### Organizers

Central questions in ecology that directly impinge on applications involve an understanding of spatial aspects of natural systems. While much of classical population and community ecology made assumptions about spatial homogeneity of systems, a large body of theory has developed over the past several decades that provide both key results and general framework for taking account of spatial factors as they affect population structure, community composition, and landscape-level structure. Some of the most critical questions that affect our ability to project the future trends of natural systems, and particularly how human actions impact these systems, must take account of spatial factors. This workshop will provide an entree to a variety of questions of ecological interest that rely upon interesting mathematics, and lead to problems that have had, as yet, relatively little mathematical analysis. The intent of the workshop is to provide an overview of some of the areas of spatial ecology that lead to interesting mathematics.

The themes of the workshop are framed at different levels of organization:

Population Level:

- How do underlying spatial heterogeneities affect population dynamics?
- How much of the observed spatial structure in populations is due to biotic versus abiotic factors?

Community Level:

- How do underlying spatial heterogeneities affect community dynamics?
- How much of spatial structure in communities is due to biotic versus abiotic factors?
- The above questions are to be addressed both within and between trophic levels.
- How do spatial aspects of systems affect disease dynamics?

Landscape Level:

- How do the spatial aspects of ecological systems affect natural resource management issues?
- How do social choice criteria interface with ecological spatial dynamics for systems in which there is the potential of human control?
- Can we manage natural systems, e.g, under what circumstances can we expect to be successful in determining the impact of human actions given uncertainties about our models and the stochasticity inherent in natural systems driven by abiotic factors?

### Accepted Speakers

Monday, March 13, 2006 | |
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Time | Session |

09:30 AM 10:30 AM | Andrew (Sandy) Liebhold - Spatial Dynamics of Forest Insect Outbreaks: The Role of Movement, Stochasticity and Habitat Heterogeneity There is a long history in the use of forest insect populations as model systems in the study of animal population dynamics. In this talk, I will provide an overview of how we have extended these studies to explore the spatial dynamics of forest insect populations. Much of this work has been motivated by the availability of digital maps that document the geographical extent of outbreaks of several forest insect species over successive years over large geographical regions. While the most striking temporal pattern evident in these data is the existence of periodicity in the presence of regional outbreaks, the most striking characteristic of the spatial dynamics of virtually all species is spatial synchrony. The term spatial synchrony refers to coincident changes in abundance among geographically disjunct populations. The ubiquitous presence of spatial synchrony provides an enticing challenge for population ecologists because this behavior may be caused by several different types of processes, most notably by a small amount of dispersal among populations or by the impact of a small but synchronous random effect, such as variation in weather. By comparing patterns of spatial synchrony among various species with varying dispersal capabilities, we have concluded that regional stochastic effects are the most likely cause of the ubiquitous synchrony in dynamics. However there is also evidence that long-distance dispersal can also greatly impact patterns of spatial dynamics as well. For example, populations of the larch budmoth in the European Alps exhibit recurring outbreak waves that move from west to east. We feel that the most likely cause of these population waves is an interaction between habitat heterogeneity (landscape connectivity) and the dominant reaction-diffusion processes that affect populations. We have also investigated how habitat heterogeneity can impact the synchronizing affect of regional stochasticity. Specifically, geographical variation in density-dependent population processes (caused by variation in habitat quality and other habitat characters) can greatly dilute the synchronizing effect of regional stochasticity. Geographical variation in habitat quality has probably received too little attention because it is one of the major determinants of observed patterns of spatial dynamics. Joint work with Ottar N. Bj?rnstad and Derek M. Johnson. |

01:30 PM 02:30 PM | Danny Grunbaum - Finding the Fudge Factor: Effective functional response curves for spatially and temporally heterogeneous ecological systems Simplified ordinary differential equation models of ecological systems have provided most of our theoretical understanding of consumer-resource dynamics. These models represent a macroscopic "mean field'' perspective that usually lumps together individuals differing in size, stage, mobility, physiological condition, genetic identity, micro-environment and many other details. One payoff of this simplification is a high degree of analytical and numerical tractability. Another is a clear experimental path to estimating regulatory mechanisms such as functional response curves. Functional responses are used in ODE models to estimate mean trophic rates as functions of mean resource and consumer densities. However, in most ecological systems resources and consumers are very heterogeneous in time and space. In heterogeneous landscapes, a given quantity of resource can be distributed in many ways, some of which result in higher consumption rates than others by specific types of consumers. This implies that functional responses cannot be functions only of mean resource and consumer densities. They may nonetheless be functions of mean densities along with a small number of other parameters. In this talk I will present a dimensional analysis that suggests what the other parameters might be, and how they might be used to derive ODE approximations for mobile consumers of heterogeneous resources using effective functional response curves. |

