Coevolution of competing systems: local cooperation and global inhibition

0.52 MB·English

by  Jose M. Albornoz

EPJ manuscript No.
(will be inserted by the editor)
Coevolution of competing systems: local cooperation and global
inhibition
Jos´e M. Albornoz1,2 and Antonio Parravano2
1 Departamento de Electro´nica y Comunicaciones, Facultad de Ingenier´ıa, Universidad de Los Andes, La Hechicera, M´erida,
M´erida 5251, Venezuela.
2 Centro de F´ısica Fundamental, Facultad de Ciencias, Universidad deLos Andes,Apartado Postal 26, La Hechicera, M´erida,
0
M´erida 5251, Venezuela.
1
0
2 Received: date/ Revised version: date

n
a Abstract. Usingasetofheterogeneouscompetingsystemswithintra-systemcooperationandinter-system
J aggression, weshow howthecoevolution ofthesystem parameters (degreeof organization and conditions
9 for aggression) depends on the rate of supply of resources S˙. The model consists of a number of units
1 groupedintosystemsthatcompetefortheresourceS;withineachsystemseveralunitscanbeaggregated
into cooperative arrangements whose size is a measure of the degree of organization in the system. Ag-

] gression takes place when the systems release inhibitors that impair the performance of other systems.
O
Using a mean field approximation we show that i) even in the case of identical systems there are stable
A inhomogeneoussolutions,ii)asystemsteadilyproducinginhibitorsneedslargeperturbationstoleavethis
regime,andiii)aggression maygivecomparativeadvantages.Adiscretemodelisusedinordertoexamine
.
n how the particular configuration of the units within a system determines its performance in the presence
i ofaggression.Wefindthatfull-scale,onesidedaggressionisonlyprofitableforless-organizedsystems,and
l
n that systems with a mixture of degrees of organization exhibit robustness against aggression. By using a
[ geneticalgorithm wefindthat,intermsofthefull-occupation resource supplyrateS˙F,thecoevolution of
2 the set of systems displays the following behavior: i) for S˙ < S˙F/10 aggressions are irrelevant and most
v systemsexhibitahighdegreeoforganization; ii)ForS˙F/10<S˙ <S˙F/3aggressions arefrequent,making
1 systems with a low degree of organization competitive; iii) for S˙F/3 < S˙ < S˙F/2 the systems display
4 global evolutive transitions between periods of calm (few aggressions and high degree of organization)
7 and periods of belligerence (frequent aggressions and low degree of organization); iv) for S˙ > S˙F/2 the
2 periodsofaggressionbecomesprogressivelyrarerandshorter.Finally,whenS˙ approachesS˙F theselection
. pressure on the cooperativity and the aggression between systems disappears. This kind of model can be
4
useful to analyse the interplay of the cooperation/competition processes that can be found in some so-
0
cial, economic, ecological and biochemical systems; as an illustration we refer tothe competition between
9
0 drug-selling gangs.
:
v PACS. 89.65.-s Social and economic systems – 87.23.Kg Dynamics of evolution – 87.18.-h Biological
i
X complexity
r
a
1 Introduction to social norms based on a n-person Prisoner’s dilemma.
In this game the players can defect (getting a payoff and
There are many situations in which the global perfor- hurting each of the other players) but can be punished
mance of a set of competing systems, measured in some if seen by another player. The evolution of the player’s
way, is below its maximum possible value due to compe- strategy (the boldness and the vengefulness) is driven by
tition among the systems. Generally, in complex systems the selectionof the strategiesgiving the best scores;how-
the competing components use different strategies that ever, the maximum global score (no one defects) is rarely
may consume part of the available resources in order to observed.
reduce the performance of the others. In these situations There are several instances in which there is an inter-
thereisaglobalcostbutthecollateraleffectistofavorthe play between risk and profit. In some cases, the balance
emergence of new strategies that tend to increase the di- betweenriskandprofitdependsonthedegreeoforganiza-
versityandcomplexityinthesesystems.Thisisthecasein tion within a competing system: a greaterdegree of orga-
some social and economic systems [1,2,3,4,5] and in cer- nizationusuallyresultsinagreaterprofitbutontheother
tain ecological and biochemical systems [6,7,8,9,10]. For handitmayalsoincreasepotentiallossesincaseofanag-
example,Axelrod[1,2]considersanevolutionaryapproach gression from a competing system. Consider for instance
2 Jos´e M. Albornoz, Antonio Parravano: Coevolution of competing systems: local cooperation and global inhibition
the various drug-selling gangs that coexist in the poorest sionis modelledby the releaseofinhibitors,whichreduce
neighborhoods of a large city [11,12]. The gangs compete theefficiencyby blockingthe unitduringaperiodoftime
to sell drugs to a limited number of customers (the re- as in the case of the occupation by an inhibitor of the
source).The customersprefer to buy in the safest,easiest active site where a substrate is bound to an enzyme [14].
andquickestpossiblewayandthegangstrytosellasmuch Theperformance(i.e.theproductionrate)ofasystemde-
as they can by using strategies to attract the largest pos- pends on its comparative ability to acquire the resources,
siblefractionoftheconsumersinthe city.Themodelpre- whichinturndependsontheconfigurationoftheunitsin
sentedhereconsiderstwotypesofstrategies:intra-system thesystem.Inabsenceofinhibitors,theproductionrateis
cooperationand inter-systemaggression.In the gangsex- greater in systems where units cooperate than in systems
ample, the degree of intra-gang cooperation can be asso- where units work in isolation. However, when inhibitors
ciatedtothe variouswaysinwhichthegangmembersare arepresent,the number ofunits blockedper inhibitor de-
grouped.Thedealerscandistributethemselvesinasmany pends on the working configuration of the units. In this
intersections as dealersin the gang,howeverbusy corners way, the effect of an aggression will be more damaging
(as well as shopping centers) where several dealers oper- to systems with a greater degree of organization (greater
ate at the same time offer an environment in which cus- number of units working in cooperation).
