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arXiv:1107.4214v1 [cond-mat.stat-mech] 21 Jul 2011

Competitive Brownian and L´ evy walkers

E. Heinsalu,1,2E. Hern´ andez-Garc´ ıa,1and C. L´ opez1

1IFISC, Instituto de F´ ısica Interdisciplinar y Sistemas Complejos (CSIC-UIB), E-07122 Palma de Mallorca, Spain

2National Institute of Chemical Physics and Biophysics, R¨ avala 10, Tallinn 15042, Estonia

(Dated: July 19, 2011)

Biological models where individuals of the same species perform a two-dimensional Markovian

random walk and undergo reproduction and death are studied. Spatial motion is either normal

diffusion characterized by Gaussian jumps (Brownian bugs) or superdiffusion characterized by L´ evy

flights (L´ evy bugs). Competitive interactions between the individuals are considered in three differ-

ent situations: a) no interaction, b) global interaction in which birth and death rates are influenced

by all individuals in the system, and c) reproduction and death rates of an individual depend on

the number of individuals in a neighborhood (finite-range nonlocal interaction). We find strong

differences between the globally and the finite-range nonlocally interacting systems. In the former

one single or few-cluster configurations are achieved with the spatial distribution of the particles tied

to the type of diffusion. In the L´ evy case long tails arise in some properties characterizing the shape

and dynamics of clusters, which turn to be much short-ranged under Brownian diffusion. Under

non-local finite-range interactions periodic patterns appear with periodicity set by the interaction

range. This length acts as a cut-off limiting the influence of the long L´ evy jumps, so that spatial

configurations under both types of diffusion are now more similar. The process of family mixing is

also considered.

PACS numbers: 05.40.-a, 05.40.Fb, 87.18.Hf, 87.23.Cc

I. INTRODUCTION

Birth and death are the most relevant processes in de-

termining the dynamics of biological populations which

in the context of statistical physics can be modeled us-

ing interacting particle models where particle number is

changing in time. As it is understood by now, birth and

death processes are also responsible for clustering mech-

anisms in systems where random-walking individuals un-

dergo reproduction and death. As a result, aggregation

of organisms can occur even in simple models where birth

and death processes are combined with spatial diffusion.

In fact, in the most simple Brownian bug model, where

particles reproduce and die with the same probability and

undergo Brownian motion [1–3], clustering of particles

was observed. In this model the clustering is produced

simply by the reproductive correlations (the offspring is

born at the same location of the parent) and by the irre-

versibility of the death process.

Taking into account another central ingredient that is

present in ecological systems, namely, the competition

with other individuals in the neighborhood for resources,

the formation of periodic spatial structures was observed

in Refs. [4–6].In these nonlocally interacting Brown-

ian bug models it was assumed that the reproduction

probability depends on the number of other organisms

in the neighborhood. In Ref. [7] nonlocally interacting

L´ evy bugs, i.e., reproducing and dying organisms that

undergo L´ evy flights, were studied. This type of motion

is relevant to model cell migration [8], biological search-

ing strategies [9, 10], bacteria dynamics [11], or pattern

formation of mussels [12]. In Ref. [7] it was shown that

the formation of a periodic pattern is robust with respect

to the type of spatial motion that the particles perform.

The periodic arrangement of clusters in these nonlocally

interacting bug models is a consequence of the compet-

itive interaction and has a spatial scale determined by

the interaction range [4]. However, a deeper analysis of

the differences and similarities between the Brownian and

L´ evy cases is still missing. In particular, as shown in

[5, 13], this can be very conveniently performed by con-

sidering the limit of the interaction distance approaching

the system size (global interaction), since a unique cluster

appears which helps to understand and characterize the

cluster properties, and the fluctuations of the population

size.

In the present paper we report on differences between

the systems of Brownian and L´ evy bugs, in the situations

of global and non-local interactions, as well as in the non-

interacting case. In addition, results on the dependence

of population on diffusion, and mixing of families of par-

ticles are presented for the finite-range interaction case.

The paper is organized as follows: in Sec. II we describe

the models to be analyzed. In Sec. III the noninteracting

bug systems are studied. The infinite competition range

where each particle is competing with all the others is an-

alyzed in Sec. IV. Finally, the nonlocally interacting (i.e.

with a finite interaction range) models are investigated

in Sec. V.

II.MODEL AND NUMERICAL ALGORITHM

We consider a system consisting initially of N0point-

like particles, which could be thought as being biological

organisms or bugs, placed randomly in a two-dimensional

L×L square domain with periodic boundary conditions.

Except when explicitly stated, we take L = 1, so that

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2

lengths are measured in units of system size. The par-

ticles diffuse, reproduce at rate ri

i = 1,...,N, and N ≡ N(t) is the number of particles in

the system at time t. The numerical algorithm used to

evolve the system follows the one suggested in Ref. [14].

The following sequence of steps is repeated until the final

simulation time is reached:

We first compute the random time τ after which the

next demographic event (birth or death) will occur. For

this we need to determine the total birth and death rates,

b, and die at rate ri

d;

Btot=

N

?

i=1

ri

b,Dtot=

N

?

i=1

ri

d, (1)

and compute also the total rate

Rtot= Btot+ Dtot=

N

?

i=1

(ri

b+ ri

d). (2)

For the random times τ we choose an exponential prob-

ability density with the complementary cumulative dis-

tribution

p(τ) = exp(−τ/˜ τ) (3)

so that values of τ could be generated from τ = −˜ τ ln(ξ0),

where ξ0is a uniform random number on (0,1) [15]. The

characteristic time or time-scale parameter ˜ τ = ?τ? is

determined by the total rate:

˜ τ = R−1

tot.(4)

After the random time τ, a particle i, chosen among

all the N(t) bugs, either reproduces or disappears. With

probability Btot/Rtotthe event is reproduction and with

probability Dtot/Rtot it is death.

choosing a particular particle i is weighted proportion-

ally to its contribution to the corresponding total rate.

