Colonization–extinction dynamics of epixylic lichens along a decay gradient in a dynamic landscape

Metapopulation models are often used for understanding and predicting species dynamics in fragmented landscapes. Several models have been proposed depending on e.g. the relative importance of patch dynamics on the metapopulation dynamics. Dead wood is a dynamic substrate patch, and species that are confined to such patches have experienced a high degree of habitat loss in managed forests. Little is, however, known about how the population dynamics of epixylic species are affected by the fast dynamics of their substrate patches. We quantified the effect of local patch conditions and metapopulation processes on colonizations and extinctions of epixylic lichen species in a managed boreal forest landscape. This was done by twice surveying seven lichen metapopulations on 293 stumps in 30 stands of ages covering the duration of the dynamic patches (stumps). We also investigated the relative importance of local stochastic extinctions from stumps that remained available, and deterministic extinctions due to stump surface disappearance. We found importance of a decay gradient, surrounding metapopulation size, and local population sizes, in driving the colonization–extinction dynamics of epixylic lichens. The species were sorted along the stump decay gradient. Increasing surrounding metapopulation size was associated with increased colonization rates, and increasing local population size decreased lichen extinction rates. Finally, both local stochastic extinctions and deterministic extinctions due to patch disappearance occur, confirming that the long‐term persistence of epixylic lichens depends on colonization rates that compensate for stochastic population extinctions as well as deterministic extinctions.     

Reduced population size can induce quick evolution of inbreeding depression in the invasive ladybird <Emphasis Type="Italic">Harmonia axyridis</Emphasis>

Understanding biological invasion is currently one of the main scientific challenges for ecologists. The introduction process is crucial for the success of an invasion, especially when it involves a demographic bottleneck. A small introduced population is expected to face a higher risk of extinction before the first stage of invasion is complete if inbreeding depression, caused by the expression of deleterious alleles, is important. Changes in mating regimes or in population size can induce the evolution of deleterious allele frequencies, either by selection or by drift, possibly resulting in the purging or the fixation of such alleles within the population. The harlequin ladybird Harmonia axyridis became invasive on several continents following a scenario including at least one event of demographic bottleneck. Although native populations suffered from severe inbreeding depression, it was greatly reduced in invasive ones suggesting that deleterious alleles were purged during the invasion process. In this study, we performed an experiment designed to manipulate the effective population size of H. axyridis across successive generations to mimic contrasting introduction events. We used the measurement of two fitness-related phenotypic traits in order to test (1) if inbreeding depression can evolve at the time-scale of an invasion; and (2) if the changes in inbreeding depression following a bottleneck in laboratory conditions are compatible with the purging of deleterious alleles observed in this species. We found that two generations of very low population size are enough to induce a substantial change in inbreeding depression. Although the genetic changes mostly consisted in fixation of deleterious alleles, purging did also occur, sometimes simultaneously with fixation.     

The Probability of Fixation in Populations of Changing Size

The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size.     

Allele fixation in a dynamic metapopulation: Founder effects vs refuge effects

The fixation of mutant alleles has been studied with models assuming various spatial population structures. In these models, the structure of the metapopulation that we call the “landscape” (number, size and connectivity of subpopulations) is often static. However, natural populations are subject to repetitive population size variations, fragmentation and secondary contacts at different spatiotemporal scales due to geological, climatic and ecological processes. In this paper, we examine how such dynamic landscapes can alter mutant fixation probability and time to fixation. We consider three stochastic landscape dynamics: (i) the population is subject to repetitive bottlenecks, (ii) to the repeated alternation of fragmentation and fusion of demes with a constant population carrying capacity, (iii) idem with a variable carrying capacity. We show by deriving a variance, a coalescent and a harmonic mean population effective size, and with simulations that these landscape dynamics generate repetitive founder effects which counteract selection, thereby decreasing the fixation probability of an advantageous mutant but accelerate fixation when it occurs. For models (ii) and (iii), we also highlight an antagonistic “refuge effect” which can strongly delay mutant fixation. The predominance of either founder effects or refuge effects determines the time to fixation and mainly depends on the characteristic time scales of the landscape dynamics.     

