Speed Selection in Coupled Fisher Waves
Martin R. Evans
SUPA, School of Physics and Astronomy, University of Edinburgh, U.K.
September 13, 2013
Collaborators:
Juan Venegas-Ortiz, Rosalind Allen
M. R. Evans
Speed Selection in Coupled Fisher Waves
Plan: Speed Selection in Coupled Fisher Waves
Plan
I Recap of Fisher equation, travelling waves + speed
selection principle
II Coupled Fisher waves + new speed selection
mechanism
References:
J. Venegas-Ortiz, R. J. Allen and M.R. Evans (2013)
M. R. Evans
Speed Selection in Coupled Fisher Waves
I Fisher-KPP Equation (1d)
Population density u(x, t) obeys
∂u
∂2u
= D 2 + αu(K − u)
∂t
∂x
Fisher (1937) - The wave of advance of advantageous gene
Kolmogorov, Petrovsky, Piskunov (1937) Study of the diffusion
equation with growth of the quantity of matter and its application
to a biological problem.
Skellam (1951) Random dispersals in theoretical populations.
Basically a nonlinear diffusion equation - nonlinearity is growth with
saturation
Linear growth rate αK , Carrying capacity K
M. R. Evans
Speed Selection in Coupled Fisher Waves
I Fisher-KPP Equation (1d)
Population density u(x, t) obeys
∂u
∂2u
= D 2 + αu(K − u)
∂t
∂x
Fisher (1937) - The wave of advance of advantageous gene
Kolmogorov, Petrovsky, Piskunov (1937) Study of the diffusion
equation with growth of the quantity of matter and its application
to a biological problem.
Skellam (1951) Random dispersals in theoretical populations.
Basically a nonlinear diffusion equation - nonlinearity is growth with
saturation
Linear growth rate αK , Carrying capacity K
Appears ubiquitously in: disordered systems such as Directed
Polymers in random media; nonequilibrium systems such as directed
percolation; computer science search problems etc
M. R. Evans
Speed Selection in Coupled Fisher Waves
Population density
Travelling Wave Solutions
Front
Tip
Space
Nonlinear wave with finite width travelling at speed v .
Stable f.p. solution u = K invades unstable f.p. solution u = 0
M. R. Evans
Speed Selection in Coupled Fisher Waves
Population density
Travelling Wave Solutions
Front
Tip
Space
Nonlinear wave with finite width travelling at speed v .
Stable f.p. solution u = K invades unstable f.p. solution u = 0
Speed selection principle
The selected speed of the front is determined by tip dynamics and
initial conditions
M. R. Evans
Speed Selection in Coupled Fisher Waves
Physical Examples
Ordering Dynamics e.g. Time-dependent Landau-Ginzburg equation
∂
∂m
= D∇2 m −
f (m)
∂t
∂m
f (m) = −am2 + bm4
Contact Process
1
→ 0
rate
1
01 → 11
rate λ
10 → 11
rate λ
Mean field for density ρi
∂ρi
∂t
∂ρ
→
∂t
= −ρi + λ(1 − ρi )ρi+1 + λρi−1 (1 − ρi )
' λ
∂2ρ
+ (2λ − 1)ρ − 2λρ2
∂x 2
Generally stable phase (fixed point) invades unstable phase (fixed
point)
M. R. Evans
Speed Selection in Coupled Fisher Waves
Speed Selection Principle (physics approach) I
Simple Linear Analysis: linearise at tip u 1
∂2u
∂u
= D 2 + αuK
∂t
∂x
M. R. Evans
Speed Selection in Coupled Fisher Waves
Speed Selection Principle (physics approach) I
Simple Linear Analysis: linearise at tip u 1
∂2u
∂u
= D 2 + αuK
∂t
∂x
Try wave solution u(x, t) = f (z) where z = x − vt
−vf 0 = Df 00 + αKf
Exponential solutions f = e−λz where v λ = Dλ2 + αK .
