Coevolution Maintains Diversity in the Stochastic ldquoKill the Winnerrdquo Model

Chi Xue and Nigel Goldenfeld

Department of Physics and Center for the Physics of Living Cells University of Illinois at Urbana-ChampaignLoomis Laboratory of Physics 1110 West Green Street Urbana Illinois 61801-3080 USA

and Carl R Woese Institute for Genomic Biology and Institute for Universal BiologyUniversity of Illinois at Urbana-Champaign 1206 West Gregory Drive Urbana Illinois 61801 USA

(Received 8 June 2017 published 28 December 2017)

The ldquokill the winnerrdquo hypothesis is an attempt to address the problem of diversity in biology It arguesthat host-specific predators control the population of each prey preventing a winner from emerging andthus maintaining the coexistence of all species in the system We develop a stochastic model for the kill thewinner paradigm and show that the stable coexistence state of the deterministic kill the winner model isdestroyed by demographic stochasticity through a cascade of extinction events We formulate anindividual-level stochastic model in which predator-prey coevolution promotes the high diversity ofthe ecosystem by generating a persistent population flux of species

DOI 101103PhysRevLett119268101

The high diversity of coexisting species in most ecosys-tems has been a major puzzle for more than 50 years In aclassic paper Hutchinson articulated the so-called paradoxof the plankton for the case ofmarine ecosystems [12]Whydo many species of plankton that feed on the same nutrientscoexist somehow avoiding the phenomenon of competitiveexclusion [3] where one species outcompetes all the othersThe various tentative resolutions of the paradox can be

divided into two classes [4ndash6] In the first the ecosystem isargued to have failed to reach a fixed point equilibrium statein which the competitive exclusion principle applies due totemporal orand spatial factors For example the timeneeded for the system to reach equilibrium might be muchlonger than the time over which the system undergoessignificant changes in its boundary conditions such asweather [7] Spatial heterogeneity can increase the globaldiversity of the system by maintaining local patches thateach obey the competitive exclusion principle but globallysupport the coexistence of multiple species [89] (foranother perspective see [10]) In the second class ofresolutions interactions such as predation in conjunctionwith competitive exclusion promote the coexistence ofspecies through time-dependent or stochastic steady states[11ndash13] One widely celebrated example of this behaviorwhich is seen in both natural ecosystems as well as somelaboratory systems such as chemostats [1415] is thecontinual succession of different community membersknown as ldquokill the winnerrdquo (KTW) dynamics [1216]This has been frequently revisited and expanded in thecontext of marine systems [1718] and is related to theJanzen-Connell hypothesis [1920] for tree biodiversityIn the KTW hypothesis [121617] there are two groups

of resource consumers for example bacteria and planktonThe plankton community generally has a lower efficiencyof resource usage than bacteria They remain in the system

only because a protozoan consumes the bacteria non-selectively and thus limits the bacterial population leavingroom for plankton to thrive Inside the bacterial communitydifferent strains have distinct growth rates They coexistwith no dominating winners due to host-specific virusescontrolling the corresponding strains This results in twolayers of coexistence through KTW dynamics (bacteria-plankton coexistence and bacterial strain coexistence)nested like Russian dolls [17]The original KTW model [1216] was formulated as

deterministic Lotka-Volterra-type equations for the speciesbiomass concentrations The high diversity of the system isexhibited in the steady state where multiple species coexistwith positive biomass these calculations assume that thesystem is spatially homogeneous and that the number ofindividuals is large enough so that it is valid to use acontinuous density to describe the population Howeverthis is not appropriate when the population is finite becauselarge fluctuations are able to drive the system towardsextinction an outcome that cannot be captured by acontinuous density that is allowed to become arbitrarilysmall [2122] Requiring the population size to be integer-valued leads inexorably to shot noise referred to in theecological context as demographic stochasticityThe purpose of this Letter is to explore the effect of

demographic stochasticity on the KTW paradigm anddemonstrate that the stochasticity causes the coexistencesteady state in the deterministic KTWmodel to break downthrough a cascade of extinctions leading to a loss ofdiversity This cascade can be avoided by allowing thepredators and prey to coevolve by mutation We propose astochastic model of the coevolution and show that itgenerically maintains the diversity of the ecosystem evenin the absence of spatial extension Our results for KTWmodels complement earlier findings that mutation controls

PRL 119 268101 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

29 DECEMBER 2017

0031-9007=17=119(26)=268101(5) 268101-1 copy 2017 American Physical Society

the diversity of strategies in more abstract models ofecosystems idealized as evolutionary games such as theprisonerrsquos dilemma [23] Our results strongly suggest thatdiversity reflects the dynamical interplay between ecologi-cal and evolutionary processes and is driven by how far thesystem is from an equilibrium ecological state (as couldbe quantified by deviations from detailed balance) Thesurprisingly deep role of demographic stochasticity uncov-ered here is consistent with earlier demonstrations thatindividual-level minimal models capture a wide variety ofecological phenomena including large-amplitude persis-tent population cycles [24] anomalous phase shifts due tothe emergence of mutant subpopulations [2526] spatialpatterns [2728] and even reversals of the direction ofselection [29] without requiring overly detailed modelingof interspecies interactionsModelmdashThe key component of the KTW hypothesis is

that for each resource competitor there is a correspondingpredator that can prevent it from becoming a dominantwinner The Russian-doll-like hierarchy is not essentiallyimportant for the basic idea Thus we focus on only asingle layer of KTW interaction the host-specific viralinfection and ignore the multilevel structureWe write down the individual reactions for a simplified

system ofm pairs of prey and predators which we will taketo be bacteria and viruses (phages) as follows

Xi⟶bi

2Xi Xi thorn Xj⟶eij

Xj eth1aTHORN

Yi thorn Xi⟶pi ethβi thorn 1THORNYi Yi⟶

di empty eth1bTHORN

All rates are positive i j frac14 1 2hellip m are strain indicesBacterial individuals Xi have strain-specific growth rate biThey compete with each other for an implicit resource withstrength eij Viruses of the ith strain Yi infect the corre-sponding host Xi with rate pi and burst size βi and decay tonothing empty with rate di These reactions form the minimalgeneralized KTW modelBelow are the corresponding mean-field rate equations

_Bi frac14 biBi minusXm

jfrac141

eijBiBj minus piBiVi eth2aTHORN

_Vi frac14 βipiBiVi minus diVi eth2bTHORN

The dot operator stands for the time derivative Bi and Virepresent the densities of the ith bacterial and viral strainsrespectively The competition matrix eij is sometimes takenas random [30] or more realistically as arising fromevolutionary dynamics such as in a food web [31] TheKTW model describes situations where diversity is main-tained by predation with a secondary contribution from thecompetition eij Here we are interested in the sensitivity of

KTW to stochasticity and the conclusions are not changedif we set eij to a constant value e for simplicityEquation (2) has a nonzero steady state as shown below

Bi frac14

diβipi

Vi frac14

1

pi

bi minus e

Xm

jfrac141

Bj

eth3THORN

We require all Bi and V

i to be positive which limits theparameters to satisfy bi gt e

Pmjfrac141 dj=βjpj forall i Linear

stability analysis shows that the steady state Eq (3) isexponentially stable with all eigenvalues of the linearstability matrix having negative real parts as long as thequantity xi equiv βip2

i Bi V

i frac14 diethbi minus e

Pmjfrac141 dj=βjpjTHORN is dis-

tinct for each i The steady state can be either a focus or anode depending on whether the eigenvalues have nonzeroimaginary parts or not The parameters used in this Letterresult in the steady state being a focus but the conclusionalso applies to the node caseIn Fig 1 we show in the first row the time series of prey

and predator densities obtained from a numerical evolutionof Eq (2) for m frac14 10 pairs of bacteria and phages Species

