Invasion by Extremes: Population Spread with Variation in Dispersal and Reproduction

Abstract

For populations having dispersal described by fat-tailed kernels (kernels with tails that are not exponentially bounded), asymptotic population spread rates cannot be estimated by traditional models because these models predict continually accelerating (asymptotically infinite) invasion. The impossible predictions come from the fact that the fat-tailed kernels fitted to dispersal data have a quality (nondiscrete individuals and, thus, no moment-generating function) that never applies to data. Real organisms produce finite (and random) numbers of offspring; thus, an empirical moment-generating function can always be determined. Using an alternative method to estimate spread rates in terms of extreme dispersal events, we show that finite estimates can be derived for fat-tailed kernels, and we demonstrate how variable reproduction modifies these rates. Whereas the traditional models define spread rate as the speed of an advancing front describing the expected density of individuals, our alternative definition for spread rate is the expected velocity for the location of the furthest-forward individual in the population. The asymptotic wave speed for a constant net reproductive rate R0 is approximated as (1/T)(piuR)/2)(1/2) m yr(-1), where T is generation time, and u is a distance parameter (m2) of Clark et al.'s 2Dt model having shape parameter p = 1. From fitted dispersal kernels with fat tails and infinite variance, we derive finite rates of spread and a simple method for numerical estimation. Fitted kernels, with infinite variance, yield distributions of rates of spread that are asymptotically normal and, thus, have finite moments. Variable reproduction can profoundly affect rates of spread. By incorporating the variance in reproduction that results from variable life span, we estimate much lower rates than predicted by the standard approach, which assumes a constant net reproductive rate. Using basic life-history data for trees, we show these estimated rates to be lower than expected from previous analytical models and as interpreted from paleorecords of forest spread at the end of the Pleistocene. Our results suggest reexamination of past rates of spread and the potential for future response to climate change.

The habitat and conduit functions of roads in the spread of three invasive plant species

“…If long-range dispersal events are relatively frequent (''fat tails'' of the dispersal kernels; Kot et al 1996), longrange dispersal may control the rate of range expansion making road networks irrelevant (Pearson and Dawson 2005;Christen and Matlack 2006). Dispersal may also be occuring within a site, but regional invasion is not necessarily an extension of local dispersal processes (Shigasada et al 1995;Clark et al 2001).…”

Inferences from Spatial Population Genetics

“…The question that one must address prior to statistical analysis is how important such long-distance immigrants are for population processes. For example, the speed of range expansions is known to be affected by the most extreme long-distance migrants in a way generally not characterized by the σ parameter (Mollison, 1977;Clark et al, 2001), so if one is interested in characterizing such processes, not only it will be difficult to estimate σ but this may be irrelevant. One the other hand, some processes of local adaptation (as may lead to allele frequency clines, for example) are not very sensitive to long-distance migrants, and then approximations ignoring them are not only adequate but required to formulate good statistical inferences.…”

Aerial Dispersal and Multiple-Scale Spread of Epidemic Disease

“…The form of a pathogen's dispersal gradient strongly affects the expected pattern of disease spread (Kot et al, 1996;Clark et al, 2001;Hastings et al, 2005). A disease that spreads through close proximity or contact between hosts is likely to have an exponentially bound dispersal function and to proliferate as a traveling wave phenomenon (e.g., Grenfell et al, 2001;Cummings et al, 2004).…”