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 diseas e foci of head blight of wheat in plots (100 to 2,500 m(2)) originating fro m small areas (1 to 16 m(2)) inoculated with Gibberella zeae-colonized corn kernels. These anisotropic, asymmetrical foci arose from ascospores produc ed in perithecia. The model is Z= expC-(AX(2) + BY2 + CXY + DX + EY + F)], in which Z = the incidence of seed or spikelet infection at point (X,Y) loc ated in the plot, exp = the exponential function, X = the abscissa or spati al coordinate of the point along a given axis (approximately parallel to th e average wind vector during the period of spore release in these experimen ts), Y = the ordinate or spatial coordinate of the point along the axis per pendicular 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) + BY2 + CXY + DX + EY + F, C = the bilinear coeffici ent, 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 gene ralized two-dimensional Gaussian model was tested on data from a circular o r isotropic focus, an elliptical or anisotropic focus with two axes of symm etry, and two anisotropic foci with one and zero axis of symmetry. Its good ness-of-fit (r(2) and adjusted r(2)) was compared with the inverse power, m odified inverse power, exponential, and classical Gaussian models. Submodel s using only the linear terms, only the quadratic terms, or combinations se lected from stepwise regression procedures using various probabilities to e nter and to stay and a procedure maximizing the adjusted r(2) were also con sidered. Spatial analysis of the residuals was performed using Geary's c co efficient at the first distance class. For the circular and elliptical foci , our model provided a fit similar to the modified inverse power and expone ntial models. However, for anisotropic foci with one or zero axis of symmet ry arising from ascospores influenced by wind direction, the generalized tw o-dimensional Gaussian model provided a better fit. For these anisotropic f oci, the linear term X but not the quadratic term X-2 was generally retaine d in the model, indicating an exponential gradient in the direction paralle l to the wind. In all models, the quadratic term Y-2 was retained, along wi th Y in some cases, indicating that the gradient in the direction roughly p erpendicular 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 ani sotropy of foci from wind-dispersed spores.