Ecosystems with mutually exclusive interactions self-organize to a state of high diversity

Diversity and ecotones in a model ecosystems of sessile species

Sessile species compete for space and accessible light, with directed interactions evident in one species overgrowing another and with multispecies systems characterized by nontransitive relationships. Such patterns are observed in coral reefs or lichens on rock surfaces. Open systems with episodic invasions of such species have been predicted to exhibit a stable high-diversity state when the interaction probability is below a certain critical threshold. Here, we explore this metastable high-diversity state and find that the diversity in the high-diversity state scales with the square root of the system area. When introducing two different environments, we predict a hugely increased diversity along mutual environment border. Further, the presence of spatially segregated environments is predicted to allow for increased robustness of the high-diversity state.

Eco-evolutionary cyclic dominance among predators, prey, and parasites

Predator-prey interactions are one of ecology's central research themes, but with many interdisciplinary implications across the social and natural sciences. Here we consider an often-overlooked species in these interactions, namely parasites. We first show that a simple predator-prey-parasite model, inspired by the classical Lotka-Volterra equations, fails to produce a stable coexistence of all three species, thus failing to provide a biologically realistic outcome. To improve this, we introduce free space as a relevant eco-evolutionary component in a new mathematical model that uses a game-theoretical payoff matrix to describe a more realistic setup. We then show that the consideration of free space stabilizes the dynamics by means of cyclic dominance that emerges between the three species. We determine the parameter regions of coexistence as well as the types of bifurcations leading to it by means of analytical derivations as well as by means of numerical simulations. We conclude that the consideration of free space as a finite resource reveals the limits of biodiversity in predator-prey-parasite interactions, and it may also help us in the determination of factors that promote a healthy biota.

Emerging diversity in a population of evolving intransitive dice

Exploiting the mathematical curiosity of intransitive dice, we present a simple theoretical model for coevolution that captures scales ranging from the genome of the individual to the system-wide emergence of species diversity. We study a set of evolving agents that interact competitively in a closed system, in which both the dynamics of mutations and competitive advantage emerge directly from interpreting a genome as the sides of a die. The model demonstrates sympatric speciation where new species evolve from existing ones while in contact with the entire ecosystem. Allowing free mutations both in the genomes and the mutation rates, we find, in contrast to hierarchical models of fitness, the emergence of a metastable state of finite mutation rate and diversity.

Community formation in wealth-mediated thermodynamic strategy evolution

We study a dynamical system defined by a repeated game on a 1D lattice, in which the players keep track of their gross payoffs over time in a bank. Strategy updates are governed by a Boltzmann distribution, which depends on the neighborhood bank values associated with each strategy, relative to a temperature scale, which defines the random fluctuations. Players with higher bank values are, thus, less likely to change strategy than players with a lower bank value. For a parameterized rock-paper-scissors game, we derive a condition under which communities of a given strategy form with either fixed or drifting boundaries. We show the effect of a temperature increase on the underlying system and identify surprising properties of this model through numerical simulations.

Persistent homology and the shape of evolutionary games

For nearly three decades, spatial games have produced a wealth of insights to the study of behavior and its relation to population structure. However, as different rules and factors are added or altered, the dynamics of spatial models often become increasingly complicated to interpret. To tackle this problem, we introduce persistent homology as a rigorous framework that can be used to both define and compute higher-order features of data in a manner which is invariant to parameter choices, robust to noise, and independent of human observation. Our work demonstrates its relevance for spatial games by showing how topological features of simulation data that persist over different spatial scales reflect the stability of strategies in 2D lattice games. To do so, we analyze the persistent homology of scenarios from two games: a Prisoner's Dilemma and a SIRS epidemic model. The experimental results show how the method accurately detects features that correspond to real aspects of the game dynamics. Unlike other tools that study dynamics of spatial systems, persistent homology can tell us something meaningful about population structure while remaining neutral about the underlying structure itself. Regardless of game complexity, since strategies either succeed or fail to conform to shapes of a certain topology there is much potential for the method to provide novel insights for a wide variety of spatially extended systems in biology, social science, and physics.

