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.     

Logistic growth in the presence of non-white environmental noise

A method is given for studying realistic random fluctuations in the carrying capacity of the logistic population growth model. This method is then applied using an environmental noise based on a Poisson process, and the time-dependent moments of the population probability density calculated. These moments are expressed in terms of a parameter obtained by dividing the correlation time of the environmental fluctuations by the characteristic response time of the population. When this quotient is large (very slow fluctuations tracked by the population) or small (very rapid fluctuations which are averaged), exact solutions are obtained for the probability density itself. It is also shown that at equilibrium, the average population sizes given by these two exact solutions bound all other cases.Numerical simulations confirm these developments and point to a trade-off between population stability and average population size. Additional simulations show that the probability of becoming extinct in a given time is greatest for populations intermediate between tracking and averaging the carrying capacity fluctuations. In addition to specifying when environmental noise can be ignored, these results indicate the direction in which growth parameters evolve in a fluctuating environment.     

Host–parasitoid spatial models: the interplay of demographic stochasticity and dynamics

Host–parasitoid metapopulation models have typically been deterministic models formulated with population numbers as a continuous variable. Spatial heterogeneity in local population abundance is a typical (and often essential) feature of these models and means that, even when average population density is high, some patches have small population sizes. In addition, large temporal population fluctuations are characteristic of many of these models, and this also results in periodically small local population sizes. Whenever population abundances are small, demographic stochasticity can become important in several ways. To investigate this problem, we have reformulated a deterministic, host–parasitoid metapopulation as an integer-based model in which encounters between hosts and parasitoids, and the fecundity of individuals are modelled as stochastic processes. This has a number of important consequences: (1) stochastic fluctuations at small population sizes tend to be amplified by the dynamics to cause massive population variability, i.e. the demographic stochasticity has a destabilizing effect; (2) the spatial patterns of local abundance observed in the deterministic counterpart are largely maintained (although the area of ''spatial chaos'' is extended); (3) at small population sizes, dispersal by discrete individuals leads to a smaller fraction of new patches being colonized, so that parasitoids with small dispersal rates have a greater tendency for extinction and higher dispersal rates have a larger competitive advantage; and (4) competing parasitoids that could coexist in the deterministic model due to spatial segregation cannot now coexist for any combination of parameters.     

Uncertainty and predictability in population dynamics of a bitrophic ecological model: Mixed-mode oscillations,bistability and sensitivity to parameters

We consider a two-trophic ecological model comprising of two predators competing for their common prey. We cast the model into the framework of a singular perturbed system of equations in one fast variable (prey population density) and two slow variables (predator population densities), mimicking the common observation that the per-capita productivity rate decreases from bottom to top along the trophic levels in Nature. We assume that both predators exhibit Holling II functional response with one of the predators (territorial) having a density dependent mortality rate. Depending on the system parameters, the model exhibits small, intermediate and/or large fluctuations in the population densities. The large fluctuations correspond to periodic population outbreaks followed by collapses (commonly known as cycles of “boom and bust”). The small fluctuations arise due to a singular Hopf bifurcation in the system, and are ecologically more desirable. However, more interestingly, the system exhibits mixed-mode oscillations (which are concatenations of the large amplitude oscillations and the small amplitude oscillations) that indicate the adaptability of the species to prolong the time gap between successive cycles of boom and bust. Numerical simulations are carried out to demonstrate the extreme sensitivity of the system to initial conditions (chaos and bistability of limit cycles are observed) as well as to the system parameters (here we only show the sensitivity to the density dependent mortality rate of the territorial predator). This model throws light at the uncertainties in long term behaviors that are associated with a real ecological system. We show that even very small changes in the system parameters due to natural or human-induced causes can lead to a complete different ecological phenomenon, thus affecting the predictability of the density of the prey population. In this paper, we explain the mechanisms behind the irregular fluctuations in the population sizes in an attempt to understand the dynamics occurring in a natural population and also comment on the inherent uncertainties associated with the system.     

