A study of the impact of environmental stochasticity on extinction probabilities by Monte Carlo integration

An environmental process was characterized by a stationary second order autogressive process with Gaussian noise. This process was then linked to survivorship and reproductive success by logistic transformations. The sensitivity of extinction probabilities to variations in the parameters of the environmental process was studied by computer experiments in Monte Carlo integration. Against the background of the rather limited number of fertility and mortality levels studied in these experiments, the extinction probabilities were demonstrated to be quite sensitive to variations in the parameters of the environmental process. Although more extensive experiments will need to be carried out, those conducted so far suggest that concerted efforts should be made to model those environmental factors that are critical to the survivability of an endangered species in assessing its chances for continued existence.     

Adaptive topography of fluctuating selection in a Mendelian population

An adaptive topography is derived for a large randomly mating diploid population under weak density-independent selection in a fluctuating environment. Assuming a stationary distribution of environmental states with no temporal autocorrelation, a diffusion approximation for population size and allele frequency, p, reveals that the expected change in p involves the gradient with respect to p of the stochastic intrinsic rate of increase (the density-independent long-run growth rate), r = r - sigma 2 e/2, where r is the mean Malthusian fitness in the average environment and is the environmental variance in population growth rate. The expected relative fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its fitness with population growth rate. The influence of fitness correlation between genotypes is illustrated by an analysis of the Haldane-Jayakar model of fluctuating selection on a single diallelic locus, and on two loci with additive effects on a quantitative character.     

Estimating environmental effects on population dynamics: consequences of observation error

Within the paradigm of population dynamics a central task is to identify environmental factors affecting population change and to estimate the strength of these effects. We here investigate the impact of observation errors in measurements of population densities on estimates of environmental effects. Adding observation errors may change the autocorrelation of a population time series with potential consequences for estimates of effects of autocorrelated environmental covariates. Using Monte Carlo simulations, we compare the performance of maximum likelihood estimates from three stochastic versions of the Gompertz model (log–linear first order autoregressive model), assuming 1) process error only, 2) observation error only, and 3) both process and observation error (the linear state–space model on log‐scale). We also simulated population dynamics using the Ricker model, and evaluated the corresponding maximum likelihood estimates for process error models. When there is observation error in the data and the considered environmental variable is strongly autocorrelated, its estimated effect is likely to be biased when using process error models. The environmental effect is overestimated when the sign of the autocorrelations of the intrinsic dynamics and the environment are the same and underestimated when the signs differ. With non‐autocorrelated environmental covariates, process error models produce fairly exact point estimates as well as reliable confidence intervals for environmental effects. In all scenarios, observation error models produce unbiased estimates with reasonable precision, but confidence intervals derived from the likelihood profiles are far too optimistic if there is process error present. The safest approach is to use state–space models in presence of observation error. These are factors worthwhile to consider when interpreting earlier empirical results on population time series, and in future studies, we recommend choosing carefully the modelling approach with respect to intrinsic population dynamics and covariate autocorrelation.     

Using reproductive value to estimate key parameters in density-independent age-structured populations

The dynamics of reproductive value are used to provide a simple derivation of Tuljapurkar's approximation for the long-run growth rate and environmental variance of lnN, in a density-independent age-structured population in a random environment. With no environmental autocorrelation, the dynamics of total population size, N, generally shows time lags and autocorrelation caused by life history, which may strongly bias estimates of environmental variance obtained by ignoring age structure. In contrast, the total reproductive value, V, is Markovian and obeys a first-order autoregressive process. This suggests a simple method for estimating the environmental variance, and avoiding potentially large bias due to age-structure fluctuations, by converting a multivariate time series of age structure to a univariate time series of lnV. We illustrate the method by estimating the long-run growth rate and the environmental variance in an exponentially growing population of Bighorn Sheep.     

