Biological control of pathozone behaviour and disease dynamics of Rhizoctonia solani by Trichoderma viride

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1 New Phytol. (1997), 136, Biological control of pathozone behaviour and disease dynamics of Rhizoctonia solani by Trichoderma viride BYD.J. BAILEY* AND C.A. GILLIGAN Department of Plant Sciences, University of Cambridge, Downing Street, Cambridge, CB2 3EA, UK {Received 13 January 1997; accepted 2 May 1997) SUMMARY This paper presents and tests a method to scale up from the dynamics of infection and disease on single plants in order to predict the behaviour of epidemics in populations of plants, in the presence of a biological control agent. Specifically, we quantify the effect of the antagonistic fungus, Trichoderma viride, Pers ex. Gray, on the pathozone dynamics of the damping-off fungus, Rhizoctonia solani Kuhn on radish. The results from these individual-based experiments are used to predict the progress of an epidemic and the results are compared with experimental epidemics in microcosms. The addition of T. viride close to a germinating radish plant reduced the extent of the pathozone influence from 22-6 to 13-8 mm. Trichoderma viride also inhibited the evolution of infection efficiency of R. solani. The evolution of infection efficiency over time is described by a simple non-linear model for the probability of infection with distance, in which certain of the parameters vary with time. By combining this with a model based on conditional probabilities for the location and infectivity of randomly dispersed propagules within the pathozone, we were able to scale up and predict the disease progress of R. solani in a population of radish plants, and the extent of control effected by T. viride. Disease progress rose progressively to a maximum of 42 % diseased plants in the unprotected crop compared with a sigmoidal approach to 13 % in the protected crop. We show that these properties are consistent with a monomolecular function for primary infection with a temporallyvarying rate parameter. We also show that the central region of the pathozone, where the joint probabilities of occurrence of a randomly located propagule and its ability to infect are maximal, has a large influence on the sensitivity of epidemics to pathozone dynamics. This is an important example of interpreting population behaviour of an epidemic from the infection and disease of single plants. Key words: Epidemiology, modelling, rhizosphere, infection dynamics, population scaling. R. solani from the host, in which certain parameters INTRODUCTION., rrni ^i u U'lv r - f ^- vary with time. Thus the probability of infection. It has long been recognized that successful control of P(r, t), depends on distance (r) and time (t): disease may be effected by changing the behaviour of. f\ f( \ f)(f)\ fungal plant pathogens in the rhizosphere or pathozone. Yet little information is available to describe where e(t) are the time varying parameters. The this behaviour or the quantitative effects of these model for P(r, t) is obtained from placement experiresponses on the dynamics of epidemic development, ments (Henis & Ben-Yephet, 1970; Gilligan, 1980; In this paper we show how placement of the Gilligan & Bailey, 1997), in which replicates combiological control agent, Trichoderma viride Pers ex. prise single plants challenged with propagules of Gray into the pathozone of the host plant, Raphanus inoculum placed at known distances from hypocotyls sativus L. affects the pathozone behaviour of Rhizo- of radish. We also show that P(r, t) can be used to ctonia solani Kuhn. The evolution of infection predict disease progress curves in the presence or efficiency(gilligan, 1985; Gilligan & Bailey, 1997) is absence of T. viride in a population of radish summarized by a non-linear model for the changing seedlings exposed to randomly dispersed inoculum probability of infection with distance of inoculum of of R. solani, thus enabling scaling up from individual to population behaviour. The rhizosphere is commonl^^ acknowledged as * To whom correspondence should be addressed.. r-i i- j - n ju.ti djb21@cus cam.ac.uk the region of soil surrounding and mfluenced by the

