Integrodifference Equation Model Research Articles

Spread and persistence for integro-difference equations with shifting habitat and strong Allee effect.

We study an integro-difference equation model that describes the spatial dynamics of a species with a strong Allee effect in a shifting habitat. We examine the case of a shifting semi-infinite bad habitat connected to a semi-infinite good habitat. In this case we rigorously establish species persistence (non-persistence) if the habitat shift speed is less (greater) than the asymptotic spreading speed of the species in the good habitat. We also examine the case of a finite shifting patch of hospitable habitat, and find that the habitat shift speed must be less than the asymptotic spreading speed associated with the habitat and there is a critical patch size for species persistence. Spreading speeds and traveling waves are established to address species persistence. Our numerical simulations demonstrate the theoretical results and show the dependence of the critical patch size on the shift speed.

Introduction for special issue: integro-difference equation models in ecology

Introduction for special issue: integro-difference equation models in ecology

Evolution of dispersal by memory and learning in integrodifference equation models

In this paper, we develop an integrodifference equation model that incorporates spatial memory and learning so that each year, a fraction of the population use the same dispersal kernel as the previous year, and the remaining individuals return to where they bred or were born. In temporally static environments, the equilibrium of the system corresponds to an ideal free dispersal strategy, which is evolutionarily stable. We prove local stability of this equilibrium in a special case, and we observe convergence towards this equilibrium in numerical computations. When there are periodic or stochastic temporal changes in the environment, the population is less able to match the environment, but is able to do so to some extent depending on the parameters. Overall, the mechanism proposed in this model shows a possible way for the dispersal kernel of a population to evolve towards an ideal free dispersal kernel.

Take me for a ride: Herbivores can facilitate plant reinvasions.

Herbivores shape plant invasions through impacts on demography and dispersal, yet only demographic mechanisms are well understood. Although herbivores negatively impact demography by definition, they can affect dispersal either negatively (e.g., seed consumption), or positively (e.g., caching). Exploring the nuances of how herbivores influence spatial spread will improve the forecasting of plant movement on the landscape. Here, we aim to understand how herbivores impact how fast plant populations spread through varying impacts on plant demography and dispersal. We strive to determine whether, and under what conditions, we see net positive effects of herbivores, in order to find scenarios where herbivores can help to promote spread. We draw on classic invasion theory to develop a stage-structured integrodifference equation model that incorporates herbivore impacts on plant demography and dispersal. We simulate seven herbivore "syndromes" (combinations of demographic and/or dispersal effects) drawn from the literature to understand how increasing herbivore pressure alters plant spreading speed. We find that herbivores with solely negative effects on plant demography or dispersal always slow plant spreading speed, and that the speed slows monotonically as herbivore pressure increases. However, we also find that plant spreading speed can be hump shaped with respect to herbivore pressure: plants spread faster in the presence of herbivores (for low herbivore pressure) and then slower (for high herbivore pressure). This result is robust, occurring across all syndromes in which herbivores have a positive effect on plant dispersal, and is a sign that the positive effects of herbivores on dispersal can outweigh their negative effects on demography. For all syndromes we find that sufficiently high herbivore pressure results in population collapse. Thus, our findings show that herbivores can speed up or slow down plant spread. These insights allow for a greater understanding of how to slow invasions, facilitate native species recolonization, and shape range shifts with global change.

Ideal free dispersal in integrodifference models.

In this paper, we use an integrodifference equation model and pairwise invasion analysis to find what dispersal strategies are evolutionarily stable strategies (also known as evolutionarily steady or ESS) when there is spatial heterogeneity and possibly seasonal variation in habitat suitability. In that case there are both advantages and disadvantages of dispersing. We begin with the case where all spatial locations can support a viable population, and then consider the case where there are non-viable regions in the habitat. If the viable regions vary seasonally, and the viable regions in summer and winter do not overlap, dispersal may really be necessary for sustaining a population. Our findings generally align with previous findings in the literature that were based on other modeling frameworks, namely that dispersal strategies associated with ideal free distributions are evolutionarily stable. In the case where only part of the habitat can sustain a population, we show that a partial occupation ideal free distribution that occupies only the viable region is associated with a dispersal strategy that is evolutionarily stable. As in some previous works, the proofs of these results make use of properties of line sum symmetric functions, which are analogous to those of line sum symmetric matrices but applied to integral operators.

