Dynamics of populations with individual variation in dispersal on bounded domains

Most classical models for the movement of organisms assume that all individuals have the same patterns and rates of movement (for example, diffusion with a fixed diffusion coefficient) but there is empirical evidence that movement rates and patterns may vary among different individuals. A simple way to capture variation in dispersal that has been suggested in the ecological literature is to allow individuals to switch between two distinct dispersal modes. We study models for populations whose members can switch between two different nonzero rates of diffusion and whose local population dynamics are subject to density dependence of logistic type. The resulting models are reaction–diffusion systems that can be cooperative at some population densities and competitive at others. We assume that the focal population inhabits a bounded region and study how its overall dynamics depend on the parameters describing switching rates and local population dynamics. (Traveling waves and spread rates have been studied for similar models in the context of biological invasions.) The analytic methods include ideas and results from reaction–diffusion theory, semi-dynamical systems, and bifurcation/continuation theory.     

Population and evolutionary dynamics in spatially structured seasonally varying environments

Increasingly imperative objectives in ecology are to understand and forecast population dynamic and evolutionary responses to seasonal environmental variation and change. Such population and evolutionary dynamics result from immediate and lagged responses of all key life‐history traits, and resulting demographic rates that affect population growth rate, to seasonal environmental conditions and population density. However, existing population dynamic and eco‐evolutionary theory and models have not yet fully encompassed within‐individual and among‐individual variation, covariation, structure and heterogeneity, and ongoing evolution, in a critical life‐history trait that allows individuals to respond to seasonal environmental conditions: seasonal migration. Meanwhile, empirical studies aided by new animal‐tracking technologies are increasingly demonstrating substantial within‐population variation in the occurrence and form of migration versus year‐round residence, generating diverse forms of ‘partial migration’ spanning diverse species, habitats and spatial scales. Such partially migratory systems form a continuum between the extreme scenarios of full migration and full year‐round residence, and are commonplace in nature. Here, we first review basic scenarios of partial migration and associated models designed to identify conditions that facilitate the maintenance of migratory polymorphism. We highlight that such models have been fundamental to the development of partial migration theory, but are spatially and demographically simplistic compared to the rich bodies of population dynamic theory and models that consider spatially structured populations with dispersal but no migration, or consider populations experiencing strong seasonality and full obligate migration. Second, to provide an overarching conceptual framework for spatio‐temporal population dynamics, we define a ‘partially migratory meta‐population’ system as a spatially structured set of locations that can be occupied by different sets of resident and migrant individuals in different seasons, and where locations that can support reproduction can also be linked by dispersal. We outline key forms of within‐individual and among‐individual variation and structure in migration that could arise within such systems and interact with variation in individual survival, reproduction and dispersal to create complex population dynamics and evolutionary responses across locations, seasons, years and generations. Third, we review approaches by which population dynamic and eco‐evolutionary models could be developed to test hypotheses regarding the dynamics and persistence of partially migratory meta‐populations given diverse forms of seasonal environmental variation and change, and to forecast system‐specific dynamics. To demonstrate one such approach, we use an evolutionary individual‐based model to illustrate that multiple forms of partial migration can readily co‐exist in a simple spatially structured landscape. Finally, we summarise recent empirical studies that demonstrate key components of demographic structure in partial migration, and demonstrate diverse associations with reproduction and survival. We thereby identify key theoretical and empirical knowledge gaps that remain, and consider multiple complementary approaches by which these gaps can be filled in order to elucidate population dynamic and eco‐evolutionary responses to spatio‐temporal seasonal environmental variation and change.     

Species with more volatile population dynamics are differentially impacted by weather

Climatic variation has been invoked as an explanation of population dynamics for a variety of taxa. Much work investigating the link between climatic forcings and population fluctuation uses single-taxon case studies. Here, we conduct comparative analyses of a multi-decadal dataset describing population dynamics of 50 co-occurring butterfly species at 10 sites in Northern California. Specifically, we explore the potential commonality of response to weather among species that encompass a gradient of population dynamics via a hierarchical Bayesian modelling framework. Results of this analysis demonstrate that certain weather conditions impact volatile, or irruptive, species differently as compared with relatively stable species. Notably, precipitation-related variables, including indices of the El Niño Southern Oscillation, have a more pronounced impact on the most volatile species. We hypothesize that these variables influence vegetation resource availability, and thus indirectly influence population dynamics of volatile taxa. As one of the first studies to show a common influence of weather among taxa with similar population dynamics, the results presented here suggest new lines of research in the field of biotic–abiotic interactions.     