0 25 50 75 100t

(a)

0

001

002

003

popu

latio

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nsity

0 1000 2000 3000001500250035

0 25 50 75 100t

(b)

0

020

040

060

popu

latio

n de

nsity

0 1000 2000 3000030405

0 25 50 75 100t

(c)

0

005

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latio

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0 25 50 75 100t

(d)

0

050

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200

250

popu

latio

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nsity

FIG 1 Population density time series obtained from thegeneralized KTW framework with ten bacterium-phage pairsThe left column is for bacteria and the right for viruses The firstrow shows the result from a numerical evolution of the deter-ministic generalized KTW equations with species densitiesinitially perturbed randomly away from the steady state Theparameters are b frac14 eth075 08 085 09 095 1 105 11 11512THORN piequivpfrac142 βi equiv β frac14 10 di equiv d frac14 05 and eij equiv e frac14 01The insets show the long time behavior which demonstrates thatthe steady state is a focus For readability only the decays of B2

and V2 are shown The second row presents a typical simulationresult of the stochastic version of the generalized KTW modelusing the same set of parameters The system size is C frac14 1000and populations are initialized with the steady state value

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268101-2

densities are initially perturbed away from the steady stateAs shown in the figure insets species densities decay backto the steady state at long times confirming the result of thelinear stability analysis The oscillatory behavior at shorttime scales demonstrates the steady state to be a focusTo reveal the effect of demographic noise we also

conduct the stochastic simulation of the correspondingindividual level reactions (1) with the same parameter setusing the Gillespie algorithm [32] The resultant speciesdensity time series are shown in the second row in Fig 1 Incontrast to the deterministic behavior of oscillatory decayspecies go extinct in a short time Bacterial strains becomeextinct due to a random fluctuation this consequentiallytriggers the extinction of the corresponding viral strainsdue to a lack of food The number of species monotonicallydecreases in the process and the system diversity under-goes a cascadeThe reason that the stable deterministic steady state of

the generalized KTW model cannot be maintained in thepresence of demographic stochasticity lies in the fact thatspecies populations in the stochastic model are all finiteand the probability of the population reaching zero due to arandom fluctuation is always nonzeroEcosystems have evolved many potential mechanisms to

get around the path to extinction as introduced at thebeginning of the Letter Here we discuss one possibilityPrey and predator coevolve with each other so that fitmutants are constantly being introduced into the systemthus preventing the elimination of the species Specificallyprey improve their phenotypic traits (eg strengthening theshell) to escape from predators and predators also adjusttheir corresponding traits (eg sharpening the claws) tocatch prey This coevolutionary arms race has been welldocumented in many systems [3334] Previous theoreticalstudies focused on the dynamics of the traits of prey andpredator groups [35ndash37] and the structure of the predationnetwork [38] under different coevolving modes Here westudy how coevolution affects the diversity of the host-specific predation systemCoevolution modelmdashWe modify the stochastic general-

ized KTWmodel (1) by adding in the following two sets ofreactions to describe mutations of the prey Xi from strain ito i 1 and similarly those of the predator Yi

Xi⟶μ1=2Xi1 Yi⟶

μ2=2Yi1 eth4THORN

We assume that the mutation rates are strain independentand one individual can mutate into its two neighbor strainswith the same rate μ1=2 for bacteria or μ2=2 for viruses Weset the boundary condition to be open so that mutations outof the index set f1 2hellip mg are ignored We will refer toEqs (1) and (4) as the coevolving KTW (CKTW) modelFor sufficiently high mutation rates the absorbing

extinction state in the generalized KTW model can beavoided in the sense that a strain can reemerge as mutants

are generated from its neighbor relatives after itspopulation drops to zero Therefore mutation can stimulatea flux of population through different strains and promotecoexistenceWe define the diversity of the system in the CKTW

model using the Shannon entropy S frac14 minusP

mifrac141 fi ln fi

Here fi is the fraction of the ith bacterial (viral) strain inthe entire bacterial (viral) community The expressionreaches the maximum when all strains coexist at theirdeterministic steady state Eq (3) and the minimum 0when only one strain exists We score S frac14 minus1 if either thebacterial or viral community goes extinctWe present population density time series in Fig 2 and

the dependence of prey diversity on the mutation rates inFig 3 We set μ1 frac14 μ2 equiv μ for simplification The diversityof the prey community is calculated at the end of thediversity time series shown in the inset in Fig 3(a) after thesystem has gone through the transient region Although inprinciple species in a stochastic system will always goextinct at a time exponentially long depending on thepopulation size [22] this extinction time scale is notrelevant in our simulation and we thus focus on the systemstate in the long steady region before the destined collapseFor a small enough mutation rate (population time series

not shown) the entire community can become extinctbefore mutants can emerge and the system still collapsesdemonstrated by the diversity time series in the inset inFig 3(a) as in the generalized KTW model This corre-sponds to region I in Fig 3(a) For intermediate mutationrates most strains stay near extinction driven by

0 25 50 75 100t

(a)

0

010

020

030

040

popu

latio

n de

nsity

0 25 50 75 100t

(b)

0

10

20

30

40

50

popu

latio

n de

nsity

0 25 50 75 100t

(c)

0

002

004

006

008

010

popu

latio

n de

nsity

0 25 50 75 100t

(d)

0

02

04

06

08

10

12

popu

latio

n de

nsity

FIG 2 Population density time series in the stochastic co-evolving KTW model The left column is for bacteria and theright for viruses The system size is C frac14 1000 and the mutationrates are set to be equal μ1 frac14 μ2 equiv μ Other rates are the same asthose in Fig 1 The upper and lower rows show the cases of lowand high mutation rates μ frac14 0015 and μ frac14 1 respectively

PRL 119 268101 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

29 DECEMBER 2017

268101-3

demographic noise while some mutants can grow to bedominant if they happen to confront only a few predatorswhen first emerging Subsequently the predator populationexpands feeding on the dominating winners thus reducingthe winner population and allowing the next dominator togrow In this way we see that winner populations spikealternatively in the time series as in the first row in Fig 2Near the onset of coexistence the diversity has a largedeviation and is very sensitive to the mutation rate asshown in region II in Fig 3(a) The large deviation is alsoseen in the diversity time series in the inset For a largemutation rate the coevolution-driven population flow isfast enough to compensate for the demographic fluctua-tions All strains remain near the steady state and no onecan win over others as shown by the population time seriesin the second row in Fig 2 The diversity slowlyapproaches the maximum with small deviations as dem-onstrated in region III in Fig 3(a) For an extremely largemutation rate (figures not shown) we cannot view themutation as a perturbation to the ecological populationdynamics Species populations deviate from the mean-fieldsteady state Eq (3) Specifically under the boundary

condition in our model in which the mutations out ofthe species space f1 2hellip mg are effectively individualdeath the population leaks through the boundary andeventually reaches zeros at extremely large mutation ratesAccording to the above discussion we show three phasesof dynamics as illustrated in Fig 3(b) the extinction phaseat a low mutation rate the winner-alternating phase at anintermediate mutation rate and the coexisting phase at ahigh mutation rateOpen system modelmdashSo far we have preassigned a

