Relating dispersal and range expansion of California sea otters

Linking dispersal and range expansion of invasive species has long challenged theoretical and quantitative ecologists. Subtle differences in dispersal can yield large differences in geographic spread, with speeds ranging from constant to rapidly increasing. We developed a stage-structured integrodifference equation (IDE) model of the California sea otter range expansion that occurred between 1914 and 1986. The non-spatial model, a linear matrix population model, was coupled to a suite of candidate dispersal kernels to form stage-structured IDEs. Demographic and dispersal parameters were estimated independent of range expansion data. Using a single dispersal parameter, alpha, we examined how well these stage-structured IDEs related small scale demographic and dispersal processes with geographic population expansion. The parameter alpha was estimated by fitting the kernels to dispersal data and by fitting the IDE model to range expansion data. For all kernels, the alpha estimate from range expansion data fell within the 95% confidence intervals of the alpha estimate from dispersal data. The IDE models with exponentially bounded kernels predicted invasion velocities that were captured within the 95% confidence bounds on the observed northbound invasion velocity. However, the exponentially bounded kernels yielded range expansions that were in poor qualitative agreement with range expansion data. An IDE model with fat (exponentially unbounded) tails and accelerating spatial spread yielded the best qualitative match. This model explained 94% and 97% of the variation in northbound and southbound range expansions when fit to range expansion data. These otters may have been fat-tailed accelerating invaders or they may have followed a piece-wise linear spread first over kelp forests and then over sandy habitats. Further, habitat-specific dispersal data could resolve these explanations.     

Families of discrete kernels for modeling dispersal

Integer lattices are important theoretical landscapes for studying the consequences of dispersal and spatial population structure, but convenient dispersal kernels able to represent important features of dispersal in nature have been lacking for lattices. Because leptokurtic (centrally peaked and long-tailed) kernels are common in nature and have important effects in models, of particular interest are families of dispersal kernels in which the degree of leptokurtosis can be varied parametrically. Here we develop families of kernels on integer lattices with several important properties. The degree of leptokurtosis can be varied parametrically from near 0 (the Gaussian value) to infinity. These kernels are all asymptotically radially symmetric. (Exact radial symmetry is impossible on lattices except in one dimension.) They have separate parameters for shape and scale, and their lower order moments and Fourier transforms are given by simple formulae. In most cases, the kernel families that we develop are closed under convolution so that multiple steps of a kernel remain within the same family. Included in these families are kernels with asymptotic power function tails, which have provided good fits to some observations from nature. These kernel families are constructed by randomizing convolutions of stepping-stone kernels and have interpretations in terms of population heterogeneity and heterogeneous physical processes.     

Why trees migrate so fast: confronting theory with dispersal biology and the paleorecord

Reid's paradox describes the fact that classical models cannot account for the rapid (10(2)-10(3) m yr-1) spread of trees at the end of the Pleistocene. I use field estimates of seed dispersal with an integrodifference equation and simulation models of population growth to show that dispersal data are compatible with rapid spread. Dispersal estimates lay to rest the possibility that rapid spread occurred by diffusion. The integrodifference model predicts that, if the seed shadow has a long 'fat' tail, then rapid spread is possible, despite short average dispersal distances. It further predicts that velocity is more sensitive to life history than is classical diffusion. Application of such models is frustrated because the tail of the seed shadow cannot be fitted to data. However, the data can be used to test a 'long-distance' hypothesis against alternative ('local') models of dispersal using Akaike's Information Criterion and likelihood ratio tests. Tests show that data are consistent with >10% of seed dispersed as a long (10(2) m) fat-tailed kernel. Models based on such kernels predict spread as rapid as that inferred from the pollen record. If fat-tailed dispersal explains these rapid rates, then it is surprising not to see large differences in velocities among taxa with contrasting life histories. The inference of rapid spread, together with lack of obvious life-history effects, suggests velocities may have not reached their potentials, being stalled by rates of climate change, geography, or both.     