tomers can buy drugs in an easy and quick way; besides, In real systems the strategies used by its components
the presence of other customers gives the impression of a tosurviveandreproduceareingeneralverysophisticated
relatively safe place, hence increasing the probability of and are the result of evolution. Models like the one pre-
a successful transaction. Inter-system aggressions can be sentedhere help to understand how environmentalcondi-
associatedwith the violence between gangs.In our model tions and interaction between the systems drive the evo-
anaggressionhasacostfortheaggressorandcausesmore lution of the model parameters; in our case, the fraction
damage to units working in cooperation than to isolated ofunits workinginisolationandthe conditionsthatmust
units. The shooting of a rival drug-dealer immediately be satisfied for the release of inhibitors.
stops drug sales at the crime scene. During the next days The modelispresentedinSec.2,both ina continuous
orweeksthepoliceispatrollingthearea,scaringawayha- aswellas ina discrete version.In Sec.3 we considera set
bitual customers who chooseto buy drugs in safer places. ofcompetingsystemsincludingfourtypesofsystemswith
Iftheshootingoccursinabusycorner,whereseveraldrug- differentcooperatingconfigurations;twocasesareconsid-
dealersoperatesimultaneously,incomelossespershooting ered: when the inhibitors are introduced by an external
arelargercomparedtotheone-dealerone-cornercase.On source and when the inhibitors are released by the sys-
the other hand, the attacking gang pays a cost in several tems. In Sec. 4 a genetic algorithm is used to allow the
ways: money to buy guns and/or hire mercenary aggres- evolution of the systems’ strategies when the units oper-
sors, demand of dealers for higher wages because of the ate either in isolation or in cooperative arrangements of
added risk, etc. This particular example illustrates the four units each. Conclusions are given in Sec. 5.
kind of competition/cooperation mechanisms considered
in this paper that are common to certain socioeconomic,
ecological, and biochemical systems. Our analysis consid- 2 The model
ersthesegeneralaspects;howeverwewillrefertothedrug-
gang example in order to illustrate the relation between ConsiderasetofNg systems,eachonewiththesamenum-
ourgeneral(nonspecific)modelandaconcrete,real-world ber Ne of units. The Ng ×Ne units compete to acquire
situation. the availableresources(hereafter knownas the substrate)
that are supplied at a rate S˙. Each unit can be in one of
The aim of this work is to formulate a model to anal- four states: i) idle (ready to acquire either a substrate or
ysethecoevolutionofthestrategiesofsystemscompeting aninhibitor),ii)busy(processingasubstrateduringape-
forlimitedresources[13];specifically,westudy the coevo- riodoftimeτ ),iii)recovering(duringaperiodoftime τ
p r
lution of local intra-system cooperation and global inter- necessarytoreachtheidlestateafterreleasingaproduct),
system aggression. Our model consist of Ng ×Ne identi- oriv)inhibited(unabletoacquireasubstrateduringape-
cal units organized in Ng systems with Ne elements per riodoftimeτI afteracquiringaninhibitor).Therefore,the
system. This type of systems is found in cells, where en- dutycycleofanon-inhibitedunitisτ =τ +τ .Theunits
p r
zymesaregroupedwithinspecializedorganelles;similarly, inasystemcanbe organizedincooperativearrangements
in the industrial sector, machines are grouped in facto- of µ units; such arrangements increase the efficiency of
ries,or, as in the gang example, drug dealers are grouped the units to acquire either substrates or inhibitors. When
in gangs. The systems compete to acquire the resources κ of the µ units in a cooperative arrangement are busy
(substrates, raw material or customers) supplied to the the remaining µ−κ idle units increase their probability
system at a rate S˙ and at the same time are allowed to to bind to a substrate (or to a inhibitor) by a factor α−κ
produce inhibitors thatreduce the performanceof others. relative to the isolated case, where 1/α (0 < α < 1) is
The cooperation among associated units increases their theefficiencyofthe cooperation.Ifaunitisinhibited,the
efficiencytoacquiretheresourceswhentheyarearranged other idle units in the cooperative arrangement are also
in oligomers, as in the case of enzymes [14]; the number inhibited.
ofunits cooperatinginagivenconfigurationcanbe inter- In the following two subsections we present a continu-
preted as a measure of its degree of organization.Aggres- ousandadiscreteversionofthe model.Inthe continuous
Jos´e M. Albornoz, Antonio Parravano: Coevolution of competing systems: local cooperation and global inhibition 3
version (Sec. 2.1) we use a mean field approach to obtain provides F as a function of S, I, and the model parame-
i
the stationary solutions for the average state of the N ters α, p , µ , τ , and τ . In stationary conditions
g 0 i i I,i
systems.These stationarysolutionsareusefulto visualize
themainpropertiesofthemodel,howevertheycannotbe S˙
S = (5)
used to follow the dynamics of the individualunits nor to
N p F
e i i
describe the general case of inhomogeneous intra-system P
cooperativity.Thediscreteversionofthe model(Sec. 2.2) and
allows to follow the evolution of the state of each unit, of I˙
i
eachcooperativearrangement,andthe productionrateof I = P . (6)
N p F
e i i
each system. It must be borne in mind that the continu- P
ousmodelpresentedheredoesnotrepresentallthedetails Finally, the rate of inhibitor production is assumed
contained in the discrete model; nevertheless its station- to depend on the total production rate G˙i = P˙i +I˙i as
ary solutions shed light on the essential properties of the shown in Fig. 1. Inhibitors are produced only when the
type of systems we want to study. production rate is in the range P˙cri,i/2 < P˙i < P˙cri,i,
where P˙ is the critical production rate below which
cri,i
system i releases inhibitors. Since an aggression has an
2.1 Mean Field approximation: stationary solutions associated cost, a system must achieve a minimum per-
formance level in order to release an inhibitor; therefore
Inordertoshowsomeofthebasicpropertiesofthemodel, inhibitors will not be releasedif P˙ <P˙ /2.Within the
i cri,i
we find the stationary solutions when the internal struc- P˙ /2 < P˙ < P˙ range the rate of inhibitor produc-
cri,i i cri,i
ture of each system is described by the fraction of idle
tionisproportionaltoP˙;thatis,I˙=βP˙.Notethat:i)In
units F in system i (1 ≤ i ≤ N ). The recovering state
i g the range P˙ /2 < G˙ < (1+β)P˙ /2 the production
of each unit is neglected so that the duty cycle is τ =τ . cri,i i cri,i
p
We approximatethe total production rate G˙ of products rateis regulatedtothe valueP˙i =P˙cri,i/2;thatis,within
P plus inhibitors I in system i as i thisrangeanychangeinS˙ willproduceachangeinI˙i,but
not in P˙ . ii) In the range P˙ < G˙ < (1+β)P˙ the
i cri,i i cri,i
G˙ =P˙ +I˙ =N F p S, (1) system adopts one of two distinct regimes: one in which
i i i e i i
P˙ =G˙ andI˙ =0,andtheotherinwhichP˙ =G˙ /(1+β)
i i i i i
where S is the stationary number of substrates that can and I˙ =G˙ β/(1+β). These two regimes result in a hys-
i i
be bound by the N ×N units, N F is number of idle
g e e i teresisbehavior,asfollows:a)startingintheupperbranch
units in system i, and
anddecreasingS˙,noinhibitorsareproduceduntilthepro-
p = p0 ≃ p0 (2) ductionratereachesthecriticalvalueP˙cri,i,b)startingin
i αhκii α(µi−1)(1−Fi) the lower branch and increasing S˙, the system produces
inhibitors until the production rate reaches the critical
is the probability (per unit time) that an idle unit binds
value (1+β)P˙ .
