## Abstract

Using game theory, we provide mathematical proof that if a species of asexually reproducing microbes is not characterized by maximum variability in competitive abilities among its individual organisms, then that species is vulnerable to replacement by competitors. Furthermore, we prove that such maximally variable species are neutral towards each other in competition for limited resources; they coexist. Our proof is constructive: given one species which does not possess maximum variability, we construct a second species with the same (or lower) mean competitive ability which can outcompete the first, in the sense that its expected value in competition is positive, whereas the expected value of the non-maximally variable species is negative. Our results point towards the mechanistic underpinnings for the frequent observations that (1) microbes are characterized by large intra-specific variability and that (2) the number of extant microbe species is very large.

## Keywords

Microbes Ecology Game theory Competition Phenotypic variability Fitness## Introduction

The problem addressed here is the inexplicably high taxonomic and phenotypic diversity observed among and within microbe species (e.g., plankton). Microbes are the key engines of fundamental biological processes and show immense phenotypic variability within species in terms of physiological, demographic, or morphological characteristics (i.e., traits). Henceforth, we call this characteristic *intra-specific variability*. A fundamental conundrum arises because selection should favor individuals that maximize fitness (i.e., winner takes it all) and thus minimize intra-specific variation. This conundrum is well documented in the literature. Enabled by recent advances in measurement capacity, key traits subject to selection show surprising diversity. A non-exhaustive list of examples of highly variable traits includes elemental composition, morphology, physiology, and behavior (Boyd et al. 2013; Moal et al. 1987; Rynearson and Virginia Armbrust 2004; Fredrickson et al. 2011; Kiørboe 2013; Menden-Deuer and Montalbano 2015). Such intra-specific variability has even been documented in marine metazoa (Morozov et al. 2013). Moreover, demonstrable variation among clonal individuals has shown intra-specific variability enhances species dispersal rates, cell encounters, and population distributions (Menden-Deuer 2010). Recent empirical data confirm that variations in motility at the strain level manifest themselves in different population distributions (Harvey et al. 2015). Such intra-specific variability may enhance species survival and coexistence and has been suggested to be adaptive based on model simulations (Menden-Deuer and Rowlett 2014). Observations of intra-specific variability in marine microbes are not restricted to artificial laboratory settings. Satellite observations show the distribution of a species of photosynthetic plankton persisting for months, over hundreds of kilometers along the North American west coast (Du et al. 2011; White et al. 2014). Laboratory experimentation on this plankton species documents intra-specific variability in physiology and movement behaviors (Menden-Deuer and Montalbano 2015) that provide explanatory power for the species persistence documented with satellite observations. Thus, in the laboratory and ocean immense intra-specific, variability is demonstrable but difficult to explain theoretically.

Beyond the tremendous intra-specific physiological and behavioral variability discussed above, there is also an astonishing observable taxonomic diversity within marine microbes. This vast diversity of species occupies all branches on the tree of life (Rynearson and Palenik 2011). Recent investigations have shown that although thousands of species have already been described and discovered, many if not most marine microbes are yet to be identified (de Vargas et al. 2015; Worden et al. 2015). Although the number of planktonic species is unknown, the likely number of species exceeds most if not all other groupings of organisms. Using data collected globally, researchers estimated that there may be around 150,000 eukaryotic plankton species (de Vargas et al. 2015). Moreover, if prokaryotic species are to be included, there are likely millions of species of marine microbes and their total number may not be knowable (Ward 2002).

Here, we introduce game theory for microbes, which aims to understand the mechanistic underpinnings that yield the observable diversity in species and behaviors. The model interpolates between microscopic, individual level competition and macroscopic, species level survival. Justified by observations, species are characterized by intra-specific variability, manifested in different distributions of their competitive abilities. To avoid trivial outcomes, the mean competitive ability is restricted to have a maximum value of 1/2. This can also be interpreted as assuming species are equally limited by energy and resources, when the species is assessed cumulatively. Species do vary in their trait distributions, which imparts different degrees of intra-specific variability from degenerate (= invariant) distributions to the uniform distribution with maximal variability. Species are represented by specific distributions that do not change over time but individuals’ competitive abilities are chosen at random within each fixed cumulative distribution. Randomization of individual competitive abilities over time within a species represents competition in a dynamic system, where species are characterized by multiple physiological, behavioral, and morphological traits, that can impart changing degrees of fitness as a function of the particular abiotic and biotic conditions.

