Dispersal patterns, dispersal mechanisms, and invasion wave speeds for invasive thistles

Understanding and predicting population spread rates is an important problem in basic and applied ecology. In this article, we link estimates of invasion wave speeds to species traits and environmental conditions. We present detailed field studies of wind dispersal and compare nonparametric (i.e., data-based) and mechanistic (fluid dynamics model-based) dispersal kernel and spread rate estimates for two important invasive weeds, Carduus nutans and Carduus acanthoides. A high-effort trapping design revealed highly leptokurtic dispersal distributions, with seeds caught up to 96 m from the source, far further than mean dispersal distances (approx. 2 m). Nonparametric wave speed estimates are highly sensitive to sampling effort. Mechanistic estimates are insensitive to sampling because they are obtained from independent data and more useful because they are based on the dispersal mechanism. Over a wide range of realistic conditions, mechanistic spread rate estimates were most sensitive to high winds and low seed settling velocities. The combination of integrodifference equations and mechanistic dispersal models is a powerful tool for estimating invasion spread rates and for linking these estimates to characteristics of the species and the environment.     

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.     

Invasion by extremes: population spread with variation in dispersal and reproduction

For populations having dispersal described by fat-tailed kernels (kernels with tails that are not exponentially bounded), asymptotic population spread rates cannot be estimated by traditional models because these models predict continually accelerating (asymptotically infinite) invasion. The impossible predictions come from the fact that the fat-tailed kernels fitted to dispersal data have a quality (nondiscrete individuals and, thus, no moment-generating function) that never applies to data. Real organisms produce finite (and random) numbers of offspring; thus, an empirical moment-generating function can always be determined. Using an alternative method to estimate spread rates in terms of extreme dispersal events, we show that finite estimates can be derived for fat-tailed kernels, and we demonstrate how variable reproduction modifies these rates. Whereas the traditional models define spread rate as the speed of an advancing front describing the expected density of individuals, our alternative definition for spread rate is the expected velocity for the location of the furthest-forward individual in the population. The asymptotic wave speed for a constant net reproductive rate R0 is approximated as (1/T)(piuR)/2)(1/2) m yr(-1), where T is generation time, and u is a distance parameter (m2) of Clark et al.'s 2Dt model having shape parameter p = 1. From fitted dispersal kernels with fat tails and infinite variance, we derive finite rates of spread and a simple method for numerical estimation. Fitted kernels, with infinite variance, yield distributions of rates of spread that are asymptotically normal and, thus, have finite moments. Variable reproduction can profoundly affect rates of spread. By incorporating the variance in reproduction that results from variable life span, we estimate much lower rates than predicted by the standard approach, which assumes a constant net reproductive rate. Using basic life-history data for trees, we show these estimated rates to be lower than expected from previous analytical models and as interpreted from paleorecords of forest spread at the end of the Pleistocene. Our results suggest reexamination of past rates of spread and the potential for future response to climate change.     

Dispersal and competition models for plants

New models for seed dispersal and competition between plant species are formulated and analyzed. The models are integrodifference equations, discrete in time and continuous in space, and have applications to annual and perennial species. The spread or invasion of a single plant species into a geographic region is investigated by studying the travelling wave solutions of these equations. Travelling wave solutions are shown to exist in the single-species models and are compared numerically. The asymptotic wave speed is calculated for various parameter values. The single-species integrodifference equations are extended to a model for two competing annual plants. Competition in the two-species model is based on a difference equation model developed by Pakes and Maller [26]. The two-species model with competition and dispersal yields a system of integrodifference equations. The effects of competition on the travelling wave solutions of invading plant species is investigated numerically.     

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.     

Do Spatially-Implicit Estimates of Neutral Migration Comply with Seed Dispersal Data in Tropical Forests?