In the case of reproduction, the new bug is located at the

same position (xi,yi) as the parent particle i. Finally,

all the particles perform a jump of random length ℓ in

a random direction characterized by an angle uniformly

distributed on (0,2π) (ℓ and the direction of the jump are

independent for each particle). The new present time is

t + τ and the process is repeated.

When particles undergo normal diffusion (Brownian

bugs), a Gaussian jump-length probability density func-

tion is used,

The probability of

ϕ(ℓ) =

1

˜ℓ√2πexp

?

−ℓ2

2˜ℓ2

?

,(5)

with variance ?ℓ2? =˜ℓ2;˜ℓ is the space-scale parame-

ter. The random jump length ℓ can be computed from

ℓ =˜ℓξG, where ξGis sampled from the standard Gaus-

sian distribution with average 0 and standard deviation

1. Since we draw the angle specifying the direction of the

jump from the interval (0,2π), we can neglect the sign

of ℓ. Note that the random walk defined in this way is

not exactly the same as the one in which the walker per-

forms jumps extracted from a two-dimensional Gaussian

distribution, but it also leads to normal diffusion and al-

lows a more direct comparison with the L´ evy case. The

corresponding diffusion coefficient can be estimated as

κ = ?ℓ2?/(2?τ?) =˜ℓ2/(2˜ τ). (6)

As we choose to fix the value of κ, and the demographic

rates, then the space-scale parameter is determined by

√2κ˜ τ =

˜ℓ =

?

2κ/Rtot.(7)

In order to simulate the system where the particles

undergo superdiffusive L´ evy flights (L´ evy bugs) one can

use a symmetric L´ evy-type probability density function

for the jump size, behaving asymptotically as [16, 17]

ϕµ(ℓ) ≈˜ℓµ|ℓ|−µ−1,

with the L´ evy index 0 < µ < 2. For all L´ evy-type proba-

bility density functions with µ < 2 the variance diverges,

?ℓ2? = ∞, leading to the occurrence of extremely long

jumps, and typical trajectories are self-similar, showing

at all scales clusters of shorter jumps interspersed with

long excursions.For 0 < a < µ < 2 fractional mo-

ments ?|ℓ|a? are finite. For the L´ evy index in the range

1 < µ < 2 the value of ?|ℓ|? is finite. The complementary

cumulative distribution corresponding to (8) behaves as

ℓ → ±∞(|ℓ| ≫˜ℓ)(8)

Pµ(ℓ) ≈ µ−1(|ℓ|/˜ℓ)−µ,

As a simple form of complementary cumulative distri-

bution function which behaves asymptotically as (9), we

use

ℓ → ±∞.(9)

Pµ(ℓ) = (1 + b1/µ|ℓ|/˜ℓ)−µ,(10)

where b = [Γ(1 − µ/2)Γ(µ/2)]/Γ(µ), and we have re-

stricted to ℓ ≥ 0 since, as before, we can neglect the

sign of ℓ because the direction of the jump is assigned by

drawing an angle on (0,2π). One can generate a random

step-length ℓ by inverting (10):

ℓ =˜ℓ(ξ−1/µ

0

− 1)

b1/µ

.(11)

with ξ0being a uniform random variable on the unit in-

terval. Now the diffusion coefficient (6) is infinite, but

one can define a generalized diffusion coefficient in terms

of the scales˜ℓ and ˜ τ as [16, 17]

κµ=˜ℓµ/(2˜ τ).(12)

Therefore, in the case of the L´ evy flights, when fixing the

value of κµ, the space-scale parameter is:

˜ℓ = (2κµ˜ τ)1/µ= (2κµ/Rtot)1/µ.(13)

As we consider the particles to be point-like, the spa-

tial dynamics does not include any interaction between

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3

the bugs. The interaction is instead taken into account

through reproduction and death rates, which we assume

to be affected by competitive interactions.

If the birth and death rates of a particle are influenced

by the number of other bugs within a certain radius R,

one talks about a nonlocal interaction of finite range. In

the present paper we assume that the birth and death

rates of the i-th particle depend linearly on the number

of neighbors in the interaction range [4],

ri

ri

b= max?0,rb0− αNi

R

?,

?.

(14)

d= max?0,rd0+ βNi

Ris the number of particles which are at a dis-

tance smaller than R from particle i, the parameters rb0

and rd0are the zero-neighbor birth and death rates, and

the parameters α and β determine how ri

on the neighborhood. For positive values of α and β,

the more neighbors a particle has within the radius R,

the smaller is the probability for reproduction and the

larger is the probability that the bug does not survive,

which could arise from competition for resources. The

function max() enforces the positivity of the rates. The

periodicity of the domain is taken into account by using

the minimum image convention when calculating the dis-

tance dijbetween two particles i and j; to this aim, also

the condition R < L/2 must be fulfilled (we generally

choose R ≪ L).