fixed number of predator-prey pairs in the system A morerealistic approach is to let the system be open and evolve byitself to establish however many species there can beAs mutants take on new traits the population spreads in

the trait space This expansion usually is associated with atrade-off in the fitness [35] The further the trait is from theorigin the lower the growth rate becomes We model thistrade-off effect assuming a 1D trait space by setting up Mspecies and assigning the highest birth rate to the specieswith indexM=2 and decreasing the birth rate as the speciesindex goes fromM=2 to 1 and fromM=2 toM The specieswith indexM=2 is at the center of the trait space and then isthe origin of the trait expansion Species 1 and M have thelowest birth rates that are almost 0 and further mutation ofthe two will result in mutants with negative birth rateswhich cannot grow and are thus excluded from the modelThe species space f1 2hellipMg contains all possiblespecies that can potentially exist in the system Howeverunder conditions of resource limitation formulated by thecompetition strength e only a few with relatively highgrowth rates out ofM can eventually be established in thesystem The number of species that manage to thrivecorresponds to m in the previous modelsSee Supplemental Material [39] for the stochastic sim-

ulation parameters and the resultant population time seriesand diversity dependence on the mutation rate Eventhough the number of established pairs varies with timeand the population leaks out of the region deterministicallyallowed by the carrying capacity the system still exhibitsthree phases depending on the mutation rate similar to theCKTW model with a fixed number of speciesDiscussionmdashIn contrast to the model in Ref [18] where

killing winners is exerted externally we focus on theintrinsically established KTW of the system In the inter-mediate and fast mutation regions of the CKTWmodel theecological and evolutionary dynamics are coupled to eachother and occur on the same time scale This type ofcoupling can most easily be observed in microbial systemsin which organisms have a high mutation frequency[264041] Recent work has shown clearly the existenceof genomic islands where genomes of different strains varyin loci thought to be associated with phage resistance [42]Both host-specific predation and mutation are important ingenerating the observed diversity of the bacterial genomeThe minimal CKTW model can in principle describe the

10-4 10-3 10-2 10-1 100-10

-05

0

05

10

15

20

25

S

I II III

0 200 400

t

-1

0

1

2

S

(a)

mutationrate

low intermediate high

extinctionwinners

alternatingall species coexisting

(b)

FIG 3 (a) The main figure shows the prey diversity S definedin the main text as a function of the mutation rate μ1 frac14 μ2 equiv μFor each value of μ we conduct 100 replicates and calculate thediversity values at the end of the simulations represented by thegray dots with the blue one being their mean The inset showsdiversity time series at mutation rates from the three regions withμ frac14 0 0015 and 1 respectively For this particular set ofparameters the mean-field generalized KTW equations give anequal bacterial strain concentration at the steady state and themaximum diversity in the corresponding CKTW model is lnm(b) A descriptive phase diagram of the dynamics with themutation rate as the tuning parameter

PRL 119 268101 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

29 DECEMBER 2017

268101-4

diversity in the above system For example by controllingthe mutation rate through an inducible promoter usingmolecular techniques pioneered in Ref [43] we envisage afast bacterium-phage coevolution experiment to test thepredicted phase diagramIn addition to inevitable simplification of biological

details both the generalized KTW and the coevolvingKTW models assume that the system is well mixedignoring any spatial dispersion Consequently they cannotcapture the reservoir effect [44] present in an ecosystemwhich means that for any local community organisms inits surrounding environment can move into it keeping itsupplied and refreshed Specifically even if a species goesextinct in a local community it can be reseeded there by thesurrounding reservoir Well-mixed models should bethought of as describing not the entire system but a muchsmaller correlation volume in which local demographicstochasticity can be significant [2745]

This work was supported by the National Aeronauticsand Space Administration Astrobiology Institute (NAI)under Cooperative Agreement No NNA13AA91A issuedthrough the Science Mission Directorate C X was parti-ally supported by the L S Edelheit Family BiologicalPhysics Fellowship from the Department of PhysicsUniversity of Illinois at Urbana-Champaign We wouldlike to thank Tron Frede Thingstad Selina Varingge Hong-YanShih and Brendan Bohannan for helpful discussions

nigeluiucedu[1] G E Hutchinson Am Nat 95 137 (1961)[2] J S Clark M Dietze S Chakraborty P K Agarwal I

Ibanez S LaDeau and M Wolosin Ecol Lett 10 647(2007)

[3] G Hardin Science 131 1292 (1960)[4] J B Wilson N Z J Ecol 13 17 (1990)[5] S Roy and J Chattopadhyay Ecol Complex 4 26 (2007)[6] P Chesson Annu Rev Ecol Syst 31 343 (2000)[7] R Levins Am Nat 114 765 (1979)[8] P Richerson R Armstrong and C R Goldman Proc Natl

Acad Sci USA 67 1710 (1970)[9] S A Levin Am Nat 108 207 (1974)

[10] J W Fox Trends Ecol Evol 28 86 (2013)[11] M Scheffer S Rinaldi J Huisman and F J Weissing

Hydrobiologia 491 9 (2003)[12] T Thingstad and R Lignell Aquatic Microbial Ecology 13

19 (1997)[13] K Vetsigian R Jajoo and R Kishony PLoS Biol 9

e1001184 (2011)[14] A Fernaacutendez S Huang S Seston J Xing R Hickey C

Criddle and J Tiedje Appl Environ Microbiol 65 3697(1999)

[15] S Louca and M Doebeli Environ Microbiol 19 3863(2017)

[16] T F Thingstad Limnol Oceanogr 45 1320 (2000)[17] CWinter T Bouvier M GWeinbauer and T F Thingstad

Microbiol Mol Biol Rev 74 42 (2010)[18] S Maslov and K Sneppen Sci Rep 7 39642 (2017)[19] D H Janzen Am Nat 104 501 (1970)[20] J H Connell in Dynamics of Population edited by P J

den Boer and G R Gradwell (PUDOC Wageningen TheNetherlands 1971) pp 298ndash312

[21] O Ovaskainen and B Meerson Trends Ecol Evol 25 643(2010)

[22] O Gottesman and B Meerson Phys Rev E 85 021140(2012)

[23] D F Toupo D G Rand and S H Strogatz Int J Bifur-cation Chaos Appl Sci Eng 24 1430035 (2014)

[24] A J McKane and T J Newman Phys Rev Lett 94218102 (2005)

[25] T Yoshida L E Jones S P Ellner G F Fussmann andN G Hairston Nature (London) 424 303 (2003)

[26] H-Y Shih and N Goldenfeld Phys Rev E 90 050702(2014)

[27] U C Taumluber J Phys A 45 405002 (2012)[28] T Butler and N Goldenfeld Phys Rev E 80 030902

(2009)[29] GW A Constable T Rogers A J McKane and C E

Tarnita Proc Natl Acad Sci USA 113 E4745 (2016)[30] R M May Nature (London) 238 413 (1972)[31] A J McKane Eur Phys J B 38 287 (2004)[32] D T Gillespie J Comput Phys 22 403 (1976)[33] S T Chisholm G Coaker B Day and B J Staskawicz

