Clark J S, Lewis M, Horvath L
University Program in Ecology and Department of Biology, Duke University, Durham, North Carolina 27708, USA.
Am Nat. 2001 May;157(5):537-54. doi: 10.1086/319934.
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.
对于具有由肥尾核（尾部不是指数有界的核）描述的扩散的种群，传统模型无法估计渐近种群扩散率，因为这些模型预测入侵会持续加速（渐近无限）。这些不可能的预测源于这样一个事实，即拟合扩散数据的肥尾核具有一种性质（非离散个体，因此没有矩生成函数），而这种性质永远不适用于数据。真实的生物体产生有限（且随机）数量的后代；因此，总能确定一个经验矩生成函数。通过使用一种替代方法，根据极端扩散事件来估计扩散率，我们表明可以为肥尾核得出有限估计值，并且我们展示了可变繁殖如何改变这些速率。传统模型将扩散率定义为描述个体预期密度的前进前沿的速度，而我们对扩散率的替代定义是种群中最前沿个体位置的预期速度。对于恒定的净繁殖率(R_0)，渐近波速近似为((1/T)(\pi uR)/2)^{(1/2)} m yr^{-1})，其中(T)是世代时间，(u)是克拉克等人的二维模型的距离参数（(m^2)），形状参数(p = 1)。从具有肥尾和无限方差的拟合扩散核中，我们得出有限的扩散率以及一种简单的数值估计方法。具有无限方差的拟合核产生的扩散率分布渐近正态，因此具有有限矩。可变繁殖会深刻影响扩散率。通过纳入由可变寿命导致的繁殖方差，我们估计的速率比标准方法预测的要低得多，标准方法假设净繁殖率恒定。利用树木的基本生活史数据，我们表明这些估计速率低于先前分析模型的预期以及从更新世末期森林扩散的古记录所解释的速率。我们的结果表明需要重新审视过去的扩散速率以及未来对气候变化响应的潜力。
