Counterintuitive properties of the fixation time in network-structured populations

Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth–death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations.     

Aspiration dynamics of multi-player games in finite populations

On studying strategy update rules in the framework of evolutionary game theory, one can differentiate between imitation processes and aspiration-driven dynamics. In the former case, individuals imitate the strategy of a more successful peer. In the latter case, individuals adjust their strategies based on a comparison of their pay-offs from the evolutionary game to a value they aspire, called the level of aspiration. Unlike imitation processes of pairwise comparison, aspiration-driven updates do not require additional information about the strategic environment and can thus be interpreted as being more spontaneous. Recent work has mainly focused on understanding how aspiration dynamics alter the evolutionary outcome in structured populations. However, the baseline case for understanding strategy selection is the well-mixed population case, which is still lacking sufficient understanding. We explore how aspiration-driven strategy-update dynamics under imperfect rationality influence the average abundance of a strategy in multi-player evolutionary games with two strategies. We analytically derive a condition under which a strategy is more abundant than the other in the weak selection limiting case. This approach has a long-standing history in evolutionary games and is mostly applied for its mathematical approachability. Hence, we also explore strong selection numerically, which shows that our weak selection condition is a robust predictor of the average abundance of a strategy. The condition turns out to differ from that of a wide class of imitation dynamics, as long as the game is not dyadic. Therefore, a strategy favoured under imitation dynamics can be disfavoured under aspiration dynamics. This does not require any population structure, and thus highlights the intrinsic difference between imitation and aspiration dynamics.     

Evolution on neutral networks accelerates the ticking rate of the molecular clock

Large sets of genotypes give rise to the same phenotype, because phenotypic expression is highly redundant. Accordingly, a population can accept mutations without altering its phenotype, as long as the genotype mutates into another one on the same set. By linking every pair of genotypes that are mutually accessible through mutation, genotypes organize themselves into neutral networks (NNs). These networks are known to be heterogeneous and assortative, and these properties affect the evolutionary dynamics of the population. By studying the dynamics of populations on NNs with arbitrary topology, we analyse the effect of assortativity, of NN (phenotype) fitness and of network size. We find that the probability that the population leaves the network is smaller the longer the time spent on it. This progressive ‘phenotypic entrapment’ entails a systematic increase in the overdispersion of the process with time and an acceleration in the fixation rate of neutral mutations. We also quantify the variation of these effects with the size of the phenotype and with its fitness relative to that of neighbouring alternatives.     

Stochasticity and time delays in gene expression and evolutionary game theory

We discuss effects of stochasticity and time delays in simple models of population dynamics. In social-type models, where individuals react to the information concerning the state of the population at some earlier time, sufficiently large time delays may cause oscillations. In biological-type models, where some changes already take place in the population at an earlier time, oscillations might not be present for any time delay. We illustrate this idea in models of delayed random walks, gene expression, and population dynamics of evolutionary game theory.     

Learning enables adaptation in cooperation for multi-player stochastic games

Interactions among individuals in natural populations often occur in a dynamically changing environment. Understanding the role of environmental variation in population dynamics has long been a central topic in theoretical ecology and population biology. However, the key question of how individuals, in the middle of challenging social dilemmas (e.g. the ‘tragedy of the commons’), modulate their behaviours to adapt to the fluctuation of the environment has not yet been addressed satisfactorily. Using evolutionary game theory, we develop a framework of stochastic games that incorporates the adaptive mechanism of reinforcement learning to investigate whether cooperative behaviours can evolve in the ever-changing group interaction environment. When the action choices of players are just slightly influenced by past reinforcements, we construct an analytical condition to determine whether cooperation can be favoured over defection. Intuitively, this condition reveals why and how the environment can mediate cooperative dilemmas. Under our model architecture, we also compare this learning mechanism with two non-learning decision rules, and we find that learning significantly improves the propensity for cooperation in weak social dilemmas, and, in sharp contrast, hinders cooperation in strong social dilemmas. Our results suggest that in complex social–ecological dilemmas, learning enables the adaptation of individuals to varying environments.     

