New J. Phys. 17 (2015)103035 doi:10.1088/1367-2630/17/10/103035
PAPER
Frequency-dependent selection at rough expanding fronts
Jan-Timm Kuhr and Holger Stark
Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
E-mail: [email protected]
Keywords: range expansion, surface growth, evolutionary game theory, non-equilibrium phase transition, kinetic roughening, evolution
Abstract
Microbial colonies are experimental model systems for studying the colonization of new territory by
biological species through range expansion. We study a generalization of the two-species Eden model,
which incorporates local frequency-dependent selection, in order to analyze how social interactions
between two species influence surface roughness of growing microbial colonies. The model includes
several classical scenarios from game theory. Wethen concentrate on an expanding public goods
game, where either cooperators or defectors take over the front depending on the system parameters.
We analyze in detail the critical behavior of the nonequilibrium phase transition between global
cooperation and defection and thereby identify a new universality class of phase transitions dealing
with absorbing states. At the transition, the number of boundaries separating sectors decays with a
novel power law in time and their superdiffusive motion crosses over from Eden scaling to a nearly
ballistic regime. In parallel, the width of the front initially obeys Eden roughening and, at later times,
passes over to selective roughening.
1. Introduction
Living species are usually confined to their territory, a spatial region defined by geographical borders, climate, or
other environmental constraints. Uninhabited regions are colonized through range expansion, where
individuals reproduce and disperse at the front of their territory [1]. This process is seen in biological invasions
[2], as a result of shifting climate zones [3–5], during colonizations in our own species’history [6–8], tumor
growth [1,9,10], and biofilm growth [11–13]. Evidently, expansions occur on very different spatial
(micrometers to 10
7
m)and temporal (hours to millennia)scales.
In this article we aim to characterize range expansion under the influence of short-range ‘social interactions’
of individuals at the front. Such interactions are present if success in reproduction depends on the presence of
nearby individuals of the own and/or other species. Here, we set up a model for the expanding front based on
evolutionary game theory [14–16]and investigate its roughening dynamics for two interacting species. Besides
exploring an interesting non-equilibrium growth process, we hope to contribute to interpreting experiments on
range expansion in multi-species colonies of simple organisms.
In experiments, microbial growth is excellently suited to study range expansion and other processes in
population dynamics and evolution such as spatial spread of infections and adaptation to an environment (see
for example [17]). Microbes reproduce fast, their environment and genotype can be controlled, and
experimental conditions are easily reproducible. Grown in a Petri dish, the spatial patterns of single-species
microbial colonies have long been a rich field of study [18–23]. The observed patterns crucially depend on
motility, availability of nutrients, growth medium, and adhesion between cells, to name but a few. However,
even under conditions of negligible motility and abundant nutrients a colony’s front is rough and has interesting
statistical properties [24–26].
Multi-species colonies are composed of more than one species and show additional intriguing features, even
if the species are identical except for a marker [27]. During reproduction they keep their marker but compete
with other species for space at the front. Thereby, sectors of single species form, which are separated by boundaries.
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26 June 2015
REVISED
1 September 2015
ACCEPTED FOR PUBLICATION
9 September 2015
PUBLISHED
16 October 2015
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© 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
Their statistical and dynamic properties are determined by the evolving roughness of the expanding
front [28,29].
Usually, when two or more species of microbes live in a common environment, they influence each other
during reproduction. In particular, reproductive success of any species, also called its fitness, depends on the
population sizes of all the species. This constitutes ‘social interactions’between the species commonly referred to
as frequency-dependent selection. Research in the field has initiated a wealth of fascinating experiments [10,30–
36]either in well-mixed liquid culture without any spatial order [32,37]or in the Petri dish, where the
populations are spatially structured [38,39]. Many of the experimental observations can be discussed within the
framework of evolutionary game theory [14–16]. For example, light has been shed on a long-standing theoretical
question in evolution [14,15,40–42]: Why do individuals cooperate if non-cooperators can exploit them?
Literature emphasizes the importance of a population to be structured in groups [15,43–47], for example, by
spatial distance. While within a single group cooperators are always inferior to non-cooperating defectors, if the
latter interact only with their neighborhood, distant large groups of cooperators will ultimately outcompete
defectors. Some models also stress the central role of demographic fluctuations and of populations growing in
size [48,49]. Both, experiments and theory, explain the advantage of cooperators during colony growth by their
ability to locally advance faster [39,50,51], vividly termed ‘survival of the fastest’[39].
Cooperation between between nearby cells is often mediated by some biochemical compound (apublic good)
which the microbes release into the extracellular environment. This compound then promotes reproduction of
neighboring cells. In general, a released substance may act beneficial or detrimental to other individuals, also
depending on their species, and implies some cost to the producer. Examples include secretion of digestive
invertase to break down sucrose [37,52–54], siderophores to scavenge iron from the environment [32,55,56],
polymers which support biofilms [43,57,58], release of toxins [59,60](sometimes through lysis [61,62]),
surfactants which facilitate swarming [63], and the exchange of amino acids [38].
This plethora of biochemical compounds, released by cells and affecting nearby cells, implies a wealth of
specific features, which certainly are not covered by a single model. However, since the released biomolecules
usually mediate short-range interactions between individuals, properties on large scales should be independent
of microscopic details. Hence, we formulate a simple model which captures the essence of an interaction while
ignoring complicated details. The classical Eden model [64,65], a simple growth process on a lattice, has been
used successfully to mimic growing cell colonies. It generates a cluster (the colony), the surface of which exhibits
scaling properties also found for expanding fronts of microbial colonies [66,67]. Extended to two identically
growing but still distinguishable species, it generates sectors occupied by a single species only [29,68]. Indeed,
this behavior is found for two-species microbial colonies [27]. Moreover, boundaries between sectors move
superdiffusively as in the experiments.
In this article we explore a generalization of the two-species Eden model, which incorporates local
frequency-dependent selection. We thereby aim to analyze how social interactions influence surface roughness
of growing microbial colonies. We set up an expanding public goods game, where either cooperators or
defectors take over the front depending on the system parameters [14–16]. Right at the transition the front
displays critical behavior, which we analyze in detail. In particular, we establish that our model belongs to a new
universality class of phase transitions dealing with absorbing states. At the transition, the number of boundaries
separating sectors decays with a novel power law in time and their superdiffusive motion crosses over from Eden
scaling to a nearly ballistic regime. In parallel, the width of the front initially obeys Eden roughening and, at later
times, passes over to what we call selective roughening.
The remainder of this article is organized as follows. To analyze multi-species microbial colony growth, we
introduce the Eden model with frequency-dependent selection in section 2and analyze its phenomenology in
section 3. We then concentrate on the expanding public goods game with its social dilemma in section 4and
analyze the critical behavior at the transition between long-term cooperation and long-term defection by
applying statistical analysis. Finally, we discuss and summarize our findings in section 5.
