### Organizers

In many situations it is adequate to assume that systems are homogeneously mixing and to take the limit of large populations, but in a number of cases the spatial distribution of individuals changes the behavior of the system. This workshop will focus on the impact of these effects on a wide variety of systems ranging from the scale of microbes to populations of plants and animals on a local and global scale. The workshop will bring together people who prove theoretical results about models, those use numerical and simulation results in their analysis, and involve a number of participants who work closely with biologists to analyze data. In this way we seek to stimulate the development, analysis, and application of new models.

### Accepted Speakers

Monday, April 16, 2012 | |
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Time | Session |

08:30 AM | Shuttle to MBI |

08:45 AM 09:15 AM | Breakfast |

09:15 AM 09:30 AM | Welcome, overview of workshop, and introductions: Marty Golubitsky |

09:30 AM 10:30 AM | |

09:30 AM 10:30 AM | Steve Krone - Particle Systems and Reaction-Diffusion Equations: connecting micro and macro models This will be something of an introductory talk that considers two types of spatial models used in population biology, and connections between them. Interacting particle systems can be thought of as "microscopic" level descriptions of populations, including interactions between discrete individuals and stochasticity. Reaction-diffusion equations provide deterministic models that can be thought of as "macroscopic" versions of particle systems through scaling limits. We will discuss the basic ideas behind this connection, treat a few examples, and try to understand the extent to which the two types of models predict the same behavior. |

10:30 AM 11:00 AM | Break |

11:00 AM 12:00 PM | J. Theodore Cox - Survival and coexistence for a class of stochastic spatial models We present a method for obtaining survival and coexistence results for a class of interacting particle systems. This class includes: a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, a model for the evolution of cooperation of Ohtsuki, Hauert, Lieberman and Nowak, and a continuous time version of a non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath and Levin. Each of these, for a range of parameter values, can viewed as a "voter model perturbation," meaning the dynamics are "close" to the dynamics of the voter model, a simple, neutral competition model. The voter model is mathematically tractable because of its dual process, a system of coalescing random walks. We show that when space and time are rescaled appropriately the particle density converges to a solution of a reaction diffusion equation. Analysis of this equation leads in some cases to asymptotically sharp survival and coexistence results, which are qualitatively different from the (pure) voter model case. This work with Rick Durrett and Ed Perkins is closely related to earlier work of Durrett and Neuhaueser on models with rapid stirring. |

11:00 AM 12:00 PM | |

12:00 PM 02:00 PM | Lunch Break |

02:00 PM 03:00 PM | |

02:00 PM 03:00 PM | David Hiebeler - Biological Dispersal Strategies of Internet Worms For the past decade, Internet worms (a type of malicious software similar to a virus) spreading through networks have been using biological strategies, such as hierarchical dispersal and adaptive strategies, to spread more efficiently among susceptible computers. There is a direct analogy between susceptible computers on the Internet and susceptible hosts in community-structured populations. Our measurements show that the Internet is an incredibly clustered heterogeneous environment when measured according to the dispersal strategy used by worms. We have used these measurements to build an epidemiological simulation model of the entire Internet (4.29 billion hosts, with roughly 2 million susceptible) efficient enough to run on an ordinary desktop computer. A worm which would have a basic reproduction ratio far less than one and therefore be quite unsuccessful at spreading using simple random dispersal strategies can be very successful by exploiting the large variance or clustering of vulnerable computers among subnetworks in the Internet. With the new Internet addressing scheme (IPv6) currently being rolled out, these issues will be amplified by many orders of magnitude. |

03:00 PM 03:30 PM | Break |

03:30 PM 04:40 PM | |

03:30 PM 04:40 PM | Mark Lewis - First passage time in complex environments: Connecting random walks to functional responses In this talk I will outline first passage time analysis for animals undertaking complex movement patterns while searching for prey. I will extend the analysis to complex heterogeneous environments to assess the effects of man-made linear landscape features on functional responses in wolves searching for elk. We developed a mechanistic first passage time model, based on an anisotropic elliptic partial differential equation, and used this to explore how wolf movement responses to seismic lines influence the encounter rate of the wolves with their prey. (This work is joint with Hannah McKenzie, Evelyn Merrill and Ray Spiteri) |

05:00 PM 07:00 PM | Reception and Poster Session in MBI Lounge |

07:00 PM | Shuttle pick-up from MBI |

Tuesday, April 17, 2012 | |
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Time | Session |

