Emergence of a complex and stable network in a model ecosystem with extinction and mutation

Coexistence of many species under a random competition-colonization trade-off

The competition-colonization (CC) trade-off is a well-studied coexistence mechanism for metacommunities. In this setting, it is believed that the coexistence of all species requires their traits to satisfy restrictive conditions limiting their similarity. To investigate whether diverse metacommunities can assemble in a CC trade-off model, we study their assembly from a probabilistic perspective. From a pool of species with parameters (corresponding to traits) sampled at random, we compute the probability that any number of species coexist and characterize the set of species that emerges through assembly. Remarkably, almost exactly half of the species in a large pool typically coexist, with no saturation as the size of the pool grows, and with little dependence on the underlying distribution of traits. Through a mix of analytical results and simulations, we show that this unlimited niche packing emerges as assembly actively moves communities toward overdispersed configurations in niche space. Our findings also apply to a realistic assembly scenario where species invade one at a time from a fixed regional pool. When diversity arises de novo in the metacommunity, richness still grows without bound, but more slowly. Together, our results suggest that the CC trade-off can support the robust emergence of diverse communities, even when coexistence of the full species pool is exceedingly unlikely.

Adaptation of metabolite leakiness leads to symbiotic chemical exchange and to a resilient microbial ecosystem

Microbial communities display remarkable diversity, facilitated by the secretion of chemicals that can create new niches. However, it is unclear why cells often secrete even essential metabolites after evolution. Based on theoretical results indicating that cells can enhance their own growth rate by leaking even essential metabolites, we show that such "leaker" cells can establish an asymmetric form of mutualism with "consumer" cells that consume the leaked chemicals: the consumer cells benefit from the uptake of the secreted metabolites, while the leaker cells also benefit from such consumption, as it reduces the metabolite accumulation in the environment and thereby enables further secretion, resulting in frequency-dependent coexistence of multiple microbial species. As supported by extensive simulations, such symbiotic relationships generally evolve when each species has a complex reaction network and adapts its leakiness to optimize its own growth rate under crowded conditions and nutrient limitations. Accordingly, symbiotic ecosystems with diverse cell species that leak and exchange many metabolites with each other are shaped by cell-level adaptation of leakiness of metabolites. Moreover, the resultant ecosystems with entangled metabolite exchange are resilient against structural and environmental perturbations. Thus, we present a theory for the origin of resilient ecosystems with diverse microbes mediated by secretion and exchange of essential chemicals.

The balance of interaction types determines the assembly and stability of ecological communities

What determines the assembly and stability of complex communities is a central question in ecology. Past work has suggested that mutualistic interactions are inherently destabilizing. However, this conclusion relies on the assumption that benefits from mutualisms never stop increasing. Furthermore, almost all theoretical work focuses on the internal (asymptotic) stability of communities assembled all at once. Here, we present a model with saturating benefits from mutualisms and sequentially assembled communities. We show that such communities are internally stable for any level of diversity and any combination of species interaction types. External stability, or resistance to invasion, is thus an important but overlooked measure of stability. We demonstrate that the balance of different interaction types governs community dynamics. A higher fraction of mutualistic interactions can increase the external stability and diversity of communities as well as species persistence, if mutualistic interactions tend to provide unique benefits. Ecological selection increases the prevalence of mutualisms, and limits on biodiversity emerge from species interactions. Our results help resolve long-standing debates on the stability, saturation and diversity of communities.

Population size changes and extinction risk of populations driven by mutant interactors

Spontaneous random mutations are an important source of variation in populations. Many evolutionary models consider mutants with a fixed fitness, chosen from a fitness distribution without considering microscopic interactions among the residents and mutants. Here, we go beyond this and consider "mutant interactors," which lead to new interactions between the residents and invading mutants that can affect the population size and the extinction risk of populations. We model microscopic interactions between individuals by using a dynamic interaction matrix, the dimension of which increases with the emergence of a new mutant and decreases with extinction. The new interaction parameters of the mutant follow a probability distribution around the payoff entries of its ancestor. These new interactions can drive the population away from the previous equilibrium and lead to changes in the population size. Thus, the population size is an evolving property rather than an externally controlled variable. We calculate the average population size of our stochastic system over time and quantify the extinction risk of the population by the mean time to extinction.