either to a substrate or to an inhibitor. In this expression cri,i
WearenowinterestedinthestationarysolutionsP˙ (S˙)
theparameterp istheprobability(perunitsubstrateand i
0
unit time) that an isolated idle unit binds to a substrate and I˙i(S˙). An iterative procedure is employed to obtain
(or to an inhibitor), α controls the degree of cooperation them: starting with an initial guess value for I, the value
betweenthecomponentsofacooperativearrangement,µi of Fi in eachsystem is obtained through eqs. (4) and (5),
is the number ofunits in the cooperativearrangementsof whileG˙ isobtainedbyusingeq.(1);thenfunctionG˙ (P˙ )
i i i
system i, and hκii ≃ (µi −1)(1−Fi) approximates the is used to obtain new values for I˙i. Equation (6) is used
average number of busy units in a cooperative arrange- to obtain a new value of I, and the procedure is repeated
ment in system i. Note that the number of units µ per until convergence is achieved. To deal with the bi-valued
cooperative arrangement is assumed to be constant in a regionof G˙ (P˙ ) the procedure is repeated considering all
i i
givensystem,butcandifferfromsystemtosystem.These possible combinations of the systems’ behavior (i.e. the
approximations are overcome in the discrete model (Sec. inhibitor production being ON or OFF in the bi-valued
2.2). region).
The stationary fraction of occupied units Focc,i and Figure 2 shows the equilibrium solutions in the plane
inhibited units Finh,i in system i is approximated as [P˙/S˙, S˙] for two systems, labeled A and B. For the cho-
senparametervalues,theproductionregulatedregimesof
F =SF p τ and F =IF p µ τ . (3)
occ,i i i inh,i i i i I,i the two systemspartially overlap,as wellasthe bi-valued
regimes. Note that all possible combinations occur: the
whereτ isthetimeduringwhichacooperativearrange-
I,i
symbolA↑(A↓)indicatesthatsystemAisproducing(not
ment in system i is disabled once one of its units binds
producing) inhibitors. Eventwo identical systems (for in-
an inhibitor, and I is the stationary number of inhibitors
stance, the case considered in Fig. 2 but with µ = µ
available to be bound. Since F = 1 − F − F , the A B
occ inh and P˙ = P˙ ) can display heterogeneous behavior
equation cri,A cri,B
inthe rangeP˙ <G˙ <(1+β)P˙ .Inthis rangetwoho-
cri cri
1−Fiα(µi−1)(1−Fi) mogeneous(A↑&B↑andA↓&B↓)andtwoinhomogeneous
=Sτ +Iµ τ (4)
i I,i (A↑&B↓ and A↓&B↑) solutions can be distinguished in
F p
i 0
4 Jos´e M. Albornoz, Antonio Parravano: Coevolution of competing systems: local cooperation and global inhibition
Fig.2. NormalizedtotalproductionrateP˙/S˙ =(P˙A+P˙B)/S˙
Fig. 1. The production rate P˙i as function of the total rate asafunctionofthesupplyrateofsubstrateS˙ fortwosystems
production G˙i =P˙i+I˙i.The inhibitorproduction is I˙i=βP˙i labeledAandB.Thetwosystemsareidentical,exceptforthe
forP˙cri,i/2<P˙i <P˙cri,i,whereas I˙i=0forP˙i <P˙cri,i/2and sizeofcooperativearrangementsµandthecriticalproduction
P˙i >P˙cri,i.ForP˙cri,i/2<G˙i <(1+β)P˙cri,i/2theproduction rate P˙cri, which are are greater in system B. The parameter
rateisregulatedtothevalueP˙cri,i/2andtheinhibitorproduc- values are β = 0.2, p0 = 0.1, α = 0.5, τA = τB = 1, τI,A =
tionisI˙i =G˙i−P˙cri,i/2.IntherangeP˙cri,i<G˙i <(1+β)P˙cri,i τI,B =5, µA =1, µB =2, P˙cri,A=0.6, and P˙cri,B =0.7.
theinhibitor production is bi-valued.
efficiency), whereas system B converts all the processed
Fig. 2. This behavior is a consequence of the hysteresis
substrate into products. Moreover, the idle units in sys-
in G˙ (P˙ ), which in turn is characteristic of real systems
i i tem B are more efficient at acquiring substrates (µ > 1
B
withcompetition,cooperation,andaggression:onceacon-
and α < 1) than the idle units in systen A (µ = 1),
A
flict arises it tends to persist over time, and a substantial
andthereforeF <F always.However,systemB is also
B A
change in the conditions is required to extinguish it [16].
moreefficientbindinginhibitorsandadditionallyµ (>1)
B
We now focus in a particular case in which system A
units are blocked per bound inhibitor. Therefore, there is
is constituted by single, non cooperative units (i.e. µA = acriticalvalueofS˙ forwhichthefractionofblockedunits
1) with the capacity to produce inhibitors (I˙A = βP˙A), in system B is large enough to reduce its production rate
whereas system B is composed by cooperative arrange- below that of system A. Figure 3 shows the normalized
ments (i.e. µB > 1) that never produce inhibitors (I˙B = critical value S˙c(µB)/Ne of the substrate supply rate at
0). In this case, the fraction of idle units in the systems which the production rates of the two systems are the
satisfy the equations same (i.e. P˙ = P˙ = S˙/(2+β)). For S˙ > S˙ the pro-
A B c
duction rate of system A is larger than that of system B.