Cell 124 803 (2006)[34] A Buckling and P B Rainey Proc R Soc B 269 931

(2002)[35] U Dieckmann P Marrow and R Law J Theor Biol 176

91 (1995)[36] A Agrawal and C M Lively Evol Ecol Res 4 91 (2002)[37] J S Weitz H Hartman and S A Levin Proc Natl Acad

Sci USA 102 9535 (2005)[38] S J Beckett and H T Williams Interface Focus 3

20130033 (2013)[39] See Supplemental Material at httplinkapsorg

supplemental101103PhysRevLett119268101 for simu-lations of an open coevolving ecosystem It providesdetailed stochastic simulation parameters and the resultantpopulation time series and diversity dependence on themutation rate

[40] B J Bohannan and R E Lenski Ecology 78 2303 (1997)[41] T Yoshida S P Ellner L E Jones B J Bohannan R E

Lenski and N G Hairston Jr PLoS Biol 5 e235 (2007)[42] F Rodriguez-Valera A-B Martin-Cuadrado B Rodriguez-

Brito L Pašić T F Thingstad F Rohwer and A MiraNat Rev Microbiol 7 828 (2009)

[43] N H Kim G Lee N A Sherer K M Martini NGoldenfeld and T E Kuhlman Proc Natl Acad SciUSA 113 7278 (2016)

[44] A Shmida and S Ellner Vegetatio 58 29 (1984)[45] T Butler and N Goldenfeld Phys Rev E 84 011112

(2011)

PRL 119 268101 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

29 DECEMBER 2017

268101-5

densities are initially perturbed away from the steady stateAs shown in the figure insets species densities decay backto the steady state at long times confirming the result of thelinear stability analysis The oscillatory behavior at shorttime scales demonstrates the steady state to be a focusTo reveal the effect of demographic noise we also

conduct the stochastic simulation of the correspondingindividual level reactions (1) with the same parameter setusing the Gillespie algorithm [32] The resultant speciesdensity time series are shown in the second row in Fig 1 Incontrast to the deterministic behavior of oscillatory decayspecies go extinct in a short time Bacterial strains becomeextinct due to a random fluctuation this consequentiallytriggers the extinction of the corresponding viral strainsdue to a lack of food The number of species monotonicallydecreases in the process and the system diversity under-goes a cascadeThe reason that the stable deterministic steady state of

the generalized KTW model cannot be maintained in thepresence of demographic stochasticity lies in the fact thatspecies populations in the stochastic model are all finiteand the probability of the population reaching zero due to arandom fluctuation is always nonzeroEcosystems have evolved many potential mechanisms to

get around the path to extinction as introduced at thebeginning of the Letter Here we discuss one possibilityPrey and predator coevolve with each other so that fitmutants are constantly being introduced into the systemthus preventing the elimination of the species Specificallyprey improve their phenotypic traits (eg strengthening theshell) to escape from predators and predators also adjusttheir corresponding traits (eg sharpening the claws) tocatch prey This coevolutionary arms race has been welldocumented in many systems [3334] Previous theoreticalstudies focused on the dynamics of the traits of prey andpredator groups [35ndash37] and the structure of the predationnetwork [38] under different coevolving modes Here westudy how coevolution affects the diversity of the host-specific predation systemCoevolution modelmdashWe modify the stochastic general-

ized KTWmodel (1) by adding in the following two sets ofreactions to describe mutations of the prey Xi from strain ito i 1 and similarly those of the predator Yi

Xi⟶μ1=2Xi1 Yi⟶

μ2=2Yi1 eth4THORN

We assume that the mutation rates are strain independentand one individual can mutate into its two neighbor strainswith the same rate μ1=2 for bacteria or μ2=2 for viruses Weset the boundary condition to be open so that mutations outof the index set f1 2hellip mg are ignored We will refer toEqs (1) and (4) as the coevolving KTW (CKTW) modelFor sufficiently high mutation rates the absorbing

extinction state in the generalized KTW model can beavoided in the sense that a strain can reemerge as mutants

are generated from its neighbor relatives after itspopulation drops to zero Therefore mutation can stimulatea flux of population through different strains and promotecoexistenceWe define the diversity of the system in the CKTW

model using the Shannon entropy S frac14 minusP

mifrac141 fi ln fi

Here fi is the fraction of the ith bacterial (viral) strain inthe entire bacterial (viral) community The expressionreaches the maximum when all strains coexist at theirdeterministic steady state Eq (3) and the minimum 0when only one strain exists We score S frac14 minus1 if either thebacterial or viral community goes extinctWe present population density time series in Fig 2 and

the dependence of prey diversity on the mutation rates inFig 3 We set μ1 frac14 μ2 equiv μ for simplification The diversityof the prey community is calculated at the end of thediversity time series shown in the inset in Fig 3(a) after thesystem has gone through the transient region Although inprinciple species in a stochastic system will always goextinct at a time exponentially long depending on thepopulation size [22] this extinction time scale is notrelevant in our simulation and we thus focus on the systemstate in the long steady region before the destined collapseFor a small enough mutation rate (population time series

not shown) the entire community can become extinctbefore mutants can emerge and the system still collapsesdemonstrated by the diversity time series in the inset inFig 3(a) as in the generalized KTW model This corre-sponds to region I in Fig 3(a) For intermediate mutationrates most strains stay near extinction driven by

0 25 50 75 100t

(a)

0

010

020

030

040

popu

latio

n de

nsity

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(b)

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latio

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latio

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FIG 2 Population density time series in the stochastic co-evolving KTW model The left column is for bacteria and theright for viruses The system size is C frac14 1000 and the mutationrates are set to be equal μ1 frac14 μ2 equiv μ Other rates are the same asthose in Fig 1 The upper and lower rows show the cases of lowand high mutation rates μ frac14 0015 and μ frac14 1 respectively

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268101-3

demographic noise while some mutants can grow to bedominant if they happen to confront only a few predatorswhen first emerging Subsequently the predator populationexpands feeding on the dominating winners thus reducingthe winner population and allowing the next dominator togrow In this way we see that winner populations spikealternatively in the time series as in the first row in Fig 2Near the onset of coexistence the diversity has a largedeviation and is very sensitive to the mutation rate asshown in region II in Fig 3(a) The large deviation is alsoseen in the diversity time series in the inset For a largemutation rate the coevolution-driven population flow isfast enough to compensate for the demographic fluctua-tions All strains remain near the steady state and no onecan win over others as shown by the population time seriesin the second row in Fig 2 The diversity slowlyapproaches the maximum with small deviations as dem-onstrated in region III in Fig 3(a) For an extremely largemutation rate (figures not shown) we cannot view themutation as a perturbation to the ecological populationdynamics Species populations deviate from the mean-fieldsteady state Eq (3) Specifically under the boundary

condition in our model in which the mutations out ofthe species space f1 2hellip mg are effectively individualdeath the population leaks through the boundary andeventually reaches zeros at extremely large mutation ratesAccording to the above discussion we show three phasesof dynamics as illustrated in Fig 3(b) the extinction phaseat a low mutation rate the winner-alternating phase at anintermediate mutation rate and the coexisting phase at ahigh mutation rateOpen system modelmdashSo far we have preassigned a

fixed number of predator-prey pairs in the system A morerealistic approach is to let the system be open and evolve byitself to establish however many species there can beAs mutants take on new traits the population spreads in