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.

Ants improve the reproduction of inferior morphs to maintain a polymorphism in symbiont aphids

Identifying stable polymorphisms is essential for understanding biodiversity. Distinctive polymorphisms are rare in nature because a superior morph should dominate a population. In addition to the three known mechanisms for polymorphism persistence, we recently reported a fourth mechanism: protection of the polymorphism by symbionts. Attending ants preferentially protect polymorphic aphid colonies consisting of green and red morphs. Here, we show that attending ants manipulate the reproductive rate of their preferred green morphs to equal that of the red morphs, leading to the persistence of the polymorphism within the colonies. We could not, however, explain how the ants maintained the polymorphism in aphid colonies regardless of inter-morph competition. Manipulation by symbionts may be important for the maintenance of polymorphisms and the resulting biodiversity in certain symbiotic systems.

Evolution models with extremal dynamics

The random-neighbor version of the Bak-Sneppen biological evolution model is reproduced, along with an analogous model of random replicators, the latter eventually experiencing topology changes. In the absence of topology changes, both types of models self-organize to a critical state. Species extinctions in the replicator system degenerates the self-organization to a random walk, as does vanishing of species interaction for the BS-model. A replicator model with speciation is introduced, experiencing dramatic topology changes. It produces a variety of features, but self-organizes to a possibly critical state only in a few special cases. Speciation-extinction dynamics interfering with self-organization, biological macroevolution probably is not a self-organized critical system.

A new dimension: Evolutionary food web dynamics in two dimensional trait space

Species within a habitat are not uniformly distributed. However this aspect of community structure, which is fundamental to many conservation activities, is neglected in the majority of models of food web assembly. To address this issue, we introduce a model which incorporates a second dimension, which can be interpreted as space, into the trait space used in evolutionary food web models. Our results show that the additional trait axis allows the emergence of communities with a much greater range of network structures, similar to the diversity observed in real ecological communities. Moreover, the network properties of the food webs obtained are in good agreement with those of empirical food webs. Community emergence follows a consistent pattern with spread along the second trait axis occurring before the assembly of higher trophic levels. Communities can reach either a static final structure, or constantly evolve. We observe that the relative importance of competition and predation is a key determinant of the network structure and the evolutionary dynamics. The latter are driven by the interaction-competition and predation-between small groups of species. The model remains sufficiently simple that we are able to identify the factors, and mechanisms, which determine the final community state.

Emergence of structured communities through evolutionary dynamics

Species-rich communities, in which many competing species coexist in a single trophic level, are quite frequent in nature, but pose a formidable theoretical challenge. In particular, it is known that complex competitive systems become unstable and unfeasible when the number of species is large. Recently, many studies have attributed the stability of natural communities to the structure of the interspecific interaction network, yet the nature of such structures and the underlying mechanisms responsible for them remain open questions. Here we introduce an evolutionary model, based on the generic Lotka-Volterra competitive framework, from which a stable, structured, diverse community emerges spontaneously. The modular structure of the competition matrix reflects the phylogeny of the community, in agreement with the hierarchial taxonomic classification. Closely related species tend to have stronger niche overlap and weaker fitness differences, as opposed to pairs of species from different modules. The competitive-relatedness hypothesis and the idea of emergent neutrality are discussed in the context of this evolutionary model.

Symmetry of interactions rules in incompletely connected random replicator ecosystems

The evolution of an incompletely connected system of species with speciation and extinction is investigated in terms of random replicators. It is found that evolving random replicator systems with speciation do become large and complex, depending on speciation parameters. Antisymmetric interactions result in large systems, whereas systems with symmetric interactions remain small. A co-dominating feature is within-species interaction pressure: large within-species interaction increases species diversity. Average fitness evolves in all systems, however symmetry and connectivity evolve in small systems only. Newcomers get extinct almost immediately in symmetric systems. The distribution in species lifetimes is determined for antisymmetric systems. The replicator systems investigated do not show any sign of self-organized criticality. The generalized Lotka-Volterra system is shown to be a tedious way of implementing the replicator system.

