Integrodifference Equations Research Articles

Analysis of Integrodifference Equations with a Separable Dispersal Kernel

Integrodifference equations are a class of infinite-dimensional dynamical systems in discrete time that have recently received great attention as mathematical models of population dynamics in spatial ecology. The dispersal of individuals between generations is described by a ‘dispersal kernel’, a probability density function for the distance that an individual moves within a season. Previous authors recognized that the dynamics are reduced to a finite-dimensional problem when the dispersal kernel is separable. We prove some open questions from their work on the dynamics of a single population and then extend the idea to investigate the dynamics of two spatially distributed species in (i) a competitive relation, and (ii) a predator-prey relation. In all cases, we discuss how the dynamics of the population(s) depend on the amount of suitable space that is available to them. We find a number of bifurcations, such as period-doubling sequences and Naimark-Sacker bifurcations, which we illustrate through simulations.

On nonlocal Robin boundary value problems for Riemann\u2013Liouville fractional Hahn integrodifference equation

In this paper, we study a nonlocal Robin boundary value problem for fractional Hahn integrodifference equation. Our problem contains three fractional Hahn difference operators and a fractional Hahn integral with different numbers of q, omega and order. The existence and uniqueness result is proved by using the Banach fixed point theorem. In addition, the existence of at least one solution is obtained by using Schauder’s fixed point theorem.

Travelling waves and paradoxical effects in a discrete-time growth-dispersal model

• The Beverton–Holt model with pulse control within each generation has been developed. • The traveling wave, speed and shape have been studied. • The effects of efficacy and timing of pesticide application on dynamics are studied. • The influence of dynamic complexity on pest control and spread are analyzed. • Threshold conditions for occurrence of paradoxical effects have been discussed. Integrodifference equations have been widely used to describe the spatiotemporal dynamics of a population. Here, we first extended the discrete Beverton–Holt model, to model the effects of an instantaneous control measure within each generation on a population’s growth. Then we investigated the effects of instantaneous killing rate and timing of pesticide application on a pest’s dynamics, together with the effects of a spatial factor on the existence of a travelling wave solution, its spreading speed and asymptotic stability, and the occurrence of paradoxical effects. The main results indicate that the instantaneous killing rate significantly influences the existence of a travelling wave and its speed of spread, while the timing of pesticide applications does not. In order to address how such dynamic complexities affect the above results, we included an interruption constant into the model which can generate chaotic dynamics. The results indicate that the pesticide efficacy not only affects the existence of a travelling wave and its speed of spread, but it can also produce paradoxical effects, i.e. low pesticide efficacy can result in an increase, not a decrease, of the stable population size, while high pesticide efficacy could inhibit the pests growth. Furthermore, different patterns of stable population sizes between odd and even generations can be produced. The results reveal that the pest control tactics should be designed according to the spatial characteristics and the pest generation.

Integrodifference equations in the presence of climate change: persistence criterion, travelling waves and inside dynamics.

To understand the effects that the climate change has on the evolution of species as well as the genetic consequences, we analyze an integrodifference equation (IDE) models for a reproducing and dispersing population in a spatio-temporal heterogeneous environment described by a shifting climate envelope. Our analysis on the IDE focuses on the persistence criterion, travelling wave solutions, and the inside dynamics. First, the persistence criterion, characterizing the global dynamics of the IDE, is established in terms of the basic reproduction number. In the case of persistence, a unique travelling wave is found to govern the global dynamics. The effects of the size and the shifting speed of the climate envelope on the basic reproduction number, and hence, on the persistence criterion, are also investigated. In particular, the critical domain size and the critical shifting speed are found in certain cases. Numerical simulations are performed to complement the theoretical results. In the case of persistence, we separate the travelling wave and general solutions into spatially distinct neutral fractions to study the inside dynamics. It is shown that each neutral genetic fraction rearranges itself spatially so as to asymptotically achieve the profile of the travelling wave. To measure the genetic diversity of the population density we calculate the Shannon diversity index and related indices, and use these to illustrate how diversity changes with underlying parameters.

On nonlocal Dirichlet boundary value problem for sequential Caputo fractional Hahn integrodifference equations

The aim of this work is to study a nonlocal Dirichlet boundary value problem for sequential Caputo fractional Hahn integrodifference equation. The problem contains two fractional Hahn difference operators and a fractional Hahn integral with different numbers of order. We use the Banach fixed point theorem to prove the existence and uniqueness of the solution. In particular, the existence of at least one solution is presented by using the Schauder fixed point theorem.