Solve for λ
v ± (v 2 − 4αDK )1/2
λ=
2D
Real if
√
v > v ∗ = 2 αDK
M. R. Evans
Speed Selection in Coupled Fisher Waves
Speed Selection Principle (physics approach) I
Simple Linear Analysis: linearise at tip u 1
∂2u
∂u
= D 2 + αuK
∂t
∂x
Try wave solution u(x, t) = f (z) where z = x − vt
−vf 0 = Df 00 + αKf
Exponential solutions f = e−λz where v λ = Dλ2 + αK .
Solve for λ
v ± (v 2 − 4αDK )1/2
λ=
2D
Real if
√
v > v ∗ = 2 αDK
Speed selection principle
The marginal speed v ∗ is selected for sufficiently steep initial
conditions.
M. R. Evans
Speed Selection in Coupled Fisher Waves
Speed Selection Principle II:
More involved Linear Analysis
∂u
∂2u
= D 2 + αuK
∂t
∂x
Solve for initial condition u(x, 0) = e−λx for x ≥ 0:
h
√ i
u(x, t) = (1/2)e−λ(x−v (λ)t) 1 + erf (x − 2Dλt)/(2 Dt)
where
v (λ) = Dλ +
M. R. Evans
αK
.
λ
Speed Selection in Coupled Fisher Waves
Speed Selection Principle II:
More involved Linear Analysis
∂u
∂2u
= D 2 + αuK
∂t
∂x
Solve for initial condition u(x, 0) = e−λx for x ≥ 0:
h
√ i
u(x, t) = (1/2)e−λ(x−v (λ)t) 1 + erf (x − 2Dλt)/(2 Dt)
where
v (λ) = Dλ +
Far tip x 2Dλt
Use erf(z) ' 1 for z 1 ⇒
Thus the tip speed is v (λ)
αK
.
λ
u(x, t) ' exp [−λ(x − v (λ)t]
M. R. Evans
Speed Selection in Coupled Fisher Waves
Speed Selection Principle II:
More involved Linear Analysis
∂u
∂2u
= D 2 + αuK
∂t
∂x
Solve for initial condition u(x, 0) = e−λx for x ≥ 0:
h
√ i
u(x, t) = (1/2)e−λ(x−v (λ)t) 1 + erf (x − 2Dλt)/(2 Dt)
where
v (λ) = Dλ +
Far tip x 2Dλt
Use erf(z) ' 1 for z 1 ⇒
Thus the tip speed is v (λ)
αK
.
λ
u(x, t) ' exp [−λ(x − v (λ)t]
Front x 2Dλt
2
e−z
Use erf(−z) ' −1 + √ for z 1
z π
⇒ u ' exp −λ∗ (x − v ∗ t) − (x − v ∗ t)2 /(4Dt)
where v ∗ = 2(DαK )1/2 .
M. R. Evans
Speed Selection in Coupled Fisher Waves
Speed Selection Principle III:
Crossover point between far tip and front regimes x = 2Dλt.
If λ > λ∗ = v ∗ /2D (steep i.c.), then the crossover point advances
faster than the tip, and “front” has speed v ∗ .
If λ < λ∗ , the front catches up with crossover point which falls behind
tip, the speed of the front will be controlled by the tip v (λ).
M. R. Evans
Speed Selection in Coupled Fisher Waves
Speed Selection Principle III:
Crossover point between far tip and front regimes x = 2Dλt.
If λ > λ∗ = v ∗ /2D (steep i.c.), then the crossover point advances
faster than the tip, and “front” has speed v ∗ .
If λ < λ∗ , the front catches up with crossover point which falls behind
tip, the speed of the front will be controlled by the tip v (λ).
Speed selection principle
∗
If the initial profile decays faster than u(x, 0) ∼ e−λ x , where
λ∗ = v ∗ /2D,
√ then wave travels at marginal speed
v = v ∗ = 2 αDK
If the initial profile decays less steeply, say as e−λx where λ < λ∗ ,
the wave advances at faster speed v (λ) = Dλ + αK /λ.