Population persistence time under intermittent control in stochastic environments

If there exists a critical population size above which environmental degradation becomes serious, the population should be suppressed or reduced upon reaching that level. Since population size control is accompanied by costs, a reduction in control frequency may be preferable from an economic viewpoint. Although this can be realized by decreasing the population size drastically in each control, such management may result in increased population extinction probability according to environmental stochasticity. The effects of population management on both mean population persistence time and management cost were analyzed theoretically using a diffusion process. The model showed the functional forms of both mean persistence time and control frequency explicitly; these decreased with an increasing number of individuals removed from the population in each control operation. Based on the analysis, indices representing management costs are proposed. Mean persistence time is generally an increasing function of the cost indices. Nevertheless, if the cost of each control increases with the number of individuals removed, even the most conservative management practice (continuous control) may not be overly expensive.     

A neutral model with fluctuating population size and its effective size

We consider a diffusion model with neutral alleles whose population size is fluctuating randomly. For this model, the effects of fluctuation of population size on the effective size are investigated. The effective size defined by the equilibrium average heterozygosity is larger than the harmonic mean of population size but smaller than the arithmetic mean of population size. To see explicitly the effects of fluctuation of population size on the effective size, we investigate a special case where population size fluctuates between two distinct states. In some cases, the effective size is very different from the harmonic mean. For this concrete model, we also obtain the stationary distribution of the average heterozygosity. Asymptotic behavior of the effective size is obtained when the population size is large and/or autocorrelation of the fluctuation is weak or strong.     

Expected relative fitness and the adaptive topography of fluctuating selection

Wright's adaptive topography describes gene frequency evolution as a maximization of mean fitness in a constant environment. I extended this to a fluctuating environment by unifying theories of stochastic demography and fluctuating selection, assuming small or moderate fluctuations in demographic rates with a stationary distribution, and weak selection among the types. The demography of a large population, composed of haploid genotypes at a single locus or normally distributed phenotypes, can then be approximated as a diffusion process and transformed to produce the dynamics of population size, N, and gene frequency, p, or mean phenotype, . The expected evolution of p or is a product of genetic variability and the gradient of the long-run growth rate of the population, , with respect to p or . This shows that the expected evolution maximizes , the mean Malthusian fitness in the average environment minus half the environmental variance in population growth rate. Thus, as a function of p or represents an adaptive topography that, despite environmental fluctuations, does not change with time. The haploid model is dominated by environmental stochasticity, so the expected maximization is not realized. Different constraints on quantitative genetic variability, and stabilizing selection in the average environment, allow evolution of the mean phenotype to undergo a stochastic maximization of . Although the expected evolution maximizes the long-run growth rate of the population, for a genotype or phenotype the long-run growth rate is not a valid measure of fitness in a fluctuating environment. The haploid and quantitative character models both reveal that the expected relative fitness of a type is its Malthusian fitness in the average environment minus the environmental covariance between its growth rate and that of the population.     

Joint Effects of Habitat Heterogeneity and Species’ Life-History Traits on Population Dynamics in Spatially Structured Landscapes

Both habitat heterogeneity and species’ life-history traits play important roles in driving population dynamics, yet there is little scientific consensus around the combined effect of these two factors on populations in complex landscapes. Using a spatially explicit agent-based model, we explored how interactions between habitat spatial structure (defined here as the scale of spatial autocorrelation in habitat quality) and species life-history strategies (defined here by species environmental tolerance and movement capacity) affect population dynamics in spatially heterogeneous landscapes. We compared the responses of four hypothetical species with different life-history traits to four landscape scenarios differing in the scale of spatial autocorrelation in habitat quality. The results showed that the population size of all hypothetical species exhibited a substantial increase as the scale of spatial autocorrelation in habitat quality increased, yet the pattern of population increase was shaped by species’ movement capacity. The increasing scale of spatial autocorrelation in habitat quality promoted the resource share of individuals, but had little effect on the mean mortality rate of individuals. Species’ movement capacity also determined the proportion of individuals in high-quality cells as well as the proportion of individuals experiencing competition in response to increased spatial autocorrelation in habitat quality. Positive correlations between the resource share of individuals and the proportion of individuals experiencing competition indicate that large-scale spatial autocorrelation in habitat quality may mask the density-dependent effect on populations through increasing the resource share of individuals, especially for species with low mobility. These findings suggest that low-mobility species may be more sensitive to habitat spatial heterogeneity in spatially structured landscapes. In addition, localized movement in combination with spatial autocorrelation may increase the population size, despite increased density effects.     

