Eco-evolutionary dynamics of a population with randomly switching carrying capacity

Hypochaos prevents tragedy of the commons in discrete-time eco-evolutionary game dynamics

While quite a few recent papers have explored game-resource feedback using the framework of evolutionary game theory, almost all the studies are confined to using time-continuous dynamical equations. Moreover, in such literature, the effect of ubiquitous chaos in the resulting eco-evolutionary dynamics is rather missing. Here, we present a deterministic eco-evolutionary discrete-time dynamics in generation-wise non-overlapping population of two types of harvesters-one harvesting at a faster rate than the other-consuming a self-renewing resource capable of showing chaotic dynamics. In the light of our finding that sometimes chaos is confined exclusively to either the dynamics of the resource or that of the consumer fractions, an interesting scenario is realized: The resource state can keep oscillating chaotically, and hence, it does not vanish to result in the tragedy of the commons-extinction of the resource due to selfish indiscriminate exploitation-and yet the consumer population, whose dynamics depends directly on the state of the resource, may end up being composed exclusively of defectors, i.e., high harvesters. This appears non-intuitive because it is well known that prevention of tragedy of the commons usually requires substantial cooperation to be present.

Coupled environmental and demographic fluctuations shape the evolution of cooperative antimicrobial resistance

There is a pressing need to better understand how microbial populations respond to antimicrobial drugs, and to find mechanisms to possibly eradicate antimicrobial-resistant cells. The inactivation of antimicrobials by resistant microbes can often be viewed as a cooperative behaviour leading to the coexistence of resistant and sensitive cells in large populations and static environments. This picture is, however, greatly altered by the fluctuations arising in volatile environments, in which microbial communities commonly evolve. Here, we study the eco-evolutionary dynamics of a population consisting of an antimicrobial-resistant strain and microbes sensitive to antimicrobial drugs in a time-fluctuating environment, modelled by a carrying capacity randomly switching between states of abundance and scarcity. We assume that antimicrobial resistance (AMR) is a shared public good when the number of resistant cells exceeds a certain threshold. Eco-evolutionary dynamics is thus characterised by demographic noise (birth and death events) coupled to environmental fluctuations which can cause population bottlenecks. By combining analytical and computational means, we determine the environmental conditions for the long-lived coexistence and fixation of both strains, and characterise a fluctuation-driven AMR eradication mechanism, where resistant microbes experience bottlenecks leading to extinction. We also discuss the possible applications of our findings to laboratory-controlled experiments.

Microbial interactions in theory and practice: when are measurements compatible with models?

Most predictive models of ecosystem dynamics are based on interactions between organisms: their influence on each other's growth and death. We review here how theoretical approaches are used to extract interaction measurements from experimental data in microbiology, particularly focusing on the generalised Lotka-Volterra (gLV) framework. Though widely used, we argue that the gLV model should be avoided for estimating interactions in batch culture - the most common, simplest and cheapest in vitro approach to culturing microbes. Fortunately, alternative approaches offer a way out of this conundrum. Firstly, on the experimental side, alternatives such as the serial-transfer and chemostat systems more closely match the theoretical assumptions of the gLV model. Secondly, on the theoretical side, explicit organism-environment interaction models can be used to study the dynamics of batch-culture systems. We hope that our recommendations will increase the tractability of microbial model systems for experimentalists and theoreticians alike.

Periodic temporal environmental variations induce coexistence in resource competition models

Natural ecosystems, in particular on the microbial scale, are inhabited by a large number of species. The population size of each species is affected by interactions of individuals with each other and by spatial and temporal changes in environmental conditions, such as resource abundance. Here, we use a generic population dynamics model to study how, and under what conditions, a periodic temporal environmental variation can alter an ecosystem's composition and biodiversity. We demonstrate that using timescale separation allows one to qualitatively predict the long-term population dynamics of interacting species in varying environments. We show that the notion of Tilman's R* rule, a well-known principle that applies for constant environments, can be extended to periodically varying environments if the timescale of environmental changes (e.g., seasonal variations) is much faster than the timescale of population growth (doubling time in bacteria). When these timescales are similar, our analysis shows that a varying environment deters the system from reaching a steady state, and stable coexistence between multiple species becomes possible. Our results posit that biodiversity can in part be attributed to natural environmental variations.

