A physical theory of focus development in plant disease

M.W. Zawolek

Research output: Thesisinternal PhD, WU


Chapter 1. The 'diffusion theory' of focus development in plant disease is introduced. Foci develop in space and time. The theory applies primarily to air-borne fungal diseases of the foliage.

Chapter 2. The contents of the present volume are outlined.

Chapter 3. The 'diffusion theory' of focus development, intended to model phytopathologically interesting phenomena, emerges from a simple set of assumptions summarizing existing knowledge of plant pathologists. Using relatively easy and clear inferences, supported by methods used in mathematics and physics, this knowledge leads to a system of two partial differential equations (3.45), (3.46). These equations represent the 'diffusion theory, in mathematical terms.

Chapter 4. As any other new theory, the 'diffusion theory, must be validated by comparing it to models known from the literature and to experimental results. The 'diffusion theory' is a theoretical construct, permitting the development of a dynamic simulation model here indicated as the 'diffusion model,. The 'diffusion theory, is validated by comparing the 'diffusion model' to the model of Minogue and Fry and to EPIMUL of Kampmeijer and Zadoks. The various models show good qualitative and fair quantitative consistency. The quantitative differences are due to different assumptions about spore dispersal and deposition mechanisms. More important, the 'diffusion theory' was successfully validated by comparing its predictions to experimental data from yellow stripe rust ( Puccinia striiformis ) on wheat and from downy mildew ( Peronospora farinosa ) on spinach.

Chapter 5. Having derived the 'diffusion model,, it is necessary to determine its general behaviour for various parameter values. A method of sensitivity analysis, new to phytopathology, allows a detailed examination of linear, quadratic and mixed influences of parameters on responses of the 'diffusion model,. The analysis indicated a few phytopathologically interesting relationships. As the 'diffusion theory, attempts to describe the reality of plant disease dispersal in foci, the theory may lead to new hypotheses susceptible to experimental verification.

Chapter 6. The 'diffusion theory' is based on an idealized picture of spore movement; spore motion is purely random at an infinitesimally small scale of time and space. Reality is different. Therefore, the 'diffusion theory, is compared to the 'telegrapher's theory', more complex and derived from different assumptions about spore motion. Comparison of the two theories does not indicate differences of practical importance, so that the diffusion approximation seems to be adequate for phytopathological applications.

Chapter 7. Wind is an important meteorological factor, affecting the development of air-borne plant disease. The extension of the 'diffusion theory' to situations with a prevailing wind direction is made. The extended 'diffusion theory, allows to build a 'diffusion model, which simulates focus development under the influence of wind. The results of computer simulations show a good qualitative consistency with experimental data from brown leaf rust ( Puccinia recondita ) on wheat. In both cases, two phenomena were observed, a shift of the centre of the focus in the prevailing wind direction and, simultaneously, a certain 'inertial of the centre of the focus.

Chapter 8. The 'diffusion theory,, being formulated in terms of partial differential equations, can easily be extended by modification of parameters and/or equations. Thus, the appropriate 'diffusion model' for any specific situation can be created on the basis of the 'diffusion theory'. Application of the 'diffusion theory, by means of a numerical solution of the apropriate 'diffusion model, allows to study various characteristics of focal epidemics. Possibilities are discussed to combine the 'diffusion theory, with some of the methods of computer simulation.

Chapter 9. Some results are given of models simulating phytopathologically interesting situations. These models allow to explain some real-life phenomena; the generation of daughter foci, the calculation of the 'cryptic error' in plant breeding trials, and the interaction between two different mechanisms of spore dispersal.

Chapter 10. The book concludes with a general discussion. It shows the place of the 'diffusion theory' among other models in plant pathology and it proposes some improvements and further developments of the theory.

Final remark. Mathematically, the 'diffusion theory, is a special case of a more encompassing family of models considered by Diekmann and Thieme. The general results from the Diekmann-Thieme theory have been applied by Van den Bosch et al. in a phytopathological context using what amounts to a special limiting variant of the 'diffusion theory'. These results only relate to the behaviour of the focal front for a large period of time in an area which is homogeneously planted in all directions. The strength of the 'diffusion model,, as implemented, is its ability to deal also with short periods and with environments which vary in time and/or space.

Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Zadoks, J.C., Promotor
  • Metz, J.A.J., Co-promotor, External person
Award date31 May 1989
Place of PublicationWageningen
Publication statusPublished - 1989


  • plant pests
  • plant diseases
  • epidemiology
  • distribution
  • models
  • theory
  • science


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