Why is cyclic dominance so rare?

Natural populations can contain multiple types of coexisting individuals. How does natural selection maintain such diversity within and across populations? A popular theoretical basis for the maintenance of diversity is cyclic dominance, illustrated by the rock-paper-scissor game. However, it appears difficult to find cyclic dominance in nature. Why is this the case? Focusing on continuously produced novel mutations, we theoretically addressed the rareness of cyclic dominance. We developed a model of an evolving population and studied the formation of cyclic dominance. Our results showed that the chance for cyclic dominance to emerge is lower when the newly introduced type is similar to existing types compared to the introduction of an unrelated type. This suggests that cyclic dominance is more likely to evolve through the assembly of unrelated types whereas it rarely evolves within a community of similar types.

Lying on networks: The role of structure and topology in promoting honesty

Lies can have a negating impact on governments, companies, and the society as a whole. Understanding the dynamics of lying is therefore of crucial importance across different fields of research. While lying has been studied before in well-mixed populations, it is a fact that real interactions are rarely well-mixed. Indeed, they are usually structured and thus best described by networks. Here we therefore use the Monte Carlo method to study the evolution of lying in the sender-receiver game in a one-parameter family of networks, systematically covering complete networks, small-world networks, and one-dimensional rings. We show that lies that benefit the sender at a cost to the receiver, the so-called black lies, are less likely to proliferate on networks than they do in well-mixed populations. Honesty is thus more likely to evolve, but only when the benefit for the sender is smaller than the cost for the receiver. Moreover, this effect is particularly strong in small-world networks, but less so in the one-dimensional ring. For lies that favor the receiver at a cost to the sender, the so-called altruistic white lies, we show that honesty is also more likely to evolve than it is in well-mixed populations. But contrary to black lies, this effect is more expressed in the one-dimensional ring, whereas in small-world networks it is present only when the cost to the sender is greater than the benefit for the receiver. Last, for lies that benefit both the sender and the receiver, the so-called Pareto white lies, we show that the network structure actually favors the evolution of lying, but this only occurs when the benefit for the sender is slightly greater than the benefit for the receiver. In this case again the small-world topology acts as an amplifier of the effect, while other network topologies fail to do the same. In addition to these main results we discuss several other findings, which together show clearly that the structure of interactions and the overall topology of the network critically determine the dynamics of lying.

Trade-off shapes diversity in eco-evolutionary dynamics

We introduce an Interaction- and Trade-off-based Eco-Evolutionary Model (ITEEM), in which species are competing in a well-mixed system, and their evolution in interaction trait space is subject to a life-history trade-off between replication rate and competitive ability. We demonstrate that the shape of the trade-off has a fundamental impact on eco-evolutionary dynamics, as it imposes four phases of diversity, including a sharp phase transition. Despite its minimalism, ITEEM produces a remarkable range of patterns of eco-evolutionary dynamics that are observed in experimental and natural systems. Most notably we find self-organization towards structured communities with high and sustained diversity, in which competing species form interaction cycles similar to rock-paper-scissors games.

Microbial mutualism at a distance: The role of geometry in diffusive exchanges

The exchange of diffusive metabolites is known to control the spatial patterns formed by microbial populations, as revealed by recent studies in the laboratory. However, the matrices used, such as agarose pads, lack the structured geometry of many natural microbial habitats, including in the soil or on the surfaces of plants or animals. Here we address the important question of how such geometry may control diffusive exchanges and microbial interaction. We model mathematically mutualistic interactions within a minimal unit of structure: two growing reservoirs linked by a diffusive channel through which metabolites are exchanged. The model is applied to study a synthetic mutualism, experimentally parametrized on a model algal-bacterial co-culture. Analytical and numerical solutions of the model predict conditions for the successful establishment of remote mutualisms, and how this depends, often counterintuitively, on diffusion geometry. We connect our findings to understanding complex behavior in synthetic and naturally occurring microbial communities.

