Stability of two-species communities: Drift, environmental stochasticity, storage effect and selection

Species coexistence as an emergent effect of interacting mechanisms

Although extensively studied, the maintenance of biodiversity remains a highly debated and investigated topic of contemporary research in ecology. Several studies have quantified the contributions of various coexistence mechanisms to biodiversity. However, often stochastic individual-level interactions are abstracted away, or mechanisms are studied in isolation. The intertwined nature and reciprocal influences between mechanisms, as they arise from individual-level interactions, are therefore rarely considered. We propose a novel mechanistic simulation model grounded in neutral theory to capture and quantify emergent effects arising from such mechanism interactions. Three coexistence mechanisms are supported: storage effect, intransitivity, and resource partitioning. We show that basic neutral dynamics and related models of isolated mechanisms can be replicated. Beyond that, we observe difficult to predict, yet significant emergent effects for mechanism combinations. In some cases, coexistence times could be extended more than tenfold compared to the individual mechanisms' performances. Our findings suggest that studies of individual coexistence mechanisms might be insufficient and indeed misleading for quantifying their overall impact on biodiversity. The particular combination of mechanisms and their interactions appear to be of vital importance.

Processes governing species richness in communities exposed to temporal environmental stochasticity: A review and synthesis of modelling approaches

Research into the processes governing species richness has often assumed that the environment is fixed, whereas realistic environments are often characterised by random fluctuations over time. This temporal environmental stochasticity (TES) changes the demographic rates of species populations, with cascading effects on community dynamics and species richness. Theoretical and applied studies have used process-based mathematical models to determine how TES affects species richness, but under a variety of frameworks. Here, we critically review such studies to synthesise their findings and draw general conclusions. We first provide a broad mathematical framework encompassing the different ways in which TES has been modelled. We then review studies that have analysed models with TES under the assumption of negligible interspecific interactions, such that a community is conceptualised as the sum of independent species populations. These analyses have highlighted how TES can reduce species richness by increasing the frequency at which a species becomes rare and therefore prone to extinction. Next, we review studies that have relaxed the assumption of negligible interspecific interactions. To simplify the corresponding models and make them analytically tractable, such studies have used mean-field theory to derive fixed parameters representing the typical strength of interspecific interactions under TES. The resulting analyses have highlighted community-level effects that determine how TES affects species richness, for species that compete for a common limiting resource. With short temporal correlations of environmental conditions, a non-linear averaging effect of interspecific competition strength over time gives an increase in species richness. In contrast, with long temporal correlations of environmental conditions, strong selection favouring the fittest species between changes in environmental conditions results in a decrease in species richness. We compare such results with those from invasion analysis, which examines invasion growth rates (IGRs) instead of species richness directly. Qualitative differences sometimes arise because the IGR is the expected growth rate of a species when it is rare, which does not capture the variation around this mean or the probability of the species becoming rare. Our review elucidates key processes that have been found to mediate the negative and positive effects of TES on species richness, and by doing so highlights key areas for future research.

Extinction time distributions of populations and genotypes

Ultimately, the eventual extinction of any biological population is an inevitable outcome. While extensive research has focused on the average time it takes for a population to go extinct under various circumstances, there has been limited exploration of the distributions of extinction times and the likelihood of significant fluctuations. Recently, Hathcock and Strogatz [D. Hathcock and S. H. Strogatz, Phys. Rev. Lett. 128, 218301 (2022)0031-900710.1103/PhysRevLett.128.218301] identified Gumbel statistics as a universal asymptotic distribution for extinction-prone dynamics in a stable environment. In this study we aim to provide a comprehensive survey of this problem by examining a range of plausible scenarios, including extinction-prone, marginal (neutral), and stable dynamics. We consider the influence of demographic stochasticity, which arises from the inherent randomness of the birth-death process, as well as cases where stochasticity originates from the more pronounced effect of random environmental variations. Our work proposes several generic criteria that can be used for the classification of experimental and empirical systems, thereby enhancing our ability to discern the mechanisms governing extinction dynamics. Employing these criteria can help clarify the underlying mechanisms driving extinction processes.

Effect of compositional fluctuation on the survival of bet-hedging species

Understanding the coexistence of diverse species in a changing environment is an important problem in community ecology. Bet-hedging is a strategy that helps species survive in such changing environments. However, studies of bet-hedging have often focused on the expected long-term growth rate of the species by itself, neglecting competition with other coexisting species. Here we study the extinction risk of a bet-hedging species in competition with others. We show that there are three contributions to the extinction risk. The first is the usual demographic fluctuation due to stochastic reproduction and selection processes in finite populations. The second, due to the fluctuation of population growth rate caused by environmental changes, may actually reduce the extinction risk for small populations. Besides those two, we reveal a third contribution, which is unique to bet-hedging species that diversify into multiple phenotypes: The phenotype composition of the population will fluctuate over time, resulting in increased extinction risk. We compare such compositional fluctuation to the demographic and environmental contributions, showing how they have different effects on the extinction risk depending on the population size, generation overlap, and environmental correlation.

