Mobility promotes and jeopardizes biodiversity
in rock-paper-scissors games
Tobias Reichenbach, Mauro Mobilia, and Erwin Frey
Arnold-Sommerfeld-Center for Theoretical Physics (ASC) and Center for Nanoscience
(CeNS), Department of Physics, Ludwig-Maximilians-Universität München
Satellite conference
“Evolution and Game Theory”
at ECCS07, Dresden, 4.+5.October 2007
Colicinogenic Bacteria1
Three strands of E.coli:
C: colicin-producing
S: sensitive
Different environments:
Plate versus Flask
R: resistant
Spatial structure
matters!
Patterns?
Transition from
Plate to Flask?
1
B. Kerr et al., Nature 418 171 (2002)
May-Leonard model
Species A, B, C and empty sites ⊘
Cyclic dominance:
Birth:
µ
σ
AB → A⊘
σ
BC → B⊘
σ
CA → C ⊘
c
A⊘ → AA
µ
B⊘ → BB
µ
A⊘ → CC
1
Rate equations (without spatial structure):
◮
Coexistence fixed point is
unstable
◮
Trajectories spiral outwards on
an invariant manifold
◮
Heteroclinic orbits
yC
yB
0
b
1
yA
1
a
→ Loss of biodiversity
May-Leonard model on a lattice (N = L2 )2
Add migration:
ǫ
A⊘ → ⊘A
ǫ
AB → BA
...
=⇒ macroscopic diffusion D =
1
ǫ
2L2
While stochastic effects dominate the system’s appearance at small
scales, regular spiral waves form at larger scales
2
T. Reichenbach, M. Mobilia, and E. Frey, Nature 448, 1046 (2007)
Rotating spiral waves
D = 3 × 10−5 :
D = 3 × 10−4 :
Biodiversity is lost above critical diffusivity
a
b
Extinction probability, Pext
3 × 10−6
3 × 10−4 Mc
3 × 10−5
1
Biodiversity
0.8
0.6
0.4
0.2
Uniformity
0
1 × 10
−5
1 × 10
Mobility, M
−4
Mc 1 × 10−3
Pext = {Prob. of only one species after time t ∼ N}
M
Mathematical Description3
Lattice simulations:
Look for a description in terms of local
densities
s(r, t) = (a(r, t), b(r, t), c(r, t))
Continuum limit:
Stochastic partial differential equations:
∂t si (r, t) = D∆si + Ai (s) + Ci (s)ξi
→
Numerically
(www.xmds.org)
SPDE:
solvable
Results from SPDE agree with lattice
simulations (look at correlation functions)
3
T. Reichenbach, M. Mobilia, and E. Frey, accepted at Phys. Rev. Lett.
Complex Ginzburg-Landau Equation
In the vicinity of the reactive fixed point, the PDE can be written
as a complex Ginzburg-Landau equation:
∂t z = D∆z + c1 z − (1 − ic3 )|z|2 z
⇒ Analytic expressions for velocity, frequency, and wavelength of
the spirals
Compare analytic results to results from SPDE:
Wavelength:
(wavelength) λ̃
√
(velocity) v ∗ / D
Velocity:
1
0.5
0.25
0.125
0.008
0.04
0.2
1
5
25
(reproduction rate) µ
128
64
32
0.008
0.04
0.2
1
5
µ
25
State diagram
Numerical observation: there is a universal critical value λC
above which the system looses biodiversity
Critical mobility, Mc
1 × 10−3
Uniformity
1 × 10−4
Biodiversity
1 × 10−5
0.01
0
Reproduction rate, µ
blue: Lattice Simulations
red: CGLE
black: SPDE
100
Conclusions
◮
Mixed populations: Fluctuations due to finite system size lead
to uniformity
◮
Local interactions and diffusion: pattern formation and
biodiversity
◮
There is a threshold value for the diffusion constant above
which biodiversity is lost
◮
Analytic description possible via Complex Ginzburg Landau
Equation
T. Reichenbach, M. Mobilia, and E. Frey,
- Phys. Rev. E 74, 051907 (2006)
- Nature 448, 1046 (2007)
- arXiv:0710.0383 [q-bio.PE], accepted at Phys. Rev. Lett.
Correlation functions
In space: correlation length:
In time: oscillations:
0.2
0.1
0
0.15
0.1
lcorr
(correlation) gaa (r)
autocorrelation gaa (t)
0.2
0.1
0.01
0.05
1e-06
1e-05
0.0001
(diffusion) D
0
-0.1
0
100
200
0
(time) t
0.2
(distance) r
red: lattice simulations
blue: numerical solution of SPDE
Scaling of the correlation length: lcorr ∼
0.1
√
D
0.3
Convergence to the continuum limit (where SPDE
hold)
(correlation length) lcorr
L = 100(ǫ = 0.2)
L = 300(ǫ = 1.8)
0.2
L = 500(ǫ = 5)
ǫ = 2DL2
Continuum description
works already for ǫ ≥ 5
(β = γ = 1)
0.1
0
1
10
ǫ = 2DM
Stochastic versus deterministic PDE
Stoch. PDE
Det. PDE
In the SPDE, noise is ∼
1/L, thus tends to zero
when L → ∞.
Neglect noise → deterministic PDE
◮
◮
Deterministic PDE yield
correct wavelength and
frequency of the spirals
Noise breaks large spirals
up into a structure of
short entangled spirals
(correlation) gaa (r)
0.15
0.1
0.05
0
-0.05
0
0.1
0.2
(distance) r
0.3