09:00 AM 10:00 AM | Alan Hastings - Spatial population dynamics and uncertainty in Tribolium: Lab Experiments and Models In joint work with Brett Melbourne we have studied highly replicated spatial population dynamics of flour beetles in a lab setting. I will describe the results of experiments on single species and spatial spread, and corresponding models. The models have to incorporate stochasticity of different forms to provide a good match to the data. In particular, demographic heterogeneity, fixed differences among individuals, are critical for understanding the dynamics. |

10:30 AM 11:30 AM | Marissa Baskett - The role of gene flow in rapid evolutionary response to global change Dispersal and the resulting genetic exchange between populations in spatially heterogeneous environments is typically expected to impede adaptation to local conditions. However, theory suggests some cases where this paradigm breaks down, such as when dispersal provides demographic support and gene flow enhances adaptive capacity to populations experiencing variable population sizes or environmental shifts. A current major driver of environmental change is anthropogenic activities, where humans can both be a source of environmental heterogeneity in space that selects on traits within populations experiencing exchange and a source of environmental shifts in time to which populations must adapt for local persistence. I will present a series of models exploring the potential for a beneficial versus detrimental role of gene flow given anthropogenically-driven global change. First, I will present a model of coral adaptation to climate change, where, given dispersal between populations experiencing different thermal stress, the potential for propagule input to enhance recovery from stressful events outweighs the potential for gene flow to impede adaptation to local thermal conditions. Second, I will present a model of exchange between salmon hatchery and wild populations, where the fitness and demographic consequences of domestication selection in the hatchery critically depend on the relative timing of natural selection, hatchery release, and density dependence in the life cycle. Both of these examples illustrate how a basic science understanding of gene flow can inform conservation management and how models of evolutionary response to global change can inform a basic science understanding of the adaptive role of gene flow. |

01:30 PM 02:30 PM | Sebastian Schreiber - Persistence and spatial spread in the face of uncertainty This talk will review three recent results about persistence of spatially structured populations and the spatial spread of populations in the presence of stochasticity. For the first part of this talk, I discuss the relationship between attractors of deterministic models and quasi-stationary distributions of their stochastic, finite population counterpoints i.e. models accounting for demographic stochasticity. These results shed some insight into when persistence should be observed over long time frames despite extinction being inevitable. An application to the coupled-Ricker model will be given. For the second part of the talk, I present results on stochastic persistence and boundedness for stochastic models accounting for environmental (but not demographic) noise. Stochastic boundedness asserts that asymptotically the population process tends to remain in compact sets. In contrast, stochastic persistence requires that the population process tends to be "repelled" by some "extinction set." Using these results, I will illustrate how environmental noise coupled with dispersal can rescue locally extinction prone populations. For the final part of the talk, I present invasion speed formulas for models combining state-structured local demography (e.g., an integral or matrix projection model) with general dispersal kernels, and stationary temporal variation in both local demography and dispersal kernels. Using these results, I will show that random temporal variability in dispersal can accelerate population spread. More surprisingly, demographic variability can further accelerate spread if it is positively correlated with dispersal variability. |

03:00 PM 03:30 PM | Rebecca Tyson - Post-Harvest Diseases of Apples: From Spore Dispersal to Epidemiology Postharvest diseases, especially those caused by fungi, can cause considerable damage to harvested apples in controlled atmosphere storage. Fungicides are used to control the disease, but resistance to fungicides is increasing and there is pressure by consumers and ecologists to reduce reliance on chemical controls. There is some evidence that physical conditions related to orchard management are predictive of postharvest disease incidence, and so the first line of defense against postharvest disease should involve best practices in orchards. In this work, we develop and analyse mathematical models to understand the dispersal of spores in the orchard, the initial infection level of fruit entering storage, and the epidemiology of the disease once the apples are in storage. We focus on conditions in the Okanagan Valley, where summers are dry and fungal spore presence is generally low. This leads to a mathematical problem where we are attempting to quantitatively and deterministically evaluate conditions surrounding rare events, that is, infection of fruit, and the fundamental stochasticity of the problem is crucial. |