A universal transition in the robustness of evolving open systems

Can the structure of a system that consists of many elements interacting with each other grow in complexity when new elements are added to it? This is an essential question for understanding various real, open, complex systems, such as living organisms, ecosystems, and social systems. Using a very simple model, this study demonstrates that such systems can grow only when the elements have a moderate number of interactions on average. This behaviour comes from a balance between two opposing effects: although an increase in the number of interactions makes each individual element more robust against disturbances, it also increases the net impact of the loss of any element on the system.

When do evolutionary food web models generate complex networks?

Evolutionary foodweb models are used to build food webs by the repeated addition of new species. Population dynamics leads to the extinction or establishment of a newly added species, and possibly to the extinction of other species. The food web structure that emerges after some time is a highly nontrivial result of the evolutionary and dynamical rules. We investigate the evolutionary food web model introduced by Loeuille and Loreau (2005), which characterizes species by their body mass as the only evolving trait. Our goal is to find the reasons behind the model's remarkable robustness and its capability to generate various and stable networks. In contrast to other evolutionary food web models, this model requires neither adaptive foraging nor allometric scaling of metabolic rates with body mass in order to produce complex networks that do not eventually collapse to trivial structures. Our study shows that this is essentially due to the fact that the difference in niche value between predator and prey as well as the feeding range are constrained so that they remain within narrow limits under evolution. Furthermore, competition between similar species is sufficiently strong, so that a trophic level can accommodate several species. We discuss the implications of these findings and argue that the conditions that stabilize other evolutionary food web models have similar effects because they also prevent the occurrence of extreme specialists or extreme generalists that have in general a higher fitness than species with a moderate niche width.

Emergence of stable polymorphisms driven by evolutionary games between mutants

Under neutrality, polymorphisms are maintained through the balance between mutation and drift. Under selection, a variety of mechanisms may be involved in the maintenance of polymorphisms, for example, sexual selection or host-parasite coevolution on the population level or heterozygote advantage in diploid individuals. Here we address the emergence of polymorphisms in a population of interacting haploid individuals. In our model, each mutation generates a new evolutionary game characterized by a payoff matrix with an additional row and an additional column. Hence, in general, the fitness of new mutations is frequency-dependent rather than constant. This dynamical process is distinct from the sequential fixation of advantageous traits and naturally leads to the emergence of polymorphisms under selection. It causes substantially higher diversity than observed under the established models of neutral or frequency-independent selection. Our framework allows for the coexistence of an arbitrary number of types, but predicts an intermediate average diversity.

SPECIES-AREA RELATIONSHIP FOR POWER-LAW SPECIES ABUNDANCE DISTRIBUTION

We studied the mathematical relations between species abundance distributions (SADs) and species-area relationships (SARs) and found that a power-law SAR can be generally derived from a power-law SAD without a special assumption such as the "canonical hypothesis". In the present analysis, an SAR-exponent is obtained as a function of an SAD-exponent for a finite number of species. We also studied the inverse problem, from SARs to SADs, and found that a power-SAD can be derived from a power-SAR under the condition that the functional form of the corresponding SAD is invariant for changes in the number of species. We also discuss general relationships among lognormal SADs, the broken-stick model (exponential SADs), linear SARs and logarithmic SARs. These results suggest the existence of a common mechanism for SADs and SARs, which could prove a useful tool for theoretical and experimental studies on biodiversity and species coexistence.

Random walk in genome space: a key ingredient of intermittent dynamics of community assembly on evolutionary time scales

Community assembly is studied using individual-based multispecies models. The models have stochastic population dynamics with mutation, migration, and extinction of species. Mutants appear as a result of mutation of the resident species, while migrants have no correlation with the resident species. It is found that the dynamics of community assembly with mutations are quite different from the case with migrations. In contrast to mutation models, which show intermittent dynamics of quasi-steady states interrupted by sudden reorganizations of the community, migration models show smooth and gradual renewal of the community. As a consequence, instead of the 1/f diversity fluctuations found for the mutation models, 1/f(2), random-walk like fluctuations are observed for the migration models. In addition, a characteristic species-lifetime distribution is found: a power law that is cut off by a "skewed" distribution in the long-lifetime regime. The latter has a longer tail than a simple exponential function, which indicates an age-dependent species-mortality function. Since this characteristic profile has been observed, both in fossil data and in several other mathematical models, we conclude that it is a universal feature of macroevolution.