2 360 D. y. Bailey and C. A. Gilligan presence of a root (Lynch, 1990). The pathozone is dextrose agar overlain with a membrane filter, 4-7 cm the volume of soil surrounding a subterranean plant in diameter and 45/im pore size (Ref ; organ within which the centre of the propagule must Schleicher & Schull, Germany). Mycelial disc lie if it is to have a finite chance of infecting the organ inoculum was selected in favour of chopped potato (Gilligan, 1985). The pathozone is therefore closely soil inoculum used by Benson & Baker (1974) and related to the rhizosphere but, whilst the dimensions Gilligan & Simons (1987) because it is markedly less of the former are dictated principally by the variable in germination and colony growth (Gilligan properties of the root and surrounding soil, the & Bailey, 1997). This reduced variability improves major components responsible for the size and shape the chances of detecting differences between treatof the pathozone also depend on the properties of ments that affect the pathozone profile of the inoculum (Gilligan & Bailey, 1997). The pathozone probability of infection (and disease) for inoculum is characterized not only by the furthest extent from placed at different distances from the host, which infection can occur but also by a decrease in Inoculum of T. viride was prepared by adding 10 g the probability of infection as the distance between of blue poppy seed, Papaver somniferum L. and 5 ml inoculum and host increases (Gilligan, 1985). Vari- of reverse osmosis water to a 100 ml conical flask, ous authors have quantified the furthest extent of the The fiask was sterilized by autoclaving for 1 h at pathozone. These include Henis & Ben-Yephet 121 C, then incubated for 3 d at 23 C after adding (1970) for R. solani on Phaseolus vulgaris; Brown & 1 ml of a 1-0 x 10^ conidial spore suspension, ob- Hornby (1971), Gilligan (1980) and Wildermuth, tained from a 3-wk-old culture of T. umvie grown on Warcup & Rovira (1984) for GaeMmawwomj^ces^ram- potato dextrose agar. Following incubation, and m?5 Sacc. Arx & Olivier on wheat; Reynolds, Benson immediately before use, the inoculum was passed & Bruck (1985) for Phytophthora spp. on Fraser Fir through a 0*5 mm mesh sieve and a sample plated out and Punja & Grogan (1981) for Sclerotium rolfsii to confirm 100% viability of colonized seeds. Curzi on Phaseolus spp. Gilligan & Simons (1987) quantified the change in probability of infection, _,.. ^.. -ii- r r. 1 Placement experiment termed infection etticiency, with distance tor K. solani on radish and G. graminis on wheat at a single time Placement experiments were used to compare the after inoculation. All of these, however, provide static effects of T. viride on the probability of infection by snapshots of infection efficiency up to a single time R. solani. Single propagules (colonized poppy seed) but they fail to show the dynamics. The pathozone of T. viride were introduced alongside germinating changes with time as inoculum germinates and grows seeds of radish in soil packs containing single or moves towards the host, while tbe susceptibility of propagules of JR. solani placed at 0, 2, 5, 10, 15 or the host, especially for a damping-off fungus such as 20 mm from the seed. Both seed and inoculum were i?..so/(3m, changes. These dynamical processes affect positioned at a depth of 5 mm. There were 15 the rate of spread of an epidemic from initial replicates per treatment arranged in a fully-randominoculum (Gilligan, 1994) and might be affected by ized design which included a full set of replicated the presence of a biological control agent. Small controls containing uncolonized, sterile poppy seed, differences of a biological control agent on primary Soil packs were prepared by adding 60 g of clean infection (from initial inoculum) can become ampli- quartz sand (Grade mm and 10% moisture fied by subsequent multiplication due to secondary by weight) to 10-cm lengths of 5-cm-wide clear (plant to plant) infection and so have profound Layflat tubing (Layfiat Tubing, Isle of Wight) effects on the course of an epidemic (Kleczkowski, sealed at each end with three staples. After adding Bailey & Gilligan, 1996). Here we space plants in inoculum and seed, the packs were sealed in clear populations far enough apart to eliminate secondary plastic containers and incubated at 23 C for 9 d infection in order to test how well the dynamical (16 h light and 8 h dark). After emergence, plants behaviour of a pathogen exposed to a biological were visually assessed for disease symptoms (lodging control agent within the pathozone can predict due to damping-oft) on days 5, 6, 7, 8 and 9 after epidemic behaviour. The prediction of epidemics incubation. These data were used to estimate from pathozone behaviour in relation to primary and pathozone profiles for each treatment at each time of secondary infection is discussed elsewhere (Klecz- observation, kowski, Gilligan & Bailey, 1997). Population experiment MATERIALS AND METHODS,,,..,.,., i he progress of disease m a population of radish ^, plants exposed to primary infection by R. solani in Inoculum,, ^.., i the presence or absence of 1. viride was examined m Inoculum oi R. solani (isolate R5) was prepared by replicated microcosms. Forty radish seeds were sown cutting l-o-mm diameter discs of mycelium from the 4 cm apart in seed trays measuring 32 cm by 20 cm growing edge of a 6-d-old colony grown on potato and containing 3 kg of sand (Grade "

3 Pathozone dynamics and biocontrol (b) R. solani % + foodbase \ (c) R. solani + T. viride + foodbase Day6 A V0"" e e ro CD CO o > X! CD X2 O Distance (mm) Figure 1. Profiles describing changes in the probability of disease with distance between the host and (a) Rhizoctonia solani, (b) R. solani with an additional foodbase (poppy seed) placed alongside the host, (c) R. solani with Trichoderma viride colonized poppy seed placed alongside the host, at different times after introduction of the inoculum. Data were fitted individually with a critical exponential function, P(r) = (d^ + d^r) exp( 6^ r). 10% moisture by weight). Propagules of the pathogen (mycelial disc inoculum) were placed at random locations at a rate of 100 discs per tray. In half the trays, each radish seed was protected from disease by a single propagule of T. viride (colonized poppy seed) carefully positioned beside the seed at the time of sowing in such a way as to minimize displacement during germination of the host. Propagules of R. solani and T. viride were arranged at a common depth just below the surface. This essentially restricted the dynamics of parasite and antagonist interaction to a plane as the R. solani grew preferentially over the sand surface. There were three replicates per treatment. Individual trays were covered with clear plastic lids and incubated at 23 C (16 h light, 8 h dark). The numbers of diseased (post-emergent, dampedoff) plants were recorded at daily intervals for 10 days following infestation. The possibility of secondary infection (i.e. spread from plant to plant) was minimized by the comparatively large distance between plants and by removal of plants immediately after they were diagnosed as diseased. Modelling changes in pathozone profile over time Preliminary investigation (Gilligan & Bailey, 1997) showed that the profiles for the probability of infection with distance were non-monotonic with a maximum chance of infection for inoculum close to but not touching the host (Fig. 1). The pathozone profiles for the probability of infection at a single time, P(r), were empirically described by a critical exponential model of the form, -d^r). (1)