M. R. Evans
Speed Selection in Coupled Fisher Waves
Propagation of coupled Fisher-KPP waves
Motivation
Horizontal gene transfer within bacterial population
More generally acquisation of trait ⇒ two sub populations NA and
NB : those with trait and those with without
M. R. Evans
Speed Selection in Coupled Fisher Waves
Propagation of coupled Fisher-KPP waves
Motivation
Horizontal gene transfer within bacterial population
More generally acquisation of trait ⇒ two sub populations NA and
NB : those with trait and those with without
∂ 2 NA
∂NA
= D
+ αNA (K − NT ) − βNA + γNA NB ,
∂t
∂x 2
∂NB
∂ 2 NB
(2)
= D
+ αNB (K − NT ) + βNA − γNA NB
∂t
∂x 2
NT = NA + NB is the total population density
(1)
New parameters
γ rate of horizontal transmission of trait
M. R. Evans
β loss rate of trait
Speed Selection in Coupled Fisher Waves
Propagation of coupled Fisher-KPP waves
Motivation
Horizontal gene transfer within bacterial population
More generally acquisation of trait ⇒ two sub populations NA and
NB : those with trait and those with without
∂ 2 NA
∂NA
= D
+ αNA (K − NT ) − βNA + γNA NB ,
∂t
∂x 2
∂NB
∂ 2 NB
(2)
= D
+ αNB (K − NT ) + βNA − γNA NB
∂t
∂x 2
NT = NA + NB is the total population density
(1)
New parameters
γ rate of horizontal transmission of trait
β loss rate of trait
Summing (1,2) gives a standard Fisher-KPP equation:
(3)
∂NT
∂ 2 NT
=D
+ αNT (K − NT ) ,
∂t
∂x 2
M. R. Evans
√
with speed vT = 2 αKD
Speed Selection in Coupled Fisher Waves
Propagation of coupled Fisher-KPP waves
∂NA
∂t
∂NT
∂t
∂ 2 NA
+ αNA (K − NT ) − βNA + γNA NB ,
∂x 2
∂ 2 NT
= D
+ αNT (K − NT )
∂x 2
= D
Population density
Numerics reveal
vtot
vc
Space
i.e. an invading population wave which does not carry the trait in turn
being invaded by a ‘trait wave’ with different speed
M. R. Evans
Speed Selection in Coupled Fisher Waves
Fixed point Structure
∂NA
∂t
∂NT
∂t
∂ 2 NA
+ αNA (K − NT ) − βNA + γNA NB ,
∂x 2
∂ 2 NT
= D
+ αNT (K − NT )
∂x 2
= D
Choose intermediate transmission rate α > γ > β/K .
There are three homogeneous density, steady-state solutions
(NT∗ , NA∗ ) with dynamical stabilities:
The fixed point (NT∗ , NA∗ ) = (0, 0) is unstable.
(NT∗ , NA∗ ) = (K , 0) is a saddle point.
(NT∗ , NA∗ ) = (K , K − β/γ) is stable
M. R. Evans
Speed Selection in Coupled Fisher Waves
Fixed point Structure
∂NA
∂t
∂NT
∂t
∂ 2 NA
+ αNA (K − NT ) − βNA + γNA NB ,
∂x 2
∂ 2 NT
= D
+ αNT (K − NT )
∂x 2
= D
Choose intermediate transmission rate α > γ > β/K .
There are three homogeneous density, steady-state solutions
(NT∗ , NA∗ ) with dynamical stabilities:
The fixed point (NT∗ , NA∗ ) = (0, 0) is unstable.
(NT∗ , NA∗ ) = (K , 0) is a saddle point.
(NT∗ , NA∗ ) = (K , K − β/γ) is stable
The coupled travelling waves we observe correspond to a
homogeneous spatial domain of (0, 0) being invaded by the solution
(K , 0) being invaded by the solution (K , K − β/γ) (i.e. the trait wave).
M. R. Evans
Speed Selection in Coupled Fisher Waves
Effect of different initial conditions
Spread of trait in established Population
Initially NT = K throughout the domain (and remains so)
x/10000
4
vs
0
t/100000
M. R. Evans
1
Speed Selection in Coupled Fisher Waves
Effect of different initial conditions
Spread of trait in established Population
Initially NT = K throughout the domain (and remains so)
x/10000
4
vs
0
t/100000
1
Spread of trait in expanding population
Domain initially empty NT = 0.