Asynchronization of local population dynamics and persistence of a metapopulation: a lesson from an endangered composite plant, Aster kantoensis

Fragmentation of a large habitat makes local populations less linked to others, and a whole population structure changes to a metapopulation. The smaller a local population is, the more strengthened extinction factors become. Then, frequent extinctions of local populations threaten persistence of the metapopulation unless recolonizations occur rapidly enough after local extinctions. Spatially structured models have been more widely used for predicting future population dynamics and for assessing the extinction risk of a metapopulation. In this article, we first review such spatially structured models that have been applied to conservation biology, focusing on effects of asynchronization among local population dynamics on persistence of the whole metapopulation. Second, we introduce our ongoing project on extinction risk assessment of an endangered composite biennial plant, Aster kantoensis, in the riverside habitat, based on a lattice model for describing its spatiotemporal population dynamics. The model predicted that the extinction risk of A. kantoensis depends on both the frequency of flood occurrence and the time to coverage of a local habitat by other competitively stronger perennials. Finally, we present a measure (Hassell and Pacala's CV ²) for quantifying the effect of asynchronization among local population dynamics on the persistence of a whole metapopulation in conservation ecology. Received: January 12, 2000 / Accepted: February 8, 2000     

Selection in a subdivided population with local extinction and recolonization

In a subdivided population, local extinction and subsequent recolonization affect the fate of alleles. Of particular interest is the interaction of this force with natural selection. The effect of selection can be weakened by this additional source of stochastic change in allele frequency. The behavior of a selected allele in such a population is shown to be equivalent to that of an allele with a different selection coefficient in an unstructured population with a different size. This equivalence allows use of established results for panmictic populations to predict such quantities as fixation probabilities and mean times to fixation. The magnitude of the quantity N(e)s(e), which determines fixation probability, is decreased by extinction and recolonization. Thus deleterious alleles are more likely to fix, and advantageous alleles less likely to do so, in the presence of extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations.     

Understanding extinction debts: spatio–temporal scales,mechanisms and a roadmap for future research

Extinction debt refers to delayed species extinctions expected as a consequence of ecosystem perturbation. Quantifying such extinctions and investigating long‐term consequences of perturbations has proven challenging, because perturbations are not isolated and occur across various spatial and temporal scales, from local habitat losses to global warming. Additionally, the relative importance of eco‐evolutionary processes varies across scales, because levels of ecological organization, i.e. individuals, (meta)populations and (meta)communities, respond hierarchically to perturbations. To summarize our current knowledge of the scales and mechanisms influencing extinction debts, we reviewed recent empirical, theoretical and methodological studies addressing either the spatio–temporal scales of extinction debts or the eco‐evolutionary mechanisms delaying extinctions. Extinction debts were detected across a range of ecosystems and taxonomic groups, with estimates ranging from 9 to 90% of current species richness. The duration over which debts have been sustained varies from 5 to 570 yr, and projections of the total period required to settle a debt can extend to 1000 yr. Reported causes of delayed extinctions are 1) life‐history traits that prolong individual survival, and 2) population and metapopulation dynamics that maintain populations under deteriorated conditions. Other potential factors that may extend survival time such as microevolutionary dynamics, or delayed extinctions of interaction partners, have rarely been analyzed. Therefore, we propose a roadmap for future research with three key avenues: 1) the microevolutionary dynamics of extinction processes, 2) the disjunctive loss of interacting species and 3) the impact of multiple regimes of perturbation on the payment of debts. For their ability to integrate processes occurring at different levels of ecological organization, we highlight mechanistic simulation models as tools to address these knowledge gaps and to deepen our understanding of extinction dynamics.     