If the birth and death rates of a particle are instead

influenced by all the other particles in the system, i.e.,

R

(15)

Here Ni

band ri

ddepend

ri

ri

b≡ rb= max{0,rb0− α[N(t) − 1]} ,

d≡ rd= max{0,rd0+ β[N(t) − 1]} ,

which are formally Eqs. (14), (15) with R = L (so that

Ni

R= N(t) − 1) then one talks about global interaction.

In the case the rates of the demographic events are the

same for all the particles and assume constant values,

(16)

(17)

ri

b≡ rb= rb0,ri

d≡ rd= rd0, (18)

which is equivalent to α = β = 0 in Eqs. (14), (15), bugs

are noninteracting.

In the following we discuss the Brownian and L´ evy bug

systems when particles do not influence each other and

when inter-particle interaction occurs, either global or

nonlocal. Although we formally maintain the parameter

β in Eqs. (15) and (17), in our numerical examples we

restrict to β = 0.

III.SIMPLE BUG MODELS WITH NO

INTERACTION

A.Noninteracting Brownian bugs

The simple Brownian bug model with no interaction,

i.e., when the birth and death rates of the particles are

given by Eq. (18), has been studied and discussed in var-

ious works [1, 2]. The ensemble average of the total pop-

ulation size follows

?N(t)? = N0exp[∆(t − t0)], (19)

independently of the diffusivity of the particles; it only

depends on the difference ∆ = rb−rd. If the birth rate is

larger than the death rate, ∆ > 0, the total population

generally explodes exponentially, though there is a finite

probability for extinction that depends on the initial size

of the population and decreases with increasing ∆. If

the death rate is larger than the birth rate, ∆ < 0 the

extinction of the population takes place with probabil-

ity 1. If birth and death are equally probable, ∆ = 0,

then the average over many realizations is ?N(t)? = N0

and the average lifetime is infinite. However, in single

realizations the fluctuations in the number of particles

are huge leading to fast extinction in some runs. In fact,

there exists a typical lifetime proportional to N0, defined

as the time for which the fluctuations become as large as

the mean value, beyond which the population is extinct

with probability close to 1 [1].

As a surprising effect, in the systems where the nonin-

teracting Brownian bugs undergo death and reproduction

with equal probabilities, spatial clustering of the particles

was observed in single realizations [1–3]. A typical time

evolution of such a system is illustrated in Fig. 1a. We

note that in all figures presenting the spatial configura-

tions of the bugs, we have divided the particles according

to their initial position into nine groups characterized by

different colors as in Fig. 1 at time t = 0; if reproduction

takes place, the newborn particle assumes the same color

as the parent. From Fig. 1a one can see that many small

clusters form some time after starting from a uniform

initial distribution. The occurrence of the clustering is

related to the fact that in the case of reproduction the

new particle is located at the same position as the par-

ent. Due to the fluctuations and irreversibility of death

the number of clusters decreases in time, until there will

be a single cluster consisting of particles coming from a

single ancestor. There are constant, spontaneous, short-

lived break-offs from the main cluster, which are always

located near it. The center of mass of such a cluster un-

dergoes a motion similar to that of a single bug [1] and

its linear width fluctuates with a typical value propor-

tional to√N0[1]. Furthermore, the larger the diffusion

coefficient κ, the wider is the cluster (notice that when

simulating the system numerically, if the diffusivity be-

comes so large that the jump lengths become comparable

to the system size, one needs to take a larger simulation

box). Finally, due to the fluctuations also the last cluster

disappears.

B. Noninteracting L´ evy bugs

In the case of noninteracting L´ evy bugs, the number of

particles still follows Eq. (19), independently of the L´ evy

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4

t = 0

(a)

0.5

1

y

t = 100

t = 1000

t = 10000

(b)

0 0.51

x

0

0.5

1

y

0.51

x

0.51

x

0.51

x

FIG. 1: Simple bug models with no interaction; spatial configuration of particles at different times t: (a) Brownian bugs with

κ = 10−6and (b) L´ evy bugs with κµ = 10−5and µ = 1. Reproduction and death occur with equal probability, rb= rd= 0.1,

and the initial number of particles is N0 = 1000. Particles are colored with the color their ancestors bear in the panel at t = 0.

index µ, and also the cluster formation observed in the

case of Brownian bugs takes place. Now, however, as par-

ticles can perform long jumps, there are also small clus-

ters continuously appearing and disappearing far from

the main clusters (Fig. 1b). The smaller the value of µ

the more anomalous the system, i.e., the larger is the

probability for long jumps and therefore there are more

flash-clusters. When the number of clusters has already

decreased to one, due to the long jumps and fluctuation

of the number of particles, new clusters that are placed

far from the central cluster can appear in the system also

for some time and often the disappearance of the main

cluster takes place whereas another new central cluster

appears somewhere else. As a result the center of mass

undergoes anomalous diffusion as single bugs do. The

value of the L´ evy index µ influences also the linear size

of the main clusters: the smaller is µ, the more compact

are the clusters, although also more particle jumps to

long distances occur. The influence of the value of κµ

is similar as in the case of Brownian bugs, i.e., a larger

value of the anomalous diffusion coefficient results in a

larger linear size of the clusters.