the trait space This expansion usually is associated with atrade-off in the fitness [35] The further the trait is from theorigin the lower the growth rate becomes We model thistrade-off effect assuming a 1D trait space by setting up Mspecies and assigning the highest birth rate to the specieswith indexM=2 and decreasing the birth rate as the speciesindex goes fromM=2 to 1 and fromM=2 toM The specieswith indexM=2 is at the center of the trait space and then isthe origin of the trait expansion Species 1 and M have thelowest birth rates that are almost 0 and further mutation ofthe two will result in mutants with negative birth rateswhich cannot grow and are thus excluded from the modelThe species space f1 2hellipMg contains all possiblespecies that can potentially exist in the system Howeverunder conditions of resource limitation formulated by thecompetition strength e only a few with relatively highgrowth rates out ofM can eventually be established in thesystem The number of species that manage to thrivecorresponds to m in the previous modelsSee Supplemental Material [39] for the stochastic sim-

ulation parameters and the resultant population time seriesand diversity dependence on the mutation rate Eventhough the number of established pairs varies with timeand the population leaks out of the region deterministicallyallowed by the carrying capacity the system still exhibitsthree phases depending on the mutation rate similar to theCKTW model with a fixed number of speciesDiscussionmdashIn contrast to the model in Ref [18] where

killing winners is exerted externally we focus on theintrinsically established KTW of the system In the inter-mediate and fast mutation regions of the CKTWmodel theecological and evolutionary dynamics are coupled to eachother and occur on the same time scale This type ofcoupling can most easily be observed in microbial systemsin which organisms have a high mutation frequency[264041] Recent work has shown clearly the existenceof genomic islands where genomes of different strains varyin loci thought to be associated with phage resistance [42]Both host-specific predation and mutation are important ingenerating the observed diversity of the bacterial genomeThe minimal CKTW model can in principle describe the

10-4 10-3 10-2 10-1 100-10

-05

0

05

10

15

20

25

S

I II III

0 200 400

t

-1

0

1

2

S

(a)

mutationrate

low intermediate high

extinctionwinners

alternatingall species coexisting

(b)

FIG 3 (a) The main figure shows the prey diversity S definedin the main text as a function of the mutation rate μ1 frac14 μ2 equiv μFor each value of μ we conduct 100 replicates and calculate thediversity values at the end of the simulations represented by thegray dots with the blue one being their mean The inset showsdiversity time series at mutation rates from the three regions withμ frac14 0 0015 and 1 respectively For this particular set ofparameters the mean-field generalized KTW equations give anequal bacterial strain concentration at the steady state and themaximum diversity in the corresponding CKTW model is lnm(b) A descriptive phase diagram of the dynamics with themutation rate as the tuning parameter

PRL 119 268101 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

29 DECEMBER 2017

268101-4

diversity in the above system For example by controllingthe mutation rate through an inducible promoter usingmolecular techniques pioneered in Ref [43] we envisage afast bacterium-phage coevolution experiment to test thepredicted phase diagramIn addition to inevitable simplification of biological

details both the generalized KTW and the coevolvingKTW models assume that the system is well mixedignoring any spatial dispersion Consequently they cannotcapture the reservoir effect [44] present in an ecosystemwhich means that for any local community organisms inits surrounding environment can move into it keeping itsupplied and refreshed Specifically even if a species goesextinct in a local community it can be reseeded there by thesurrounding reservoir Well-mixed models should bethought of as describing not the entire system but a muchsmaller correlation volume in which local demographicstochasticity can be significant [2745]

This work was supported by the National Aeronauticsand Space Administration Astrobiology Institute (NAI)under Cooperative Agreement No NNA13AA91A issuedthrough the Science Mission Directorate C X was parti-ally supported by the L S Edelheit Family BiologicalPhysics Fellowship from the Department of PhysicsUniversity of Illinois at Urbana-Champaign We wouldlike to thank Tron Frede Thingstad Selina Varingge Hong-YanShih and Brendan Bohannan for helpful discussions

nigeluiucedu[1] G E Hutchinson Am Nat 95 137 (1961)[2] J S Clark M Dietze S Chakraborty P K Agarwal I

Ibanez S LaDeau and M Wolosin Ecol Lett 10 647(2007)

[3] G Hardin Science 131 1292 (1960)[4] J B Wilson N Z J Ecol 13 17 (1990)[5] S Roy and J Chattopadhyay Ecol Complex 4 26 (2007)[6] P Chesson Annu Rev Ecol Syst 31 343 (2000)[7] R Levins Am Nat 114 765 (1979)[8] P Richerson R Armstrong and C R Goldman Proc Natl

Acad Sci USA 67 1710 (1970)[9] S A Levin Am Nat 108 207 (1974)

[10] J W Fox Trends Ecol Evol 28 86 (2013)[11] M Scheffer S Rinaldi J Huisman and F J Weissing

Hydrobiologia 491 9 (2003)[12] T Thingstad and R Lignell Aquatic Microbial Ecology 13

19 (1997)[13] K Vetsigian R Jajoo and R Kishony PLoS Biol 9

e1001184 (2011)[14] A Fernaacutendez S Huang S Seston J Xing R Hickey C

Criddle and J Tiedje Appl Environ Microbiol 65 3697(1999)

[15] S Louca and M Doebeli Environ Microbiol 19 3863(2017)

[16] T F Thingstad Limnol Oceanogr 45 1320 (2000)[17] CWinter T Bouvier M GWeinbauer and T F Thingstad

Microbiol Mol Biol Rev 74 42 (2010)[18] S Maslov and K Sneppen Sci Rep 7 39642 (2017)[19] D H Janzen Am Nat 104 501 (1970)[20] J H Connell in Dynamics of Population edited by P J

den Boer and G R Gradwell (PUDOC Wageningen TheNetherlands 1971) pp 298ndash312

[21] O Ovaskainen and B Meerson Trends Ecol Evol 25 643(2010)

[22] O Gottesman and B Meerson Phys Rev E 85 021140(2012)

[23] D F Toupo D G Rand and S H Strogatz Int J Bifur-cation Chaos Appl Sci Eng 24 1430035 (2014)

[24] A J McKane and T J Newman Phys Rev Lett 94218102 (2005)

[25] T Yoshida L E Jones S P Ellner G F Fussmann andN G Hairston Nature (London) 424 303 (2003)

[26] H-Y Shih and N Goldenfeld Phys Rev E 90 050702(2014)

[27] U C Taumluber J Phys A 45 405002 (2012)[28] T Butler and N Goldenfeld Phys Rev E 80 030902

(2009)[29] GW A Constable T Rogers A J McKane and C E

Tarnita Proc Natl Acad Sci USA 113 E4745 (2016)[30] R M May Nature (London) 238 413 (1972)[31] A J McKane Eur Phys J B 38 287 (2004)[32] D T Gillespie J Comput Phys 22 403 (1976)[33] S T Chisholm G Coaker B Day and B J Staskawicz

Cell 124 803 (2006)[34] A Buckling and P B Rainey Proc R Soc B 269 931

(2002)[35] U Dieckmann P Marrow and R Law J Theor Biol 176

91 (1995)[36] A Agrawal and C M Lively Evol Ecol Res 4 91 (2002)[37] J S Weitz H Hartman and S A Levin Proc Natl Acad