Enhanced robustness of evolving open systems by the bidirectionality of interactions between elements

Living organisms, ecosystems, and social systems are examples of complex systems in which robustness against inclusion of new elements is an essential feature. A recently proposed simple model has revealed a general mechanism by which such systems can become robust against inclusion of elements with totally random interactions when the elements have a moderate number of links. The interaction is, however, in many systems often intrinsically bidirectional like for mutual symbiosis and competition in ecology. This study reports the strong reinforcement effect of the bidirectionality of the interactions on the robustness of evolving systems. We show that the system with purely bidirectional interactions can grow with twofold average degree, in comparison with the purely unidirectional system. This drastic shift of the transition point comes from the reinforcement of each node, not from a change in structure of the emergent system. For systems with partially bidirectional interactions we find that the regime of the growing phase gets expanded. In the dense interaction regime, there exists an optimum proportion of bidirectional interactions for the growth rate at around 1/3. In the sparsely connected systems, small but finite fraction of bidirectional links can change the system's behaviour from non-growing to growing.

Deterministic extinction by mixing in cyclically competing species

We consider a cyclically competing species model on a ring with global mixing at finite rate, which corresponds to the well-known Lotka-Volterra equation in the limit of infinite mixing rate. Within a perturbation analysis of the model from the infinite mixing rate, we provide analytical evidence that extinction occurs deterministically at sufficiently large but finite values of the mixing rate for any species number N≥3. Further, by focusing on the cases of rather small species numbers, we discuss numerical results concerning the trajectories toward such deterministic extinction, including global bifurcations caused by changing the mixing rate.

Models of life: epigenetics, diversity and cycles

This review emphasizes aspects of biology that can be understood through repeated applications of simple causal rules. The selected topics include perspectives on gene regulation, phage lambda development, epigenetics, microbial ecology, as well as model approaches to diversity and to punctuated equilibrium in evolution. Two outstanding features are repeatedly described. One is the minimal number of rules to sustain specific states of complex systems for a long time. The other is the collapse of such states and the subsequent dynamical cycle of situations that restitute the system to a potentially new metastable state.

Characterization of phase transitions in a model ecosystem of sessile species

We consider a model ecosystem of sessile species competing for space. In particular, we consider the system introduced by Mathiesen et al. [J. Mathiesen, N. Mitarai, K. Sneppen, and A. Trusina, Phys. Rev. Lett. 107, 188101 (2011)PRLTAO0031-900710.1103/PhysRevLett.107.188101] where species compete according to a fixed interaction network with links determined by a Bernoulli process. In the limit of a small introduction rate of new species, the model exhibits a discontinuous transition from a high-diversity state to a low-diversity state as the interaction probability between species, γ, is increased from zero. Here we explore the effects of finite introduction rates and system size on the phase transition by utilizing efficient parallel computing. We find that the low state appears for γ>γ_{c}. As γ is increased further, the high state approaches the low state, suggesting the possibility that the two states merge at a high γ. We find that the fraction of time spent in the high state becomes longer with higher introduction rates, but the availability of the two states is rather insensitive to the value of the introduction rate. Furthermore, we establish a relation between the introduction rate and the system size, which preserves the probability for the system to remain in the high-diversity state.

Biodiversity in models of cyclic dominance is preserved by heterogeneity in site-specific invasion rates

Global, population-wide oscillations in models of cyclic dominance may result in the collapse of biodiversity due to the accidental extinction of one species in the loop. Previous research has shown that such oscillations can emerge if the interaction network has small-world properties, and more generally, because of long-range interactions among individuals or because of mobility. But although these features are all common in nature, global oscillations are rarely observed in actual biological systems. This begets the question what is the missing ingredient that would prevent local oscillations to synchronize across the population to form global oscillations. Here we show that, although heterogeneous species-specific invasion rates fail to have a noticeable impact on species coexistence, randomness in site-specific invasion rates successfully hinders the emergence of global oscillations and thus preserves biodiversity. Our model takes into account that the environment is often not uniform but rather spatially heterogeneous, which may influence the success of microscopic dynamics locally. This prevents the synchronization of locally emerging oscillations, and ultimately results in a phenomenon where one type of randomness is used to mitigate the adverse effects of other types of randomness in the system.

