Integrodifference Equations Research Articles

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.

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<alpha</=2. Fractional derivative models are used in physics to model anomalous super-diffusion, where a cloud of particles spreads faster than the classical diffusion model predicts. This paper also establishes a connection between the new RD model and a corresponding ID equation with a heavy tail dispersal kernel. The general theory developed here accommodates a wide variety of infinitely divisible dispersal kernels that adapt to any scale. Each one corresponds to a generalised RD model with a different dispersal operator. The connection established here between RD and ID equations can also be exploited to generate convergent numerical solutions of RD equations along with explicit error bounds.

Harvesting control in an integrodifference population model with concave growth term

We consider the harvest of a certain proportion of a population that is modeled by an integrodifference equation, which is discrete in time and continuous in the space variable. The dispersal of the population is modeled by an integral of the growth function evaluated at the current population density against a kernel function. A concave growth function is used. In our model, growth occurs first, then dispersal and lastly harvesting control before the next generation. With the goal of maximizing the discounted profit stream, the optimal control is characterized by an optimality system. Illustrative examples are computed numerically.

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.

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, α , we examined how well these stage-structured IDEs related small scale demographic and dispersal processes with geographic population expansion. The parameter α was estimated by fitting the kernels to dispersal data and by fitting the IDE model to range expansion data. For all kernels, the α estimate from range expansion data fell within the 95% confidence intervals of the α 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.

Backscattering of stationary radiation in time-dependent random media: average intensity and intensity fluctuations

We consider backscattering of stationary radiation in a random medium whose wavespeed fluctuations depend on time and on space. We modify a previous derivation of the equations that govern the range-evolution of the spectra of the ensemble-averaged forward-and back-propagating components of the field and their second-order statistics, and extend the approach to treat the fourth-order statistics. The latter are governed by integro-difference equations that account for the broadening of the signal spectra due to the time-dependence of the random fluctuations. In the quasi-monochromatic regime, where spectra owing to a monochromatic excitation remain confined to a narrow band over extensive ranges, the integro-difference equations transform into ordinary differential equations that govern the time-dependence of the quantities of interest. We use this simplification to track the power fluxes and their fluctuations (scintillation) in a one-dimensionally stratified slab, where the wave-speed fluctuations depend on the range-coordinate normal to the planes of stratification, and also to treat modal propagation in a duct, where the wave-speed fluctuations depend on all three spatial dimensions. The results suggest that a Gaussian equilibrium is approached at large ranges, on a suitably defined backscattering scale that depends on the medium parameters and the geometry.

On the Estimation of Dispersal Kernels from Individual Mark-Recapture Data

We present a new method for estimating a distribution of dispersal displacements (a dispersal kernel) from mark-recapture data. One conventional method of calculating the dispersal kernel assumes that the distribution of displacements are Gaussian (e.g. resulting from a diffusion process) and that individuals remain within sampled areas. The first assumption prohibits an analysis of dispersal data that do not exhibit the Gaussian distribution (a common situation); the second assumption leads to underestimation of dispersal distance because individuals that disperse outside of sampling areas are never recaptured. Our method eliminates these two assumptions. In addition, the method can also accommodate mortality during a sampling period. This new method uses integrodifference equations to express the probability of spatial mark-recapture data; associated dispersal, survival, and recapture parameters are then estimated using a maximum likelihood method. We examined the accuracy of the estimators by applying the method to simulated data sets. Our method suggests designs for future mark-recapture experiments.

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

ABSTRACTThe 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.

A Kernel-Based Spatio-Temporal Dynamical Model for Nowcasting Weather Radar Reflectivities

A good short-period forecast of heavy rainfall is essential for many meteorological and hydrological applications. Traditional deterministic and stochastic nowcasting methodologies have been inadequate in their characterization of pixelwise rainfall reflectivity propagation, intensity, and uncertainty. The methodology presented herein uses an approach that efficiently parameterizes spatio-temporal dynamic models in terms of integro-difference equations within a hierarchical framework. The approach accounts for the uncertainty in the prediction and provides relevant distributional information concerning the nowcast. An application is presented that shows the effectiveness of the technique and its potential for nowcasting weather radar reflectivities.