Spreading Phenomena in Integrodifference Equations with Nonmonotone Growth Functions

Integrodifference equations are discrete-time cousins of reaction-diffusion equations. Like their continuous-time counterparts, they are used to model spreading phenomena in ecology and other sciences. Unlike their continuous-time counterparts, even scalar integrodifference equations can exhibit nonmonotone dynamics. Few authors studied the existence of spreading speeds and traveling waves in the nonmonotone case; previous numerical simulations indicated the existence of traveling two-cycles. Our numerical observations indicate the presence of several spreading speeds and multiple traveling wave profiles in these equations. We generalize the concept of a spreading speed to encompass this situation and prove the existence of such generalized spreading speeds and associated traveling waves in the corresponding second-iterate operator. Our numerical simulations let us conjecture that these spreading speeds could be linearly determined. We prove the existence of bistable traveling waves in a related second-iterate operator. We relate our results to the existence of stacked waves and to dynamical stabilization.

Gaussian Two-Armed Bandit and Optimization of Batch Data Processing

We consider the minimax setting for the two-armed bandit problem with normally distributed incomes having a priori unknown mathematical expectations and variances. This setting naturally arises in optimization of batch data processing where two alternative processing methods are available with different a priori unknown efficiencies. During the control process, it is required to determine the most efficient method and ensure its predominant application. We use the main theorem of game theory to search for minimax strategy and minimax risk as Bayesian ones corresponding to the worst-case prior distribution. To find them, a recursive integro-difference equation is obtained. We show that batch data processing almost does not increase the minimax risk if the number of batches is large enough.

The dispersal success and persistence of populations with asymmetric dispersal

Asymmetric dispersal is a common trait among populations, often attributed to heterogeneity and stochasticity in both environment and demography. The cumulative effects of population dispersal in space and time have been described with some success by Van Kirk and Lewis’s average dispersal success approximation (Bull Math Biol 59(1): 107–137 1997), but this approximation has been demonstrated to perform poorly when applied to asymmetric dispersal. Here we provide a comparison of different characterizations of dispersal success and demonstrate how to capture the effects of asymmetric dispersal. We apply these different methods to a variety of integrodifference equation population models with asymmetric dispersal, and examine the methods’ effectiveness in approximating key ecological traits of the models, such as the critical patch size and the critical speed of climate change for population persistence.

Stochastic dispersal increases the rate of upstream spread: A case study with green crabs on the northwest Atlantic coast.

Dispersal heterogeneity is an important process that can compensate for downstream advection, enabling aquatic organisms to persist or spread upstream. Our main focus was the effect of year-to-year variation in larval dispersal on invasion spread rate. We used the green crab, Carcinus maenas, as a case study. This species was first introduced over 200 years ago to the east coast of North America, and once established has maintained a relatively consistent spread rate against the dominant current. We used a stage-structured, integro-difference equation model that couples a demographic matrix for population growth and dispersal kernels for spread of individuals within a season. The kernel describing larval dispersal, the main dispersive stage, was mechanistically modeled to include both drift and settlement rate components. It was parameterized using a 3-dimensional hydrodynamic model of the Gulf of St Lawrence, which enabled us to incorporate larval behavior, namely vertical swimming. Dispersal heterogeneity was modeled at two temporal scales: within the larval period (months) and over the adult lifespan (years). The kernel models variation within the larval period. To model the variation among years, we allowed the kernel parameters to vary by year. Results indicated that when dispersal parameters vary with time, knowledge of the time-averaged dispersal process is insufficient for determining the upstream spread rate of the population. Rather upstream spread is possible over a number of years when incorporating the yearly variation, even when there are only a few “good years” featured by some upstream dispersal among many “bad years” featured by only downstream dispersal. Accounting for annual variations in dispersal in population models is important to enhance understanding of spatial dynamics and population spread rates. Our developed model also provides a good platform to link the modeling of larval behavior and demography to large-scale hydrodynamic models.

Nonlocal boundary value problems for second-order nonlinear Hahn integro-difference equations with integral boundary conditions

In this paper, we study a boundary value problem for second-order nonlinear Hahn integro-difference equations with nonlocal integral boundary conditions. Our problem contains two Hahn difference operators and a Hahn integral. The existence and uniqueness of solutions is obtained by using the Banach fixed point theorem, and the existence of at least one solution is established by using the Leray-Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem. Illustrative examples are also presented to show the applicability of our results.