03:00 PM 04:00 PM | Claudia Neuhauser - Effect of Symbiotic Interactions on Plant Community Structure in Spatial Habitats Optimal foraging and habitat selection theories that are based on non-spatial, deterministic models predict evolution towards generalist strategies in fine-grained habitats and towards specialization in coarse-grained habitats. In addition, coevolutionary processes appear to favor extreme specialization among parasites. We introduce a spatially explicit, stochastic model that confirms the effect of habitat coarseness on specialization in the absence of coevolutionary processes. To understand the effects of coevolutionary processes, we introduce feedback between hosts and their symbionts into our spatially explicit, stochastic model. We find that mutualists modify their habitat so that it becomes coarse-grained, and parasites modify their habitat so that it becomes fine-grained, suggesting that the lifestyle of the symbiont prevents habitat types from becoming extreme. This is joint work with Nicolas Lanchier, University of Minnesota. |

Tuesday, March 14, 2006 | |
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Time | Session |

09:00 AM 10:00 AM | Nanako Shigesada - How is Spatial Dynamics of Invasion Influenced by Fragmentation Range expansions of invading species in homogeneous environments have been extensively studied since the pioneer works by Fisher (1937) and Skellam (1951). However, environments for living organisms are often fragmented by natural or artificial habitat destruction. Here we focus on how such environmental fragmentation affects the range expansion of invading species. We consider a single-species invasion in heterogeneous environments that are generated by segmenting an original favorable habitat into regularly striped, island-like, corridor-like, or randomly patched pattern. To deal with range expansion in such fragmented environments, we modify Fisher's equation by assuming that the intrinsic growth rate and diffusion coefficient vary depending on habitat properties. By examining the traveling periodic wave (TPW) speed in the striped environment, we first derive the ray speed in a parametric form, from which the envelope of the expanding range can be predicted. The envelopes show varieties of patterns, nearly circular, oval-like, spindle-like or vanishing in the extreme case, depending on parameter values. By deriving the formula for the ray speed, we discuss how the pattern and speed of the range expansion are affected by the size of fragmentation, and the qualities of favorable and unfavorable habitats. Secondly, we numerically solve extended Fisher's equation for island-like, corridor-like, and randomly patched environments with an initial distribution localized at the origin. The model is analyzed to examine how the spread of organisms is influenced by the patterns of habitat fragmentation, and which type of fragmentation is more favorable for species survival. |

10:30 AM 11:30 AM | Mark Kot - Integrodifference Equations, Invasions, and Branching Random Walks Biological invasions often have dramatic ecological and economic consequences. Thus, there is keen interest in models that correctly predict rates of spread of invading organisms. In this talk, I discuss the formulation and analysis of integrodifference equations, link deterministic integrodifference equations to stochastic branching random walks, and show how these models shed light on the rate of spread of invading organisms. |

01:00 PM 02:00 PM | Frithjof Lutscher - Life in the Flow: Persistence, invasion and competition in rivers The question how populations in rivers can persist despite flow- induced washout has been termed the "drift paradox". More generally, systems with unidirectional flow and flow-induced wash-out include rivers, plug-flow reactors, prevailing wind directions, and climate- change models. A first simple model in the form of a reaction-advection-diffusion equation explored persistence criteria by looking at the minimal domain size (Speirs and Gurney (2001), Ecology). Starting from this simple model, I will report on several extensions, namely: vertical structure in the population (drift and benthic state), spatial heterogeneity and the influence on channel geometry, effects of resource gradients, and competition of two species. I will focus on the minimal domain size, on speeds of upstream invasions, and on spatially mediated coexistence. |

Wednesday, March 15, 2006 | |
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Time | Session |

Thursday, March 16, 2006 | |
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Time | Session |