When Can Herbivores Slow or Reverse the Spread of an Invading Plant? A Test Case from Mount St. Helens

Here we study the spatial dynamics of a coinvading consumer-resource pair. We present a theoretical treatment with extensive empirical data from a long-studied field system in which native herbivorous insects attack a population of lupine plants recolonizing a primary successional landscape created by the 1980 volcanic eruption of Mount St. Helens. Using detailed data on the life history and interaction strengths of the lupine and one of its herbivores, we develop a system of integrodifference equations to study plant-herbivore invasion dynamics. Our analyses yield several new insights into the spatial dynamics of coinvasions. In particular, we demonstrate that aspects of plant population growth and the intensity of herbivory under low-density conditions can determine whether the plant population spreads across a landscape or is prevented from doing so by the herbivore. In addition, we characterize the existence of threshold levels of spatial extent and/or temporal advantage for the plant that together define critical values of "invasion momentum," beyond which herbivores are unable to reverse a plant invasion. We conclude by discussing the implications of our findings for successional dynamics and the use of biological control agents to limit the spread of pest species.

Computing the Continuous-spectrum Linearised Bounded Standing Wave on a Plane Bed of Arbitrary Slope

The problem is revisited of continuous-spectrum small-amplitude bounded standing waves over a plane beach of arbitrary incline. Three documented approaches are compared and discussed vis-??-vis computational difficulties and this reveals, in particular, that the computation in established models, for particular beach angles, is optimised by seeking solutions to a functional integro-difference equation in a L2 space. This simplification is found to be impossible for general beach angles and developed instead is an alternative method initially based on Cauchy Principal value convergence of the inversion integral but improved (by linear combination of solutions) and developed into a comprehensive computational package which is validated by three fundamental tests. A single f77 routine is provided for the user through a web-link. Comparisons with established but hitherto un-computed methods are found numerically to favour the present development. Examples presented include both very steep and very shallow beaches and both potentials and velocities are calculated and displayed throughout the water column.

A Taylor collocation method for the approximate solution of general linear Fredholm–Volterra integro-difference equations with mixed argument

In this paper, a Taylor collocation method is developed to find an approximate solution of high-order linear Fredholm–Volterra integro-difference equations with variable coefficients under the mixed conditions. The solution is obtained in terms of Taylor polynomial near any point. Also, examples are presented which illustrate the partinent features of the method and the results are discussed.

Optimal harvesting in an integrodifference population model

AbstractWe consider the harvest of a certain proportion of a population that is modelled by an integrodifference equation. This model is discrete in time and continuous in the space variable. The dispersal of the population is modelled by an integral of the population density against a kernel function. The control is the harvest, and the goal is to maximize the profit. The optimal control is characterized by introducing an adjoint function. This paper gives the first optimal control result in integrodifference equations. Numerical results and interpretations are given for four different kernels. Copyright © 2005 John Wiley &amp; Sons, Ltd.

A mathematical model for dispersal of an annual plant population with a seed bank

In order to describe the spatial dispersion of a population of annual plants with a seed bank, we have formulated a delay integrodifference equation model. The plant population growth is considered as being dependent on the density of seedlings that spring from seeds of the two previous years. We have analyzed the effects of the seed bank on the total population size, on the population spatial distribution as well as on the population velocity of invasion. It is showed that the seed bank can significantly alter the dynamics of annual plants, since it can have both a stabilizing and a destabilizing effect on the total population and can also homogenize the spatial distribution of the plants.

