PHYSICAL REVIEW E 72, 011912 共2005兲

Phase transitions in multiplicative competitive processes
Hideaki Shimazaki
Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

Ernst Niebur
Department of Neuroscience, Zanvyl Krieger Mind/Brain Institute, School of Medicine, Johns Hopkins University,
Baltimore, Maryland 21218, USA
共Received 4 April 2003; revised manuscript received 23 March 2005; published 20 July 2005兲
We introduce a discrete multiplicative process as a generic model of competition. Players with different
abilities successively join the game and compete for finite resources. Emergence of dominant players andevolutionary development occur as a phase transition. The competitive dynamics underlying this transition isunderstood from a formal analogy to statistical mechanics. The theory is applicable to bacterial competition,
predicting novel population dynamics near criticality.
DOI: 10.1103/PhysRevE.72.011912

PACS number共s兲: 87.23.Kg, 05.45.⫺a, 03.75.Nt

Competition occurs when two or more players such asorganisms, individuals, or companies strive for common but
limited resources. It plays a significant role in biological andsocial activities, and is the basis of evolution. Most naturalcompetition processes allow the introduction of new players,which is a hallmark of an open, nonequilibrium system. In
this contribution, we introduce an irreversible discrete multiplicative process with normalization at each time step as a
generic model of competition. Players with different abilities
successively join the game and compete for finite resources.
The model shows macroscopically observable changes in its
behavior; at a singularity in the statistical distribution of theplayers’ abilities, certain players become dominant over allothers. The emergence of dominant players and the evolutionary development of the system occur as a transition fromstationary to nonstationary state of the multiplicative process. We analyze the phase transition in the mathematicalframework of Bose-Einstein condensation 共BEC兲, although,of course, systems modeled are classical and not quantum
mechanical. The same approach has been applied successfully to models of complex networks 关1兴 and ecosystems 关2兴that behave analogously to a Bose gas. We show that thisapproach is applicable to bacterial competition, providing
surprising insights and predictions to their dynamics.
Before we present the model, we first introduce a generalframework for how our multiplicative competition model isrelated to a statistical mechanics concept. Let ␸共⑀兲 be a function that satisfies the following conditions for an arbitrary C2
density function g共⑀兲共艌0兲 defined on ⑀ 苸 关0 , ⑀max兴:冕冕冕d⑀g共⑀兲 = 1,共2兲d⑀g共⑀兲e−␤⑀␸共⑀兲 = M .

共3兲

All terms on the right-hand sides of these equations and ␤ in
Eq. 共3兲 are positive constants. Then ␸共⑀兲 is given byy0,
1 − e−␤⑀−␣

共4兲

where e−␣ = m / M. For, dividing Eq. 共2兲 by y 0, then subtracting Eq. 共1兲, yields

冕d⑀g共⑀兲

␸共⑀兲 − y 0
= N,
y0

共5兲

where N = m / y 0. Multiplying Eq. 共3兲 by N / M and subtracting
Eq. 共5兲 from the result, the fundamental lemma of the calculus of variation then yields Eq. 共4兲. Hence, the so-called occupation number in Eq. 共5兲,n共⑀兲 =

1␸共⑀兲 − y 0= ␤⑀+␣
,
y0
e
−1

共6兲

becomes the Bose distribution. From Eq. 共4兲, such a function
␸共⑀兲 may be obtained from the sum of a geometric progression with ratio e−␤⑀−␣ ⬍ 1. This motivates the analysis of thefollowing multiplicative process.
The competition we introduce is defined by three conditions at each time step. 共i兲 Players compete for a fixed total
amount of resources. 共ii兲 The resource gained by a player is
proportional to the player’s innate ability and to its resource
gained at the previous time step. 共iii兲 New players join thegame, each with the same initial resources. The only exception is the first player 共pioneer兲, who starts the game with all
the resources available. These rules are summarized in a
simple multiplicative process