Controlling the Mean Time to Extinction in Populations of Bacteria

Populations of ecological systems generally have demographic fluctuations due to birth and death processes. At the same time, they are exposed to changing environments. We studied populations composed of two phenotypes of bacteria and analyzed the impact that both types of fluctuations have on the mean time to extinction of the entire population if extinction is the final fate. Our results are based on Gillespie simulations and on the WKB approach applied to classical stochastic systems, here in certain limiting cases. As a function of the frequency of environmental changes, we observe a non-monotonic dependence of the mean time to extinction. Its dependencies on other system parameters are also explored. This allows the control of the mean time to extinction to be as large or as small as possible, depending on whether extinction should be avoided or is desired from the perspective of bacteria or the perspective of hosts to which the bacteria are deleterious.

Evolution in fluctuating environments: A generic modular approach

Evolutionary processes take place in fluctuating environments, where carrying capacities and selective forces vary over time. The fate of a mutant type and the persistence time of polymorphic states were studied in some specific cases of varying environments, but a generic methodology is still lacking. Here, we present such a general analytic framework. We first identify a set of elementary building blocks, a few basic demographic processes like logistic or exponential growth, competition at equilibrium, sudden decline, and so on. For each of these elementary blocks, we evaluate the mean and the variance of the changes in the frequency of the mutant population. Finally, we show how to find the relevant terms of the diffusion equation for each arbitrary combination of these blocks. Armed with this technique one may calculate easily the quantities that govern the evolutionary dynamics, like the chance of ultimate fixation, the time to absorption, and the time to fixation.

Eco-evolutionary dynamics of multigames with mutations

Most environments favor defection over cooperation due to natural selection. Nonetheless, the emergence of cooperation is omnipresent in many biological, social, and economic systems, quite contrary to the well-celebrated Darwinian theory of evolution. Much research has been devoted to better understanding how and why cooperation persists among self-interested individuals despite their competition for limited resources. Here we go beyond a single social dilemma since individuals usually encounter various social challenges. In particular, we propose and study a mathematical model incorporating both the prisoner’s dilemma and the snowdrift game. We further extend this model by considering ecological signatures like mutation and selfless one-sided contribution of altruist free space. The nonlinear evolutionary dynamics that results from these upgrades offer a broader range of equilibrium outcomes, and it also often favors cooperation over defection. With the help of analytical and numerical calculations, our theoretical model sheds light on the mechanisms that maintain biodiversity, and it helps to explain the evolution of social order in human societies.

Identification of Valerate as Carrying Capacity Modulator by Analyzing Lactiplantibacillus plantarum Colonization of Colonic Microbiota in vitro

Humans ingest many microorganisms, which may colonize and interact with the resident gut microbiota. However, extensive knowledge about host-independent microbe-microbe interactions is lacking. Here, we investigated such colonization process using a derivative of the model probiotic Lactiplantibacillus plantarum WCFS1 into continuously cultivated gut microbiota in the intestinal PolyFermS fermentation model inoculated with five independently immobilized human adult fecal microbiota. L. plantarum successfully colonized and organized itself spatially in the planktonic, that is, the reactor effluent, and sessile, that is, reactor biofilm, fractions of distinct human adult microbiota. The microbiota carrying capacity for L. plantarum was independent of L. plantarum introduction dose and second supplementation. Adult microbiota (n = 3) dominated by Prevotella and Ruminoccocus exhibited a higher carrying capacity than microbiota (n = 2) dominated by Bacteroides with 10⁵ and 10³ CFU/ml of L. plantarum, respectively. Cultivation of human adult microbiota over 3 months resulted in decreased carrying capacity and correlated positively with richness and evenness, suggesting enhanced resistance toward colonizers. Our analyses ultimately allowed us to identify the fermentation metabolite valerate as a modulator to increase the carrying capacity in a microbiota-independent manner. In conclusion, by uncoupling microbe-microbe interactions from host factors, we showed that L. plantarum colonizes the in vitro colonic community in a microbiota-dependent manner. We were further able to demonstrate that L. plantarum colonization levels were not susceptible to the introduction parameters dose and repeated administration but to microbiota features. Such knowledge is relevant in gaining a deeper ecological understanding of colonizer-microbiota interactions and developing robust probiotic strategies.