Effects of demographic stochasticity on biological community assembly on evolutionary time scales

We study the effects of demographic stochasticity on the long-term dynamics of biological coevolution models of community assembly. The noise is induced in order to check the validity of deterministic population dynamics. While mutualistic communities show little dependence on the stochastic population fluctuations, predator-prey models show strong dependence on the stochasticity, indicating the relevance of the finiteness of the populations. For a predator-prey model, the noise causes drastic decreases in diversity and total population size. The communities that emerge under influence of the noise consist of species strongly coupled with each other and have stronger linear stability around the fixed-point populations than the corresponding noiseless model. The dynamics on evolutionary time scales for the predator-prey model are also altered by the noise. Approximate 1/f fluctuations are observed with noise, while 1/f2 fluctuations are found for the model without demographic noise.

Rank abundance relations in evolutionary dynamics of random replicators

We present a nonequilibrium statistical mechanics description of rank abundance relations (RAR) in random community models of ecology. Specifically, we study a multispecies replicator system with quenched random interaction matrices. We here consider symmetric interactions as well as asymmetric and antisymmetric cases. RARs are obtained analytically via a generating functional analysis, describing fixed-point states of the system in terms of a small set of order parameters, and in dependence on the symmetry or otherwise of interactions and on the productivity of the community. Our work is an extension of Tokita [Phys. Rev. Lett. 93, 178102 (2004)], where the case of symmetric interactions was considered within an equilibrium setup. The species abundance distribution in our model come out as truncated normal distributions or transformations thereof and, in some case, are similar to left-skewed distributions observed in ecology. We also discuss the interaction structure of the resulting food-web of stable species at stationarity, cases of heterogeneous cooperation pressures as well as effects of finite system size and of higher-order interactions.

Self-optimization, community stability, and fluctuations in two individual-based models of biological coevolution

We compare and contrast the long-time dynamical properties of two individual-based models of biological coevolution. Selection occurs via multispecies, stochastic population dynamics with reproduction probabilities that depend nonlinearly on the population densities of all species resident in the community. New species are introduced through mutation. Both models are amenable to exact linear stability analysis, and we compare the analytic results with large-scale kinetic Monte Carlo simulations, obtaining the population size as a function of an average interspecies interaction strength. Over time, the models self-optimize through mutation and selection to approximately maximize a community potential function, subject only to constraints internal to the particular model. If the interspecies interactions are randomly distributed on an interval including positive values, the system evolves toward self-sustaining, mutualistic communities. In contrast, for the predator-prey case the matrix of interactions is antisymmetric, and a nonzero population size must be sustained by an external resource. Time series of the diversity and population size for both models show approximate 1/f noise and power-law distributions for the lifetimes of communities and species. For the mutualistic model, these two lifetime distributions have the same exponent, while their exponents are different for the predator-prey model. The difference is probably due to greater resilience toward mass extinctions in the food-web like communities produced by the predator-prey model.

Diversity as a product of inter-specific interactions

We demonstrate diversification rather than optimization for highly interacting organisms in a well-mixed biological system by means of a simple model of coevolution. We find the cause to be the complex network of interactions formed, allowing species that are less well adapted to an environment to succeed, instead of the 'best' species. This diversification can be considered as the construction of many coevolutionary niches by the network of interactions between species. The model predictions are discussed in relation to experimental work on dense communities of the bacteria Escherichia coli, which may coexist with their own mutants under certain conditions. We find that diversification only occurs above a certain threshold interaction strength, below which competitive exclusion occurs.