Zealots tame oscillations in the spatial rock-paper-scissors game

The rock-paper-scissors game is a paradigmatic model for biodiversity, with applications ranging from microbial populations to human societies. Research has shown, however, that mobility jeopardizes biodiversity by promoting the formation of spiral waves, especially if there is no conservation law in place for the total number of competing players. First, we show that even if such a conservation law applies, mobility still jeopardizes biodiversity in the spatial rock-paper-scissors game if only a small fraction of links of the square lattice is randomly rewired. Secondly, we show that zealots are very effective in taming the amplitude of oscillations that emerge due to mobility and/or interaction randomness, and this regardless of whether the later is quenched or annealed. While even a tiny fraction of zealots brings significant benefits, at 5% occupancy zealots practically destroy all oscillations regardless of the intensity of mobility, and regardless of the type and strength of randomness in the interaction structure. Interestingly, by annealed randomness the impact of zealots is qualitatively the same as by mobility, which highlights that fast diffusion does not necessarily destroy the coexistence of species, and that zealotry thus helps to recover the stable mean-field solution. Our results strengthen the important role of zealots in models of cyclic dominance, and they reveal fascinating evolutionary outcomes in structured populations that are a unique consequence of such uncompromising behavior.

Transition matrix model for evolutionary game dynamics

We study an evolutionary game model based on a transition matrix approach, in which the total change in the proportion of a population playing a given strategy is summed directly over contributions from all other strategies. This general approach combines aspects of the traditional replicator model, such as preserving unpopulated strategies, with mutation-type dynamics, which allow for nonzero switching to unpopulated strategies, in terms of a single transition function. Under certain conditions, this model yields an endemic population playing non-Nash-equilibrium strategies. In addition, a Hopf bifurcation with a limit cycle may occur in the generalized rock-scissors-paper game, unlike the replicator equation. Nonetheless, many of the Folk Theorem results are shown to hold for this model.

Three is much more than two in coarsening dynamics of cyclic competitions

The classical game of rock-paper-scissors has inspired experiments and spatial model systems that address the robustness of biological diversity. In particular, the game nicely illustrates that cyclic interactions allow multiple strategies to coexist for long-time intervals. When formulated in terms of a one-dimensional cellular automata, the spatial distribution of strategies exhibits coarsening with algebraically growing domain size over time, while the two-dimensional version allows domains to break and thereby opens the possibility for long-time coexistence. We consider a quasi-one-dimensional implementation of the cyclic competition, and study the long-term dynamics as a function of rare invasions between parallel linear ecosystems. We find that increasing the complexity from two to three parallel subsystems allows a transition from complete coarsening to an active steady state where the domain size stays finite. We further find that this transition happens irrespective of whether the update is done in parallel for all sites simultaneously or done randomly in sequential order. In both cases, the active state is characterized by localized bursts of dislocations, followed by longer periods of coarsening. In the case of the parallel dynamics, we find that there is another phase transition between the active steady state and the coarsening state within the three-line system when the invasion rate between the subsystems is varied. We identify the critical parameter for this transition and show that the density of active boundaries has critical exponents that are consistent with the directed percolation universality class. On the other hand, numerical simulations with the random sequential dynamics suggest that the system may exhibit an active steady state as long as the invasion rate is finite.

Cyclic dominance in evolutionary games: a review

Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rock-paper-scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg-Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.