Game-environment feedback dynamics in growing population: Effect of finite carrying capacity

The tragedy of the commons (TOC) is an unfortunate situation where a shared resource is exhausted due to uncontrolled exploitation by the selfish individuals of a population. Recently, the paradigmatic replicator equation has been used in conjunction with a phenomenological equation for the state of the shared resource to gain insight into the influence of the games on the TOC. The replicator equation, by construction, models a fixed infinite population undergoing microevolution. Thus, it is unable to capture any effect of the population growth and the carrying capacity of the population although the TOC is expected to be dependent on the size of the population. Therefore, in this paper, we present a mathematical framework that incorporates the density dependent payoffs and the logistic growth of the population in the eco-evolutionary dynamics modeling the game-resource feedback. We discover a bistability in the dynamics: a finite carrying capacity can either avert or cause the TOC depending on the initial states of the resource and the initial fraction of cooperators. In fact, depending on the type of strategic game-theoretic interaction, a finite carrying capacity can either avert or cause the TOC when it is exactly the opposite for the corresponding case with infinite carrying capacity.

Exclusion of the fittest predicts microbial community diversity in fluctuating environments

Microorganisms live in environments that inevitably fluctuate between mild and harsh conditions. As harsh conditions may cause extinctions, the rate at which fluctuations occur can shape microbial communities and their diversity, but we still lack an intuition on how. Here, we build a mathematical model describing two microbial species living in an environment where substrate supplies randomly switch between abundant and scarce. We then vary the rate of switching as well as different properties of the interacting species, and measure the probability of the weaker species driving the stronger one extinct. We find that this probability increases with the strength of demographic noise under harsh conditions and peaks at either low, high, or intermediate switching rates depending on both species' ability to withstand the harsh environment. This complex relationship shows why finding patterns between environmental fluctuations and diversity has historically been difficult. In parameter ranges where the fittest species was most likely to be excluded, however, the beta diversity in larger communities also peaked. In sum, how environmental fluctuations affect interactions between a few species pairs predicts their effect on the beta diversity of the whole community.

Beyond the adiabatic limit in systems with fast environments: A τ-leaping algorithm

We propose a τ-leaping simulation algorithm for stochastic systems subject to fast environmental changes. Similar to conventional τ-leaping the algorithm proceeds in discrete time steps, but as a principal addition it captures environmental noise beyond the adiabatic limit. The key idea is to treat the input rates for the τ-leaping as (clipped) Gaussian random variables with first and second moments constructed from the environmental process. In this way, each step of the algorithm retains environmental stochasticity to subleading order in the timescale separation between system and environment. We test the algorithm on several toy examples with discrete and continuous environmental states and find good performance in the regime of fast environmental dynamics. At the same time, the algorithm requires significantly less computing time than full simulations of the combined system and environment. In this context we also discuss several methods for the simulation of stochastic population dynamics in time-varying environments with continuous states.