共1兲d⑀g共⑀兲␸共⑀兲 = m + y 0 ,1539-3755/2005/72共1兲/011912共4兲/$23.00

␸共⑀兲 =

y i共t + 1兲 = aib共t兲y i共t兲,

共7兲

where y i共t兲 is the gain of the ith player at time t and ai is its
ability, a positive and time-independent random variable chosen from a distribution ␳共a兲. The term b共t兲 is a normalization
factor, which models the limited resources. For simplicity,
we first assume that l, the number of newly introduced players at every time step, is 1. This means we consider t + 1
simultaneous equations 共i = 0 , . . . , t兲 at time t. The normalization factor is then defined by b共t兲 = m / M t, where M t is givenby011912-1©2005 The American Physical Society

PHYSICAL REVIEW E 72, 011912 共2005兲

H. SHIMAZAKI AND E. NIEBURt

M t = 兺 a j y j共t兲.

共8兲

j=0

This assures that the amount of resources distributed amongt players at time t is m for t ⬎ 0. The initial value of a new
player is y i共i兲 = y 0, except for the pioneer whose initial value
is y 0共0兲 = m + y 0. Therefore, the total resources distributed
among all players are limited to m + y 0 at every time step.
Due to the normalization, we let a 苸 关amin , 1兴, where amin
⬎ 0, without loss of generality.
We now consider the time evolution of players except forthe pioneer. The gain of the ith player at time t is given by
y i共t兲 = ⍀i共t兲y 0, where

冉兿 冊
t−1

⍀i共t兲 = at−iit⬘=ib共t⬘兲 .

共9兲The dynamics given by ⍀i共t兲 shows distinct phases determined by the shape of the ability distribution. To see this, weparametrize ␳共a兲 by the inverse temperature ␤ 共=1 / T兲
through a = e−␤⑀. The random variable ⑀ 苸 关0 , ⑀max兴 is chosenfrom g共⑀兲, a state density function which now defines thesystem. We now systematically change ␳共a兲关=g共⑀兲兩d⑀ / da兩兴by fixing g共⑀兲 and changing ␤. If g共⑀兲 is a monotonicallyincreasing function, which we will assume, high temperatures mean that the distribution of the players’ abilities isrestricted to a small range—i.e., all players have relatively
similar abilities—while low temperature is synonymous withoccasional appearance of superior players relative to others.
From Eq. 共9兲, we haveln ⍀i = 共t − i兲兵− ␤⑀i − 具ln b典其,共10兲

b共t 兲 = 兺tt−1
ln b共t⬘兲 ⬃ 共t − i兲具ln b典. By assuming
using ln ⌸tt−1
⬘=i ⬘
⬘=i

stationarity of M t, discussed below, we let its time average be
具ln b典 = − ␣ ,共11兲

where ␣ is a time-independent constant. From Eqs. 共10兲 and
共11兲, we obtain y i共t兲 = e共−␤⑀i−␣兲共t−i兲y 0. Given e−␤⑀i−␣ ⬍ 1, thecumulative gain ␸t共⑀i兲 = 兺tt⬘=iy i共t⬘兲 converges to ␸共⑀i兲 in Eq.
共4兲. Hence, the normalization of ␸t共⑀i兲 by the initial value y 0,nt共⑀i兲 = 兵␸t共⑀i兲 − y 0其 / y 0, approaches the Bose distribution in
the thermodynamic limit 共t → ⬁兲. Indeed, in this limit,
lim M t =

t→⬁

冕

⬁

d⑀g共⑀兲 兺 e−␤⑀y共t⬘兲,

共12兲

t⬘=0

where 兺t⬁⬘=0y共t⬘兲 = ␸共⑀兲. As the assumption of stationarity
yields limt→⬁ ln b共t兲 = 具ln b典, we have limt→⬁ m / M t = e−␣from Eq. 共11兲. By substituting this into Eq. 共12兲, we obtain aself-consistent equation

