Many ways to stay in the game: Individual variability maintains high biodiversity in planktonic micro-organisms
MMANY WAYS TO STAY IN THE GAMEINDIVIDUAL VARIABILITY MAINTAINS HIGH BIODIVERSITYIN PLANKTONIC MICRO-ORGANISMS
SUSANNE MENDEN-DEUERGRADUATE SCHOOL OF OCEANOGRAPHYUNIVERSITY OF RHODE ISLANDAND JULIE ROWLETTMAX PLANCK INSTITUT F ¨UR MATHEMATIK, BONN
Abstract.
In apparent contradiction to competition theory, the number ofknown, co-existing plankton species far exceeds their explicable biodiversity -a discrepancy termed the
Paradox of the Plankton.
We introduce a new game-theoretic model for competing micro-organisms in which one player consists ofall organisms of one species. The stable points for the population dynamics inour model, known as strategic behavior distributions (SBDs), are probabilitydistributions of behaviors across all organisms which imply a stable popula-tion of the species as a whole. We find that intra-specific variability is thekey characteristic that ultimately allows co-existence because the outcomesof competitions between individuals with variable competitive abilities is un-predictable. Our simulations based on the theoretical model show that up to100 species can coexist for at least 10000 generations, and that even smallpopulation sizes or species with inferior competitive ability can survive whenthere is intra-specific variability. In nature, this variability can be observed asniche differentiation, variability in environmental and ecological factors, andvariability of individual behaviors or physiology. Therefore previous specificexplanations of the paradox are consistent with and provide specific examplesof our suggestion that individual variability is the mechanism which solves theparadox. Introduction
Planktonic organisms are integral to the functioning of the global ecosystem,generating 50 % of the organic C and breathable O [1], fuelling fisheries, and serv-ing as key agents in major biogeochemical cycles [2]. Understanding and predictingthe abundance, distribution, and diversity of plankton is critical to predicting theirglobally important geochemical footprint and the effects of changing environmentalconditions.The number of coexisting planktonic species [3] far exceeds the expected andexplicable number based on competition theory [4]. This has been termed theParadox of the Plankton [5]. There have been important contributions demonstrat-ing that specific factors can enhance coexistence of multiple species (e.g. tradeoffs[6], competition for multiple resources [7], chaotic fluid motion [8], localized ratherthan regional competition [9]). However, these conditions are not met all of thetime and are not inherent to the species, thus not subject to natural selection. Key words and phrases. paradox of the plankton, biodiversity, population dynamics, gametheory, variability, geometric measure theory. a r X i v : . [ q - b i o . P E ] D ec S. MENDEN-DEUER AND J. ROWLETT
There is evidence for high variability in a range of physiological, demographic,and morphological traits among phylogenetically distinct microbes and within asingle species [10]. Phenotypic plasticity in important traits is commonly observedin microbial plankton, including in global, inter- and intra specific patterns of tem-perature regulation [11], [12]; and responses to elevated pCO2 concentrations [13].Species even show and retain possibly maladaptive traits, reflecting ancient, ratherthan current environmental conditions [14]. Plasticity in traits was identified asa key characteristic to adapting to changing or novel conditions and has been as-sociated with elevated success in range expansion of invasive species [15]. Pho-totrophic plankton display considerable variability in important traits, includingtolerance of environmental conditions [16], elemental composition [17], and growthrate [18]. A comprehensive study of intra-strain variability in one phytoplank-ton species showed distinguishing characteristics in cell size, maximum growth andphotosynthesis rates, tolerance of low salinities, resource use, and toxicity [10].Molecular analyses discovered that such variability is genetically rooted [19], [20].Intra-specific variability is further documented in [21]. The maintenance of indi-vidual variability may be supported by the complexity of cellular morphology [22],[23].Motility patterns are also observed to be variable on the individual level; forexample a small fraction of a population will move away from a limiting resourceor fail to avoid predators [24], [25], [26]. These same observations of individuallyvariable motility patterns appear to be strategic on the population level for exampleby fleeing predators [25] or by using vertical swimming patterns to optimize theacquisition of light and nutrients [27].We created a new game theoretic model for competing micro-organisms to pro-vide the theoretical foundation for the observed individual variability. Previousgame theoretic models such as [28] assume that interspecies rankings are fixed andidentical across all individuals. This assumption