4
x/10000
vs
0
vc
t/100000
M. R. Evans
1
Speed Selection in Coupled Fisher Waves
Summary of different speeds I
√
vT = 2 DαK speed of total population wave
vs speed of trait wave invading saturated population (NT = K )
vc speed of trait wave invading expandng population
M. R. Evans
Speed Selection in Coupled Fisher Waves
Summary of different speeds I
√
vT = 2 DαK speed of total population wave
vs speed of trait wave invading saturated population (NT = K )
vc speed of trait wave invading expandng population
also we have
vtip speed of far tip of trait wave (NA , NT 1)
M. R. Evans
Speed Selection in Coupled Fisher Waves
Summary of different speeds I
√
vT = 2 DαK speed of total population wave
vs speed of trait wave invading saturated population (NT = K )
vc speed of trait wave invading expandng population
also we have
vtip speed of far tip of trait wave (NA , NT 1)
vs : Set NT = K yields F-KPP equation
∂NA
∂ 2 NA
β
=D
+
γN
K
−
−
N
.
A
A
∂t
∂x 2
γ
p
Thus usual speed selection predicts vs = 2 D(γK − β)
M. R. Evans
Speed Selection in Coupled Fisher Waves
Summary of different speeds I
√
vT = 2 DαK speed of total population wave
vs speed of trait wave invading saturated population (NT = K )
vc speed of trait wave invading expandng population
also we have
vtip speed of far tip of trait wave (NA , NT 1)
vs : Set NT = K yields F-KPP equation
∂NA
∂ 2 NA
β
=D
+
γN
K
−
−
N
.
A
A
∂t
∂x 2
γ
p
Thus usual speed selection predicts vs = 2 D(γK − β)
vtip : Set NT , NA 1 and linearise
∂NA
∂ 2 NA
=D
+ NA (αK − β) .
∂t
∂x 2
p
Thus usual speed selection predicts vtip = 2 D(αK − β)
M. R. Evans
Speed Selection in Coupled Fisher Waves
Summary of different speeds II
√
vT = 2 DαK speed of total population wave
p
vs = 2 D(γK − β) speed of trait wave invading saturated
population (NT = K )
p
vtip = 2 D(αK − β) speed of far tip of trait wave (NA , NT 1)
M. R. Evans
Speed Selection in Coupled Fisher Waves
Summary of different speeds II
√
vT = 2 DαK speed of total population wave
p
vs = 2 D(γK − β) speed of trait wave invading saturated
population (NT = K )
p
vtip = 2 D(αK − β) speed of far tip of trait wave (NA , NT 1)
We show that
Speed of trait wave invading expandng population
vtip (vT − vtip )
vc = vT −
vT −
q
2 − v2
(vT − vtip )2 + vtip
s
N.B.
vT > vtip > vc > vs
M. R. Evans
Speed Selection in Coupled Fisher Waves
Determination of vc : the kink in the trait wave
Population density
1
NT
0.8
0.4
vc
0.2
0
(a)
3000
NT
zL
2
NT
NA
-467
vtot
zR
6000
5000
x
-401
3
NA
4000
Log[Population Density]
-52
2000
Population density x 10
vtot
NA
0.6
vtip
-534
-601
1
5100
-668
(c)
5105
5110
5115
x
5120
5125
5130
5500
(b)
5550
5600
x
5650
5700
Far tip at x >> vT t
Kink at x = vT t
M. R. Evans
Speed Selection in Coupled Fisher Waves
5750
Speed selection mechanism for the trait wave
Ahead of kink
Linearise NA , NB 1 to get ∂NA /∂t = D(∂ 2 NA /∂x 2 ) + NA (αK − β)
p
Then vtip = 2 D(αK − β) is selected and defining zR = x − vT t we
get
NA (zR , t) ∼ exp[−vtip (zR + (vT − vtip )t)/(2D)
M. R. Evans
Speed Selection in Coupled Fisher Waves
Speed selection mechanism for the trait wave
Ahead of kink
Linearise NA , NB 1 to get ∂NA /∂t = D(∂ 2 NA /∂x 2 ) + NA (αK − β)
p
Then vtip = 2 D(αK − β) is selected and defining zR = x − vT t we
get
NA (zR , t) ∼ exp[−vtip (zR + (vT − vtip )t)/(2D)
Behind kink
Write
NA (zL , t) ∼ exp[−at + bzL ]
,
where zL = vT t − x and a and b are unknown constants such that
vc = vT − a/b
M. R. Evans
Speed Selection in Coupled Fisher Waves
Speed selection mechanism for the trait wave
Ahead of kink
Linearise NA , NB 1 to get ∂NA /∂t = D(∂ 2 NA /∂x 2 ) + NA (αK − β)
p
Then vtip = 2 D(αK − β) is selected and defining zR = x − vT t we
get
NA (zR , t) ∼ exp[−vtip (zR + (vT − vtip )t)/(2D)
Behind kink
Write
NA (zL , t) ∼ exp[−at + bzL ]
,
where zL = vT t − x and a and b are unknown constants such that
vc = vT − a/b
By demanding that our two asymptotic expressions must match at the
kink: i.e. NA (zR = 0, t) = NA (zL = 0, t), we can determine
a = vtip (vT − vtip )/(2D)
.