Extinction conditions for isolated populations affected by environmental stochasticity

We determine the critical patch size below which extinction occurs for populations living in one-dimensional habitats surrounded by completely hostile environments in the presence of environmental fluctuations. The population dynamics is reformulated in terms of a stochastic reaction–diffusion equation and is reduced to a deterministic equation that incorporates the systematic contributions of the noise. We obtain bifurcation diagrams and relations for the mean population density at the stationary state, the critical patch size, and the mean number of individuals in the habitat. The effect of the noise differs, depending on whether it affects the net growth rate or the intraspecific competition term. Fluctuations in the net growth rate decrease the critical patch size, whereas fluctuations in the competition term do not change the critical patch size. We compare our analytical results with numerical solutions of the stochastic partial differential equations and show that our procedure proves useful in dealing with reaction–diffusion equations with multiplicative noise.     

Predicting population genetic change in an autocorrelated random environment: Insights from a large automated experiment

Most natural environments exhibit a substantial component of random variation, with a degree of temporal autocorrelation that defines the color of environmental noise. Such environmental fluctuations cause random fluctuations in natural selection, affecting the predictability of evolution. But despite long-standing theoretical interest in population genetics in stochastic environments, there is a dearth of empirical estimation of underlying parameters of this theory. More importantly, it is still an open question whether evolution in fluctuating environments can be predicted indirectly using simpler measures, which combine environmental time series with population estimates in constant environments. Here we address these questions by using an automated experimental evolution approach. We used a liquid-handling robot to expose over a hundred lines of the micro-alga Dunaliella salina to randomly fluctuating salinity over a continuous range, with controlled mean, variance, and autocorrelation. We then tracked the frequencies of two competing strains through amplicon sequencing of nuclear and choloroplastic barcode sequences. We show that the magnitude of environmental fluctuations (determined by their variance), but also their predictability (determined by their autocorrelation), had large impacts on the average selection coefficient. The variance in frequency change, which quantifies randomness in population genetics, was substantially higher in a fluctuating environment. The reaction norm of selection coefficients against constant salinity yielded accurate predictions for the mean selection coefficient in a fluctuating environment. This selection reaction norm was in turn well predicted by environmental tolerance curves, with population growth rate against salinity. However, both the selection reaction norm and tolerance curves underestimated the variance in selection caused by random environmental fluctuations. Overall, our results provide exceptional insights into the prospects for understanding and predicting genetic evolution in randomly fluctuating environments.     

Effects of spatial autocorrelation,natal philopatry and phenotypic plasticity on the heritability of laying date

We investigated the effect of spatial autocorrelation on heritability (h²) estimates of laying date and clutch size in a population of great tits Parus major. We found that h² of laying date, but not clutch size, declined significantly with increasing distance between the nestbox of mothers and daughters. This decline was caused by a decreasing effect of spatial autocorrelation in laying date, rather than by the existence of genotype–environment interactions (GEI). After correcting for the effect of spatial autocorrelation, h² of laying date was low (0.16 ± 0.07), but significant, and surprisingly consistent with increasing distance between parental and offspring environments. The h² of clutch size was not much affected by spatial autocorrelation. Most previously published estimates of the heritability of laying date include various degrees of common environment effects, which can bias estimates both upwards and downwards. We suggest that using techniques that take spatial autocorrelation into account might be a fruitful approach to estimate h² of traits that show a high degree of plasticity.     