Modelling population redistribution in a leaf beetle: an evaluation of alternative dispersal functions

1. Dispersal is a fundamental ecological process, so spatial models require realistic dispersal kernels. We compare five different forms for the dispersal kernel of the tansy beetle Chrysolina graminis moving between patches of its host-plant (tansy Tanacetum vulgare) in a riparian landscape. 2. Multi-patch mark-recapture data were collected every 2 weeks over 2 years within a large network of patches and from 2226 beetles. Dispersal was common (28.4% of 880 recaptures after a fortnight) and was more likely over longer intervals, out of small patches, for females and during flooding. Interpatch movement rates did not differ between years and exhibited no density dependence. Dispersal distances were similar for males and females, in both years and over all intervals, with a median dispersal distance of just 9.8 m, although a maximum of 856 m was recorded. 3. A model of dispersal, where patches competed for dispersers based on their size and distance from the beetle's source patch (scaled by the dispersal kernel) was fitted to the field data with a maximum likelihood procedure and each of five alternative kernels. The best fitting had relatively extended tails of long-distance dispersal, while Gaussian and negative exponential kernels performed worst. 4. The model suggests that females disperse more commonly than males and that both are strongly attracted to large patches but do not differ between years, which are consistent with the empirical results. Model-predicted emigration and immigration rates and dispersal phenologies match those observed, suggesting that the model captured the major drivers of tansy beetle dispersal. 5. Although negative exponential and Gaussian kernels are widely used for their simplicity, we suggest that these should not be the models of automatic choice, and that fat-tailed kernels with relatively higher proportions of long-distance dispersal may be more realistic.     

Fat-tailed gene flow in the dioecious canopy tree species Fraxinus mandshurica var. japonica revealed by microsatellites

Pollen flow, seed dispersal and individual reproductive success can be simultaneously estimated from the genotypes of adults and offspring using stochastic models. Using four polymorphic microsatellite loci, gene flow of the wind-pollinated and wind-seed-dispersed dioecious tree species, Fraxinus mandshurica var. japonica, was quantified in a riparian forest, in northern Japan. In a 10.5-ha plot, 74 female adults, 76 male adults and 292 current-year seedlings were mapped and genotyped, together with 200 seeds. To estimate dispersal kernels of pollen and seeds, we applied normal, exponential power, Weibull, bivariate t-distribution kernels, and two-component models consisting of two normal distribution functions, one with a small and one with a large variance. A two-component pollen flow model with a small contribution (26.1%) from short-distance dispersal (sigma = 7.2 m), and the rest from long-distance flow (sigma = 209.9 m), was chosen for the best-fitting model. The average distance that integrated pollen flows inside and outside the study plot was estimated to be 196.8 m. Tree size and flowering intensity affected reproduction, and there appeared to be critical values that distinguished reproductively successful and unsuccessful adults. In contrast, the gene flow model that estimated both pollen and seed dispersal from established seedlings resulted in extensive seed dispersal, and the expected spatial genetic structures did not satisfactorily fit with the observations, even for the selected model. Our results advanced small-scale individual-based parentage analysis for quantifying fat-tailed gene flow in wind-mediated species, but also clarified its limitations and suggested future possibilities for gene flow studies.     

Modeling Responses of Leafy Spurge Dispersal to Control Strategies

Leafy spurge （Euphorbia esula L.） has substantial negative effects on grassland biodiversity, productivity, and economic benefit in North America. To predict these negative impacts, we need an appropriate plant-spread model which can simulate the response of an invading population to different control strategies. In this study, using a stochastic map lattice approach we generated a spatially explicitly stochastic process-based model to simulate dispersal trajectories of leafy spurge under various control scenarios. The model integrated dispersal curve, propagule pressure, and population growth of leafy spurge at local and short-temporal scales to capture spread features of leafy spurge at large spatial and long-temporal scales. Our results suggested that narrow-, medium-, and fat-tailed kernels did not differ in their ability to predict spread, in contrast to previous works. For all kernels, Allee effects were significantly present and could explain the lag phase （three decades） before leafy spurge spread accelerated. When simulating from the initial stage of introduction, Allee effects were critical in predicting spread rate of leafy spurge, because the prediction could be seriously affected by the low density period of leafy spurge community. No Allee effects models were not able to simulate spread rate well in this circumstance. When applying control strategies to the current distribution, Allee effects could stop the spread of leafy spurge; no Allee effects models, however, were able to slow but not stop the spread. The presence of Allee effects had significant ramifications on the efficiencies of control strategies. For both Allee and no Allee effects models, the later that control strategies were implemented, the more effort had to be input to achieve similar control results.     

