Schedule for: 16w5121 - Integrodifference Equations in Ecology: 30 years and counting

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

White pine blister rust (Cronartium ribicola) is a fungal pathogen that attacks and harms five-needle, soft, or white pines. This rust has a complex life cycle that includes two hosts and numerous spore stages. In this talk, I will introduce you to a series of increasingly complex models for the spread of this disease, and I will show you how we can use integrodifference equations, exponential transforms, and the method of steepest descent to estimate the pathogen's rate of spread.

In this talk, I will discuss how species mobility and competitive interactions determine species invasion speeds. I will analytically investigate invasion speeds in a two-species competition model for the case where one species spreads into its rival that is still in the process of expanding its spatial range. I will also present some preliminary results regarding invasion speeds in a three-species competition model. Our results show that different species have different invasion speeds, and different parts of a species distribution may have different propagation speeds.

Reproductive skew is ubiquitous - a few individuals have many offspring before death, while many individuals have few offspring. Why is that? My talk will be about using IDEs to answer questions like this about individuals and their paths through life, illustrated by applications mostly to plants. IDEs arose in ecology as spatial- or trait-structured population models, and most research has focused on population phenomena such as persistence, spread rates and spatial patterns. But empirically-derived IDEs are often based on observations and properties of individuals, such as how survival and fecundity depend on individual size and features of their local environment. In such models, we can shift our focus to the individual stochasticity (live, die, grow, shrink, move, stay?) that underlies the model but is averaged over to get a deterministic population IDE. Empirical applications, and theory (for age-structured populations, so far) suggest that luck plays a surprisingly large role in determining which individuals join the lucky few, even when there is substantial persistent heterogeneity among individuals.

For a large set of biologically reasonable conditions, scalar integrodifference equations generate invasions with a constant asymptotic invasion speed. Spread data from both laboratory experiments and field observations frequently show temporal variation in invasion speed; this variability is often attributed to environmental variability or demographic stochasticity. My colleagues (L. Sullivan, A. Shaw, B. Li, and T. E. X. Miller) and I have discovered three ways that persistent variability in invasion speed can be produced in scalar, deterministic, spatially homogeneous, and temporally constant integrodifference models, and thus offer a potential alternative explanation for some of the invasion variability observed in nature. An essential component of each mechanism is a low-density Allee effect. I will present these theoretical results and discuss some implications for empirical studies.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Globally, habitat boundaries are shifting directionally at a pace that will likely increase under climate change. Predicting the trajectories of ecotones – areas of habitat interface – requires basic understanding of the mechanisms by which they arise and their responsiveness to global change drivers. Ecological theory for spreading populations, which emphasises the coupled roles of density-dependent demography and dispersal, provides a lens for understanding of ecotone dynamics but has been under-used for this purpose. Building upon theory for single-species invasion, I will develop a model of “collision” between invasion waves of two foundation species and use the model to study ecotone formation and change. I will connect the model to empirical data from ecotones defined by creosote bush and black grama grass, two foundation species of Chihuahuan desert habitats. Expansion of shrublands into grasslands is a global phenomenon that can reduce biodiversity and ecosystem services, but its underlying mechanisms are poorly understood. I suggest that IDEs provides a powerful framework for understanding and predicting the dynamics of habitat boundaries based on the combined influences of demography, dispersal, and competition.

We investigate the inside dynamics of integrodifference equations to understand the genetic consequences of a population with non overlapping generations undergoing a range expansion. To obtain the inside dynamics, we decompose the solution into several neutral genetic components. The inside dynamics are then given by the spatio-temporal evolution of the neutral genetic components. We consider thin-tailed dispersal kernels and a variety of per capita growth functions in order to classify the traveling wave solutions as either pushed or pulled fronts. We find that pulled fronts are synonymous with the founder effect in population genetics. Growth functions with overcompensation are shown not to promote genetic diversity in the expanding population. Growth functions with a strong Allee effect cause the traveling wave solution to be a pushed front preserving the genetic variation in the population. We show that the contribution of each neutral fraction can be computed by a simple formula dependent on the initial distribution of the neutral fractions, the traveling wave solution, and the asymptotic spreading speed.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

There have been extensive investigations on monostable traveling waves and spreading speeds for various monotone evolution systems. In this talk,  I will introduce the general theory of bistable traveling waves (i.e., connecting two stable equilibria) for monotone discrete-time recursive systems. The developed theory establishes, for the first time, the relation between  monostable and bistable systems about their propagation phenomena. A dynamical systems approach to the global stability of such a wave will also be presented. As an application example,  I will further discuss an integrodifference  equation model of  two species competition in spatial ecology.