2. Eden model with frequency-dependent selection
In this work we employ a lattice model (see figure 1)to analyze range expansion at rough fronts under the
influence of frequency-dependent selection. We set up a cellular automaton on a two-dimensional hexagonal
1
lattice of transverse extension Land an infinite longitudinal extension. Periodic boundary conditions are applied
in the transverse direction. The state s
{
}of the system at time tis specified by the state variables
s
ij,of lattice sites
ij,.
(
)Consider a system with two species (extension to more species is straightforward). For any time t, any site
1
On a square lattice it is impossible toenclose a cluster Awithin a cluster B, which only contains nearest neighbor sites of cluster A.Ona
hexagonal lattice this is possible.
2
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
(i,j)is either empty (
s
0
ij,=)or occupied by an individual of either species 1 (
s
1
ij,=)or 2 (
s
2
ij,=). All
individuals which have at leastone free nearest neighbor site can reproduce. Toperform a reproduction step, we
choose one of these fecund individuals with a probability proportional to its reproduction rate (see below)and a
new individual of the same species is placed with equal probability on one of the free neighboring sites.
In contrast to the Eden model [64]and some of its two-species generalizations [29,69], reproduction rates in
our model depend on the states of the nearest-neighbor sites. Let n
1
and n
2
denote the number of nearest
neighbors of species 1 and 2, respectively, then the reproduction rate of an individual at lattice site (i,j)is
b
s
bn n s
bn n s
0 if has no free neighbors,
if 1,
if 2.
1
ij
ij
ij
ij
,
,
1
012 ,
2
012 ,
()
⎧
⎨
⎪
⎩
⎪
=++ =
++ =
Here,
b
1
0
and b2
0are the respective contributions to the reproduction rates of species 1 and 2, which are
independent of the states of their nearest neighbors. Frequency-dependent selection is introduced through the
parameters
,
,
,
and
.
With the reproduction rates bij,we implement a random sequential update of the system using a simplified
version of the Gillespie algorithm [70]. The overall reproduction rate of the population is bb
ij ijtot ,,
≔åand an
individual at site ij,
(
)is selected to reproduce with probability bb.
ij,tot
We then choose one of the empty
nearest-neighbor sites of the reproducing individual at random and place there a new individual of the same
species. This implies that there are no mutations. Since the mean time until the next reproduction event is
b,
tot
1-
we update time by
t
tb
tot1
+-after each reproduction event. We assume that individuals do not die and that
they are immobile. Therefore, any site with
s
0
ij,¹remains in its specific state indefinitely. As initial condition
we occupy all sites of an initial line randomly, but in equal parts, with species 1 and 2, if not stated otherwise.
The formulated model generalizes version C of the Eden model, introduced by Jullien and Botet [71],toa
two-species system. We already applied a similar model to range expansion without frequency-dependent
selection but included the possibility of mutations [69].If
,
,
,
and
are zero, our model reduces to that of
Saito and Müller-Krumbhaar [29], however they used a square lattice. Since diffusion is not included in the
model, configurations and patterns behind the front are frozen. This corresponds to observations in microbial
experiments on range expansion [27,62]. Our model does not include adhesion between cells, which
contributes to surface tension and can thus smooth the surface of a bacterial colony [23]. It also ignores the
vertical extension of bacterial colonies since the horizontal extension isusually larger by several orders of
magnitude.
In game theory the parameters
,
,
,
and
from equation (1)define the payoff matrix of a two-strategy
game [14–16]. Different scenarios, some well known in game theory, are implemented if we set these parameters
accordingly.
Figure 1. Two-species Eden model with frequency-dependent selection on a hexagonal lattice. The bacterial colony grows from the
bottom line (lattice sites with narrow black edge)of length L, where individuals of species 1 (blue)and species 2 (red)occupy the lattice
sites. The colony expands into the empty, infinitely-extended half-space. Individuals capable of reproduction (indicated by bold
edges), have at least one empty lattice site as a nearest neighbor. In a reproduction event, one of these empty neighboring sites (i,j)is
chosen with equal probability and the reproducing individual changes the corresponding state
s
ij,to its own state 1 or 2. Each
individual has its own reproduction rate
b
ij,
given in equation (1). For example, the reproduction rate of the individual at site
3, 3(
)
(bold black edge)is
bb32.
3,3 2
0
=++
Along the transverse direction periodic boundary conditions apply.
3
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
3. Phenomenology
We now describe some generic examples of our model for growing microbial colonies (see figure 2), which
emerge for typical parameter settings, and discuss their characteristic features. We then concentrate on so-called
social dilemmas, where one species (defectors)exploits the other one (cooperators). Due to the spatial extent of
our system, cooperators are able to outcompete defectors in a defined parameter region. This is in contrast to a
single group, where all members interact with each other and, therefore, cooperators are always inferior to
defectors.
In the simplest case selection is frequency-independent,
0,
====
and both species reproduce
with the same rate,
bb
0
10
2
=
(see figure 2(a)). For this ‘neutral’setting we observe roughening of the front typical
for the Eden model [64,65]. Simultaneously, sectors composed of a single species merge and thereby coarsen
[29,68]. This inherent process happens according to the following scenario. If the tips of two advancing
boundary lines meet, they annihilate and the enclosed sector loses contact to the front. Consequently, the
number of boundaries and sectors can only decrease. Sectors repeatedly coarsen as they merge in these events.
When all boundaries have vanished, the front ‘has fixed’to a single species, which keeps on expanding. In finite
systems,
L,
<¥fixation to a single species always occurs since in our stochastic model there is always a finite
rate at which boundaries annihilate. Hence, twoabsorbing states exist. Eventually, the expanding front willfix
either to species 1or species 2.
In figure 2(b), species 2 has a larger reproduction rate,
bb,
0
20
1
>
and therefore a constant selective advantage.
As the front expands faster at locations where it is composed of species 2, the roughness of the front increases.
Indentations of the front usually are caused by sectors of species 1, whereas species 2 creates bulges.
Furthermore, boundaries are biased such that sectors of species 2 widen while sectors of species 1 shrink
laterally. Hence, sectors of species 2 merge andcoarsen quickly. Eventually, the expanding front will fixto
species 2, which has almost happened in figure 2(b).
Figure 2. Growth patterns of our model for different parameters, which correspond to typical settings. Time is always t=15 and
lattice size is L=200. Cooperators are depicted in blue and defectors in red. If not stated otherwise bb1,
2
02
0
==
0,
====
and the initial ratio of both species is 1:1. (a)Neutral growth, (b)selective advantage for species 2 (
b1.5
2
0
=
),
which occupies 10% of initial sites, (c)coordination game (1
==
),(d)snowdrift game (1
==
), and (e)public goods
game (
0.1, 1.1
==
).
4
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
If the reproduction rates depend on the state of nearest neighbors (frequency-dependent selection), new
patterns arise. In this article we are mainly interested in frequency-dependent selection and therefore set
bb1
1
02
0
==
from here on. We now discuss some interesting cases, see figures 2(c)and (d), which correspond to
well-known settings in game theory [14–16].