Food webs: Experts consuming families of experts

Food webs of habitats as diverse as lakes or desert valleys are known to exhibit common "food-web patterns", but the detailed mechanisms generating these structures have remained unclear. By employing a stochastic, dynamical model, we show that many aspects of the structure of predatory food webs can be understood as the traces of an evolutionary history where newly evolving species avoid direct competition with their relatives. The tendency to avoid sharing natural enemies (apparent competition) with related species is considerably weaker. Thus, "experts consuming families of experts" can be identified as the main underlying food-web pattern. We report the results of a systematic, quantitative model validation showing that the model is surprisingly accurate.

The species–area relationship and evolution

Models relating to the species-area curve usually assume the existence of species, and are concerned mainly with ecological timescales. We examine an individual-based model of co-evolution on a spatial lattice based on the tangled nature model in which species are emergent structures, and show that reproduction, mutation and dispersion by diffusion, with interaction via genotype space, produces power-law species-area relations as observed in ecological measurements at medium scales. We find that long-lasting co-evolutionary habitats form, allowing high diversity levels in a spatially homogenous system.

Nonlocal competition and logistic growth: patterns, defects, and fronts

Logistic growth of diffusing reactants on spatial domains with long-range competition is studied. The bifurcations cascade involved in the transition from the homogeneous state to a spatially modulated stable solution is presented, and a distinction is made between a modulated phase, dominated by single or few wave numbers, and the spiky phase, where localized colonies are separated by depleted region. The characteristic defects in the periodic structure are presented for each phase, together with the invasion dynamics in the case of local initiation. It is shown that the basic length scale that controls the bifurcation is the width of the Fisher front, and that the total population grows as this width decreases. A mix of analytic results and extensive numerical simulations yields a comprehensive examination of the possible phases for logistic growth in the presence of nonlocal competition.

Some properties of the speciation model for food-web structure-mechanisms for degree distributions and intervality

We present a mathematical analysis of the speciation model for food-web structure, which had in previous work been shown to yield a good description of empirical data of food-web topology. The degree distributions of the network are derived. Properties of the speciation model are compared to those of other models that successfully describe empirical data. It is argued that the speciation model unifies the underlying ideas of previous theories. In particular, it offers a mechanistic explanation for the success of the niche model of Williams and Martinez and the frequent observation of intervality in empirical food webs.

Food-web formation with recursive evolutionary branching

A reaction-diffusion model describing the evolutionary dynamics of a food-web was constructed. In this model, predator-prey relationships among organisms were determined by their position in a two-dimensional phenotype space defined by two traits: as prey and as predator. The mutation process is expressed with a diffusion process of biomass in the phenotype space. Numerical simulation of this model showed co-evolutionary dynamics of isolated phenotypic clusters, including various types of evolutionary branching, which were classified into branching as prey, branching as predators, and co-evolutionary branching of both prey and predators. A complex food-web develops with recursive evolutionary branching from a single phenotypic cluster. Biodiversity peaks at the medium strength of the predator-prey interaction, where the food-web is maintained at medium biomass by a balanced frequency between evolutionary branching and extinction.

The impact of nonlinear functional responses on the long-term evolution of food web structure

We investigate the long-term web structure emerging in evolutionary food web models when different types of functional responses are used. We find that large and complex webs with several trophic layers arise only if the population dynamics is such that it allows predators to focus on their best prey species. This can be achieved using modified Lotka-Volterra or Holling/Beddington functional responses with effective couplings that depend on the predator's efficiency at exploiting the prey, or a ratio-dependent functional response with adaptive foraging. In contrast, if standard Lotka-Volterra or Holling/Beddington functional responses are used, long-term evolution generates webs with almost all species being basal, and with additionally many links between these species. Interestingly, in all cases studied, a large proportion of weak links result naturally from the evolution of the food webs.

Species abundance patterns in complex evolutionary dynamics

An analytic theory of species abundance patterns (SAPs) in biological networks is presented. The theory is based on multispecies replicator dynamics equivalent to the Lotka-Volterra equation, with diverse interspecies interactions. Various SAPs observed in nature are derived from a single parameter. The abundance distribution is formed like a widely observed left-skewed lognormal distribution. As the model has a general form, the result can be applied to similar patterns in other complex biological networks, e.g., gene expression.