A generalized two-dimensional Gaussian model of disease foci of head blight of wheat caused by Gibberella zeae

A generalized two-dimensional Gaussian model is proposed to describe disease foci of head blight of wheat in plots (100 to 2,500 m(2)) originating from small areas (1 to 16 m(2)) inoculated with Gibberella zeae-colonized corn kernels. These anisotropic, asymmetrical foci arose from ascospores produced in perithecia. The model is Z = exp[-AX(2) + BY(2) + CXY + DX + EY + F)], in which Z = the incidence of seed or spikelet infection at point (X,Y) located in the plot, exp = the exponential function, X = the abscissa or spatial coordinate of the point along a given axis (approximately parallel to the average wind vector during the period of spore release in these experiments), Y = the ordinate or spatial coordinate of the point along the axis perpendicular to the X axis (approximately perpendicular to the wind direction in these experiments), A and B = the quadratic coefficients of the second-order polynomial AX(2) + BY(2) + CXY + DX + EY + F, C = the bilinear coefficient, D and E = the linear coefficients, and exp(-F) = the incidence of seed or spikelet infection at the focus peak in which X = 0 and Y = 0. The generalized two-dimensional Gaussian model was tested on data from a circular or isotropic focus, an elliptical or anisotropic focus with two axes of symmetry, and two anisotropic foci with one and zero axis of symmetry. Its goodness-of-fit (r(2) and adjusted r(2)) was compared with the inverse power, modified inverse power, exponential, and classical Gaussian models. Submodels using only the linear terms, only the quadratic terms, or combinations selected from stepwise regression procedures using various probabilities to enter and to stay and a procedure maximizing the adjusted r(2) were also considered. Spatial analysis of the residuals was performed using Geary's c coefficient at the first distance class. For the circular and elliptical foci, our model provided a fit similar to the modified inverse power and exponential models. However, for anisotropic foci with one or zero axis of symmetry arising from ascospores influenced by wind direction, the generalized two-dimensional Gaussian model provided a better fit. For these anisotropic foci, the linear term X but not the quadratic term X(2) was generally retained in the model, indicating an exponential gradient in the direction parallel to the wind. In all models, the quadratic term Y(2) was retained, along with Y in some cases, indicating that the gradient in the direction roughly perpendicular to the wind was Gaussian or Gaussian-exponential in shape. The bilinear term XY provided an indication of the orientation of the focus in relation to the axes of the sampling grid. This model has the versatility and parameters (quadratic, bilinear, and linear) to better describe the anisotropy of foci from wind-dispersed spores.