Campylobacter jejuni colonization and transmission in broiler chickens: a modelling perspective.

Campylobacter jejuni is one of the most common causes of acute enteritis in the developed world. The consumption of contaminated poultry, where C. jejuni is believed to be a commensal organism, is a major risk factor. However, the dynamics of this colonization process in commercially reared chickens is still poorly understood. Quantification of these dynamics of infection at an individual level is vital to understand transmission within populations and formulate new control strategies. There are multiple potential routes of introduction of C. jejuni into a commercial flock. Introduction is followed by a rapid increase in environmental levels of C. jejuni and the level of colonization of individual broilers. Recent experimental and epidemiological evidence suggest that the celerity of this process could be masking a complex pattern of colonization and extinction of bacterial strains within individual hosts. Despite the rapidity of colonization, experimental transmission studies exhibit a highly variable and unexplained delay time in the initial stages of the process. We review past models of transmission of C. jejuni in broilers and consider simple modifications, motivated by the plausible biological mechanisms of clearance and latency, which could account for this delay. We show how simple mathematical models can be used to guide the focus of experimental studies by providing testable predictions based on our hypotheses. We conclude by suggesting that competition experiments could be used to further understand the dynamics and mechanisms underlying the colonization process. The population models for such competition processes have been extensively studied in other ecological and evolutionary contexts. However, C. jejuni can potentially adapt phenotypically through phase variation in gene expression, leading to unification of ecological and evolutionary time-scales. For a theoretician, the colonization dynamics of C. jejuni offer an experimental system to explore these 'phylodynamics', the synthesis of population dynamics and evolutionary biology.     

A probabilistic framework for windows of opportunity: the role of temporal variability in critical transitions

The establishment of young organisms in harsh environments often requires a window of opportunity (WoO). That is, a short time window in which environmental conditions drop long enough below the hostile average level, giving the organism time to develop tolerance and transition into stable existence. It has been suggested that this kind of establishment dynamics is a noise-induced transition between two alternate states. Understanding how temporal variability (i.e. noise) in environmental conditions affects establishment of organisms is therefore key, yet not well understood or included explicitly in the WoO framework. In this paper, we develop a coherent theoretical framework for understanding when the WoO open or close based on simple dichotomous environmental variation. We reveal that understanding of the intrinsic timescales of both the developing organism and the environment is fundamental to predict if organisms can or cannot establish. These insights have allowed us to develop statistical laws for predicting establishment probabilities based on the period and variance of the fluctuations in naturally variable environments. Based on this framework, we now get a clear understanding of how changes in the timing and magnitude of climate variability or management can mediate establishment chances.     

The emergence of cooperation from a single mutant during microbial life cycles

Cooperative behaviour is widespread in nature, even though cooperating individuals always run the risk of being exploited by free-riders. Population structure effectively promotes cooperation given that a threshold in the level of cooperation was already reached. However, the question how cooperation can emerge from a single mutant, which cannot rely on a benefit provided by other cooperators, is still puzzling. Here, we investigate this question for a well-defined but generic situation based on typical life cycles of microbial populations where individuals regularly form new colonies followed by growth phases. We analyse two evolutionary mechanisms favouring cooperative behaviour and study their strength depending on the inoculation size and the length of a life cycle. In particular, we find that population bottlenecks followed by exponential growth phases strongly increase the survival and fixation probabilities of a single cooperator in a free-riding population.     