Dynamics of populations on the verge of extinction

Theoretical considerations suggest that extinction in dispersal-limited populations is necessarily a threshold-like process that is analogous to a critical phase transition in physics. We use this analogy to find robust, common features in the dynamics of extinctions, and suggest early warning signals which may indicate that a population is endangered. As the critical threshold of extinction is approached, the population spontaneously fragments into discrete subpopulations and, consequently, density regulation fails. The population size declines and its spatial variance diverges according to scaling laws. Therefore, we can make robust predictions exactly in the range where prognosis is vital, on the verge of extinction.     

Extinction and fixation times with dominance and inbreeding

The effect of partial inbreeding on extinction and fixation times of a selected allele with partial dominance is studied using a diffusion model. Asymptotic approximations are obtained for large populations and the accuracy of the approximations was found to increase with inbreeding level. They show that inbreeding reduces extinction and fixation times compared to random mating at least by a factor 1+F, where F is Wright’s fixation index. The reduction of extinction and fixation times due to inbreeding is stronger for strong selection and if alleles are either highly recessive or highly dominant. This bears implications for the effect of inbreeding on the signature of selective sweeps. These findings extend previous results obtained for random mating populations and help clarifying previous simulation and numerical results on the effect of inbreeding on the dynamics of selected alleles.     

The influence of local extinctions on the probability of fixation of chromosomal rearrangements

We investigated the influence of local extinctions in a subdivided population on the probability of fixation of an initially rare allele, for different migration rates. The selective regimes considered were strict underdominance, meiotic drive, and underdominance associated with meiotic drive. We show that local extinctions can increase the probability of fixation of initially rare alleles in underdominant loci for relatively high migration rates, even when both homozygotes have the same fitness. This increase is due to drift during founder events. On the contrary, local extinctions decrease the probability of fixation of alleles favoured by meiotic drive. For a locus where both meiotic drive and underdominance act, the effect of local extinctions depends on the relative strength of the two selective regimes and the initial frequency of the rare allele. For parameter values such that the rare allele is initially selected against, local extinctions decrease the probability of fixation for low migration rates while they cause an increase for moderate migration rates. When the parameter values are such that the rare allele should always be favoured by selection, local extinctions always decrease the probability of fixation. In this latter case, we show the existence of an optimal migration rate which maximizes the probability of fixation.     

A complete classification of Darwinian extinction in ecological interactions

The evolution of a population by individual-level natural selection can result in the population's extinction. Selection causes the spread of phenotypes with higher relative fitness, but at the same time, selection can also indirectly produce changes in the physical, biotic, or genotypical environment through population interactions (e.g., environment modification, interspecific interactions, and genomic conflict). Because fitness is environment dependent, this can cause mean fitness to decrease, resulting in extinction. I call this process "Darwinian extinction." Examples of Darwinian extinction include a variety of dynamics and modes of extinction, but the variation is constrained. I determine the complete classification of possible dynamics and modes of Darwinian extinction due to ecological interactions, using bifurcation theory and models with ecological and evolutionary changes occurring on different timescales. This classification is also extended to extinctions due to interactions within the population. The mode of extinction may be either sudden or gradual (requiring additional stochastic processes), and each mode has specific types of dynamics associated with it. Darwinian extinction is a robust and normal phenomenon, and this reasonably complete classification can help us understand more thoroughly its role in nature.     

Dynamics of neutral and selected alleles when the offspring distribution is skewed

We analyze the dynamics of two alternative alleles in a simple model of a population that allows for large family sizes in the distribution of offspring number. This population model was first introduced by Eldon and Wakeley, who described the backward-time genealogical relationships among sampled individuals, assuming neutrality. We study the corresponding forward-time dynamics of allele frequencies, with or without selection. We derive a continuum approximation, analogous to Kimura's diffusion approximation, and we describe three distinct regimes of behavior that correspond to distinct regimes in the coalescent processes of Eldon and Wakeley. We demonstrate that the effect of selection is strongly amplified in the Eldon-Wakeley model, compared to the Wright-Fisher model with the same variance effective population size. Remarkably, an advantageous allele can even be guaranteed to fix in the Eldon-Wakeley model, despite the presence of genetic drift. We compute the selection coefficient required for such behavior in populations of Pacific oysters, based on estimates of their family sizes. Our analysis underscores that populations with the same effective population size may nevertheless experience radically different forms of genetic drift, depending on the reproductive mechanism, with significant consequences for the resulting allele dynamics.     

