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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 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.
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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 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.
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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 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.
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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 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.
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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 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.
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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 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.
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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 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.
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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 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.
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Ben Kerr
No description available
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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 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.
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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, 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.
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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 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.
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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 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.
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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 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