Propagation of invasive plant species in the presence of a road

Invasive plant species pose a significant threat to biodiversity and the economy, yet their management is often resource-intensive and expensive, and further research is required to make control measures more efficient. Evidence suggests that roads can have an important effect on the spread of invasive plant species, although little is known about the underlying mechanisms at play. We have developed a novel mathematical model to analyse the impact of roads on the propagation of invasive plants. The integro-difference equation model is formulated for stage-structured population and incorporates a road sub-domain in the spatial domain. The results of our study reveal, that, depending on the definition of the growth function in the model, there are three distinct types of behaviour in front of the road. Roads can act as barriers to invasion, lead to a formation of a beachhead in front of the road, or act as corridors allowing the invasive species to invade the domain in front of the road. Analytical and computational findings on how roads can impact the spread of invasive species show that a small change in conditions of the environment favouring the invasive species can change the case for the road, allowing the invasive species to invade the domain in front of the road where it previously could not spread.

Predicting spatially heterogeneous invasive spread: Pyracantha angustifolia invading a dry Andean valley in northern Argentina

Understanding the drivers of invasive species spread is key to designing optimal management programmes for controlling them. Population models, parameterized from demographic and dispersal data, are useful for simulating invasion when long-term observations are lacking, which is particularly important when invasions are occurring rapidly. We aimed to understand how the spread of Pyracantha angustifolia (an invasive ornamental shrub) is influenced by heterogeneity in local conditions within a dry inter-montane valley in northwestern Argentina that is currently experiencing rapid urban expansion and concomitant reduction in cattle farming. Field surveys demonstrated that individuals growing in shrublands have a higher fecundity than those in grassland and rocky habitats. Furthermore, grazing pressure of cattle and horses substantially reduces fecundity and impacts the relationship between age and fecundity. We incorporated our field-estimated habitat and age-specific fecundities into both analytical integro-difference equation models and individual-based models to predict rates of spread across each habitat type in the presence or absence of grazing pressure. Results indicate that the rate of spread would be substantially higher in shrublands (by up to 33%) and highlight that grazing pressure can, through direct consumption of seeds by livestock, substantially depress the rate of spread (by up to 53%). These results suggest that shrubland areas in the valley are most vulnerable to invasion and that grazing may help to reduce the impact of Pyracantha. This suggests that a synergism between urban expansion and reduction in domestic livestock density contributes to the spread of an invasive species in this environment.

Lagrangian Integro-Difference Equation Model for Precipitation Nowcasting

Abstract Delivering reliable nowcasts (short-range forecasts) of severe rainfall and the resulting flash floods is important in densely populated urban areas. The conventional method is advection-based extrapolation of radar echoes. However, during rapidly evolving convective rainfall this so-called Lagrangian persistence (LP) approach is limited to deterministic and very short-range nowcasts. To address these limitations in the 1-h time range, a novel extension of LP, called Lagrangian Integro-Difference equation model with Autoregression (LINDA), is proposed. The model consists of five components: 1) identification of rain cells, 2) advection, 3) autoregressive process describing growth and decay of the cells, 4) convolution describing loss of predictability at small scales, and 5) stochastic perturbations to simulate forecast uncertainty. Advection is separated from the other components that are applied in the Lagrangian coordinates. The reliability of LINDA is evaluated using the NEXRAD WSR-88D radar that covers the Dallas–Fort Worth metropolitan area, as well as the NEXRAD mosaic covering the continental United States. This is done with two different configurations: LINDA-D for deterministic and LINDA-P for probabilistic nowcasts. The validation dataset consists of 11 rainfall events during 2018–20. For predicting moderate to heavy rainfall (5–20 mm h−1), LINDA outperforms the previously proposed LP-based approaches. The most significant improvement is seen for the ETS and POD statistics with the 5 mm h−1 threshold. For 30-min nowcasts, they show 15% and 16% increases, respectively, to the second-best method and 48% and 34% increases compared to LP. For the 5 mm h−1 threshold, the increase in the relative operating characteristic (ROC) skill score of 30-min nowcasts from the second-best method is 10%. Significance Statement Delivering reliable forecasts of severe rainfall for the next few hours has a major societal importance. This is particularly true for densely populated urban areas, where flash floods can cause property damage and loss of lives. Such forecasts are conventionally produced by direct extrapolation of weather radar measurements. However, for intense localized rainfall this approach has low prediction ability beyond 30 min. To extend this limit, we propose a novel method that combines machine vision with a statistical model for growth and decay of rainfall. The method is designed for predicting highly localized rain cells and bands. In addition, a stochastic extension for producing probabilistic forecasts is developed. Using several verification metrics, we demonstrate that for predicting moderate to heavy rainfall (5–20 mm h−1), the proposed method has significantly improved forecast skill compared to the reference methods. The evaluation is done by using the NEXRAD WSR-88D that covers the Dallas–Fort Worth urban metroplex with a population of over 7 million. Demonstration of the applicability of LINDA in a larger domain is done by using the NEXRAD radar network that covers the continental United States.