excludes individual variability.In the lattice version of a multi-species Lotka-Volterra competition model in [29],the mathematical equilibrium is the exclusion of all but one superior species. Inthat model co-existence is only possible if interactions are local rather than global.However, interactions in an ocean or lake environment do not remain local, and theexperimental data of [9] showed that competing species without individual variabil-ity which are periodically mixed will cease to co-exist.Our new theoretical model for competition among micro-organisms and the re-sulting population dynamics incorporates the following key features: (1) individualorganisms can draw from a distribution of behaviors rather than expressing a static,predetermined behavior; (2) microbial populations consist of multiple, clonal indi-viduals; (3) the survival of a species is a cumulative function of the success across itsindividuals. Our approach based on individual variability is inherent to a speciesand therefore subject to natural selection. We find that incorporation of intra-specific variability results in the co-existence of numerous species, even when pop-ulation sizes are low or competitively inferior behaviors are incorporated. Thus, wesuggest that intra-specific variability is the mechanism that explains the paradoxof the plankton.This work is organized as follows. In §
2, we present the mathematical back-ground and the theoretical model. This model suggests that if there is variabilityacross individuals, then there is an infinite set of stable points for the population
ANY WAYS TO STAY IN THE GAME 3 dynamics. These stable points, known as strategic behavior distributions (SBDs) represent all the different probability distributions of strategies across individu-als which guarantee survival of the species as a whole. Using a game to modelinterspecies competition, this abundance of SBDs means that there are many dif-ferent ways to stay in the game, biologically corresponding to many different waysfor species to co-exist, as long as there is some variability among individuals andresources allow. In case of insufficient resources, this means that there are many dif-ferent ways for all species to fare equally well. We tested the theoretical model andits predictions in § §
4. We summarize the methods used in our simulations andmathematical proof in §
5. 2.
Game-theoretic Model
Game Theory Preliminaries.
In a classic game theory sense, players areviewed as individuals. A specific move in a game is known as a pure strategy.
If aplayer has only one pure strategy, then he cannot affect the outcome of competition,so his role is marginalized. If a player has at least two pure strategies, then a mixedstrategy is a probability distribution across that player’s pure strategies. Takefor example the Rock-Paper-Scissors (RPS) game. A mixed strategy is a list ofthree numbers between zero and one which add up to one. These numbers are theprobability of executing Rock, Paper, or Scissors.We extended the definition of a player as consisting of many individuals allbelonging to a single species. An n -player game models competition between n species. In our model, a mixed strategy for a player (which we define to be awhole species), is a probability distribution of that player’s pure strategies acrossits individuals. For example, in RPS, this would be the probability that a randomlyselected individual draws Rock, or Paper, or Scissors. Each player has an associatedpayoff function ( ℘ i ) for i = 1 , . . . , n which depends on the (mixed) strategies of allplayers.To model competition amongst different species of micro-organisms we use an n -player non-cooperative game. Such a game is based on the absence of coalitionsor communication between players. The i th player has a set of m i pure strategies,each of which can be identified with one of the standard unit vectors in R m i . Theset of mixed strategies corresponds to a probability distribution over pure strategiesand can be identified with the convex subset S i ∼ = x = ( x , . . . , x m i ) ∈ R m i : x j ≥ ∀ j, m i (cid:88) j =1 x j = 1 . The total strategy space over all players can then be identified with S ∼ = n (cid:89) i =1 S i ⊂ R N , N = n (cid:88) i =1 m i . Each player has an associated payoff function ℘ i : S → R . The payoff functions arelinear in the strategy of the respective player. That is, if all other players’ strategies S. MENDEN-DEUER AND J. ROWLETT are fixed, the payoff function is a linear function from S i → R . Typically, the payofffunction is assumed to be continuous on S . In [30], Nash proved that for every suchgame, there exists at least one “equilibrium strategy,” which in the following senseis the “best” strategy for all players. For s ∈ S , let ℘ i ( s ; σ ; i ) denote the payoff forthe strategy in which the i th player’s strategy according to s is replaced by σ ∈ S i .An equilibrium strategy s satisfies ℘ i ( s ) ≥ ℘ i ( s ; σ i ; i ) ∀ σ i ∈ S i , ∀ i = 1 , . . . , n. In other words, if a single player changes his strategy, he cannot increase his payoff.
