• Modelling;
  • soil-borne pathogens;
  • competition;
  • parasitism;
  • survival;
  • disease control


Nonlinear models were fitted to selected data sets, obtained from published papers, in which inoculum of a fungal plant pathogen or microbial antagonist of a pathogen increased and then decreased with time. Conventional models that have been widely used in botanical epidemiology, such as the logistic, monomolecular and Weibull functions, could not be fitted to these non-monotonic curves. Two models, the over-damped in which Y=A+B exp (α1t +C exp (α2t), and the critically damped in which Y=A+ (B+Ct) exp (–λt), were fitted: Y is a measure of the pathogen or antagonist, t is time, A is an asymptote, B and C are locational parameters and α1, 2 and λ are rate parameters. These models are standard solutions of a second order differential equation. They have a single cycle, involving a rise and subsequent decline in the population density of the micro-organism which ultimately tends to an asymptotic equilibrium. The over-damped model was successful only in fitting large data-sets and was used to describe the population dynamics of bacterial colonization of live- and heat-killed conidia of Cochliobolus sativus (Ito & Kurib.). Drechs. ex Dastur as well as the population dynamics of Laetisaria arvalis Burdsall in untreated and in pasteurized soil with and without subsequent reinfestation with Pythium ultimum Trow. The critical damping model was used to describe pre-emergence damping-off of Rhizoctonia solani Kühn in successive crops of radish, the population dynamics of P. ultimum and low-temperature Pythium spp. in untreated soil and pasteurized soil with L. arvalis, the population dynamics of Trichoderma viride Pers. ex S. F. Gray in plant growth media and, similarly, of Macrophomina phaseolina (Maub.) Ashby in soil and, lastly, the colonization of organic fragments by P. ultimum with and without antagonism from P. nunn Lifshitz, Stanghellini & Baker. The selected model was first fitted to the empirical progress curve for each treatment. Subsequently one or more common parameters were constrained to fit all treatments while the remaining parameters were fitted separately. None of the treatments affected the rate parameters but treatments did affect the asymptote and locational parameters. The shapes of the models are discussed in relation to initial amounts of inoculum, the initial rate of increase of inoculum and the time to achieve maximum levels of disease or microbial biomass. The over-damping model is shown to represent the solution of a set of simple linked differential equations for two interacting populations. The derivation of these elementary models is discussed in relation to the study of biological control of plant pathogens.