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.

Towards a heuristic understanding of the storage effect

The storage effect is a general explanation for coexistence in a variable environment. Unfortunately, the storage effect is poorly understood, in part because the generality of the storage effect precludes an interpretation that is simultaneously simple, intuitive and correct. Here, we explicate the storage effect by dividing one of its key conditions-covariance between environment and competition-into two pieces, namely that there must be a strong causal relationship between environment and competition, and that the effects of the environment do not change too quickly. This finer-grained definition can explain a number of previous results, including (1) that the storage effect promotes annual plant coexistence when the germination rate fluctuates, but not when the seed yield fluctuates, (2) that the storage effect is more likely to be induced by resource competition than the apparent competition, and (3) why the storage effect arises readily in models with either stage structure or environmental autocorrelation. Additionally, our expanded definition suggests two novel mechanisms by which the temporal storage effect can arise-transgenerational plasticity and causal chains of environmental variables-thus suggesting that the storage effect is a more common phenomenon than previously thought.

Fixation in the stochastic Lotka-Volterra model with small fitness trade-offs

We study the probability of fixation in a stochastic two-species competition model. By identifying a naturally occurring fast timescale, we derive an approximation to the associated backward Kolmogorov equation that allows us to obtain an explicit closed form solution for the probability of fixation of either species. We use our result to study fitness tradeoff strategies and show that, despite some tradeoffs having nearly negligible effects on the corresponding deterministic dynamics, they can have large implications for the outcome of the stochastic system.

How the storage effect and the number of temporal niches affect biodiversity in stochastic and seasonal environments

Temporal environmental variations affect diversity in communities of competing populations. In particular, the covariance between competition and environment is known to facilitate invasions of rare species via the storage effect. Here we present a quantitative study of the effects of temporal variations in two-species and in diverse communities. Four scenarios are compared: environmental variations may be either periodic (seasonal) or stochastic, and the dynamics may support the storage effect (global competition) or not (local competition). In two-species communities, coexistence is quantified via the mean time to absorption, and we show that stochastic variations yield shorter persistence time because they allow for rare sequences of bad years. In diverse communities, where the steady-state reflects a colonization-extinction equilibrium, the actual number of temporal niches is shown to play a crucial role. When this number is large, the same trends hold: storage effect and periodic variations increase both species richness and the evenness of the community. Surprisingly, when the number of temporal niches is small global competition acts to decrease species richness and evenness, as it focuses the competition to specific periods, thus increasing the effective fitness differences.

One of the major challenges of community ecology and population genetics is the understanding of the factors that protect biodiversity. Surprisingly, in many generic cases temporal environmental variations (and the abundance fluctuations associated with it) promote the coexistence of competing species and facilitate genetic polymorphism. Here we present a detailed and quantitative comparison between the stabilizing (and the destabilizing) effects of periodic (seasonal) and stochastic temporal variations. When the number of species is small, we show that persistence times under periodic variations are much longer than the persistence times in a stochastic environment. However, environmental variations facilitate coexistence only when the number of temporal niches is larger than the number of species, whereas in the opposite case the same mechanism acts to increase competition and to decrease species richness. Since it is reasonable to expect the number of temporal niches under seasonal variations to be typically smaller than the corresponding number in stochastic environments, stochastic variations provide a more plausible explanation for the apparent stability of high-diversity assemblages.

Competition with abundance-dependent fitness and the dynamics of heterogeneous populations in fluctuating environment

Species competition takes place in a fluctuating environment, so the selective forces on different populations vary through time. In many realistic situations the mean fitness and the amplitude of its temporal variations are abundance-dependent. Here we present a theory of two-species competition with abundance-dependent stochastic fitness variations and solve for the chance of ultimate fixation, the time to absorption and the time to fixation. We then examine the ability of this two-species system to serve as an effective model for high-diversity assemblages and to account for the presence of an intra-specific differential response to environmental variations. The effective model is shown to capture the main features of competition between composite 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.