Sensitivity analysis of transient population dynamics

Short-term, transient population dynamics can differ in important ways from long-term asymptotic dynamics. Just as perturbation analysis (sensitivity and elasticity) of the asymptotic growth rate reveals the effects of the vital rates on long-term growth, the perturbation analysis of transient dynamics can reveal the determinants of short-term patterns. In this article, I present a completely new approach to transient sensitivity and elasticity analysis, using methods from matrix calculus. Unlike previous methods, this approach applies not only to linear time-invariant models but also to time-varying, subsidized, stochastic, nonlinear and spatial models. It is computationally simple, and does not require calculation of eigenvalues or eigenvectors. The method is presented along with applications to plant and animal populations.     

Application of network methods for understanding evolutionary dynamics in discrete habitats

In populations occupying discrete habitat patches, gene flow between habitat patches may form an intricate population structure. In such structures, the evolutionary dynamics resulting from interaction of gene‐flow patterns with other evolutionary forces may be exceedingly complex. Several models describing gene flow between discrete habitat patches have been presented in the population‐genetics literature; however, these models have usually addressed relatively simple settings of habitable patches and have stopped short of providing general methodologies for addressing nontrivial gene‐flow patterns. In the last decades, network theory – a branch of discrete mathematics concerned with complex interactions between discrete elements – has been applied to address several problems in population genetics by modelling gene flow between habitat patches using networks. Here, we present the idea and concepts of modelling complex gene flows in discrete habitats using networks. Our goal is to raise awareness to existing network theory applications in molecular ecology studies, as well as to outline the current and potential contribution of network methods to the understanding of evolutionary dynamics in discrete habitats. We review the main branches of network theory that have been, or that we believe potentially could be, applied to population genetics and molecular ecology research. We address applications to theoretical modelling and to empirical population‐genetic studies, and we highlight future directions for extending the integration of network science with molecular ecology.     

种群统计随机性和环境随机性对种群绝灭的影响

影响种群绝灭的随机干扰可分为种群统计随机性、环境随机性和随机灾害三大类。在相对稳定的环境条件下和相对较短的时间内,以前两类随机干扰对种群绝灭的影响为生态学家关注的焦点。但是,由于自然种群动态及其影响因子的复杂特征,进一步深入研究随机干扰对种群绝灭的作用在理论上和实践上都必须发展新的技术手段。本文回顾了种群统计随机性与环境随机性的概念起源与发展,系统阐述了其分析方法。归纳了两类随机性在种群绝灭研究中的应用范围、作用方式和特点的异同和区别方法。各类随机作用与种群动态之间关系的理论研究与对种群绝灭机理的实践研究紧密相关。根据理论模型模拟和自然种群实际分析两方面的研究现状,作者提出了进一步深入研究随机作用与种群非线性动态方法的策略。指出了随机干扰影响种群绝灭过程的研究的方向：更多的研究将从单纯的定性分析随机干扰对种群动力学简单性质的作用,转向结合特定的种群非线性动态特征和各类随机力作用特点具体分析绝灭极端动态的成因,以期做出精确的预测。     

Transient sensitivities of non-indigenous shrub species indicate complicated invasion dynamics

As biological invasions increasingly affect natural systems, the need for methods that can quantify the processes responsible for invasion success has increased. Further, methods should be geared to the formulation of management strategies. Demographic analyses are designed to explore the causes and properties of population change. Matrix population models, a commonly used technique for demographic analysis, have been applied to the analysis of stage-structured populations. However, most commonly, analyses have focused on long-term outcomes dynamics (ergodic dynamics). The methods available for analysis of matrix population models have recently been extended to facilitate analysis of the transient dynamics most important to invasion analysis. In this paper we analyze the transient population dynamics of three invasive shrubs and compare them to ergodic dynamics. Cytisus scoparius, Clidemia hirta, and Ardisia elliptica come from different parts of the world and are all now found in the United States of America. They also have published transition matrices that measure the probabilities that any one life-history stage will transition to another over an annual time step. These matrices have been estimated from multi-year data collected from plots in various environments. Our comparative study of transient and ergodic dynamics of invasive shrubs shows that, for all the considered shrub species, there was a clear difference between the sensitivities drawn from these two approaches. The transient sensitivities of earlier life-history transitions showed magnified importance relative to ergodic sensitivities. This was especially true of A. elliptica for which the stable population structure was most different from the starting structure analyzed in detail here. For other species, as stable population structures were heavily weighted towards early stages, the differences in the importance of early transitions transiently and ergodically were less dramatic. Late life transitions showed magnified importance in areas towards the center of the invasion or in older invasion areas. Finally, populations with shorter estimated generation times show greater transient sensitivity to early life-history stages; but the pattern was complex and varied according to species, and was also observed across other life-history transitions. Overall, the ambiguity and complexity of the results highlight the power of considering transient population dynamics for invading species, as well as the importance of specific biological and ecological knowledge of the invading species. Although there may be commonalities across invasions, important decisions on control or inference on population dynamics should treat invasions as individual, unique events.     