09:00 AM 10:00 AM | Otso Ovaskainen - 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 a systematic perturbation expansion of the underlying stochastic differential equations. The method allows one to analyze the spatial and stochastic model in an asymptotically (as interaction range tends to infinity) exact manner, in principle up to any order. Comparison with simulations show that the results are not only asymptotically correct but often good also when interactions are due to a few interacting neighbours only. As an example, we apply the method to study (i) metapopulation dynamics in a correlated and dynamic landscape, (ii) the effects of habitat loss and fragmentation, and (iii) the effects of space and stochasticity on a community of competing plant species. |

01:30 PM 02:30 PM | Michael Neubert - Spatial Bioeconomic Models and Fisheries Management Most analyses of spatial fisheries models assume a single owner whose goal is the maximization of sustainable yield. These analyses ignore the redistribution of fishing effort in response to economics and regulation. We will describe a simple, spatial, bioeconomic model that accounts for the open-access nature of most marine fisheries. We have used the model to find the maximum sustainable economic rent that can be obtained using various policy instruments (including taxes on aggregate effort, taxes on aggregate catch, effort quotas and catch quotas). We contrast these solutions to the rent-maximizing distribution of effort employed by a sole owner and to the distribution of effort in unregulated open access (when all profits are dissipated). In many cases, the solution contains unexploited regions in space. The locations of the unexploited regions, and the potential sustainable rent that results, depends upon the policy instrument employed. |

Friday, March 17, 2006 | |
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Time | Session |

09:00 AM 10:00 AM | Lou Gross - Spatial Modeling for Natural Resource Management: Invasions, IBMs and Big Science A great variety of very practical issues in natural resource management involve spatial aspects of natural systems. Indeed, one of the most commonly applied computational tools by managers are geographic information systems (GIS) which provide a spatial view of data and the potential implications of management actions. Despite their prominence, GIS have very limited capability for either the dynamic modeling familiar to mathematical ecologists, or for linkage with optimization methods for dynamic control. I will start by discussing invasive species management from a very simple spatially-implicit model, expand this to a more realistic model for spatial control of invasive plants with application Lygodium macrophyllum in south Florida, mention some spatial control aspects of individual-based models, and end with some lessons learned from a long-term, complicated modeling project for Everglades restoration. |

Name | Affiliation | |
---|---|---|

Allen, Linda | linda.j.allen@ttu.edu | Mathematics and Statistics, Texas Tech University |

Banerjee, Chandrani | chandrani12.banerjee@ttu.edu | Mathematics and Statistics, Texas Tech University |

Banerjee, Sandip | sandip.banerjee@helsinki.fi | Biological and Environmental Sciences, University of Helsinki |

Best, Janet | ||

Bevers, Michael | mbevers@fs.fed.us | Rocky Mountain Research Station, USDA Forest Service |

Bolker, Ben | Zoology, University of Florida | |

Buckley, Lauren | lbuckley@santafe.edu | Sante Fe Institute |

Calder, Catherine | calder@stat.ohio-state.edu | Statistics, The Ohio State University |

Cantrell, Steve | rsc@math.miami.edu | Mathematics, University of Miami |

Chan, Benjamin | rtd1@cornell.edu | Mathematics, Cornell University |

Cosner, Chris | c.cosner@math.miami.edu | Mathematics, University of Miami |

Cressie, Noel | Statistics, The Ohio State University | |

Cuddington, Kim | Biological Sciences, Ohio University | |

Djordjevic, Marko | mdjordjevic@mbi.osu.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Eftimie, Raluca | reftimie@math.ualberta.ca | Mathematics, University of Alberta |

Enciso, German | Mathematical Biosciences Institute (MBI), The Ohio State University | |

Fagan, Bill | bfagan@glue.umd.edu | Biology, College of Business and Management |

Giuggioli, Luca | lgiuggio@princeton.edu | Physics, University of New Mexico |

Goel, Pranay | goelpra@helix.nih.gov | Mathematical Biosciences Institute (MBI), The Ohio State University |

Grajdeanu, Paula | Mathematical Biosciences Institute (MBI), The Ohio State University | |

Gross, Louis | Ecology & Evolutionary Biology & Mathematics, University of Tennessee - Knoxville | |

Grunbaum, Danny | grunbaum@ocean.washington.edu | Biology, University of Washington |