Complex evolutionary dynamics due to punishment and free space in ecological multigames

The concurrence of ecological and evolutionary processes often arises as an integral part of various biological and social systems. We here study eco-evolutionary dynamics by adopting two paradigmatic metaphors of social dilemmas with contrasting outcomes. We use the Prisoner’s Dilemma and Snowdrift games as the backbone of the proposed mathematical model. Since cooperation is a costly proposition in the face of the Darwinian theory of evolution, we go beyond the traditional framework by introducing punishment as an additional strategy. Punishers bare an additional cost from their own resources to try and discourage or prohibit free-riding from selfish defectors. Our model also incorporates the ecological signature of free space, which has an altruistic-like impact because it allows others to replicate and potentially thrive. We show that the consideration of these factors has broad implications for better understanding the emergent complex evolutionary dynamics. In particular, we report the simultaneous presence of different subpopulations through the spontaneous emergence of cyclic dominance, and we determine various stationary points using traditional game-theoretic concepts and stability analysis.

Understanding evolutionary and ecological dynamics using a continuum limit

Continuum limits in the form of stochastic differential equations are typically used in theoretical population genetics to account for genetic drift or more generally, inherent randomness of the model. In evolutionary game theory and theoretical ecology, however, this method is used less frequently to study demographic stochasticity. Here, we review the use of continuum limits in ecology and evolution. Starting with an individual-based model, we derive a large population size limit, a (stochastic) differential equation which is called continuum limit. By example of the Wright-Fisher diffusion, we outline how to compute the stationary distribution, the fixation probability of a certain type, and the mean extinction time using the continuum limit. In the context of the logistic growth equation, we approximate the quasi-stationary distribution in a finite population.

Switching environments, synchronous sex, and the evolution of mating types

While facultative sex is common in sexually reproducing species, for reasons of tractability most mathematical models assume that such sex is asynchronous in the population. In this paper, we develop a model of switching environments to instead capture the effect of an entire population transitioning synchronously between sexual and asexual modes of reproduction. We use this model to investigate the evolution of the number of self-incompatible mating types in finite populations, which empirically can range from two to thousands. When environmental switching is fast, we recover the results of earlier studies that implicitly assumed populations were engaged in asynchronous sexual reproduction. However when the environment switches slowly, we see deviations from previous asynchronous theory, including a lower number of mating types at equilibrium and bimodality in the stationary distribution of mating types. We provide analytic approximations for both the fast and slow switching regimes, as well as a numerical scheme based on the Kolmogorov equations for the system to quickly evaluate the model dynamics at intermediate parameters. Our approach exploits properties of integer partitions in number theory. We also demonstrate how additional biological processes such as selective sweeps can be accounted for in this switching environment framework, showing that beneficial mutations can further erode mating type diversity in synchronous facultatively sexual populations.

Taming the diffusion approximation through a controlling-factor WKB method

The diffusion approximation (DA) is widely used in the analysis of stochastic population dynamics, from population genetics to ecology and evolution. The DA is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied. This failure becomes severe in situations where the direction of selection switches sign. Here we employ the WKB (Wentzel-Kramers-Brillouin) large-deviations method, which requires only the logarithm of a given quantity to be smooth over its state space. Combining the WKB scheme with asymptotic matching techniques, we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations, applicable for much wider regimes of parameters. We also introduce a scalable (independent of population size) WKB-based numerical technique. The method is applied to a central problem in population genetics and evolution, finding the chance of ultimate fixation in a zero-sum, two-types competition.

Comparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environments

Deterministic continuum models formulated as nonlocal partial differential equations for the evolutionary dynamics of populations structured by phenotypic traits have been used recently to address open questions concerning the adaptation of asexual species to periodically fluctuating environmental conditions. These models are usually defined on the basis of population-scale phenomenological assumptions and cannot capture adaptive phenomena that are driven by stochastic variability in the evolutionary paths of single individuals. In light of these considerations, in this paper we develop a stochastic individual-based model for the coevolution of two competing phenotype-structured cell populations that are exposed to time-varying nutrient levels and undergo spontaneous, heritable phenotypic changes with different probabilities. Here, the evolution of every cell is described by a set of rules that result in a discrete-time branching random walk on the space of phenotypic states, and nutrient levels are governed by a difference equation in which a sink term models nutrient consumption by the cells. We formally show that the deterministic continuum counterpart of this model comprises a system of nonlocal partial differential equations for the cell population density functions coupled with an ordinary differential equation for the nutrient concentration. We compare the individual-based model and its continuum analog, focusing on scenarios whereby the predictions of the two models differ. The results obtained clarify the conditions under which significant differences between the two models can emerge due to bottleneck effects that bring about both lower regularity of the density functions of the two populations and more pronounced demographic stochasticity. In particular, bottleneck effects emerge in the presence of lower probabilities of phenotypic variation and are more apparent when the two populations are characterized by lower fitness initial mean phenotypes and smaller initial levels of phenotypic heterogeneity. The emergence of these effects, and thus the agreement between the two modeling approaches, is also dependent on the initial proportions of the two populations. As an illustrative example, we demonstrate the implications of these results in the context of the mathematical modeling of the early stage of metastatic colonization of distant organs.