Consequences of Dispersal Heterogeneity for Population Spread and Persistence

Dispersal heterogeneity is increasingly being observed in ecological populations and has long been suspected as an explanation for observations of non-Gaussian dispersal. Recent empirical and theoretical studies have begun to confirm this. Using an integro-difference model, we allow an individual’s diffusivity to be drawn from a trait distribution and derive a general relationship between the dispersal kernel’s moments and those of the underlying heterogeneous trait distribution. We show that dispersal heterogeneity causes dispersal kernels to appear leptokurtic, increases the population’s spread rate, and lowers the critical reproductive rate required for persistence in the face of advection. Wavespeed has been shown previously to be determined largely by the form of the dispersal kernel tail. We qualify this by showing that when reproduction is low, the precise shape of the tail is less important than the first few dispersal moments such as variance and kurtosis. If the reproductive rate is large, a dispersal kernel’s asymptotic tail has a greater influence over wavespeed, implying that estimating the prevalence of traits which correlate with long-range dispersal is critical. The presence of multiple dispersal behaviors has previously been characterized in terms of long-range versus short-range dispersal, and it has been found that rare long-range dispersal essentially determines wavespeed. We discuss this finding and place it within a general context of dispersal heterogeneity showing that the dispersal behavior with the highest average dispersal distance does not always determine wavespeed.     

A Short Note on Short Dispersal Events

We study how the speed of spread for an integrodifference equation depends on the dispersal pattern of individuals. When the dispersal kernel has finite variance, the central limit theorem states that convolutions of the kernel with itself will approach a suitably chosen Gaussian distribution. Despite this fact, the speed of spread cannot be obtained from the Gaussian approximation. We give several examples and explanations for this fact. We then use the kurtosis of the kernel to derive an improved approximation that shows a very good fit to all the kernels tested. We apply the theory to one well-studied data set of dispersal of Drosophila pseudoobscura and to two one-parameter families of theoretical dispersal kernels. In particular, we find kernels that, despite having compact support, have a faster speed of spread than the Gaussian kernel.     

Recolonisation by diffusion can generate increasing rates of spread

Diffusion is one of the most frequently used assumptions to explain dispersal. Diffusion models and in particular reaction-diffusion equations usually lead to solutions moving at constant speeds, too slow compared to observations. As early as 1899, Reid had found that the rate of spread of tree species migrating to northern environments at the beginning of the Holocene was too fast to be explained by diffusive dispersal. Rapid spreading is generally explained using long distance dispersal events, modelled through integro-differential equations (IDEs) with exponentially unbounded (EU) kernels, i.e. decaying slower than any exponential. We show here that classical reaction-diffusion models of the Fisher-Kolmogorov-Petrovsky-Piskunov type can produce patterns of colonisation very similar to those of IDEs, if the initial population is EU at the beginning of the considered colonisation event. Many similarities between reaction-diffusion models with EU initial data and IDEs with EU kernels are found; in particular comparable accelerating rates of spread and flattening of the solutions. There was previously no systematic mathematical theory for such reaction-diffusion models with EU initial data. Yet, EU initial data can easily be understood as consequences of colonisation-retraction events and lead to fast spreading and accelerating rates of spread without the long distance hypothesis.     

Saddle-Point Approximations,Integrodifference Equations,and Invasions

Invasion, the growth in numbers and spatial spread of a population over time, is a fundamental process in ecology. Governments and businesses expend vast sums to prevent and control invasions of pests and pestilences and to promote invasions of endangered species and biological control agents. Many mathematical models of biological invasions use nonlinear integrodifference equations to describe the growth and dispersal processes and to predict the speed of invasion fronts. Linear models have received less attention, perhaps because they are difficult to simulate for large times. In this paper, we use the saddle-point method, alias the method of steepest descent, to derive asymptotic approximations for the solutions of linear integrodifference equations. We work through five examples, for Gaussian, Laplace, and uniform dispersal kernels in one dimension and for asymmetric Gaussian and radially symmetric Laplace kernels in two dimensions. Our approximations are extremely close to the exact solutions, even for intermediate times. We also employ an empirical saddle-point approximation to predict densities using dispersal data. We use our approximations to examine the effects of censored dispersal data on estimates of invasion speed and population density.     