Sci USA 102 9535 (2005)[38] S J Beckett and H T Williams Interface Focus 3

20130033 (2013)[39] See Supplemental Material at httplinkapsorg

supplemental101103PhysRevLett119268101 for simu-lations of an open coevolving ecosystem It providesdetailed stochastic simulation parameters and the resultantpopulation time series and diversity dependence on themutation rate

[40] B J Bohannan and R E Lenski Ecology 78 2303 (1997)[41] T Yoshida S P Ellner L E Jones B J Bohannan R E

Lenski and N G Hairston Jr PLoS Biol 5 e235 (2007)[42] F Rodriguez-Valera A-B Martin-Cuadrado B Rodriguez-

Brito L Pašić T F Thingstad F Rohwer and A MiraNat Rev Microbiol 7 828 (2009)

[43] N H Kim G Lee N A Sherer K M Martini NGoldenfeld and T E Kuhlman Proc Natl Acad SciUSA 113 7278 (2016)

[44] A Shmida and S Ellner Vegetatio 58 29 (1984)[45] T Butler and N Goldenfeld Phys Rev E 84 011112

(2011)

PRL 119 268101 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

29 DECEMBER 2017

268101-5

demographic noise while some mutants can grow to bedominant if they happen to confront only a few predatorswhen first emerging Subsequently the predator populationexpands feeding on the dominating winners thus reducingthe winner population and allowing the next dominator togrow In this way we see that winner populations spikealternatively in the time series as in the first row in Fig 2Near the onset of coexistence the diversity has a largedeviation and is very sensitive to the mutation rate asshown in region II in Fig 3(a) The large deviation is alsoseen in the diversity time series in the inset For a largemutation rate the coevolution-driven population flow isfast enough to compensate for the demographic fluctua-tions All strains remain near the steady state and no onecan win over others as shown by the population time seriesin the second row in Fig 2 The diversity slowlyapproaches the maximum with small deviations as dem-onstrated in region III in Fig 3(a) For an extremely largemutation rate (figures not shown) we cannot view themutation as a perturbation to the ecological populationdynamics Species populations deviate from the mean-fieldsteady state Eq (3) Specifically under the boundary

condition in our model in which the mutations out ofthe species space f1 2hellip mg are effectively individualdeath the population leaks through the boundary andeventually reaches zeros at extremely large mutation ratesAccording to the above discussion we show three phasesof dynamics as illustrated in Fig 3(b) the extinction phaseat a low mutation rate the winner-alternating phase at anintermediate mutation rate and the coexisting phase at ahigh mutation rateOpen system modelmdashSo far we have preassigned a

fixed number of predator-prey pairs in the system A morerealistic approach is to let the system be open and evolve byitself to establish however many species there can beAs mutants take on new traits the population spreads in

the trait space This expansion usually is associated with atrade-off in the fitness [35] The further the trait is from theorigin the lower the growth rate becomes We model thistrade-off effect assuming a 1D trait space by setting up Mspecies and assigning the highest birth rate to the specieswith indexM=2 and decreasing the birth rate as the speciesindex goes fromM=2 to 1 and fromM=2 toM The specieswith indexM=2 is at the center of the trait space and then isthe origin of the trait expansion Species 1 and M have thelowest birth rates that are almost 0 and further mutation ofthe two will result in mutants with negative birth rateswhich cannot grow and are thus excluded from the modelThe species space f1 2hellipMg contains all possiblespecies that can potentially exist in the system Howeverunder conditions of resource limitation formulated by thecompetition strength e only a few with relatively highgrowth rates out ofM can eventually be established in thesystem The number of species that manage to thrivecorresponds to m in the previous modelsSee Supplemental Material [39] for the stochastic sim-

ulation parameters and the resultant population time seriesand diversity dependence on the mutation rate Eventhough the number of established pairs varies with timeand the population leaks out of the region deterministicallyallowed by the carrying capacity the system still exhibitsthree phases depending on the mutation rate similar to theCKTW model with a fixed number of speciesDiscussionmdashIn contrast to the model in Ref [18] where

killing winners is exerted externally we focus on theintrinsically established KTW of the system In the inter-mediate and fast mutation regions of the CKTWmodel theecological and evolutionary dynamics are coupled to eachother and occur on the same time scale This type ofcoupling can most easily be observed in microbial systemsin which organisms have a high mutation frequency[264041] Recent work has shown clearly the existenceof genomic islands where genomes of different strains varyin loci thought to be associated with phage resistance [42]Both host-specific predation and mutation are important ingenerating the observed diversity of the bacterial genomeThe minimal CKTW model can in principle describe the

10-4 10-3 10-2 10-1 100-10

-05

0

05

10

15

20

25

S

I II III

0 200 400

t

-1

0

1

2

S

(a)

mutationrate

low intermediate high

extinctionwinners

alternatingall species coexisting

(b)

FIG 3 (a) The main figure shows the prey diversity S definedin the main text as a function of the mutation rate μ1 frac14 μ2 equiv μFor each value of μ we conduct 100 replicates and calculate thediversity values at the end of the simulations represented by thegray dots with the blue one being their mean The inset showsdiversity time series at mutation rates from the three regions withμ frac14 0 0015 and 1 respectively For this particular set ofparameters the mean-field generalized KTW equations give anequal bacterial strain concentration at the steady state and themaximum diversity in the corresponding CKTW model is lnm(b) A descriptive phase diagram of the dynamics with themutation rate as the tuning parameter

PRL 119 268101 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

29 DECEMBER 2017

268101-4

diversity in the above system For example by controllingthe mutation rate through an inducible promoter usingmolecular techniques pioneered in Ref [43] we envisage afast bacterium-phage coevolution experiment to test thepredicted phase diagramIn addition to inevitable simplification of biological

details both the generalized KTW and the coevolvingKTW models assume that the system is well mixedignoring any spatial dispersion Consequently they cannotcapture the reservoir effect [44] present in an ecosystemwhich means that for any local community organisms inits surrounding environment can move into it keeping itsupplied and refreshed Specifically even if a species goesextinct in a local community it can be reseeded there by thesurrounding reservoir Well-mixed models should bethought of as describing not the entire system but a muchsmaller correlation volume in which local demographicstochasticity can be significant [2745]

This work was supported by the National Aeronauticsand Space Administration Astrobiology Institute (NAI)under Cooperative Agreement No NNA13AA91A issuedthrough the Science Mission Directorate C X was parti-ally supported by the L S Edelheit Family BiologicalPhysics Fellowship from the Department of PhysicsUniversity of Illinois at Urbana-Champaign We wouldlike to thank Tron Frede Thingstad Selina Varingge Hong-YanShih and Brendan Bohannan for helpful discussions

nigeluiucedu[1] G E Hutchinson Am Nat 95 137 (1961)[2] J S Clark M Dietze S Chakraborty P K Agarwal I

Ibanez S LaDeau and M Wolosin Ecol Lett 10 647(2007)

[3] G Hardin Science 131 1292 (1960)[4] J B Wilson N Z J Ecol 13 17 (1990)[5] S Roy and J Chattopadhyay Ecol Complex 4 26 (2007)[6] P Chesson Annu Rev Ecol Syst 31 343 (2000)[7] R Levins Am Nat 114 765 (1979)[8] P Richerson R Armstrong and C R Goldman Proc Natl

Acad Sci USA 67 1710 (1970)[9] S A Levin Am Nat 108 207 (1974)