I will present a simple model for the dynamics of an ecosystem engineer, Spartina, and discuss properties of its solution. The model formulation will apply to a wide range of similar problems.

The basic ideas of optimal control of terms in integrodifference equation models to achieve a goal will be presented. One example of harvesting will show the importance of the order of events in the optimal control results. A most recent example about the management of gypsy moths will be given.

In this talk, I will try to present an overview on the recent progress made in the description of propagation phenomena in nonlocal reaction diffusion equations. Starting with epidemiological data that motivate the study of nonlocal reaction diffusion equation, I will present past and recent results for some homogeneous nonlocal models . Then, I will move to what we know in heterogeneous situations and conclude with some research perspectives.

Conservation managers often need to set general guidelines for habitat management and species recovery. Typical metrics for planning include knowing the minimum area needed to support a population and the rate at which a population will expand through heterogeneous landscapes. I will show examples of how we have combined IDE models with demography and movement data to estimate these metrics for butterfly populations of conservation concern. I will discuss some implications of these case studies for conservation, and also their implications for making theoretical models more relevant to conservation planning.

The distribution of many tree species is strongly determined by the behavior and range of vertebrate dispersers, particularly birds. Many models for seed dispersal exist, and are built around the assumption that seeds undergo a random walk while they are being carried by vertebrates, either in the digestive tract or during the process of seed storage (caching). We use a PDF of seed handling (caching and digesting) times to model non-constant seed settling during dispersal, and model the random component of seed movement using ecological diffusion, in which animals make movement choices based purely on local habitat type instead of population gradients. Spatial variability in habitat directly affects the movement of dispersers and leads to anisotropic dispersal kernels. For birds, which can easily move many kilometers, habitat changes on the scale of tens of meters can viewed as rapidly varying. We introduce multiple scales and apply the method of homogenization to determine leading order solutions for the seed digestion kernel (SDK). Using an integrodifference equation (IDE) model for adult trees, we investigate the rate of forest migration. The existing theory for predicting spread rates in IDE does not apply when dispersal kernels are anisotropic. However, in statistically stationary landscapes the homogenized SDK is isotropic on large scales and depends only on harmonically averaged motilities and averaged rates of digestion/caching. We show that speeds arising as a consistency condition in large-scale dynamics accurately predict rates of invasion for the spatially variable system.

Characterising the spread of biological populations is crucial in responding to both species invasions and the shifting of habitat under climate change. Spreading speeds can be studied through mathematical models such as the discrete-time integro-difference equation (IDE) framework. The usual approach in implementing IDE models has been to ignore spatial variation in the demographic and dispersal parameters and to assume that these are spatially homogeneous. On the other hand, real landscapes are rarely spatially uniform with environmental variation being very important in determining biological spread. In this talk we will demonstrate two new analytical methods for approximating the wavespeed of species invasion for different types of heterogeneous landscapes, which depend on landscape and dispersal scales. Finally, we will show a new efficient simulation technique for predicting species spread in 2D heterogeneous landscapes. Our technique, an adaptive algorithm, leads to an order of magnitude improvement in computational efficiency (RAM and CPU) over non-adaptive algorithms. We will demonstrate the user-friendly interface which is adaptable to many types of large scale IDE problems. Our algorithm enables the investigation of scientific questions that would have previously been impossible due to computational resource constraints, and allows large simulations to be carried out on a desktop or laptop rather than on a computer cluster.

Climate change is leading to shifts in the geographic locations over which species can persist. Projections of climate change can be characterised in terms of the rate of this shift in climate space -summarised as the velocity of climate change. IDEs are a useful approach to combine knowledge of species' demography and dispersal and the resulting projection of a species' rate of population spread - the wavespeed - can be compared against the velocity of climate change. If the wavespeed is less than the climate velocity, the species may not be able to track climate change. While this calculation involves simplifying assumptions, the wavespeed vs climate velocity comparison gives a straightforward warning about which species may suffer most under climate change. In making such comparisons, we are interested in calculating wavespeed for a wide variety of species. Unfortunately, there are demography and dispersal data for very few species. We have therefore used approaches including data synthesis, mechanistic dispersal models and statistical creation of "virtual species" to provide a more complete picture of variation in spread rates and risks from climate change.