Neutral community models have shown that limited migration can have a pervasive influence on the taxonomic composition of local communities even when all individuals are assumed of equivalent ecological fitness. Notably, the spatially implicit neutral theory yields a single parameter I for the immigration-drift equilibrium in a local community. In the case of plants, seed dispersal is considered as a defining moment of the immigration process and has attracted empirical and theoretical work. In this paper, we consider a version of the immigration parameter I depending on dispersal limitation from the neighbourhood of a community. Seed dispersal distance is alternatively modelled using a distribution that decreases quickly in the tails (thin-tailed Gaussian kernel) and another that enhances the chance of dispersal events over very long distances (heavily fat-tailed Cauchy kernel). Our analysis highlights two contrasting situations, where I is either mainly sensitive to community size (related to ecological drift) under the heavily fat-tailed kernel or mainly sensitive to dispersal distance under the thin-tailed kernel. We review dispersal distances of rainforest trees from field studies and assess the consistency between published estimates of I based on spatially-implicit models and the predictions of the kernel-based model in tropical forest plots. Most estimates of I were derived from large plots (10–50 ha) and were too large to be accounted for by a Cauchy kernel. Conversely, a fraction of the estimates based on multiple smaller plots (1 ha) appeared too small to be consistent with reported ranges of dispersal distances in tropical forests. Very large estimates may reflect within-plot habitat heterogeneity or estimation problems, while the smallest estimates likely imply other factors inhibiting migration beyond dispersal limitation. Our study underscores the need for interpreting I as an integrative index of migration limitation which, besides the limited seed dispersal, possibly includes habitat filtering or fragmentation.     

Sex-biased dispersal and the speed of two-sex invasions

Population models that combine demography and dispersal are important tools for forecasting the spatial spread of biological invasions. Current models describe the dynamics of only one sex (typically females). Such models cannot account for the sex-related biases in dispersal and mating behavior that are typical of many animal species. In this article, we construct a two-sex integrodifference equation model that overcomes these limitations. We derive an explicit formula for the invasion speed from the model and use it to show that sex-biased dispersal may significantly increase or decrease the invasion speed by skewing the operational sex ratio at the invasion's low-density leading edge. Which of these possible outcomes occurs depends sensitively on complex interactions among the direction of dispersal bias, the magnitude of bias, and the relative contributions of females and males to local population growth.     

Multi-scale methods predict invasion speeds in variable landscapes

Spread rates of invasive plant species depend heavily on variable seed/seedling survivorships over various habitat types as well as on variability in seed dispersal induced by rapid transport of propagules in open areas and slow transport in vegetated areas. The ability to capture spatial variability in seed survivorship and dispersal is crucial to accurately predict the rate of spread of plants in real world landscapes. However, current analytic methods for predicting spread rates are not suited for arbitrary, spatially heterogeneous systems. Here, we analyze invasion rates of the invasive plant Phragmites australis (common reed) over variable wetland landscapes. Phragmites is one of the most pervasive perennial grasses, outcompeting native vegetation, providing poor wildlife habitat, and proving difficult to eradicate across its invasive range in North America. Phragmites spreads sexually via seeds and asexually via underground (rhizomes) and aboveground (stolons) stems. We construct a structured integrodifference equation model of the Phragmites life cycle capturing variable seed survivorship in a seed bank, sexual and asexual recruitment into a juvenile age class, and differential competition among all classes with adults. The demographic model is coupled with a homogenized ecological diffusion/settling seed dispersal model that allows for seed deposition that varies with habitat type. The dispersal kernel we develop does not require local normalization and can be implemented efficiently using standard computational techniques. The model generates a traveling wave of isolated patches, establishing only in suitable habitats. We use the method of multiple scales to predict invasion speed as a solvability condition at large scales and test the predictions numerically. Accurate predictions are generated for a wide range of landscape parameters, indicating that invasion speeds can be understood in landscapes of arbitrary structure using this approach.     