Genetic population structure as indirect measure of dispersal ability in a Lake Tanganyika cichlid

Diversification and speciation processes are influenced by intrinsic (ecological specialization, dispersal) and extrinsic (habitat structure and instability) factors, but the effect of ecological characteristics on dispersal is difficult to assess. This study uses mitochondrial control region sequences to investigate the population structure and demographic history of the endemic Lake Tanganyika cichlid Neolamprologus caudopunctatus with a preference for the rock-sand interface along two stretches of continuous, rocky shoreline, and across a sandy bay representing a potential dispersal barrier. Populations along uninterrupted habitat were not differentiated; whereas, the sandy bay separated two reciprocally monophyletic clades. The split between the two clades between 170,000 and 260,000 years BP coincides with a period of rising water level following a major lowstand, and indicates that clades remained isolated throughout subsequent lake level fluctuations. Low long-term effective population sizes were inferred from modest genetic diversity estimates, and may be due to recent population expansions starting from small population sizes 45,000–60,000 years BP. Comparisons with available data from specialized rock-dwelling species of the␣same area suggest that habitat structure and lake level fluctuations determine phylogeographic patterns on large scales, while fine-scale population structure and demography are modulated by species-specific ecologies.     

Rainbow trout: a population simulation based on individual responses to varying environmental and demographic parameters

Synopsis An age-structured simulation model of the growth and population dynamics of a migratory rainbow trout population is presented. The model includes all principal life-history intervals and incorporates the food density-temperature relationships of salmonid growth efficiency proposed by Brett et al. (1969) and Shelbourn et al. (1973). Population size, mean weight, and biomass are adjusted and output monthly over time in age, sex, and location categories; a simulation run may continue for as long as 100 years. A variety of environmental and and biological parameters are utilized in the simulation which can be altered as a user option. Simulation results compare favorably with field data. Sensitivity analyses illustrate the importance of age-sex specific maturation ratios, age-class strength fluctuations, and natural mortality rates in determining population size.     

Generalizations of the Moran effect explaining spatial synchrony in population fluctuations

The Moran effect for populations separated in space states that the autocorrelations in the population fluctuations equal the autocorrelation in environmental noise, assuming the same linear density regulation in all populations. Here we generalize the Moran effect to include also nonlinear density regulation with spatial heterogeneity in local population dynamics as well as in the effects of environmental covariates by deriving a simple expression for the correlation between the sizes of two populations, using diffusion approximation to the theta-logistic model. In general, spatial variation in parameters describing the dynamics reduces population synchrony. We also show that the contribution of a covariate to spatial synchrony depends strongly on spatial heterogeneity in the covariate or in its effect on local dynamics. These analyses show exactly how spatial environmental covariation can synchronize fluctuations of spatially segregated populations with no interchange of individuals even if the dynamics are nonlinear.     

Coupling Demographic and Genetic Variability from Archived Collections of European Anchovy (Engraulis encrasicolus)

It is well known that temporal fluctuations in small populations deeply influence evolutionary potential. Less well known is whether fluctuations can influence the evolutionary potentials of species with large census sizes. Here, we estimated genetic population parameters from as survey of polymorphic microsatellite DNA loci in archived otoliths from Adriatic European anchovy (Engraulis encrasicolus), a fish with large census sizes that supports numerous local fisheries. Stocks have fluctuated greatly over the past few decades, and the Adriatic fishery collapsed in 1987. Our results show a significant reduction of mean genetic parameters as a consequence of the population collapse. In addition, estimates of effective population size (N_e) are much smaller than those expected in a fishes with large population census sizes (N_c). Estimates of N_e indicate low effective population sizes, even before the population collapse. The ratio N_e/N_e ranged between 10⁻⁶ and 10⁻⁸, indicating a large discrepancy between the anchovy gene pool and population census size. Therefore, anchovy populations may be more vulnerable to fishery effort and environmental change than previously thought.     