In coordination games (
and
>>
), see figure 2(c), the front expands slower near boundaries than
in the centers of sectors. Therefore, indentations in the front are typically found where boundaries currently are
or recently have been. After sectors have coarsened for some time, most individuals are located inside sectors.
Therefore, most of them only have neighbors of their own kind, which raises the average reproduction rate and
the overall front advances faster.
In snowdrift games (
and
<<
), see figure 2(d), the front expands faster near boundaries. They
annihilate less frequently as compared to figure 2(c)since narrow sectors grow faster. Boundaries are also
strongly twisted and associated with bulges of the front.
In this article we are mostly interested in social dilemmas where one species (called cooperators)raises the
reproduction rate of all neighbors regardless of their species. The increased reproduction rate is called a public
good in game theory, since it is of benefit to all nearby individuals, but it also costs resources. In contrast, the
other species (called defectors)takes advantage of the public good for its own reproduction, but does not
contribute to the reproduction of its neighbors in the same way. Defectors save resources for their own
reproduction and therefore have an advantage. In this scenario defectors do not at all contribute to reproduction
of their neighbors and, therefore, we set 0
==
and just vary
and .
According to equation (1), these two
parameters increase the respective reproduction rates of cooperators and defectors if they have cooperating
neighbors. In figure 2(e)we present a setting, where species 2 (defectors)rapidly takes over large parts of the
front. Defectors benefit from the initially large number of boundaries, where they take advantage of nearby
cooperators, and conquer most of the front. Only cooperators, living in sufficiently large sectors, can keep up
with the front during this early period and may then take over the front, depending on the parameter values.
In a situation like this, it is not a priori clear if the front eventually fixes either to cooperators or to defectors.
Depending on the values of
and ,
cooperators can either outrun defectors and,from their advanced position
at the front, overgrow their competitors laterally, see figure 3(a). Or, defectors cover cooperators with a thin layer
and thereby take over the front, see figure 3(b). Close to the transition between both scenarios, the front displays
increasing roughness since both species are able to take over while their fronts grow with different speeds. To
characterize this transition quantitatively, we performed extensive simulations and applied methods from
surface roughening [65,72–74]and the theory of phase transitions dealing with absorbing states [74,75].
4. Expanding public goods game: critical behavior
In this section we quantitatively analyze the transition between long-term cooperation and long-term defection
for an expanding public goods game. As the transition is approached, several observables show critical scaling
[74,75]. Following our earlier work [69], we perform finite-size scaling to localize the transition. Furthermore,
we determine critical exponents and thereby establish a new universality class for the transition between the two
adsorbing states. In the vicinity of the transition we also study the dynamics of the sector boundaries including
the decline of their mean number during coarsening and their superdiffusive motion as well as the roughening of
the expanding front.
Figure 3. Schematics of possible scenarios in an expandingpublic goods game. Depending on the parameters
,
,
,
and
of
equation (1), cooperators (C, blue)and defectors (D, red)advance with different speeds, as indicated by arrows. (a)Cooperators
outrun the trailing defectors. From an advanced position along the front cooperators can then expand laterally and take over the front.
(b)A thin layer of defectors keeps up with the cooperators’sector and, eventually, completely covers the cooperators.
5
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
4.1. Finite-size scaling and phase diagram
As boundaries merge, sectors coarsen and the system progresses towards one of the two absorbing states. At
finite system sizes Lthis is a stochastic process and both adsorbing states are reached with a certain probability.
However, in the thermodynamic limit,
L,
¥the magnitude of fluctuations relative to the mean value goes to
zero and one of the absorbing states is reached with certainty. We now use the method of finite-size scaling to
determine the transition point between both states [75].
In figure 4we present the probability P
fix
that the front fixes to cooperators and plot it versus
for several
lattice sizes Lat
0.1.
=
We distinguish two regimes: one where cooperators dominate (
P
L,
fix
1
2
()>)and
one where defectors take over. We locate the transition point at L
12
(
)
by
P
LL,.
fix 1 2
1
2
(())=
P
L,
fix (
)
is a
monotonically decreasing function in
of sigmoidal shape. We use a Fermi function to fit it and extract the
transition point, see figure 4. Other sigmoid functions (e.g. Hill equation, hyperbolic tangent)gave qualitatively
identical results, but with reduced fit quality and hence lower accuracy. As
L,
¥P
fix
converges to a step
function, since in infinite systems the absorbing states are reached with certainty. The step is positioned at
Llim .
cL12
≔()
¥ From the theory of critical scaling applied to absorbing states, we expect that close to the
critical point c
the states of the lattice sites are correlated on the transverse distance .
x
^Approaching ,
c
x
^
diverges as
,
∣
∣Dn-
^
where c
≔
D
-
is the distance to the critical point and
n
^
is a critical exponent. For finite
systems, an absorbing state is reached if
L.2∣∣ ()x»~D
n
^-^
The transition occurs at L
12
(
)
and rearranging equation (2), we obtain
LAL.3
c12 1
() ( ) ()»-
n
^
The characteristic length Ais related to the microscopic length scale, which here is the lattice constant, and
details of our model. It is not important to the following analysis. The inset of figure 4shows the best fit of our
data to equation (3), which yields the critical exponent
4.2
n
»
^and the critical point 1.5
8
c
»at
0.1.
=
Note that the parameter
c
1-
of our Fermi fit function gives the width of the transition region in .
Within this
region transverse fluctuations in the boundaries are typically larger than or comparable to L, from which
cL
1
~
n
^follows. This is indeed observed andcorroborates our scaling analysis (data not shown).
The above procedure can be repeated for different values of
to map out the phase diagram (see figure 5).
One realizes that the benefit of cooperators from their own species,
,
has a much more pronounced influence
on the final state than the defectors’benefit from cooperating neighbors, .
This makes sense, since at large
times
t
1the front contains large single-species sectors. Hence, the number of sector boundaries N
b
, where
defectors can benefit from cooperators, is small:
N
L.
b
Therefore, almost all cooperators have cooperating
neighbors, while only a few defectors have this advantage. This is an example of ‘preferential assortment’, where
the benefit of cooperation is almost entirely available to other cooperators [15,32,33,45–47,76,77]. So, for a
wide range of parameter combinationscooperators can indeed outcompete defectors. However, forlarge
Figure 4. Probability P
fix
of the front to fix to cooperators plotted versus
for several system sizes Lat 0.1.
=Error bars give the
standard error of the mean for each data point. Lines are fits of the data points to the Fermi function, c
1
exp 1 ,
12
[(( )) ]-+
where
1
2
and care fit parameters. Inset: The transition point
L
12
(
)
relative to the fitted critical value
1.58
c
»
(blackcrosses)
follows a power law in
L
:¥
LAL
c12 14.
2
() ( )
-=
(black line).
6
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
enough
the dynamics at the boundaries still determines the final state of the front and defectors outcompete
cooperators.