03:40 PM 04:10 PM | Allison Shaw - Evolution of movement behavior and information usage in seasonal environments Migration is a widely used strategy for dealing with seasonal environments, yet little work has been done to understand what ultimate factors drive migration. Here I will present joint work with Iain Couzin, where we have developed a spatially explicit, individual-based model in which we can evolve behavior rules via simulations under a wide range of ecological conditions to answer two questions. First, under what types of ecological conditions can an individual maximize its fitness by migrating (versus being a resident)? Second, what types of information do individuals use to guide their movement? We find that different types of migration can evolve, depending on the ecological conditions and availability of information. |

04:20 PM 04:50 PM | Ruth Baker - Models of cellular migration for cells of different shapes and sizes Continuum, partial differential equation models are often used to describe the collective motion of cell populations, with various types of motility represented by the choice of diffusion coefficient, and cell proliferation captured by the source terms. Previously, the choice of diffusion coefficient has been largely arbitrary, with the decision to choose a particular linear or nonlinear form generally based on calibration arguments rather than making any physical connection with the underlying individual-level properties of the cell motility mechanism. In this talk I will discuss a series of individual-level models, which account for important cell properties such as varying cell shape and volume exclusion, and their corresponding population-level partial differential equation formulations. I will demonstrate the ability of these models to predict the population-level response of a cell spreading problem for both proliferative and non-proliferative cases. I will also discuss the potential of the models to predict long time travelling wave invasion rates. |

Wednesday, April 18, 2012 | |
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Time | Session |

09:00 AM 10:00 AM | John Novembre - Inference of migration and dispersal in spatial population genetic models Spatial patterns of genetic variation are clearly indicative of past dispersal and migration processes, but performing formal inference with spatial models in population genetics has been challenging and fairly limited. In this lecture I will overview several areas of recent progress, some using model-based approaches and others using informal exploratory approaches. Particular attention will be given to the insights that can be gained from the spatial distribution of rare variants, as well as spatial assignment approaches. The examples will include data from humans and migratory birds. |

10:10 AM 10:40 AM | Ben Kerr - Experimental ecology and evolution in metapopulations No description available |

11:10 AM 12:10 PM | Steven Evans - Go forth and multiply? Organisms reproduce in environments that vary in both time and space. Even if an individual currently resides in a region that is typically quite favorable, it may be optimal for it to "not put all its eggs in the one basket" and disperse some of its off spring to locations that are usually less favorable because the eff ect of unexpectedly poor conditions in one location may be o set by fortuitously good ones in another. I will describe joint work with Peter Ralph and Sebastian Schreiber (both at University of California, Davis) and Arnab Sen (Cambridge) that combines stochastic diff erential equations, random dynamical systems, and even a little elementary group representation theory to explore the eff ects of diff erent dispersal strategies. |

Thursday, April 19, 2012 | |
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Time | Session |

09:00 AM 10:00 AM | Simon Levin - Collective motion and collective decision-making There is a long history of research on the mathematical modeling of animal populations, largely built on diffusion models. The classical literature, however, is inadequate to explain observed spatial patterning, or foraging and anti-predator behavior, because animals actively aggregate. This lecture will discuss models of animal aggregation, and the role of leadership in collective motion. It will also explore models of the evolution of collective behavior, and implications for the optimal design of robotic networks of interacting sensors, with particular application to marine systems. |

10:30 AM 11:30 AM | Mercedes Pascual - Forest fires, cholera epidemics and spatial stochastic systems with critical transitions No description available. |

01:30 PM 02:30 PM | Daniel Grunbaum - Secondary characteristics of spatially and temporally heterogeneous populations Most ecological interactions occur in the context of fine-scale spatial and temporal heterogeneity, a.k.a., "patchiness". In many relevant ecological applications, the primary data sources (e.g. remote sensing), the primary predictive modeling approaches (e.g. biogeochemical or resource management models) and the most important ecological outcomes (e.g., total or "mean-field" populations) do not resolve or explicitly depend on fine-scale patchiness, but nonetheless are strongly affected by unresolved patch dynamics. In this talk, I will consider large-scale characteristics of interacting populations in which spatial and temporal heterogeneity is either imposed by external environmental forcing or arises autonomously from social interactions such as schooling and swarming. In many such populations, secondary population characteristics emerge that operate over larger spatio-temporal scales than primary patch dynamics and that strongly affect ecological outcomes. I will discuss some examples in which analysis of these secondary characteristics may improve interpretation and prediction of unresolved patch dynamics in data and models. |