IV. GLOBAL INTERACTION

A. Formation of a cluster

Let us now investigate the behavior of the Brownian

and L´ evy bug systems in the case of global interaction,

i.e., birth and death rates of the particles are given by

Eqs. (16), (17). The time evolutions of the globally in-

teracting Brownian and L´ evy bug systems are illustrated

by Fig. 2a and 2b, respectively. In both systems we start

from N0= 500 particles uniformly distributed in the sim-

ulation area and choose for the parameters characteriz-

ing death and birth rates the following values: rb0= 1,

rd0 = 0.1, α = 0.02, β = 0. As in the noninteracting

case, the final state of the dynamics is complete extinc-

tion, since there is always a nonvanishing probability for

a fluctuation strong enough to produce that. However, if

the number of particles in the system is large this hap-

pens at very long times [18]. Then, there is a long-lived

quasistable state for which the average number of par-

ticles ?N(t)? can be estimated from the condition that

death and birth are equally probable, ri

there,

b= ri

d. From

?N(t)? =

∆0

α + β+ 1,(20)

where ∆0= rb0− rd0. We have restricted to parameter

values so that the max functions in Eqs. (16-17) do not

operate. Since we have chosen N0> ?N(t)? = 46 in Fig.

2, death is more probable at small times and the num-

ber of particles decreases rapidly. Approximately at time

t = 30 the number of particles has reached the value at

which death and birth become in average equally proba-

ble and after this time particle number fluctuates around

that value; parameters of the birth and the death rates

can be chosen so that these fluctuations are weak. At

this time small clusters start to form due to the repro-

ductive pair correlations. As in the case of noninteracting

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5

t = 0

(a)

0.5

1

y

t = 20

t = 50

t = 100

t = 500

(b)

00.51

x

0

0.5

1

y

0.51

x

0.51

x

0.51

x

0.51

x

FIG. 2: Globally interacting bug models; spatial configuration of particles at different times t: (a) Brownian bugs with κ = 10−5

and (b) L´ evy bugs with κµ = 10−4and µ = 1. The parameters in the reproduction and death rates are: rb0 = 1, rd0 = 0.1,

α = 0.02, β = 0. Particles are colored with the color their ancestors bear in the panel at t = 0.

bugs, fluctuations and irreversibility of death makes the

number of clusters to decrease in time, although now the

fluctuations of the particle content of the different clus-

ters are correlated to keep the total number close to the

value given by Eq. (20) and the process is faster. Finally

a single cluster consisting of particles coming from the

same ancestor remains (as stated before, it will also dis-

appear at very long times due to finite-size fluctuation

effects) though there are also now spontaneous short-

lived break-offs from the central cluster as in the case

of noninteracting bugs. The center of mass of such a

cluster is moving in space and its linear size is a fluctu-

ating quantity. The clustering of the globally interacting

particles was quantitatively discussed in Ref. [5] for the

one-dimensional Brownian bug system.

B.Fluctuations of the number of bugs

As indicated by Eq. (20), for given values of α and β,

the average number of particles in the system with global

interaction depends soley on the difference ∆0= rb0−rd0.

It is independent of the concrete values of rb0and rd0, as

well as of the value of κ or κµand µ; in fact, it does not

even depend on whether the system consists of Brown-

ian or L´ evy bugs. Nevertheless, fluctuations of the num-

ber of particles do indeed depend on the values of rb0

and rd0, even if the difference ∆0, and thus the average

number of particles, has the same value. To illustrate

this, let us calculate from the simulations time series the

probability distribution of the number of particles in the

globally interacting Brownian and L´ evy bug systems. As

can be seen from Fig. 3, for a given value of ∆0, larger

values of rb0and rd0lead to larger fluctuations. This is a

simple consequence of the Poisson character of the birth

and death events for which fluctuations in each of the

instantaneous rates are proportional to the mean rates.

For larger rates particle number distribution gets broader

implying that there is an enhanced probability that par-

ticle number becomes zero at some moment, after which

bugs become extinct (remember that what is in fact plot

in Fig. 3 is the numerical particle number distribution in

the long-lived metastable state before extinction). For

the present case with ∆0 = 0.9 and α = 0.02, β = 0,

rate values above the ones shown in Fig. 3 (i.e. rb0> 2,

rd0> 1.1) lead to observable extinction after some tenths

of thousands of steps. An ecological implication of this

could be the following: one can think of two colonies of

organisms of the same type, having both the same equi-

librium size determined for example by the size of the

living area. Now if in one of the systems the population

has no enemies and the natural death rate is low, but

in the other the death rate is higher due to the presence

of a predator, then the latter system will more proba-

bly go to extinction sooner due to the presence of larger

fluctuations.

C.The average cluster shape, cluster width, and

center of mass motion

Let us keep in the following α = 0.02, β = 0 and

rb0 = 1, rd0 = 0.1 [the same parameter values as in

Fig. 2 and in Fig. 3 for curve (a)] and study the behavior

of the cluster formed in the case of a system with global

interaction defined by Eqs. (16)-(17). As mentioned, even

after the transition from an uniform distribution of bugs

to a single cluster (and before eventual extinction at large

times), at some moment the system can consist actually

of more than one cluster. In such cases we define that

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0

0.05

0.1

0.15

0.2

20406080

p

N

(a)

(b)

(c)

FIG. 3: (Color online) Probability distribution of the number

of particles in globally interacting bug systems. The results

are numerically obtained from the time series of the particle

number in the very long-lived state before the fluctuations

lead the system to the extinction. For all the curves α =

0.02, β = 0, and the rate difference is ∆0 = rb0− rd0 = 0.9,

but the rates rb0, rd0 assume different values: (a) rb0 = 1,

rd0 = 0.1; (b) rb0 = 1.5, rd0 = 0.6; (c) rb0 = 2, rd0 = 1.1.