4.2. Critical exponents of the phase transition
In the previous section 4.1 we already encountered the critical exponent
.
n
^
We now continue to determine
further critical exponents of the phase transition. They are universal, which means independent of microscopic
details. The exponents only depend on the dimension of the system, the number of components of the order
parameter, and symmetries of the model [74,75]. Our system has properties similar to ‘compact directed
percolation’(CDP)[78]. This is a stochastic process with a flat front, which also has two distinct absorbing states.
Using this similarity, weproceed by determining critical exponents, whichare known for CDP [75], and
compare both models.
At the critical transition,
,
c
=
none of the two species has an advantage. Heterogeneous fronts,
composed of more than one sector, exist for long times before the front fixes to one of the absorbing single-
species states. This can be quantified by the mean time to fixation, t
fix
, presented in figure 6. The data show that
the fixation time has a maximum, the position and value of which grow with system size.
Along the longitudinal direction, in which the front propagates, states are correlated on the longitudinal
distance .
x
As before, we expect it to scale like
∣
∣D
n-
close to the transition. Since the front propagates with a
mean velocity,
x
is proportional to a correlation time. Close to c
this time becomes very long, which is known
as critical slowing down. Substituting equation (2)into ,∣∣
x
~Dn-
we find the scaling relation
L.4()x~
nn
^
We expect the mean fixation time to be proportional to the correlation time .x~Therefore, in the inset of
figure 6we plot t
fix
rescaled by
L
nn
^
versus Δrescaled by
L
.
1n-
^
All curves of the main plot collapse ona single
master curve for
1.59 0.03
,
c
=
and critical exponents 4.2 0.1
n
=
^and
3.5 0.1.
n
=
The values of
c
and
n
^
are in good agreement with our fit to equation (3). So, fixation of the front is determined by the
characteristic time
L.5
fix 0.83 0.05
()tx~~
Two more critical exponents right at the transition are related to the survival probability of one species or
state, which initially occupies a single site while all the other sites are occupied by the other state. We choose a
single cooperator site in a line of defectors and determine the probability P
C
(t)that after time tthere are still
cooperators at the front and also calculate the average number of cooperator sites at the front,
N
t.
C()In figure 7
we plot both quantities versus time for different defector benefit.
At the transition situated between
1.55
=
and 1.6, we find that both P
C
and N
C
(see inset)decay with power laws in time:
P
tt
C
()~
bn-¢
and
N
tt.
C
()~
-Q
The respective best fits yield
2.2 0.1bn
¢=
and
1.3 0.1.
Q
=
Using our result for
,
n
we
find
7.7 0.6.b¢=
Figure 5. Phase diagram of the two-species public goods game with range expansion. Parameter regimes, where the expanding front
fixes either to cooperators or defectors in large systems with
L
,¥
are indicated by blue and red shade, respectively. Red crosses are
from simulations with system size L=1000, where defectors always outcompeted cooperators (
P
0
fix =)while blue dots identify
events where cooperators survived (
P
0
fix >). The black diamonds indicate critical points
c
for
L
¥
determined from finite-size
scaling (see equation (3)and figure 4). Note that in finite systems defectors have an advantage in a larger parameter region.
7
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
In general, in phase transitions to absorbing states the critical exponent βgoverns the stationary density of
‘active sites’, when approaching the transition [75]. In our case, the ‘active’sites can either be cooperators or
defectors. Since the stationary state is either an all cooperator or an all defector front, the density c
∣∣~-b
jumps from 0 to 1 and henceβis 0. Our results for all the critical exponents are summarized in table 1together
with the exponents of some related universality classes. We also give the critical exponents for the neutral
transition point
0,
==
which were obtained by similar means (data not shown). For this case the scaling
of the time to fixation
,
fix
t
and therefore the ratio 32
,n
n=
^
can be determined from the annihilation
dynamics of the boundaries (see section 4.3).
The neutral system 0
==
is symmetric in the behavior of cooperators and defectors and therefore
closely related to CDP, where active and inactive sites are interchangeable. It may therefore be called a ‘rough
CDP’. The roughness of the front evolves through kinetic roughening and is identical to the one-species Eden
model. When the transition occurs at non-zero
and ,
the symmetry between the two species is missing.
Indeed, cooperator and defector sectors expand differently, which we regard as the reason for the different
Figure 6. Mean time to fixation t
fix
as a function of defector benefit
for various system sizes Lat 0.1.
=The inset depicts the
rescaled fixation time, where
.
c
D= -
All data collapse on a single master curve for critical exponents 3.5 0.1n=
and
4.2 0.1n=
^
and critical point
1.59 0.03.
c
=
Accordingly, t
fix
grows with Land the position of its maximum approaches
c
for
L
.¥
Figure 7. Survival probability P
C
of cooperators starting from a single site plotted versus time for several values of
.
Other parameters
are L=1000 and 0.1.
=For
T,
c
>
P
C
decreases exponentially, while for
T
c
<
the front fixes to cooperators with a non-zero
probability. At the transition point ,
c
the survival probability decays in time with apower law with exponent
2.2 0.1.bn
¢=
Inset: The number of cooperator sites at the front, N
C
, decays exponentially in time for
T
c
>
and is non-monotonic for T.
c
<At
the transition N
C
decreases with a power law with critical exponent 1.3 0.1.
Q
=
8
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
critical exponents. We study surface roughening of the expanding fronts in more detail in section 4.4. When
approaching the neutral point 0
==
along the phase transition line, we expect to find crossover behavior.
Following [79]we determined the crossover exponent fby mapping out the phase transition line close to the
neutral point and by fitting with cc
~
f
(data not shown). We obtained 1.6 0.1.
f
=
The set of critical exponents determines the universality class of a phase transition. To our knowledge no
other non-equilibrium transition has been found, which shares the same set of exponents. Hence, the transition
between long-term cooperation and long-term defection in our expanding public goods game with frequency
dependent selection constitutes a new universality class.
4.3. Dynamics of boundaries
In this section we investigate the dynamics of the boundaries which separates sectors of cooperators and
defectors from each other. In rough fronts thelocal front orientation is tilted against themain growth direction.
When the front grows further, this tilt directs the movement of boundaries [27,29]. Ultimately, when two of
them meet, they annihilate. In figure 8we plot their mean number N
b
versus time for several values of the
defector benefit.
Right at the transition (dashed black line),N
b
shows a power law decay. We now discuss the
different regimes in figure 8.