Biphasic range expansions with short- and long-distance dispersal

Long-distance dispersal (LDD) has long been recognized as a key factor in determining rates of spread in biological invasions. Two approaches for incorporating LDD in mathematical models of spread are mixed dispersal and heavy-tailed dispersal. In this paper, I analyze integrodifference equation (IDE) models with mixed-dispersal kernels and fat-tailed (a subset of the heavy-tailed class) dispersal kernels to study how short- and long-distance dispersal contribute to the spread of invasive species. I show that both approaches can lead to biphasic range expansions, where an invasion has two distinct phases of spread. In the initial phase of spread, the invasion is controlled by short-distance dispersal. Long-distance dispersal boosts the speed of spread during the ultimate phase, and can have significant effects even when the probability of LDD is vanishingly small. For fat-tailed kernels, I introduce a method of characterizing the “shoulder” of a dispersal kernel, which separates the peak and tail.

Spatiotemporal Modelling using Integro-Difference Equations with Bivariate Stable Kernels

SummaryAn integro-difference equation can be represented as a hierarchical spatiotemporal dynamic model using appropriate parameterizations. The dynamics of the process defined by an integro-difference equation depends on the choice of a bivariate kernel distribution, where more flexible shapes generally result in more flexible models. Under a Bayesian modelling framework, we consider the use of the stable family of distributions for the kernel, as they are infinitely divisible and offer a variety of tail behaviours, orientations and skewness. Many of the attributes of the bivariate stable distribution are controlled by a measure, which we model using a flexible Bernstein polynomial basis prior. The method is the first attempt to incorporate non-Gaussian kernels in a two-dimensional integro-difference equation model and will be shown to improve prediction over the Gaussian kernel model for a data set of Pacific sea surface temperatures.

Should I Stay or Should I Go: Partially Sedentary Populations Can Outperform Fully Dispersing Populations in Response to Climate-Induced Range Shifts

Global mean temperatures have increased by 0.72 ^circ C since the 1950s, and climate warming is resulting in geographical shifts in the range limits of many species. Climate velocity is estimated to be 0.42 km/year, and if a species fails to adapt to the new climate, it must track the location of its climatically constrained niche in order to survive. Dispersal has an important role to play in enabling a population to shift is geographical range limits, but many species are partially sedentary, with only a fraction of the population dispersing each year. We ask, can partially sedentary populations keep pace with climate or will such populations be more vulnerable to extinction? Through the development of a moving-habitat integrodifference equation model, we show that, provided climate velocity is not too large, partially sedentary populations can outperform fully dispersing populations in one of two ways: (i) by persisting at climate speeds where a fully dispersing population cannot, and (ii) exhibiting higher population densities. Moreover, we find that positive density-dependent dispersal can further improve the likelihood a population can persist. Our results highlight the positive role that non-dispersers may play in mitigating the effects of overdispersal and facilitating population persistence in a warming world.