Warming increases the spread of an invasive thistle

Background

Global warming and shifted precipitation regimes increasingly affect species abundances and distributions worldwide. Despite a large literature on species'' physiological, phenological, growth, and reproductive responses to such climate change, dispersal is rarely examined. Our study aims to test whether the dispersal ability of a non-native, wind-dispersed plant species is affected by climate change, and to quantify the ramifications for future invasion spread rates.

Methodology/Principal Findings

We experimentally increased temperature and precipitation in a two-cohort, factorial field study (n = 80). We found an overwhelming warming effect on plant life history: warming not only improved emergence, survival, and reproduction of the thistle Carduus nutans, but also elevated plant height, which increased seed dispersal distances. Using spatial population models, we demonstrate that these empirical warming effects on demographic vital rates, and dispersal parameters, greatly exacerbate spatial spread. Predicted levels of elevated winter precipitation decreased seed production per capitulum, but this only slightly offset the warming effect on spread. Using a spread rate decomposition technique (c*-LTRE), we also found that plant height-mediated changes in dispersal contribute most to increased spread rate under climate change.

Conclusions/Significance

We found that both dispersal and spread of this wind-dispersed plant species were strongly impacted by climate change. Dispersal responses to climate change can improve, or diminish, a species'' ability to track climate change spatially, and should not be overlooked. Methods that combine both demographic and dispersal responses thus will be an invaluable complement to projections of suitable habitat under climate change.     

Invasion speeds for structured populations in fluctuating environments

We live in a time where climate models predict future increases in environmental variability and biological invasions are becoming increasingly frequent. A key to developing effective responses to biological invasions in increasingly variable environments will be estimates of their rates of spatial spread and the associated uncertainty of these estimates. Using stochastic, stage-structured, integrodifference equation models, we show analytically that invasion speeds are asymptotically normally distributed with a variance that decreases in time. We apply our methods to a simple juvenile–adult model with stochastic variation in reproduction and an illustrative example with published data for the perennial herb, Calathea ovandensis. These examples buttressed by additional analysis reveal that increased variability in vital rates simultaneously slow down invasions yet generate greater uncertainty about rates of spatial spread. Moreover, while temporal autocorrelations in vital rates inflate variability in invasion speeds, the effect of these autocorrelations on the average invasion speed can be positive or negative depending on life history traits and how well vital rates “remember” the past.     

种子的长距离风传播模型研究进展

 植物种子的长距离传播在物种迁移、生物入侵、保护生物学等领域有重要的生态和进化意义。种子传播有很多方式,开阔草原等地区的草本植物和许多热带和温带的树木都是通过风传播种子的。风传播的方式最适合进行种子长距离传播现象的模拟研究。种子的风传播模型是传播生态研究的一个重要领域,尤其是种子的长距离风传播模型,对于外来入侵植物的扩散和破碎化景观中植物种群的基因交流等生态过程研究举足轻重,然而国内鲜见这方面的研究成果。本文综述了种子长距离风传播现象研究的背景和意义,分析了风传播种子模型的基本形式和构成原理,并分别就现象模型和机理模型的相关研究进展进行了总结,同时指出了未来发展的几个重要方向。种子的风传播模型可以分为现象模型和机理模型两类,现象模型按种子传播核心的形式包括短尾模型、偏峰长尾模型和混合传播核心模型,后两者对于长距离传播数据的模拟可以取得很好的效果。机理模型按照模拟机制可分为欧拉对流扩散模型和拉格郎日随机模型两类。本文重点介绍了种子的长距离风传播现象的形成机理和两类机理模型的参数构成和处理方式。适合种子脱落的天气和适合传播的天气的同步性可能是形成种子长距离风传播的一个重要前提,林缘和地表存在的上升气流及大风和暴风中形成的速度梯度都可能对于种子的长距离传播有重要的作用。机理模型的操作因子主要包括生物方面的因子、气象方面的因子和地形方面的因子。同时对目前几个应用比较成功的机理模型进行了简要的介绍和评价,包括倾斜羽毛模型、对流-扩散-下降模型、无掩蔽模型、背景模型、WINDISPER及其改进模型和PAPPUS模型。最后指出,目前在风传播种子的长距离模型研究中,对草本植物种子的传播模拟的投入明显不如树木种子的长距离传播模拟,对于破碎化景观中种子长距离的风传播的研究还存在很大的差距,而对提高机理模型预测能力的高分辨率物理环境数据输入技术的需求则为多学科交叉提供了很好的机会。     