For neutral systems, where frequency-dependent selection is absent (
0
==
), any inclination of the
front is created by stochastic surface or Eden roughening [65,72–74]. The surface undulations obey KPZ-scaling
[80]and thereby drive the decay of N
b
[27,29]. Boundaries move superdiffusively along the front with a mean-
square displacement proportional to
t
43
[28]. On average, they annihilate after having traveled the mean
distance L/
N
b
between the boundaries, for which they need the time
LN .
b32
()~
Hence, boundaries annihilate
with a rate proportional to
N
,
b
32which implies
NNN.6
bbb
32
˙()~-
Table 1. Critical exponents for the phase transition to the absorbing states (either long-term global defection or long-term global coopera-
tion)for the expanding public goods game with frequency dependent selection. The neutral case
0,
==
which has different scaling, is
also included. For comparison we also give the exponents for CDP [75,78]and for an earlier work on rough range expansion [69].
n
^
n
β
b
¢
Θ
,0
>
4.2±0.1 3.5 ±0.1 0 7.7±0.6 1.3 ±0.1
0
== 0.75 ±0.05 1.1 ±0.1 0 0.9±0.2 0.02 ±0.01
1dCDP 1 2 0 1 0
[69]1.6±0.1 1.6 ±0.1 0.50 ±0.02 0.51 ±0.07 0.32 ±0.02
Figure 8. Mean number of sector boundaries N
b
plotted as a function of time for several values of
.
Other parameters are L=1000
and 0.1.
=Solid black line:
N
tt,
b23
()~
-
as predicted in [29]for neutral growth (
0
==
). At large times the simulated
curve deviates from the power law due to finite system size. Dashed black line: Close to the critical point
1.6,
c
=»
we find a
power-law decay
N
tt.
b2.5
()~
-
9
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
So, the number of boundaries decreases as
Nt t ,7
b23
() ()~
-
as already observed by Saito and Müller-Krumbhaar [29]. This power law is excellently reproduced by our
simulations in the case of neutral growth,
0,
==
as illustrated by the solid black line in figure 8.
Equation (7)implies that for 0
==
the time to fixation scales like
L
32
since this is the time after which all
initially present boundaries with mean number
N
tL0
b
()=~
have annihilated.
For
0
¹
and
0,
¹
species reproduce with different rates. Hence, the fronts of two neighboring sectors
(occupied by different species)advance with different speeds. This influences the tilt of the front orientation, in
addition to stochastic roughening in neutral systems, and thereby the movement of the separating boundary.
Thus, we do not expect equation(6)to be valid.
Indeed, figure 8reveals different regimes for the mean number of boundaries N
b
. For
,
c
>
N
b
decays
exponentially in time in line with the exponential decay of the survival probability P
C
in figure 7and similar to
the case of selective advantage in [29]. For
c
<
boundaries annihilate less frequently. Narrow defector sectors
persist in the front dominated by cooperators since almost all individuals in the defector sectors have
cooperating neighbors. This results in the upward curvature of the curves in figure 8. However, the defector
sectors cannot expand laterally and ultimately loose contact to the front due to random fluctuations, and N
b
declines exponentially. At c
the number of boundaries decreases with a power law
N
t
,
b
~
c-where a new
exponent
2.50 0.05c»
appears. This power law implies that the number of boundaries declines from the
initial value
N
tL0
b
()=~
to the order of 1 in the coarsening time
LL .8
coarse 1 0.40 0.01
()t~»
c
Comparing with equation (5)reveals
1
.cnn<
^
This suggests that for large systems fixing the front to one
species takes much longer than coarsening to a few sectors. Hence, the few remaining boundaries move
differently compared to early times since they have to annihilate to fix the front to a single species.
To check if this is the case, we employ the initial condition, where the front is composed of only two sectors
of size
L
2each, separated by two boundaries. We quantify the boundaries’random motion by monitoring the
temporal evolution of the standard deviation for the transverse distance ,
ℓ
^
ttt.9
2
ℓℓ()≔ () () ()
⎡
⎣⎤
⎦
s-
^^
We subtract the mean distance
tℓ()
á
ñ
^
to take care of any transient drift, when the front relaxes from its initially
flat to the rough shape, and an expected small drift if
is not exactly
.
c
From figure 9we see that, for early times, σgrows like a power law,
t
,s
~h
with 0.71 0.05.
h
=This is
consistent with meandering boundaries induced by Eden roughening,
t
23
s
~
[27–29]. We expect such a
behavior since the roughness of the front has not fully developed yet. For large systems and later times we find a
crossover to
t
s
~
h¢with
0.9 0.1.
h
¢=
This confirms our earlier statement that at late times the few
boundaries remaining after coarsening move differently. Indeed, they show an even stronger superdiffusive
Figure 9. Time evolution of two boundaries with initial distance
L
2
in a system close to criticality (
1.6
=
)and at 0.1.
=
Standard deviation σof the transverse distance plotted versus time tfor several system sizes L. The boundaries move superdiffusively.
Initially,
t
s
~
h
with 0.71 0.05,
h
=and consistent with Eden scaling (
t
,
23bold black line). At later times a crossover to
t
s
~
h¢
with 0.9 0.1
h
¢=
occurs (dashed black line).
10
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
motion than Eden scaling, which is associated with the long-lived and pronounced surface undulations in
systems with frequency-dependent selection. To illustrate these findings, a system close to criticality isgiven in
figure 10.
4.4. Surface roughening of the expanding front
We now discuss the surface roughness or undulations of the expanding front and compare our results to the
classical Eden model. We measure the roughness of a front by calculating its width wfor system size Land at
time t:
wLt Lhit ht,1,. 10
i
L
1
212
()≔ ()¯() ( )
⎛
⎝
⎜⎡
⎣⎤
⎦
⎞
⎠
⎟
å-
=
Here,
hi t,()
is the longitudinal position of the front at its transverse site iand
ht
¯()
is the mean position
ht Lhit
1,. 11
i
L
1
¯()≔ ( ) ( )
å
=
In the following we will only consider systems where the front has not yet fixed to one species, for parameters
close to the phase transition. From equation (5)we know that the time to fixation diverges with system size L. So,
in order to extrapolate to
L,
¥we only sample from realizations where the front has not yet fixed to a single
species. Figure 11 plots the width
w
Lt,()
close to the critical point at
1.6.
c
=»
Initially, the roughness of
the front grows like in the original Eden model [64]:
w
t
,
~gwhere the growth exponent 13
g
=belongs to the
KPZ-universality class [65]. At intermediate times a new regime sets in, where
w
t~
g¢increases with enhanced
growth exponent 1.3 0.1.
g
¢»This marks the transition from Eden roughening to ‘selective roughening’.
Here, the typical shape of the front is determined by advancing cooperator sectors and trailing defector sectors
(see figure 10). On length scales smaller than the typical extension of a sector one still observes Eden-like
roughening. On larger length scales undulations of the front are clearly due to sectors occupied by different
species. The resulting large undulations ultimately drive the accelerated increase of the front’s width w. The
crossover to this regime happens at time ,t
´which increases with system size Las figure 11 shows. This makes
sense since we expect selective roughening to dominate over Eden roughening when the lateral extension of
sectors is comparable to L, i.e.,
L.
coarse 1
tt~~
c
´
Due to Eden roughening the width at the crossover is
w
LL.