Intraspecific variability in fluctuating environments: mechanisms of impact on species diversity

Recent studies have found considerable trait variations within species. The effect of such intraspecific trait variability (ITV) on the stability, coexistence, and diversity of ecological communities received considerable attention and in many models it was shown to impede coexistence and decrease species diversity. Here we present a numerical study of the effect of genetically inherited ITV on species persistence and diversity in a temporally fluctuating environment. Two mechanisms are identified. First, ITV buffers populations against varying environmental conditions (portfolio effect) and reduces variation in abundances. Second, the interplay between ITV and environmental variations tends to increase the mean fitness of diverse populations. The first mechanism promotes persistence and tends to increase species richness, while the second reduces the chance of a rare species population (which is usually homogeneous) to invade, thus decreasing species richness. We show that for large communities the portfolio effect is dominant, leading to ITV promoting species persistence and richness.

Adapt or Perish: Evolutionary Rescue in a Gradually Deteriorating Environment

We investigate the evolutionary rescue of a microbial population in a gradually deteriorating environment, through a combination of analytical calculations and stochastic simulations. We consider a population destined for extinction in the absence of mutants, which can survive only if mutants sufficiently adapted to the new environment arise and fix. We show that mutants that appear later during the environment deterioration have a higher probability to fix. The rescue probability of the population increases with a sigmoidal shape when the product of the carrying capacity and of the mutation probability increases. Furthermore, we find that rescue becomes more likely for smaller population sizes and/or mutation probabilities if the environment degradation is slower, which illustrates the key impact of the rapidity of environment degradation on the fate of a population. We also show that our main conclusions are robust across various types of adaptive mutants, including specialist and generalist ones, as well as mutants modeling antimicrobial resistance evolution. We further express the average time of appearance of the mutants that do rescue the population and the average extinction time of those that do not. Our methods can be applied to other situations with continuously variable fitnesses and population sizes, and our analytical predictions are valid in the weak-to-moderate mutation regime.

On the Simpson index for the Wright–Fisher process with random selection and immigration

Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects. Our object of study will be the Simpson index which measures the level of diversity of the population, one of the key parameters for ecologists who study for example, forest dynamics. Following ecological motivations, we will consider, here, the case, where there are various species with fitness and immigration parameters being random processes (and thus time evolving). The Simpson index is difficult to evaluate when the population is large, except in the neutral (no selection) case, because it has no closed formula. Our approach relies on the large population limit in the “weak” selection case, and thus to give a procedure which enables us to approximate, with controlled rate, the expectation of the Simpson index at fixed time. We will also study the long time behavior (invariant measure and convergence speed towards equilibrium) of the Wright–Fisher process in a simplified setting, allowing us to get a full picture for the approximation of the expectation of the Simpson index.

Resist or perish: Fate of a microbial population subjected to a periodic presence of antimicrobial

The evolution of antimicrobial resistance can be strongly affected by variations of antimicrobial concentration. Here, we study the impact of periodic alternations of absence and presence of antimicrobial on resistance evolution in a microbial population, using a stochastic model that includes variations of both population composition and size, and fully incorporates stochastic population extinctions. We show that fast alternations of presence and absence of antimicrobial are inefficient to eradicate the microbial population and strongly favor the establishment of resistance, unless the antimicrobial increases enough the death rate. We further demonstrate that if the period of alternations is longer than a threshold value, the microbial population goes extinct upon the first addition of antimicrobial, if it is not rescued by resistance. We express the probability that the population is eradicated upon the first addition of antimicrobial, assuming rare mutations. Rescue by resistance can happen either if resistant mutants preexist, or if they appear after antimicrobial is added to the environment. Importantly, the latter case is fully prevented by perfect biostatic antimicrobials that completely stop division of sensitive microorganisms. By contrast, we show that the parameter regime where treatment is efficient is larger for biocidal drugs than for biostatic drugs. This sheds light on the respective merits of different antimicrobial modes of action.

Neutral competition in a deterministically changing environment: Revisiting continuum approaches

Environmental variation can play an important role in ecological competition by influencing the relative advantage between competing species. Here, we consider such effects by extending a classical, competitive Moran model to incorporate an environment that fluctuates periodically in time. We adapt methods from work on these classical models to investigate the effects of the magnitude and frequency of environmental fluctuations on two important population statistics: the probability of fixation and the mean time to fixation. In particular, we find that for small frequencies, the system behaves similar to a system with a constant fitness difference between the two species, and for large frequencies, the system behaves similar to a neutrally competitive model. Most interestingly, the system exhibits nontrivial behavior for intermediate frequencies. We conclude by showing that our results agree quite well with recent theoretical work on competitive models with a stochastically changing environment, and discuss how the methods we develop ease the mathematical analysis required to study such models.