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.     

Life history and population size variability in a relict plant. Different routes towards long-term persistence

A central tenet of conservation biology is that population size affects the persistence of populations. However, many narrow endemic species combine small population ranges and sizes with long persistence, thereby challenging this tenet. I examined the performance of three different-sized populations of Petrocoptis pseudoviscosa (Caryophyllaceae), a palaeoendemic rupicolous herb distributed along a small valley in the Spanish Pyrenees. Reproductive and demographic parameters were recorded over 6 years, and deterministic and stochastic matrix models were constructed to explore population dynamics and extinction risk. Populations differed greatly in structure, fecundity, recruitment, survival rate, and life span. Strong differentiation in life-history parameters and their temporal variability resulted in differential population vulnerability under current conditions and simulated global changes such as habitat fragmentation or higher climatic fluctuations. This study provides insights into the capacity of narrow endemics to survive both at extreme environmental conditions and at small population sizes. When dealing with species conservation, the population size–extinction risk relationship may be too simplistic for ancient, ecologically restricted organisms, and some knowledge of life history may be most important to assess their future.     

Species synchrony and its drivers: neutral and nonneutral community dynamics in fluctuating environments

Independent species fluctuations are commonly used as a null hypothesis to test the role of competition and niche differences between species in community stability. This hypothesis, however, is unrealistic because it ignores the forces that contribute to synchronization of population dynamics. Here we present a mechanistic neutral model that describes the dynamics of a community of equivalent species under the joint influence of density dependence, environmental forcing, and demographic stochasticity. We also introduce a new standardized measure of species synchrony in multispecies communities. We show that the per capita population growth rates of equivalent species are strongly synchronized, especially when endogenous population dynamics are cyclic or chaotic, while their long-term fluctuations in population sizes are desynchronized by ecological drift. We then generalize our model to nonneutral dynamics by incorporating temporal and nontemporal forms of niche differentiation. Niche differentiation consistently decreases the synchrony of species per capita population growth rates, while its effects on the synchrony of population sizes are more complex. Comparing the observed synchrony of species per capita population growth rates with that predicted by the neutral model potentially provides a simple test of deterministic asynchrony in a community.     

Allee effects in stochastic populations

The Allee effect, or inverse density dependence at low population sizes, could seriously impact preservation and management of biological populations. The mounting evidence for widespread Allee effects has lately inspired theoretical studies of how Allee effects alter population dynamics. However, the recent mathematical models of Allee effects have been missing another important force prevalent at low population sizes: stochasticity. In this paper, the combination of Allee effects and stochasticity is studied using diffusion processes, a type of general stochastic population model that accommodates both demographic and environmental stochastic fluctuations. Including an Allee effect in a conventional deterministic population model typically produces an unstable equilibrium at a low population size, a critical population level below which extinction is certain. In a stochastic version of such a model, the probability of reaching a lower size a before reaching an upper size b , when considered as a function of initial population size, has an inflection point at the underlying deterministic unstable equilibrium. The inflection point represents a threshold in the probabilistic prospects for the population and is independent of the type of stochastic fluctuations in the model. In particular, models containing demographic noise alone (absent Allee effects) do not display this threshold behavior, even though demographic noise is considered an "extinction vortex". The results in this paper provide a new understanding of the interplay of stochastic and deterministic forces in ecological populations.     

The spatial scale of population fluctuations and quasi-extinction risk

Using a spatially homogeneous population model with migration (random individual dispersal) and spatially autocorrelated environmental noise, we show how migration and local density regulation affect the spatial scale of fluctuations in the log of population sizes as well as the 1-yr differences in these. The difference between the squares of these two spatial scales of population fluctuations does not depend on the spatial scale of the noise but only on migration rate and strength of local density regulation. We also show how migration, local density regulation, and spatially correlated environmental noise affect the realized population process at a specific location. As the migration increases, the realized local density regulation and the expected population size increase, while the realized environmental noise decreases. This approach also enables us to analyze the dynamics of the total population size within quadrats of different sizes. The risk of local quasi extinction is strongly reduced by increasing quadrat size or migration rate, while an increase in environmental stochasticity or spatial correlation in the environmental noise increases the risk of quasi extinction.     