We have rigorous mathematical proof (see Appendices A & B) that according to the rules governed by this game theory, only those species that possess maximal variability among individual organisms have a unique quality: there is no other species with the same (or lower) mean competitive ability which has positive expected value in competition. In other words, there is no species that can outcompete the maximally variable species. On the other hand, for any species which does not possess maximal variability among individual organisms, in the proof of Theorem 1, we explicitly construct a species with the same (or lower) mean competitive ability which has positive expected value in competition, whereas the original species has negative expected value. The maximally variable species is the only species, which does not have such vulnerability. Moreover, in Proposition 1, we prove that such maximally variable species are neutral towards each other in competition for limited resources; they coexist.

Over time, if species tend toward maximum variability, then they simultaneously tend towards neutral coexistence. In this sense, we provide a plausible mathematical explanation for two widely documented but poorly understood characteristics of microbes: (1) enormous intra-specific variability and (2) tremendous biodiversity.

This work is organized as follows. In “Game theory for microbes”, we introduce our game theoretic model for microbes. This sets the stage for our mathematical results, which are presented in “Theoretical model and main results”. We end with “Concluding remarks”. The complete, rigorous mathematical proofs of the results presented in “Theoretical model and main results” are contained in Appendices A and B.

## Game theory for microbes

In the game theoretic approach to examine competition on the level of species, we assume that each species is comprised of several individual organisms. Each individual has a competitive ability, which is not necessarily constant, but may change from one round of competition to the next. The competitive ability (CA) is represented by a number between 0 and 1. We assume that the species as a whole has a mean competitive ability (MCA), which does not exceed 1/2. The range of both the competitive abilities as well as the mean competitive abilities was chosen arbitrarily and our results are independent of the choice of specific values. At each round of competition, individual organisms are randomly put in pairs. The organism with the higher CA doubles, while the organism with the lower CA perishes; in case of equal CAs, both organisms simply persist. The magnitude of difference in CA between competitors is irrelevant.

A key, and we think novel, feature of our model is that the individual CAs are variable, depending on the distribution of CAs for a given species. Thus, the assembly of the individuals in a species represent the distribution of CAs, and their collective mean CA is equal to the species’ MCA. This situation is representative of genetically identical individuals that temporarily differ in competitive ability, due to for example their recent exposure to resources or metabolic demands, the particular state in their life cycle and such. Finally, competition in a dynamic system requires the nature of the competition to vary, for example, competition may be with respect to resource acquisition in one round, and predator defence in another. Thus, individuals that were superior relative to the mean in one round may be inferior within the next round.

This model applies to any arbitrary number of simultaneously competing species, with arbitrary (not necessarily equal) population sizes. Each species does not compete in a series of tournaments against all its competitors, but rather, it is competing against all other species simultaneously. So, our model shows how each species will fare in competition with any arbitrary number of other species.

### Same-same but different

For each MCA value, there are numerous ways to assign the individual CAs, which have this same MCA value. We term this “same-same but different.” We recall the definition of strategic behavior distribution (SBD) from Menden-Deuer and Rowlett (2014) specified to this setting. Here, an SBD is defined to be a particular way to assign CAs to individuals such that the MCA is equal to a fixed value. Interestingly, SBDs having the same MCA, nonetheless have dramatically different ecological repercussions depending on the particular characteristics of the SBD.

Largely, in order to reduce complexity, it is common to assume that variation among individuals and even species can adequately be represented by a mean or maximal response (e.g., Eppley (1972)). In this case, that would correspond to all individuals being equal and having CA always equal to the MCA. We refer to this degenerate SBD as the invariant SBD. We shall see in the proof of our main result that it is easily susceptible to invasion by a competing SBD, which also has MCA less than or equal to 1/2.

In general, we use here SBDs, which are neither purely deterministic nor purely probabilistic. The SBDs are not purely deterministic, because the CA of each individual is in general non-constant. For example, we define the bimodal SBD to select half of the organisms of the species to have CA = 1, and the remaining half to have CA = 0. Any odd-man-out gets 0 or 1 with probability 1/2. However, this selection occurs at each round of competition, so an organism which had CA = 0 in the first round of competition could have CA = 1 in the second round of competition. Moreover, for an organism, which had CA = 1 and succeeded in a round of competition and thereby replicates, both it and its clone could have either CA = 1 or CA = 0 in the next round of competition. Thus, the model is not deterministic. On the other hand, the model is also not purely probabilistic in the sense that each individual is not assigned its CA = 1 or 0 each with probability 1/2 , with the probabilities of each individual being independent. If that were the case, then it would be possible, but unlikely, that at a specific round of competition, all individuals had CA = 0. In our model, as long as the number of individuals is at least two, this is impossible. Thus, the SBDs we use are neither purely deterministic nor purely probabilistic. This fluidity of differences among individuals requires the species under consideration to be an asexually reproducing microbial species. This restricts our results to microbes, because asexual reproduction will quickly regenerate individuals with identical genetic capacity within the SBD of the species, and thus offset losses due to an inferior CA.