0.13
t~~ »
ggc
´´Indeed, rescaling width and time with
w
´and ,t
´respectively, collapses all data in
figure 11 onto a single master curve, as the inset demonstrates. Note that for finite Lthe fronts of all systems
eventually fix to one species. The fronts then show classical Eden roughening without any selective roughening.
Any contribution of selective roughening to the overall width wdecays and the width saturates according to the
classical law,
w
t
sat 12
~[65].
To conclude, surface roughening close to criticality occurs in two regimes. Until crossover time ,t
´one
observes Eden roughening, whereas for times larger than
t
´
selective roughening occurs until the front fixes to
one species. The dynamics of the width of the front is summarized by
Figure 10. Realization of a system close to criticality (
0.1,
=
1.55
=)before the front fixes to one species. Most cooperator
sectors (blue)are rapidly outcompeted by their defecting neighbors (red), but a single cooperator sector survives. Boundaries grow
persistently in one direction for long times, superimposed by fluctuations on smaller length scales. This is indicative for the dynamics
of sector boundaries close to ballistic scaling, which is expected at late times. The width of the front is dominated by selective
roughening: the cooperator sector advances and the defector sector trails behind.
11
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
wLt ttL
tL t
,,
.
12
1
()
() ()
⎪
⎪
⎧
⎨
⎩
t
t
~~
gc
ggg c
´
¢-¢´
5. Summary and conclusion
In this work we studied a generalized Eden model, where two species compete with each other at the rough
expanding front. Individuals of the two species influence each other by frequency-dependent selection, which
acts between nearest neighbors. We analyzed the evolutionary dynamics at the expanding front, where single-
species sectors form and coarsen. Ultimately, the front fixes to one species, which we identify with an absorbing
state of our model.
In its general form the model can implement several scenarios including selective advantage, and also well-
known game theoretical settings like the snow drift game orthe coordination game. Each of themcreates distinct
patterns, which should be analyzed in detail in future work. For the prominent example of a public goods game,
we find that cooperators prevail in a wide parameter regime, as expected for a spatial version of a social dilemma
[15,45–47]. For other parameter values defectors take over the front, as usual.
We identify the transition between long-term cooperation and long-term defection as a nonequilibrium
critical phase transition between two absorbing states. The set of critical exponents (see table 1), which we
determined by analyzing critical and finite size scaling, shows that the phase transition belongs to a new
universality class. We attribute this result to the fact that the front in our model is rough and not flat as in usual
absorbing states. Close to the critical transition the front’s roughness exhibits a crossover in time from slow Eden
roughening to fastselective roughening. Strong roughening hasalso been observedat phase transitionsin a
related model by Lavrentovich and Nelson [81].
Our present work does not include creation of boundaries by mutations or other means. Therefore, sectors
are compact, a property shared with regular CDP with a flat front [78]. Note however, that we found exponents
different from that of the flat CDP universality class, see table 1. This is not astonishing, since not only do we treat
rough fronts, but there also exists no symmetry when we exchange defectors and cooperators (besides at
0
==
), whereas there is an exact symmetry upon exchange of active and inactive sites in CDP.
For continuous phase transitions the critical exponents and the transverse dimension dof the system are
usually related to each other by hyperscaling relations [74,75], at least below an upper critical dimension d
c
. For
systems with several absorbing states a so-called generalized hyperscaling relation exists [82]
d.13()nn bb=Q++
¢
^
This relation ‘holds for almost all universality classes of absorbing phase transitions below their upper critical
dimension’as Henkel et al state in [75], however, without giving any counterexample. The critical exponents
Figure 11. The width wof the front plotted versus time for several system sizes Lclose to the critical point at
1.6.
c
=»
Inset: after
rescaling time tby
L
1
t~
c
´and width wby
w
L,~gc
´the curves collapse onto a single master curve. The solid black line indicates
Eden roughening with
w
t,
13
~whereas the dashed black line shows selective roughening with
w
t1.
3
~for
t
.t
´
Note that the
saturation of wat large times is due to the finite system size and the fact that we only sample from realizations where the front has not
yet fixed to a single species (see main text for details).
12
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
determined in our present work (see table 1)violate the above equation for d=1. It might therefore be that our
model has a critical dimension d
c
smaller than 1. An alternative explanation may be that the enhanced roughness
of the front leads to an effective dimension larger than one similar to models of surface growth, which create
fronts with fractal properties [65]. Interestingly, in an earlier work [69]we also found enhanced roughness in the
expanding front at the phase transition. There, the generalized hyperscaling relation was fulfilled within the
margin of error of the critical exponents.
The roughness of the front correlates with superdiffusive motion of the boundaries separating sectors. Two
factors contribute to the movement of the boundaries on long length scales. On the one hand, the direct
competition between the species on either side of the boundaries pushes them towards the sector composed of
the more slowly reproducing species. On the other hand, boundaries follow the local tilt of the front. In the
public goods game cooperator sectors are advanced, while defector sectors lag behind. Near the critical
transition, defectors outcompete their direct cooperating neighbors but the front is tilted towards sectors filled
by defectors, so the two factors move the boundaries in opposite directions. At the phase transition both effects
cancel and the front fixes with equal probability to either species. The strong roughening correlates with
superdiffusive motion of the boundaries with nearly ballistic scaling.
Accordingly, whether a species takes overthe expanding front is determined bytwo contributions: its
reproduction rate relative to its competitor and its position relative to the average front position. The influence
of different reproduction rates of neighboring species can directly be compared and is summarized in the phrase
‘survival of the fittest’. The position at the front determines the available space for progeny, which then have the
opportunity to expand sidewards. This is illustrated by the phrase ‘survival of the fastest’[39].
In our model the number of sectors only decreases. It does not include experiments with mutually beneficial
interactions between different species, which do not generate sectors[38,83,84]. In future extensions of our
model this may be remedied by including motility of individuals [85,86], by allowing reproduction to more
distant lattice sites [81], or by increasing the maximal number of individuals per lattice site from one. Moreover,
it is worthwhile to consider interactions ranging beyond nearest neighbors, since biomolecules, released by
individual microorganisms, may diffuse in the extracellular medium [87–89]. In the public goods game scenario
this would stabilize narrow sectors of defectors so that they do not lose contact to the front.
In general, range expansion of multiple species will develop enhanced roughness at the growing front. As we
demonstrated here, the corresponding models have new and interesting statistical properties. From a biological
point of view, roughness is important. It affects the territories that different species occupy and thereby their
evolutionary success through the strong random motion of sector boundaries. This may also be relevant for
range expansion in a real environment and not just in a test tube. To better understand the properties and
consequences of rough expanding fronts, further theoretical work is needed. At the same time further
experiments should look for the fingerprint of roughness in microbial colony growth.
Acknowledgments
We thank the research training group GRK 1558 funded by Deutsche Forschungsgemeinschaft for financial
support. We further thank Erwin Frey, Maria Eckl, Florian Gartner, and Raphaela Geßele for discussion and
collaboration on a related model.