Sex difference and Allee effects shape the dynamics of sex‐structured invasions

The rate at which a population grows and spreads can depend on individual behaviour and interactions with others. In many species with two sexes, males and females differ in key life-history traits (e.g. growth, survival and dispersal), which can scale up to affect population rates of growth and spread. In sexually reproducing species, the mechanics of locating mates and reproducing successfully introduce further complications for predicting the invasion speed (spread rate), as both can change nonlinearly with density. Most models of population spread are based on one sex, or include limited aspects of sex differences. Here we ask whether and how the dynamics of finding mates interact with sex-specific life-history traits to influence the rate of population spread. We present a hybrid approach for modelling invasions of populations with two sexes that links individual-level mating behaviour (in an individual-based model) to population-level dynamics (in an integrodifference equationmodel). We find that limiting the amount of time during which individuals can search for mates causes a demographic Allee effect which can slow, delay, or even prevent an invasion. Furthermore, any sex-based asymmetries in life history or behaviour (skewed sex ratio, sex-biased dispersal, and sex-specific mating behaviours) amplify these effects. In contrast, allowing individuals to mate more than once ameliorates these effects, enabling polygynandrous populations to invade under conditions where monogamously mating populations would fail to establish. We show that details of individuals' mating behaviour can impact the rate of population spread. Based on our results, we propose a stricter definition of a mate-finding Allee effect, which is not met by the commonly used minimum mating function. Our modelling approach, which links individual- and population-level dynamics in a single model, may be useful for exploring other aspects of individual behaviour that are thought to impact the rate of population spread.

Speeding up the simulation of population spread models

Summary Simulating spatially explicit population models to predict population spread allows environmental managers to make better‐informed decisions. Accurate simulation requires high spatial resolution, which, using existing techniques, can require prohibitively large amounts of computational resources (RAM, CPU, etc). We developed and implemented a novel algorithm for the simulation of integro‐difference equations (IDEs) modelling population spread, including stage structure, which uses adaptive mesh refinement. We measured the accuracy of the adaptive algorithm by comparing the results of simulations using the adaptive and a standard non‐adaptive algorithm. The relative error of the population's spatial extent was low (<0·05) for a range of parameter values. Comparing efficiency, we found that our algorithm used up to 10 times less CPU time and RAM than the non‐adaptive algorithm. Our approach provides large improvements in efficiency without significant loss of accuracy, so it enables faster simulation of IDEs and simulation at scales and at resolutions that have not been previously feasible. As an example, we simulate the spread of a hypothetical species over the UK at a resolution of 25 m. We provide our implementation of the algorithm as a user‐friendly executable application.

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.

Barnacles vs bullies: modelling biocontrol of the invasive European green crab using a castrating barnacle parasite

Invasive species raise concern around the globe, and much empirical and theoretical research effort has been devoted to their management. Integrodifference equations are theoretical tools that have been used to understand the spatiotemporal process of a species invasion, with the potential to yield insight into the possible biological control measures. We develop a system of integrodifference equations to explore the potential release of a castrating barnacle parasite Sacculina carcini to control spread and abundance of an invasive species, Carcinus maenas, the European green crab. We find that the parasite does not completely eradicate the green crab population, but has the potential to reduce its density. Our model suggests that the crab population is likely to outrun the spread of the parasite, causing two waves of invasion travelling at different speeds. By performing a sensitivity analysis, we investigate the effects of the demographic parameters on the speed of invasion. To conclude, we discuss the predicted outcomes for the European green crab, and other non-target hosts, of using the castrating barnacle as a biocontrol agent.

Moving forward: insights and applications of moving‐habitat models for climate change ecology

Summary Predicting and managing species’ responses to climate change is one of the most significant challenges of our time. Tools are needed to address problems associated with novel climatic conditions, biotic interactions and greater climate velocities. We present a spatially explicit moving‐habitat model (MHM) and demonstrate its versatility in tackling critical questions in climate change research, including dispersal in multiple spatial dimensions, population stage structure, interspecific interactions, asymmetric range shifts, Allee effects and the presence of infectious diseases. The model utilizes integrodifference equations to track changes in population density over time in a habitat that is moving. The model is quite flexible and can accommodate variation in demography, dispersal patterns, biotic interaction and stochasticity in the velocity of climate change. The methods provide a general mechanistic understanding of the underlying ecological processes that drive a system. Field data can be readily incorporated into the model to give insight into specific populations of interest and inform management decisions. Synthesis. Moving‐habitat models unite ecological theory, data‐centred modelling and conservation decision support under a single framework. Their ability to generate testable hypotheses, incorporate data and inform best management practices proves that these models provide a valuable framework for climate change biologists.