Below, to determine the success or lack thereof of defined SBDs, we use this game theoretic model to compute the expected value of SBDs in competition. A positive expected value implies population growth. A negative expected value implies population decrease. An expected value of 0 implies stable population abundance, neither increase or decrease.

## Theoretical model and main results

*M*above is an integer greater than or equal to 6. Here, it is convenient to introduce the notation

*A*, comprised of

*N*

_{A}total individual organisms, for 0 ≤

*x*

_{j}≤ 1, we define

*n*

_{A}(

*x*

_{j}) so that

*x*

_{j}. We are interested in the expected value of

*A*in competition with species

*B*. In fact, our model works equally well if we consider

*B*to be all other species, which compete with

*A*for limited resources, because we do not require the total number of organisms,

*N*

_{A}and

*N*

_{B}, respectively, to be equal. We do assume that

*N*

_{A}and

*N*

_{B}are both positive.

### Expected value in competition

*A*, which have competitive ability equal to

*x*

_{i}compete with an inferior organism by first counting all of the organisms of species

*B*, which have lower CA:

*B*,

*N*

_{A}and

*N*

_{B}are equal, some organisms might not compete at all. If

*N*

_{A}<

*N*

_{B}then only

*N*

_{A}organisms out of the total

*N*

_{B}organisms in species

*B*actually compete, so the probability that an organism from species

*B*competes is \(\frac {N_{A}}{N_{B}}\). In this case, all organisms from species

*A*compete. If

*N*

_{B}<

*N*

_{A}, then only

*N*

_{B}organisms out of the total

*N*

_{A}organisms in species

*A*compete, so the probability that an organism from species

*A*competes is \(\frac {N_{B}}{N_{A}}\). In this case, all organisms from species

*B*compete. So, in general, we define as follows:

*A*actually competes is as follows:

*B*actually competes is as follows:

*A*with competitive ability

*x*

_{i}is as follows:

*A*in competition with species

*B*is then calculated by summing the expected value for all competitive abilities:

### The uniform SBD

Our first result shows that a certain SBD, known as the *uniform SBD*, is neutral in competition to all other SBDs, which have the same MCA as that of the uniform SBD. At the same time, we also show that the uniform SBD has positive expected value in competition with any SBD, whose MCA is less than the MCA of the uniform SBD.

### **Proposition 1**

*Let*

*A be a species with MCA at most*\(\frac {1}{2}\).

*Let*

*U be the uniform SBD, which is the unique SBD having*

*n*

_{A}(

*x*

_{i})

*equal and*

*positive for all*

*i*= 0, … ,

*M*.

*Then,*\(\operatorname {MCA}(U) = \frac {1}{2}\),

*and*

*with equality if and only if the MCA of species A is equal to*\(\frac {1}{2}\).

*We note that this is independent of the number of organisms in species A and species U*.

#### Logical argumentation

*E*[

*A*,

*U*] to write out this expected value; see Eq. 3 in Appendix A. At first, the expression may look complicated, but using the definition of the uniform SBD, as done in Eq. 4 of Appendix A, the expression simplifies. With these mathematical manipulations and resulting simplifications, the expression for the expected value of species

*A*in competition with the uniform species

*U*becomes

*N*

_{c}is the number of competing organisms, which we recall is the smaller of

*N*

_{A}and

*N*

_{U}. The number

*M*is how we have defined the competitive values, as the numbers \(0, \frac {1}{M}, \frac 2 M\), and up to \(1 = \frac M M\). In the parentheses, we see 2MCA(

*A*). By the constraint on the MCA, we know that \(\operatorname {MCA} (A) \leq \frac {1}{2}\), so 2MCA(

*A*) ≤ 1. Consequently,

*E*[

*A*,

*U*] ≤ 0. Next, if \(\operatorname {MCA}(A) < \frac {1}{2}\), then

*A*) − 1 = 0, and consequently the expression for

*E*[

*A*,

*U*] is zero.

### Non-exploitable strategies

*A*, must satisfy for all other strategies,

*B*,

### **Definition 1**

### **Lemma 1**

*Every ESS is also an NES in our setting.*

#### Logical argumentation

*A*is an ESS. By the definition of an ESS,

*E*[

*A*,

*A*] ≥

*E*[

*B*,

*A*] for all strategies,

*B*. Due to the zero-sum dynamic,

*E*[

*A*,

*A*] = 0. So, this means that

*E*[

*B*,

*A*] ≤ 0, it follows that −

*E*[

*B*,

*A*] ≥ 0. Therefore,

*B*. We use the zero-sum dynamic once more to see that

*A*to be an NES holds true. Hence, we see that every ESS is also an NES, but we shall see below that not every NES is an ESS.