Guan, Bo | guan@math.ohio-state.edu | Mathematics, The Ohio State University |

Guttal, Vishwesha | vguttal@princeton.edu | Physics, The Ohio State University |

Hambrock, Richard | hambrock@math.ohio-state.edu | Department of Mathematics, The Ohio State University |

Hameed, Jaffar | js415@msstate.edu | Department of Mathematics and Statistics, Mississippi State University |

Hartvigsen, Gregg | hartvig@geneseo.edu | Biology Department, College at Geneseo, SUNY |

Herrera, Guillermo | gherrera@bowdoin.edu | Department of Economics, Bowdoin College |

Hoy, Casey | hoy.1@osu.edu | Entomology, The Ohio State University |

Joshi, Hem Raj | joshi@xavier.edu | Department of Mathematics and CS, Xavier University of Louisiana |

Just, Winfried | Math, Ohio University | |

Kang, Sanghoon | kangs@ornl.gov | Environmental Sciences Division, Oak Ridge National Laboratory |

Kang, Yun | yun.kang@asu.edu | Math, Arizona State University |

Kenkre, V.M. (Nitant) | kenkre@unm.edu | Physics, University of New Mexico |

Knight, Kathleen | Laca0023@umn.edu | Ecology, Evolution and Behavior, University of Minnesota |

Kot, Mark | kot@amath.washington.edu | Applied Mathematics, University of Washington |

Lanchier, Nicolas | Nicolas.Lanchier@univ-rouen.fr | Mathematiques, Universit'e de Rouen (Haute-Normandie) |

Lee, Sung duck | sdlee@chungbuk.ac.kr | Information and Statistics, Chungbuk National University |

Lenhart, Suzanne | lenhart@math.utk.edu | Mathematics Department, University of Tennessee |

Leung, Tony | Anthony.Leung@uc.edu | Department of Mathematical Sciences, University of Cincinnati |

Li, Bai-Lian Larry | bai-lian.li@ucr.edu | Botany & Plant Sciences, University of California, Riverside |

Liebhold, Andrew | aliebhold@fs.fed.us | US Forest ServiceNorthern Research Station, Northern Research Station |

Lim, Sookkyung | Mathematical Biosciences Institute (MBI), The Ohio State University | |

Lin, Shili | lin.328@osu.edu | Statistics, The Ohio State University |

Loladze, Irakli | iloladze@math.unl.edu | MBI, The Ohio State University |

Lou, Yuan | lou@math.ohio-state.edu | Mathematics, The Ohio State University |

Lutscher, Frithjof | flutsche@uottawa.ca | Mathematics and Statistics, University of Ottawa |

Martinez, Salome | samartin@dim.uchile.cl | Ingenieria Matematica, Universidad de Chile |

McCormack, Robert | robert.k.mccormack@ttu.edu | Mathematics and Statistics, Texas Tech University |

Miller, Adam | miller@math.ucdavis.edu | Math, University of California, Davis |

Mischaikow, Konstantin | mischaik@math.rutgers.edu | Ctr. Dynamical Systems & Nonlinear Studies, Georgia Institute of Technology |

Neubert, Mike | mneubert@whoi.edu | |

Neuhauser, Claudia | neuha001@umn.edu | Ecology, Evolution and Behavior, University of Minnesota |

Nevai, Andrew | Mathematical Biosciences Institute (MBI), The Ohio State University | |

North, Ace | ace.north@helsinki.fi | Biological and Environmental Sciences, University of Helsinki |

Ovaskainen, Otso | otso.ovaskainen@helsinki.fi | Biological and Environmental Sciences, University of Helsinki |

Park, Jung Joon | park.824@osu.edu | OARDC, The Ohio State University |

Passino, Kevin | Electrical and Computer Eng, The Ohio State University | |

Pol, Diego | dpol@mbi.osu.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Rayfield, Bronwyn | bronwynrayfield@zoo.utoronto.ca | Department of Zoology, University of Toronto |

Rodriguez, Susana | rodriguez.219@osu.edu | EEOB, The Ohio State University |

Schugart, Richard | richard.schugart@wku.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Shi, Junping | jxshix@wm.edu | Mathematics, College of William and Mary |

