The evolution of resistance to two-toxin pyramid transgenic crops.

Pyramid transgenic crops that express two Bacillus thuringiensis (Bt) toxins hold great potential for reducing insect damage and slowing the evolution of resistance to the toxins. Here, we analyzed a suite of models for pyramid Bt crops to illustrate factors that should be considered when implementing the high dose-refuge strategy for resistance management; this strategy involves the high expression of toxins in Bt plants and use of non-Bt plants as refuges. Although resistance evolution to pyramid Bt varieties should in general be slower, resistance to pyramid Bt varieties is nonetheless driven by the same evolutionary processes as single Bt-toxin varieties. The main advantage of pyramid varieties is the low survival of insects heterozygous for resistance alleles. We show that there are two modes of resistance evolution. When populations of purely susceptible insects persist, leading to density dependence, the speed of resistance evolution changes slowly with the proportion of refuges. However, once the proportion of non-Bt plants crosses the threshold below which a susceptible population cannot persist, the speed of resistance evolution increases rapidly. This suggests that adaptive management be used to guarantee persistence of susceptible populations. We compared the use of seed mixtures in which Bt and non-Bt plants are sown in the same fields to the use of spatial refuges. As found for single Bt varieties, seed mixtures can speed resistance evolution if larvae move among plants. Devising optimal management plans for deploying spatial refuges is difficult because they depend on crop rotation patterns, whether males or females have limited dispersal, and other characteristics. Nonetheless, the effects of spatial refuges on resistance evolution can be understood by considering the three mechanisms determining the rate of resistance evolution: the force of selection (the proportion of insects killed by Bt), assortative mating (deviations of the proportion of heterozygotes from Hardy-Weinberg equilibrium at the total population level), and male mating success (when males carrying resistance alleles find fewer mates). Of these three, assortative mating is often the least important, even though this mechanism is the most frequently cited explanation for the efficacy of the high dose-refuge strategy.

Weak population regulation in ecological time series.

How strongly natural populations are regulated has a long history of debate in ecology. Here, we discuss concepts of population regulation appropriate for stochastic population dynamics. We then analyse two large collections of data sets with autoregressive-moving average (ARMA) models, using model selection techniques to find best-fitting models. We estimated two metrics of population regulation: the characteristic return rate of populations to stationarity and the variability of the stationary distribution (the long-term distribution of population abundance). Empirically, longer time series were more likely to show weakly regulated population dynamics. For data sets of length > or = 20, more than 35% had characteristic return times > 6 years, and more than 29% had stationary distributions whose coefficients of variation were more than two times greater than would be the case if they were maximally regulated. These results suggest that many natural populations are weakly regulated.

Analysis of ecological time series with ARMA(p,q) models.

Autoregressive moving average (ARMA) models are useful statistical tools to examine the dynamical characteristics of ecological time-series data. Here, we illustrate the utility and challenges of applying ARMA (p,q) models, where p is the dimension of the autoregressive component of the model, and q is the dimension of the moving average component. We focus on parameter estimation and model selection, comparing both maximum likelihood (ML) and restricted maximum likelihood (REML) parameter estimation. While REML estimation performs better (has less bias) than ML estimation for ARMA (p,q) models with p = 1 (as has been found previously), for models with p > 1 the performance of the estimators is complicated by multimodal likelihood functions. The resulting difficulties in estimation lead to our recommendation that likelihood functions be routinely investigated when applying ARMA (p,q) models. To aid this investigation, we provide MATLAB and R code for the ML and REML likelihood functions. We further explore the consequences of measurement error, showing how it can be explicitly and implicitly incorporated into estimation. In addition to parameter estimation, we also examine model selection for identifying the correct model dimensions (p and q). Finally, we estimate the characteristic return rate of the stochastic process to its stationary distribution, a quantity that describes a key property of population dynamics, and investigate bias that results from both estimation and model selection. While fitting ARMA models to ecological time series with complex dynamics has challenges, these challenges can be surmounted, making ARMA a useful and broadly applicable approach.