冕1d⑀g共⑀兲 ␤⑀+␣
= N,
e
−1

共13兲

where N = m / y 0. Note that we can generalize Eq. 共7兲 suchthat we allow l 共艌1兲 new players to join the competition
with initial value y 0 at each step. From a similar argument,
we obtain Eq. 共13兲, where N = m / y 0l. Thus the deduced form,

FIG. 1. 共a兲 Normalized cumulative gain nt共⑀i兲 for m = 5 and T⬍ Tc 共left兲, T ⬇ Tc 共center兲, and T ⬎ Tc 共right兲. The solid line is theBose distribution with ␣ from Eq. 共11兲. The plus sign indicates apioneer. Plots of the last ten entrants were excluded. 共b兲 Numericalcalculation of 兩␣兩 with ␣ from Eq. 共11兲, averaged over 500 trials.
Symbols ⫹, 䊊, and ⫻ indicate m = 5, 10, and 20. Change in the signof ␣ indicates the transition. Triangles on the abscissa are the transition temperatures from Eq. 共17兲; Tc ⬇ 3.33, 6.67, and 13.3. Shownin inset is the dependence of Tc on exponent ␴ for m = 20 共dots兲 andthe analytical solution from Eq. 共17兲, solid line. 共c兲 Cumulative
occupation by the most capable player was calculated from␸t共⑀min兲 / 兺 j␸t共⑀ j兲 at t = 200. Symbols as in 共b兲. Dashed line, occupation by pioneer for m = 20. In all simulations, y 0 = 1, l = 1, ␴ = 3,
and ⑀max = 1.

Eq. 共13兲, of the competition under stationarity assumption
shows a formal analogy to those obtained for a quantum gas.
The normalization factor ␣ plays the role of keeping the total
resources gained by all players constant at every time step, in
analogy to the chemical potential of a quantum gas which is
introduced for the conservation of particle number. According to this reasoning, condensation of resources to a single
player analogous to BEC is expected at low temperature
where ␣ vanishes.
To study this prediction, we simulate the multiplicative
process, Eq. 共7兲, adopting a standard density functiong共⑀兲 = C␴⑀␴−1 ,共14兲␴where C␴ = ␴ / ⑀max
共␴ ⬎ 1兲. We find two distinct phases for the

distribution of the normalized cumulative gain nt共⑀i兲
= 兵␸t共⑀i兲 − y 0其 / y 0. At high T, nt共⑀i兲 obeys the Bose distribution 关Fig. 1共a兲兴. At low T, the coordinated distribution breaks
down: players with low energy 共high ability兲 dominate alarge fraction of the cumulative gain. This condensate exists
only below a critical point ␣ = 0. For, if ␣ is negative, ␸共⑀i兲
of the player with ⑀i ⬍ −␣ / ␤ does not converge. Its normalized cumulative gain cannot be incorporated into the integral
in Eq. 共13兲; rather, it has to be added as an extra term. The

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PHYSICAL REVIEW E 72, 011912 共2005兲

PHASE TRANSITIONS IN MULTIPLICATIVE…

critical temperature for this transition can be predicted in theusual way: Change of variables y = ␤⑀ and use of ␣ = 0 in Eq.
共13兲 yields
N=␴
共␤⑀max兲␴

冕

␤⑀max

0

y ␴−1
dy.
ey − 1

共15兲

By approximating the upper limit of integral in Eq. 共15兲 byinfinity, the critical temperature Tc共=1 / ␤c兲 is given by
−1
−1/␴
T共1兲
,
c ⬃ ⑀max兵N ⌫共␴ + 1兲␨共␴兲其

共16兲

for N Ⰶ 1. When N Ⰷ 1, this approximation is not valid because of the high critical temperature; instead, from ey ⯝ 1+ y and Eq. 共15兲, one obtains−1
T共2兲
c ⬃ ⑀max共1 − ␴ 兲N.