1−FA = S˙ [τ + τIFAβ ], (7) For low values of µB the condition P˙A = P˙B is reached
FA Neσ σ(1+β) for high values of S˙, and therefore, low values of F . For
B
F ≪1, eqs. (7-10) give
and B
1−F S˙ µ τ F β S˙ 2+β
Bα(µB−1)(1−FB) = [τ + B I A ], (8) c ≈ . (11)
FB Neσ σ(1+β) Ne τ +µB2τ+Iββ
where
FB For larger values of µB, the critical value S˙c must be ob-
σ =FA+ α(µB−1)(1−FB). (9) tained numerically. As shown in Fig. 3, S˙c decreases as
µ , β or τ increase or α increases.
F andF canbeobtainedbysolvingeqs.(7)and(8) B I
A B In order to illustrate how the distribution of perfor-
forgivenparametersS˙,β,α,µ ,τ andτ .Theproduction
B I manceschangeswheninhibitorsarereleased,wenowper-
rates are then given by
form a comparison between the previous case and a case
S˙F S˙F wherenone of the systems releaseinhibitors.Fig.4 shows
P˙A = σ(1+Aβ) and P˙B = σα(µB−1)B(1−FB). (10) ttihoennoofrSm˙/aNlizewdhpernodsyusctteiomnAracteosnvPe˙ArtsanadfrPa˙cBtioans βa ofufnicts-
e
SystemAalwaysusesafractionβoftheprocessedsub- productionintoinhibitors(A↑&B↓,continuouscurves),as
strate to release inhibitors (thus reducing its production well as the production rates P˙ and P˙ when none of
A,0 B,0
Jos´e M. Albornoz, Antonio Parravano: Coevolution of competing systems: local cooperation and global inhibition 5
Fig. 3. The normalized critical value S˙c/Ne of the substrate Fig. 4. Normalized production rates of systems A and B.
input rate at which P˙A =P˙B when system A converts a frac- The continuous curves correspond to the case when system A
tion β of its products into inhibitors and its units do not co- producesinhibitors(βA =0.2). Thedashedcurvescorrespond
operate (i.e. µA =1). For substrate supply rates greater than to the case when there is no inhibitor production. In the two
S˙c the production rate of system A is larger than that of sys- cases p0 = 0.1, α = 0.5, τA = τB = 1, τI,A = τI,B = 5,
temB.Thecontinuouscurvecorrespondstoparametervalues µA = 1 and µB = 4. The four vertical dashed lines indicate
β =0.2, p0 =0.1, α=0.5, τA =τB =1, τI,A =τI,B =5 and thevaluesS˙A,S˙B,S˙R,andS˙catwhichP˙A=P˙A,0,P˙B =P˙B,0,
µA =1.Thedashedcurvesshowstheeffectofvaryingβ,τI or P˙B,0/P˙A,0 =P˙B/P˙A, and P˙A=P˙B, respectively.
α.
its production and,evenmore,canincreaseits rankingin
the systems’ performance distribution. This comparative
the systems produce inhibitors (A↓&B↓, dashed curves).
Note that depending on the value of S˙, the production of advantageisrelevantwhentheproductionrankingamong
thesystemsdeterminesthechancesforsurvivalandrepro-
inhibitorsbysystemAcanincreasetheproductionofsys-
tem B (S˙ < S˙ ), can increase the production of system duction.
B
Note that if we add a third system C, composed by
A(S˙ >S˙ ),candecreasethe ratioP˙ /P˙ ofthe systems
A B A non-cooperative units (µ = 1 as system A) that never
production (S˙ >S˙R), or can make the production of sys- releasesinhibitors(I˙ =0CassystemB),itwillalwaysper-
temAlargerthanthatofsystemB(S˙ >S˙ ).Forthecase C
c form better than system A. This kind of “opportunism”
shown in Fig. 4, S˙A = 0.62, S˙B = 0.53, S˙R = 0.59, and is relevant for the coevolution of the system parameters
S˙c =0.74.Thesevaluesdependonthe modelparameters: under selection pressure studied in Sec. 4.
for example, if α = 0.75 and β = 0.2 then S˙ = 0.60,
A
S˙ =0.11, S˙ =0.46, and S˙ =0.61, whereas, if α=0.5
B R c
and β = 0.1 then S˙A = 0.75, S˙B = 0.64, S˙R = 0.72, and 2.2 Discrete non stationary model
S˙ =1.03.
