Anomalously slow attrition times for asymmetric populations with internal group dynamics
Zhenyuan Zhao, Juan Camilo Bohorquez, Alex Dixon, Neil F. Johnson
AAnomalously slow attrition times for asymmetric populationswith internal group dynamics
Zhenyuan Zhao , Juan Camilo Bohorquez , Alex Dixon & Neil F. Johnson Department of Physics, University of Miami, Coral Gables, FL 33126, U.S.A. Department of Industrial Engineering, Universidad de Los Andes, Bogota, Colombia Department of Physics, Clarendon Laboratory, Cambridge University, Cambridge CB3 0HE, U.K. (Dated: November 4, 2018)The many-body dynamics exhibited by living objects include group formation within a population,and the non-equilibrium process of attrition between two opposing populations due to competitionor conflict. We show analytically and numerically that the combination of these two dynamicalprocesses generates an attrition duration T whose nonlinear dependence on population asymmetry x is in stark contrast to standard mass-action theories. A minority population experiences a longersurvival time than two equally balanced populations, irrespective of whether the majority populationadopts such internal grouping or not. Adding a third population with pre-defined group sizes allows T ( x ) to be tailored. Our findings compare favorably to real-world observations. Predator-prey systems have been widely studied bymany disciplines, including physics[1]. Outside the few-particle limit, mean-field mass action equations such asLotka-Volterra provide a reasonable description of theaverage and steady-state behavior, i.e. dN A ( t ) /dt = f ( N A ( t ) , N B ( t )) and dN B ( t ) /dt = g ( N A ( t ) , N B ( t ))where N A ( t ) and N B ( t ) are the A and B popula-tion at time t . Such population-level descriptions ofliving systems do not explicitly account for the well-known phenomenon of intra-population group (e.g. clus-ter) formation[4], leading to intense debate concern-ing the best choice of functional response terms for f ( N A ( t ) , N B ( t )) and g ( N A ( t ) , N B ( t )) in order to par-tially mimic such effects[2]. Analogous mass-action equa-tions have been used to model the interesting non-equilibrium process of attrition (i.e. reduction in pop-ulation size) as a result of competition or conflict be-tween two predator populations in colonies of ants, chim-panzees, birds, Internet worms, commercial companiesand humans[3] in the absence of replenishment. The termattrition just means that ‘beaten’ objects become inert(i.e. they stop being involved) not that they are necessar-ily destroyed. The combined effects of intra-populationgrouping dynamics and inter-population attrition dy-namics have received suprisingly little attention[4, 5],despite the fact that grouping and attrition are sowidespread[3, 4] and the fact that their coexisting dy-namics generate an intriguing non-equilibrium manybody problem.In this Letter, we consider explicitly the effect of intra-population grouping dynamics on the duration T of at-trition between two opposing populations A and B . Instark contrast to the standard mass-action theories, wefind that T exhibits a maximum for highly asymmetricpredator-predator systems in the absence of populationreplenishment (Fig. 1(a)). This non-monotonic depen-dence of T on population asymmetry is remarkably insen-sitive to whether the majority (i.e. larger) population ex-hibits internal grouping or not (Fig. 1(b)). We show how T can be manipulated by the addition of a third popula-tion which blocks encounters involving smaller fragments.Within the physics community, Redner and co-workershad considered a related problem of conflict within aclustering population in one dimension, and highlightedintriguing connections to a more general class of rough-ening phenomena in physics[5]. Eguiluz and Zimmermanand others had considered an infinite-range one popu-lation coalescence-fragmentation model as a simplifiedversion of human opinion formation[6], while Galam andothers have considered interesting models involving com-petition and conflict[7, 8]. Our work focuses instead onthe consequences of coexisting grouping and attrition onthe duration T (i.e. the survival time of the minoritypopulation).The intra-population group dynamics in our model aredriven by essentially the same coalescence-fragmentationprocesses as Ref. [6], while the inter-population attritionprocess is essentially the same as Ref. [5] (Fig. 1(c)).We have checked that our main conclusions are robustto a variety of reasonable generalizations (e.g. randomlyselecting groups independent of group size, or attritionbeyond a simple cluster subtraction rule[5], or allowingfor a limited number of new recruits over time) and toa reasonably wide range of parameter space. Our modelcombines the following specific mechanisms: It is welldocumented that groups of objects (e.g. animals, peo-ple) may suddenly scatter in all directions (i.e. completefragmentation) when its members sense danger, simplyout of fear[4] or in order to confuse a predator[4]. (Cu-riously, clusters of inanimate objects such as doubly-ionized Argon atoms and animal Hox genes, also exhibitsuch complete fragmentation[9]). Since a sense of dangercan arise at any time, our model randomly selects a can-didate group for fragmentation at each timestep, withprobability proportional to its size since larger groupshave more members and hence are increasingly likely tospot danger or be themselves spotted[4]. With proba-bility ν A or ν B for groups of type A or B , the group a r X i v : . [ c ond - m a t . s o f t ] O c t !" &’()*+%",$$$$$"&",-.%$ /’*".0&$$ ! A N A ( ) T ! " ! "&3 ! "&3 T T %0",)4%)&%)$$ 6*"2()&5".0&$$ +&5)*"%.0&$$7&+.",$ ! $808’,".0&$$ N A ( ) !9 FIG. 1: (a) Duration T of attrition (or equivalently, theextinction time or survival time of the smaller population) asa function of initial A population N A (0) and fragmentationprobability ν A ( ν B = 0 . N A (0) + N B (0) =1000. Qualitative features are unchanged by varying ν B . (b)Black curve: same as Fig. 1(a) with ν A = ν B = 0 .