Fixation of new alleles and the extinction of small populations: drift load, beneficial alleles, and sexual selection

With a small effective population size, random genetic drift is more important than selection in determining the fate of new alleles. Small populations therefore accumulate deleterious mutations. Left unchecked, the effect of these fixed alleles is to reduce the reproductive capacity of a species, eventually to the point of extinction. New beneficial mutations, if fixed by selection, can restore some of this lost fitness. This paper derives the overall change in fitness due to fixation of new deleterious and beneficial alleles, as a function of the distribution of effects of new mutations and the effective population size. There is a critical effective size below which a population will on average decline in fitness, but above which beneficial mutations allow the population to persist. With reasonable estimates of the relevant parameters, this critical effective size is likely to be a few hundred. Furthermore, sexual selection can act to reduce the fixation probability of deleterious new mutations and increase the probability of fixing new beneficial mutations. Sexual selection can therefore reduce the risk of extinction of small populations.     

Selection, subdivision and extinction and recolonization

In a subdivided population, the interaction between natural selection and stochastic change in allele frequency is affected by the occurrence of local extinction and subsequent recolonization. The relative importance of selection can be diminished by this additional source of stochastic change in allele frequency. Results are presented for subdivided populations with extinction and recolonization where there is more than one founding allele after extinction, where these may tend to come from the same source deme, where the number of founding alleles is variable or the founders make unequal contributions, and where there is dominance for fitness or local frequency dependence. The behavior of a selected allele in a subdivided population is in all these situations approximately the same as that of an allele with different selection parameters in an unstructured population with a different size. The magnitude of the quantity N(e)s(e), which determines fixation probability in the case of genic selection, is always decreased by extinction and recolonization, so that deleterious alleles are more likely to fix and advantageous alleles less likely to do so. The importance of dominance or frequency dependence is also altered by extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations.     

Complete Numerical Solution of the Diffusion Equation of Random Genetic Drift

A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size.     

Probabilistic measures of persistence and extinction in measles (meta)populations

Persistence and extinction are fundamental processes in ecological systems that are difficult to accurately measure due to stochasticity and incomplete observation. Moreover, these processes operate on multiple scales, from individual populations to metapopulations. Here, we examine an extensive new data set of measles case reports and associated demographics in pre‐vaccine era US cities, alongside a classic England & Wales data set. We first infer the per‐population quasi‐continuous distribution of log incidence. We then use stochastic, spatially implicit metapopulation models to explore the frequency of rescue events and apparent extinctions. We show that, unlike critical community size, the inferred distributions account for observational processes, allowing direct comparisons between metapopulations. The inferred distributions scale with population size. We use these scalings to estimate extinction boundary probabilities. We compare these predictions with measurements in individual populations and random aggregates of populations, highlighting the importance of medium‐sized populations in metapopulation persistence.     

The route to extinction in variable environments

Estimating the extinction risk of natural populations is not only an urgent problem in conservation biology but also involves some profound aspects of population dynamics. Apart from the obvious case of a continuous decrease in a population's carrying capacity, understanding the extinction process necessarily includes environmental and demographic stochasticity. Here, we build from first principles two stochastic, single-population models that can account for various routes to extinction via demographic and environmental variability. The Ricker model of population dynamics generates extinctions from either low or high (around or above carrying capacity) population densities, primarily depending on the growth parameter r . Since extinctions from high densities seem 'unnatural', there is either something wrong with the model or with our intuition. Suitable data are scarce. Environmental variability has its strongest influence on extinction risk via per capita birth rates and is only marginally influencing that risk via per capita death rates if the growth parameter is high. The distribution of the environmental noise and the stochastic structure of the model have quantitative, but not qualitative effects on the estimates of extinction risks. We conclude that to determine the route to extinction and to estimate the extinction risk require a careful choice of both the deterministic component of the population model (e.g., under- or over-compensation) and the structure of the demographic and environmental variabilities.     

