Oecologia (Berlin) (1983) 57: 322-327 Oecologia 9 Springer-Verlag 1983 Local movement in herbivorous insects: applying a passive diffusion model to mark-recapture field experiments P.M. Kareiva Division of Biology, Brown University, Providence, RI 02912, USA Summary. A simple passive diffusion model is used to analyze the local within-habitat dispersal of twelve species of herbivorous insects. The data comprise field mark-recap- ture studies in relatively homogeneous habitats. For eight of the species, the cumulative frequency distributions of dispersal distances are consistent with a model of movement by passive diffusion. The observed departures from passive diffusion indicate the directions in which we need to modify our mathematical descriptions of movement if we are to develop realistic models of population dynamics and dis- persal. The analyses also synthesize in a standard way the relative dispersal rates of several ecologically similar species. The variation both within and between species in diffusion coefficients is striking - certainly sufficient to generate sig- nificant consequences for population dynamics and interac- tions. Introduction Although naturalists have long recognized the importance of dispersal among spatially-distributed populations (Elton 1949), only recently have theoretical ecologists begun to illuminate the dynamical consequences of dispersal (Levin 1974, 1981; Okubo 1980). Multispecies models that do in- corporate movement have drawn attention to the profound influence that rates of population flux can have on the dy- namics and persistence of species (Levin 1974; McMurtrie 1978). The analyses reported in this paper address two ques- tions raised by widely-cited models of spatially-distributed populations: 1. To what extent does passive diffusion (the standard mathematical expression for movement in most models) ap- proximate the local movement of insects in nature? 2. How much variation in dispersal rates exists among species (in this case herbivores) sharing similar ecological roles? In pursuing these two questions, I have drawn together movement data from twelve different species, all to be ana- lyzed according to the same protocol. Such standardized sets of dispersal estimates are rare. More importantly, I attempt with these analyses to evaluate theory (i.e., diffu- sion models) as systematically as possible. The application of diffusion models to real data has a rich tradition; but unlike the analyses reported in this paper, most previous work has concerned the geographic spread of populations (e.g. Skellam 1951; Long 1977), laboratory studies (e.g. Broadbent and Kendall 1953), or Drosophila trapline studies (e.g. Crumpacker and Williams 1973; Dobzhansky and Wright 1943; Johnston and Heed 1975; Richardson 1970). My approach to studying local movements in natural settings follows the pioneering investigations of Aikman and Hewitt (1972), Lamb et al. (1971) and especially Demp- ster (1957). Analyses The simplest model of movement assumes that the environ- ment is constant and homogeneous, that all individuals are identical, and that individuals move randomly. These as- sumptions lead to the well-known diffusion equation aN(x, y, t)_D Fa2N + a2N-] at T] (1) in which "N" is population density, "t" is time, "x, y" are the spatial coordinates, and "D "' is the diffusion coeffi- cient. When the arena for movement is an infinite plane, the solution to (1) that corresponds to a point release of marked individuals is a bivariate normal distribution cen- tered at the position of release. The variance of this distribu- tion in any dimension is 2Dt, where t=0 represents the time of release (see Okubo 1980 for a full development of the model). The important point is that observed fre- quency distributions of net displacements for organisms can be compared to a probability distribution representing a solution to (1) - the consistency of observed behavior with the diffusion model can be tested via this comparison. Two statistical approaches have been typically used to make this comparison, each approach essentially tests whether net movements are normally distributed about their release point: (i) Normalcy is evaluated by determining whether the kurtosis of an observed movement distribution differs sig- nificantly from the 3.0 expected for a normal curve (see Dobzhansky and Wright 1943) (ii) Or normalcy may be evaluated by examining the regression model Y=A exp(-BX2), where Y represents the density of insects that have moved a distance X. This regres- sion equation is used because it has the form of a normal curve (for examples, see Freeman 1977; Taylor 1980). Solutions to equation (1) specify more information than simply normalcy of data; they also specify the exact vari-