Multiple density dependence in two sub-populations of the amphipod Monoporeia affinis: a potential for alternative equilibria

Possible mechanisms for differences in population densities and dynamics were investigated in the amphipod Monoporeia affinis at two deep sites in the northern Bothnian Sea. The two sites were sampled yearly for 10 years. Average sizes, growth and mortality of the different age-classes were estimated from the cohort structure of the two populations. Laboratory experiments also investigated the ability of the common predatory isopod Saduria entomon to cause densitydependent (DD) mortality of the prey M. affinis. At site A, 43 m depth, the average density of M. affinis was twice as high as at site B, 81 m depth. The fluctuations in density were asynchronous between the two sites. Recruitment and subadult sizes of Monoporeia affinis were density dependent at both sites. The main functional difference between the two populations seemed to be the DD mortality of the 1 + cohort that occurred only at the low-density site B. A corresponding DD mortality was found in the predation experiments at densities of 1 + m. affinis corresponding to those found at site B. The potential importance of the predator was also indicated by a significant negative correlation between the biomass of S. entomon and the rate of change in M. affinis density in the field. The similarities in the abiotic factors between the two sites suggested that differences in carrying capacity should be small. The results could be explained by the predation regulation hypothesis for the low-density population at site B, while at site A M. affinis seemed to be regulated by intra-specific competition and limited by predation. It is suggested that in this simple predator-prey system there is potential for the existence of alternative equilibria.     

Effect of population size on the estimation of QTL: a test using resistance to barley stripe rust

The limited population sizes used in many quantitative trait locus (QTL) detection experiments can lead to underestimation of QTL number, overestimation of QTL effects, and failure to quantify QTL interactions. We used the barley/barley stripe rust pathosystem to evaluate the effect of population size on the estimation of QTL parameters. We generated a large (n=409) population of doubled haploid lines derived from the cross of two inbred lines, BCD47 and Baronesse. This population was evaluated for barley stripe rust severity in the Toluca Valley, Mexico, and in Washington State, USA, under field conditions. BCD47 was the principal donor of resistance QTL alleles, but the susceptible parent also contributed some resistance alleles. The major QTL, located on the long arm of chromosome 4H, close to the Mlo gene, accounted for up to 34% of the phenotypic variance. Subpopulations of different sizes were generated using three methods—resampling, selective genotyping, and selective phenotyping—to evaluate the effect of population size on the estimation of QTL parameters. In all cases, the number of QTL detected increased with population size. QTL with large effects were detected even in small populations, but QTL with small effects were detected only by increasing population size. Selective genotyping and/or selective phenotyping approaches could be effective strategies for reducing the costs associated with conducting QTL analysis in large populations. The method of choice will depend on the relative costs of genotyping versus phenotyping. Electronic Supplementary Material Supplementary material is available for this article at     

Ecological–genetic approach in modeling the natural evolution of a population: Prospects and special aspects of verification