Theorem 1 (Nash [30]) . For any n -person non-cooperative game such that the pay-off functions are linear in the strategy of each player and are continuous functionson the total strategy space there exists at least one equilibrium strategy. Although an equilibrium strategy is in a certain sense the best possible strategy,there may be many other strategies such that the corresponding payoff to all playersis identical to their payoffs according to an equilibrium strategy. This means thatthe feedback for such strategies is identical to the feedback for the equilibriumstrategy. We propose to consider the level sets of the total payoff function, that isthe sets of strategies so that each player has constant feedback within this set ofstrategies. This means that varying strategies within a level set of the total payofffunction doesn’t change the payoff to any of the players, so if the players’ strategiesvary within this set, the resulting feedback is unaffected. It turns out that theselevel sets of the total payoff function are in general quite large.2.2.
Population Dynamics.
The success or failure of micro-organisms to acquirenecessary resources can be identified by the rate of population increase or decrease.So, we propose to use the payoffs in a game theoretic model to define the populationdynamics: positive payoff yields population increase, negative payoff yields popu-lation decrease. A strategy for a species is a probability distribution of behaviorsacross all individuals of that species. The payoff is a cumulative function mea-suring the average success over all individuals, so the change in population shouldalso be proportional to the current population. This is how we came to define thedifferential equation for the population dynamics of the i th species,(2.1) p (cid:48) i ( t ) = p i ( t ) ℘ i ( s ) . Above p i ( t ) is the population of the i th species at time t , p (cid:48) i ( t ) is the instantaneouschange in the population at time t , and ℘ i ( s ) is the payoff to the i th player for thegame according to the strategy s over all players. Definition 1.
A strategic behavior distribution (SBD) is a rest point for the equa-tion (2.1), that is a strategy s such that ℘ i ( s ) = 0 , for each i = 1 , , . . . , n . Theset of all SBDs is therefore the zero-level set of the total payoff function. The novelty here is our application of geometric measure theory to game theoryand subsequently to the population dynamics of micro-organisms. The mathemat-ical core of this work is the following result.
Theorem 2.
For an n -player game, assume that each player has at least twopure strategies and that the payoff function is Lipschitz continuous. Let N denotethe total number of pure strategies across all players. Assume at least one of thefollowing holds: (1) the game is zero-sum and/or (2) at least one player has three ANY WAYS TO STAY IN THE GAME 5 or more pure strategies. If (1) holds, then let k = N − n + 1 ; if only (2) holdsthen let k = N − n . With respect to k -dimensional Hausdorff measure, the levelsets of the total payoff function ℘ := ( ℘ , ℘ , . . . , ℘ n ) k -almost always have positive k -dimensional Hausdorff measure. The proof of the theorem is provided in §
5. The following principle is the bio-logical interpretation of our mathematical theorem.
Principle 1.
If each species of competing micro-organisms has variability acrossindividuals, then if there exists one SBD, there is in a generic sense (almost always)an infinite set of SBDs. Moreover, for a specific set of respective feedbacks to allspecies, there are almost always infinitely many different behavior distributions overthe individuals of each species which yield those same respective feedbacks to eachspecies.
This principle is depicted in Figures 3 and 4, which show that although an equi-librium strategy may be a single point, there is an entire triangular region or line, respectively, containing infinitely many different strategies which all give feedbackto the competing species which is identical to their feedback at the equilibriumpoint. We propose that the large level sets of the total payoff function support thelarge biodiversity observed in planktonic microbes.3. Model Simulations
We implemented computer simulations of competing species using a symmetric,two-player, zero-sum game with two pure strategies, W and L, such that W (win)dominates L (lose) (see Fig. 4). In a natural environment, competition occursbetween individual organisms rather than whole populations. For the theorem andour general principle, we note that games are neither required to be symmetric norzero-sum; we made this assumption for computation convenience.At each round of competition, each individual was randomly assigned to competewith an individual belonging to a different species. The individual’s competitiveability was randomly assigned, represented by a number between 0 and 1, selectedfrom the particular SBD for its species (Fig. 5). The three possible outcomeswere (1) draw: identical values for both individuals, (2) win: larger value thancompetitor, or (3) lose: lower value than competitor. A draw has both individualsremaining, a win results in the doubling of that individual, whereas a lose resultsin the removal of that individual. To assess species co-existence these competitionswere repeated for a number of individuals, species and durations, as specified below.Species survival was then assessed across all its individuals.3.1.
Model simulation results.