The overlapping curves correspond to Brownian and L´ evy bug

systems; the distributions do not depend on this nor on the

values of κ, κµ or µ in this globally interacting case.

all the particles in the system belong to the same cluster,

even though in the L´ evy case the distance between the

particles (subclusters) can be rather large. In order to

avoid the boundary effects as much as possible, in Figs. 4-

7 the linear size of the simulation area was taken as L =

1000 and to have enough statistics simulations were run

until t = 5 × 108.

Let us start by analyzing the average shape of the clus-

ter. The average cluster, ρ(x,y), is obtained setting at

each time the origin in the center of mass of the cluster

(distances under the periodic boundary conditions are

computed from the minimum image convention) and av-

eraging over a long time (after the transition from uni-

form distribution to one single cluster but before long-

time extinction). A one-dimensional cut of it (say across

x for y = 0, i.e., ρ(x) ≡ ρ(x,y = 0)) is shown in Figs. 4

and 5. For the case of Brownian bugs, the tail of the aver-

age cluster is approximately exponential. A pair distribu-

tion function, which should be related but not identical to

the average cluster discussed here, was analytically calcu-

lated in Ref. [14] for a globally interacting Brownian bug

model of our type but in which total extinction was for-

bidden. This quantity also displayed a fast decaying tail.

In the case of L´ evy bugs the tail of ρ(x) follows instead

a power law, ρ(x) ∼ x−(2+µ), see Fig. 4b, arising from

the long jumps. Note that, in the present case of circular

symmetry, the relation ρ(x,y)dxdy = R(r)(2π)−1drdθ

of ρ(x,y) with the radial density of the average cluster,

R(r), where r and θ are the polar coordinates centered

at the cluster center, implies ρ(x) = R(r = |x|)(2π|x|)−1,

10-5

10-3

10-1

10

103

105

-0.5-0.250

x

0.250.5

ρ

(a)

LB, µ = 0.5

1

1.5

BB

10-1

10

103

0.010.10.5

ρ

x

(b)

µ = 0.5

1

1.5

FIG. 4: (Color online) (a) ρ(x), the cross-section of the two-

dimensional particle density of the average cluster in semi-

log scale; comparison between the Brownian and L´ evy bug

systems; κ = 10−5, κµ = 10−5, rb0 = 1, rd0 = 0.1, α = 0.02,

β = 0. (b) The tails of ρ(x) in log-log scale in the case of

the L´ evy bug systems for different values of µ. Solid lines

correspond to fitting curves ∝ x−(2+µ).

so that the asymptotic behavior of the radial density is

R(r) ∼ r−(1+µ). This is the same asymptotic behavior

as the individual radial jumps in (8) and it is also the

asymptotic tail of the probability of displacement from

the original position of nonreproducing particles moving

by L´ evy flights [17]. We note also that, for κ = κµ, the

central part of ρ(x) is narrower and higher in the L´ evy

than in the Brownian bug system, and it is narrower and

higher the smaller the value of µ (see Fig. 4a). This is

a somehow counterintuitive effect of the L´ evy motion on

clusters, already commented in the noninteracting case:

increasing anomalous diffusion (smaller µ) induces larger

jumps and longer tails, but at small scales it acts as mak-

ing the cluster more compact.

The influence of the diffusivity is similar in both sys-

tems: the larger is the value of κ or κµthe more spread

is the average cluster (see Fig. 5).

one-dimensional case it was shown in Ref. [5] that clus-

For the Brownian

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7

ter width is essentially the distance associated to the

Brownian walk during the lifetime of a bug and their

descendants. Thus, the width increases as κ1/2. In the

L´ evy case, defining the distance associated to the walk

is more subtle, since higher moments of displacements

diverge.But the behavior of lower ones and dimen-

sional analysis indicate that typical displacements during

a lifetime scale as κ1/µ

µ , and then this should determine

the width of ρ(x) or R(r) (i.e., the spatial dependence

should occur only through the combinations [xκ−1/µ

[rκ−1/µ

µ

]). Imposing additionally that the total number of

particles in the average cluster in this global interaction

case does not depend on particle motion or distribution,

and it is thus independent on the value of κµ we have

R(r) = κ−1/µ

µ

F(rκ−1/µ

µ

), or

µ

] or

ρ(x) =

1

κ2/µ

µ

G

?

x

κ1/µ

µ

?

,(21)

with G(u) = F(u)/u. The analogous scaling form for the

average cluster in the Gaussian diffusion case is

ρ(x) =1

κG

?x

κ1/2

?

.(22)

The insets in Fig. 5 show the validity of these scaling

forms.

In addition to the average cluster shape, which gives

information on the cluster width, it is also interesting to

study the fluctuations of the cluster width in time. We

characterize cluster width at each instant of time by the

standard deviation of the particle positions with respect

to the center of mass of the cluster at that time. Then,

a probability density π(σ) is constructed from the values

of σ at different times. Figure 6 shows that in the case

of globally interacting Brownian bugs the distribution of

σ is short-tailed. In the case of globally interacting L´ evy

bugs, in contrast, the distribution for σ is characterized

by tails with a power law decay with exponent −(1+µ).

