Pattern Formation in the Gray-Scott Model
Pouya Bastani - pbastani@math.sfu.ca
Department of Mathematics - Simon Fraser University
This determines the solution inside the spike. Far-field condition: Vj → 0
as |y| → ∞. For Uj , use matching to the outer region where v is negligible.
Bifurcation Diagram
Goal
0.25
Seen from outer region, the spike is like a delta function:
0.2
Numerical and analytical study of a pattern forming system through
• linear stability analysis and bifurcation
uv 2 → C0δ(x) + "C1δ(x) + · · ·
0.15
( ∞
1
C0 =
U0V02 dy
A −∞
F
• amplitude equations, exact and approximate solutions in special cases
0.1
Introduction
The Gray-Scott model is an example of a reaction-diffusion system describing
the following irreversible chemical reaction:
0
0.01
ut = Du∇2u − uv 2 + F (1 − u)
vt = Dv ∇2v + uv 2 − (F + k)v
D diffusion rates
k reaction rate of second equation
F feed rate of U into the system
Patterns: travelling waves, spot annihilation and self-replication, spatiotemporal chaos, mixed spot-stripe, labyrinthine stripes
0.02
0.03
0.04
0.05
0.06
0.07
k
In the case of equal diffusivity, variables and constant can be rescaled so that
u.. = uv 2 − λ(1 − u)
γv .. = v − uv 2
Numerical Simulations
U + 2V −→ 3V
V −→ P
In dimensionless form, concentrations of the reactants is given by
−∞
(2U0V0V1 + V02V1) dy
Exact Solutions
0.05
0
C1 =
( ∞
Domain size: 2.5 × 2.5
Diffusion constants: Du = 2Dv = 2 × 10−5
Number of grid points: 256 × 256
Boundary condition: periodic
Total running time: 10000
Step size: 30
Numerical Scheme: ETDRK4
Initial condition: (1, 0) everywhere except a small square at the center
perturbed to ( 12 , 41 ) with ±1% random noise
F = 0.03, k = 0.062
Relevance: animals such as leopard and zebra exhibit such patterns
Homoclinic orbits: u(−x) = u(x) v(−x) = v(x)
Heteroclinic orbits: ũ(−x) = −ũ(x) ṽ(−x) = −ṽ(x)
where ũ(x) = u(x) − u(0) and ṽ(x) = v(x) − v(0). For homoclinic orbits,
u.(0) = v .(0) = 0. Adding above equations and using p ≡ u − 1 + γv
p.. − λp = v(1 − λγ)
p(0) = u(0) − 1 + γv(0)
p.(0) = 0
Special case: λγ = 1 and p(0) = 0. By uniqueness p(x) ≡ 0 so that
u(x) = 1 − γv(x)
Eliminating u in the above system yields a second order ODE
γv .. = v(1 − v + γv 2)
which has a first integral given by
)
*
γ .2 1 2
2
γ 2
.
E(v, v ) = (v ) − v 1 − v + v
2
2
3
2
Phase plane (v, v .) for γ < 29 (left) and γ > 29 (right)
Spots: regions of high v and low u outside of which u ≈ 1 and v ≈ 0
Self-replication: Spots grow when there is high U flux to the center.
When insufficient amount of U reaches the center, due to large radius, the
spots separate and pieces move away to access more U .
F = 0.037, k = 0.06
Steady States
1) Homoclinic orbit (spike): 0 ≤ γ ≤ 29
Spatially uniform equilibrium solutions: ∇ ≡ 0:
3
v(x) =
√
1 + Qcosh(x/ γ)
0 = −uv 2 + F (1 − u)
0 = +uv 2 − (F + k)v
"
1
a
v(x) = (1 ± 1 − 4γ) −
2γ
1 + bcosh(cx)
3) Heteroclinic trajectory (kink): γ = 92
,
)
*3
2x
v(x) =
1 + tanh √
2
2 2
Saddle node bifurcation:
kc = −F +
2
F,
9γ
1−
2
2) Homoclinic orbit (valley): 92 < γ < 14
Trivial fixed point: (u, v) = (1, 0) for all values of F and k
Other fixed points: (u±, v∓) for F ≥ 4(F + k)2
!
#
"
u± = 12 1 ± 1 − 4γ 2F
F +k
#
!
