¹ Department of Statistics and Actuarial Science, The University of Iowa, Iowa City, IA 52242, USA ² Department of Statistics, London School of Economics \& the University of Hong Kong, Hong Kong, China ³ Centre for Ecological and Evolutionary Synthesis (CEES), Department of Biology, University of Oslo, P.O. Box 1066 Blindern, N-0316 Oslo, Norway

The study of the rodent fluctuations of the North was initiated in its modern form with Elton’s pioneering work. Many scientific studies have been designed to collect yearly rodent abundance data, but the resulting time series are generally subject to at least two “problems”: being short and non-linear. We explore the use of the continuous threshold autoregressive (TAR) models for analyzing such data. In the simplest case, the continuous TAR models are additive autoregressive models, being piecewise linear in one lag, and linear in all other lags. The location of the slope change is called the threshold parameter. The continuous TAR models for rodent abundance data can be derived from a general prey-predator model under some simplifying assumptions. The lag in which the threshold is located sheds important insights on the structure of the prey-predator system. We propose to assess the uncertainty on the location of the threshold via a new bootstrap called the nearest block bootstrap (NBB) which combines the methods of moving block bootstrap and the nearest neighbor bootstrap. The NBB assumes an underlying finite-order time-homogeneous Markov process. Essentially, the NBB bootstraps blocks of random block sizes, with each block being drawn from a non-parametric estimate of the future distribution given the realized past bootstrap series. We illustrate the methods by simulations and on a particular rodent abundance time series from Kilpisj{\"{a}}rvi, Northern Finland.

This work was supported by US National Science Foundation (Grant No. CMG-0620789), the Research Grants Council of Hong Kong (Grant No. HKU7036/068) and the Engineering and Physical Sciences Research Council of UK (Grant No. EP/C549058/1)