Climate change is known to have a strong impact on population distribution, forcing some species to migrate northward in order to follow their favorable habitat. In this presentation, we investigate the impact of climate change on the persistence of the population, when the latter grows and disperses at different times. In this framework we consider integro-difference equations, with a habitat that is shifting at a given speed at each generation. We also include in our model the variability of the environment from one generation to the next, considering a shifting speed and a growth function that are chosen randomly at each generation. We will start by detailing the model and its assumptions. Then we will explain how to characterise the persistence of the population at large time. Finally we study the effect of variability on the persistence of some butterfly population using numerical simulations.

By reputation, IDEs are difficult to simulate. They are, however, extremely useful and beneficial for ecologists and resource managers addressing pressing ecological and environmental challenges such as climate change. The challenge is how to make our research and the advances that come from this workshop relevant and accessible to the average ecologist and resource manager that would benefit from these tools. In this talk, I will show how we have used RStudio and Shiny to build interactive web applications that allow users to engage with and gain a greater appreciation of IDEs. The tools presented are relatively easy to use with a basic knowledge of R. I use an IDE model developed to evaluate and predict the effects of climate change on species to demonstrate the power of this tool.

Dynamic changes in habitat sizes and locations have important consequences for their residents. In this talk, I will present two integrodifference equation models. The first model describes the dynamics of a population whose habitat shifts under climate warming. The second model describes a population whose habitat expands and contracts seasonally. For both models, I will show that the persistence metric for compact integral operators on a bounded domain extends to that of an unbounded domain. I will then present the ecological insight gained from the mathematical analysis. For the first model, I will demonstrate that the population will lag behind the shifting habitat and carry a “climatic debt”, especially when the climate warming is accelerating. For the second model, I will explain how the concept of critical habitat size is extended to that of a "lower minimal limit size" in a two-season scenario. 

Climate change is moving many species’ suitable habitats, forcing populations to either track their habitats using dispersal, adapt in place, or face extinction. Most mathematical models of species’ climate responses focus on asymptotic population dynamics. In contrast, most management decisions occur on the time scale of short-term (transient) dynamics. Transient behavior can often differ greatly from asymptotic dynamics, leading to poor management decisions. In this talk, I will use an integrodifference equation to explore the transient dynamics of a population tracking its moving habitat. Although transient dynamics are usually studied in the context of non-spatial models of multiple species or stage-structured populations, I will show that (and why) transient dynamics can occur in unstructured, single-species spatial models. Using the endangered Fender’s blue butterfly subspecies (lcaricia icarioides fenderi) as a model species, I will highlight three results with strong management implications. First, transient dynamics can only occur under a certain set of biological and environmental conditions, which can serve as indicators of possible transients. Second, the model pinpoints the optimal location to place individuals within the habitat so that short-term survival is greatest. Third, the model offers an explanation for critical slowing-down—the observation that systems near a tipping point often exhibit long transient dynamics. I will close with a discussion of future extensions, such as transients in stage-structured integrodifference equations.

The theory of spreading speeds and travelling waves in integrodifference equations is well established for the "mono-stable" case: the growth function has an unstable state at zero and a positive, globally stable state. When the growth function shows overcompensation, the positive state can be destabilized and a stable two-cycle can emerge in the non-spatial dynamics. What do the resulting spatial spread dynamics look like in this case? Previous simulations seemed to reveal a travelling profile in the form of a two-cycle, with or without spatial oscillations. The existence of a travelling wave solution has been proven, but its shape and stability remain unclear. In this talk, I will show simulations that suggest that there are several travelling profiles at different speeds. I will establish corresponding generalizations of the concept of a spreading speed and prove the existence of such speeds and travelling waves in the second-iterate operator. I conjecture that rather than a travelling two-cycle for the next-generation operator, one observes a pair of stacked fronts for the second-iterate operator. I will relate the observations to the phenomenon of dynamic stabilization.

Individual animals acquire information from their surroundings, gain and store knowledge from experience, and use what they have learned to navigate through dynamic landscapes. I will first discuss a mathematical model of animal perception in dynamic landscapes (this is formulated as a partial integro-differential equation featuring non-local advection) and then present some biological results that frame challenges in the modeling of spatial memory. 

The correct model for a one-allele system always has a small tail which has support infinitely far away. This is most easily seen for a random walk model, but the idea also works for the integral equation model which is the subject of the conference. This means that models for single alleles concentrated at two far-apart places cannot be be isolated from each other, but will eventually produce competition.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.