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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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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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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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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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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