3. Reddashed curve: A contains rigid units (size 10) while B featuresinternal dynamical grouping (i.e. clustering) as explained intext. Green curve: both A and B comprise rigid units (size10). (c) Events in our model. Two groups of same type cancoalesce, e.g. 6+4 = 10. Individual groups can fragment, e.g.6 → ×
1. Two groups of opposite type interact, e.g. groupof size 6 − fragments completely. If it doesn’t fragment, a secondgroup is randomly selected with probability again pro-portional to size, since any subsequent coalescence andattrition events will likely be initiated by pairwise in-teraction between individual members in the two groupsand hence the probability will depend on the number ofmembers. If the group is of the same type (i.e. A or B ) the two groups coalesce, mimicking the observationthat groups may try to build up their size to increasetheir security[4]. ‘Coalescence’ can simply mean that twogroups act in a coordinated way, not necessarily that theyare physically joined. If of opposite type, their interac-tion leaves an A or B group of size | s A − s B | if s A > s B or s B > s A respectively (or zero is s A = s B ) where s A and s B are A and B group sizes (Fig. 1(c)). Other formsof attrition rule (e.g. stochastic) can yield similar re-sults. At time t , populations A and B hence comprise n As ( t ) and n Bs ( t ) groups (i.e. clusters) of size s , where (cid:80) sn As ( t ) = N A ( t ) and (cid:80) sn Bs ( t ) = N B ( t ). Interactionsare distance-independent as in Ref. [6] since we are in-terested in systems where messages can be transmittedover arbitrary distances (e.g. modern human communi-cations). Bird calls and chimpanzee interactions in com-plex tree canope structures can also mimic this setup, asmay the increasingly longer-range awareness that arisesin larger animal, fish, bird and insect groups[4]. Figure 1(a) shows our numerical and analytic results(Eq. (2)) for the duration T . The initial condition forthe numerical simulations comprises isolated individuals,however the curve is insensitive to these initial conditionssince the initial group (i.e. cluster) formation times actlike a small additive term. The present two-populationnumerical implementation is a straightforward general-ization of the one-population version discussed in Ref.[6]. The excellent agreement suggests that our analytictreatment of internal grouping using a time-averaged in-teraction term may have wider application within non-equilibrium many body systems. Following the modelmechanisms discussed above, the probability Q AB thatany A cluster is selected and interacts with a B clusteris the sum over all s of the probability for an A clus-ter of size s to interact with any B cluster, which gives(1 − ν A ) N A ( t ) N B ( t ) / [ N A ( t ) + N B ( t )] . The probability Q BA is similar, with ν A replaced by ν B . After an inter-action, A and B are reduced by the size of the smallestinteracting cluster, whose average value c (i.e. average in-teraction size) is well approximated by unity plus a smalllinear correction term 0 . x ) − (1 − x )(1 − ν A )(1 − ν B )because clusters are generally very small over a large pro-portion of T . Note c is not formally the same as the av-erage cluster size – in part because interactions do notoccur at every timestep and the entire system is actuallytime-dependent – but they tend to take on similar val-ues. Employing constant c , the populations after i inter-actions become N A ( t ) = N A (0) − ic , N B ( t ) = N B (0) − ic .The probability for an interaction between A and B clus-ters after i previous interactions is Q ( i ) = Q AB + Q BA and hence Q ( i ) = ( N A (0) − ic )( N B (0) − ic )( N A (0) + N B (0) − ic ) (2 − ν A − ν B ) (1)To reduce N A ( t ) and N B ( t ) by c takes 1 /Q ( i ) timestepson average. The time to reduce one population to zerois the sum of the timesteps required for each interac-tion, until the population is eliminated. Supposing B isthe smaller population, it requires N B (0) /c interactionsto eliminate it, hence the final interaction happens after N B (0) /c − T to elimi-nate the smaller population B is therefore (cid:80) Q − ( i ) with i running from 0 to N B (0) c −