Many ways to stay in the game: individual variability maintains high biodiversity in planktonic microorganisms

In apparent contradiction to competition theory, the number of known, coexisting plankton species far exceeds their explicable biodiversity—a discrepancy termed the Paradox of the Plankton. We introduce a new game-theoretic model for competing microorganisms in which one player consists of all organisms of one species. The stable points for the population dynamics in our model, known as strategic behaviour distributions (SBDs), are probability distributions of behaviours across all organisms which imply a stable population of the species as a whole. We find that intra-specific variability is the key characteristic that ultimately allows coexistence because the outcomes of competitions between individuals with variable competitive abilities are unpredictable. Our simulations based on the theoretical model show that up to 100 species can coexist for at least 10 000 generations, and that even small population sizes or species with inferior competitive ability can survive when there is intra-specific variability. In nature, this variability can be observed as niche differentiation, variability in environmental and ecological factors, and variability of individual behaviours or physiology. Therefore, previous specific explanations of the paradox are consistent with and provide specific examples of our suggestion that individual variability is the mechanism which solves the paradox.     

Persistent unstable equilibria and the grace period in dynamic models of environmental change

Models of the interactions between population, economy and environment often contain nonlinear functional relationships and variables that vary at different speeds. These properties foster apparent unpredictabilities in the system behavior. We identify a class of deterministic models on the interaction of demographic, economic and environmental interactions in which catastrophic changes in environmental quality can take place, but involve a delay between passing a critical threshold level of pollution and the final collapse of the environment. We denoted this delay as the 'environmental grace period'. We illustrate the usefulness of geometric singular perturbation theory and local bifurcation theory to analyse such models. In particular, we show how it is possible to obtain analytic expressions for: (1) the level of emissions above which environmental deterioration begins; (2) the time it takes from reaching the critical level of emissions to the beginning of rapid environmental deterioration and (3) the level of emissions at the time that rapid deterioration begins. Because our results are analytic, they make the outcomes of demo- graphic, economic and environmental interactions more predictable and, therefore, poten- tially more manageable.     

Which games are growing bacterial populations playing?

Microbial communities display complex population dynamics, both in frequency and absolute density. Evolutionary game theory provides a natural approach to analyse and model this complexity by studying the detailed interactions among players, including competition and conflict, cooperation and coexistence. Classic evolutionary game theory models typically assume constant population size, which often does not hold for microbial populations. Here, we explicitly take into account population growth with frequency-dependent growth parameters, as observed in our experimental system. We study the in vitro population dynamics of the two commensal bacteria (Curvibacter sp. (AEP1.3) and Duganella sp. (C1.2)) that synergistically protect the metazoan host Hydra vulgaris (AEP) from fungal infection. The frequency-dependent, nonlinear growth rates observed in our experiments indicate that the interactions among bacteria in co-culture are beyond the simple case of direct competition or, equivalently, pairwise games. This is in agreement with the synergistic effect of anti-fungal activity observed in vivo. Our analysis provides new insight into the minimal degree of complexity needed to appropriately understand and predict coexistence or extinction events in this kind of microbial community dynamics. Our approach extends the understanding of microbial communities and points to novel experiments.     

Frequency-dependent fitness induces multistability in coevolutionary dynamics

Evolution is simultaneously driven by a number of processes such as mutation, competition and random sampling. Understanding which of these processes is dominating the collective evolutionary dynamics in dependence on system properties is a fundamental aim of theoretical research. Recent works quantitatively studied coevolutionary dynamics of competing species with a focus on linearly frequency-dependent interactions, derived from a game-theoretic viewpoint. However, several aspects of evolutionary dynamics, e.g. limited resources, may induce effectively nonlinear frequency dependencies. Here we study the impact of nonlinear frequency dependence on evolutionary dynamics in a model class that covers linear frequency dependence as a special case. We focus on the simplest non-trivial setting of two genotypes and analyse the co-action of nonlinear frequency dependence with asymmetric mutation rates. We find that their co-action may induce novel metastable states as well as stochastic switching dynamics between them. Our results reveal how the different mechanisms of mutation, selection and genetic drift contribute to the dynamics and the emergence of metastable states, suggesting that multistability is a generic feature in systems with frequency-dependent fitness.     