This means that in the latter case the cluster width can

undergo arbitrarily large fluctuations in time. We note

that the tails in π(σ) decay with the same exponent as the

radial density R(r) of the average cluster, thus suggest-

ing that the tails of the average cluster are produced by

the large fluctuations in the width of the instantaneous

clusters (which in fact include splitting events).

The individual motion of bugs drives the behavior of

the center of mass of the system. Figure 7 depicts the

probability density p(∆CM) of the jump lengths ∆CM

performed by the center of mass each time the particle

motion step is executed in the globally interacting Brow-

nian and L´ evy bug systems. For Brownian bugs it is

short-tailed. In fact, from the arguments in Ref. [5], the

center of mass motion of the cluster for globally interact-

ing Brownian bugs is characterized by a Brownian pro-

cess with the same diffusion coefficient as the individual

particles. In the case of L´ evy bugs the probability density

of the jump lengths of the center of mass is described by a

FIG. 5: (Color online) ρ(x), the cross-section of the two-

dimensional particle density of the average cluster in semi-

log scale for different values of diffusivity: (a) Brownian bugs

and (b) L´ evy bugs with µ = 1. Other parameters are as in

Figs. 2 and 4. The insets check the correctness of the scaling

forms (22) (with G(u) = κρ and u = x/κ1/2) and (21) (with

G(u) = κ2/µ

µ ρ and u = x/κ1/µ).

distribution with a power-law tail with exponent −(1+µ),

i.e., the center of mass of the cluster formed in the case

of globally interacting L´ evy bugs undergoes jumps that

follow asymptotically the same law as the single bugs,

Eq. (8), and as the radial tails of the average cluster.

This reflects the fact commented previously that, due to

the long jumps of the L´ evy bugs, additional clusters far

from the main one appear from time to time, strongly

displacing the center of mass of the system. Due to the

fluctuations it is even possible that the cluster that used

to be the main cluster disappears and a new main clus-

ter forms somewhere else. As a result the center of mass

motion undergoes the same type of superdiffusion as the

individual particles of the system.

Extending the arguments for the Brownian bugs [5]

(which were themselves adapted from the ones in [1]) to

the L´ evy case one can heuristically show that the dis-

tributions of σ and ∆CM are related. To this aim one

makes the approximation that the number of particles in

the system is constant, say N, instead of being constant

on average. The center of mass receives a positive contri-

Page 8

8

10-7

10-5

10-3

10-1

10-2

10-1

110

π(σ)

σ

LB, µ = 0.5

1

1.5

BB

FIG. 6: (Color online) Probability density π(σ) of the stan-

dard deviation σ of the particle positions with respect to the

center of mass of the cluster in the Brownian and L´ evy bug

systems; κ = 10−5, κµ = 10−5. The distribution is obtained

averaging over a long time. The curves corresponding to the

L´ evy bug systems are well fitted by ∝ σ−(1+µ)(not shown).

Other parameters are as in Figs. 2, 4, and 5.

10-8

10-6

10-4

10-2

1

10-5

10-4

10-3

10-2

∆CM

10-1

110

p

LB, µ = 0.5

1

1.5

BB

FIG. 7: (Color online) Probability density p(∆CM) of the

jump lengths of the center of mass in the Brownian and L´ evy

bug systems. Same parameter values as in Fig. 6. The tails

of the curves corresponding to L´ evy bugs are well fitted by

∝ ∆−(1+µ)

CM

(not shown).

bution from the new particles appearing (at location ? xi)

due to the reproduction between diffusion steps (say at

time ti), a negative contribution from the particles dis-

appearing during that time (say from position ? xjat time

tj), and the direct contribution from the L´ evy jumps?ℓk

of all particles present at the diffusion step:

?∆CM=

1

N

?

i∈B

? xi(ti) −1

N

?

j∈D

? xj(tj) +1

N

N

?

k=1

?ℓk. (23)

B and D denote the sets of particles that have been

(a)

0

0.5

1

y

(b)

(c)

0 0.51

x

0

0.5

1

y

(d)

0 0.51

x

FIG. 8: Interacting Brownian bug model with R = 0.1, rb0=

1, rd0 = 0.1 and α = 0.02, β = 0. Spatial configuration

of particles at time 45000 in systems with different diffusion

coefficients: (a) κ = 10−5, (b) κ = 2×10−5, (c) κ = 4×10−5,

(d) κ = 10−4. The initial configuration of particles is the

same as in Figs. 1 and 2 at time t = 0.

born or dead, respectively, between diffusion steps. The

two first terms can combined in a single one?S ≈

N−1?n

is the displacement between a pair of these particles, one

just born and the other just disappeared, sampled inside

the same cluster. Then the modulus of each σp should

be of the order of the cluster width σ, which fluctuates in

time with probability tails ruled by an exponent −(1+µ).

This contribution in Eq. (23) gives the motion of the cen-

ter of mass due to the birth and death processes. The

contribution from the individual particle jumps is in the

last term in (23), which is a sum of L´ evy jumps of param-

eter µ so that the tails of the probability density are char-

acterized by a decay with the same exponent −(1 + µ).

