Stochastic Spatial Games
Erwin Frey
Arnold Sommerfeld Center for Theoretical Physics
& Center of Nanoscience
Ludwig–Maximilians–Universität München
Microbial laboratory
communities
B. Kerr et al., Nature 418, 171 (2002)
Colicinogenic Bacteria
Toxin producing (colicinogenic
(colicinogenic)) E.coli (C) carry a ‘col’
col’
plasmid: genes for colicin,
colicin, colicin specific immunity
proteins, lysis protein
ColicinColicin-sensitive bacteria (S)
ColicinColicin-resistant bacteria (R) are mutations of S with
altered cell membrane proteins that bind and translocate
cocilin
C
R outgrows C:
no cost for ‘col
‘col’’
C kills S
R
S
S outgrows R: better
nutrient uptake
•1
Spatial structure matters!
Questions
¾ What determines uniformity or diversity
of a microbial population?
¾ What is the role of
o
o
o
o
migration, competition & growth
stochasticity (“intrinsic noise”)
initial conditions
spatial heterogeneities
Well mixed Populations
•2
The RockRock-SiccorsSiccors-Paper Game
Consider a fixed population of
N individuals in a wellwell-mixed
environment (“urn model”)
Describe the outcome of the game by “reactions”:
kC
A+ B → A+ A
kA
B+C →
B+B
k
B
C+A →
C +C
Cyclic competition between three species A, B, C
T. Reichenbach,
Reichenbach, M. Mobilia and EF, PRE (2006)
Deterministic Evolution
Rate equations:
absorbing fixed point
a& = a ( kC b − k B c)
b& = b(k Ac − kC a )
c& = c(k B a − k Ab)
reactive (center) fixed point
a +b + c =1
Constant of motion:
K = a(t ) k A b(t ) k B c(t ) kC
cyclic trajectories around a
neutrally stable fixed point
coexistence
Stochastic Evolution
¾ processes are probabilistic
¾ intrinsic fluctuations in a, b, c
stochasticity
causes loss of
coexistence
¾ K no constant of motion
¾ neutrally stable cycles!
¾ “random walk” on phase portrait
Stochastic description in
terms of a probability density:
P (a, b, c; t ) = P (x, t )
•3
Diffusion approximation
Master equation (gain & loss balance)
∂ t P ( x, t ) = ∑ [P (x + δx, t ) wx +δx→x − P (x, t ) wx→x +δx ]
δx
KramersKramers-Moyal expansion:
δx = 1 / N << 1 for large population size N
FokkerFokker-Planck equation:
∂ t P (x, t ) = − ∂ i [Ai (x) P(x, t )] +
[
1
∂ i ∂ j Bij ( x) P (x, t )
2
deterministic drift
(rate equations)
]
diffusion
(scales as 1/N)
1/N)
How to derive the coefficients in the FPE
Reaction drift term = expected step size
A A (x) = ∑ δx A wx→x +δx = δx A =
δx
1
(ab − ac) N
N
“Diffusion term” = variance due to fluctuations
2
1
1
BAA (x) = δx Aδx A =   (ab + ac) N ~
N
N
• Reaction kinetics results in drift (in phase space)
• The effect of intrinsic noise is of the order 1/N
Extinction Probability
The typical extinction time T scales as N (random walk)
•4
Spatial Models
Local Interaction Rules
selection
A
B
reproduction
0
C
birth
Example:
dominance
exchange
MayMay-Leonard Model (well mixed)
Species A,B,C and empty sites 0
Cyclic dominance (γ
(γ)
Birth (β
(β)
AB → A0
BC → B 0
CA → C 0
A0 → AA
B 0 → BB
C 0 → CC
Without spatial structure
coexistence fixed point is
unstable
loss of biodiversity
•5
MayMay-Leonard Model on a Lattice (N=L2)
Add migration (ε
(ε)
macroscopic
diffusion
A0 → 0 A
AB → BA
...
D=
ε
2L2
D = 3 × 10 −4
D = 3 ×10 −5
Stability of Biodiversity?
¾ For well mixed populations biodiversity is lost!
¾ Is there a critical value for the diffusivity D below
which biodiversity is maintained?
¾ If yes, what is the spatial structure of the
population?
¾ If yes, in what sense?