c
Thestationarysolutionsobtainedaboveshowthatour We now consider a discrete non stationary model that
choiceofmechanismsleads to amodelthat displayssome avoids severalapproximations adopted in the Mean Field
of the distinctive behaviors of a set of systems in compe- stationary model described in Sec. 2.1. Specifically: 1) In
tition for a resource, where each system has the possibil- the discrete model the systems can have an inhomoge-
ity to increase its efficiency by intra-system cooperation neous internal structure; i.e. a system can be composed
and/ortotoreducetheperformanceofitscompetitorsby ofcooperativearrangementsof varioussizes.2) The state
investing part of its profits to inflict aggression. That is, of each unit, and therefore the state of its neighbors in a
i) Diversity: Even in the case of identical systems there cooperative arrangement, is known at any time. This is
are stable inhomogeneous solutions. ii) Hysteresis: in the a relevant difference with respect to the Mean Field sta-
bi-valuatedregion,a systemsteadily producinginhibitors tionary model because the probability of an idle unit in a
needs large perturbations to leave this regime and vicev- cooperative arrangement binding to a substrate depends
ersa. iii) Aggression may give comparative advantages: onthenumberofbusyneighborsinthearrangement:once
when one or more systems are producing inhibitors, the anyidleunitbindstoasubstrate,theotherunitsinthear-
total production is always below the maximum possible rangementincreasetheirprobabilitytobindtosubstrates;
value. However, the distribution of profits can change in therefore some cooperative arrangement tend to be fully
suchawaythatasystemproducinginhibitorscanincrease busy while others remain fully idle. In contrast, in the
6 Jos´e M. Albornoz, Antonio Parravano: Coevolution of competing systems: local cooperation and global inhibition
meanfield approximationthe mean number of busy units or decreased (if α > 1) by a factor α−κi,j relative to the
hκi wasapproximatedas (µ−1)(1−F). This approxima- isolated case (κ = 0 or α = 1). When an unit binds an
i,j
tion neglects the fact that the inhibition of a cooperative inhibitor,itremainsinoperativeduringτ iterations.Ifan
I
arrangementdecreasesF butdoesnotincreasetheproba- unit is in its inhibited phase (−τ ≤ φ ≤ 0) the binding
I
bilityofanidleunitbindingtoasubstrate.3)Substrates, probabilities are p = q = 0 for the idle units that co-
i,j i,j
productsandinhibitorsarediscrete.4)Asystemfulfilling operate with the inhibited unit; however the cooperating
the conditions to release inhibitors at a given time can units that were processing a substrate continue normally
convert a product into an inhibitor only if a product was theirwaytowardstheidlestate.Inthefollowingweadopt
produced at that time. α<1.
As in the Mean Field stationary model, there is a set
The configurationofthe cooperativeunits inasystem
of N systems, each one possesing the same number N
g e remains fixed during the simulation; that is, if the units
of units, and the N ×N units compete to acquire the
g e
j and j+1 of system i are set as a cooperative arrange-
availableresourcesthataresuppliedatarateS˙.However,
ment these two units remain in cooperation during the
now the state of each unit is characterized by an integer
complete simulation of one generation. For system i the
phase variable φ (t) where 1 ≤ i ≤ N indicates the
i,j g configurationischaracterizedbythethreeparametersM ,
system, 1 ≤ j ≤ N enumerates the units in the system, i
e D ,andT giving,respectively,the numberofunits work-
and t is a discrete time counter. The phase of unit (i,j) i i
ing in isolation (Monomers), the number of cooperating
evolves in a similar way as the phase of the stochastic
arrangements of two units (Dimers), and the number of
automata originally proposed by Mikhailov and Hess [17,
18,19,20], but allowing for two additional processes: (i) arrangements of four units (Tetramers); all possible con-
the unit can bind either to a substrate or to an inhibitor figurations satisfy Mi+2Di+4Ti =Ne.
and (ii) units can work in cooperation with other units, At the beginning of a given iteration S˙ substrates are
thusmodifyingtheirabilitytobindtoasubstrateortoan
added and the following procedure is executed: (a) the
inhibitor. The idle state corresponds to the phase φ = 0,
phases of the busy and the inhibited units are increased
wheretheunitisreadyto acquireeitherasubstrateoran
byonestep.(b)anunit(i,j)isselectedatrandomamong
inhibitor. When the unit acquires a substrate the phase
those in their idle state provided it is not associated to
changes to φ=1; afterwards the phase value is increased
an inhibited unit in a cooperative arrangement; (c) the
in each time step until the unit reaches the maximum
selected unit starts its processing cycle with probability
phase value φ = τ; once this happens the unit returns to
p ,isinhibitedwithprobabilityq ,orremainsidlewith
itsoriginalidlestateφ=0.Aproductisreleasedatafixed i,j i,j
probability 1−p −q . If the unit starts its processing
phase1<τ <τ.Whentheunitacquiresaninhibitorthe i,j i,j
p
phase changes from φ=0 to the negative value φ=−τI; cycle the number of substrates NS decreases by one unit.
afterwardsthephasevalueisincreasedbyoneunitineach If an inhibitor is bound, the total number of inhibitors
time step until the unit reaches its idle state φ = 0. The NI decreases in one unit. If the unit (i,j) does not leave
algorithm to iterate the phase of unit (i,j) is then the idle state, it is not selected again during the present
iteration.Steps(b)and(c)arerepeateduntilN +N =0,
S I
φi,j(t+1)= oruntilalltheunitsthatwereidleatthebeginningofthe
φi,j(t)+1 if φi,j(t)6=0, iteration t are selected.

10 iiff φφii,,jj((tt))==τ0withprobabilitypi,j, (12) t byThtehenuNmbuerniotsf pinrodsyuscttesmreileaisseddeninotaedgiavesnP˙it(etr)atainond
0−τI iiff φφii,,jj((tt))==00wwiitthhpprroobbaabbiilliittyy1qi−,j,pi,j−qi,j. dthuecttiootnalraptreoedpuecrtiuonnitraatnedaspPe˙r=duPtyN1cgyP˙cil.eTinhesymseitaemn pirois-
The probabilities pi,j and qi,j are given by denoted as υi = τP˙i/Ne. υi is a measure of the perfor-
mance of system i, which in turn depends on the par-
p =p N /ακi,j and q =p N /ακi,j. (13)
i,j 0 S i,j 0 I ticular configuration of the units within the system. The
correspondingmeanproductionrateforallsystemsisυ =
During a single iteration t the number of substrates N
aInndEiqn.hi(b1i3t)ortshNeIpcahraamngeetseraspt0hereyparreesebnotusntdhebypirdolbeaubniliittsSy. t(i1o/nNrga)tPe iN1sgPυ˙i==NτPe˙N/Ng/eτN,gt.hSeninc0e≤thυe ≤ma1x.imInumabspernocdeuoc-f
that an isolated idle unit binds either to a substrate or inhibitors, the maximum production is approached when
to an inhibitor when only one substrate or one inhibitor the binding probability is p ≫ 1/τ. If NeNg > τS˙ then,
is available to be bound. The parameter α > 0 controls thefractionofunitsintheiridlestateis∼1−τS˙/(N N )
e g
the degree of cooperation between units and the expo- and in average P˙ = S˙. However, if the configuration of
nent κ is the number of busy units that cooperate with
i,j cooperatingarrangementsintheN systemsisinhomoge-
g
unit j in system i. When all the units in a cooperative neous, then the distribution of the production rates P˙ is
arrangementare in their idle state, κ =0 and therefore i
i,j also inhomogeneous.
there is no cooperation; that is, the probability to bind
to a substrate is the same as if the units were working Theinhibitorscanbeintroducedbyanexternalsource
in isolation. When κi,j units in a cooperative structure atarateI˙ext(t)orcanbereleasedintotheenvironmentby
are busy (1 ≤ φ ≤ τ) the probabilities p and q of some systems at a rate I˙(t). In Secs. 3 and 4 we describe
i,j i,j i
the idle units in the structure are increased (if α < 1) theconditionsasystemmustfulfilltoreleaseaninhibitor.
