**Bjornstad,** Ottar - Pennsylvania State University

Title of Presentation: "Spatial Contact Networks and Timing of Outbreaks in Epidemic Metapopulations: Theory, Data and Statistics."

Infectious diseases provide a particularly clear illustration of the spatio-temporal underpinnings of consumer-resource dynamics. The paradigm is the locally unstable, oscillatory dynamics of extremely contagious, directly transmitted, immunizing infections caused by morbilliviruses for which more or less irregular epidemics are interspersed by prolonged periods of local extinction of the parasite. Spatial transmission and 'recolonization' in such systems are ultimately tied to movement by the hosts. The network of spatial spread may therefore be related to the transportation network within the host metapopulation. I discuss two critical issues in the spatiotemporal dynamics of host-pathogen interactions. First, how do rates of movement of hosts between patches affect the timing and predictability of outbreaks? Second, how can we reconstruct topology of transportation networks from outbreak data? I address the first using stochastic epidemic models, and the second the models' associated hazard likelihoods. The theory and methods are discussed with particular reference to measles. I propose a gravity model for the spatial transmission networks and discuss how recurrent epidemics may either be periodic and relatively predictable or erratic and unpredictable depending on the strength of spatial transmission.

**Fagan,** William - University of Maryland

Title of Presentation: "Riverine Landscapes: Exploring Connectivity, Extinction Risk, and Restoration in an Alternative Geometry."

Riverine landscapes differ in fundamental ways from terrestrial ones. Of particular note is that the dendritic patch geometry of river networks leads to inherently asymmetrical opportunities for connections among patches. I have been exploring what happens when spatial ecological processes such as dispersal and habitat fragmentation play out in riverine systems, with a particular emphasis on understanding how these processes influence species’ regional distributions and local occupancy patterns. Here I will discuss two aspects of fragmentation (or alternatively, connectivity) that are relevant to the spatial ecology of riverine systems. First, I will present evidence that demonstrates, for a diverse assemblage of native fish species, how interspecific variation in range fragmentation has translated directly into variation in realized extinctions (both locally and at larger spatial scales). Second, I will discuss results from an ecological economics study of dam removal for restoring river connectivity, where the focus is on the tradeoff curves that emerge when one considers alternative strategies for breaching dams by hierarchically clustered groups rather than on a case-by-case basis. The specific study systems involved are native fishes of the Sonoran Desert in the lower Colorado River system and Pacific salmon in the Willamette River, Oregon.

**Hanski,** Ilkka - University of Helsinki

Title of Presentation: "Towards Integration of Spatial Ecology, Landscape Ecology, and Evolutionary Ecology."

I discuss with examples the need for better integration of spatial ecology, landscape ecology, and evolutionary ecology. Theoretical research in spatial ecology is often inadequately linked with empirical research, and vice versa, because of disparate ways of conceiving space and dissimilar questions asked. Metapopulation models that explicitly describe the influence of landscape structure on spatial turnover of populations have been helpful in forging a more united framework to study the dynamics and persistence of metapopulations in highly fragmented landscapes. I give examples and outline further research questions. Linking ecological and evolutionary dynamics in spatially realistic settings is necessary for addressing questions about the responses of species in changing environments. For instance, to what extent are the ecological responses of species to global environmental changes amplified by evolutionary responses?

**Hillen,** Thomas - University of Alberta

Title of Presentation: "Volume Effects in Chemotaxis."

In the literature on chemotaxis modeling the existence of finite time blow-up solutions has been a major focus. If, however, the finite volume of individuals is introduced into the model, then solutions will no longer blow-up but exist globally in time. In addition they show interesting pattern formations.

Painter and I classified the volume effects into(i) "volume filling", if the finite cell size is incorporated, (ii) "quorum sensing" for cells that release a repulsive chemical signal, and (iii) "finite sampling radius", to model the measurement of a chemical signal on the cells outer membrane.

The volume-filling model has been discussed in a paper by Hillen and Painter in 2000. I will use this talk to review the results on the volume filling effect (i), and then I will focus on the quorum-sensing effect (ii) and the finite sampling radius (iii).

I will show results on global in time existence and on pattern formation. Moreover, I will pose some open problems.

**Levin,** Simon - Princeton University

Title of Presentation: "Spatial ecology: Ecological Theory and Economic Challenges."

Space fundamentally alters ecological and evolutionary interactions, and creates challenges for foraging by humans and other organisms. This lecture will review both classical and modern ecological theory of the maintenance of biodiversity, and discuss challenges for harvesting theory and conservation biology.

**Mischaikow,** Konstantin - Georgia Institute of Technology

Title of Presentation: "Assigning Numbers to Patterns: Topology of Complex Structures."

In spatially explicit systems the combination of reaction and dispersal lead to the development and evolution of patterns. Conversely, observed geometric structures in physical systems can give insight into the underlying processes and dynamics. However, for complicated patterns characterizing this information can become extremely difficult or costly. I will describe recent work that makes use of topological invariants to analyze time varying patterns. I will also argue that these invariants are relatively easy to compute and by their very nature extract consistent and robust geometric features. To emphasize the applicability of these techniques, most of the results I will talk about are based on experimental data of a weakly turbulent state of Rayleigh-Benard convection known a spiral defect chaos and numerical simulations of the Cahn-Hilliard equation and FitzHugh-Nagumo equations. In particular, I will show how we use the topological characterization to

(a) distinguish the asymptotic states as a function of a control parameter,

(b) compute a entropy for the dynamics,

(c) determine that we are observing stochastic as opposed to chaotic evolution of the patterns,

(d) observe subtle breaks in the symmetry of the system,

(e) measure interior as opposed to boundary phenomena, and

(f) potentially characterize the dispersal mechanism.