[10] J W Fox Trends Ecol Evol 28 86 (2013)[11] M Scheffer S Rinaldi J Huisman and F J Weissing

Hydrobiologia 491 9 (2003)[12] T Thingstad and R Lignell Aquatic Microbial Ecology 13

19 (1997)[13] K Vetsigian R Jajoo and R Kishony PLoS Biol 9

e1001184 (2011)[14] A Fernaacutendez S Huang S Seston J Xing R Hickey C

Criddle and J Tiedje Appl Environ Microbiol 65 3697(1999)

[15] S Louca and M Doebeli Environ Microbiol 19 3863(2017)

[16] T F Thingstad Limnol Oceanogr 45 1320 (2000)[17] CWinter T Bouvier M GWeinbauer and T F Thingstad

Microbiol Mol Biol Rev 74 42 (2010)[18] S Maslov and K Sneppen Sci Rep 7 39642 (2017)[19] D H Janzen Am Nat 104 501 (1970)[20] J H Connell in Dynamics of Population edited by P J

den Boer and G R Gradwell (PUDOC Wageningen TheNetherlands 1971) pp 298ndash312

[21] O Ovaskainen and B Meerson Trends Ecol Evol 25 643(2010)

[22] O Gottesman and B Meerson Phys Rev E 85 021140(2012)

[23] D F Toupo D G Rand and S H Strogatz Int J Bifur-cation Chaos Appl Sci Eng 24 1430035 (2014)

[24] A J McKane and T J Newman Phys Rev Lett 94218102 (2005)

[25] T Yoshida L E Jones S P Ellner G F Fussmann andN G Hairston Nature (London) 424 303 (2003)

[26] H-Y Shih and N Goldenfeld Phys Rev E 90 050702(2014)

[27] U C Taumluber J Phys A 45 405002 (2012)[28] T Butler and N Goldenfeld Phys Rev E 80 030902

(2009)[29] GW A Constable T Rogers A J McKane and C E

Tarnita Proc Natl Acad Sci USA 113 E4745 (2016)[30] R M May Nature (London) 238 413 (1972)[31] A J McKane Eur Phys J B 38 287 (2004)[32] D T Gillespie J Comput Phys 22 403 (1976)[33] S T Chisholm G Coaker B Day and B J Staskawicz

Cell 124 803 (2006)[34] A Buckling and P B Rainey Proc R Soc B 269 931

(2002)[35] U Dieckmann P Marrow and R Law J Theor Biol 176

91 (1995)[36] A Agrawal and C M Lively Evol Ecol Res 4 91 (2002)[37] J S Weitz H Hartman and S A Levin Proc Natl Acad

Sci USA 102 9535 (2005)[38] S J Beckett and H T Williams Interface Focus 3

20130033 (2013)[39] See Supplemental Material at httplinkapsorg

supplemental101103PhysRevLett119268101 for simu-lations of an open coevolving ecosystem It providesdetailed stochastic simulation parameters and the resultantpopulation time series and diversity dependence on themutation rate

[40] B J Bohannan and R E Lenski Ecology 78 2303 (1997)[41] T Yoshida S P Ellner L E Jones B J Bohannan R E

Lenski and N G Hairston Jr PLoS Biol 5 e235 (2007)[42] F Rodriguez-Valera A-B Martin-Cuadrado B Rodriguez-

Brito L Pašić T F Thingstad F Rohwer and A MiraNat Rev Microbiol 7 828 (2009)

[43] N H Kim G Lee N A Sherer K M Martini NGoldenfeld and T E Kuhlman Proc Natl Acad SciUSA 113 7278 (2016)

[44] A Shmida and S Ellner Vegetatio 58 29 (1984)[45] T Butler and N Goldenfeld Phys Rev E 84 011112

(2011)

PRL 119 268101 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

29 DECEMBER 2017

268101-5

diversity in the above system For example by controllingthe mutation rate through an inducible promoter usingmolecular techniques pioneered in Ref [43] we envisage afast bacterium-phage coevolution experiment to test thepredicted phase diagramIn addition to inevitable simplification of biological

details both the generalized KTW and the coevolvingKTW models assume that the system is well mixedignoring any spatial dispersion Consequently they cannotcapture the reservoir effect [44] present in an ecosystemwhich means that for any local community organisms inits surrounding environment can move into it keeping itsupplied and refreshed Specifically even if a species goesextinct in a local community it can be reseeded there by thesurrounding reservoir Well-mixed models should bethought of as describing not the entire system but a muchsmaller correlation volume in which local demographicstochasticity can be significant [2745]

This work was supported by the National Aeronauticsand Space Administration Astrobiology Institute (NAI)under Cooperative Agreement No NNA13AA91A issuedthrough the Science Mission Directorate C X was parti-ally supported by the L S Edelheit Family BiologicalPhysics Fellowship from the Department of PhysicsUniversity of Illinois at Urbana-Champaign We wouldlike to thank Tron Frede Thingstad Selina Varingge Hong-YanShih and Brendan Bohannan for helpful discussions

nigeluiucedu[1] G E Hutchinson Am Nat 95 137 (1961)[2] J S Clark M Dietze S Chakraborty P K Agarwal I

Ibanez S LaDeau and M Wolosin Ecol Lett 10 647(2007)

[3] G Hardin Science 131 1292 (1960)[4] J B Wilson N Z J Ecol 13 17 (1990)[5] S Roy and J Chattopadhyay Ecol Complex 4 26 (2007)[6] P Chesson Annu Rev Ecol Syst 31 343 (2000)[7] R Levins Am Nat 114 765 (1979)[8] P Richerson R Armstrong and C R Goldman Proc Natl

Acad Sci USA 67 1710 (1970)[9] S A Levin Am Nat 108 207 (1974)

[10] J W Fox Trends Ecol Evol 28 86 (2013)[11] M Scheffer S Rinaldi J Huisman and F J Weissing

Hydrobiologia 491 9 (2003)[12] T Thingstad and R Lignell Aquatic Microbial Ecology 13

19 (1997)[13] K Vetsigian R Jajoo and R Kishony PLoS Biol 9

e1001184 (2011)[14] A Fernaacutendez S Huang S Seston J Xing R Hickey C

Criddle and J Tiedje Appl Environ Microbiol 65 3697(1999)

[15] S Louca and M Doebeli Environ Microbiol 19 3863(2017)

[16] T F Thingstad Limnol Oceanogr 45 1320 (2000)[17] CWinter T Bouvier M GWeinbauer and T F Thingstad

Microbiol Mol Biol Rev 74 42 (2010)[18] S Maslov and K Sneppen Sci Rep 7 39642 (2017)[19] D H Janzen Am Nat 104 501 (1970)[20] J H Connell in Dynamics of Population edited by P J

den Boer and G R Gradwell (PUDOC Wageningen TheNetherlands 1971) pp 298ndash312

[21] O Ovaskainen and B Meerson Trends Ecol Evol 25 643(2010)

[22] O Gottesman and B Meerson Phys Rev E 85 021140(2012)

[23] D F Toupo D G Rand and S H Strogatz Int J Bifur-cation Chaos Appl Sci Eng 24 1430035 (2014)

[24] A J McKane and T J Newman Phys Rev Lett 94218102 (2005)

[25] T Yoshida L E Jones S P Ellner G F Fussmann andN G Hairston Nature (London) 424 303 (2003)

[26] H-Y Shih and N Goldenfeld Phys Rev E 90 050702(2014)