Transient population dynamics impede restoration and may promote ecosystem transformation after disturbance

The apparent failure of ecosystems to recover from increasingly widespread disturbance is a global concern. Despite growing focus on factors inhibiting resilience and restoration, we still know very little about how demographic and population processes influence recovery. Using inverse and forward demographic modelling of 531 post‐fire sagebrush populations across the western US, we show that demographic processes during recovery from seeds do not initially lead to population growth but rather to years of population decline, low density, and risk of extirpation after disturbance and restoration, even at sites with potential to support long‐term, stable populations. Changes in population structure, and resulting transient population dynamics, lead to a > 50% decline in population growth rate after disturbance and significant reductions in population density. Our results indicate that demographic processes influence the recovery of ecosystems from disturbance and that demographic analyses can be used by resource managers to anticipate ecological transformation risk.     

The usefulness of sensitivity analysis for predicting the effects of cat predation on the population dynamics of their avian prey

Sensitivity analyses of population projection matrix (PPM) models are often used to identify life-history perturbations that will most influence a population's future dynamics. Sensitivities are linear extrapolations of the relationship between a population's growth rate and perturbations to its demographic parameters. Their effectiveness depends on the validity of the assumption of linearity. Here we assess whether sensitivity analysis is an appropriate tool to investigate the effect of predation by cats on the population growth rates of their avian prey. We assess whether predation by cats leads to non-linear effects on population growth and compare population growth rates predicted by sensitivity analysis with those predicted by a non-linear simulation. For a two-stage, age-classified House Sparrow Passer domesticus PPM slight non-linearity arose when PPM elements were perturbed, but perturbation to the vital rates underlying the matrix elements had a linear impact on population growth rate. We found a similar effect with a slightly larger three-stage, age-classified PPM for a Winter Wren Troglodytes troglodytes population perturbed by cat predation. For some avian species, predation by cats may cause linear or only slightly nonlinear impacts on population growth rates. For these species, sensitivity analysis appears to be a useful conservation tool. However, further work on multiple perturbations to avian prey species with more complicated life histories and higher-dimension PPM models is required.     

Sensitivity Analysis of Reactive Ecological Dynamics

Ecological systems with asymptotically stable equilibria may exhibit significant transient dynamics following perturbations. In some cases, these transient dynamics include the possibility of excursions away from the equilibrium before the eventual return; systems that exhibit such amplification of perturbations are called reactive. Reactivity is a common property of ecological systems, and the amplification can be large and long-lasting. The transient response of a reactive ecosystem depends on the parameters of the underlying model. To investigate this dependence, we develop sensitivity analyses for indices of transient dynamics (reactivity, the amplification envelope, and the optimal perturbation) in both continuous- and discrete-time models written in matrix form. The sensitivity calculations require expressions, some of them new, for the derivatives of equilibria, eigenvalues, singular values, and singular vectors, obtained using matrix calculus. Sensitivity analysis provides a quantitative framework for investigating the mechanisms leading to transient growth. We apply the methodology to a predator-prey model and a size-structured food web model. The results suggest predator-driven and prey-driven mechanisms for transient amplification resulting from multispecies interactions.     

Nutrient enrichment alters dynamics in experimental plant populations

Controlling weed populations requires an understanding of their underlying population dynamics which can be achieved through a combination of model development and long-term studies. In this paper, we develop models based on long-term data from experimental populations of the weedy annual plant Cardamine pensylvanica. Four replicate populations of C. pensylvanica were grown in growth chambers under three different nutrient levels but with all other environmental conditions held constant. We analyze the resulting time series using generalized additive models and perform stability analyses using Lyapunov exponents. Further, we test whether the proposed mechanism, delayed density dependence caused by maternal effects, is operating in our system by experimentally manipulating maternal density and assessing the resulting offspring quality. Our results show that that increasing the frequency of nutrients causes plant population dynamics to shift from stable to damped 2-point oscillations to longer cycles. This shift in population dynamics is due to a shift at high nutrients from populations being regulated by first order density feedbacks to being regulated by both first order and second order density feedbacks. A consequence of these first order and second order feedbacks was an increase in cycle lengths as demonstrated by the presence of complex eigenvalues. A short-term experiment confirmed that when grown under high nutrients, the density of maternal plants strongly affected offspring size, providing a mechanism whereby these second order density feedbacks could operate. Our results demonstrate that increasing nutrient frequency results in a qualitative shift in dynamics from stable to longer cycles.     