03:40 PM 04:10 PM | Daniel Remenik - Voter model perturbations and the evolution of the dispersal distance The problem of how often to disperse in a randomly fluctuating environment has long been investigated, primarily using patch models with uniform dispersal. Here, we consider the problem of choice of seed size for plants in a stable environment when there is a trade off between survivability and dispersal range. For this we analyze a stochastic spatial model to study the competition of different dispersal strategies. Most work on such systems has been done by simulation or non-rigorous methods such as pair approximation. I will describe a model based on the general voter model perturbations recently studied by Cox, Durrett, and Perkins (2011) which allows us to rigorously and explicitly compute evolutionarily stable strategies. A main difficulty in this case is to extend the earlier work in three or more dimensions to the more complicated two-dimensional case, which is the natural setting for this problem. This is joint work with Rick Durrett. |

04:20 PM 04:50 PM | Bard Ermentrout - Weak and Slow: Spatial patterns in a heterogeneous environment I look at the interactions between heterogeneities and delayed negative feedback in systems which admit stationary persistent structures. The former can cause pinning and stabilize neutrally stable dynamics while the latter can induce several types of dynamics instabilities and motion. I show that the time-scale of the negative feedback and the amplitude of the heterogeneities interact to produce qualitatively different sequences of bifurcations. The models are motivated by dynamics of neurons in the rodent hippocampus during navigation. This work is joint with Rodica Curtu, Carina Curto, and Vladimir Itskov. |

Friday, April 20, 2012 | |
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Time | Session |

09:00 AM 10:00 AM | Nicolas Lanchier - Two-strategy games on the lattice In the seventies, biologists Maynard Smith and Price used concepts from game theory to describe animal conflicts. Their work is at the origin of the popular framework of evolutionary game theory. Space is another component that has been identified as a key factor in how communities are shaped. Spatial game models are therefore of primary interest for biologists and sociologists. There is however a lack of analytical results in this field. The objective of this talk is to explore the framework analytically through a simple spatial game model based on interacting particle systems (agent-based models). Our results indicate that the behavior of this process strongly differs from the one of its non-spatial mean-field approximation, which reveals the importance of space in game theoretic interactions. |

10:30 AM 11:30 AM | Rick Durrett - Evolving voter model In the evolving voter model we choose oriented edges (x,y) at random. If the two individuals have the same opinion, nothing happens. If not, x imitates y with probability 1-α, and otherwise severs the connection with y and picks a new neighbor at random (i) from the graph, or (ii) from those with the same opinion as x. Despite the similarity of the rules, the two models have much different phase transitions. This is one example from a large nonrigorous literature on systems where the network structure and the states of the individual in it coevolve |

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

Aydogmus, Ozgur | aydogmus@iastate.edu | Mathematics, Iowa State University |

Baker, Ruth | ruth.baker@maths.ox.ac.uk | Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford |

Baskett, Marissa | mlbaskett@ucdavis.edu | Environmental Science and Policy, University of California, Davis |

Bessonov, Mariya | myb@math.cornell.edu | Mathematics, Cornell University |

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

Chatterjee, Shirshendu | shirshendu@nyu.edu | Mathematics, New York University |

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

Cox, Ted | jtcox@syr.edu | Mathematics, Syracuse University |

Cressie, Noel | ncressie@stat.ohio-state.edu | Department of Statistics, The Ohio State University |

Dawes, Adriana | atdawes@math.ualberta.ca | Department of Mathematics / Department of Molecular Genetics, The Ohio State University |

Dawson, Donald | ddawson@math.carleton.ca | School of Mathematics and Statistics, Carleton University |

Durrett, Rick | rtd@math.duke.edu | Department of Mathematics, Duke University |

Engblom, Stefan | stefane@it.uu.se | Information Technology, Uppsala University |

Ermentrout, Bard | bard@pitt.edu | Department of Mathematics, University of Pittsburgh |

Evans, Steven | evans@stat.berkeley.edu | Statistics and Mathematics (joint), University of California, Berkeley |

Flegg, Mark | flegg@maths.ox.ac.uk | Mathematical Institute, University of Oxford |

Grunbaum, Daniel | grunbaum@ocean.washington.edu | Department of Biology and School of Oceanography, University of Washington |