**Neubert,** Michael - Woods Hole Oceanographic Institution

Title of Presentation: "Spatial Bioeconomic Problems and Fisheries Management."

It may seem obvious that the management of fish stocks should have a spatial component. After all, fish live in spatially heterogeneous environments. Indeed a principal activity of fisheries oceanographers is to quantify dispersal rates, habitat heterogeneity, and spatial variability in stock size. Nevertheless, bioeconomic models in fisheries management are, almost exclusively, non-spatial. This is partly due to the fact the analysis of bioeconomic models often involves the maximization of yield or profit, and spatiotemporal optimization problems are difficult. The need to improve fisheries management is so urgent, however, that economists and ecologists are beginning to use spatial models more frequently. In this lecture, I will present a series of simple, spatially-explicit, bioeconomic models and discuss the results of their analysis. In particular, I will highlight how the results inform the current debate about the usefulness of marine reserves in fisheries management.

**Shi,** Junping - College of William and Mary

Title of Presentation: "Bistability, Allee Effect and Threshold Manifold in Structured Population Models."

In a population model with Allee effect, threshold dynamic exists such that extinction occurs when the initial population is below the threshold, and persistence occurs when it is above the threshold. In a structured population model such as reaction-diffusion equation or nonlinear matrix equation, the threshold is related to the order of the phase space, and very often, the exact number of equilibrium states of the system. I will review the development of related theories on 1) global bifurcation diagrams of equilibrium solutions; 2) existence and properties of threshold manifold; and various examples from population biology.

**Tilman, **David - University of Minnesota

Title of Presentation: "Niche Tradeoffs or Neutrality? A General Theory of Community Assembly, Diversity, and Species Abundances."

A new theory, stochastic niche theory (Tilman 2004, PNAS), resolves many of the differences between neutral theory and classical tradeoff-based niche theories of resource competition and community structure. In stochastic niche theory, invading species become established only if propagules can survive stochastic mortality while growing to maturity on the resources left unconsumed by established species. Comparisons of stochastic niche theory and neutral theory suggest that neutrality is highly unlikely to explain patterns observed in species diversity, species relative abundances, and community assembly. Neutrality requires high immigration rates to maintain diversity, and predicts a rapid diversity collapse to a single species should fragmentation eliminate immigration, whereas niche models predict more realistic diversity losses. Spatially explicit simulations of the fates of neutral and niche-differentiated species show that niche differentiation is a much more important mechanism than neutrality for the maintenance of diversity in spatially heterogeneous habitats. In addition, stochastic niche theory generates realistic distributions of species relative abundances for which, contrary to neutral theory but consistent with numerous observations, there are strong correlations among species traits, species abundances, and environmental conditions.

Co-Authors: Daniel Klein, Williams College and Peter Hawthorne, University of Minnesota

**Wu,** Jianhong - York University

Title of Presentation: "Weaving Delays and Dispersal in Ecology and Epidemiology."

The talk will give a short survey of the current progress in modeling time lag involved in the self-regulatory mechanism of biological systems consisting of species that moves randomly in space, and in modelling the interaction of time lags and spatial movement of individuals for the spatial spreads of diseases. The modelling process leads naturally to many infinite dimensional dynamical systems including parabolic and hyperbolic partial functional differential equations with delayed and non-local nonlinearities. Recent work on nonlinear dynamics and global attractors will be reported, and challenges in terms of numerical analysis and fitting biological data will be discussed.

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**Amarasekare, **Priyanga - University of Chicago

Title of Presentation: "Spatial Competitive Coexistence in Patchy Environments: A Comparative Analysis."

I present a mathematical framework for spatial competitive coexistence that allows for comparative analysis of multiple mechanisms. The basis for comparison is mechanisms that operate in spatially homogeneous competitive environments (e.g., life history trade-offs) vs. mechanisms that operate in spatially heterogeneous competitive environments (e.g., source sink dynamics). This comparative approach leads to several new insights about spatial coexistence. First, spatial variation in the expression of a life history trade-off leads to a unique regional pattern that cannot be predicted by considering trade-offs or source-sink dynamics alone. This result represents an instance where spatial heterogeneity constrains rather than promotes coexistence, and illustrates the kind of counterintuitive emergent properties that arise due to interactions between different classes of mechanisms. Second, the analysis distinguishes between situations where dispersal mortality is not necessary for coexistence and those where such mortality is essential for coexistence because it preserves spatial variation in the strength of competition.

**Auger,** Pierre - UMR CNRS 5558 and IRD GEODES

Title of Presentation: "Macroscopic Behaviour of Host-parasitoid Spatial Population Dynamics."

Time series for populations of the same species at different locations show significant synchronism. This spatial synchrony can be observed for many different species of insects, fishes, birds and mammals. Although it decreases with the distance between populations, this synchrony effect can be observed for important distances, from kilometers to thousands of kilometers. In order to explain this spatial synchrony, two main causes have been investigated. The first one is known as the Moran effect resulting from environmental correlations. A second cause is dispersal. The aim of this work is to study the effects of dispersal frequency on spatial synchrony of a host-parasitoid metapopulation. We consider a square two-dimensional grid of spatial patches.We show that above a threshold of the dispersal frequency, the dynamics of the metapopulation can be described by a macroscopic model governing the total insect population densities on the grid.

**Ault,** Jerald S. and Jiangang Luo - University of Miami, Rosenstiel School of Marine and Atmospheric Science

Title of Presentation: "A Spatial Dynamic Multispecies Model to Assess Population Risks from Exploitation and Environmental Changes in Coastal Ecosystems."

