Invasive species can be heavily influenced by spatial heterogeneity in the landscape through which they are spreading. This heterogeneity may be endogenous (that is, patchiness in the distribution of the invader itself, or heterogeneity in other species due directly to interactions with the invader) or it may be exogenous (such as patchy resources or heterogenous climatic conditions). I will discuss two studies, each dealing with one of these types of heterogeneity. First, I will present a model for spatial spread of an invasive seaweed that suggests different native grazers can have very different effects on invasion success by generating unique patterns of exogenous heterogeneity. This difference is due to the spatial distribution of disturbed patches, which when cleared of native seaweeds become available for establishment by the invader, that are characteristic of different grazers. Second, I will discuss the effects of population asynchrony (endogenous heterogeneity) on population stability and persistence. For this second topic, I will draw on models that are commonly applied to insect populations.
Transportation networks play a crucial role in human mobility, the exchange of goods, and the spread of invasive species. With 90% of world trade carried by sea, global shipping provides one of the most important modes of transportation. Shipping also constitutes the world largest transportation vector for marine bioinvasion, transferring accidentally numerous species around the world. Here we use information about the itineraries of 16,363 cargo ships during the year 2007 to construct a network of shipping connections between ports. We perform a statistical analysis of the network topology, and we show that the network possesses a heavy-tailed distribution for the connectivity of ports with systematic differences between ship types. Our analysis improves current assumptions based on gravity models of ship movements, an important step towards understanding global patterns of shipping mobility and trade. Coupling the shipping network with biogeography and environmental conditions at the ports, we quantify the likelihood of invasion by the exchange of ballast water. Our model allows to identify high risk invasion routes, hot spots of bioinvasion, and major source regions from which bioinvasion is likely to occur. Our predictions agree with observations in the field at various locations in the world. Finally, motivated from the invasion process, we present a conceptional model for the spread of a binary variable (here: invaded or non-invaded port) on a complex network. Despite its simplicity, the model exhibits complex dynamics and shows many properties that set it apart from similar models of epidemic spread or cascading failures on complex networks.
Applied ecologists tend to dislike the biological simplicity of wavespeed models and will often use complex simulation models to address real-world problems. Using Neubert & Caswell's stage-structured formulation of wavespeed models I present a number of examples of their use in tackling applied ecological problems and illustrate the utility arising from both their relative simplicity and the supporting mathematical theory. We have used these models to investigate plant re-introductions, determinants of range shift under climate change and control methods for invasives. Despite their simplicity, wavespeed models can mirror reality adequately. The assumptions are clear and allow one both to examine extremes (best and worst case scenarios) and the well-developed sensitivity theory enables examination of uncertainty and critical drivers of population spread. Of particular utility are the methods for incorporating dispersal, especially through empirical estimation of the dispersal moment generating function which allows complex dispersal data to be used. I show how we have used measured dispersal data and kernels generated by mechanistic dispersal models. Finally, I return to the simplicity issue by formally comparing the predictions and utility of wavespeed models with those of simulation models in investigating spread and control of an invasive plant.
Invasion speeds can be calculated from matrix integrodifference equation models that incorporate stage-specific demography and dispersal. These models also permit the calculation of the sensitivity and elasticity of invasion speed to changes in demographic and dispersal parameters. Such calculations have been used to understand the factors determining invasion speed and to explore possible tactics to manage invasive species. This talk will extend these calculations to temporally varying environments (periodic and stochastic). Periodic models can describe seasonal variation within a year, or can be used to study the frequency of occurrence of events (e.g., floods, fires) on interannual time scales. Stochastic models can incorporate variances, covariances, and temporal autocorrelation of parameters. The invasion speed in a variable environment is calculated from a growth rate which is in turn calculated from a periodic or stochastic product of moment-generating function matrices. Sensitivity analysis provides expressions that include stochastic, periodic, and time-invariant environments as special cases.
Spread of exotics presents an emerging set of challenges for mathematical biologists, as studies shift from descriptions of past or on-going invasions to analysis of increasingly detailed data on what defines a potential invader's capacity to grow and compete in novel environments. Mathematical studies of invasive species traditionally focused on rates of spread. Models concern arrival at a distant site, conditional on the distribution in the recent past, emphasizing dispersal and colonization capacity. Applications have been most successful when applied to past or contemporary invasions, where an average rate of spread can be 'inverted' to infer processes that might explain movement. These models are typically not consulted for predicting future invasions, in part due to recognition that successful invasion depends on poorly understood interactions that control growth and reproduction once immigrants arrive in a new site.