Analytic Approximation of Invasion Wave Amplitude Predicts Severity of Insect Outbreaks

Outbreaks of phytophagous forest insects are largely driven by host demographics and spatial effects of dispersal. We develop a structured integrodifference equation (IDE) outbreak model that tracks the demographics of sedentary hosts under insect infestation pressure. The model is appropriate for a spectrum of pests attacking the later age classes of long-lived hosts, including mountain pine beetle (MPB), spruce budworm, and spruce beetle, which, among them are responsible for more forest damage than fire. The model generates a train of periodic waves of infestation. We approximate the IDE with a partial differential equation and search for traveling wave solutions. The resulting ordinary differential equation predicts the shape of an outbreak wave profile and peak infestation as functions of wavefront speed, which can be calculated analytically. This culminates in the derivation of an explicit approximation of invasion wave amplitude based on net reproductive rate of the infesting insect and its host searching efficiency. Results are compared with observations taken during a recent MPB outbreak in the northern US Rocky Mountains.

Existence Results for Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems

The existence and uniqueness results of two fractional Hahn difference boundary value problems are studied. The first problem is a Riemann-Liouville fractional Hahn difference boundary value problem for fractional Hahn integrodifference equations. The second is a fractional Hahn integral boundary value problem for Caputo fractional Hahn difference equations. The Banach fixed-point theorem and the Schauder fixed-point theorem are used as tools to prove the existence and uniqueness of solution of the problems.

Bayesian non-parametric modeling for integro-difference equations

Integro-difference equations (IDEs) provide a flexible framework for dynamic modeling of spatio-temporal data. The choice of kernel in an IDE model relates directly to the underlying physical process modeled, and it can affect model fit and predictive accuracy. We introduce Bayesian non-parametric methods to the IDE literature as a means to allow flexibility in modeling the kernel. We propose a mixture of normal distributions for the IDE kernel, built from a spatial Dirichlet process for the mixing distribution, which can model kernels with shapes that change with location. This allows the IDE model to capture non-stationarity with respect to location and to reflect a changing physical process across the domain. We address computational concerns for inference that leverage the use of Hermite polynomials as a basis for the representation of the process and the IDE kernel, and incorporate Hamiltonian Markov chain Monte Carlo steps in the posterior simulation method. An example with synthetic data demonstrates that the model can successfully capture location-dependent dynamics. Moreover, using a data set of ozone pressure, we show that the spatial Dirichlet process mixture model outperforms several alternative models for the IDE kernel, including the state of the art in the IDE literature, that is, a Gaussian kernel with location-dependent parameters.

Flexible integro-difference equation modeling for spatio-temporal data

The choice of kernel in an integro-difference equation (IDE) approach to model spatio-temporal data is studied. By using approximations to stochastic partial differential equations, it is shown that higher order cumulants and tail behavior of the kernel affect how an IDE process evolves over time. The asymmetric Laplace and the family of stable distributions are presented as alternatives to the Gaussian kernel. The asymmetric Laplace has an extra parameter controlling skewness, whereas the class of stable distributions includes parameters controlling both tail behavior and skewness. Simulations show that failing to account for kernel shape may lead to poor predictions from the model. For an illustration with real data, the IDE model with flexible kernels is applied to ozone pressure measurements collected biweekly by radiosonde at varying altitudes. The results obtained with the different kernel families are compared and it is confirmed that better model prediction may be achieved by electing to use a more flexible kernel.

Spatio-temporal dynamic model and parallelized ensemble Kalman filter for precipitation data

This paper presents a spatiotemporal dynamic model which allows Bayesian inference of precipitation states in some Venezuelan meteorological stations. One of the limitations that are reported in digital databases is the reliability of the records and the lack of information for certain days, weeks, or months. To complete the missing data, the Gibbs algorithm, a Markov Chain Monte Carlo (MCMC) procedure, was used. A feature of precipitation series is that their distribution contains discrete and continuous components implying complicated dynamics. A model is proposed based on a discrete representation of a stochastic integro-difference equation. Given the difficulty of obtaining explicit analytical expressions for the predictive posterior distribution, approximations were obtained using a sequential Monte Carlo algorithm called the parallelized ensemble Kalman filter. The proposed method permits the completion of the missing data in the series where required and secondly allows the splitting of a large database into smaller ones for separate evaluation and eventual combination of the individual results. The objective is to reduce the dimension and computational cost in order to obtain models that are able to describe the reality in real time. It was shown that the obtained models are able to predict spatially and temporally the states of rainfall for at least three to four days quickly, efficiently and accurately. Three methods of statistical validation were used to evaluate the performance of the model and showed no significant discrepancies. Speedup and efficiency factors were calculated to compare the speed of calculation using the parallelized ensemble Kalman filter algorithm with the speed of the sequential version. The improvement in speed for four pthread executions was greatest.