### The unique NES

Our main theorem proves that (1) there is no ESS in our setting, but (2) the uniform SBD is the unique NES.

### **Theorem 1**

*Let*

*A be a species comprised of*

*N*≥ 4

*individual organisms and which has MCA less than or equal to*1/2.

*Assume*

*that*

*M*≥ 6.

*Then, there are*

*two mutually exclusive possibilities*:

- 1.
*A is the uniform SBD.* - 2.
*There is an SBD for a species**B comprised of at most**N individuals which has MCA less than or equal to*1/2*and which has positive expected value in competition with species A*.

*Therefore, we conclude:*

- 1.
*There is no SBD for species with at least**N*≥ 4*individual organisms and which has MCA less than or equal to*1/2*which is an ESS.* - 2.
*The unique NES is the uniform SBD.*

### Strategy of the proof

We have already demonstrated for the uniform SBD, *U*, that *E*[*B*, *U*] ≤ 0 for any SBD, *B*, which has MCA less than or equal to \(\frac 1 2\). So, the task at hand is to understand species, *A*, which are *not uniform*. We therefore begin by assuming that *A* is *not* the uniform SBD. We then proceed with several straightforward steps.

#### First simplification

We begin by making a simplifying observation. If \(\operatorname {MCA}(A) < \frac {1}{2}\), then we have shown that the uniform species has positive expected value in competition with species *A*. So, in that case, the theorem is proven. To complete the proof, we can henceforth assume that \(\operatorname {MCA}(A) = \frac {1}{2}\).

#### Pairs of CAs

*n*

_{A}(

*x*

_{0}) and

*n*

_{A}(

*x*

_{M}). There are three possibilities: either

*n*

_{A}(

*x*

_{0}) >

*n*

_{A}(

*x*

_{M}) ,

*n*

_{A}(

*x*

_{0}) <

*n*

_{A}(

*x*

_{M}), or

*n*

_{A}(

*x*

_{0}) =

*n*

_{A}(

*x*

_{1}).

- 1.In the first case, if
*n*_{A}(*x*_{0}) >*n*_{A}(*x*_{M}), we define species*B*to have*n*_{B}(*x*_{0}) =*N*/2 and*n*_{B}(*x*_{M}) =*N*/2, where*N*is the total number of organisms as in the theorem. Then, we compute the expected value of*B*in competition versus*A*in Eq. 8 of “B.2 Pairs of CAs”,So, in this case, we have found a species$$E[B,A] = \frac{n_{A} (0) - n_{A} (1)}{2} > 0.$$*B*which has MCA at most \(\frac {1}{2}\) (in fact equal to \(\frac {1}{2}\)), and the same number of organisms as species*A*, but which has positive expected value in competition versus species*A*. - 2.In the second case,
*n*_{A}(*x*_{0}) <*n*_{A}(*x*_{M}). Here, we define*B*withWe compute in “B.2 Pairs of CAs” that$$n_{B} (x_{j}) := \left\{\begin{array}{ll} 0 & x_{j} = 0 \\ 0 & x_{j} = 1 \\ \frac{N}{M-1} & \text{ otherwise.} \end{array}\right. $$since$$E[B,A] = \frac{1}{M-1} \left( n_{A}(x_{M}) - n_{A}(x_{0}) \right) > 0,$$*n*_{A}(*x*_{M}) >*n*_{A}(*x*_{0}). - 3.
In the third case,

*n*_{A}(*x*_{0}) =*n*_{A}(*x*_{M}). In this case, we do not do anything, but instead proceed to look at*n*_{A}(*x*_{1}) and*n*_{A}(*x*_{M− 1}).

#### Pair-by-pair

*n*

_{A}(

*x*

_{1}) and

*n*

_{A}(

*x*

_{M− 1}). There are the same three cases as above. In the first and second cases, we can define a species

*B*, which has \(\operatorname {MCA}(B) \leq \frac {1}{2}\), and the total number of organisms of species

*B*does not exceed

*N*. If \(n_{A}\left (\frac {1}{M} \right ) > n_{A} \left (1 - \frac {1}{M} \right )\), we define the species

*B*to have \(n_{B} (x_{1}) = \frac N 2\) and \(n_{B} (x_{M-1}) = \frac N 2\). Then, we compute in “B.3 Pair-by-pair” that the expected value of

*B*in competition versus

*A*is as follows:

*n*

_{A}(

*x*

_{1}) <

*n*

_{A}(

*x*

_{M− 1}), then we define

*B*is

*N*, and the MCA of species

*B*is \(\frac {1}{2}\). We compute there that

*n*

_{A}(

*x*

_{1}) =

*n*

_{A}(

*x*

_{M− 1}), we do not do anything, just look at the next pair, looking at

*n*

_{A}(

*x*

_{2}) and

*n*

_{A}(

*x*

_{M− 2}). In this way, we continue pair-by-pair.