These heuristic arguments imply that the modulus ∆CM

will also be distributed with long tails characterized by

an exponent −(1 + µ), as observed.

p=1? σpby considering that the two sets have ap-

proximately the same number of particles, n. ? σp= ? xi−? xj

V.NONLOCAL INTERACTION

A.Formation of a periodic pattern

In Refs. [4, 6, 7] on the nonlocally interacting Brow-

nian and L´ evy bugs it was assumed that the birth and

death rates of the i-th particle are given by Eqs. (14),

(15). In the case of Brownian bugs, for small enough dif-

fusion coefficient and large enough ∆0, the occurrence of

a periodic pattern consisting of clusters that are arranged

Page 9

9

(a)

0

0.5

1

y

(b)

(c)

00.51

x

0

0.5

1

y

(d)

00.51

x

FIG. 9: Interacting L´ evy bug model with R = 0.1, rb0 = 1,

rd0= 0.1 and α = 0.02, β = 0 (same parameters as in Fig. 8

for Brownian bugs). The spatial configuration of particles

at time 45000 in systems with different generalized diffusion

coefficients and anomalous exponent: (a) κµ = 10−4, µ = 1;

(b) κµ = 10−3, µ = 1; (c) κµ = 10−4, µ = 1.5; (d) κµ =

5×10−5, µ = 1.5. The initial configuration of particles is the

same as in Figs. 1 and 2 at time t = 0.

in a hexagonal lattice was observed (see Fig. 8a-c) [4, 6].

For large values of the diffusion coefficient such periodic

pattern is replaced by a more homogeneous distribution

of particles (Fig. 8d). In the case of L´ evy bugs, since the

diffusion coefficient (6) is infinite, one could expect that

the spatial distribution will not reveal a periodic pattern;

however, as shown in Ref. [7], for proper parameters pe-

riodic cluster arrangements do indeed occur (see Fig. 9).

The reason for the divergence of the diffusion coefficient

in the L´ evy case is in the statistical weight of large jumps.

These large jumps have some influence on the character-

istics of the pattern formed, but the relevant structure is

ruled mainly by the interactions between particles. In the

L´ evy bug system however, at variance with the Brownian

case, even at small values of κµthere are many solitary

particles appearing for short time periods in the space be-

tween the periodically arranged clusters due to the large

jumps [7], c.f. Figs. 8a and 9a. However, the periodic-

ity of the pattern is of the order of R (the interaction

range) in both systems, being only slightly influenced by

κ or κµand µ, as demonstrated in Refs. [4, 7] through a

mean-field theory calculation.

In Ref. [7] also the two-dimensional particle density of

the average cluster, obtained by setting the origin at the

center of mass of each cluster forming the periodic pat-

tern and averaging over all the clusters in the simulation

area and over time, was studied. In both, Brownian and

L´ evy bug systems the central part of the average cluster,

where most of the particles are concentrated, was well

fitted by a Gaussian function, but the way the particle

density decreases when moving away from the center of

mass of the cluster is rather different. In the Brownian

case a Gaussian decay provides a good approximation,

whereas in the L´ evy case it is close to exponential. The

comparison with the systems with global interaction, dis-

cussed in Sec. IVC, reveals therefore that the interaction

range R turns the exponential decay into Gaussian and

the power law decay into exponential.

For a given value of diffusion coefficient, there exists

a critical value of ∆0 below which the system gets ex-

tinct, independently of α [4]. Above this critical value,

for every α the increase of ∆0results in the increase of

the average number of particles, but the pattern forma-

tion is not much influenced. The latter is, however, true

solely if ∆0increases through the increase of rb0and the

death rate is low. Namely, as in the case of global in-

teraction discussed in Sec. IVB, an increase of the death

rate, though accompanied by a compensating increase

of birth rate, leads to larger fluctuations in the particle

number. In numerical simulations we have observed that

the larger are the fluctuations in the number of particles,

the more difficult is the formation of the periodic pattern,

and finally the particles do not arrange in the periodic

pattern but in random clusters (see also Ref. [14]). This

effect may in fact make difficult to observe the periodic

clustering phenomenon in real competitive biological sys-

tems.

In the following we keep for the parameters in the birth

and death rate the same values as in the case of global

interaction, i.e., rb0= 1, rd0= 0.1, α = 0.02, β = 0. For

these parameter values the number of particles fluctuates

only weakly around the mean value. Differently from the

case of global interaction, now the average number of par-

ticles in the system is influenced not only by the birth and

death rates, but also by the diffusion, i.e., in the case of

Brownian bugs by κ and in the case of L´ evy bugs by κµ

and µ, see Fig. 10. The smaller is κ, κµor µ, the larger

the particle number. At the same time Figs. 8 and 9 in-

dicate that by decreasing κ or κµthe linear width of the

clusters becomes smaller, the particle density in the clus-

ters higher, and the density between the clusters lower

(c.f. Sec. IVC and see also Ref. [7]). Somehow counter-

intuitively, the effect of decreasing µ seems to have the

same effects, as commented above for the noninteracting

and global cases. Furthermore, the value of κ or κµand

µ seems to weakly influence the number of clusters in the

system: In Figs. and 8 and 9 smaller values lead to larger

number of clusters. This observation is not explained by

the linear instability analysis of [4, 7].

B. Mixing of different families

It is interesting to study the evolution of the system

also regarding the disappearance or survival of the dif-

ferent groups, by dividing initially the particles into dif-

ferent families and following their descent. In the case

Page 10

10

1400

(a)

1800

2200

2600

0

2⋅10-54⋅10-56⋅10-58⋅10-5

κ

10-4

〈 N 〉

2000

(b)

2400

2800

3200

11.21.41.6 1.8

〈 N 〉

µ

κµ = 10-5

5⋅10-5

10-4

FIG. 10: (Color online) a): Average number of particles ver-

sus diffusion coefficient in the system with Gaussian jumps.

b): Average number of particles versus anomalous exponent

µ in the system with L´ evy jumps for various values of the

anomalous diffusion coefficient. Other parameters as in Figs.