A spatial model for the spread of invading organisms subject to competition

 A spatially explicit integrodifference equation model is studied for the spread of an invading organism against an established competitor. Provided the invader is initially confined to a bounded region, the invasion spreads asymptotically as a travelling wave whose speed depends on the strength of the competitive interaction and on the dispersal characteristics of the invader. Even an inferior, but established, competitor can significantly reduce the invasion speed. The invasion speed is also influenced by the exact shape of the dispersal kernel (especially the thickness of the tail) as well as the mean dispersal distance for each generation. Received 10 April 1996; received in revised form 21 August 1996     

The genetic signature of rapid range expansions: How dispersal,growth and invasion speed impact heterozygosity and allele surfing

As researchers collect spatiotemporal population and genetic data in tandem, models that connect demography and dispersal to genetics are increasingly relevant. The dominant spatiotemporal model of invasion genetics is the stepping-stone model which represents a gradual range expansion in which individuals jump to uncolonized locations one step at a time. However, many range expansions occur quickly as individuals disperse far from currently colonized regions. For these types of expansion, stepping-stone models are inappropriate. To more accurately reflect wider dispersal in many organisms, we created kernel-based models of invasion genetics based on integrodifference equations. Classic theory relating to integrodifference equations suggests that the speed of range expansions is a function of population growth and dispersal. In our simulations, populations that expanded at the same speed but with spread rates driven by dispersal retained more heterozygosity along axes of expansion than range expansions with rates of spread that were driven primarily by population growth. To investigate surfing we introduced mutant alleles in wave fronts of simulated range expansions. In our models based on random mating, surfing alleles remained at relatively low frequencies and surfed less often compared to previous results based on stepping-stone simulations with asexual reproduction.     

A modelling approach to estimate the effect of exotic pollinators on exotic weed population dynamics: bumblebees and broom in Australia

The role of mutualisms in contributing to species invasions is rarely considered, inhibiting effective risk analysis and management options. Potential ecological consequences of invasion of non‐native pollinators include increased pollination and seed set of invasive plants, with subsequent impacts on population growth rates and rates of spread. We outline a quantitative approach for evaluating the impact of a proposed introduction of an invasive pollinator on existing weed population dynamics and demonstrate the use of this approach on a relatively data‐rich case study: the impacts on Cytisus scoparius (Scotch broom) from proposed introduction of Bombus terrestris. Three models have been used to assess population growth (matrix model), spread speed (integrodifference equation), and equilibrium occupancy (lattice model) for C. scoparius. We use available demographic data for an Australian population to parameterize two of these models. Increased seed set due to more efficient pollination resulted in a higher population growth rate in the density‐independent matrix model, whereas simulations of enhanced pollination scenarios had a negligible effect on equilibrium weed occupancy in the lattice model. This is attributed to strong microsite limitation of recruitment in invasive C. scoparius populations observed in Australia and incorporated in the lattice model. A lack of information regarding secondary ant dispersal of C. scoparius prevents us from parameterizing the integrodifference equation model for Australia, but studies of invasive populations in California suggest that spread speed will also increase with higher seed set. For microsite‐limited C. scoparius populations, increased seed set has minimal effects on equilibrium site occupancy. However, for density‐independent rapidly invading populations, increased seed set is likely to lead to higher growth rates and spread speeds. The impacts of introduced pollinators on native flora and fauna and the potential for promoting range expansion in pollinator‐limited ‘sleeper weeds’ also remain substantial risks.     

Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

Reproduction-Dispersal equations, called reaction-diffusion equations in the physics literature, model the growth and spreading of biological species. Integro-Difference equations were introduced to address the shortcomings of this model, since the dispersal of invasive species is often more widespread than what the classical RD model predicts. In this paper, we extend the RD model, replacing the classical second derivative dispersal term by a fractional derivative of order 1相似文献   

Seed dispersal curves: Behavior of the tail of the distribution

Summary Seed dispersal is important both to plant fitness and to plant population structure. We suggest that the tail of the seed dispersal curve is at least as important biologically as the modal portion of the curve, and we present a relatively simple, four-parameter model, based on diffusion principles, for the tail of the seed distribution. This model includes two types of qualitative behavior: algebraic tails (which tend to be longer and have greater reach) and exponential tails (which are shorter and have less reach). We have selected 68 data sets from the literature, each giving a seed shadow that could be categorized statistically as (1) exponential, (2) algebraic, (3) neither, or (4) both models fit adequately. Algebraic shapes for seed-shadow tails were common in this sample, and tail behavior was not generally specific to a particular dispersal mode. This result may suggest that algebraic tails are generally favored by selection and can be achieved by several means, but limitations of existing data sets and of statistical methodology preclude final judgement. Smaller complete samples of seed distances would provide a better basis for the analysis of tails than do the present form of data sets (consisting of counts of seeds in discrete distance categories).     

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.     

Assessment of spatial discordance of primary and effective seed dispersal of European beech (Fagus sylvatica L.) by ecological and genetic methods

Spatial discordance between primary and effective dispersal in plant populations indicates that postdispersal processes erase the seed rain signal in recruitment patterns. Five different models were used to test the spatial concordance of the primary and effective dispersal patterns in a European beech (Fagus sylvatica) population from central Spain. An ecological method was based on classical inverse modelling (SSS), using the number of seed/seedlings as input data. Genetic models were based on direct kernel fitting of mother‐to‐offspring distances estimated by a parentage analysis or were spatially explicit models based on the genotype frequencies of offspring (competing sources model and Moran‐Clark's Model). A fully integrated mixed model was based on inverse modelling, but used the number of genotypes as input data (gene shadow model). The potential sources of error and limitations of each seed dispersal estimation method are discussed. The mean dispersal distances for seeds and saplings estimated with these five methods were higher than those obtained by previous estimations for European beech forests. All the methods show strong discordance between primary and effective dispersal kernel parameters, and for dispersal directionality. While seed rain was released mostly under the canopy, saplings were established far from mother trees. This discordant pattern may be the result of the action of secondary dispersal by animals or density‐dependent effects; that is, the Janzen‐Connell effect.     

A mathematical model for weed dispersal and control

Mathematical models for weed dispersal and control are developed, analyzed and numerically simulated. A model incorporating periodic control, e.g. herbicide application, is derived for a plant population in a spatially homogeneous setting. The model is extended to a spatially heterogeneous population where plant dispersal is incorporated. The dispersal and control model involves integrodifference equations, discrete in time and continuous in space. The models are analyzed to determine values of the control parameter that prevent weed spread. The effects of the control on travelling wave solutions are investigated numerically.     

The Reaction of the Medium in Relation to Root Formation in Coleus

Background: As seed dispersal can vary among years and individuals, studies that focus on a single year or on a few individuals may lead to erroneous conclusions.

Aims: To study temporal and spatial intraspecific variation of seed dispersal in Scrophularia canina, a widespread species with capsule-type fruit.

Methods: Primary seed dispersal was quantified by placing traps in each cardinal direction around 10 individuals during two consecutive years. We correlated several seed shadow parameters (modal dispersal distance, kurtosis, skewness, percentiles, slope, and seed percentage beneath the plant canopy) with three plant features (maximum height, lateral spread and seed production).

Results: Scrophularia canina dispersed their seeds by boleochory, giving rise to a typical leptokurtic curve, but behaving as a barochorous species, because about 90% of seeds landed beneath the plant canopy. Temporal dispersal in S. canina included several seed waves associated with maximum wind speeds. Plant lateral spread was significantly positively correlated with seed percentiles and percentage of seeds beneath the plant canopy regardless of year. A seed production effect was only evident when both years were considered together.

Conclusions: Although time-consuming, investigation of the dispersal process for more than 1 year provides more realistic information on seed dispersal. Lateral spread is the main plant feature determining seed shadow.     

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.