From pairwise to group interactions in games of cyclic dominance

We study the rock-paper-scissors game in structured populations, where the invasion rates determine individual payoffs that govern the process of strategy change. The traditional version of the game is recovered if the payoffs for each potential invasion stem from a single pairwise interaction. However, the transformation of invasion rates to payoffs also allows the usage of larger interaction ranges. In addition to the traditional pairwise interaction, we therefore consider simultaneous interactions with all nearest neighbors, as well as with all nearest and next-nearest neighbors, thus effectively going from single pair to group interactions in games of cyclic dominance. We show that differences in the interaction range affect not only the stationary fractions of strategies but also their relations of dominance. The transition from pairwise to group interactions can thus decelerate and even revert the direction of the invasion between the competing strategies. Like in evolutionary social dilemmas, in games of cyclic dominance, too, the indirect multipoint interactions that are due to group interactions hence play a pivotal role. Our results indicate that, in addition to the invasion rates, the interaction range is at least as important for the maintenance of biodiversity among cyclically competing strategies.

Speciation, diversification, and coexistence of sessile species that compete for space

Speciation, diversification, and competition between species challenge the stability of complex ecosystems. Laboratory experiments often focus on one or two species competing under conditions where they may grow exponentially. Field studies, in contrast, emphasize multi-species communities characterized by many types of ecological interactions. A general problem is to understand conditions that support a dynamically maintained coexistence of many species in an ecosystem over a long time span. In the present paper we propose a lattice model of multiple competing and evolving sessile species. When allowing the interspecies interactions to mutate, we obtain coexistence of many species in a complex ecosystem, provided that there is a cost for each interaction. The diversity reached by the model incorporating speciation is found to be substantially higher than in the case when entirely new species appear due to immigration from outside of the considered ecosystem. The species self-organize their spatial distribution through competitive interactions to create many patches, implicitly protecting each other from competitively superior species, and speciation in each patch leads the system to high diversity. We also show that species that exist a long time tend to have a relatively small population, as this allows them to avoid encounter with competitive invaders.

Geometry for a penguin-albatross rookery

We introduce a simple ecological model describing the spatial organization of two interacting populations whose individuals are indifferent to conspecifics and avoid the proximity to heterospecifics. At small population densities Φ a nontrivial structure is observed where clusters of individuals arrange into a rhomboidal bipartite network with an average degree of 4. For Φ → 0 the length scale, order parameter, and susceptibility of the network exhibit power-law divergences compatible with hyperscaling, suggesting the existence of a zero-density nontrivial critical point. At larger densities a critical threshold Φ(c) is identified above which the evolution toward a partially ordered configuration is prevented and the system becomes jammed in a fully mixed state.

Disturbance accelerates the transition from low- to high-diversity state in a model ecosystem

The effect of disturbance on a model ecosystem of sessile and mutually competitive species [Mathiesen et al., Phys. Rev. Lett. 107, 188101 (2011); Mitarai et al., Phys. Rev. E 86, 011929 (2012)] is studied. The disturbance stochastically removes individuals from the system, and the created empty sites are recolonized by neighboring species. We show that the stable high-diversity state, maintained by occasional cyclic species interactions that create isolated patches of metapopulations, is robust against small disturbance. We further demonstrate that finite disturbance can accelerate the transition from the low- to high-diversity state by helping the creation of small patches through diffusion of boundaries between species with standoff relations.

Global attractors and extinction dynamics of cyclically competing species

Transitions to absorbing states are of fundamental importance in nonequilibrium physics as well as ecology. In ecology, absorbing states correspond to the extinction of species. We here study the spatial population dynamics of three cyclically interacting species. The interaction scheme comprises both direct competition between species as in the cyclic Lotka-Volterra model, and separated selection and reproduction processes as in the May-Leonard model. We show that the dynamic processes leading to the transient maintenance of biodiversity are closely linked to attractors of the nonlinear dynamics for the overall species' concentrations. The characteristics of these global attractors change qualitatively at certain threshold values of the mobility and depend on the relative strength of the different types of competition between species. They give information about the scaling of extinction times with the system size and thereby the stability of biodiversity. We define an effective free energy as the negative logarithm of the probability to find the system in a specific global state before reaching one of the absorbing states. The global attractors then correspond to minima of this effective energy landscape and determine the most probable values for the species' global concentrations. As in equilibrium thermodynamics, qualitative changes in the effective free energy landscape indicate and characterize the underlying nonequilibrium phase transitions. We provide the complete phase diagrams for the population dynamics and give a comprehensive analysis of the spatio-temporal dynamics and routes to extinction in the respective phases.