In tournaments matching 2 species, persistenceof both species was observed when each species was characterized by an SBD(Fig. 5). Although extinctions did occur due to stochastic fluctuations when pop-ulation size was small ( (cid:46)
100 individuals), persistence of 2 species was consistentlyobserved over dozens of generations at larger population sizes (Fig. 1 (A)). Ourmodel accurately reproduced rapid (within <
10 generations) species extinctionwhen individual competitive abilities were identical across the entire species, con-sistent with the competitive exclusion principle.Extension of the simulations to include up to 100 species showed that consistentpersistence for 1000 generations was observed with population sizes of 500 or more
S. MENDEN-DEUER AND J. ROWLETT
Individuals per species (n) D u r a ti on ( t )
100 1000 10000 10 100 500 1000 020406080100 (a) (b)(c) (d)(e)
Figure 1.
The outcomes of the competition model. (A) showsthe mean over 100 repetitions of the randomized model simulationof two competing species with the vertical axis showing the aver-aged percent probability that two species co-exist. Probability ofcoexistence of both species increased with simulation duration andindividuals per species. Co-existence for at least 1000 generationswas observed in at least 90% of the simulations when there wereinitially at least 1000 individuals per species. (B), (C), (D), and(E) show the number of individuals for 6, 10, 30, or 100 competingspecies, respectively, averaged over 30 repetitions. Initial start-ing abundances were 10, 100, or 1000 individuals per species. Theshaded areas show the variance amongst runs. No extinctions wereobserved.
ANY WAYS TO STAY IN THE GAME 7 individuals (Fig. 1 (B)–(E)). Irrespective of the initial number of species, approxi-mately 100 individuals were sufficient for total species diversity to persist in at least80% of the replicated runs. Competitive exclusion, that is, total extinction of allbut one species was only observed when there were fewer than 10 individuals perspecies. Species persistence did not imply lack of variation in relative abundance.Depending on the number of species and individuals, there were considerable fluc-tuations in the abundance, and some species could be rare, but they persisted forat least 1000 competitions.The upper limits of the simulation durations, number of species, and populationsizes were determined by the availability of CPUs due to the computationally inten-sive nature of modelling individual-based interactions. However, while technicallychallenging, simulating larger population sizes would only further support the pro-posed ideas. Larger population sizes would provide an even greater buffer againstextinction; the probability of extinction decreases with increasing population size,as shown in Figure 1A. It is therefore not implied that extinctions would be ob-served when more species and/or individuals are evaluated simultaneously but infact that extinctions would be even more unlikely. Simulation of population sizes of10 individuals provides an example of probable cell-cell encounters within a timeframe (minutes to hours) relevant to individual cells, within a volume of sub-ml tomls.3.2. Lattice Grid Simulations.
Landscape ecology has identified the importanceof spatial variability in maintaining species diversity [31]. Restricting competitionto proximate competitors has been shown to maintain species diversity in microbes[9]. To examine the effect of spatial structure in maintaining species diversity, weformulated a spatially explicit lattice model following RPS dynamics. Spatial vari-ability was mimicked by allowing species to compete either locally or globally, thatis either only with individuals directly adjacent or throughout the entire matrix.Individual variability was mimicked by allowing individuals of each of three speciesto vary the strategy in each step by selecting either Rock, Paper, or Scissors. Suchvariability in behavior has been shown, for example in sensitivity, resistance, andtoxicity of bacterial strains [32]. Constant behavior types corresponded to threespecies: one species whose individuals always drew Rock, a species whose individ-uals always drew Scissors, and a species whose individuals always drew Paper.Our model confirmed the results of [9], that when each species expresses con-stant behaviors, local competition maintains species diversity longer than globalcompetition (Fig. 2). Irrespective of whether competition was global or local, nospecies extinctions were observed when individuals had variable strategies (Fig. 2).4.
Discussion
Representing species through many individuals with variable competitive abil-ities brought inter-species competition from an aggregate, species level to the in-dividual level. Our results show that high species diversity is explicable when theunderlying behavior distributions allow for variability across individuals, and con-versely, that SBDs maintain species diversity. This variability across individualsmay be intrinsic, corresponding to inherent behavioral heterogeneity, as well as ex-trinsic, resulting from environmental, ecological, spatial, or temporal variability. Itis of course well known that identical species can co-exist in a neutral sense. Thekey here is that the large (infinite) number of different
SBDs allow many different
S. MENDEN-DEUER AND J. ROWLETT
50 100 150 200 25050100150200250 (a)
50 100 150 200 25050100150200250 (b)
50 100 150 200 25050100150200250 (c)
50 100 150 200 25050100150200250 (d)
Duration [t] P r ob . s p ec i e s r e m a i n [ % ] Global constantGlobal strategicLocal constantLocal strategic (e)
Figure 2.