#### Second simplification

*n*

_{A}(

*x*

_{k}) ≠

*n*

_{A}(

*x*

_{M−k}) for some

*k*= 0, 1, …. Once we reach such a pair, we show in “B.4 Second simplification” how to construct the species

*B*, which has

*E*[

*B*,

*A*] > 0. So, in all such cases, the theorem has been proven. It therefore only remains to consider the case when

*k*.

#### Exploiting non-uniformity

We are now reduced to the case where *n*_{A}(*x*_{k}) = *n*_{A}(*x*_{M−k}) holds for all *k*. That means that the SBD is symmetric around the competitive ability \(\frac {1}{2}\). However, we assumed that *A* is *not-uniform*. This means that the numbers *n*_{A}(*x*_{k}) cannot all be identical. So, some of the *n*_{A}(*x*_{k}) are larger, and some are smaller. It is precisely this lack of uniformity which allows us to find a weakness and exploit it with species *B*. The details of this process are contained in “B.5 Exploiting non-uniformity”.

## Concluding remarks

Evolution, or survival of the fittest, in the case of microbes must be approached from a different perspective in comparison to macro-organisms. Rather than assessing the fitness of individual microbes and applying evolution to individuals interpreted as representing a species, we propose that it is more fitting to consider cohorts of individuals that collectively but not individually represent the entire species. Individual microbes do not always behave in a way that maximizes fitness, nor does loss of an individual necessarily result in the loss of the genotype, because asexual reproduction yields multiple clones and can rapidly increase the abundance of specific clones or replace lost individuals representing a particular clone. Thus, loss of individuals can have no consequence for the species as a whole, suggesting a very different risk distribution compared to sexually reproducing organisms, where the species as a whole may nonetheless survive and remain variable with respect to their trait distribution. Thus, we analyze evolutionary fitness of species by interpolating between the individual-level competitions and the species-level growth or attrition.

As we are investigating competition among species, if one species has a positive expected value, implying population growth, the other species has a negative expected value, implying population decrease. Presuming that species evolve so as to minimize vulnerability to invasion and attrition by other species, rather than to maximize fitness, then all species of microbes which survive would evolve to the uniform SBD. Obviously, the rate of change in the SBD will be influenced by the environmental conditions, the presence of co-occurring species and to what degree SBDs are similar or overlap with co-occurring species (e.g., character displacement). Irrespective of any external factors, the uniform SBD is the unique SBD, which contains the maximal variability among its individuals. Moreover, it is the unique SBD, which is not vulnerable to replacement by species with other SBDs with the same (or lower) MCA. There are two mutually exclusive options: evolve towards the uniform SBD or risk vulnerability to invasion.

Interestingly, the uniform SBD is also neutral in competition with all other SBDs, which have the same mean competitive ability. Thus, our second result shows that the uniform SBD coexists with all other SBDs, which have the same MCA. Hence, it predicts that microbes, which evolve to the uniform SBD, simultaneously evolve to coexist. We therefore have discovered a plausible mathematical explanation for two prevalent characteristics of microbes, which until now have been inexplicable in the context of competition theory: (1) maximal variability of traits among individuals of a single species and (2) coexistence of virtually limitless different species of microbes. For multi-cellular organisms, intra-specific variability has been suggested to be an important characteristic in the management of biodiversity (Des Roches et al. 2018). Our results imply that a quantification of the trait distributions of microbial organisms is essential to understand processes that drive ecosystem functioning, including ocean ecosystems and the global carbon cycle.

## Notes

### Acknowledgements

Julie Rowlett is grateful to Ksenia Fedosova, Philip Gerlee, and Jeffrey Steif for stimulating discussions and insightful comments. Many thanks to Medet Nursultanov for carefully checking the proofs and for suggesting the SBDs defined in Eqs. 18 and 20. We are also grateful to Edvin Wedlin for suggesting the term “non-exploitable strategy,” NES. Constructive feedback by three anonymous reviewers has improved a prior version of this manuscript.

### Funding information

Susanne Menden-Deuer was supported through the National Aeronautics and Space Administration (award #NNX15AL2G).

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## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.