This study presents a complex approach for modeling the natural evolution of a population in terms of population number and dynamics of the genetic structure. A set of dynamic models that consider various types of natural selection was applied to describe possible mechanisms underlying the formation of existing genetic variations in litter sizes in coastal, inland, and farmed arctic fox populations (Alopex lagopus, family Canidae, order Carnivora). The r–K selection model for uniform population and the models with natural selection were assessed on various life cycle stages in a two-age population. The life cycle of arctic fox was fitted to the population model with two age stages. The different reproductive potentials and survivability of progeny on the early stage of life cycle were genetically determined using the model with a single diallelic gene. A monomorphism was obtained for a considered characteristic in a population of coastal arctic fox with constant food supply. Meanwhile, a polymorphism with cyclic fluctuations in population number and gene frequency was obtained in inland arctic fox populations, which could be due to cyclic fluctuations of prey. In farmed fox populations, the considered gene becomes pleiotropic (defines the survival rate of individuals on early and late stages of the life cycle) because of artificial selection performed by farmers to increase the reproductive success of breeders. The application of an appropriate model (with selection by pleiotropic gene) can be used to determine the elimination rate of low litter size alleles from the farmed populations. The possible applications of the proposed models for formulating and solving optimal control tasks in arctic fox populations are discussed too.     

The influence of fluctuating population densities on evolutionary dynamics

The causes and consequences of fluctuating population densities are an important topic in ecological literature. Yet, the effects of such fluctuations on maintenance of variation in spatially structured populations have received little analytic treatment. We analyze what happens when two habitats coupled by migration not only differ in their trade‐offs in selection but also in their demographic stability—and show that equilibrium allele frequencies can change significantly due to ecological feedback arising from locally fluctuating population sizes. When an ecological niche exhibits such fluctuations, these drive an asymmetry in the relative impact of gene flow, and therefore, the equilibrium frequency of the locally adapted type decreases. Our results extend the classic conditions on maintenance of diversity under selection and migration by including the effect of fluctuating population densities. We find simple analytic conditions in terms of the strength of selection, immigration, and the extent of fluctuations between generations in a continent‐island model. Although weak fluctuations hardly affect coexistence, strong recurrent fluctuations lead to extinction of the type better adapted to the fluctuating niche—even if the invader is locally maladapted. There is a disadvantage to specialization to an unstable habitat, as it makes the population vulnerable to swamping from more stable habitats.     

Pattern of variation in avian population growth rates

A central question in population ecology is to understand why population growth rates differ over time. Here, we describe how the long-term growth of populations is not only influenced by parameters affecting the expected dynamics, for example form of density dependence and specific population growth rate, but is also affected by environmental and demographic stochasticity. Using long-term studies of fluctuations of bird populations, we show an interaction between the stochastic and the deterministic components of the population dynamics: high specific growth rates at small densities r(1) are typically positively correlated with the environmental variance sigma(e)(2). Furthermore, theta, a single parameter describing the form of the density regulation in the theta-logistic density-regulation model, is negatively correlated with r(1). These patterns are in turn correlated with interspecific differences in life-history characteristics. Higher specific growth rates, larger stochastic effects on the population dynamics and stronger density regulation at small densities are found in species with large clutch sizes or high adult mortality rates than in long-lived species. Unfortunately, large uncertainties in parameter estimates, as well as strong stochastic effects on the population dynamics, will often make even short-term population projections unreliable. We illustrate that the concept of population prediction interval can be useful in evaluating the consequences of these uncertainties in the population projections for the choice of management actions.     

Asexual reproduction changes predator population dynamics in a life predator–prey system

Many organisms display oscillations in population size. Theory predicts that these fluctuations can be generated by predator–prey interactions, and empirical studies using life model systems, such as a rotifer-algae community consisting of Brachionus calyciflorus as predator and Chlorella vulgaris as prey, have been successfully used for studying such dynamics. B. calyciflorus is a cyclical parthenogen (CP) and clones often differ in their sexual propensity, that is, the degree to which they engage into sexual or asexual (clonal) reproduction. Since sexual propensities can affect growth rates and population sizes, we hypothesized that this might also affect population oscillations. Here, we studied the dynamical behaviour of B. calyciflorus clones representing either CPs (regularly inducing sex) or obligate parthenogens (OPs). We found that the amplitudes of population cycles to be increased in OPs at low nutrient levels. Several other population dynamic parameters seemed unaffected. This suggests that reproductive mode might be an important additional variable to be considered in future studies of population oscillations.