The limits to population density in birds and mammals

We address two fundamental ecological questions: what are the limits to animal population density and what determines those limits? We develop simple alternative models to predict population limits in relation to body mass. A model assuming that within‐species area use increases with the square of daily travel distance broadly predicts the scaling of empirical extremes of minimum density across birds and mammals. Consistent with model predictions, the estimated density range for a given mass, ‘population scope’, is greater for birds than for mammals. However, unlike mammals and carnivorous birds, expected broad relationships between body mass and density extremes are not supported by data on herbivorous and omnivorous birds. Our results suggest that simple constraints on mobility and energy use/supply are major determinants of the scaling of density limits, but further understanding of interactions between dietary constraints and density limits are needed to predict future wildlife population responses to anthropogenic threats.     

Census error and the detection of density dependence

1. Studies aiming to identify the prevalence and nature of density dependence in ecological populations have often used statistical analysis of ecological time-series of population counts. Such time-series are also being used increasingly to parameterize models that may be used in population management. 2. If time-series contain measurement errors, tests that rely on detecting a negative relationship between log population change and population size are biased and prone to spuriously detecting density dependence (Type I error). This is because the measurement error in density for a given year appears in the corresponding change in population density, with equal magnitude but opposite sign. 3. This effect introduces bias that may invalidate comparisons of ecological data with density-independent time-series. Unless census error can be accounted for, time-series may appear to show strongly density-dependent dynamics, even though the density-dependent signal may in reality be weak or absent. 4. We distinguish two forms of census error, both of which have serious consequences for detecting density dependence. 5. First, estimates of population density are based rarely on exact counts, but on samples. Hence there exists sampling error, with the level of error depending on the method employed and the number of replicates on which the population estimate is based. 6. Secondly, the group of organisms measured is often not a truly self-contained population, but part of a wider ecological population, defined in terms of location or behaviour. Consequently, the subpopulation studied may effectively be a sample of the population and spurious density dependence may be detected in the dynamics of a single subpopulation. In this case, density dependence is detected erroneously, even if numbers within the subpopulation are censused without sampling error. 7. In order to illustrate how process variation and measurement error may be distinguished we review data sets (counts of numbers of birds by single observers) for which both census error and long-term variance in population density can be estimated. 8. Tests for density dependence need to obviate the problem that measured population sizes are typically estimates rather than exact counts. It is possible that in some cases it may be possible to test for density dependence in the presence of unknown levels of census error, for example by uncovering nonlinearities in the density response. However, it seems likely that these may lack power compared with analyses that are able to explicitly include census error and we review some recently developed methods.     

Predicting when climate‐driven phenotypic change affects population dynamics

Species' responses to climate change are variable and diverse, yet our understanding of how different responses (e.g. physiological, behavioural, demographic) relate and how they affect the parameters most relevant for conservation (e.g. population persistence) is lacking. Despite this, studies that observe changes in one type of response typically assume that effects on population dynamics will occur, perhaps fallaciously. We use a hierarchical framework to explain and test when impacts of climate on traits (e.g. phenology) affect demographic rates (e.g. reproduction) and in turn population dynamics. Using this conceptual framework, we distinguish four mechanisms that can prevent lower‐level responses from impacting population dynamics. Testable hypotheses were identified from the literature that suggest life‐history and ecological characteristics which could predict when these mechanisms are likely to be important. A quantitative example on birds illustrates how, even with limited data and without fully‐parameterized population models, new insights can be gained; differences among species in the impacts of climate‐driven phenological changes on population growth were not explained by the number of broods or density dependence. Our approach helps to predict the types of species in which climate sensitivities of phenotypic traits have strong demographic and population consequences, which is crucial for conservation prioritization of data‐deficient species.     

Food web complexity and chaotic population dynamics

In mathematical models, very simple communities consisting of three or more species frequently display chaotic dynamics which implies that long‐term predictions of the population trajectories in time are impossible. Communities in the wild tend to be more complex, but evidence for chaotic dynamics from such communities is scarce. We used supercomputing power to test the hypothesis that chaotic dynamics become less frequent in model ecosystems when their complexity increases. We determined the dynamical stability of a universe of mathematical, nonlinear food web models with varying degrees of organizational complexity. We found that the frequency of unpredictable, chaotic dynamics increases with the number of trophic levels in a food web but decreases with the degree of complexity. Our results suggest that natural food webs possess architectural properties that may intrinsically lower the likelihood of chaotic community dynamics.     

