l Churchill College, Cambridge, CB3 ODS, UK; 2 King's College, Cambridge, CB2 1ST, UK; 3 Department of Plant Sciences, University of Cambridge, Downing Street, Cambridge, CB2 3EA, UK

Background and objectives
In the quest for simple dynamic models of biological systems, several key factors have often been ignored. Traditionally, ecological and epidemiological models concentrate on the mean behaviour of systems under homogeneous conditions and with continuous interactions between populations. Real populations are, however, subject to spatial variability and temporal disturbances, such as those associated with planting and harvesting of a seasonal crop. While the traditional approach has been successful in many areas, it has not been able to deliver a rigorous mathematical description of phenomena such as persistence of plant pathogens and the related frequent failure of biological control. Given the economical importance of many plant diseases, it is essential to draw together theory and experimentation to provide biologically plausible descriptions of such processes that can be rigorously tested against data.

Using a combination of mathematical analysis, model fitting and experimental data, we investigate the population dynamics of biological control in two systems. In the first, the progress of damping-off disease in radish seedlings caused by the plant pathogen, Rhizoctonia solani, both with and without the addition of the antagonistic fungus, Trichoderma viride in replicate microcosm experiments is quantified. A simple nonlinear model is then presented which describes both the mean and variance of the experimental data [1]. In the second system, models are used to investigate the mean dynamics of biological control over many seasons. Experimental data kindly supplied by Adams and Fravel [21, are taken from a 2-year experiment (including five growing seasons) investigating the use of the mycoparasite Sporidesmium scierotivorum as a persistent biological control agent of Scierotinia minor, an economically important fungal parasite of lettuce. We investigate the factors contributing to the observed persistence of the S. minor population [3].

Results and conclusions
For the Rhizoctonia-radish system, we show that although the biological control agent, T. viride reduces the average amount of disease, the variability among replicates is greatly enhanced in the presence of the antagonist. These results are shown to be consistent with predictions from the simple model in which small differences in host susceptibility, fungal infectivity and control are amplified as the pathogen spreads from initial inoculum in soil and later from plant to plant. The differences are maintained by the interruption of disease progress as a result of a decline in host susceptibility. This behaviour has important consequences for the design and interpretation of ecological experiments and can also help account for the notorious failure of many biological control strategies. In the S. minor-S.  scierotivorum system, analytical results show that disturbances (repeated planting and harvesting of the host) allow the parasite to persist by maintaining a quantity of host tissue in the system on which the parasite can reproduce. We demonstrate that the standard epidemiological assumption of homogeneous mixing fails to predict the persistence of the parasite population, while incorporating spatial heterogeneity, by allowing for heterogeneous mixing, prevents fade-out. An alternative hypothesis for persistence involving a stepped change in rates of infection is also tested and model fitting is used to show that changes in some environmental conditions may contribute to parasite persistence.

1. Kleczkowski A, Bailey DJ, Gilligan, CA, 1996. Proceedings of the Royal Society of London Series B 263, 777-783.
2. Adams PB, Fravel DF, 1990. Phytopathology 80, 1120-1124.
3. Gubbins S, Gilligan CA, 1997. Philosophical Transactions of the Royal Society of London Series B 352, 1935-1949.