M. R. Evans
Speed Selection in Coupled Fisher Waves
Speed selection mechanism for the trait wave II
To find the remaining constant b, we linearize in the region behind the
kink, where the total population is large (NT ≈ K ), but the amplitude
of the trait wave is still small (NA 1). This gives
∂NA /∂t = D(∂ 2 NA /∂x 2 ) + NA (γK − β)
But now we need a new selection mechanism for solution
Substituting in NA (zL , t) ∼ exp[−at + bzL ] with a now determined by
matching gives a quadratic for b with root
q
2 − v2
b = (1/(2D)) vT − (vT − vtip )2 + vtip
s
.
Using
vc = vT − a/b
we finally obtain the following expression for the speed of the trait
wave:
vtip (vT − vtip )
q
vc = vT −
2 − v2
vT − (vT − vtip )2 + vtip
s
M. R. Evans
Speed Selection in Coupled Fisher Waves
Analytical prediction and simulation results for the
speed vc of the trait wave
6
wave speed
wave speed
2
1.5
1
0.5
5
4
3
2
1
0
0.2
0.4
γ
0.6
0.8
1
2
4
6
K
8
10
The black lines show the analytical result while the red circles show
simulation results for the trait wave speed
For comparison
√ the blue lines show the speed of the total population
wave vT = 2 αDK while the purple lines show the speed
of the trait
p
wave as it invades an established population, vs = 2 D(γK − β).
M. R. Evans
Speed Selection in Coupled Fisher Waves
Recap: Spread of trait in expanding population
Consider an expanding population starting with the spatial domain
initially empty.
M. R. Evans
Speed Selection in Coupled Fisher Waves
Recap: Spread of trait in expanding population
Consider an expanding population starting with the spatial domain
initially empty.
4
x/10000
vs
vc
0
1
t/100000
Invasion of Expanding Population
Clearly the trait wave advances at speed vc that is greater than the
speed vs at which it invades an established population – but still less
than the speed vT of the total population wave.
At very long times, when the total population wave is very far ahead
of the trait wave, the speed of the trait wave reverts to vs
M. R. Evans
Speed Selection in Coupled Fisher Waves
Initial-condition dependent speeding-up transition
vs
30000
0.8
x(t)
Population density
1
0.6
20000
vc
0.4
10000
0.2
(a)
0
d
200
400
x
600
800
(b)
vs
20000
40000
t
60000
80000
Initially system partially occupied by non-trait population.
M. R. Evans
Speed Selection in Coupled Fisher Waves
Summary
Summary
Fisher-KPP equation exhibits nonlinear travelling waves
We study coupled Fisher waves and find new selection
mechanism
M. R. Evans
Speed Selection in Coupled Fisher Waves
Summary
Summary
Fisher-KPP equation exhibits nonlinear travelling waves
We study coupled Fisher waves and find new selection
mechanism
Outlook
Many other generalisations to coupled Fisher waves remain to be
studied
Important to consider effect of noise
M. R. Evans
Speed Selection in Coupled Fisher Waves