JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 165.123.34.86 on Wed, 17 Jun 2015 18:34:14 UTC All use subject to JSTOR Terms and Conditions Editor's Note Invasions of new territory-by plants, animals, or genes-are an old topic in ecology, but hardly obsolete: the spread of pests such as the gypsy moth, exotic plants, recurrent and emerging infectious diseases, and genetically engineered organisms are important contemporary problems for ecology. For nearly half a . . ~~century, reaction-diffusion models have been the main analytic framework for Emphasizing9 modeling spatial spread, in part because of the well-developed mathematical theory new ideas that tells us how to compute things like the long-term rate of spread and the conditions for spatial pattern formation. In this paper, we are given the tools to to study the rate of spread for invading organisms in a very different kind of model, integrodifference equations. Unlike diffusion equations, these models can accomstimulate research modate leptokurtic (broad-tailed) dispersal patterns, and in such cases they can exhibit the accelerating rates of spread that have been observed in some invasions.in ecology Of course this does not mean that we should stop using diffusion models, but it gives us an alternative with a different set of assumptions that is likely to be more accurate when the dispersal pattern of individuals is far from the Gaussian distribution implicit in a diffusion model. AbstractModels that describe the spread of invading organisms often assume that the dispersal distances of propagules are normally distributed. In contrast, measured dispersal curves are typically leptokurtic, not normal. In this paper, we consider a class of models, integrodifference equations, that directly incorporate detailed dispersal data as well as population growth dynamics. We provide explicit formulas for the speed of invasion for compensatory growth and for different choices of the propagule redistribution kernel and apply these formulas to the spread of D. pseudoobscura. We observe that: (1) the speed of invasion of a spreading population is extremely sensitive to the precise shape of the redistribution kernel and, in particular, to the tail of the distribution; (2) fat-tailed kernels can generate accelerating invasions rather than constant-speed travelling waves; (3) normal redistribution kernels (and by inference, many reaction-diffusion models) may grossly underestimate rates of spread of invading populations in comparison with models that incorporate more realistic leptokurtic distributions; and (4) the relative superiority of different redistribution kernels depends, in general, on the precise magnitude of the net reproductive rate. The addition of...

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Elements of Mathematical Ecology provides an introduction to classical and modern mathematical models, methods, and issues in population ecology. The first part of the book is devoted to simple, unstructured population models that ignore much of the variability found in natural populations for the sake of tractability. Topics covered include density dependence, bifurcations, demographic stochasticity, time delays, population interactions (predation, competition, and mutualism), and the application of optimal control theory to the management of renewable resources. The second part of this book is devoted to structured population models, covering spatially-structured population models (with a focus on reaction-diffusion models), age-structured models, and two-sex models. Suitable for upper level students and beginning researchers in ecology, mathematical biology and applied mathematics, the volume includes numerous clear line diagrams that clarify the mathematics, relevant problems thoughout the text that aid understanding, and supplementary mathematical and historical material that enrich the main text.

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