References
[1]Murray J D 2007 Mathematical Biology :I. An Introduction vol 1 3rd edn (Berlin: Springer)
[2]Griffiths R W, Schloesser D W, Leach J H and Kovalak W P 1991 Distribution and dispersal of the zebra mussel (Dreissena polymorpha)
in the Great Lakes Region Can. J. Fish. Aquat. Sci. 48 1381–8
[3]Loarie S R, Duffy P B, Hamilton H, Asner G P, Field C B and Ackerly D D 2009 The velocity of climate change Nature 462 1052–5
[4]Chen I C, Hill J K, Ohlemuller R, Roy D B and Thomas C D 2011 Rapid range shifts of species associated with high levels of climate
warming Science 333 1024–6
[5]Peacock E et al 2015 Implications of the circumpolar genetic structure of polar bears for their conservation in a rapidly warming arctic
PLoS One 10 e112021
[6]Stringer C 2003 Human evolution: out of ethiopia Nature 423 692–5
[7]Liu H, Prugnolle F, Manica A and Balloux F 2006 A geographically explicit genetic model of worldwide human-settlement history Am.
J. Hum. Genet. 79 230–7
[8]Moreau C, Bherer C, Vezina H,Jomphe M, Labuda D and Excoffier L 2011 Deep human genealogies reveal a selective advantage to be
on an expanding wave front Science 334 1148–50
[9]Brú A, Albertos S, Luis Subiza J, García-Asenjo J L and Brú I 2003 The universal dynamics of tumor growth Biophys. J. 85 2948–61
[10]Xavier J B 2011 Social interaction in synthetic and natural microbial communities Mol. Syst. Biol. 7483
[11]Hall-Stoodley L, Costerton J W and Stoodley P 2004 Bacterial biofilms: from the natural environment to infectious diseases Nat. Rev.
Micro. 295–108
[12]Nadell C D, Xavier J B and Foster K R 2009 The sociobiology of biofilms FEMS Microbiol. Rev. 33 206–24
[13]Mitri S, Xavier J B and Foster K R 2011 Social evolution in multispecies biofilms Proc. Natl Acad. Sci. USA 108 (Suppl 2)10839–46
13
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
[14]Hofbauer J and Sigmund K 1998 Evolutionary Games and Population Dynamics 1st edn (Cambridge: Cambridge University Press)
[15]Szabó G and Fáth G 2007 Evolutionary games on graphs Phys. Rep. 446 97–216
[16]Frey E 2009 Evolutionary game theory: theoretical concepts and applications to microbial communities Physica A389 4265–98
[17]Buckling A, Maclean R C, Brockhurst M A and Colegrave N 2009 The Beagle in a bottle Nature 457 824–9
[18]Shapiro J A 1995 The significances of bacterial colony patterns BioEssays 17 597–607
[19]Ben Jacob E, Aharonov Y and Shapira Y 1999 Bacteria harnessing complexity Biofilms 1239–63
[20]Golding I, Cohen I and Ben-Jacob E 1999 Studies of sector formation in expanding bacterial colonies Europhys. Lett. 48 587–93
[21]Matsushita M, Wakita J, Itoh H, Watanabe K, Arai T, Matsuyama T, Sakaguchi H and Mimura M 1999 Formation of colony patterns by
a bacterial cell population Physica A274 190–9
[22]Matsuyama T and Matsushita M 2001 Population morphogenesis by cooperative bacteria Forma 16 307–26
[23]Nguyen B, Upadhyaya A, van Oudenaarden A and Brenner M P 2004 Elastic instability in growing yeast colonies Biophys. J. 86 2740–7
[24]Vicsek T, Cserzo M and Horvath V K 1990 Self-affine growth of bacterial colonies Physica A167 315–21
[25]Wakita J-I, Itoh H, Matsuyama T and Matsushita M 1997 Self-affinity for the growing interface of bacterial colonies J. Phys. Soc. Japan.
66 67–72
[26]Huergo M, Pasquale M, Bolzán A, Arvia A and González P 2010 Morphology and dynamic scaling analysis of cell colonies with linear
growth fronts Phys. Rev. E82 031903
[27]Hallatschek O, Hersen P, Ramanathan S and Nelson D R 2007 Genetic drift at expanding frontiers promotes gene segregation Proc. Natl
Acad. Sci. USA 104 19926–30
[28]Derrida B and Dickman R 1991 On the interfacebetween two growing Eden clusters J. Phys. A: Math. Gen. 24 L191–5
[29]Saito Y and Müller-Krumbhaar H 1995 Critical phenomena in morphology transitions of growth models with competition Phys. Rev.
Lett. 74 4325–8
[30]Crespi B J 2001The evolution of social behaviorin microorganisms Trends Ecol. Evol. 16 178–83
[31]Velicer G J 2002 Social strife in the microbial world Trends Microbiol. 11 330–7
[32]Griffin A S, West S A and Buckling A 2004 Cooperation and competition in pathogenic bacteria Nature 430 1024–7
[33]Kreft J U 2004 Conflicts of interest in biofilms Biofilms 1265–76
[34]West S A, Diggle S P, Buckling A, Gardner A and Griffin A S 2007 The social lives of microbes Annu. Rev. Ecol. Evol. Syst. 38 53–77
[35]Hibbing M E, Fuqua C, Parsek M R and Peterson S B 2010 Bacterial competition: surviving and thriving in the microbial jungle Nat.
Rev. Micro. 815–25
[36]Mitri S and Richard Foster K 2013 The genotypic view of social interactions in microbial communities Annu. Rev. Genet. 47 247–73
[37]Gore J, Youk H and van Oudenaarden A 2009 Snowdrift game dynamics and facultative cheating in yeast Nature 459 253–6
[38]Müller M JI, Neugeboren B I, Nelson D R and Murray A W 2014 Genetic drift opposes mutualism during spatial population expansion
Proc. Natl Acad. Sci. USA 111 1037–42
[39]van Dyken J D, Müller M JI, Mack K ML and Desai M M 2013 Spatial population expansion promotes the evolution of cooperation in
an experimental Prisonerʼs Dilemma Curr. Biol. 23 919–23
[40]Darwin C 1859 On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life
(London: John Murray)
[41]Axelrod R 2009 The Evolution of Cooperation revised edition (New York: Basic Books)
[42]Korolev K S 2013 The fate of cooperation during range expansions PLoS Comput. Biol. 9e1002994
[43]Rainey P B and Rainey K 2003 Evolution of cooperation and conflict in experimental bacterial populations Nature 425 72–74
[44]Chuang J S, Rivoire O and Leibler S 2009 Simpsonʼs paradox in a synthetic microbial system Science 323 272–5
[45]Nowak M A, Bonhoeffer S and May R M 1994 Spatial games and the maintenance of cooperation Proc. Natl Acad. Sci. USA 91 4877–81
[46]Ohtsuki H, Hauert C, Lieberman E and Nowak M A 2006 A simple rule for the evolution of cooperation on graphs and social networks
Nature 441 502–5
[47]Fu F, Nowak M A and Hauert C 2010 Invasion and expansion of cooperators in lattice populations: Prisonerʼs dilemma vs. snowdrift
games J. Theor. Biol. 266 358–66
[48]Melbinger A, Cremer J and Frey E 2010 Evolutionary game theory in growing populations Phys. Rev. Lett. 105 178101
[49]Cremer J, Melbinger A and Frey E 2012 Growth dynamics and the evolution of cooperation in microbial populations Sci. Rep. 2281
[50]Xavier J B and Foster K R 2007 Cooperation and conflict in microbial biofilms Proc. Natl Acad. Sci. USA 104 876–81
[51]Nadell C D, Foster K R and Xavier J B 2010 Emergence of spatial structure in cell groups and the evolution of cooperation PLoS Comput.