Jos´e M. Albornoz, Antonio Parravano: Coevolution of competing systems: local cooperation and global inhibition 7
3 Inhomogeneous configurations
ConsiderasetofN =20systems,eachoneconsistingon
g
N =120 units arranged in different cooperative configu-
e
rations:M =N for1≤i≤5;D =N /2for6≤i≤10;
i e i e
T = N /4 for 11 ≤ i ≤ 15; and M = 40, D = 20, and
i e i i
T = 10 for 16 ≤ i ≤ 20. To quantify the performance
i
ofthe ithsystemwe evaluate the averageproductionrate
hυ iperunitandperdutycycleτ insystemi.Letuslabel
i
these four types of systems as types M, D, T and MDT
and their corresponding production rates as hυi , hυi ,
M D
hυi and hυi .
T MDT
3.1 External release of inhibitors
First we consider the case when the set of systems is sup-
pliedwithaconstantrateS˙ ofsubstratesandanalysethe
dependence of hυi , hυi , hυi and hυi on the ex-
M D T MDT
ternal rate of release of inhibitors I˙ . The objective of
ext
this analysis is to determine how the degree of organiza- Fig.5. Theaverageproductionratehυiperunitandperduty
tionwithineachsystemdeterminesitsperformanceinthe cycle τ for the four types of systems M, D, T and MDT (see
text) as a function of the external rate of release of inhibitors
presence of increasing levels of aggression.
When S˙ = 1N N /τ there are enough substrates to I˙ext.TheadoptedparametersareNe =120,τ =100,τI =5τ,
maintainhalfof2theeungits busy;thiscorrespondstoapro- p0=1/τ andα=1/4. Thesubstratesupplyrateiskeptfixed
toS˙ =12.Asimulationfor200τ wasperformedforeachvalue
duction rate hυi = 1/2. The average number of free sub-
strates depends on p and on the number of cooperative of I˙ext.
0
arrangements in the system. If p = 1/τ and N = 1,
0 S
an isolated unit remains in average half of the time in
its idle state. If p is reduced, N must be increased in
0 S
the same proportion in order to maintain the units work-
ing at half velocity. If not all the units work in isolation,
the units in cooperative arrangementswill be busier than
the isolated ones (for α < 1). Therefore, in absence of
inhibitors one expects hυi > hυi > hυi , but these
T D M
relations change depending on the rate of release of in-
hibitors I˙ and on the blocking time τ . The results in
ext I
Fig. 5 correspond to the case when S˙ = 1N N /τ. As
2 e g
I˙ increases,theperformanceoftheT-systemsdecreases
ext
whereasfor the M-systems itincreases:the higher impact
of inhibition on the T-systems means that there are less
highly-organizedunitscompetingfortheresource,thereby
increasing its availability for the rest of the systems. Fol-
lowing the drug-gang analogy, this means that shootings
takingplaceinbusycornerswhereseveraldealersoperate
atthe same time force the customers to moveto other lo-
cationswheresmallergroupsofdealersoperate,improving
the profits of these dealers. However, for I˙ ≥ 1.2 even
ext
Fig. 6. The average production rates for the four types of
theperformanceoftheM-systemsdecreasesbecausemore
thanhalfoftheunits areblocked(i.e.τ I˙ µ≥ 1N N , systemsM,D,TandMDTasafunctionofthenonbelligerent
where µ ≃ 2 is the mean number of uInietxstin co2opegrate- period τnon for <hI˙exti=1. Model parameters as in Fig. 5.
ing arrangements). This result illustrates how the perfor-
manceofallsystemswillbe negativelyaffectedregardless
of their degree of organization for sufficiently high levels
ofaggression.NotethattheperformanceoftheMDTsys-
temsremainsalmostconstantforI˙ <1.2.Therefore,at
ext
leastuptoacertainintensityofaggression,diversification
in the degree of organization is a robust strategy against
aggression.
8 Jos´e M. Albornoz, Antonio Parravano: Coevolution of competing systems: local cooperation and global inhibition
In order to examine the effect of a non-stationary in-
hibitor release rate, Fig. 6 shows the average production
rates hυi , hυi , hυi and hυi as a function of the
M D T MDT
nonbelligerent period τ . The inhibitor release rate is
non
I˙ = 0 during τ −1 iterations and in the next iter-
ext non
ation the release rate is I˙ = τ ; that is, the average
non non
inhibitor releaserateis hI˙ i=1.Inordertoreduce syn-
ext
chronization effects the inhibition time was randomized
10% around its mean value τ = 5τ. The results show
I
how the sudden onset of aggression affects the systems:
those with a greater degree of organization have a better
performanceasthenonbelligerentperiodgrowslongerdue
to their greater efficiency. Note that as τ increases the
non
average production rates tend to the values correspond-
ing to I˙ = 0. When a large number of inhibitors are
ext
added in a single iteration (τ ≫ 1) many idle units
non
in all cooperating arrangements are blocked, and there-
forealltypesofsystemsreducetheirperformancesduring
τ iterations. The performance of the MDT systems re-
I
mains almost constant for a wide range of values of τ ,
non
illustrating once againthe robustness ofthe systems with
diverse degrees of organization. Fig. 7. The average production rate hυii of the20 competing
systems as a function of the critical averaged production υcri
(see text). The model parameters are the same as in Fig. 5
3.2 Internal release of inhibitors and τave =10τ.Half of the systems (black curves) have their
unitsworkinginisolation(M-systems)andtheotherhalf(grey
Consider the case in which inhibitors are produced under curves) have their units working in cooperative arrangements
certainconditions andat a givencost by the systems.We of four units(T-systems). The curvelabelled υtot corresponds
assume that at a given time an inhibitor is released into to thetotal average production and thecurve labelled σ(υtot)
theenvironmentbysystemiifthefollowingconditionsare to its standard deviation. In the region in between the two
fulfilled: (i) at least one of the units in system i has re- diagonaldashedlinesthecondition(iii)isfullfilled.TheRoman
leased a product during the present iteration t; (ii) there numberslabelthe5differentregimesdisplayedbythesystems.