[27] U C Taumluber J Phys A 45 405002 (2012)[28] T Butler and N Goldenfeld Phys Rev E 80 030902

(2009)[29] GW A Constable T Rogers A J McKane and C E

Tarnita Proc Natl Acad Sci USA 113 E4745 (2016)[30] R M May Nature (London) 238 413 (1972)[31] A J McKane Eur Phys J B 38 287 (2004)[32] D T Gillespie J Comput Phys 22 403 (1976)[33] S T Chisholm G Coaker B Day and B J Staskawicz

Cell 124 803 (2006)[34] A Buckling and P B Rainey Proc R Soc B 269 931

(2002)[35] U Dieckmann P Marrow and R Law J Theor Biol 176

91 (1995)[36] A Agrawal and C M Lively Evol Ecol Res 4 91 (2002)[37] J S Weitz H Hartman and S A Levin Proc Natl Acad

Sci USA 102 9535 (2005)[38] S J Beckett and H T Williams Interface Focus 3

20130033 (2013)[39] See Supplemental Material at httplinkapsorg

supplemental101103PhysRevLett119268101 for simu-lations of an open coevolving ecosystem It providesdetailed stochastic simulation parameters and the resultantpopulation time series and diversity dependence on themutation rate

[40] B J Bohannan and R E Lenski Ecology 78 2303 (1997)[41] T Yoshida S P Ellner L E Jones B J Bohannan R E

Lenski and N G Hairston Jr PLoS Biol 5 e235 (2007)[42] F Rodriguez-Valera A-B Martin-Cuadrado B Rodriguez-

Brito L Pašić T F Thingstad F Rohwer and A MiraNat Rev Microbiol 7 828 (2009)

[43] N H Kim G Lee N A Sherer K M Martini NGoldenfeld and T E Kuhlman Proc Natl Acad SciUSA 113 7278 (2016)

[44] A Shmida and S Ellner Vegetatio 58 29 (1984)[45] T Butler and N Goldenfeld Phys Rev E 84 011112

(2011)

PRL 119 268101 (2017) P HY S I CA L R EV I EW LE T T ER Sweek ending

29 DECEMBER 2017

268101-5

Coevolution Maintains Diversity in the Stochastic ldquoKill the Winnerrdquo ModelSupplemental Material

This Supplemental Material describes simulations ofan open coevolving ecosystem as described briefly in themain text We construct a species space to manifest thetrade-off in the birth rate due to mutation For the par-ticular set of parameters used in the simulations thereare M = 20 distinct pairs of preys and predators thatcan potentially exist in the system The birth rates areb = (005 015 025 035 045 055 065 075 085095 1 09 08 07 06 05 04 03 02 01) The 11thspecies has the highest birth rate and is the origin of traitexpansion Mutations of the first and last species gen-erating mutants with negative birth rates are excludedfrom the model Other parameters are pi equiv p = 2βi equiv β = 10 di equiv d = 05 and eij equiv e = 1 Theindividual level reactions have the same form as Eq (1)and (4) in the main text with index i = 1 2 M In the mean-field situation the carrying capacity allowsthe coexistence of 13 pairs with indices from 5 to 17while the rest 7 species are forbidden In the presenceof demographic stochasticity mutants can emerge in the

forbidden region in the species space although they cannot develop a significant population size limited by thehigh competition with other individuals The number ofcoexisting pairs can be greater that the value 13 predictedby the mean-field calculation and varies with time Asshown in the prey population time series in Fig S1(a)and (b) a small mutation rate results in the alternationof dominating winners and a large mutation rate gener-ates coexistence with much smaller fluctuations FigureS1 (c) and (d) show the distribution of prey populationacross all species as a function of the distance to the win-ner defined as the most abundant strain The red bargraph stands for a snapshot at a certain moment andthe blue one represents the average of the distributionover a long time interval Itrsquos clear that a winner standsout at low mutation rate while no one is significantlydominant at high mutation rate Figure S1(e) shows thedependence of prey diversity defined as the Shannon en-tropy on the mutation rate The three regions as seen inthe CKtW model with fixed number of pairs in the maintext are recovered

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I II III

(e)

FIG S1 Simulation results of the CKtW model with the number of coexisting species limited by the carrying capacityParameters used are b = (005 015 025 035 045 055 065 075 085 095 1 09 08 07 06 05 04 03 02 01)pi equiv p = 2 βi equiv β = 10 di equiv d = 05 and eij equiv e = 1 The system size is C = 1000 Population is initiated in the fittest speciesand expands in the species space (a) and (b) are prey population time series for small mutation rate micro1 = micro2 equiv micro = 002 andlarge mutation rate micro1 = micro2 equiv micro = 05 respectively At small mutation rate winners alternate with time and the population islocalized to the winner species At large mutation rate all species coexist and the population distribution is roughly uniformin the mean-field allowed region with some mutants leaked into the forbidden species (c) and (d) show the prey populationdistribution across the species as a function of index distance from the winner strain constructed from (a) and (b) respectivelyThe red bar graph is calculated at t = 4995 after the transient regime The blue one is the distribution averaged over 501snapshots uniformly sampled between t = 2495 and t = 4995 For reference the mean-field steady state predicts that specieswith indices from 5 to 17 coexist with equal abundance and that other species have zero population The center bar at 0distance is the population fraction of the most abundant strain Itrsquos clear that a winner dominates at low mutation rate butnot at the high one (e) The dependence of prey Shannon entropy on the mutation rate defined in the same way as in themain text At low mutation the system collapses due to extinction at intermediate mutation diversity increases rapidly withthe rate at high mutation diversity stays near the maximum given by the deterministic steady state

• Xue NG coevolution yields diversity in stochastic KtW PRL 2017
• v16_supplemental_material

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latio

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I II III

(e)

FIG S1 Simulation results of the CKtW model with the number of coexisting species limited by the carrying capacityParameters used are b = (005 015 025 035 045 055 065 075 085 095 1 09 08 07 06 05 04 03 02 01)pi equiv p = 2 βi equiv β = 10 di equiv d = 05 and eij equiv e = 1 The system size is C = 1000 Population is initiated in the fittest speciesand expands in the species space (a) and (b) are prey population time series for small mutation rate micro1 = micro2 equiv micro = 002 andlarge mutation rate micro1 = micro2 equiv micro = 05 respectively At small mutation rate winners alternate with time and the population islocalized to the winner species At large mutation rate all species coexist and the population distribution is roughly uniformin the mean-field allowed region with some mutants leaked into the forbidden species (c) and (d) show the prey populationdistribution across the species as a function of index distance from the winner strain constructed from (a) and (b) respectivelyThe red bar graph is calculated at t = 4995 after the transient regime The blue one is the distribution averaged over 501snapshots uniformly sampled between t = 2495 and t = 4995 For reference the mean-field steady state predicts that specieswith indices from 5 to 17 coexist with equal abundance and that other species have zero population The center bar at 0distance is the population fraction of the most abundant strain Itrsquos clear that a winner dominates at low mutation rate butnot at the high one (e) The dependence of prey Shannon entropy on the mutation rate defined in the same way as in themain text At low mutation the system collapses due to extinction at intermediate mutation diversity increases rapidly withthe rate at high mutation diversity stays near the maximum given by the deterministic steady state

• Xue NG coevolution yields diversity in stochastic KtW PRL 2017
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