎠ ⎟ 2
−∞
∞
∫ dz − f (U)uback
ufront
∫ dU =12
ddz
dUdz
⎛
⎝ ⎜
⎞
⎠ ⎟ 2
= 0−∞
∞
∫
−c dUdz
⎛
⎝ ⎜
⎞
⎠ ⎟ 2
−∞
∞
∫ dz = f (U)uback
ufront
∫ dU
Multiply by dU/dz,
uback ufront
z
c
integrate,
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The wave speed:
![Page 17: Traveling Waves in a bistable Reaction-Diffusion Systemkeshet/MCB2012/SlidesLEK/TravWaves_ppt.pdf · 2012. 4. 12. · Bistable.ode! Interpretation:! Each trajectory describes how](https://reader035.vdocuments.mx/reader035/viewer/2022070213/610af709f83f56203f759ba4/html5/thumbnails/17.jpg)
Finally..
z
u=1
u=0
f(u)=u(1-u)(u- α)
![Page 18: Traveling Waves in a bistable Reaction-Diffusion Systemkeshet/MCB2012/SlidesLEK/TravWaves_ppt.pdf · 2012. 4. 12. · Bistable.ode! Interpretation:! Each trajectory describes how](https://reader035.vdocuments.mx/reader035/viewer/2022070213/610af709f83f56203f759ba4/html5/thumbnails/18.jpg)
Result!
But what does this tell us????
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Which way does it move?
1 α
f(u)
1 α
f(u)
Always positive
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Which way does it move?
1 α
f(u)
1 α
f(u)
> 0 < 0
c > 0 c < 0
![Page 21: Traveling Waves in a bistable Reaction-Diffusion Systemkeshet/MCB2012/SlidesLEK/TravWaves_ppt.pdf · 2012. 4. 12. · Bistable.ode! Interpretation:! Each trajectory describes how](https://reader035.vdocuments.mx/reader035/viewer/2022070213/610af709f83f56203f759ba4/html5/thumbnails/21.jpg)
Which way does it move?
1
f(u)
1
f(u)
c > 0 c < 0
![Page 22: Traveling Waves in a bistable Reaction-Diffusion Systemkeshet/MCB2012/SlidesLEK/TravWaves_ppt.pdf · 2012. 4. 12. · Bistable.ode! Interpretation:! Each trajectory describes how](https://reader035.vdocuments.mx/reader035/viewer/2022070213/610af709f83f56203f759ba4/html5/thumbnails/22.jpg)
Which way does it move?
1
f(u)
1
f(u)
c > 0 c < 0
![Page 23: Traveling Waves in a bistable Reaction-Diffusion Systemkeshet/MCB2012/SlidesLEK/TravWaves_ppt.pdf · 2012. 4. 12. · Bistable.ode! Interpretation:! Each trajectory describes how](https://reader035.vdocuments.mx/reader035/viewer/2022070213/610af709f83f56203f759ba4/html5/thumbnails/23.jpg)
We can now answer the question:
Under what conditions would the wave stop?
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Wave stops:
€
0 = c = − f (U)0
1
∫ dU dUdz
⎛
⎝ ⎜
⎞
⎠ ⎟ 2
−∞
∞
∫ dz
f (U)0
1
∫ dU = 0
Speed is zero if:
i.e.: if :
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Geometry of stalled wave:
1 α
f(u)
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Maxwell condition:
1 α
f(u)
€
f (U)0
1
∫ dU = 0
“Equal areas”
“Maxwell condition”:
€
f (U)0
α
∫ dU = − f (U)α
1
∫ dU
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Can we get an explicit solution for the wave speed?
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Explicit solution? In some special cases, e.g. f(u) cubic or piecewise linear, can calculate wave speed fully by this method.
(See Keener & Sneyd p 274, Murray p 305)
€
f (u) = au(u −1)(α − u) 0 <α <1For
Speed of the wave is:
€
c =a2(1− 2α)
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Implications: Wave moves right (c>0) if α<1/2 Moves left (c<0) if α>1/2
Wave stops (c=0) precisely for one value of the parameter, α=1/2
€
f (u) = au(u −1)(α − u) 0 <α <1