Deep integro-difference equation models for spatio-temporal forecasting

Integro-difference equation (IDE) models describe the conditional dependence between the spatial process at a future time point and the process at the present time point through an integral operator. Nonlinearity or temporal dependence in the dynamics is often captured by allowing the operator parameters to vary temporally, or by re-fitting a model with a temporally-invariant linear operator in a sliding window. Both procedures tend to be excellent for prediction purposes over small time horizons, but are generally time-consuming and, crucially, do not provide a global prior model for the temporally-varying dynamics that is realistic. Here, we tackle these two issues by using a deep convolution neural network (CNN) in a hierarchical statistical IDE framework, where the CNN is designed to extract process dynamics from the process’ most recent behaviour. Once the CNN is fitted, probabilistic forecasting can be done extremely quickly online using an ensemble Kalman filter with no requirement for repeated parameter estimation. We conduct an experiment where we train the model using 13 years of daily sea-surface temperature data in the North Atlantic Ocean. Forecasts are seen to be accurate and calibrated. A key advantage of our approach is that the CNN provides a global prior model for the dynamics that is realistic, interpretable, and computationally efficient. We show the versatility of the approach by successfully producing 10-minute nowcasts of weather radar reflectivities in Sydney using the same model that was trained on daily sea-surface temperature data in the North Atlantic Ocean.

MATHEMATICAL MODELLING OF THE EVOLUTION OF HUMAN BEHAVIOURS AND STRATEGIC CHOICE

Evolutionary anthropology seeks answers to the eternal philosophical question: how have we come to be? Studies in evolutionary anthropology attempt to explain the evolution of uniquely human behaviours by investigating the mechanisms that drive different trajectories. In this thesis dynamic mathematical models are built to investigate the tradeoffs defining divergent evolutionary pathways of strategic choice. This is done by comparing evolutionary equilibria at great ape-like and hunter-gatherer-like life histories. One investigation considers the evolution of pair bonding in humans. Mate guarding that pays off in increased likelihood of obtaining paternities outperforms strategies of paternal care at hunter-gatherer-like male-biased sex ratios while multiple mating is the evolutionary equilibrium at chimpanzee-like female-biased sex ratios. This demonstrates the promise of mate guarding as a pathway to social monogamy and links male reproductive strategies to the grandmother hypothesis through sex ratio dependence. Competition and care as male reproductive strategies are then investigated more generally in an ordinary differential equation model. Equilibria depend on life history and benefit parameters. An integro-difference equation model that explores the effect of non-parental imitation on the coevolution of low fertility at high socio-economic status is also given, applying Darwinian selection to a contemporary case of cultural evolution. This thesis shows that competitive strategies often pay off at human-like parameters despite the possibility of increased offspring survivorship through care. However, coexistence or persistence of a non-competitive strategy may also occur under specific parameters. Uncovering drivers of the evolution of different male reproductive strategies is important for guiding further research and shedding light on why we are so different from other great apes.

Skewed temperature dependence affects range and abundance in a warming world

Population growth metrics such as R0 are usually asymmetric functions of temperature, with cold-skewed curves arising when the positive effects of a temperature increase outweigh the negative effects, and warm-skewed curves arising in the opposite case. Classically, cold-skewed curves are interpreted as more beneficial to a species under climate warming, because cold-skewness implies increased population growth over a larger proportion of the species's fundamental thermal niche than warm-skewness. However, inference based on the shape of the fitness curve alone, and without considering the synergistic effects of net reproduction, density and dispersal, may yield an incomplete understanding of climate change impacts. We formulate a moving-habitat integrodifference equation model to evaluate how fitness curve skewness affects species' range size and abundance during climate warming. In contrast to classic interpretations, we find that climate warming adversely affects populations with cold-skewed fitness curves, positively affects populations with warm-skewed curves and has relatively little or mixed effects on populations with symmetric curves. Our results highlight the synergistic effects of fitness curve skewness, spatially heterogeneous densities and dispersal in climate change impact analyses, and that the common approach of mapping changes only in R0 may be misleading.

Visual Enhancement of Touch and Body Ownership: an ERP study

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.

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.

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.

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.

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.

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.