We developed a generalized spatial dynamic age-structured multispecies ecosystem model for understanding fishery production dynamics by linking bioenergetic principles of physiology, population ecology, and community trophodynamics to regional finite-element hydrodynamic circulation models. Animal movement is based on a search of an environmental-habitat feature vector that maximizes cohort production dynamics. We implemented a numerical version of the mathematical model and used scientific data visualization to display real-time results. We discuss two applications to regional-scale ecosystems that study the space-time behavior of recruitment and predator-prey production dynamics, and fishery and environmental impacts on cohorts of: (1) spotted seatrout ( __Cynoscion__ __nebulosus__) and pink shrimp ( __Penaeus__ __duorarum__) in the tropical waters of Biscayne Bay, Florida; and, (2) Atlantic menhaden ( __Brevoortia__ __tyrannus__) and striped bass ( __Marone__ __saxatilis__) in temperate waters along the US Atlantic coast.

**Aumann,** Craig - University of Alberta

Title of Presentation: "How Transient Patches Affect Population Dynamics: The Case of Hypoxia and Blue Crabs."

Transient hypoxic patches, where patch characteristics change quickly relative to the timescale of the population or community involved, may have important consequences for the population dynamics of many estuarine species. For commercially important blue crabs (Callinectes sapidus), one hypothesis is that temporary reductions in habitat caused by hypoxia increase rates of cannibalism. A second hypothesis is that crab population dynamics are a result of food limitation caused by hypoxia-induced mortality of the benthos. To assess these alternative hypotheses and to determine whether population dynamics under transient patches differs from static patches, we developed a spatially explicit individual-based model of blue crabs in an hierarchical framework to connect the autoecology of crabs with the spatial and temporal dynamics of their physical and biological environments. The primary scenario considered examined the interactive effects of hypoxic extent versus static and transient patch types. Static patches resulted in populations limited by egg production/recruitment while transient patches lead to populations limited by the effects of cannibalism/patch interactions. Crab survivorship was greatest for scenarios with the largest hypoxic extents which also had the lowest prey abundance. Nearly all crab mortality was accounted for by aggression - not starvation. We conclude that crab populations in typical estuaries are primarily controlled by increased density-dependent cannibalism resulting from temporary habitat loss and not food limitation.

**Britton,** Nick - University of Bath

Title of Presentation: "Traveling Waves in Systems of Integrodifference Equations."

**Calabrese,** Justin - University of Maryland

Title of Presentation: "Species Diversity in a Patchily Distributed Bark Beetle Assemblage: Towards a Synthesis of Aggregation Theory and Neutral Theory."

The study of species that utilize discrete, ephemeral, patchily distributed resources has focused on how intraspecific aggregation mediates coexistence in the face of potentially strong asymmetric competitive interactions, while paying comparatively little attention to species abundance. In contrast, neutral theory has sought to explain species abundance patterns in a range of assemblages by assuming that individuals or species are equivalent and that competition is symmetric or non-existent. Though the equivalence assumption is likely to be false in most cases, recent advances in neutral theory suggest that many non-neutral mechanisms can lead to "functional equivalence" among species or individuals in an assemblage, and therefore facilitate neutral dynamics. Here we explore the possibility that intraspecific aggregation may decouple interspecific competition from species abundance patterns in a discrete resource system, resulting in an apparently neutral assemblage. We focus on an assemblage on wood boring beetles that breed in the fallen petioles of Cecropia insignis trees in Costa Rica. We utilize a hypothesis testing approach to demonstrate, through a suite of analyses, that species within the Cecropia assemblage are strongly aggregated and that species diversity patterns at the assemblage level are consistent with the predictions of a neutral model. Spatially patchy, discrete resource systems provide excellent opportunities to explore how aggregative processes may lead to the creation of functionally neutral assemblages.

**Caswell,** Hal - Woods Hole Oceanographic Institution

Title of Presentation: "Demography and Dispersal"

Populations grow or shrink in size by demography. They spread in space by dispersal. What happens when you combine the two?

(co-authors Christine Hunter, Mike Neubert)

**Chesson, **Peter - University of California

Title of Presentation: "Quantifying and Testing Mechanisms in Spatial Ecology."

The complexity of spatial models leads to critical challenges in understanding how these models work, developing theory based on model results, and testing theory in the field. Many of the most important results in spatial models can be traced to interactions between nonlinearities in the dynamical equations and variation in space in population densities or environmental factors. In spatial models of multispecies competition, study of such interactions has led to partitioning of landscape level population growth rates into contributions from three coexistence mechanisms, termed the spatial storage effect, fitness-density covariance and nonlinear competitive variance. Each of these mechanisms can be viewed as a particular way in which spatial niches can promote coexistence in variable landscapes. These mechanisms can be quantified in similar ways in models (both analytically and by simulation) and in the field. This quantification leads 1) to improved understanding of the operation of mechanisms through the ability to dissect out contributing components, 2) to a powerful way of analyzing models numerically and by simulation, and 3) promises field tests of species coexistence mechanisms of exceptional power and confidence.

**Crone,** Elizabeth - University of Montana

Title of Presentation: "Mechanisms of Synchronous Mast-seeding in Trees and Wildflowers."