Steady state of homozygosity and Gst for the island model

We examine homozygosity and G(st) for a subdivided population governed by the finite island model. Assuming an infinite allele model and strong mutation we show that the steady state distributions of G(st) and homozygosity have asymptotic expansions in the mutation rate. We use this observation to derive asymptotic expansions for various moments of homozygosity and to derive rigorous formulas for the mean and variance of G(st). We show that G(st) approximately 1/(1+2Nm), similarly to the well known formula of Wright for the infinite island model, and that the variance of G(st) goes to zero as mutation increases.     

Local Replicator Dynamics: A Simple Link Between Deterministic and Stochastic Models of Evolutionary Game Theory

Classical replicator dynamics assumes that individuals play their games and adopt new strategies on a global level: Each player interacts with a representative sample of the population and if a strategy yields a payoff above the average, then it is expected to spread. In this article, we connect evolutionary models for infinite and finite populations: While the population itself is infinite, interactions and reproduction occurs in random groups of size N. Surprisingly, the resulting dynamics simplifies to the traditional replicator system with a slightly modified payoff matrix. The qualitative results, however, mirror the findings for finite populations, in which strategies are selected according to a probabilistic Moran process. In particular, we derive a one-third law that holds for any population size. In this way, we show that the deterministic replicator equation in an infinite population can be used to study the Moran process in a finite population and vice versa. We apply the results to three examples to shed light on the evolution of cooperation in the iterated prisoner’s dilemma, on risk aversion in coordination games and on the maintenance of dominated strategies.     

Additions To The Flora Of Shetland

ABSTRACT

Background: Pallenis spinosa (Asteraceae) produces both winged and wingless achenes. Both achene morphs are non-dormant and show a similar embryo size, rendering dispersal ability as their only apparent functional difference.

Aims: We studied morph-specific release and spatial dispersal patterns to ascertain whether the common view of seed dimorphism as a mixed strategy, that is functionally fully differentiated morphs, is appropriate for this system.

Methods: For three years, at the onset of achene release, in early autumn, we placed achene traps at different distances from source plants, censusing achene arrival at 3–4 day intervals. We constructed morph-specific dispersal kernels and related release intensity to prevailing meteorological conditions in census intervals. Selected kernel models were used to describe dispersal effects of observed changes in the proportion of winged achenes (p_w) in successive released fractions.

Results: Achene release extended up to early-mid winter, peaking in rainy, windy intervals. Throughout the season, p_w decreased progressively. Unexpectedly, the wingless morph produced the longest dispersal tails and it only showed ability for fat-tailed dispersal. Consequently, maximum dispersal distances steadily increased throughout the season.

Conclusions: Achene dimorphism in P. spinosa appears to allow a within-season continuous reshaping of the seed-dispersal kernel instead of representing a mixed strategy.     

Long-distance seed dispersal in plant populations

Long-distance seed dispersal influences many key aspects of the biology of plants, including spread of invasive species, metapopulation dynamics, and diversity and dynamics in plant communities. However, because long-distance seed dispersal is inherently hard to measure, there are few data sets that characterize the tails of seed dispersal curves. This paper is structured around two lines of argument. First, we argue that long-distance seed dispersal is of critical importance and, hence, that we must collect better data from the tails of seed dispersal curves. To make the case for the importance of long-distance seed dispersal, we review existing data and models of long-distance seed dispersal, focusing on situations in which seeds that travel long distances have a critical impact (colonization of islands, Holocene migrations, response to global change, metapopulation biology). Second, we argue that genetic methods provide a broadly applicable way to monitor long-distance seed dispersal; to place this argument in context, we review genetic estimates of plant migration rates. At present, several promising genetic approaches for estimating long-distance seed dispersal are under active development, including assignment methods, likelihood methods, genealogical methods, and genealogical/demographic methods. We close the paper by discussing important but as yet largely unexplored areas for future research.     