Complementary to the challenge of predicting spread using assumptions about future establishment and growth is the need to quantify competition and its interaction with the environment once immigrants arrive. This second challenge includes the traditional goal of 'defining the niche,' specifically asking whether or not an invader can outcompete residents in at least some of the available niche space. Models have addressed this problem for some time, but have tended to focus on crude descriptions of life history characteristics, e.g., traits that allow for rapid growth and high dispersal and/or correlations between patterns of abundance and regional climate. New opportunities for mathematical biologists are presented by the increasingly detailed empirical studies of invasion, including spatio-temporal observational data sets and field experiments where residents and invaders are allowed to directly compete.
In this talk I discuss emerging tools for quantifying the relationships between competition and the environment that determine the capacity to survive and grow, mature, and produce offspring. I address how observational and experimental data can be analyzed to determine the climate/competition interactions for large numbers of tree species, one of which (Ailanthus altissima) is has begun to invade only in the last decade. I point out how the large number of processes, both observed and not, and the complexity of the interactions, provide a rich set of problems for mathematical ecologists.
The logistic model is the quintessential model of invasion. The fundamental idea of the logistic is "something replacing something else": vegetative cover replacing empty space, biomass replacing nutrient, infected individuals replacing uninfected ones, DVD households replacing VHS households. Biological mechanisms producing logistic growth are analogous to an autocatalytic chemical reaction converting a substrate to a product.
Given data on the abundance of a growing population, ecologists have handled the statistics of fitting the logistic model in a variety of ad hoc ways. Ecology textbooks are largely silent about estimating the unknown quantities in the logistic. In fact, different sources of variability in the data, that is, sources of departures of data from a logistic-predicted trajectory, would lead by statistical principles to different statistical methods for estimating parameters and predicting future outcomes.
Here I examine a particular stochastic version of the logistic model. The version is a diffusion process with environmental-type noise. The equilibrium (carrying capacity) is no longer a point equilibrium but rather is a gamma probability distribution. Many statistical properties of the model can be derived as formulas. With simulations, I evaluate an approximation, based on singular perturbation, for the full transition probability distribution of the process. The approximation turns out to be accurate and quite helpful for fitting the model to time series data. With the transition distribution in hand, incorporating sampling variability and estimating parameters with data cloning is straightforward. The model has the convenient property that the time intervals between observations can be unequal. Various examples that use the model for statistical analysis of population time series are presented.
How far invasive species will spread and what sets the geographic range limits of such species are issues of great interest to biologists and resource managers alike. However, scientists lack strong mechanistic understanding of the factors that set geographic range limits in empirical systems, especially in animals, and there exists a clear need for detailed case studies of this subject. The evergreen bagworm Thyridopteryx ephemeraeformis (Lepidoptera: Psychidae) is a major pest of cedars, arborvitae, junipers, and other landscape trees throughout much of North America. Across dozens of populations spread over six degrees of latitude in the American Midwest, female mating success of this bagworm declines from ~100% to ~0% near the edge of the species range. When coupled with additional latitudinal declines in fecundity and in egg and pupal survivorship, a spatial gradient of bagworm reproductive success emerges. This gradient is associated with a progressive decline in local abundance and an increase in the risk of local population extinction, up to a latitudinal threshold where extremely low female fitness meshes spatially with the species' geographic range boundary. The reduction in fitness of female bagworms near the geographic range limit, which concords with the abundant centre hypothesis from biogeography, provides a concrete, empirical example of how an Allee effect may interact with other demographic factors to induce a geographic range limit. We develop a coupled system of nonautonomous differential equations to explore the relative roles of reproductive asynchrony, temperature, and other factors in generating this Allee effect.
I will discuss models that include the kinds of spatial dynamics and control and life history appropriate for this plant. The ideas will focus on developing simple analytic models and using approaches that into account not only control but restoration and bio-economic aspects. Mathematical tools will include linear and quadratic programming in addition to optimization and other approaches to try and obtain general rules.