Synchronous, episodic, mast-seeding is increasingly recognized as common in plant populations. Traditionally, ecologists have focused on explaining causes of mast-seeding at two levels: economies of scale, such as decreased pollen limitation or seed predation, and weather or climate correlates, that could act as proximate cues for synchronous flowering. Recent mathematical models show that synchronous mast-seeding could also result from resource allocation and storage within individual plants, coupled by pollen limitation in low-flowering years. From a theoretical perspective, these models imply that synchronous, episodic fluctuations in reproduction could occur in the absence of external environmental variation. From a practical perspective, these models imply that variation in resources that limit reproduction would not necessarily correlate with fluctuations in seed output. I explore the importance of each implication for two mast-seeding plants: Astragalus scaphoides (Bitterroot milkvetch), a perennial wildflower for which individual and population processes are sufficient to cause alternate-year flowering in a constant environment; and Pinus albicaulis (whitebark pine), a tree for which predicting mast years is an important management question, that managers have not been able to solve using traditional statistical models.

*KEY WORDS:* Ardisia elliptica; biological invasions; dispersal kernel; integrodifference equation; invasion rate; matrix models

**Diekmann,** Odo - Utrecht University

Title of Presentation: "Can a Species Keep Pace with a Shifting Climate?"

Due to climate changes, the favorable habitat for a species may crumble away in the South while expanding into the North. Will a species be able to move along? How does the answer depend on domain size, speed of climate shift, individual movement?

In the context of a caricatural diffusion model one can answer these questions. The talk is based on joint work with Kees Nagelkerk, Henri Berestycki and Paul Zegeling and it overlaps considerably with work done by Potapov and Lewis.

**Doebeli,** Michael - University of British Columbia

Title of Presentation: "Adaptive speciation in spatially structured populations."

Understanding the evolutionary origin and maintenance of diversity is a fundamental biological problem. Generation of diversity occurs when an ancestral lineage splits into two descending lineages, a process called speciation. The past years have seen a shift in our understanding of this process, which has shaken the foundations of one of the most cherished beliefs in evolutionary biology: that speciation is no more than a by-product of geographical isolation. I review some of the theoretical work showing that adaptive speciation - speciation directly favored by natural selection rather than incidental speciation following geographical isolation - is a plausible evolutionary process. This work is based on the mathematical framework of adaptive dynamics, and in particular on the phenomenon of evolutionary branching due to frequency-dependent ecological interactions in well-mixed populations. I then describe extensions of this theory to spatially structured populations. We have shown that spatial structure can facilitate adaptive speciation due to competitive interactions, and we are currently investigating adaptive speciation in spatially structured predator-prey models.

**Donahue,** Megan - California State University Los Angeles

Title of Presentation: "Scaling-up Population Dynamics Using Scale Transition Theory."

A key insight from recent advances in spatial theory is that spatial heterogeneity changes population dynamics by interacting with nonlinear population processes. Scale transition theory is a method for scaling from local to regional population dynamics by combining local, nonlinear population models with regional spatial variation in population parameters. The approach has three critical steps: identifying and fitting local nonlinear population models, characterizing spatial variation in population parameters, and using a second-order approximation to predict dynamics at the regional scale. I describe this approach to spatial scaling and its application to an intertidal system. Marine systems are characterized by high variability in larval supply and strong post-settlement density dependence. I apply scale transition theory to the population dynamics of an intertidal porcelain crab, Petrolisthes cinctipes, to illustrate how local population processes interact with spatial variation to change regional population dynamics.

**Fitzgibbon,** William - University of Houston

Title of Presentation: "Semi Linear Parabolic Systems Modeling the Spread of Disease Between Populations with Non-coincient Habitats."

We are concerned with mathematical models describing the circulation of infectious diseases spatially among two or more populations. Beginning with the work of Bailey it has become common to describe this process with systems of ordinary differential equations. Further work has used weakly coupled semi linear parabolic systems similar to reaction diffusion systems to describe the spatio-temporal spread of the diseases. Here most work has been limited to populations confined to a single habitat. However, one can also consider diseases that are circulating between populations confined to overlapping but non-coincident habitants. Here the modeling produces spatially coupled semi linear parabolic systems. In this case the resulting equations present a much formidable analytic challenge. In the case at hand we shall illustrate the foregoing considerations with three successively more complex systems that model that the vector transmission of a disease between two host populations.

**Griffith,** Daniel A. - Department of Geography & Regional Studies & CESP Research Fellow for Spring 2005, University of Miami

Title of Presentation: "Spatial Modeling in Ecology: Principal Coordinate and Spatial Filter Analyses Exploiting Relative Location Information."

Spatial structuring of ecological communities may originate from common environmental variables underlying them and/or from spatial interaction processes occurring within them. In order to assess the importance of these two sources, spatial relationships need to be introduced explicitly into statistical models describing ecological communities. Two approaches that capture this category of spatial relationships are the principal coordinates of neighbor matrices method proposed by Legendre and his colleagues, and spatial filtering based upon geographic connectivity matrices developed by Griffith and his colleagues. In both cases, the goal is to create spatial predictors that can be easily incorporated into conventional regression models, in order to avoid resorting to the family of spatial autoregressive and trend surface (i.e., gradient) models that have become popular amongst a number of quantitative spatial scientists. One important advantage of these two approaches is that they provide a flexible tool that allows the full range of general and generalized linear modeling theory to be applied to ecological and geographical problems in the presence of non-zero spatial autocorrelation.

This presentation outlines the conceptual framework upon which principal coordinates of neighbor matrices methodology and spatial filtering methodology are built, demonstrating how these methods are linked to spatial autocorrelation. Paralleling principal components analysis in multivariate statistics, these methods begin by diagonalizing a n-by-n geographic structure matrix, extracting eigenvectors that represent orthogonal and uncorrelated map pattern components of spatial autocorrelation. These eigenvectors, themselves, can be used directly as synthetic explanatory variables in general and generalized linear models. Comparisons between the two methodologies are illustrated with the georeferenced Oribatid mite dataset that already has been analyzed by Borcard and Legendre.