Shigesada, Nanako | nshigesa@mail.doshisha.ac.jp | Faculty of Culture and Information Science, Doshisha University |

Srinivasan, Partha | p.srinivasan35@csuohio.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Stigler, Brandy | bstigler@mbi.osu.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Tian, Paul | tianjj@mbi.osu.edu | Mathematical Biosciences Institute (MBI), The Ohio State University |

Vadrevu, Krishna | vadrevu.2@osu.edu | Entomology, The Ohio State University |

Van Tassell, Jason | van-tassell.1@osu.edu | Evolution, Ecology and Organismal Biology, The Ohio State University |

Weiss, Howard | weiss@math.psu.edu | Mathematics, Pennsylvania State University |

Wilson, Will | wgw@duke.edu | Biology, Duke University |

Yakubu, Abdul-Aziz | ayakubu@Howard.edu | Mathematics, Howard University |

A great variety of very practical issues in natural resource management involve spatial aspects of natural systems. Indeed, one of the most commonly applied computational tools by managers are geographic information systems (GIS) which provide a spatial view of data and the potential implications of management actions. Despite their prominence, GIS have very limited capability for either the dynamic modeling familiar to mathematical ecologists, or for linkage with optimization methods for dynamic control. I will start by discussing invasive species management from a very simple spatially-implicit model, expand this to a more realistic model for spatial control of invasive plants with application Lygodium macrophyllum in south Florida, mention some spatial control aspects of individual-based models, and end with some lessons learned from a long-term, complicated modeling project for Everglades restoration.

Simplified ordinary differential equation models of ecological systems have provided most of our theoretical understanding of consumer-resource dynamics. These models represent a macroscopic "mean field'' perspective that usually lumps together individuals differing in size, stage, mobility, physiological condition, genetic identity, micro-environment and many other details. One payoff of this simplification is a high degree of analytical and numerical tractability. Another is a clear experimental path to estimating regulatory mechanisms such as functional response curves. Functional responses are used in ODE models to estimate mean trophic rates as functions of mean resource and consumer densities. However, in most ecological systems resources and consumers are very heterogeneous in time and space. In heterogeneous landscapes, a given quantity of resource can be distributed in many ways, some of which result in higher consumption rates than others by specific types of consumers. This implies that functional responses cannot be functions only of mean resource and consumer densities. They may nonetheless be functions of mean densities along with a small number of other parameters. In this talk I will present a dimensional analysis that suggests what the other parameters might be, and how they might be used to derive ODE approximations for mobile consumers of heterogeneous resources using effective functional response curves.

Biological invasions often have dramatic ecological and economic consequences. Thus, there is keen interest in models that correctly predict rates of spread of invading organisms. In this talk, I discuss the formulation and analysis of integrodifference equations, link deterministic integrodifference equations to stochastic branching random walks, and show how these models shed light on the rate of spread of invading organisms.

There is a long history in the use of forest insect populations as model systems in the study of animal population dynamics. In this talk, I will provide an overview of how we have extended these studies to explore the spatial dynamics of forest insect populations. Much of this work has been motivated by the availability of digital maps that document the geographical extent of outbreaks of several forest insect species over successive years over large geographical regions. While the most striking temporal pattern evident in these data is the existence of periodicity in the presence of regional outbreaks, the most striking characteristic of the spatial dynamics of virtually all species is spatial synchrony. The term spatial synchrony refers to coincident changes in abundance among geographically disjunct populations. The ubiquitous presence of spatial synchrony provides an enticing challenge for population ecologists because this behavior may be caused by several different types of processes, most notably by a small amount of dispersal among populations or by the impact of a small but synchronous random effect, such as variation in weather. By comparing patterns of spatial synchrony among various species with varying dispersal capabilities, we have concluded that regional stochastic effects are the most likely cause of the ubiquitous synchrony in dynamics. However there is also evidence that long-distance dispersal can also greatly impact patterns of spatial dynamics as well. For example, populations of the larch budmoth in the European Alps exhibit recurring outbreak waves that move from west to east. We feel that the most likely cause of these population waves is an interaction between habitat heterogeneity (landscape connectivity) and the dominant reaction-diffusion processes that affect populations. We have also investigated how habitat heterogeneity can impact the synchronizing affect of regional stochasticity. Specifically, geographical variation in density-dependent population processes (caused by variation in habitat quality and other habitat characters) can greatly dilute the synchronizing effect of regional stochasticity. Geographical variation in habitat quality has probably received too little attention because it is one of the major determinants of observed patterns of spatial dynamics.