Relating coupled map lattices to integro-difference equations: dispersal-driven instabilities in coupled map lattices

Recently there has been a great deal of interest within the ecological community about the interactions of local populations that are coupled only by dispersal. Models have been developed to consider such scenarios but the theory needed to validate model outcomes has been somewhat lacking. In this paper, we present theory which can be used to understand these types of interaction when population exhibit discrete time dynamics. In particular, we consider a spatial extension to discrete-time models, known as coupled map lattices (CMLs) which are discrete in space. We introduce a general form of the CML and link this to integro-difference equations via a special redistribution kernel. General conditions are then derived for dispersal-driven instabilities. We then apply this theory to two discrete-time models; a predator–prey model and a host–pathogen model.

A new polynomial approach for solving difference and Fredholm integro-difference equations with mixed argument

The purpose of this study is to give a Taylor polynomial approximation for the solution of high-order linear Fredholm difference equations with variable coefficients and mixed argument under the mixed conditions about any point. For this purpose, a Taylor matrix method is introduced. This method is based on first taking the truncated Taylor expansions of the functions in the difference equation and then substituting their matrix forms into the given equation. Hence the result matrix equation can be solved and the unknown Taylor coefficients can be found approximately. Also, examples that illustrate the pertinent features of the method are presented and the results are discussed.

When Can Herbivores Slow or Reverse the Spread of an Invading Plant? A Test Case from Mount St. Helens

Here we study the spatial dynamics of a coinvading consumer‐resource pair. We present a theoretical treatment with extensive empirical data from a long‐studied field system in which native herbivorous insects attack a population of lupine plants recolonizing a primary successional landscape created by the 1980 volcanic eruption of Mount St. Helens. Using detailed data on the life history and interaction strengths of the lupine and one of its herbivores, we develop a system of integrodifference equations to study plant‐herbivore invasion dynamics. Our analyses yield several new insights into the spatial dynamics of coinvasions. In particular, we demonstrate that aspects of plant population growth and the intensity of herbivory under low‐density conditions can determine whether the plant population spreads across a landscape or is prevented from doing so by the herbivore. In addition, we characterize the existence of threshold levels of spatial extent and/or temporal advantage for the plant that together define critical values of “invasion momentum,” beyond which herbivores are unable to reverse a plant invasion. We conclude by discussing the implications of our findings for successional dynamics and the use of biological control agents to limit the spread of pest species.

Stochasticity, invasions, and branching random walks

We link deterministic integrodifference equations to stochastic, individual-based simulations by means of branching random walks. Using standard methods, we determine speeds of invasion for both average densities and furthest-forward individuals. For density-independent branching random walks, demographic stochasticity can produce extinction. Demographic stochasticity does not, however, reduce the overall asymptotic speed of invasion or preclude continually accelerating invasions.

Projecting Rates of Spread for Invasive Species

All else being equal, the faster an invading species spreads, the more dangerous its invasion. The projection of spread rate therefore ought to be a central part of the determination of invasion risk. Originally formulated in the 1970s to describe the spatial spread of advantageous alleles, integrodifference equation (IDE) models have since been co-opted by population biologists to describe the spread of populations. More recently, they have been modified to include population structure and environmental variability. We review how IDE models are formulated, how they are parameterized, and how they can be analyzed to project spread rates and the sensitivity of those rates to changes in model parameters. For illustrative purposes, we apply these models to Cytisus scoparius, a large shrub in the legume family that is considered a noxious invasive species in eastern and western North America, Chile, Australia, and New Zealand.

Spatially-explicit matrix models. A mathematical analysis of stage-structured integrodifference equations.

This paper is concerned with mathematical analysis of the 'critical domain-size' problem for structured populations. Space is introduced explicitly into matrix models for stage-structured populations. Movement of individuals is described by means of a dispersal kernel. The mathematical analysis investigates conditions for existence, stability and uniqueness of equilibrium solutions as well as some bifurcation behaviors. These mathematical results are linked to species persistence or extinction in connected habitats of different sizes or fragmented habitats; hence the framework is given for application of such models to ecology. Several approximations which reduce the complexity of integrodifference equations are given. A simple example is worked out to illustrate the analytical results and to compare the behavior of the integrodifference model to that of the approximations.