1. Using (cid:80) n i = γ + ψ ( n +1),where γ is the Euler-Mascheroni constant and ψ is thedigamma function, and (cid:80) na +1 = (cid:80) n − (cid:80) a , we obtainthe duration: T = N A (0) − N B (0) c (2 − ν A − ν B ) (cid:104) N B (0) N A (0) − N B (0)+ (cid:2) γ + ψ (cid:16) N B (0) c + 1 (cid:17)(cid:3) (2) − (cid:2) ψ (cid:16) N A (0) c + 1 (cid:17) − ψ (cid:16) N A (0) − N B (0) c + 1 (cid:17)(cid:3)(cid:105) . When A is the smaller population, the form is identicalbut with A and B interchanged. This T expression de-pends only on the initial populations of A and B , hence T ≡ T ( x, { ν } ) for constant N , where { ν } ≡ ( ν A , ν B ).Differentiation yields a maximum T at x max (cid:39) .
788 for ν A = ν B and N = 10 , independent of { ν } . Numericalsimulations confirm that T ( x, { ν } ) ∼ t ( { ν } ) t ( x ), with t ( { ν } ) ∼ / (2 − ν A − ν B ) exactly as in Eq. (2), therebysupporting our use of a constant average interaction size c . Two factors therefore determine the duration T : oneoriginates from the grouping dynamics within a givenpopulation (i.e. t ( { ν } )), while the other originates fromthe asymmetry (i.e. t ( x )). Replacing sums by integrals,Eq. (2) can be approximated in the large populationlimit as: T cont = N A (0) − N B (0) c (2 − ν A − ν B ) (cid:104) ln N B (0)[ N A (0) − N B (0) + c ] cN A (0)+4 (cid:16) N B (0) − cN A (0) − N B (0) (cid:17)(cid:105) . (3)The peak at x max is robust to a variety of model vari-ants, and can be understood as follows: When x ∼ A and B are abundant and have a reason-ably large average size. Interactions between A and B clusters are frequent and the attrition per interaction ishigh, hence T is small. As x increases, with A beingthe larger population, an interaction between an A and B cluster is increasingly likely to eliminate the B clus-ter completely since the A cluster is increasingly likelyto be the larger cluster. However the interaction rate isdecreasing rapidly, and T increases overall. For x →
1, itmay take a long time to find a B cluster however there arevery few to find, hence T becomes smaller. Interestingly,the distribution of time-intervals between interactions of A and B clusters is approximately exponential for all x ,except near x max where it becomes approximately power-law. Note that if the attrition were to end after a givenfraction of the initial population is eliminated, the samequalitative results would still emerge since the theory isessentially invariant under overall changes of timescale.Figure 1(b) shows the results of A and/or B adopting dif-ferent internal grouping. The duration T remains essen-tially unchanged if the larger population chooses a staticinternal structure comprising rigid units of a particularsize. If the smaller population adopts such rigid units, T decreases significantly. Hence T is largely dictated bythe internal group dynamics of the minority population.If both A and B are internally rigid, T is small for all x .The top red curve in Fig. 2 compares our the-ory to empirical results for human conflicts, while thelower two blue curves show the mass-action predictions.The mass-action equations traditionally used for at-trition are[3]: (1) dN A ( t ) /dt = − u L N A ( t ) N B ( t ) and dN B ( t ) /dt = − u L N A ( t ) N B ( t ), called Lanchester’s undi-rected mass-action model; (2) dN A ( t ) /dt = − d L N B ( t )and dN B ( t ) /dt = − d L N A ( t ), called Lanchester’s directedmass-action model[3] where u L and d L are constants.‘Old’ wars are blue circles and ‘new’ wars are red tri- Colombia(1984-present)Peru(1980-1999)EI Salvador(1979-1992)Iraq(2003-present)Afghanistan(2001-present)Algeria(1991-2002)Vietnam(1958-1975)Sierra Leone(1991-2002) US Civil War(1861-1865)World War II(1939-1945) ‘new’ war (post-WWII) ‘old’ war (pre-WWII) dynamical clustering model directed fire model undirected fire model D u r a t i on Population asymmetry |x|
World War I(1914-1918)Spanish Civil War(1936-1939) !" ! " !1.)("%)/’%/0*&$ FIG. 2: Duration T of human conflicts, as function of asym-metry x between the two opposing military populations. x = | N A (0) − N B (0) | / [ N A (0)+ N B (0)]. Data are up to the endof 2008, hence final datapoints for the three ongoing wars willlie above positions shown, as indicated by arrows. Lower twoblue lines are the mass-action results. Upper red curve (i.e.Eq. (2)) generated using ν A = ν B = 0 . N A (0) + N B (0)]fixed (as in (a)). Changing ν A and ν B changes height of the-oretical peak, but leaves qualitative features unchanged. angles, with World War II labelled by both since it is anatural dividing point. Since N (cid:29)