Biol. 6e1000716
[52]Greig D and Travisano M 2004 The Prisonerʼs Dilemma and polymorphism in yeast SUC genes Proc. Biol. Sci 271 S25–6
[53]Koschwanez J H, Foster K R and Murray A W I 2011 PLoS Biol. 9e1001122
[54]sen Datta M, Korolev K S, Cvijovic I, Dudley C and Gore J 2013 Range expansion promotes cooperation in an experimental microbial
metapopulation Proc. Natl. Acad. Sci. USA 110 7354–9
[55]Pattus F and Abdallah M A 2000 Siderophores and iron-transport in microorganisms J. Chin. Chem. Soc. 47 1–20
[56]Ratledge C and Dover L G 2000 Iron metabolism inpathogenic bacteria Annu. Rev. Microbiol. 54 881–941
[57]Nadell C D and Bassler B L 2011 A fitness trade-off between local competition and dispersal in Vibrio cholerae biofilms Proc. Natl Acad.
Sci. USA 108 14181–5
[58]van Gestel J, Weissing F J, Kuipers O P and Kovács A T2014 Density of founder cells affects spatial pattern formation and cooperation
in Bacillus subtilis biofilms ISME J. 82069–79
[59]Riley M A and Gordon D M 1999 The ecological role of bacteriocins in bacterial competition Trends Microbiol. 7129–33
[60]Cornforth D M and Foster K R 2013 Competition sensing: the social side of bacterial stress responses Nat. Rev. Micro. 11 285–93
[61]Lee V T and Schneewind O 2001 Protein secretion and the pathogenesis of bacterial infections Genes Dev. 15 1725–52
[62]Weber M F, Poxleitner G, Hebisch E, Frey E and Opitz M 2014 Chemical warfare and survival strategies in bacterial range expansions
J. R. Soc. Interface 11 20140172
[63]Xavier J B, Kim W and Foster K R 2011 A molecular mechanism that stabilizes cooperative secretions in Pseudomonas aeruginosa Mol.
Microbiol. 79 166–79
[64]Eden M 1960 A two-dimensional growth process Proc. 4th Berkeley Symp. on Mathematical Statistics and Probability vol 4 pp 223–39
[65]Barabási A-L and Stanley H E 1995 Fractal concepts in surface growth 1st edn (Cambridge: Cambridge University Press)
[66]Plischke M and Rácz Z 1985 Dynamic scaling and the surface structure of Eden clusters Phys. Rev. A32 3825–8
[67]Jullien R and Botet R 1985 Scaling properties of the surface of the Eden model in D=2, 3, 4 J. Phys. A: Math. Gen. 18 2279–87
[68]Ali A and Grosskinsky S 2010 Pattern formation through genetic drift at expanding population fronts Adv. Complex Syst. 13 349–66
14
New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
[69]Kuhr J-T, Leisner M and Frey E 2011 Range expansion with mutation and selection: dynamical phase transition in a two-species Eden
model New J. Phys. 13 113013
[70]Gillespie D T 1976 A general method for numerically simulating the stochastic time evolution of coupled chemical reactions J. Comput.
Phys. 22 403–34
[71]Jullien R and Botet R 1985 Surface thickness in the Eden model Phys. Rev. Lett. 54 2055
[72]Krug J and Spohn H 1992 Kinetic roughening of growing surfaces Solids far from Equilibrium ed C Godrèche (Cambridge: Cambridge
University Press)
[73]Halpin-Healy T and Zhang Y-C 1995 Kinetic roughening phenomena, stochastic growth, directed polymers and all that, aspects of
multidisciplinary statistical mechanics Phys. Rep. 254 215–414
[74]Ódor G 2004 Universality classes in nonequilibrium lattice systems Rev. Mod. Phys. 76 663–724
[75]Henkel M, Hinrichsen H and Lübeck S 2008 Non-equilibrium Phase Transitions: Absorbing Phase Transitions vol 1 1st edn (Berlin:
Springer)
[76]Hamilton W D 1964 The genetical evolution of social behaviour. I J. Theor. Biol. 71–16
[77]Hamilton W D 1964 The genetical evolution of social behaviour. II J. Theor. Biol. 717–52
[78]Essam J W 1989 Directed compact percolation: cluster size and hyperscaling J. Phys. A: Math. Gen. 22 4927
[79]Lübeck S 2006 Crossover scaling in the Domany-Kinzel cellular automaton J. Stat. Mech. Theor. Exp. 2006 P09009
[80]Kardar M, Parisi G and Zhang Y-C 1986 Dynamic scaling of growing interfaces Phys. Rev. Lett. 56 889–92
[81]Lavrentovich M O and Nelson D R 2014 Asymmetric mutualism in two- and three-dimensional range expansions Phys. Rev. Lett. 112
138102
[82]Mendes J F F, Dickman R, Henkel M and Marques M C 1994 Generalized scaling for modes with multiple absorbing states J. Phys. A:
Math. Gen. 27 3019–28
[83]Momeni B, Brileya K A, Fields M W and Shou W 2013 Strong inter-population cooperation leads to partner intermixing in microbial
communities eLife 2e00230
[84]Kovács A T 2014 Impact of spatial distribution on the development of mutualism in microbes Front. Microbiol. 5649
[85]Reichenbach T, Mobilia M and Frey E 2007 Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games Nature 448
1046–9
[86]Gelimson A, Cremer J and Frey E 2013 Mobility, fitness collection, and the breakdown of cooperation Phys. Rev. E87 042711
[87]Julou T, Mora T, Guillon L, Croquette V, Schalk I J, Bensimon D and Desprat N 2013 Cell–cell contacts confine public goods diffusion
inside Pseudomonas aeruginosa clonal microcolonies Proc. Natl Acad. Sci. USA 110 12577–82
[88]Allen B, Gore J, Nowak M A and Bergstrom C T 2013 Spatial dilemmas of diffusible public goods eLife 2e01169
[89]Menon R and Korolev K S 2015 Public good diffusion limits microbial mutualism Phys. Rev. Lett. 114 168102
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New J. Phys. 17 (2015)103035 J-T Kuhr and H Stark