Why do some exotic species thrive so successfully once they have been introduced to a new environment? One of the reasons most frequently called upon is the enemy release hypothesis, which explains the inordinate success of introduced species by the lack of pressure from co-evolved natural enemies. In this talk, I consider the situation that the new environment changes in the sense that it does not remain free of natural enemies. More specifically, infectious diseases can follow their host and alter the fate of invasion. In particular, this may lead to a rather spontaneous invasion crash ("Now you see them, now you don't"). The mathematical models employed are reaction-diffusion systems with the feature of bistability. The results bear implications for potential biological control methods of invasive pest species.
Modeling population dynamics is essential to study ecological populations whether for maintaining the existing populations (e.g. Population Viability Analysis) or for controlling spread of invasive species. Meta-population dynamics that takes into account of immigration and emigration also plays an important role in studying spread of invasive species. Systems of ordinary differential equations are used to model growth and spread of forest insects and pests. Many of these processes evolve in continuous time and space but the data are obtained only at discrete times and discrete locations and with measurement error. Hierarchical models are a convenient way to model such imperfectly and partially observed processes.
Statistical inference for such models poses significant computational difficulties. In this paper, I review data cloning (Lele et al. 2007), a simple computational method that exploits advances in Bayesian computation, in particular the Markov Chain Monte Carlo method, to conduct statistical inference for hierarchical models. This includes (i) parameter estimation, (ii) confidence intervals, (iii) model selection, and (iv) forecasting future states and the uncertainty associated with such forecasts.
One of the basic tenets of good modeling is that complexity of the model should not exceed information in the data. The mismatch in the two can lead to parameter non-estimability. Determining estimability of the parameters in a hierarchical model is, in general, a very difficult problem. Data cloning provides a simple graphical test to not only check if the full set of parameters is estimable but also, and perhaps more importantly, if a specified function of the parameters is estimable.
I will illustrate data cloning in 1) Population Viability Analysis in the presence of measurement error, 2) Two species Leslie-Gower Competition model, and 3) Analysis of systems of differential equations arising in epidemiology and ecology.
Optimal control will discussed as a tool to find intervention strategies in models for invasive species. Examples of optimal control of discrete time systems and of systems of ordinary differential equations will be given. Control characterizations and numerical results for models with different types of control intervention actions will be presented. Applications include nascent foci and native-invasive competition examples.
In this talk I will outline a recent interdisciplinary effort to model and understand the spread of invasive copepods across lake networks in North America. This 5-year project, developed through the Canadian Aquatic Invasive Species Network (CAISN), tracked the invasion status of approximately 500 interconnected lakes in the Canadian shield. Here the invader, spiny waterflea, is spread by recreational boaters moving between the lakes. The water flea modifies the trophic structure of lake it encounters and has a major ecological impact.
To understand spread we used a stochastic gravity model, parameterized by boater survey data. In this model, the number of trips linking lakes is a random variable whose magnitude is a nonlinear function of empirically measured quantities such as lake size and distance between lakes. In our work, the gravity model was used to infer the total propagule pressure experienced by each lake. This propagule pressure was then linked to an establishment model. The establishment model estimates the probability that a propagule will establish at a given lake, depending upon local physical and chemical conditions. Fitting our hybrid model to large datasets has been an interesting challenge. As I will show, our hybrid gravity/establishement modelling approach has proved to be very effective at determining which lakes become invaded as the invasion spreads across the complex network.
This work is collaborative with Subhash Lele, Jim Muirhead, Alex Potapov and Norm Yan.
Integrodifference equations provide a very natural general framework to model the spread of invasive species if the species in question has a clearly distinct growth and dispersal phase during its life cycle. Many insect species satisfy this description, in particular where climate imposes strong seasonality.
Early applications of integrodifference equations considered homogeneous landscapes and a number of relatively simple dispersal mechanisms. I will report on some recent developments concerning heterogeneous landscapes and density-dependent dispersal. I will explain several approximations that can be used to identify important spatial scales and simplify model parametrization. The main focus of the presentation will be on spreading speeds, and, in the case of landscape heterogeneity, also on persistence conditions. Several insights about landscape alterations for spread control will be discussed. I will end with some open and challenging questions for integrodifference equations as applied to invasive species.