are no free inhibitors; and (iii) υ > υ (t) > υ /2,
cri i cri
where υ is the system production averaged during the
i
previous τ iterations and υ is the critical average condition (iii) is fullfilled. The Roman numbers label the
ave cri
production rate below which a system will release an in- 5 different regimes observed in Fig. 7. In regime (I) none
hibitor. Since an aggressionhas an associated cost, a sys- of the system production falls in the range [υ /2,υ ]
cri cri
tem must achieve a minimum level of performance in or- and therefore there is no inhibitor production; the co-
der to release an inhibitor; therefore, as in the case of operation in T-systems results in a better performance
the continuous Mean Field model, an inhibitor will not than in M-systems. In regime (II) a few M-systems re-
be released if υ (t) < υ /2. The cost of producing an leasing inhibitors (in fact, only one in Fig. 3) are able
i cri
inhibitor is one product. During a given iteration a given to substantially reduce the production of T-systems and
system may release more than one inhibitor, but in gen- to increase the production of the remaining non belliger-
eral, conditions (i) and (ii) limit the release of inhibitors ent M-systems; however, T-systems perform better than
to at most to one per iteration. This corresponds to low M-systems.Notethatthesystemisself-regulatedtoman-
intensity conflicts in the drug-gang example. Continuing tain the production of the attacking system k close to
withthisanalogy,anaggressiontakesplaceoncetheprof- υ = υ /2. This is an interesting instance of emergent
k cri
its of a particular gang fall below a given critical value, cooperation:whenthevalueofυ ismodest,someofthe
cri
howeveriftheprofitsaretoolowthegangwillnotbeable less organized systems keep their performance just above
to afford an aggression. υ /2, negatively affecting the performance of those sys-
cri
Fig.7showstheaverageproductionrateshυ iasfunc- tems with a greater degree of organization and thereby
i
tion of υ for 20 systems competing to acquire the sub- improvingtheavailabilityofsubstratefortherestoftheir
cri
strates supplied at a constant rate S˙ = 1N N /τ. Half non-belligerent,less-organizedpeers.Inregime(III)allT-
2 e g
of the systems (black lines) have their units (N = 120) systems release inhibitors. The performance of T-systems
e
workingin isolation(M-systems) andthe other half (grey isreducedbecause ofthe costofproducinginhibitors and
lines) have their units working in cooperative arrange- the blocking oftheir cooperatingunits. Inthis regimethe
ments of four units (T-systems). The curve labelled υ M-systems perform the best, but the all-systems average
tot
corresponds to all-systems average production per unit productionυ is reducedby about10 %.Inregime (IV)
sys
and the curve labelled σ(υ ) gives its standard devia- all M-systems release inhibitors and their performance is
tot
tion.Intheregioninbetweenthediagonaldashedlinesthe higher than that of T-systems; the disadvantage associ-
Jos´e M. Albornoz, Antonio Parravano: Coevolution of competing systems: local cooperation and global inhibition 9
atedtothecostofreleasinginhibitorsisover-compensated or more systems have performances which are more than
by the fact that an inhibitor is able to block four units one standard deviation above hυ i. (iii) Crossover be-
tot
in the T-systems. As in regime (III), the all-system pro- tweenarandompairofoffspringchromosomesoccurswith
duction υ is reduced by about 10%. In regime (V) the a probability p , and offspring mutate with a proba-
sys cross
T-systems sporadically release inhibitors in such a way bilityp bychangingoneoftheirchromosomedigits.
mutate
thattheiraveragedproductionrateshυ iremainsclosebut The system with the best performance is always repro-
i
below υ /2 (note that during the simulation υ fluctu- duced without any change.
cri i
ates around hυ i and sporadically υ > υ (t) > υ /2). Figure 8 shows the results for three different values of
i cri i cri
These results suggest that full-scale, one-sided aggression thesubstratesupplyrate(S˙ =1/3,1/2and2/3ofthefull
is profitableonly forless-organizedsystems,asillustrated occupation value S˙ = N N /τ). In each case the evolu-
F e g
byregime(IV):regime(III)showsthatfull-blownaggres- tion for 1500 generations is shown. The thick continuous
sion from T-systems has a detrimental effect in their own curve shows the evolution of the mean monomer fraction
performance. f = fi /N of the population, the dotted curve with
M M g
P
large fluctuations shows the monomer fraction of the sys-
tem with performance ranked at half way (f ) , the
M med
4 Evolution of system parameters dots indicate the monomer fraction of the systems that
have released inhibitors during the simulation of a gen-
eration, and the almost horizontal thin curve gives the
Thebehaviorofthesetofsystemsdependsonthearrange- all-systems averageproduction υ . As S˙ increases (from
ment of the units in each system and on the particular tot
top to bottom in Fig. 8) the average monomer fraction
conditions at which each system releases inhibitors. The
over generations f decreases, the fluctuations of the
results in Fig. 7 show that small changes in the system M
monomer fraction of the half ranked system (f ) de-
parameterscandrasticallychangethe distributionofper- M med
crease,andthenumberofsystemsreleasinginhibitorsde-
formances. In Fig. 7 we considered that the conditions to
creases. Figure 9 shows, for four particular generations,
release inhibitors were the same for all systems and that
the distribution of the parameters fi and υi for the 20
halfofthe systems werehomogeneousM-systems andthe M cri
systems. Note the diversity of system parameters values
other half homogeneous T-systems.Now we allow hetero-
inallcases.Thedistributionofparametersvaluesremains
geneity in both parameters; that is, systems are charac-
relatively stable for low and high values of S˙ (see the top
terized by the parameters (fi ,υi ), where the monomer
M cri and bottom panels in Fig. 8 and the two top panels of
fractionfi isthefractionoftheunitsworkinginisolation
M Fig.9)butforintermediatevaluesofS˙ thedistributionof
insystemi(therestworksincooperativearrangementsof
the system parameters values displays two very different
fourunits),andυi isthevalueoftheaverageproduction
cri distributions (see the center panel in Fig. 8 and the two
rate below which system i may release inhibitors. Note
botton panels of Fig. 9).