共17兲

The transition can be seen in the occupation by the mostcapable player, defined by ␸t共⑀min兲 / 兺 j␸t共⑀ j兲. As T decreases
below Tc, its occupation dramatically increases, supporting
the prediction 关Fig. 1共c兲兴.
We verified the existence of a BEC analog in a discrete
multiplicative process, as was shown in a continuous model关1兴. However, we emphasize that, as a matter of principle,
our classical dynamical system is not equivalent to a quantum gas. The most important difference is based on the following observation. The time evolution of each player’s gain
is different above and below the predicted Tc. Above Tc, thegains of all players monotonically decrease. Below Tc, not allof them show monotonic behavior, and the competitive dynamics is disordered. The observed nonequilibrium phasetransition from the ordered to disordered state occurs as a
violation of the stationarity in weighted mean ability,
M t / 共m + y 0兲. It is stationary if the gain of all players monotonically decreases to zero, which allows the argument below
Eq. 共10兲. Otherwise, if the gain of one player rises to dominate the resources, the weighted mean ability approaches theability of this one dominant player. Then, due to the replacement of the dominant player upon the entrance of a player
with higher ability, we observe an irreversible increase of theweighted mean ability, indicating that the system is now
evolving. Thus dominance and evolution are aspects of nonstationary dynamics. We emphasize that the phase transition
yielding evolution does not happen in equilibrium systems.
We now consider application of the theory to the competition of clonal strains of asexual Escherichia coli serially
propagated on glucose-limited medium. The population dynamics is most suitably described by a stochastic branchingprocess with mutation and selection. Consider the ith strain
with fitness ai, mutation rate ␩, and population size y i共t兲. Let
y i共t + 1兲 关i = 1 , . . . , Q共t兲兴 be given by a Poisson distribution
Q共t兲
aiy i共t兲. Here Q共t兲 is thewith mean ␭i = m̃共1 − ␩兲aiy i共t兲 / 兺i=0
number of mutants that were generated and that survived theinitial step since the process started. The number of mutants
produced at time t, Q共t + 1兲 − Q共t兲, is drawn from a Poisson
Q共t兲
distribution with mean l = m̃兺Q共t兲
j=0 ␩a j y j共t兲 / 兺i=0 ai y i共t兲. Notethat the average total number of cells at time t is fixed to
Q共t兲
␭i + l = m̃ due to the normalization factor in the above兺i=0
equations.

FIG. 2. A stochastic branching process with mutation and selection described in the text approaches a critical state by natural selection. Average total population is m̃ = 100. Simulation started with
100 unique strains whose fitness is drawn from a fitness distribution
with T = 1, ␴ = 3 / 2, and ⑀max = 1. The ith strain produces mutants
共mutation rate ␩ = 0.01兲 whose fitness is drawn from a fitness distribution with Ti given by Eq. 共19兲 共␥ = 2兲. 共a兲 Average temperatureof the system stays near the transition point Tc = 33. In early stages,
the average temperature increases, indicating initial adaptive evolution. 共b兲 Number of existing strains at each steps 共diversity兲 decreases during initial adaptive evolution. However, further dominance by a few species is suppressed as the system approaches near
criticality.

It is clear that our process, Eq. 共7兲, is a deterministic
approximation of this stochastic population dynamics. A
monotonically decreasing fitness distribution ␳共a兲 should be
used because most mutations are likely to be deleterious 关3兴.
We thus decided to use the same ␳共a兲 used in the analysis of
the deterministic model 关and the state density Eq. 共14兲兴 because it satisfies this basic tenet. Our results do not depend,
however, on the precise form of the state density; other parametrizations of the fitness distribution of ␳共a兲 such as theBeta distribution defined on a 苸 关0 , 1兴 yield essentially thesame results 共not shown兲.
Routes to adaptive evolution. A strong prediction of thetheory is the existence of a singular point on the emergenceof evolution. We observed the transition from stationary to
nonstationary state in the numerical simulation of the stochastic process by decreasing the temperature T. The transition point is predictable from the critical temperature Tc obtained by the deterministic theory 关Eq. 共17兲兴. Above Tc,
dominance by a capable player 共strain兲 appears. Below Tc,
the dynamics are governed by the random drift of dominantstrains. The random fluctuation of dominant strains is themost striking difference from dynamics of the deterministic
model.
Another route to generate an evolutionary development is
to increase Tc by fixing T. The critical temperature given by
Eq. 共17兲 is proportional to N, which is related to mutation
rate through ␩ = l / 共m + l兲 = 共N + 1兲−1. We thus obtainTc ⬃ ⑀max共1 − ␴−1兲共␩−1 − 1兲.