Biological invasions including the spread of infectious diseases have strong ecological and economical impacts. The perception of their often harmful effects has been continuously growing both in sciences and in the public. Mathematical modelling is a suitable method to investigate the dynamics of invasions, both supplementary to and initiating eld studies as well as control measures.
Holling-type II and III predation as well as Lotka-Volterra competition models with possible infection of the prey or one of the competitors are introduced. The interplay of local predation, intra- and interspecific competition as well as infection and diffusive spread of the populations can cause spatial and spatiotemporal pattern formation. The environmental noise may have constructive as well as destructive effects.
A plant competition- ow model is considered for conditions of invasibility of a certain model area occupied by a native species. Short-distance invasion is assumed as diusion whereas long-distance seed dispersal can be stratified diffusive or advective. The variability of the environment due to contingent landslides and artificial causes such as deforestation or weed control leads to the temporary extinction of one or both species at a randomly chosen time and spatial range. The spatiotemporal dimension of these extreme fragmentation events as well as a possible selected harvesting or infection of the invading weed turn out to be the crucial driving forces of the system dynamics.
Population models that combine demography and dispersal are important tools for forecasting the spatial spread of biological invasions. Current models describe the dynamics of only one sex (typically females). Such models cannot account for the sex-related biases in dispersal and mating behavior that are typical of many animal species. In this lecture, I will construct a two-sex integrodifference equation model that overcomes these limitations. I will derive an explicit formula for the invasion speed from the model, and use it to show that sex-biased dispersal may significantly increase or decrease the invasion speed by skewing the operational sex ratio at the invasion's low-density leading edge. Which of these possible outcomes occurs depends sensitively on complex interactions among the direction of dispersal bias, the magnitude of bias, and the relative contributions of females and males to local population growth.
Biological invasion admittedly consists of a few distinctly different stages such as exotic species introduction, establishment and geographical spread. Each of the stages has its own specific mechanisms and implications, which require application of specific research approaches. In my talk, I focus on the challenges arising during the stage of the geographical spread. A well-developed theory based on diffusion-reaction equations predicts a simple pattern of alien species spread consisting of a continuous traveling boundary or 'population front' separating the invaded and non-invaded regions. A propagating population front has been a paradigm of the invasive species spread for several decades. However, it also appears to be at odds with some observations. In some cases, the spread takes place through formation of a distinct patchy spatial structure without any continuous boundary. Perhaps the most well known and well studied example of this 'patchy invasion' is the gypsy moth spread in the USA; e.g. see www.fs.fed.us/ne/morgantown/4557/gmoth/atlas/#defoliation
In order to address this problem theoretically, I first re-examine the current views on possible mechanisms of the patchy spread and argue that the importance of the stratified diffusion may be significantly overestimated. Second, I will revisit the traditional diffusion-reaction framework and show that the patchy spread is, in fact, its inherent property in case the invasive species is affected by predation or an infectious disease and its growth is damped by the strong Allee effect. The patchy spread described by a diffusion-reaction model appears to be a scenario of alien species invasion "at the edge of extinction" and this can have important implications for the management and control of the invasive species. I will then show that patchy spread is not an exclusive property of the diffusion-reaction systems but can be observed as well in a completely different type of model such as a coupled map lattice which is capable of taking into account environmental heterogeneity. Finally, I will argue that these theoretical results taken together with the evidence from field data may result in a paradigm shift: A typical pattern of exotic species spread is a patchy invasion rather than the continuous population front propagation.
In populations with cyclic dynamics, invasions typically generate either regular (periodic) spatiotemporal oscillations, spatiotemporal chaos, or a mixture of the two. I will present examples of these behaviours in numerical simulations of invasions. I will then describe how to determine which of the behaviours will occur, as a function of ecological parameters. The characteristic invasion profile is a band of regular oscillations immediately behind the invasion front, which undergoes a subsequent transition to chaos. The key determinant of long-term behaviour is the width of the band, and I will explain how to calculate it. Specific applications of the results reveal a marked sensitivity to ecological parameters. In the light of recent evidence that climate change is having a significant effect on the demographic parameters underlying oscillatory ecological systems, this implies a potentially dramatic effect of climate change on the spatiotemporal dynamics generated by invasions.