*KEY WORDS:* ecological community; eigenvalue; eigenvector; Moran Coefficient; principal coordinates of neighbor matrices; spatial autocorrelation; spatial filter; spatial model; spatial structure.

**Gross,** Louis - University of Tennessee

Title of Presentation: "Space and Control: Linking Theory and Practice for Natural Resource Management."

Environmental problems related to natural resource management span a wide variety of spatial, temporal and biotic hierarchy levels. Resource managers are charged with local, short-term decisions regarding controlling access and types of use for sub-areas within larger natural areas, as well as much longer-term planning for entire areas as large as many thousands of hectares. Similarly, regional-scale planning for water flows, hunting regulations, preserve allocation, and forest harvesting requires stakeholders to provide input to the process, and such planning has led to numerous contentious public debates throughout the US. Moving from ecological theory to practice requires the use of computational models utilizing the best available science to analyze potential effects. To be effective, such models must be defensible scientifically, readily available to various stakeholders and extensible to allow new users to evaluate alternative plans, assumptions and choices of criteria to prioritze management actions.

Much of natural resource management can be viewed as problems in spatial control: what to do, where to do it, when to do it, and how to monitor and assess the success of the effort. The ready availability of computatational capability opens up a variety of new opportunities for spatially-explicit control methods to be made accessible for resource managers concerned with site-specific issues as well as with regional-level coordination of effort. I will describe efforts to develop spatial control as a general approach applicable to problems of invasive species management, hydrology planning, and endangered species conservation planning. I will also discuss educational issues associated with training resource managers in computational science.

**Holmes,** Elizabeth - National Marine Fisheries Service

Title of Presentation: "Diffusion Approximation Approaches For Metapopulation Viability Analysis."

Population viability analysis (PVA) assesses the rate of population decline and the risks of extinction or quasi-extinction over a defined time horizon for a population of concern. In the past decade PVA has gained broad acceptance in the conservation community as a useful tool for assessing and managing 'at risk' species. Indeed, the International Union for the Conservation of Nature (IUCN) revised Red List Criteria, probably the most widely applied set of decision rules for determining the status of at risk species, is in large part defined by metrics that require some form of PVA. Although many PVAs are focused on single populations in single sites, there are often needs for spatially explicit PVAs for metapopulations distributed among multiple sites. Unfortunately, for species of conservation concern, it is unusual to have the data needed to parameterize a detailed spatially-explicit model. One approach to the problem of limited population data is to find a diffusion approximation that correctly models the long-run stochastic properties of a complex population process. This approach has been used successfully for single population models. In this talk, I present theoretical work on a diffusion approximation for metapopulation dynamics and discuss an approach to maximum likelihood estimation for such a model. One of the main practical implications of this approach is that it is not necessary to know the multitude of parameters describing the local dynamics, dispersal levels, spatial patterns of dispersal, and spatial synchrony between local populations in order to make basic predictions about the statistical distribution of the long-term metapopulation trajectories for use in PVA. Calculation of PVA metrics for metapopulations is illustrated using data on threatened Columbia River salmon.

**Holt,** Robert - University of Florida

Title of Presentation: "Temporal Fluctuations and 'Inflation' in Metapopulations: Some Theoretical Explorations."

In spatially heterogeneous landscapes, some habitats may be persistent "sources", providing immigrants sustaining populations in unfavorable "sink" habitats (where extinction is inevitable without immigration). Recent theoretical and empirical studies of source-sink systems have demonstrated that temporal variability in the local growth rate in sinks can substantially increase average abundance, provided the variation is positively autocorrelated—in effect, temporal variation "inflates" average abundance. Here we extend these results by considering a metapopulation in which all habitat patches are sinks, occupied by a species with discrete generations that can disperse among patches. Using numerical studies, buttressed by analytic results, we show that temporal variation and limited dispersal can jointly permit the indefinite persistence of the metapopulation, and that positive autocorrelation both lowers the magnitude of variation required for persistence, and increases the average abundance of persisting metapopulations. These effects are weakened but not destroyed if variation in local growth rates is spatially synchronized, and dispersal is localized. We show that the inflationary effect is robust to a number of extensions of the basic model, including habitat fragmentation, demographic stochasticity and strong density dependence. Since ecological and environmental processes contributing to temporally variable growth rates in natural populations are typically autocorrelated, these observations may have important implications for species persistence.

**Horvitz,** Carol C. & Anthony Koop - University of Miami

Title of Presentation: "Relative Importance of Avian and Mammalian Seed Dispersers to Wavespeed of an Invasive Shrub in Everglades Nat'l Park."

The spread of species across landscapes has interested scientists across many disciplines. As a result, various different models of invasion have been developed to study the patterns and rates of invasion. Recently, Neubert and Caswell developed a model of invasion for an integrodifference equation model for wavespeed of a stage-structured population. The goal of this study was to estimate the rate of spread of the invasive shrub Ardisia elliptica and evaluate its relative sensitivity to different seed dispersal vectors. The structured-dispersal model accounted for gravity-, bird- and raccoon-dispersal. Inverse modeling was used used to parameterize Gaussian dispersal kernels from maps of seedlings. Infrequent long-distance dispersal by raccoons was very important in determining invasion rate. Furthermore, germination, growth of small stage classes and reproduction by smaller sized plants were all more important to invasion rate than population growth. The results from this study demonstrate the importance of separately modeling the contribution of different dispersal vectors and of the utility of the Neubert-Caswell model. However, this study probably underestimated rare long-distance dispersal as do many other studies of invasions.

**Klausmeier,** Christopher - Georgia Institute of Technology

Title of Presentation: "Algal Games: The Vertical Distribution of Phytoplankton in Poorly-mixed Water Columns."