Typical extinction times: Extensivity
T typical extinction time; N the size of the population:
T /N →∞
T / N → O(1)
T /N →0
super-extensive / stable
extensive / neutral / marginal
sub-extensive / unstable
•6
Diversity is lost above critical Diffusivity
Pext = Prob{only one species after t~N }
T. Reichenbach,
Reichenbach, M. Mobilia and EF, Nature (2007)
Diffusion Threshold for Biodiversity!
¾
¾
For large systems there is a well defined
critical/threshold value Dc (β,γ)
β,γ) for the
diffusivity/mobility.
Loss of biodiversity seems to be related to the
size of the spatial structures (spirals) in the
population.
Mathematical Description
Look for a description in terms of local densities
s(r, t ) = (a (r, t ), b(r, t ), c(r, t ))
∂ t si (r, t ) = D∆si + Ai (s)
Rate equations: a + b + c = ρ
a& = a[β (1 − ρ ) − γc ] = A1 (s)
b& = b[β (1 − ρ ) − γa ] = A (s)
2
c& = c[β (1 − ρ ) − γb] = A3 (s)
•7
Mathematical Description
Look for a description in terms of local densities
s(r, t ) = (a (r, t ), b(r, t ), c(r, t ))
∂ t si (r, t ) = D∆si + Ai (s)
diffusiondiffusion-reaction equation
What about “intrinsic noise”?
• diffusion produces “noise”
• reactions produce “noise”
B11 (s) =
[
1
∂ i ∂ j Bij (s) P(s, t )
2
1
a[β (1 − ρ ) + γc ]
N
]
C1 = B11
Mathematical Description
Look for a description in terms of local densities
s(r, t ) = (a (r, t ), b(r, t ), c(r, t ))
∂ t si (r, t ) = D∆si + Ai (s) + Ci (s)ξ i
ξ i (r, t )ξ j (r ' , t ' ) = δ ijδ (r − r ' )δ (t − t ' ) (white noise)
¾ stochastic partial differential equation
¾ particle exchange
¾ reactions
G
G
diffusion
drift & multiplicative noise (~N-1/2)
T. Reichenbach,
Reichenbach, M. Mobilia and EF, PRL (2007)
Convergence to continuum limit
•8
How good is such a description?
Stochastic lattice simulation
Stochastic reactionreaction-diffusion equations
Spatial Correlations
g ij (r,0) = si (r, t ) s j (0, t ) − si (r, t ) s j (0, t )
lcorr ~ D
lattice simulation
stochastic PDE
raising the diffusion constant D
increases the size of the spirals
Temporal Correlations
g ij (0, t ) = si (r, t ) s j (r,0) − si (r, t ) s j (r,0)
the rotation frequency is a function of
the reaction rates β and γ only
•9
What can one predict with it?
¾ ReactionReaction-diffusion equations are a faithful description of
the dynamics and stationary state!
¾ This allows to use the full spectrum of methods from
nonlinear dynamics
@ reactive fixed point:
• twotwo-dimensional flow
• with rotational symmetry
∂ t z = D∆z + c1 z − c2 (1 − ic3 ) | z |2 z
complex GinzburgGinzburg-Landau equation
Compare CGLE and stochastic PDE’
PDE’s
velocity of wave fronts
wavelength of spirals
• velocity and wavelength scale as D
• algebraic increase/decrease and saturation
as a function of the birth rate β
State Diagram
Dc
critical diffusion
constant
Dc saturates as β > γ
uniformity
biodiversity
smaller β strongly
reduces Dc
γ =1
β
(growth rate)
•10
Is the noise essential?
„
„
„
Stochastic Simulations
Stochastic PDE’s
Noise ~ 1 / N
SPDE vs. PDE
• SPDE for homogeneous initial state
• PDE for some initial periodic pattern:
wavelength of the spirals the same but number
depends sensitively on initial conditions.
Spatial correlation function
deterministic
stochastic
•11
Conclusions
„
„
„
„
„
Mixed populations: fluctuations due to finite
size of the system lead to uniformity.
Local interaction: pattern formation and
biodiversity.
There is a threshold value for the diffusion
constant above which biodiversity is lost.
The threshold value decreases when the
growth rate becomes lower than reaction
rate (delay).
Classification of stability according to the
scaling of T with N.
Thanks to ...
„
„
„
Tobias Reichenbach
Mauro Mobilia
Jonas Cremer
•12