Population Dynamics in a Changing Environment: Random versus Periodic Switching

Environmental changes greatly influence the evolution of populations. Here, we study the dynamics of a population of two strains, one growing slightly faster than the other, competing for resources in a time-varying binary environment modeled by a carrying capacity switching either randomly or periodically between states of abundance and scarcity. The population dynamics is characterized by demographic noise (birth and death events) coupled to a varying environment. We elucidate the similarities and differences of the evolution subject to a stochastically and periodically varying environment. Importantly, the population size distribution is generally found to be broader under intermediate and fast random switching than under periodic variations, which results in markedly different asymptotic behaviors between the fixation probability of random and periodic switching. We also determine the detailed conditions under which the fixation probability of the slow strain is maximal.

Cooperation in Microbial Populations: Theory and Experimental Model Systems

Cooperative behavior, the costly provision of benefits to others, is common across all domains of life. This review article discusses cooperative behavior in the microbial world, mediated by the exchange of extracellular products called public goods. We focus on model species for which the production of a public good and the related growth disadvantage for the producing cells are well described. To unveil the biological and ecological factors promoting the emergence and stability of cooperative traits we take an interdisciplinary perspective and review insights gained from both mathematical models and well-controlled experimental model systems. Ecologically, we include crucial aspects of the microbial life cycle into our analysis and particularly consider population structures where ensembles of local communities (subpopulations) continuously emerge, grow, and disappear again. Biologically, we explicitly consider the synthesis and regulation of public good production. The discussion of the theoretical approaches includes general evolutionary concepts, population dynamics, and evolutionary game theory. As a specific but generic biological example, we consider populations of Pseudomonas putida and its regulation and use of pyoverdines, iron scavenging molecules, as public goods. The review closes with an overview on cooperation in spatially extended systems and also provides a critical assessment of the insights gained from the experimental and theoretical studies discussed. Current challenges and important new research opportunities are discussed, including the biochemical regulation of public goods, more realistic ecological scenarios resembling native environments, cell-to-cell signaling, and multispecies communities.

Eco-evolutionary dynamics of a population with randomly switching carrying capacity

Environmental variability greatly influences the eco-evolutionary dynamics of a population, i.e. it affects how its size and composition evolve. Here, we study a well-mixed population of finite and fluctuating size whose growth is limited by a randomly switching carrying capacity. This models the environmental fluctuations between states of resources abundance and scarcity. The population consists of two strains, one growing slightly faster than the other, competing under two scenarios: one in which competition is solely for resources, and one in which the slow (cooperating) strain produces a public good (PG) that benefits also the fast (free-riding) strain. We investigate how the coupling of demographic and environmental (external) noise affects the population's eco-evolutionary dynamics. By analytical and computational means, we study the correlations between the population size and its composition, and discuss the social-dilemma-like 'eco-evolutionary game' characterizing the PG production. We determine in what conditions it is best to produce a PG; when cooperating is beneficial but outcompeted by free riding, and when the PG production is detrimental for cooperators. Within a linear noise approximation to populations of varying size, we also accurately analyse the coupled effects of demographic and environmental noise on the size distribution.