Phytoplankton often face the dilemma of living in contrasting gradients of two essential resources: light that is supplied from above and nutrients that are often supplied from below. In poorly mixed water columns, algae can be heterogeneously distributed, with thin layers of biomass found on the surface, at depth, or on the sediment surface. Here, we show that these patterns can result from intraspecific competition for light and nutrients. First, we present numerical solutions of a reaction-diffusion-taxis model of phytoplankton, nutrients, and light. We argue that motile phytoplankton can form a thin layer under poorly mixed conditions. We then analyze a related game theoretical model that treats the depth of a thin layer of phytoplankton as the strategy. The evolutionarily stable strategy is the depth at which the phytoplankton are equally limited by both resources, as long as the layer is restricted to the water column. The layer becomes shallower with an increase in the nutrient supply and deeper with an increase in the light supply. For low nutrient levels, low background attenuation, and shallow water columns, a benthic layer occurs; for intermediate nutrient levels in deep water columns, a deep chlorophyll maximum occurs; and for high nutrient levels, a surface scum occurs. These general patterns are in agreement with field observations. Thus, this model can explain many patterns of algal distribution found in poorly mixed aquatic ecosystems.

**Li,** Bingtuan - University of Louisville

Title of Presentation: "Spreading Speeds in Cooperative Systems."

Introduced species constitute a major ecological threat to existing ecosystems. One crucial measure of a species' invasiveness is the rate at which it spreads into a new environment occupied by resident species. The so-called "linear determinacy" equates spread rate in the full nonlinear model with spread rate in the system linearized about the leading edge of the invasion. In this talk I will present conditions which insure that the linear determinacy is valid for cooperative systems. The conditions are such that they can be verified for a class of reaction-diffusion models and a class of discrete-time models with spreading kernels. I will also discuss the existence of traveling wave solutions to discrete and continuous time systems, and relate the minimum traveling wave speeds to spreading speeds (Joint work with Hans Weinberger and Mark Lewis).

**Lopez-Gomez,** Julian - Universidad Complutense de Madrid

Title of Presentation: "Strategic Symbiosis in Competitive Environments."

In this talk we study how the effects of strategic local symbiosis provide us with an exceptional mechanism to increase productivity in extremely competitive environments. The most striking consequence from our analysis is that the productivity can blow-up in cooperation areas, though some of the species might become extinct elsewhere, by segregation, as a result of the aggressions received from the competitors. As a by-product, it becomes clear why strategic symbiosis effects in population dynamics, or competitive markets, help to avoid massive extinction of populations, or industrial and financial companies. In a further stage of our analysis we have detected that, in the presence of strategic symbiosis, high level aggressions might provoke dramatic increments of the complexity of the system; a mechanism that might held to explain the extraordinary bio-diversity of Earth's biosphere, as well as the complexity of global economy.

**Lou,** Yuan - Ohio State University

Title of Presentation: "Lotka-Volterra Competition Model with Small or Intermediate Diffusion Coefficients."

It is well known that for reaction-diffusion 2-species Lotka-Volterra competition models with spatially independent reaction terms, global stability of an equilibrium for the kinetic system implies global stability for the reaction-diffusion system. This is not in general true for spatially inhomogeneous models. We show here that for an important range of such models, for small enough diffusion coefficients, global convergence to an equilibrium holds for the reaction-diffusion system, if for each point in space the kinetic system has a globally attracting hyperbolic equilibrium. We will also discuss the effect of intermediate diffusion coefficients.

**Lutscher,** Frithjof - University of Alberta and University of Calgary

Title of Presentation: "Models for Stream Ecosystems."

Individuals in rivers and streams are subject to unidirectional flow that potentially influences their movement. In addition, several processes such as groundwater exchange and runoff tend to create resource gradients along rivers and streams. We present some simple models for such systems and analyze them with respect to traveling waves, critical domain size, pattern formation through biotic (competition, predation) or abiotic (flow rates) factors. Some of these simple models are complemented by research in small experimental streams.

**Magal,** Christelle - Université de Tours

Title of Presentation: "Spatio-temporal Dynamics of a Host and a Generalist Parasitoid."

This work is triggered by the need to control an invasive leafmining microlepidopteron attacking horse chestnut trees. The moth was first observed in Macedonia in 1985 and has invaded most European countries by now. It is attacked by a range of parasitoids, most of them being generalist already present in the ecosystem. Our goal was to model the system and explore the stability of the system in the search of conditions enabling a control of the pest. We followed the work of Owen & Lewis (1) and assumed that its main parasitoid has a type 2 functional response and can survive without the pest on other leafminers. Under these assumptions, we obtain a system of reaction-diffusion equations.

We first investigated the system without incorporating space. We identified at most six equilibrium points and discussed the existence and stability of each of them. We then carried out numerical simulations of the PDE and obtained the speed of propagation of the leafminer with and without parasitoids. We thereby identified the conditions in which parasitoids can reverse the leafminer advance. We also studied the effect of the diffusion rate on the propagation of leafminers. Overall, incorporation of space, combined with the polyphagy of the parasitoids, leads to a decrease of chances for coexistence, in contrast to several other models in which space promotes coexistence by enabling hosts to escape in space.

1. M.R. Owen and M.A. Lewis (2001). How Predation can Slow, Stop or Reverse a Prey Invasion. Bulletin of Mathematical Biology, 63, 655-684.

**McManus,** John W. and Felimon C. Gayanilo, Jr. - National Center for Caribbean Coral Reef Research (NCORE) - Rosenstiel School of Marine and Atmospheric Science, University of Miami.

Title of Presentation: "Predator- Prey Interdependence in a Spatially Heterogeneous Model Aquatic Ecosystem."