1, we take the end-point for the undirected mass-action model to correspondto reducing the smaller population to one instead of zero,thereby avoiding problems with a continuum descriptionof N A ( t ) and N B ( t ) near zero. Figure 2(b) offers somesupport for a recent hypothesis in the social science do-main, distinguishing between old wars in which A and B adopt traditional, fairly rigid, military structures, andnew wars in which B (and possibly A ) adopt more fluidtactics akin to our model[10]. By contrast, the ‘old’ warsare well described by both the green curve of Fig. 1(b)(i.e. rigid armies) and the traditional mass-action theo-ries (blue curves), implying that such internal group dy-namics were absent in ‘old’ wars.Figure 3(a) shows that the duration T can be manip-ulated by adding a third-party population C which canblock interactions (Fig. 3(b)). For simplicity, we assumethe N C members of C are permanently arranged into n C groups each with s C permanent members. Apart frompeacekeepers in human conflict, C could mimic the tar-geted blocking of interactions between particular physi-ological clusters. A and B undergo dynamical clusteringas before, except that if a C group is selected and itis bigger or equal to the size of the A and B clusters,the interaction is blocked and the two A and B clustersare permanently pacified (i.e. neutralized). Figure 2(a) !" *+,)*!,) -./+0 T N A ( ) n C = s C = n C = s C = n C = s C = FIG. 3: (a) Black curve as in Figs. 1(a), 1(b), 2(b) with A and B undergoing internal dynamical clustering. ν A = ν B = 0 . n C = 100 third-party groups, each of size s C = 1. Green dotted curve: n C = 1 third-party group, ofsize s C = 100. (b) Third-party blocking event. If neither A or B clusters are bigger than C cluster, then C cluster blocksthe interaction and permanently neutralizes both clusters. shows that if C comprises only a few, large groups (e.g.green dotted curve) then T decreases irrespective of theasymmetry. Having a few, large C groups means thatsome sizeable battles can be blocked, however it also al-lows the build-up of sizeable groups of both A and B which in turn makes the typical size of interactions big-ger. By contrast, if C comprises many small groups (e.g.red dashed curve) T can be much larger, showing a hugeincrease around x ∼
0. If real-time management of the C population is possible, this duration profile can be ma-nipulated even further.We stress that our model findings are not a simpleconsequence of either dilution leading to reaction slow-down, or of the specific cluster selection scheme that wechose. In our model, as in nature, opposing predatorgroups actively seek each other out at each timestep,even if their density is low, making this unlike simplechemical dilution, and hence unlike simple mass-actionequations. Regarding cluster selection, we have verifiednumerically that our main conclusions are unchanged ifwe select clusters independent of size, or use other frag-mentation schemes (e.g. binary splitting into two clus-ters). This is because the smaller population spends themajority of the conflict as very small groups or individ- uals, hence the weighting by size is not so important. Inshort, our results emerge from the interplay between pop-ulation asymmetry, the presence of clustering, and theintentional engagement between the two opposing pop-ulations. Although the specific consequences may varyby application area, we believe that related phenomenalying beyond mass-action predictions will arise in a widerange of physical, chemical, biological and social systems,whenever intra-population clustering coexists with inter-population reactions.We thank R. Denney and M. Spagat for very usefuldiscussions. [1] A. McKane and T. Newman, Phys. Rev. Lett. , 218102(2005); M. Mangel, The theoretical biologist’s toolbox (Cambridge, New York, 2006); P. Romanczuk, I.D.Couzin, and L. Schimansky-Geier,
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