Using genetic markers to estimate the pollen dispersal curve

Pollen dispersal is a critical process that shapes genetic diversity in natural populations of plants. Estimating the pollen dispersal curve can provide insight into the evolutionary dynamics of populations and is essential background for making predictions about changes induced by perturbations. Specifically, we would like to know whether the dispersal curve is exponential, thin-tailed (decreasing faster than exponential), or fat-tailed (decreasing slower than the exponential). In the latter case, rare events of long-distance dispersal will be much more likely. Here we generalize the previously developed TWOGENER method, assuming that the pollen dispersal curve belongs to particular one- or two-parameter families of dispersal curves and estimating simultaneously the parameters of the dispersal curve and the effective density of reproducing individuals in the population. We tested this method on simulated data, using an exponential power distribution, under thin-tailed, exponential and fat-tailed conditions. We find that even if our estimates show some bias and large mean squared error (MSE), we are able to estimate correctly the general trend of the curve - thin-tailed or fat-tailed - and the effective density. Moreover, the mean distance of dispersal can be correctly estimated with low bias and MSE, even if another family of dispersal curve is used for the estimation. Finally, we consider three case studies based on forest tree species. We find that dispersal is fat-tailed in all cases, and that the effective density estimated by our model is below the measured density in two of the cases. This latter result may reflect the difficulty of estimating two parameters, or it may be a biological consequence of variance in reproductive success of males in the population. Both the simulated and empirical findings demonstrate the strong potential of TWOGENER for evaluating the shape of the dispersal curve and the effective density of the population (d(e)).     

Asymptotically robust variance estimation for person‐time incidence rates

Person‐time incidence rates are frequently used in medical research. However, standard estimation theory for this measure of event occurrence is based on the assumption of independent and identically distributed (iid) exponential event times, which implies that the hazard function remains constant over time. Under this assumption and assuming independent censoring, observed person‐time incidence rate is the maximum‐likelihood estimator of the constant hazard, and asymptotic variance of the log rate can be estimated consistently by the inverse of the number of events. However, in many practical applications, the assumption of constant hazard is not very plausible. In the present paper, an average rate parameter is defined as the ratio of expected event count to the expected total time at risk. This rate parameter is equal to the hazard function under constant hazard. For inference about the average rate parameter, an asymptotically robust variance estimator of the log rate is proposed. Given some very general conditions, the robust variance estimator is consistent under arbitrary iid event times, and is also consistent or asymptotically conservative when event times are independent but nonidentically distributed. In contrast, the standard maximum‐likelihood estimator may become anticonservative under nonconstant hazard, producing confidence intervals with less‐than‐nominal asymptotic coverage. These results are derived analytically and illustrated with simulations. The two estimators are also compared in five datasets from oncology studies.     

Resource-dependent dispersal and the speed of biological invasions

Many mobile organisms exhibit resource-dependent movement in which movement rates adjust to changes in local resource densities through changes in either the probability of moving or the distance moved. Such changes may have important consequences for invasions because reductions in resources behind an invasion front may cause higher dispersal while simultaneously reducing population growth behind the front and thus lowering the number of dispersers. Intuiting how the interplay between population growth and dispersal affects invasions is difficult without mathematical models, yet most models assume dispersal rates are constant. Here we present spatial-spread models that allow for consumer-resource interactions and resource-dependent dispersal. Our results show that when resources affect the probability of dispersal, then the invasion dynamics are no different than if resources did not affect dispersal. When resources instead affect the distance dispersed, however, the invasion dynamics are strongly affected by the strength of the consumer-resource interaction, and population cycles behind the wave front lead to fluctuating rates of spread. Our results suggest that for actively dispersing invaders, invasion dynamics can be determined by species interactions. More practically, our work suggests that reducing invader densities behind the front may be a useful method of slowing an invader's rate of spread.