Agent-based models of were constructed of simple aquatic ecosystems with various levels of trophic complexity, consisting of benthic algae, herbivores and piscivores. Failure of the mean-field assumption due to spatial heterogeneity led to considerable variations from Lotka-Volterra dynamics. It was necessary to alter the growth and population characteristics of the populations at lower levels when higher trophic levels were added, in order to avoid sudden population collapses. Following such community accommodation, reduction or removal of the higher trophic level populations led to fluctuations leading to local extinctions at lower levels, except for algae which then proliferated. This presentation will report on continuing work to determine the generality of this phenomenon, and potential implications for fisheries and the need to conserve predators in the management of aquatic systems.

**Nevai,** Andrew L. - University of California, Los Angeles

Title of Presentation: "A Mathematical Model of Plant Competition for Light."

We present an analytic model to study the outcome of competition between two plant species that interact only by their shared use of sunlight. This horizontally homogeneous model accounts for the light environment created by the relative vertical placement of the species' leaves. From the description of a plant population's vertical structure and carbon budget we construct a single-species population growth equation in which the specific growth rate function is a functional of vertical leaf distribution and incorporates nearly a dozen directly measurable primary plant parameters. Direct scrutiny of the specific growth rate function reveals the qualitative manner in which each feature of plant structure and function influences population growth, persistence, and equilibrium abundance. We subsequently derive a Kolmogorov system of competition equations in which each specific growth rate function is a functional of the two species' vertical leaf distributions. As each species nullcline is defined by an implicit equation, we will describe some of the implicit methods that we use to determine their shape and relative position in phase space. In particular, we will demonstrate that, under simplifying assumptions, the nullclines intersect at most once. We will also outline how nullcline endpoint analysis enables us to divide parameter space into regions in which either species exclusion or stable coexistence occurs. Finally, we show that in some respects this model is more inclusive than it initially appears to be. By modifying the principal assumptions of the underlying one-species model, we are sometimes able to incorporate more realistic descriptions of plant structure, function, and habitat. In each case, the resulting model behavior is compared to the predictions made by the unenriched one-species model. This work is in collaboration with Richard R. Vance of UCLA.

**Nisbet,** Roger - University of California, Santa Barbara

Title of Presentation: "Population Response to Environmental Stress in Systems with Unidirectional Flow."

I present models that describe the dynamics of populations living in media with strong unidirectional flow (e.g., aquatic organisms in streams). In many such systems there is a "response length" that characterizes the distance downstream over which the impact of a point source disturbance is felt. The response to spatially distributed (non point source) disturbances can be calculated by representing it as a (Fourier) sum of components at different wavelengths. The steady state population distribution "tracks" short wavelength variation, but partly "averages" long wavelength disturbances. The onset of averaging occurs with components whose wavelength is similar in magnitude to the response length. I review the biological information required for calculation of the response length, and demonstrate its use in a study of population distributions near a coastal power plant and in studies of the dynamics of stream invertebrates.

**Olson,** Donald B. - RSMAS/MPO University of Miami

Title of Presentation: "Structured Dynamics in Fisheries and Agriculture."

The spatial dynamics of interacting populations that are explicitly structured in terms of age, resource allocation, and temporally varying conditions within the available space itself are considered with applications to fisheries resources and the development of agriculture in response to economics and climate change. The first example considers an age structured population consisting of a benthic organism with includes juvenile and adults using seabed resources and a planktonic larval stage. The populations are also structured in terms of the state of the benthic habitat and its connectivity. The model is parameterized in relationship to conch ( *Strombus gigas*) foraging on sea grass beds. The dynamics suggested strong Allee effects in the sense that larval production and settlement both require a critical mass of adult conch. The distribution of habitat is also considered for realistic sea grass distributions in the Caribbean. The application of the model to the design and implementation of marine protected areas is discussed. The second example considers the development of competing agricultural sectors in the Pampas of Argentina. The commodities considered include cattle and soybeans. These are being exploited by a a human population that is structured in terms of capital, markets, and preferred occupation. The entire system is spatially bound by climatic shifts in the wetness of the Pampas and forced by larger scale market and capital issues. The analysis considers the conditions under which one commodity can replace another. The stability of commodity interactions under different financial and climate scenarios is considered. Of particular interest is the response of the system in terms of transition towards equilibrium in comparison to the time scale of the forcing terms. The history of agriculture in the Pampas is used to consider the impact of technological and market changes on the state of agriculture in the region. Other regimes that could be considered using similar model structures are considered.

**Ou,** Chun-Hua - York University

Title of Presentation: "Existence and Uniqueness of Traveling Wavefronts in a Hyperbolic-parabolic Model with Delayed Non-local Interaction."

In this talk, we are concerned with a single species population with two age classes: immature and mature. We assume that the spatial movement of the mature individual follows the Fick’s law, but with a time delay, and the diffusion of immature population is rapid. Upon this assumption, we establish a hyperbolic equation for the model.

Our main purpose here is to study the existence and uniqueness of traveling wavefronts, in the case when the birth function b(u) possesses the property of so-called bistable nonlinearity. The main difficulty here is due to the fact that the Comparison Principle doesn’t hold any more. Consequently, we cannot directly use the squeezing technique to verify the existence as well as the uniqueness of traveling wavefronts as we have worked before. To overcome this difficulty, we introduce a new evolution system which is associated with the original hyperbolic equation and for which the squeezing technique is applicable. Finally, by studying the relation between the new system and the original hyperbolic equation, we are successful in obtaining the main result in this talk.

**Ovaskainen,** Otso - Helsinki University

Title of Presentation: "Asymptotically Exact Analysis of Stochastic and Spatial Systems."