Joint work with Ottar N. Bj?rnstad and Derek M. Johnson.

The question how populations in rivers can persist despite flow- induced washout has been termed the "drift paradox". More generally, systems with unidirectional flow and flow-induced wash-out include rivers, plug-flow reactors, prevailing wind directions, and climate- change models.

A first simple model in the form of a reaction-advection-diffusion equation explored persistence criteria by looking at the minimal domain size (Speirs and Gurney (2001), Ecology). Starting from this simple model, I will report on several extensions, namely: vertical structure in the population (drift and benthic state), spatial heterogeneity and the influence on channel geometry, effects of resource gradients, and competition of two species. I will focus on the minimal domain size, on speeds of upstream invasions, and on spatially mediated coexistence.

Most analyses of spatial fisheries models assume a single owner whose goal is the maximization of sustainable yield. These analyses ignore the redistribution of fishing effort in response to economics and regulation. We will describe a simple, spatial, bioeconomic model that accounts for the open-access nature of most marine fisheries. We have used the model to find the maximum sustainable economic rent that can be obtained using various policy instruments (including taxes on aggregate effort, taxes on aggregate catch, effort quotas and catch quotas). We contrast these solutions to the rent-maximizing distribution of effort employed by a sole owner and to the distribution of effort in unregulated open access (when all profits are dissipated). In many cases, the solution contains unexploited regions in space. The locations of the unexploited regions, and the potential sustainable rent that results, depends upon the policy instrument employed.

Optimal foraging and habitat selection theories that are based on non-spatial, deterministic models predict evolution towards generalist strategies in fine-grained habitats and towards specialization in coarse-grained habitats. In addition, coevolutionary processes appear to favor extreme specialization among parasites. We introduce a spatially explicit, stochastic model that confirms the effect of habitat coarseness on specialization in the absence of coevolutionary processes. To understand the effects of coevolutionary processes, we introduce feedback between hosts and their symbionts into our spatially explicit, stochastic model. We find that mutualists modify their habitat so that it becomes coarse-grained, and parasites modify their habitat so that it becomes fine-grained, suggesting that the lifestyle of the symbiont prevents habitat types from becoming extreme. This is joint work with Nicolas Lanchier, University of Minnesota.

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 a systematic perturbation expansion of the underlying stochastic differential equations. The method allows one to analyze the spatial and stochastic model in an asymptotically (as interaction range tends to infinity) exact manner, in principle up to any order. Comparison with simulations show that the results are not only asymptotically correct but often good also when interactions are due to a few interacting neighbours only.

As an example, we apply the method to study (i) metapopulation dynamics in a correlated and dynamic landscape, (ii) the effects of habitat loss and fragmentation, and (iii) the effects of space and stochasticity on a community of competing plant species.

Range expansions of invading species in homogeneous environments have been extensively studied since the pioneer works by Fisher (1937) and Skellam (1951). However, environments for living organisms are often fragmented by natural or artificial habitat destruction.

Here we focus on how such environmental fragmentation affects the range expansion of invading species. We consider a single-species invasion in heterogeneous environments that are generated by segmenting an original favorable habitat into regularly striped, island-like, corridor-like, or randomly patched pattern. To deal with range expansion in such fragmented environments, we modify Fisher's equation by assuming that the intrinsic growth rate and diffusion coefficient vary depending on habitat properties.

By examining the traveling periodic wave (TPW) speed in the striped environment, we first derive the ray speed in a parametric form, from which the envelope of the expanding range can be predicted. The envelopes show varieties of patterns, nearly circular, oval-like, spindle-like or vanishing in the extreme case, depending on parameter values. By deriving the formula for the ray speed, we discuss how the pattern and speed of the range expansion are affected by the size of fragmentation, and the qualities of favorable and unfavorable habitats.

Secondly, we numerically solve extended Fisher's equation for island-like, corridor-like, and randomly patched environments with an initial distribution localized at the origin. The model is analyzed to examine how the spread of organisms is influenced by the patterns of habitat fragmentation, and which type of fragmentation is more favorable for species survival.