Determination of the stationary state densities of the stochastic nonlinear dynamical systems

The stationary state probability densities appear not only in the study of dynamical systems with random vector fields, but also in the deterministic dynamical systems exhibiting chaotic behavior when the uncertainties in the initial conditions are represented with the probability densities. But since it is very hard problem to determine these densities, in this paper the new efficient method to obtain an approximate solution of Fokker–Planck–Kolmogorov equation which arises in the determination of the stationary state probability densities has been given by representing the densities with compactly supported functions. With specific choice of the compactly supported functions as piecewise multivariable polynomials which are supported on the ellipsoidal regions, the parameters to be calculated for determining the densities can be considerably decreased compared to Multi-Gaussian Closure scheme, in which the stationary densities are assumed to be the weighted average of the Gaussian densities. The main motivation to choose the compactly supported functions is that, in the chaotic dynamics the states are trapped in a specific compact subspace of the state space. The stationary state densities of two basic examples commonly considered in the literature have been estimated using the Parzen’s estimator, and the densities obtained using the newly proposed method have been compared with these estimated densities and the densities obtained with the Multi-Gaussian Closure scheme. The results indicate that the presented compactly supported piecewise polynomial scheme can be successful compared to Multi-Gaussian scheme, when the system is highly nonlinear.     

Studying the effect of environmental change on biotic evolution: past genetic contributions, current work and future directions

Evolutionary geneticists currently face a major scientific opportunity when integrating across the rapidly increasing amount of genetic data and existing biological scenarios based on ecology, fossils or climate models. Although genetic data acquisition and analysis have improved tremendously, several limitations remain. Here, we discuss the feedback between history and genetic variation in the face of environmental change with increasing taxonomic and temporal scale, as well as the major challenges that lie ahead. In particular, we focus on recent developments in two promising genetic methods, those of 'phylochronology' and 'molecular clocks'. With the advent of ancient DNA techniques, we can now directly sample the recent past. We illustrate this amazing and largely untapped utility of ancient DNA extracted from accurately dated localities with documented environmental changes. Innovative statistical analyses of these genetic data expose the direct effect of recent environmental change on genetic endurance, or maintenance of genetic variation. The 'molecular clock' (assumption of a linear relationship between genetic distance and evolutionary time) has been used extensively in phylogenetic studies to infer time and correlation between lineage divergence time and concurrent environmental change. Several studies at both population and species scale support a persuasive relationship between particular perturbation events and time of biotic divergence. However, we are still a way from gleaning an overall pattern to this relationship, which is a prerequisite to ultimately understanding the mechanisms by which past environments have shaped the evolutionary trajectory. Current obstacles include as-yet undecided reasons behind the frequent discrepancy between molecular and fossil time estimates, and the frequent lack of consideration of extensive confidence intervals around time estimates. We suggest that use and interpretation of both ancient DNA and molecular clocks is most effective when results are synthesized with palaeontological (fossil) and ecological (life history) information.     

Dynamics and Ground State Properties of Two-component Bose-Einstein Condensate in Different Hyperfine States

The dynamics of the Josephson effects for a homogeneous two-component Bose-Einstein condensate in different hyperfine states is studied by two coupled nonlinear equations, which are obtained by mean field approximation. We have shown the nonlinear Josephson effects, including Plasmon and macroscopic self-trapping. We have also studied the relaxation dynamics and the system under damping will evolve into a stationary state of two equivalent components. For the equivalent two-component condensate, Green’s function method is applied to find the excitation spectrum. We found that the excitation spectrum has two branched, of which one is the phonon excitation and one is of the single-particle form in the long wave-length limit. We have also studied the depletion of the condensate in the ground state.     

Topological characterization of deterministic chaos: enforcing orientation preservation

The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.     