The spatially explicit RPS competition model. (A)shows the initial state of the lattice model with three species col-ored red, blue, and green. (B) and (C) show the lattice after 100rounds of global competition with constant and variable strategies,respectively. (D) shows the lattice after 100 rounds of local com-petition with variable strategies. (E) shows the probabilities overtime, averaged over 30 simulations, that three species remain incoexistence.species to co-exist in a neutral sense because these SBDs imply many different waysto stay in the game.
Our model simulations showing prolonged co-existence basedon a simple game imply the same results for more complicated games and payofffunctions because the more complicated the game, and the more variability acrossindividuals, the larger the set of SBDs. This can be seen for example in Figures3 and 4. In the RPS game players each have three pure strategies, whereas in
ANY WAYS TO STAY IN THE GAME 9 the game in 4, players each have only two pure strategies. Consequently, RPS isa more complicated game, and correspondingly the set of SBDs for the RPS is awhole triangular region (a two-dimensional set), which is much larger than the setof SBDs in 4, which is a line segment (a one-dimensional set).4.1.
Previous explanations for the paradox.
There have been multiple plau-sible explanations for the paradox of the plankton, typically focused on a singlefactor such as resource competition. However, organisms simultaneously competenot just for resources, but also require defence from predation and are subject tomultiple stressors and selection factors.One explanation proposed to resolve the paradox is niche differentiation. Ifspecies compete for several resources and each possesses a ranking in terms of thesuccess of that species at obtaining each resource, competition can be modelledusing a competitive network, like a tournament. This was the approach taken in[28]. However, with this approach, they showed that an even number of speciescan never co-exist. They assumed that all individuals of a single species have iden-tical, fixed, competitive abilities. For example, if species 1 is better at obtainingresource A while species 2 is better at obtaining resource B, then every individualin species 1 is better at obtaining resource A, and every individual in species 2 isbetter at obtaining resource B. This assumption is not actually required for nichedifferentiation. Different species can be better or worse on average at obtainingspecific resources yet variable on the individual level. Moreover the success of indi-viduals may (and likely does) fluctuate over time. Niche differentiation correspondsto different species which may be superior or inferior at specific functions in sucha way that cumulatively they fare equally well and consequently co-exist. In ourmodel this simply corresponds to variable strategies and payoffs among players. Acompetitive network model corresponding to niche differentiation which allows thecompetitive ability of individuals to be variable within each species therefore fitsnicely into our theory.The number of co-existing species in a model based on niche differentiation with-out individual variation, such as [28], increases when competition is restricted toproximate neighbors. In [29] local competition or temporal-niche differentiationis suggested as the underlying mechanism which supports the large biodiversityof planktonic micro-organisms. That work is based on a lattice Lotka-Volterrapredator-prey model which again sets a fixed competitive ability uniformly forall organisms of a single species. The experimental data of [9] with bacteria ofthree fixed types showed that local competition which is periodically mixed doesnot result in coexistence (“mixed plate” experiment of [9]). Due to the dynamicand mixing environment inhabited by plankton, competition cannot remain localover long periods of time. This laboratory experiment with three bacteria strainscomprised of identical clones failed to show co-existence when bacteria were pe-riodically mixed, however the numerical simulations of [9] showed co-existence byincorporating individual variability by “sequentially picking random focal pointsand probabilistically changing their states.” Whereas localized competition mayenhance species persistence, it is not a consistent mechanism to ensure survival ifthe species cannot express individual variation. In contrast, individual variabilitywas uniformly observed to promote coexistence in our spatially explicit RPS sim-ulation. This supports the notion that individual variability is perhaps the keymechanism supporting planktonic biodiversity.
Gause’s Principle.
Our prediction may appear to contradict Gause’s com-petitive exclusion principle [33]. However, Gause observed that competitive exclu-sion does not necessarily occur if ecological factors are variable.