Environmental variation, fluctuating selection and genetic drift in subdivided populations

Although there have many studies of the population genetical consequences of environmental variation, little is known about the combined effects of genetic drift and fluctuating selection in structured populations. Here we use diffusion theory to investigate the effects of temporally and spatially varying selection on a population of haploid individuals subdivided into a large number of demes. Using a perturbation method for processes with multiple time scales, we show that as the number of demes tends to infinity, the overall frequency converges to a diffusion process that is also the diffusion approximation for a finite, panmictic population subject to temporally fluctuating selection. We find that the coefficients of this process have a complicated dependence on deme size and migration rate, and that changes in these demographic parameters can determine both the balance between the dispersive and stabilizing effects of environmental variation and whether selection favors alleles with lower or higher fitness variance.     

Importance of metapopulation dynamics to explain fish persistence in a river system

1. Habitat modification and fragmentation are key factors responsible for fish population decline worldwide. Previous assessments documented a total of 72 species extinctions for the sole class of Actinopterygii. However, global extinctions are difficult to monitor or study based on fossil records. By contrast, local extinctions occurring at the population level are easier to study. Given this context, an important question relates to whether extinction dynamics studied at the local scale can provide useful information to understand extinctions occurring at larger scales. This would be the case if local extinctions were not balanced by recolonisation as in a classic metapopulation. Our aim is thus to explain the observed regional (per basin) persistence of 252 fish populations by testing contribution of local extinction rates and more generally metapopulation dynamics components.
2. To address this aim, we used the annual extinction probability of 252 regional populations of up to 14 species inhabiting 18 coastal rivers, which became isolated c. 8,500 years ago. We specifically compared extinction probabilities obtained by seven theoretical models to investigate whether regional extinction rates (i.e. loss from a river system) were correlated to local extinction rates (i.e. loss from an occupied site) and the role of metapopulation dynamics to explain regional persistence.
3. Using empirical data, we showed the importance of variables related to metapopulation dynamics to explain extinction rates across the 18 river systems. As expected, the regional extinction rate decreased with the colonisation rate, area, metapopulation size, and percentage of occupied localities. By contrast, an inconsistent relationship emerged between regional and local extinction rates, as species with high local extinction rates were not particularly prone to regional extinction.
4. Our results provide strong support for the contribution of colonisation rates to explain persistence. Overall, our results show that the equilibrium number of occupied localities could be a good predictor of the long-term persistence of metapopulations in rivers. Finally, our results suggest the importance of connectivity to maintain sustainable populations within the river system.

     

Generalized population models and the nature of genetic drift

The Wright–Fisher model of allele dynamics forms the basis for most theoretical and applied research in population genetics. Our understanding of genetic drift, and its role in suppressing the deterministic forces of Darwinian selection has relied on the specific form of sampling inherent to the Wright–Fisher model and its diffusion limit. Here we introduce and analyze a broad class of forward-time population models that share the same mean and variance as the Wright–Fisher model, but may otherwise differ. The proposed class unifies and further generalizes a number of population-genetic processes of recent interest, including the Λ and Cannings processes. Even though these models all have the same variance effective population size, they encode a rich diversity of alternative forms of genetic drift, with significant consequences for allele dynamics. We characterize in detail the behavior of standard population-genetic quantities across this family of generalized models. Some quantities, such as heterozygosity, remain unchanged; but others, such as neutral absorption times and fixation probabilities under selection, deviate by orders of magnitude from the Wright–Fisher model. We show that generalized population models can produce startling phenomena that differ qualitatively from classical behavior — such as assured fixation of a new mutant despite the presence of genetic drift. We derive the forward-time continuum limits of the generalized processes, analogous to Kimura’s diffusion limit of the Wright–Fisher process, and we discuss their relationships to the Kingman and non-Kingman coalescents. Finally, we demonstrate that some non-diffusive, generalized models are more likely, in certain respects, than the Wright–Fisher model itself, given empirical data from Drosophila populations.