"
γ≡
1 1 ∓ 1 − 4γ 2F
F
v∓ = 2γ
1√
Q=
+
The plots below show U and V in the above 3 cases.
1
0≤F ≤
4
Approximate 1D Solution
Linear Stability Analysis
Weak interaction regime: ratio of diffusivities is small
Perturbation of steady-states by "ũ(x, y, t) and "ṽ(x, y, t):
ũt = Du∇2ũ − (v 2 + F )ũ − 2uvṽ + O(")
ṽt = Dv ∇2ṽ − (F + k − 2vu)ṽ − v 2ũ + O(")
Ansatz with amplitudes u0, v0 and wave number k1, k2:
ũ = u0eλt−i(k1x+k2y)
ṽ = v0eλt−i(k1x+k2y)
Neglect terms of order " and let κ2 = k12 + k22:
$
$
2
2
$ Du κ + v + F + λ
$
2uv
$
0 = $$
2
v
Dv + F + k − 2uv + λ $
Trivial steady state: stable for all values of k and F (λ < 0)
λ1 = −Duκ2 − F
Other steady states: uv = F + k
λ2 = −Dv κ2 − F − k
v 2 = γ1 v − F
vt = "2vxx − v + Auv 2
τ ut = Duxx − u + 1 − uv 2
vx(±l, t) = ux(±l, t) = 0
where " + 1, "2 + D and x ∈ [−l, l] (rescaled variables).
Low-feed regime: A = O("1/2). Let A = "1/2A and v = "−1/2ν.
νt = "2νxx = ν + Auv 2
τ ut = Duxx − u + 1 − 1" ν 2u
Equilibrium spike solution located at x = 0:
!x#
!x#
u∼U
ν ∼ ξw
"
"
1
where w, ξ, U are to be determined. To first order U ∼ const. Let ξ = AU
w .. − w + w 2 = 0,
w .(0) = 0,
w(y) ∼ Ce−|y| as |y| → ∞
&y '
3
2
which has the explicit solution w = 2 sech 2
Center Manifold
Weak interaction regime: Du = 1, Dv = δ 2 + 1
Traveling wave ansatz: u = u(x − ct) and v = v(x − ct)
Rescaled variables: c = δγ and x − ct = δη
u̇
ṗ
v̇
q̇
=
=
=
=
δp
δ[−δγp + uv 2 − A(1 − u)]
q
−γq − uv 2 + Bv
Two time scales: u and p are slow variables, whereas v and q are fast.
Slow subsystem: ü + δγ u̇ + F (1 − u) = 0 defined on {u, p, v = 0, q = 0}
Fast subsystem: v̈ + γ v̇ + uv 2 − (F + k)v = 0 where u is constant.
Semi-strong interaction regime: corresponds to a well-stirred reaction, in which the diffusion constants are negligibly small with ratio of order
1. In this case, the system undergoes a Hopf bifurcation when
High-feed regime: A = O(1). et v = "−1V , u = "U/A and y = "−1x
v
v
v
−F −k =0
(F + k) − ( − F ) < 0
δ
δ
δ
ie. when λ is purely imaginary. The critical feed rate is given by
%
√
√
k − 2k − (2k − k)2 − 4k 2
0 ≤ k ≤ kc
Fc =
2
where U and V are O(1). Expand in powers of A"
[1] J.K. Hale, L.A. Peletier, and W.C. Troy. Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis. SIAM
Journal of Applied Mathematics. vol. 61 no. 1 pp. 102-130 (2002).
V = V0(y) + A"V1(y) + · · ·
U = U0(y) + A"U1(y) + · · ·
Substitute in the ODE system and collect in powers of "A
[2] J.E. Pearson. Complex patterns in a simple system. Science vol. 261
pp.189-192 (1993)
V .. − V + V 2U = 0
V0.. − V0 + V02U0 = 0
U0.. − V02U0 = 0
U .. − "2U + A" − V 2U = 0
V1.. − V1 + 2V0U0V1 + V02U1 = 0
U1.. + 1 − 2V0U0V1 − V02U1 = 0
References
[3] T. Kolokolnikov, M.J. Ward, and M. Wei. Zigzag and Breakup Instabilities of Stripes and Rings in the Two-Dimensional Gray-Scott Model
Studies in Applied Mathematics vol. 116 no 1 pp. 35-95 (2006)