Hamman, Elizabeth | hamman@ufl.edu | Biology, University of Florida |

Hao, Yiping | yphao@iastate.edu | Mathematics, Iowa State University |

Hastings, Alan | amhastings@ucdavis.edu | Department of Environmental Science and Policy, University of California, Davis |

Hein, Andrew | amhein@ufl.edu | Biology, |

Hiebeler, David | hiebeler@math.umaine.edu | Mathematics & Statistics, University of Maine |

Hindes, Jason | jmh486@cornell.edu | Physics, Cornell University |

Hughes, Barry | barrydh@unimelb.edu.au | Mathematics and Statistics, University of Melbourne |

Kanarek, Andrew | andrew.kanarek@gmail.com | National Institute for Mathematical and Biological Synthesis, University of Tennessee |

Kerr, Ben | kerrb@u.washington.edu | Biology, University of Washington |

Klapper, Isaac | klapper@math.montana.edu | Department of Mathematical Sciences, Montana State University |

Krone, Steve | krone@uidaho.edu | Mathematics, University of Idaho |

Lanchier, Nicolas | lanchier@math.asu.edu | Mathematics and Statistics, Arizona State University |

Langebrake Inman, Jessica | jessica.langebrake@gmail.com | Biology, University of Florida |

Lawley, Sean | lawley@math.duke.edu | Mathematics, University of Utah |

Levin, Simon | slevin@princeton.edu | Department of Ecology & Evolutionary Biology, Princeton University |

Lewis, Mark | mlewis@math.ualberta.ca | Canada Research Chair in Mathematical Biology, University of Alberta |

Luo, Shishi | szl@math.duke.edu | Mathematics, Duke University |

Ma, Yanping | yma@lmu.edu | Mathematics, Loyola Marymount University |

Melbourne, Brett | brett.melbourne@colorado.edu | Ecology and evolutionary biology, University of Colorado |

Musgrave, Jeffrey | musgrave.jeff@gmail.com | Mathematics and Statistics, University of Ottawa |

Nolen, James | nolen@math.duke.edu | Mathematics, Duke University |

Novembre, John | jnovembre@ucla.edu | Ecology and Evolutionary Biology, University of California, Los Angeles |

Pascual, Mercedes | pascual@umich.edu | Ecology and Evolutionary Biology, University of Michigan |

Perkins, Ed | perkins@math.ubc.ca | Mathermatics, University of British Columbia |

Popovic, Lea | lpopovic@mathstat.concordia.ca | Dept of Mathematics and Statistics, Concordia University |

Remenik, Daniel | dremenik@math.utoronto.ca | Mathematics, University of Toronto |

Roth, Gregory | greg.roth51283@gmail.com | Evolution and Ecology, University of California, Davis |

Rovetti, Robert | rrovetti@lmu.edu | Mathematics, Loyola Marymount University |

Roychoudhury, Pavitra | padm3003@vandals.uidaho.edu | Mathematics, University of Idaho |

Ryan, Daniel | ryan@nimbios.org | NIMBioS, University of Tennessee |

Schertzer, Emmanuel | schertzer@math.columbia.edu | Department of Ecology & Evolutionary Biology, Princeton University |

Schreiber, Sebastian | sschreiber@ucdavis.edu | Department of Evolution and Ecology, University of California, Davis |

Shaw, Allison | akshaw@princeton.edu | Ecology and Evolutionary Biology, Princeton University |

Spardy, Lucy | mbrooks@ufl.edu | |

Tyson, Rebecca | rebecca.tyson@ubc.ca | Mathematics & Statistics, University of British Columbia, Okanagan |

Voller, Zachary | zvoller@iastate.edu | Department of Mathematics, Iowa State University |

Wang, Jing | jingw@iastate.edu | Mathematics, Iowa State University |

Wang, Chi-Jen | cjwang@iastate.edu | Mathematics, Iowa State University |

Wang, Min | mwang@iastate.edu | Mathematics & Statistics, Iowa State University |

Weiss, Howie | weiss@math.gatech.edu | Mathematics, Georgia Institute of Technology |

Wu, Jialiang | gtg337v@mail.gatech.edu | Biomedical Engineering, Georgia Institute of Technology |

Ziv, Guy | gziv@princeton.edu | Natural Capital Project, Stanford University |

Our measurements show that the Internet is an incredibly clustered heterogeneous environment when measured according to the dispersal strategy used by worms. We have used these measurements to build an epidemiological simulation model of the entire Internet (4.29 billion hosts, with roughly 2 million susceptible) efficient enough to run on an ordinary desktop computer. A worm which would have a basic reproduction ratio far less than one and therefore be quite unsuccessful at spreading using simple random dispersal strategies can be very successful by exploiting the large variance or clustering of vulnerable computers among subnetworks in the Internet. With the new Internet addressing scheme (IPv6) currently being rolled out, these issues will be amplified by many orders of magnitude.