Coexistence and survival in conservative Lotka-Volterra networks

Analyzing coexistence and survival scenarios of Lotka-Volterra (LV) networks in which the total biomass is conserved is of vital importance for the characterization of long-term dynamics of ecological communities. Here, we introduce a classification scheme for coexistence scenarios in these conservative LV models and quantify the extinction process by employing the Pfaffian of the network's interaction matrix. We illustrate our findings on global stability properties for general systems of four and five species and find a generalized scaling law for the extinction time.

Labyrinthine clustering in a spatial rock-paper-scissors ecosystem

The spatial rock-paper-scissors ecosystem, where three species interact cyclically, is a model example of how spatial structure can maintain biodiversity. We here consider such a system for a broad range of interaction rates. When one species grows very slowly, this species and its prey dominate the system by self-organizing into a labyrinthine configuration in which the third species propagates. The cluster size distributions of the two dominating species have heavy tails and the configuration is stabilized through a complex spatial feedback loop. We introduce a statistical measure that quantifies the amount of clustering in the spatial system by comparison with its mean-field approximation. Hereby, we are able to quantitatively explain how the labyrinthine configuration slows down the dynamics and stabilizes the system.

Multistability with a metastable mixed state

Complex dynamical systems often show multiple metastable states. In macroevolution, such behavior is suggested by punctuated equilibrium and discrete geological epochs. In molecular biology, bistability is found in epigenetics and in the many mutually exclusive states that a human cell can take. Sociopolitical systems can be single-party regimes or a pluralism of balancing political fractions. To introduce multistability, we suggest a model system of D mutually exclusive microstates that battle for dominance in a large system. Assuming one common intermediate state, we obtain D+1 metastable macrostates for the system, one of which is a self-reinforced mixture of all D+1 microstates. Robustness of this metastable mixed state increases with diversity D.

Emergence of diversity in a model ecosystem

The biological requirements for an ecosystem to develop and maintain species diversity are in general unknown. Here we consider a model ecosystem of sessile and mutually excluding organisms competing for space [Mathiesen et al. Phys. Rev. Lett. 107, 188101 (2011)]. Competition is controlled by an interaction network with fixed links chosen by a Bernoulli process. New species are introduced in the system at a predefined rate. In the limit of small introduction rates, the system becomes bistable and can undergo a phase transition from a state of low diversity to high diversity. We suggest that isolated patches of metapopulations formed by the collapse of cyclic relations are essential for the transition to the state of high diversity.

Clonal selection prevents tragedy of the commons when neighbors compete in a rock-paper-scissors game

The rock-paper-scissors game is a model example of the ongoing cyclic turnover typical of many ecosystems, ranging from the terrestrial and aquatic to the microbial. Here we explore the evolution of a rock-paper-scissors system where three species compete for space. The species are allowed to mutate and change the speed by which they invade one another. In the case when all species have similar mutation rates, we observe a perpetual arms race where no single species prevails. When only two species mutate, their aggressions increase indefinitely until the ecosystem collapses and only the nonmutating species survives. Finally we show that when only one species mutates, group selection removes individual predators with the fastest growth rates, causing the growth rate of the species to stabilize. We explain this group selection quantitatively.

Fracturing ranked surfaces

Discretized landscapes can be mapped onto ranked surfaces, where every element (site or bond) has a unique rank associated with its corresponding relative height. By sequentially allocating these elements according to their ranks and systematically preventing the occupation of bridges, namely elements that, if occupied, would provide global connectivity, we disclose that bridges hide a new tricritical point at an occupation fraction p = p_c, where p_c is the percolation threshold of random percolation. For any value of p in the interval p_c < p ≤ 1, our results show that the set of bridges has a fractal dimension d_BB ≈ 1.22 in two dimensions. In the limit p → 1, a self-similar fracture is revealed as a singly connected line that divides the system in two domains. We then unveil how several seemingly unrelated physical models tumble into the same universality class and also present results for higher dimensions.