Mark Kot

Dispersal data and the spread of invading organisms

Models 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 an Allee effect to an integrodifference equation may decrease the overall rate of spread. An Allee effect may also introduce a critical range; the population must surpass this spatial thresh-old in order to invade successfully. Fat-tailed kernels and Allee effects provide alternative explanations for the accelerating rates of spread observed for many invasions.

Adaptation: Statistics and a Null Model for Estimating Phylogenetic Effects

-Tests of adaptive explanations are often critically confounded by phylogenetic heritage. In this paper we propose statistics and a null model for estimating phylogenetic effects in comparative data. We apply a model-independent measure of autocorrelation (Morans I) for estimating whether cross-taxonomic trait variation is related to phylogeny. We develop a phylogenetic correlogram for assessing how autocorrelation varies with patristic distance and for judging the appropriateness and effectiveness of an autoregressive model. We then revise Cheverud et al.s (1985, Evolution, 39:1335-1351) autocorrelational model to incorporate greater flexibility in the relation between trait variation and phylogenetic distance. Finally, we analyze various comparative data sets (body weight in carnivores, clutch size in birds) and phylogenies (morphological, molecular) to illustrate some of the complications that may arise from using an autoregressive model and to explore the effects of different weighting matrices in adjusting for these complications. Although our approach has limitations, it is both effective in partitioning trait variation into adaptive and phylogenetic components and flexible in adjusting to peculiarities in taxonomic distribution. [Phylogenetic effects; phylogenetic correlation; autoregressive models; comparative methods.] The comparative method is commonly used to investigate adaptation. A researcher examines the attributes of a number of species. Statistical analyses of these data are then used to formulate and test adaptive hypotheses of life history, morphology, physiology, demography, and behavior (e.g., Clutton-Brock and Harvey, 1977; Damuth, 1981; Gittleman and Harvey, 1982; Harvey and Clutton-Brock, 1985; Gittleman, 1986a, b; Huey and Bennett, 1987). If traits are analyzed across a broad range of independently derived taxa, the resulting adaptive explanations may be quite robust (Clutton-Brock and Harvey, 1984; Huey and Bennett, 1986; Gittleman, 1989). If, however, the data reflect a highly structured phylogeny (with little statistical independence), results may be misleading (Felsenstein, 1985). To neglect phylogeny is to invite type I and type II errors (see Fig. 1). A number of techniques have been developed for removing the effects of phylogeny (see reviews in Huey, 1987; Pagel and Harvey, 1988; Gittleman, 1989; Burghardt and Gittleman, 1990). Some of these techniques are better suited for particular variables or certain evolutionary questions, and all possess limitations. Nominal or categorical data (e.g., mating system: monogamy, polygamy) may be analyzed by evaluating the agreement between the variation in a trait and an accepted phylogeny (Dobson, 1985; Greene, 1986) or by using outgroup comparisons to identify evolutionary transitions among traits (Gittleman, 1981; Ridley, 1983). For quantitative data, there are several strategies. One may avoid spurious correlation by averaging over closely related species, thereby reducing the degrees of freedom and significance of the correlation. Alternatively, one may transform the data so that phylogenetically disparate groups appear on a common scale. Even within this general framework there are several methods for evaluating the association between the ordinal or continuous values of a trait and phylogeny: (1) Nested analysis of variance partitions the total variation in a continuous character among various taxonomic levels. By selecting the taxonomic level that accounts for the greatest proportion of the total variance as the appropriate level for analysis, this method attempts to control for bias from low-level clades that are both uniform and speciesrich (Harvey and Mace, 1982; Harvey and

Discrete-time growth-dispersal models

Abstract Integrodifference equations are discrete-time models that share many of the attributes of scalar reaction-diffusion equations. At the same time, they readily exhibit period doubling and chaos. We examine the properties of some simple integrodifference equations.

Nearly one dimensional dynamics in an epidemic

The incidence of measles in New York City and Baltimore was studied using recently developed techniques in nonlinear dynamics. The data, monthly case reports for the years 1928-1963, suggest almost two dimensional, chaotic flows whose essential attributes are captured by one dimensional, unimodal maps. The effects of noise, inevitable in ecological and epidemiological systems are discussed.

Spreading disease: integro-differential equations old and new.

We investigate an integro-differential equation for a disease spread by the dispersal of infectious individuals and compare this to Mollisons [Adv. Appl. Probab. 4 (1972) 233; D. Mollison, The rate of spatial propagation of simple epidemics, in: Proc. 6th Berkeley Symp. on Math. Statist. and Prob., vol. 3, University of California Press, Berkeley, 1972, p. 579; J. R. Statist. Soc. B 39 (3) (1977) 283] model of a disease spread by non-local contacts. For symmetric kernels with moment generating functions, spreading infectives leads to faster traveling waves for low rates of transmission, but to slower traveling waves for high rates of transmission. We approximate the shape of the traveling waves for the two models using both piecewise linearization and a regular-perturbation scheme.

Speeds of invasion in a model with strong or weak Allee effects.

We study an invasion model based on a reaction-diffusion equation with an Allee effect. We use a special, piecewise-linear, population growth rate. This function allows us to obtain traveling wave solutions and to compute wave speeds for a full range of Allee effects, including weak Allee effects. Some investigators claim that linearization fails to give the correct speed of invasion if there is an Allee effect. We show that the minimum speed for a sufficiently weak Allee may, in fact, be the same as that derived by means of linearization.

Discrete-time travelling waves: Ecological examples

Integrodifference equations are discrete-time models that possess many of the attributes of continuous-time reaction-diffusion equations. They arise naturally in population biology as models for organisms with discrete nonoverlapping generations and well-defined growth and dispersal stages. I examined the varied travelling waves that arise in some simple ecologically-interesting integrodifference equations. For a scalar equation with compensatory growth, I observed only simple travelling waves. For carefully chosen redistribution kernels, one may derive the speed and approximate the shape of the observed waveforms. A model with overcompensation exhibited flip bifurcations and travelling cycles in addition to simple travelling waves. Finally, a simple predator-prey system possessed periodic wave trains and a variety of travelling waves.

Chaos in ecological systems: The coals that Newcastle forgot

Until recently, ecologists have ignored the possibility that chaos may be an important component of ecological systems. With the development of powerful new concepts and techniques for the study and detection of chaotic systems, it is becoming apparent that chaos may be widespread in nature, and may necessitate a fundamental reappraisal of many ideas in community and population ecology.

Invasion speeds in fluctuating environments

Biological invasions are increasingly frequent and have dramatic ecological and economic consequences. A key to coping with invasive species is our ability to predict their rates of spread. Traditional models of biological invasions assume that the environment is temporally constant. We examine the consequences for invasion speed of periodic and stochastic fluctuations in population growth rates and in dispersal distributions.

Effects of noise on some dynamical models in ecology

We investigate effects of random perturbations on the dynamics of one-dimensional maps (single species difference equations) and of finite dimensional flows (differential equations for n species). In particular, we study the effects of noise on the invariant measure, on the “correlation” dimension of the attractor, and on the possibility of detecting the nonlinear deterministic component by applying reconstruction techniques to the time series of population abundances. We conclude that adding noise to maps with a stable fixed-point obscures the underlying determinism. This turns out not to be the case for systems exhibiting complex periodic or chaotic motion, whose essential properties are more robust. In some cases, adding noise reveals deterministic structure which otherwise could not be observed. Simulations suggest that similar results hold for flows whose attractor is almost two-dimensional.