that in Fig. 7 we used the condition υ >υ (t)>υ /2
for the release of inhibitors, but now wcrei useiυ (t)<cυrii . Under conditions of substrate scarcity (S˙ = 6) most
i cri
For a given set of system parameters (fi ,υi ) there is a of the systems have a large fraction of their units oper-
M cri
ating in isolation, as the intensity of the aggressions (as
distribution of the performances hυ i in a simulation. A
i
changeoftheparametersfi orυi inoneofthe systems reflected by the number of systems releasing inhibitors
M cri
in the top panel of Fig. 8) makes collaborative arrays an
generally results in a redistribution of the performances
ill-advised strategy. Note that the best performing sys-
hυ i and in a change of the all-systems performance υ .
i tot
temsinthe upper-leftpanelofFig.9havehighmomomer
A genetic algorithmis implemented to analyse the co-
evolution of the system parameters (fi ,υi ) with 0 ≤ fractions and a low aggression threshold υicri. When the
fi ≤ 1 and 0 ≤ υi ≤ 1. The chromoMsomceriof each sys- substrate supply is relatively abundant (S˙ = 18) there
M cri
tem is a eight digit binary number; the first four digits are few systems releasing inhibitors in each generation,
give the 16 possible values of the momomer fraction fi and most of the systems have a large fraction of their
M
(= 0,1/15,2/15,...15/15) and the last four digits give units operating in tetrameric configurations; the upper-
the critical average production rate υi below which the right panel of Fig. 9 shows that that the best performers
cri
systemmayreleaseinhibitors.Forthefirstgenerationthe are indeed those systems with a low monomer fraction.
parameters(fi ,υi )arechosenatrandom,whilethe pa- Whenthesubstratesupplyishalfwaybetweenabundance
rameters N , MN ,cτr,i τ , τ , α and the substrate supply and scarcity (S˙ = 12), two quasistable behaviors are ob-
g e I ave
rate S˙ remain fixed for all generations. Each generation served, as reflected in the central panel of Fig. 8 and the
two bottom panels of Fig. 9: the systems switch between
consistsoft iterationsofthemodel.Attheendofagen-
sim
eration simulation the parameters (fi ,υi ) for the next periods of frequent inhibitor release (aggression periods)
M cri
generation (offspring) are set by the following procedure: and high fM values, to periods where few inhibitors are
(i) the performances hυii of the Ng systems are rankedin released(calmperiods)togetherwithlowfM values(note
decreasing order. (ii) those systems whose performances that (fM)med =0 for many generations in these quiet pe-
are more than one standard deviation above hυ i have riods).
tot
two offspring and the rest (in decreasing order of perfor- In order to examine the general trends as S˙ is in-
mance) have one offspring until the population size N is creased, we performed 5 simulations of 1500 generations
g
reached.Notethatallsystemshaveanoffspringunlessone for 24 different values of S˙ (i.e. S˙ = 1,2,...,24); the first
10 Jos´e M. Albornoz, Antonio Parravano: Coevolution of competing systems: local cooperation and global inhibition
Fig.8.Evolutionofthemonomerfractionofthepopulationduring1500generationsforthethreelabelledvaluesofthesubstrate
supplyS˙.ThethickcontinuouscurveshowstheevolutionofthemonomerfractionfM = fMi /Ng ofthepopulation,thedotted
P
curve with large fluctuations shows the monomer fraction of the system with performance ranked at half way (fM)med, the
dots indicate themonomer fraction of the systems that havereleased inhibitors during thesimulation of a generation, and the
almost horizontal thin curve shows the total average production υtot. The model parameters are Ng =20, Ne =120, τ =100,
τI =6τ,τave =10τ,p0 =1/τ, α=1/4, pcross=0.05, and pmutate=0.01.
100generationsarediscardedaswellasthefirst30τ itera- diagonallinerepresentsthemaximumaverageproduction
tionsineachgeneration.Figure10showstheaveragerate rate per unit and per cycle P˙ = S˙τ/N N . The error
max e g
ofreleaseofinhibitorsper iterationandper systemhI˙ias bars in Figs. 10, 11, and 12 correspond to one standard
afunctionofthesubstratesupplyrateS˙.Figure11shows deviation.
the averagemonomerfractioninthefivesimulations,also It can be observed in Fig. 11 that for S˙ < 2 the
as a function of the substrate supply rate S˙: the continu-
best performers are those systems with a relatively low
ous curve with error bars corresponds to the average f
M monomer fraction (∼ 20%). In this situation there are
of the mean monomer fraction f of the population, the
M manyidleunits,andthereleasedinhibitorsarenotableto
long-dash curve (f vs S˙) corresponds to the same
M,top5 reduce the already low system performance because their
averagebuttakingintoaccountonlythefivesystemswith
number is not enough to block a significant fraction of
bestperformancesineachgeneration,whiletheshort-dash
tetramers. Even when many systems release inhibitors in
curve(fM,bot5vsS˙)correspondstotheaverageforthefive small quantities (hυcrii > P˙max = S˙τ/NeNg, Fig. 12),
systems with the worst performances in each generation.
tetramericconfigurationsarethebeststrategytocompete
Figure12showstheaveragehυ iofυ = υi /N for
cri cri cri g for the few available substrates. This type of behavior is
P
eachof the 24 substrate supply rate values:the long-dash also evident in Fig. 4 for S˙ < S˙ : in this range, the per-
C
curve (hυ i) and the short-dash curve (hυ i)
cri,top5 cri,bot5 formance of system B is greater than the performance of
give respectively the corresponding averages for the five
system A in spite of the release of inhibitors. For com-
systemswithbestandworstperformances,whilethegrey
parison purposes one can assume N = 2, N = 10 and
g e