It is well known that both space and stochasticity can play central roles in ecological systems. Theoretical ecologists have developed numerous approaches that apply to spatial and stochastic systems, such as simulations, pair-approximations, and spatial moment equations. However, these approaches are heuristic in the sense that they do not give a mathematically rigorous description of the system. For example, the usage of spatial moment equations involves a choice of moment closure, different choices leading to different answers.

We have developed a new method for the analysis of continuous-space continuous-time stochastic and spatial systems that is based on the theory of distributions (to account for space) and on a systematic perturbation expansion of the underlying stochastic differential equations (to account for stochasticity). As an example, we apply the method to a metapopulation model in which the habitat patches are distributed either at random or in a correlated manner, and in which the landscape can be dynamic so that patches are created and destroyed due to a disturbance process. We assume that the species follows a spatial version of the Levins metapopulation model, in which the colonization rate of an empty patch is a sum of contributions from the occupied patches. Each occupied patch is assumed to spread its colonization effort according to a dispersal kernel, which is characterized by a length scale L that is proportional to the mean dispersal distance.

As L becomes infinitely large, colonization becomes global, and the metapopulation follows the spatially implicit Levins model. If L is finite, the behaviour of the system deviates from the Levins model. For example, we show that the equilibrium fraction of occupied patches is given by $p^*=p_0-c/L^2 + {\cal O}(L^{-3})$, where $p_0$ is the equilibrium state of the Levins model. Our method allows us to compute the constant $c$ without resorting to heuristic assumptions. Comparison with simulations show that the result is not only asymptotically (as $L\rightarrow \infty$) correct, but good also when $L$ is relatively small.

While our method is technically somewhat more involved than e.g. the moment closure method, it is able to produce asymptotically exact results. The method thus promises to be a powerful general technique to study the roles played by spatial correlations and temporal variability

**Padron,** Victor - University of Minnesota and Universidad de Los Andes

Title of Presentation: "An Evolutionarily Stable Migration Strategy Under Logistic Growth."

We consider a single-species model which is composed of several patches connected by linear migration rates and having logistic growth. A spatially varying, temporally constant environment, is introduced by the non-homogeneity of its carrying capacity. Under this condition any type of purely diffusive behavior, characterized in our model by symmetric migration rates, produces an uneven population distribution, i.e. some locations receive more individuals than can be supported by the environmental carrying capacity, while others receive less.

Using an evolutionarily stable strategy approach we show that an asymmetric migration mechanism, induced by the heterogeneous carrying capacity of the environment, will be selected. This strategy produces a much more balanced population distribution, which agrees with the postulates of the Ideal Free Distribution in foraging theory. In the case of the standard logistic growth, with constant intrinsic growth rate, this distribution maximizes the total population.

This is joint work with María Cristina Trevisan, Univeridad de Los Andes, Mérida, Venezuela.

**Potapov,** Alex - University of Alberta

Title of Presentation: "Control of a Lake System Invasion: Bioeconomics Approach."

We consider the problem of control of an alien species invasion into a system of lakes. It is assumed that invaders are transported between the lakes by boaters, and boat treatment can reduce their transport. We propose a mathematical model of the invasion process and prevention measures along with related costs. The corresponding optimal control problem has been studied analytically and numerically. The results show that, in some cases, optimal control may be the absence of any measures. Discounting (lower cost of future gain and losses) may decrease the control intensity. In presence of Allee effect the invader spread can be stopped and the optimal spatial pattern of control can be obtained.

**Schreiber,** Sebastian - The College of William and Mary

Title of Presentation: "Coevolution of Patch Choice in Host-Parasitoid Systems."

Parasitoids and hosts often live in patchy environments. From the perspective of the host, patches may vary in plant nutritional quality, plant defenses, and microclimate. From the perspective of the parasitoid, patches may vary in host abundance, host size, and microclimate. In the first part of this talk, I will discuss theoretical results about the coevolution of host and parasitoid patch preferences and under what conditions coevolution stabilizes host-parasitoid interactions.

**Thieme,** Horst and Maia Martcheva - Arizona State University

Title of Presentation: "Infinite ODE Systems Modeling Size-structured Metapopulations and Macroparasitic Diseases."

Spatially implicit metapopulation models with discrete patch-size structure and host-macroparasite models which distinguish hosts by their parasite loads lead to infinite systems of ordinary differential equations.

We develop a this-related theory in sufficient generality and also establish conditions for the solution semiflow to be dissipative, have a compact attractor for bounded sets, and be uniformly persistent. We also prove that a metapopulation dies out, if nobody emigrates from its birth patch or if empty patches are not colonized.

**Zhu,** Huaiping - York University

Title of Presentation: "Transient Dynamics in a Model for the Transmission of WNV."

We propose a system of differential equations to model the transmission of WNV between mosquitoes and birds. The global stability of the disease free equilibrium and the Hopf bifurcation of the endemic equilibrium will be discussed. However for the infections diseases like WNV, the long term behavior of the system is less important compare to the short term dynamics in terms of the control and prediction. We introduce a geometrical approach to investigate the short-term dynamics of the interaction between the mosquitoes and birds.

**Zou,** Xingfu - University of Western Ontario

Title of Presentation: "Dynamics of a Population Model with Dispersals over Three Patches."

We consider a system of nonlinear delay differential equations describing the matured population dynamics of a single species with dispersals over three patches. By studying existence of nonnegative homogeneous equilibria and their stability, and bifurcations from the equilibria, we obtain very rich dynamics for the model, including transient oscillations, phase-locked oscillations, mirror-reflecting waves and standing waves near the equilibria. We also work out formulas for determining the statibility of the bifurcating periodic solutions.

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