We propose thatthe primary mechanism driving the large variability of ecological factors amongstplankton is individual variability. When species possess variability across individ-uals, corresponding to the hypothesis in the theorem that each species has at leasttwo pure strategies, then competition between individuals is unpredictable. Thesuccess of the species is determined by the cumulative success across its individu-als. If a species is cumulatively inferior, our model predicts its extinction consistentwith the competitive exclusion principle.If however species are on average cumulatively equally matched, then both themathematical theory and our simulations predict neutral co-existence. More im-portantly, there are infinitely many different probability distributions across in-dividuals which have identical cumulative competitive abilities over time. Eachof these different probability distributions across individuals summarizes the cu-mulative characteristics of each different species. The entirety of these diversebehavior distributions correspond to an unlimited biodiversity capable of neutrallyco-existing.Our model does not imply that species will co-exist regardless of their environ-ment. Indeed, the payoff functions depend on the specific environmental conditions;we chose a simple payoff function for the purpose of simulation, but this is not re-quired for our theorem. If resources are scarce, then there may not be any SBDs,because the resources cannot support the entire population across all species. How-ever, even in this case, the mathematical theorem still applies in the sense that,if there is at least one strategy across all species with equal payoffs, then thereare (generically) infinitely many such strategies. This means that there are manydifferent ways for all species to fare equally well, whatever the circumstances maybe.Assuming equal cumulative competitive ability across all species with diverse be-havior distributions may be inappropriate over short durations. We therefore testedthe persistence of two species with unequal competitive abilities by generating SBDsfor an inferior competitor with mean competitive ability 0.25 (Fig. 5). This inferiorcompetitor was matched against a species with a bimodal SBD (Fig. 5) includingweak (0-0.1) and strong (0.9-1.0) competitors. Thus, an inferior competitor couldbe competing against an individual with either poor or excellent competitive ability,and the outcome of competitions between these two species was not predictable `apriori. Similar to the general case shown in Fig. 1, the inferior competitor persistedconsistently when the populations size was (cid:38)
Individual Variability.
There is significant evidence for individual variabil-ity among planktonic micro-organisms. For example, their spatial abundance hasbeen modelled with biased random motion [34]. Biased random motion impliesan extremely high amount of variability in the motility patterns of different indi-viduals within a single species. If the environment is chaotically mixing, which is
ANY WAYS TO STAY IN THE GAME 11 often the case for oceanic plankton, it is not possible to predict the optimal motil-ity pattern and therefore a species expressing a variety of motility patterns will bemore successful cumulatively over time than a species with a uniformly identicalmotility pattern. Plasticity in a variety of traits, not only motility, and phenotypicdiversity amongst planktonic micro-organisms has been demonstrated in [23], [22],and [13]. However, rather than expressing completely random motility patterns,[27] showed that the vertical motility patterns of phytoplankton are strategic toeffectively optimize the acquisition of light and nutrients. Phytoplankton may evenflee from predators [25] but this behavior is not observed to be uniform across allindividuals. While motility patterns may appear to be somewhat random and vari-able when assessed at the individual level, they appear to be strategic on averageover time when assessed cumulatively for a species. Our model incorporates bothindividual variability and strategy. A species need only be strategic as a whole,corresponding to one of the infinitely many SBDs, which allows unlimited variabil-ity across individuals. However, an SBD assessed cumulatively over all individuals,while not necessarily an equilibrium strategy, is sufficient to ensure survival.A natural question is, why doesn’t evolution drive species toward a Nash equi-librium strategy? The crux of the paradox of the plankton seems to be that com-petition models which may be suitable for larger organisms do not allow sufficientflexibility to be consistent with the much greater biodiversity observed in plankton(and perhaps other) micro-organisms [35], [36], [37], [38], [39]. This is what lead usto seek a model which would allow a much larger set of strategies which are goodenough to ensure survival and which allow a large variety of species to co-exist.Viewing players as entire species allows for several different probability distribu-tions across individuals each of which characterize a different species. Based onfeedback alone, there is no difference between all the different strategies in eachlevel set of the payoff function.The RPS game satisfies the hypotheses of our theorem because each player hasat least three pure strategies. Moreover, variations of that game such that it is nolonger zero-sum also satisfy the hypotheses of our theorem. For the classical RPSgame, the unique Nash equilibrium strategy is the mixed strategy to draw Rock,Paper, and Scissors each with probability 1/3. The corresponding payoff to playersis 0. The set of SBDs is the set of all mixed strategies such that the payoff toeach player is zero, and in this case has dimension 2 and can be represented by theset of points in an isosceles right triangle. One can imagine that the strategies ofplankton species can vary - including off of the set of SBDs - in such a way thattheir cumulative strategies roam around within this triangle or at least do so in atime-averaged sense. This region, which contains infinitely many different SBDs,corresponds to the large biodiversity which can co-exist; see Fig. 3. The large setof SBDs in our mathematical model, not only for this example, but for any of theinfinitely many games which satisfy the hypotheses of our theorem, allows for muchgreater biodiversity than a model which drives species toward a Nash equilibriumstrategy.4.4. Concluding Remarks.