Density dependence and stochastic variation in a newly established population of a small songbird

Models describing fluctuations in population size should include both density dependence and stochastic effects. We examine the relative contribution of variation in parameters of the expected dynamics as well as demographic and environmental stochasticity to fluctuations in a population of a small passerine bird, the pied flycatcher, that was newly established in a Dutch study area. Using the theta-logistic model of density regulation, we demonstrate that the estimated quasi-stationary distribution including demographic stochasticity is close to the stationary distribution ignoring demographic stochasticity, indicating a long expected time to extinction. We also show that the variance in the estimated quasi-stationary distribution is especially sensitive to variation in the density regulation function. Reliable population projections must therefore account for uncertainties in parameter estimates which we do by using the population prediction interval (PPI). After 2 years the width of the 90% PPI was already larger than the corresponding estimated range of variation in the quasi-stationary distribution. More precise prediction of future population size than can be derived from the quasi-stationary distribution could only be made for a time span less than about five years.     

The effective size of fluctuating populations

We consider a Wright-Fisher model whose population size is an autocorrelated stochastic process. Our interest is in the effects of autocorrelated fluctuations of the population size on the effective size. We define the inbreeding effective size and the variance effective size and show that these effective sizes are the same for this model. In the literature, it is said that the effective size is equal to the harmonic mean of population size when the size fluctuates. We will show, however, that the effective size is not the same as the harmonic mean of population size unless the fluctuations of population size are uncorrelated. The effective size is larger (resp. smaller) than the harmonic mean when the fluctuations of population size are positively (resp. negatively) autocorrelated. Further, we obtain some asymptotic expressions for effective size when the population size is very large and/or the autocorrelation of the fluctuation is very strong.     

Autocorrelation and spectrum of the modulated retinal discharge: A comparison of theoretical and experimental results

The theoretical autocorrelation of the cat ganglion cell discharge under stationary and dynamic conditions of light stimulus is compared with the autocorrelation determined experimentally. To obtain the theoretical autocorrelation, the stationary discharge is described by a stationary random point process of independent intervals equally distributed according to a gamma function, and the dynamic behaviour is described by a model defined in a previous paper. Comparison shows that the model predicts the experimental results. Finally, the power density spectrum is analysed and the relevance of the results to signal transmission by the retinal system is discussed.     

Forms of density regulation and (quasi-) stationary distributions of population sizes in birds

The theta-logistic model of density regulation is an especially flexible class of density regulation models where different forms of non-linear density regulation can be expressed by only one parameter, θ. Estimating the parameters of the theta-logistic model is, however, challenging. This is mainly due to the need for information concerning population growth at low densities as well as data on fluctuations around the carrying capacity K in order to estimate the strength of density regulation. Here we estimate parameters of the theta-logistic model for 28 populations of three species of birds that have grown from very small population sizes followed by a period of fluctuations around K. We then use these parameters to estimate the quasi-stationary distribution of population size. There were often large uncertainties in these parameters specifying the form of density regulation that were generally independent of the duration of the study period. In contrast, precision in the estimates of environmental variance increased with the length of the time series. In most of the populations, a large proportion of the probability density of the (quasi-) stationary distribution of population sizes was located at intermediate population sizes relative to K. Thus, we suggest that the (quasi-) stationary distribution of population sizes represents a useful summary statistic that in many cases provides a more robust characterisation of basic population dynamics (e.g. range of variation in population fluctuations or proportion of time spent close to K) than can be obtained from analyses of single model parameters.     

Population dynamics of an expanding passerine at the distribution margin

Individuals may be maladapted to novel environments at the species’ distribution margin. We investigated population dynamics in a marginal habitat where reproduction has been proven poor. Survival, population growth rate (λ) and its components, breeding and natal dispersal were studied in great tits Parus major breeding at the northern margin of its distribution in northern Finland. We used long term capture–mark–recapture data sets. Study area size and population density were used to explain adult survival rates. The average annual estimates of adult survival rose from 0.371 to 0.388 between the periods of 1971–1984 and 1999–2009. The estimates are slightly lower than estimates of small passerines in Europe. Low local survival rate of fledglings (0.050–0.055) probably reflects intensified emigration from this low quality area. Temporal variation in λ was large (0.498–1.856). Despite of low adult survival and recruitment rates, the mean estimates of λ (1.008 and 1.033) indicate an overall stability in the population size. Indeed, our results suggest that the immigration has an important role in the population dynamics of northern great tits. Thus the population is demographically and genetically dependent on core habitats which may cause adaptive problems due to intensive gene flow. Given those limitations, options for evolution of local adaptations in northern distribution margins are discussed.     

