Mathematical Modeling of the Glassy-winged Sharpshooter in Texas Vineyards

Xyllela fastidiosa is the bacterium found in plant xylems that causes Pierces disease (PD) and is a serious threat to Texas vineyards. The spread of PD results from xylem feeding insects migrating into vineyards after feeding on infected plants. Our goal is to temporally model the main vector that causes PD in TX by using a modified logistic growth equation. Using the insect database compiled by Dr. Forest Mitchell of Texas AgriLife Research it is determined that the Glassy-winged Sharpshooter (GWSS) is the major xylem feeding insect vectoring PD in Texas vineyards. A representative histogram of the GWSS population is created using 25 of the 40 participating vineyards that are located in the Texas Hill Country. Numerically we begin with the logistic growth equation; modifications such as a time delay parameter, harvesting term, and immigration constant are added to induce the oscillation and decay noted by the histogram. Analytically we obtained a lower boundary for parameter values by non-dimensionalizing the differential equation and focusing on the local behavior of the linearized function near the equilibrium at carrying capacity. The model has achieved a reasonable fit to the data’s highest frequency interval using these simple parameters. Further optimization techniques are being explored to obtain nonlinear parameters that will provide realistic behaviors.