The novelty of our suggestion is to recognize theimportance of individual-scale, cell-cell interactions that incorporate individual-level heterogeneity in behaviors and physiological characteristics, rather than as-sessing species, groups or aggregates based on average characteristics. Thus, thepredictability of the outcome of competitions, which is near certain when assessed
Nash EquilibriumStrategy
Figure 3.
For the RPS game, the set of SBDs can be representedby the set of all points inside an isosceles right triangle, whereasthe Nash equilibrium strategy is a single point inside this triangle.on the mean, becomes unpredictable when assessed on the individual level. Thisunpredictability results in the observed survival of individuals and persistence ofdiverse species. The mechanisms proposed here rely on species that are (1) highlydiverse in one or many of their physiological, behavioral, or morphological func-tions, and (2) reproduce rapidly, asexually. Competition is first assessed at anindividual, organismal level and then cumulatively assessed over all individuals todetermine the resulting outcome for the species as a whole.Our model for the population dynamics likely only applies to microbial organisms with large population sizes, but we have not explored the issue systematically.For clonal organisms, each individual only represents a small fraction of the totalpopulation, and thus variable, and probable map-adaptive behaviors pose no risk tothe overall gene pool. For sexually reproducing organisms, each individual is unique(although there may be great similarities among groups). Thus, we interpret ourmodel to be a poor fit for species where each organism represents a unique set ofevolved characteristics that would be lost to the population if the individual wereto perish. This risk is not inherent in clonal populations.Our model possesses the strength that it is insensitive to specific formulations;payoffs for strategies can be constant or varying among species or over time, andthe distribution of functions within a species can be constant or varying over time.The key factor is that in microbial systems each individual represents only a smallfraction of the total population, where survival of each individual is associated onlywith a small risk. Consequently, some individuals can represent extreme strategiesthat may not be `a priori suitable or advantageous but may be successful infre-quently.
ANY WAYS TO STAY IN THE GAME 13 Methods
We arrived at a key discovery based on the following simple example. Considera symmetric, two-player, zero-sum game with two pure strategies, W and L, suchthat W (win) dominates L (lose) (see Fig. 4). If the probability that players 1and 2 execute strategy W are respectively s and s , then the unique equilibriumstrategy is s = s = 1, and the payoff to both players is 0. There are infinitely manystrategies such that the payoff to each player is identical to their payoff accordingto the equilibrium strategy.
The payoff functions are ℘ ( s , s ) = s − s , and ℘ ( s , s ) = s − s . The set of SBDs is the line segment { ≤ s = s ≤ } Ifthe game changes so that strategy L dominates strategy W, the Nash equilibriumbecomes ( s = s = 0), but the set of SBDs remains unchanged (see Fig. 4).Although an SBD is not optimal in the sense of a Nash equilibrium, in this exampleit gives identical feedback to the Nash equilibrium. This example illustrates ourmathematical theorem. In this case the total payoff function is a linear function,and since the game is zero-sum, ℘ = − ℘ . Therefore, the total payoff function canbe canonically identified with a linear map from R → R , and the level sets of thisfunction have dimension one (the level sets are line segments). This shows that foreach value of the payoff functions (i.e. feedback to each competing species), thereis a whole line of strategies with those same payoff to each respective player.5.1. Proof of the Mathematical Theorem.
