Delay differential population models for the decline of Homalodisca vitripennis (Hemiptera: Cicadellidae) densities over a ten-year period

The glassy-winged sharpshooter, Homalodisca vit-ripennis, is an invasive pest which presents a major economic threat to grape industries in California, as well as Texas [24] and other wine growing regions, because it spreads a disease-causing bacterium, Xylella fastidiosa. We continue an earlier investigation [1] into a long-term phenological decline of H. vitripennisdensities by studying a system of delayed differential equations (DDEs). We analyze aggregate population data for H. vitripennisfrom a 10-year study in which bi-weekly monitoring of H. vitripennispopulation numbers significantly decreased. These data present several challenges for modelers. First, they involve truly aggregate population level sampling and hence cannot properly be treated as ordinary longitudinal time series data corresponding to individual level models. The appropriate modeling involves estimation of probability distributions for parameters rather than estimation of the dynamic parameters themselves. Moreover, our analysis reveals that the correct corresponding statistical models involve errors that are observation size dependent (e.g., relative errors should be employed in statistical models). We use these data to test whether DDEs are useful in modeling the observed H. vitripennispopulation decline. To do this, we perform an analysis of variance (ANOVA) type test comparing the glassy-winged sharpshooter (GWSS) model with delay to a model without delay. The model is fit to the aggregate H. vitripennisdata using iterative reweighted weighted least squares (IRWLS) by estimating probability densities over the delay and one of the egg developmental rate parameters. Results indicate that a positive delay provides improvement with a significance level of p <; 10
^-4
.