Monitoring kinetics of highly environment sensitive states of fluorescent molecules by modulated excitation and time-averaged fluorescence intensity recording

In this work, a concept is described for how the kinetics of photoinduced, transient, long-lived, nonfluorescent or weakly fluorescent states of fluorophore marker molecules can be extracted from the time-averaged fluorescence by using time-modulated excitation. The concept exploits the characteristic variation of the population of these states with the modulation parameters of the excitation and thereby circumvents the need for time resolution in the fluorescence detection. It combines the single-molecule sensitivity of fluorescence detection with the remarkable environmental responsiveness obtainable from long-lived transient states, yet does not in itself impose any constraints on the concentration or the fluorescence brightness of the sample molecules that can be measured. Modulation of the excitation can be performed by variation of the intensity of a stationary excitation beam in time or by repeated translations of a CW excitation beam with respect to the sample. As a first experimental verification of the approach, we have shown how the triplet-state parameters of the fluorophore rhodamine 6G in different aqueous environments can be extracted. We demonstrate that the concept is fully compatible with low time-resolution detection by a CCD camera. The concept opens for automated transient-state monitoring or imaging on a massively parallel scale and for high-throughput biomolecular screening as well as for more fundamental biomolecular studies. The concept should also be applicable to the monitoring of a range of other photoinduced nonfluorescent or weakly fluorescent transient states, from which subtle changes in the immediate microenvironment of the fluorophore marker molecules can be detected.     

Estimating epidemic coupling between populations from the time to invasion

Identifying the mechanisms by which diseases spread among populations is important for understanding and forecasting patterns of epidemics and pandemics. Estimating transmission coupling among populations is challenging because transmission events are difficult to observe in practice, and connectivity among populations is often obscured by local disease dynamics. We consider the common situation in which an epidemic is seeded in one population and later spreads to a second population. We present a method for estimating transmission coupling between the two populations, assuming they can be modelled as susceptible–infected–removed (SIR) systems. We show that the strength of coupling between the two populations can be estimated from the time taken for the disease to invade the second population. Confidence in the estimate is low if only a single invasion event has been observed, but is substantially improved if numerous independent invasion events are observed. Our analysis of this simplest, idealized scenario represents a first step toward developing and verifying methods for estimating epidemic coupling among populations in an ever-more-connected global human population.     

Testing the optimality properties of a dual antibiotic treatment in a two-locus,two-allele model

Mathematically speaking, it is self-evident that the optimal control of complex, dynamical systems with many interacting components cannot be achieved with ‘non-responsive’ control strategies that are constant through time. Although there are notable exceptions, this is usually how we design treatments with antimicrobial drugs when we give the same dose and the same antibiotic combination each day. Here, we use a frequency- and density-dependent pharmacogenetics mathematical model based on a standard, two-locus, two-allele representation of how bacteria resist antibiotics to probe the question of whether optimal antibiotic treatments might, in fact, be constant through time. The model describes the ecological and evolutionary dynamics of different sub-populations of the bacterium Escherichia coli that compete for a single limiting resource in a two-drug environment. We use in vitro evolutionary experiments to calibrate and test the model and show that antibiotic environments can support dynamically changing and heterogeneous population structures. We then demonstrate, theoretically and empirically, that the best treatment strategies should adapt through time and constant strategies are not optimal.     

Superposition states of light in a three-photon absorbing dissipative medium

We consider the process of simultaneous absorption of three photons in a medium in the presence of weak one-photon absorption. We show that in such a system stationary three-component superposition states of light may be formed in the range of small values of state amplitude (weak perturbation). This circumstance is associated with the fact that in this range of interaction the field spends more time in one of the three types of superposition states of light (constituting an ensemble of quantum trajectories of the system) than in two other types. We also show that in the range of large values of state amplitude it is possible to obtain three types of non-stationary superposition states of light. By using numerical simulation of quantum trajectories of the system we study the dynamics of the quantum entropy of the field. We calculate the Wigner functions of the field states. We also obtain analytical results for the density matrix of the steady state of the system.