共18兲

Given T, decreasing ␩ increases Tc such that T ⬍ Tc isachieved. A sufficiently low mutation rate not only prevents
deleterious offspring but is necessary for adaptive evolution

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H. SHIMAZAKI AND E. NIEBUR

PHYSICAL REVIEW E 72, 011912 共2005兲

itself. The deleterious role of high mutation rate to adaptive
evolution termed “clonal interference” was observed in experiments of E. coli competition 关4兴 and theoretically investigated elsewhere 关5兴.
Approach to the transition point by natural selection. An
important question is whether T is comparable to Tc undernatural conditions. Evolutionary progress observed in laboratory experiments of E. coli competition shows alteration oflong periods of relative stasis with short bursts of rapidchange caused by rare beneficial mutations 关6兴. Therefore,
empirically observed dynamics is consistent with model behavior at T ⬍ Tc in our construct at least during early stagesof competition. However, we argue that there exists a biologically plausible scenario in which T may eventually become comparable to Tc by internally tuning the parameter关7,8兴.
Instead of using a common fitness distribution for all
strains, it is physiologically plausible to assume that eachstrain has its unique fitness distribution. We assume that thefitness of mutants originating from the ith strain of fitness ai
is drawn from a fitness distribution characterized by the temperature Ti = 1 / ␤i. Since most mutants are deleterious, theaverage fitness produced with ␤i should be less than ai 关i.e.,
具a典 = e−␤i具␧典 ⬍ ai, where 具␧典 = ␧max␴ / 共␴ + 1兲 for the state density Eq. 共14兲兴. Henceforth, the inverse temperature ␤i isgiven by␤i = − ␥具␧典−1ln ai ,

关1兴 G. Bianconi and A. L. Barabasi, Phys. Rev. Lett. 86, 5632
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关2兴 I. Volkov, J. R. Banavar, and A. Maritan, Phys. Rev. Lett. 92,
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关3兴 R. A. Fisher, The Genetical Theory of Natural Selection 共Dover, New York, 1958兲.
关4兴 J. A. G. M. de Visser, C. W. Zeyl, P. J. Gerrish, J. L. Blanchard, and R. E. Lenski, Science 283, 404 共1999兲.

共19兲

where ␥ ⬎ 1 to satisfy 具a典 ⬍ ai.
At the beginning of adaptive evolution, strains withhigher fitness are chosen by natural selection. Dominance bystrains with high fitness increases the average temperature of
the population. Suppose that adaptive evolution achieves aneutral condition T ⬎ Tc. It is then by chance whether or nota certain strain is picked up. Since a dominant strain is prone
to produce mutants inferior to the dominant strain itself,
those deleterious strains are likely to be picked up and theaverage temperature decreases. There are ever-going cycles
of adaptive evolution, neutral state, then collapse of thedominance 共Fig. 2兲. The advantage of the dynamics that approaches criticality is rather clear. It allows initial adaptive
evolution, permanently eliminating unfavorable genotypes,
but then significantly slowing down or preventing further
evolution and dominance: close to the critical state, strains
can coexist for a substantial period of time. Diversity introduced by the dynamics near criticality is clearly advantageous for the whole ecosystem, which is exposed to global
environmental changes. We thus conjecture that this strategy
might be taken by some haploid species.
We thank H.G. Schuster and S. Shinomoto for very helpful comments. This work was supported by the Murata Overseas Scholarship Foundation 共H.S.兲 and NIH Grant No. R01
NS43188-01A1 共E.N.兲.

关5兴 P. J. Gerrish and R. E. Lenski, Genetica 共The Hague, Neth.兲
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