The uncertain blitzkrieg of Pleistocene megafauna

We investigated, using meta‐analysis of empirical data and population modelling, plausible scenarios for the cause of late Pleistocene global mammal extinctions. We also considered the rate at which these extinctions may have occurred, providing a test of the so‐called ‘blitzkrieg’ hypothesis, which postulates a rapid, anthropogenically driven, extinction event. The empirical foundation for this work was a comprehensive data base of estimated body masses of mammals, comprising 198 extinct and 433 surviving species > 5 kg, which we compiled through an extensive literature search. We used mechanistic population modelling to simulate the role of human hunting efficiency, meat off‐take, relative naivety of prey to invading humans, variation in reproductive fitness of prey and deterioration of habitat quality (due to either anthropogenic landscape burning or climate change), and explored the capacity of different modelling scenarios to recover the observed empirical relationship between body mass and extinction proneness. For the best‐fitting scenarios, we calculated the rate at which the extinction event would have occurred. All of the modelling was based on sampling randomly from a plausible range of parameters (and their interactions), which affect human and animal population demographics. Our analyses of the empirical data base revealed that the relationship between body mass and extinction risk relationship increases continuously from small‐ to large‐sized animals, with no clear ‘megafaunal’ threshold. A logistic ancova model incorporating body mass and geography (continent) explains 92% of the variation in the observed extinctions. Population modelling demonstrates that there were many plausible mechanistic scenarios capable of reproducing the empirical body mass–extinction risk relationship, such as specific targeting of large animals by humans, or various combinations of habitat change and opportunistic hunting. Yet, given the current imperfect knowledge base, it is equally impossible to use modelling to isolate definitively any single scenario to explain the observed extinctions. However, one universal prediction, which applied in all scenarios in which the empirical distribution was correctly predicted, was for the extinctions to be rapid following human arrival and for surviving fauna to be suppressed below their pre‐‘blitzkrieg’ densities. In sum, human colonization in the late Pleistocene almost certainly triggered a ‘blitzkrieg’ of the ‘megafauna’, but the operational details remain elusive.     

When stable-stage equilibrium is unlikely: integrating transient population dynamics improves asymptotic methods

Background and Aims Evaluation of population projection matrices (PPMs) that are focused on asymptotically based properties of populations is a commonly used approach to evaluate projected dynamics of managed populations. Recently, a set of tools for evaluating the properties of transient dynamics has been expanded to evaluate PPMs and to consider the dynamics of populations prior to attaining the stable-stage distribution, a state that may never be achieved in disturbed or otherwise ephemeral habitats or persistently small populations. This study re-evaluates data for a tropical orchid and examines the value of including such analyses in an integrative approach.Methods Six small populations of Lepanthes rubripetala were used as a model system and the R software package popdemo was used to produce estimates of the indices for the asymptotic growth rate (lambda), sensitivities, reactivity, first-time step attenuation, maximum amplification, maximum attenuation, maximal inertia and maximal attenuation. The response in lambda to perturbations of demographic parameters using transfer functions and multiple perturbations on growth, stasis and fecundity were also determined. The results were compared with previously published asymptotic indices.Key Results It was found that combining asymptotic and transient dynamics expands the understanding of possible population changes. Comparison of the predicted density from reactivity and first-time step attenuation with the observed change in population size in two orchid populations showed that the observed density was within the predicted range. However, transfer function analysis suggests that the traditional approach of measuring perturbation of growth rates and persistence (inertia) may be misleading and is likely to result in erroneous management decisions.Conclusions Based on the results, an integrative approach is recommended using traditional PPMs (asymptotic processes) with an evaluation of the diversity of dynamics that may arise when populations are not at a stable-stage distribution (transient processes). This method is preferable for designing rapid and efficient interventions after disturbances, and for developing strategies to establish new populations.     

Travelling wave dynamics and self-organization in a spatio-temporally structured population

We analyse spatial population dynamics showing that periodic or period-like chaotic dynamics produce self-organization structures, such as travelling waves. We suggest that self-organized patterns are associated with spatial synchrony patterns that often depend on geographical distance between subpopulations. The population dynamics also show statistical spatial autocorrelation patterns. We contrast our theoretical simulations with empirical data on annual damages in young sapling stands caused by voles. We conclude, on the basis of the periodicity, synchrony, and spatial autocorrelation patterns, and our simulation results, that vole dynamics represent travelling waves in population dynamics. We suggest that because such synchrony patterns are frequently observed in natural populations, spatial self-organization may be more common in population dynamics than reported in the literature.