Temporal population variability in local forest communities has mixed effects on tree species richness across a latitudinal gradient

Among the local processes that determine species diversity in ecological communities, fluctuation-dependent mechanisms that are mediated by temporal variability in the abundances of species populations have received significant attention. Higher temporal variability in the abundances of species populations can increase the strength of temporal niche partitioning but can also increase the risk of species extinctions, such that the net effect on species coexistence is not clear. We quantified this temporal population variability for tree species in 21 large forest plots and found much greater variability for higher latitude plots with fewer tree species. A fitted mechanistic model showed that among the forest plots, the net effect of temporal population variability on tree species coexistence was usually negative, but sometimes positive or negligible. Therefore, our results suggest that temporal variability in the abundances of species populations has no clear negative or positive contribution to the latitudinal gradient in tree species richness.

Comprehensive phase diagram for logistic populations in fluctuating environment

Population dynamics reflects an underlying birth-death process, where the rates associated with different events may depend on external environmental conditions and on the population density. A whole family of simple and popular deterministic models (such as logistic growth) supports a transcritical bifurcation point between an extinction phase and an active phase. Here we provide a comprehensive analysis of the phases of that system, taking into account both the endogenous demographic noise (random birth and death events) and the effect of environmental stochasticity that causes variations in birth and death rates. Three phases are identified: in the inactive phase the mean time to extinction T is independent of the carrying capacity N and scales logarithmically with the initial population size. In the power-law phase T∼N^{q}, and in the exponential phase T∼exp(αN). All three phases and the transitions between them are studied in detail. The breakdown of the continuum approximation is identified inside the power-law phase, and the accompanying changes in decline modes are analyzed. The applicability of the emerging picture to the analysis of ecological time series and to the management of conservation efforts is briefly discussed.

Phase Diagram for Logistic Systems under Bounded Stochasticity

Extinction is the ultimate absorbing state of any stochastic birth-death process; hence, the time to extinction is an important characteristic of any natural population. Here we consider logistic and logisticlike systems under the combined effect of demographic and bounded environmental stochasticity. Three phases are identified: an inactive phase where the mean time to extinction T increases logarithmically with the initial population size, an active phase where T grows exponentially with the carrying capacity N, and a temporal Griffiths phase, with a power-law relationship between T and N. The system supports an exponential phase only when the noise is bounded, in which case the continuum (diffusion) approximation breaks down within the Griffiths phase. This breakdown is associated with a crossover between qualitatively different survival statistics and decline modes. To study the power-law phase we present a new WKB scheme, which is applicable both in the diffusive and in the nondiffusive regime.

Noise-induced stabilization and fixation in fluctuating environment

The dynamics of a two-species community of N competing individuals are considered, with an emphasis on the role of environmental variations that affect coherently the fitness of entire populations. The chance of fixation of a mutant (or invading) population is calculated as a function of its mean relative fitness, the amplitude of fitness variations and their typical duration. We emphasize the distinction between the case of pairwise competition and the case of global competition; in the latter a noise-induced stabilization mechanism yields a higher chance of fixation for a single mutant. This distinction becomes dramatic in the weak selection regime, where the chance of fixation for a single deleterious mutant is an N-independent constant for global competition and decays like (ln N)⁻¹ in the pairwise competition case. A Wentzel-Kramers-Brillouin (WKB) technique yields a general formula for the chance of fixation of a deleterious mutant in the strong selection regime. The possibility of long-term persistence of large [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr{O}}$$\end{document}O(N)] suboptimal (and extinction-prone) populations is discussed, as well as its relevance to stochastic tunneling between fitness peaks.

Theory of time-averaged neutral dynamics with environmental stochasticity

Competition is the main driver of population dynamics, which shapes the genetic composition of populations and the assembly of ecological communities. Neutral models assume that all the individuals are equivalent and that the dynamics is governed by demographic (shot) noise, with a steady state species abundance distribution (SAD) that reflects a mutation-extinction equilibrium. Recently, many empirical and theoretical studies emphasized the importance of environmental variations that affect coherently the relative fitness of entire populations. Here we consider two generic time-averaged neutral models; in both the relative fitness of each species fluctuates independently in time but its mean is zero. The first (model A) describes a system with local competition and linear fitness dependence of the birth-death rates, while in the second (model B) the competition is global and the fitness dependence is nonlinear. Due to this nonlinearity, model B admits a noise-induced stabilization mechanism that facilitates the invasion of new mutants. A self-consistent mean-field approach is used to reduce the multispecies problem to two-species dynamics, and the large-N asymptotics of the emerging set of Fokker-Planck equations is presented and solved. Our analytic expressions are shown to fit the SADs obtained from extensive Monte Carlo simulations and from numerical solutions of the corresponding master equations.