8 and 9.

of nonlocally interacting Brownian particles, a very low

diffusion coefficient leads to the situation in which after

cluster formation the inter-cluster travel is very rare be-

cause the particles are not capable to make the jumps

from one cluster to another one, and it is also very un-

likely to arrive to the next cluster doing a multistep ran-

dom walk because death is very probable between the

clusters. Therefore, in the case of very low diffusion dif-

ferent families would remain inside their initial clusters.

If one assumes that initially each particle represents a dif-

ferent family, then only inter-cluster competition occurs

and the final number of families is equal to the number of

clusters. If instead initially particles are assigned to fam-

ilies according to large areas of initial positions (larger

than typical cluster size as done in Figs. 1 and 2 at time

t = 0), there is no family competition internal to the clus-

ters, most families survive and the clusters coming from

different families occupy approximately the territory of

the ancestors even after a long time, as can be seen from

Fig. 8a. In that case, the travel of a cluster to a new

territory away from the other clusters of the same fam-

ily can take place due to the diffusion of the cluster as a

whole during the clusters arrangement into the periodic

pattern. For larger values of κ the inter-cluster travel is

possible which leads to the conquering of new territories,

i.e., particles can be found in a region where their ances-

tors were not from, Fig. 8b. The effect is larger for larger

κ and leads to the disappearance of some families, as

can be seen from Fig. 8c. Finally, for increased diffusion,

intra-cluster competition will force all surviving particles

to be from a single family (in fact, from a single ances-

tor); which family (ancestor) wins is a random event.

The process is faster for larger diffusion. Increasing the

diffusivity further even the periodic pattern disappears,

Fig. 8d.

Figure 9 illustrates the family mixing for nonlocally

interacting L´ evy bugs. In this case the inter-cluster trav-

eling is supported by the long jumps. Differently from

the case of Brownian bugs, now the particles can reach

not only the next neighboring cluster but also clusters

far away. Consequently, a cluster originally consisting of

particles coming from one ancestor can after some time

turn into a cluster consisting of particles coming from

different families placed initially far away. Thus, intra-

cluster particle competition becomes soon competition

between families, and even if the diffusivity of the parti-

cles is very low, at the end the L´ evy bug system would

consist of particles coming from one or just a few ances-

tors. As in the case of Brownian bugs, the process of

the disappearance of families is faster the greater is the

generalized diffusion coefficient.

Besides the diffusion of a cluster as a whole during the

formation of the periodic pattern and the conquering of

new territories through the migration to and survival in

another cluster, the mixing of clusters from different fam-

ilies can take place also due to the appearance of a new

cluster if in the periodic pattern there is a dislocation.

In the case of Brownian bugs the new cluster is formed

through the splitting of an old cluster. In the case of

L´ evy bugs, instead, the new cluster can appear also far

from the original territory.

VI. CONCLUSIONS AND OUTLOOK

We have presented some detailed properties of interact-

ing particle systems in which the individuals are Brown-

ian or L´ evy random walkers which interact in a competi-

tive manner. We have seen strong differences between the

globally and the finite-range nonlocally interacting sys-

tems. In the systems with global interaction the spatial

distribution of the particles becomes tied to the type of

diffusion, Brownian or L´ evy. Typical configurations con-

sist of a single or a few clusters for both types of motion.

For the L´ evy bug systems long tails appear in the mean

cluster shape and in probability distributions of cluster

width and of jumps of the center of mass. For Brownian

Page 11

11

bug systems these quantities appear to be much shorter

ranged. This is qualitatively also the situation in the non-

interacting case, although then the effects of the particle-

number fluctuations are much stronger. Under non-local

finite-range interactions the situation is rather different.

First, single cluster configurations are generally replaced

by periodic patterns with periodicity set by the interac-

tion range R. Motion of individual clusters is severely

restricted by the presence of the neighboring clusters. In

addition, the natural spatial cut-off introduced by the in-

teraction range R seems to limit the influence of the long

L´ evy jumps, so that measures of spatial cluster shape

do not generally exhibit power laws, making spatial con-

figurations under both types of diffusion more similar.

Mixing of families and their competition is nevertheless

greatly influenced by the type of motion. This suggests

that it would be interesting to consider the influence of

different types of diffusion into competitive genetic mix-

ing processes [19].

Obtaining analytic understanding in this type of inter-

acting systems is difficult, but at least the nature of the

instability leading to pattern formation and its relevant

spatial scale have been clarified by using partial integro-

differential equation descriptions of the mean field type

[4, 7], which are useful in broader contexts [20–22]. How-

ever, from previous work in the Gaussian case [5, 6, 23], it

is known that quantities such as cluster width and struc-

ture or transition thresholds strongly depend on particle-

number fluctuations. Thus, obtaining additional results

from differential equation approaches would need the in-

clusion of effective multiplicative noise terms [24] or focus

on statistical quantities such as pair correlation functions

[2, 14, 25].

Acknowledgments

This work has been supported by the targeted financ-

ing project SF0690030s09, Estonian Science Foundation

through grant no.7466, by the Balearic Government

(E.H.), and by Spanish MICINN and FEDER through

project FISICOS (FIS2007-60327).

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