The main ingredient in the proof ofour theorem is the co-area formula in geometric measure theory due to Federer [40].One of the implications of this theorem is that a Lipschitz continuous function froma higher dimensional Euclidean space, like R m into a lower dimensional Euclideanspace, like R l , such that l < m generically has large level sets. More precisely, therank of the differential of such a function f is with respect to m − l dimensionalHausdorff measure almost-always equal to l . Consequently, the level set f − ( f ( x ))contains a subset which is a topological submanifold of R m of dimension m − l . Inthe more simple case of an affine linear function as in Figure 4, the dimension ofthe level sets is given by the rank-nullity theorem from linear algebra. For an affinelinear function f from R m → R l , there is y ∈ R l and an l × m matrix M such thatfor all x ∈ R m , f ( x ) = M x + y . Consequently the level sets are all affine linearsubsets of R m with dimension equal to the dimension of the kernel of M . Since M is l × m and m > l , by the rank-nullity theorem the dimension of this kernel is atleast m − l .Under the hypotheses of the theorem, the total payoff function ℘ = ( ℘ , ℘ , . . . , ℘ n )is canonically identified with a map from a convex N − n dimensional subset of R N into R n . This is because the sum of all components of a strategy must be equal toone, and hence the last component of each player’s mixed strategy is determinedby the previous components. Given that there are n players total, this means thatthe payoffs are determined by elements of R N − n . Since all players have at least twopure strategies, N ≥ n and so N − n ≥ n . If the game is zero-sum, then ℘ n = − n − (cid:88) k =1 ℘ k , and hence the total payoff function is canonically identified with a map from R N − n into R n − , and in this case N − n ≥ n > n −
1, and we define k := N − n + 1.If only (2) holds, then N > n , and thus N − n > n , and the total payoff is (a) Line of SBDs Nash Equilibrium (b)
Line of SBDsNash Equilibrium (c)
Figure 4. (A) shows the payoff matrix for the symmetric, two-player, zero-sum game used to simulate competition. (B) showsthe line of SBDs and the unique Nash equilibrium strategy. Player1 has a higher payoff in the darker region, and player 2 has higherpayoff in the lighter region; both players have identical payoffsalong the line of SBDs and at the Nash equilibrium strategy. (C)shows what happens if the dominant strategy changes. The lighterand darker regions exchange places, and the Nash equilibrium pointis at the other end of the line of SBDs. The SBDs however, remainunchanged.canonically identified with a map from a convex N − n dimensional subset of R N into R n . In this case k := N − n . Consequently, in all cases the total payofffunction is canonically identified with a map from a larger dimensional Euclideanspace R m for m = N − n into strictly smaller dimensional Euclidean space R l for l = m − k . By the co-area formula the level sets of the total payoff functionwith respect to k -dimensional Hausdorff measure almost always have positive k -dimensional Hausdorff measure. (cid:3) Statistical Mean of Tournament Simulations.
In each simulation, a specieswas represented by at least one and up to 10000 individuals. Each tournamentincluded several randomization parameters, including the random assignment ofbehavior values to individuals as well as randomizing the selection of competitors.To explore the robustness of these results, we varied the number of individualsper species in each type of simulation over the entire range from 1 to 10000. To
ANY WAYS TO STAY IN THE GAME 15 P r obab ili t y Strategic Behavior Value (a) P r obab ili t y Strategic Behavior Value (b) P r obab ili t y Strategic Behavior Value (c) P r obab ili t y Strategic Behavior Value (d)
Figure 5. (A), (B), and (C) show the non-normal SBDs all withmean 0.5 from which individuals’ competitive abilities were ran-domly drawn to implement the competition simulations. (D) showsthe SBD for an inferior species with mean competitive ability 0.25.eliminate the effect of this stochastic variation, we repeated each simulation of aparticular parameter combination at least 30 times and computed the mean overall runs. All simulations were run in Matlab R7.10.5.3.
Spatially Explicit Model.
We included spatial information in the tourna-ment by formulating a lattice structure in which each individual was initially as-signed a species, strategic behavior, and location at random. In this spatiallyexplicit model we additionally examined the role of local versus global competition,which has been found to be an important structuring factor [9]. The boundary con-ditions were periodic for the spatially explicit model. The time step of evaluation(t) was repeated up to 10000 times. The population abundance of each species andif spatially explicit, location, of each individual were determined at each time step.All simulations were run in Matlab R7.10.
Acknowledgments
We thank Henry Segerman and Mick Follows for critical reading of this man-uscript and the anonymous reviewers whose comments greatly improved this pa-per. We gratefully acknowledge the support of the National Science Foundation(Biological-Oceanography Award 0826205 to S. Menden-Deuer), the Max Planck
Institut f¨ur Mathematik, the Universit¨at G¨ottingen, the Leibniz Universit¨at Han-nover, and the Australian National University.
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