21st Century Science Initiative Grant: Studying Complex Systems
How is the staggering biological diversity on earth maintained? This question has puzzled ecologists and evolutionary biologists for decades because the competitive exclusion principle tells us that two ecologically similar species should compete so strongly that one is necessarily driven extinct. In an attempt to resolve this apparent contradiction, two somewhat unsatisfying frameworks have been proposed. First is the "neutral theory of biodiversity", which posits that species are indistinguishable, and that the dramatic differences in empirical species' frequencies can be explained by demographic noise. Despite the great power of this theory to fit data, most ecologists feel that the model is pathological in that no two species - let alone all species - are indistinguishable. Second is a set of "niche partitioning" theories in which species are driven to develop unique niche requirements (e.g. set of resources, predators, pathogens). The fact that species' growth can be limited by a finite number of factors makes the coexistence of very many species highly improbable in this framework.
Here, we propose a new perspective that both embraces competitive exclusion yet explains the impressive biodiversity that surrounds us. Our model is fashioned after the children's game of "rock-paper-scissors", with each "strategy" designating a single species. Rocks exclude scissors when competing in isolation, scissors exclude papers and papers exclude rocks. In a system such as this, all three species can coexist indefinitely given their "intransitive" (i.e. non-hierarchical) set of interactions.
The simple "rock-paper-scissor" system has been intensively studied, as it finds applications in a variety of fields including biology, economics and psychology. Practically no attention, however, has been paid to the many-species case. We will develop methods merging graph theory, game theory and dynamical systems to provide analytical results for a large set of interacting components. Analytic results include the number of coexisting species and their frequencies. We will then test our ability to predict community composition based on the result of pairwise competition experiments, as well as our ability to infer competitive interactions based on an analysis of empirical frequencies at stationarity.
We can build a network (technically, a tournament) in which we draw a connection from species A to species B (nodes of the network) whenever B excludes A in isolation. Repeating the process for all pairs of species generates a tournament. This network can be mapped in a game-theoretical framework in the following way: two players simultaneously and independently select a species in the tournament (according to each player's "strategy"). The player that picks the winner in pair-wise competition gets one point, while the other player loses one point. What are the optimal strategies for the two players? It turns out that this simple zero-sum (one player gains one point while the other player loses one point), symmetric (the two players are exchangeable) game has only one optimal solution (strategy), and that this strategy is the same for both players. We prove that this solution is also the expected frequency of the species composing a tournament when competing in a well-mixed environment.
Using this mapping between the pairwise competitive outcomes and the game theoretical result, we can easily solve for the expected frequencies of all species at equilibrium. These results are empirically testable. For example, we can take a given number of species, perform all the possible pairwise competition experiments, and then use these outcomes to predict the community composition when all the species are competing against each other. We can also adopt the inverse approach and, given the distribution of species' densities, infer which tournament best explains the observed community structure. We present preliminary results supporting the claim that the theory can predict species distributions that are compatible with those observed in nature.
Clearly, testing the predictions of our model would be quite challenging when the species in question are vascular plants or animals, as the experimental complexity would be overwhelming. We propose, therefore, to use bacterial communities as they provide diverse yet tractable experimental systems. Bacteria offer several advantages, including small size, rapid generation time, and the development of high-throughput robotic methods that allow analysis of temporal dynamics with ample replication of communities. Moreover, we can control environmental conditions, paving the way to sensitivity analysis that can inform us about the generality of the results.
We will focus in particular on a set of bacteria commonly associated with the plant Arabidopsis thaliana. This is a fascinating community of bacteria that includes species with diverse lifestyles and with clear evidence of interspecific competition (both through resource competition and through the production of toxins) between them. We will perform pairwise experiments to determine the competitive ability of each species of bacteria, recording the density of the two species through time. We will then move to assemblages of the same species, again determining relative densities through time.
Our ability to obtain precise time-series on complex communities opens the possibility of testing theories not only at the stationary state, but also far from equilibrium. This should facilitate major advances in that all published theories to date have been developed with a static endpoint in mind. Indeed, the protracted debates on community diversity, as well as the associated statistical models, have centered on concepts such as stability and stationarity: species reach "equilibrium" so that their relative frequencies do not change over time. Despite the fact that this simplifies the mathematical treatment, it is clear that most of the ecological action happens during the transient dynamics phase. Extinction, for example, does not wait for equilibrium to be achieved but is an ongoing process throughout community establishment. Robust statistical techniques can be used to infer the parameters of dynamic models based on time series data. In particle filtering, we have a model for the dynamical system of interest (in this case a model expressing how competition among species evolves in time) and an observation model (i.e. a model describing observations in time). The time series of species' densities can be then used to infer parameter values and the likelihood of any model-dataset couple. In this way, models that predict similar endpoints can be still be assigned very diffrent likelihoods, whenever their dynamics are different. We believe that this novel approach to model selection will bring new ideas and more statistical power to the field.
Our simple theory of biodiversity could be extended in several ways. One of the most promising is to make the outcome of competition stochastic rather than deterministic. For example, one could imagine a framework in which species A excludes B with probability p. The simple "rock-paper-scissors" game that we outline above corresponds to the case in which p is either 1 or 0 (i.e. one species always wins). Another natural case to study is that in which p = 0,5: this is exactly the neutral case. As such, the theory outlined here and the neutral theory of biodiversity can be viewed as two extremes of a spectrum of competitive hierarchies. By varying values for p, we can mimic spatial heterogeneity, seasonal variations in the competitive abilities, etc., making the theory more biologically grounded.
The theory proposed here has applications that go far beyond ecology. In fact, the same type of equation has been used to study other complex networks of competing agents, such as in population genetics (i.e. frequency of alleles in a population), economics (i.e. competition among firms), evolutionary dynamics (evolutionary game theory), and psychology (formation of "social traps"). In general, whenever there is competition among several subjects and the presence of complex trade-offs, we expect similar dynamics to arise. Our goal is to develop a theoretical framework that is grounded in its application to ecological theory, yet flexible enough to provide insights and tools relevant to a diverse range of fields.