Poor environmental tracking can make extinction risk insensitive to the colour of environmental noise

The relative importance of environmental colour for extinction risk compared with other aspects of environmental noise (mean and interannual variability) is poorly understood. Such knowledge is currently relevant, as climate change can cause the mean, variability and temporal autocorrelation of environmental variables to change. Here, we predict that the extinction risk of a shorebird population increases with the colour of a key environmental variable: winter temperature. However, the effect is weak compared with the impact of changes in the mean and interannual variability of temperature. Extinction risk was largely insensitive to noise colour, because demographic rates are poor in tracking the colour of the environment. We show that three mechanisms-which probably act in many species-can cause poor environmental tracking: (i) demographic rates that depend nonlinearly on environmental variables filter the noise colour, (ii) demographic rates typically depend on several environmental signals that do not change colour synchronously, and (iii) demographic stochasticity whitens the colour of demographic rates at low population size. We argue that the common practice of assuming perfect environmental tracking may result in overemphasizing the importance of noise colour for extinction risk. Consequently, ignoring environmental autocorrelation in population viability analysis could be less problematic than generally thought.     

On the autocorrelation of one class of non-stationary random point processes

The autocorrelation of those non-stationary random point processes that can be transformed by a time transformation into a stationary process is derived. As an application, a closed form formula is obtained for the autocorrelation of those processes where the average number of points is sinusoidally modulated and the corresponding stationary process is defined by independent intervals identically distributed according to a gamma density of integer order.     

Space versus phylogeny: disentangling phylogenetic and spatial signals in comparative data

Variation in traits across species or populations is the outcome of both environmental and historical factors. Trait variation is therefore a function of both the phylogenetic and spatial context of species. Here we introduce a method that, within a single framework, estimates the relative roles of spatial and phylogenetic variations in comparative data. The approach requires traits measured across phylogenetic units, e.g. species, the spatial occurrences of those units and a phylogeny connecting them. The method modifies the expected variance of phylogenetically independent contrasts to include both spatial and phylogenetic effects. We illustrate this approach by analysing cross-species variation in body mass, geographical range size and species-typical environmental temperature in three orders of mammals (carnivores, artiodactyls and primates). These species attributes contain highly disparate levels of phylogenetic and spatial signals, with the strongest phylogenetic autocorrelation in body size and spatial dependence in environmental temperatures and geographical range size showing mixed effects. The proposed method successfully captures these differences and in its simplest form estimates a single parameter that quantifies the relative effects of space and phylogeny. We discuss how the method may be extended to explore a range of models of evolution and spatial dependence.     

Temporal autocorrelation in host density increases establishment success of parasitoids in an experimental system

Environmental variation is classically expected to affect negatively population growth and to increase extinction risk, and it has been identified as a major determinant of establishment failures in the field. Yet, recent theoretical investigations have shown that the structure of environmental variation and more precisely the presence of positive temporal autocorrelation might alter this prediction. This is particularly likely to affect the establishment dynamics of biological control agents in the field, as host–parasitoid interactions are expected to induce temporal autocorrelation in host abundance. In the case where parasitoid populations display overcompensatory dynamics, the presence of such positive temporal autocorrelation should increase their establishment success in a variable environment. We tested this prediction in laboratory microcosms by introducing parasitoids to hosts whose abundances were manipulated to simulate uncorrelated or positively autocorrelated variations in carrying capacity. We found that environmental variability decreased population size and increased parasitoid population variance, which is classically expected to extinction risk. However, although exposed to significant environmental variation, we found that parasitoid populations experiencing positive temporal autocorrelation in host abundance were more likely to persist than populations exposed to uncorrelated variation. These results confirm that environmental variation is a key determinant of extinction dynamics that can have counterintuitive effects depending on its autocorrelation structure.