This will be something of an introductory talk that considers two types of spatial models used in population biology, and connections between them. Interacting particle systems can be thought of as "microscopic" level descriptions of populations, including interactions between discrete individuals and stochasticity. Reaction-diffusion equations provide deterministic models that can be thought of as "macroscopic" versions of particle systems through scaling limits. We will discuss the basic ideas behind this connection, treat a few examples, and try to understand the extent to which the two types of models predict the same behavior.

**Evolution of movement behavior and information usage in seasonal environments**

Allison Shaw Migration is a widely used strategy for dealing with seasonal environments, yet little work has been done to understand what ultimate factors drive migration. Here I will present joint work with Iain Couzin, where we have developed a spatially explic

**Two-strategy games on the lattice**

Nicolas Lanchier In the seventies, biologists Maynard Smith and Price used concepts from game theory to describe animal conflicts. Their work is at the origin of the popular framework of evolutionary game theory. Space is another component that has been identified as

**Experimental ecology and evolution in metapopulations**

Ben Kerr No description available

**Survival and coexistence for a class of stochastic spatial models**

Ted Cox We present a method for obtaining survival and coexistence results for a class of interacting particle systems. This class includes: a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, a model for the evolution of cooperation of Ohtsuk

**Particle Systems and Reaction-Diffusion Equations: connecting micro and macro models**

Steve Krone This will be something of an introductory talk that considers two types of spatial models used in population biology, and connections between them. Interacting particle systems can be thought of as "microscopic" level descriptions of popula

**Voter model perturbations and the evolution of the dispersal distance**

Daniel Remenik The problem of how often to disperse in a randomly fluctuating environment has long been investigated, primarily using patch models with uniform dispersal. Here, we consider the problem of choice of seed size for plants in a stable environment when t

**Models of cellular migration for cells of different shapes and sizes**

Ruth Baker Continuum, partial differential equation models are often used to describe the collective motion of cell populations, with various types of motility represented by the choice of diffusion coefficient, and cell proliferation captured by the source ter

**Go forth and multiply?**

Steven Evans Organisms reproduce in environments that vary in both time and space. Even if an individual currently resides in a region that is typically quite favorable, it may be optimal for it to "not put all its eggs in the one basket" and disperse s

**Spatial population dynamics and uncertainty in Tribolium: Lab Experiments and Models**

Alan Hastings In joint work with Brett Melbourne we have studied highly replicated spatial population dynamics of flour beetles in a lab setting. I will describe the results of experiments on single species and spatial spread, and corresponding models. The models

**Post-Harvest Diseases of Apples: From Spore Dispersal to Epidemiology**

Rebecca Tyson Postharvest diseases, especially those caused by fungi, can cause considerable damage to harvested apples in controlled atmosphere storage. Fungicides are used to control the disease, but resistance to fungicides is increasing and there is pressure b

**Collective motion and collective decision-making**

Simon Levin There is a long history of research on the mathematical modeling of animal populations, largely built on diffusion models. The classical literature, however, is inadequate to explain observed spatial patterning, or foraging and anti-predator behavior

**The role of gene flow in rapid evolutionary response to global change**

Marissa Baskett Dispersal and the resulting genetic exchange between populations in spatially heterogeneous environments is typically expected to impede adaptation to local conditions. However, theory suggests some cases where this paradigm breaks down, such as when

**Biological Dispersal Strategies of Internet Worms**

David Hiebeler For the past decade, Internet worms (a type of malicious software similar to a virus) spreading through networks have been using biological strategies, such as hierarchical dispersal and adaptive strategies, to spread more efficiently among susceptib

**Evolving voter model**

Rick Durrett In the evolving voter model we choose oriented edges (x,y) at random. If the two individuals have the same opinion, nothing happens. If not, x imitates y with probability 1-α, and otherwise severs the connection with y and picks a new n