4 362 D. y. Bailey and C. A. Gilligan The model serves as a parsimonious, non-linear description of the profile, having an intercept at r = 0, a maximum, a point of inflection and a lower asymptote of zero. The parameters of the model are interpreted mechanistically in the discussion. The model was reparameterised in the form, where K = exp ( ^3), for ease of fitting (Ross, 1987; 1990). This form greatly improved convergence. Fitting was done using GENSTAT (Anon., 1993) under an assumption of binomial errors because of the quantal nature of the variable. Model (2) was fitted separately to each of the individual pathozone profiles. This gave separate estimates of 6^, 6^ and K for 10 profiles corresponding to five sampling times (days ) with and without T. viride. The model was then tested for common parameters over the five times of observation using the methods of Ross (1987, 1990) and Gilligan (1990). One parameter, 6^, did not ^ = change over time. Trends in the remaining parameters (^1 and K) over time were examined and finally the model was fitted in the form. in which 6^ (t) and K{t) are monomolecular functions of time. This produced a non-linear response surface that describes the change in the pathozone profiles over time, in the presence and absence of T. viride. We also assessed the effect of T. viride by testing for common parameters in (2) between pairs of profiles with and without Trichoderma at each time separately. Computation of extent of pathozone influence Previous work (Gilligan, 1980; Gilligan & Simons, 1987) had defined the extent {R) of pathozone influence empirically from data as the furthest distance of inoculum from the host at which at least one host became infected in placement experiments. Here we define the extent of pathozone influence theoretically as that distance beyond which the probability of infection is < 5 %: R is solved iteratively for a given time {t) by setting P{R, t) = 0-05 in equation (3), given here in exponential form, by, (3) pathozone profile with parameters changing over time. The probability {(p) that a plant is infected by a single propagule of inoculum is given by. The equation for 0 is generalized to (p{t) by recalling {6^ (t) + 6^ R) exp (- ^3 {t) R) = 0. (4) that R, 6^ and 6^ vary with time (3, 4), giving. Prediction of disease progress from pathozone behaviour The expected progress of disease in a population of radish plants in which secondary infection was prevented was computed from pathozone profiles. The method follows that of Gilligan (1985) except that allowance was made for a non-monotonic where C, is the probability that a propagule lies within the pathozone and ^ is the probability that a randomly located propagule within the pathozone infects the host. For the xdidhh-rhizoctonia system with inoculum confined to a plane, ^ is independent of time but }Jr increases with time as mycelium originating from inoculum further out in the pathozone reaches the host. The model is first derived for a single time. We consider first a single unit of inoculum confined to a plane around a single host plant. The probability of occurrence within the pathozone, ^, is given by the proportion of the area of the plane {A) occupied by a single pathozone of radius R, A where h is the radius of the host. = fir).p(r)dr, Jo where. 2(r + h) is obtained as is the density function for occurrence of a propagule at a particular distance (strictly between r and r + Ar) in a pathozone, measured from the host surface, and P(r) = (^j + (92'') exp ( ^3 r) is the probability of infection from a distance r. Hence, for a fixed time, (5) (6) (7) -6sr)dr, (8) which after solution and further manipulation gives, <f) = B[l--{l + CR)exp{ 6R)], (9) c B{t) = C(0 = AB"-^^' 2n ut) A (11) (12) l (13). (14)

5 K Pathozone dynamics and biocontrol *2 0-4-I Time (d) Figure 2. Change in parameter values over time for pathozone profiles described by the reparameterised critical exponential function, P{r) = {6^ + d^ r) K\ in the presence (#) and absence (O) of Trichoderina viride. Fitted curves for 6^ and K were of the form a(l exp maximum and then decreased asymptotically to zero in the absence and in the presence of T. viride (Fig. la, c). The presence of T. viride decreased the probability of disease at all distances and all times. Incorporation of a food base (poppy seed) without T. viride greatly increased disease (Fig. 1 b). We conclude that the poppy seed alone was not inhibitory to R. solani and that the advantage of an additional food base close to the host was not sufficient to offset the potential for T. viride to control disease. The results for the control involving uncoionized poppy seeds are not considered further. The critical exponential model provided a good description of the pathozone profiles at each time for R. solani in the absence and presence of T. viride (Fig. la, c). Both 6^ and K{= exp( ^g)) increased asymptotically with time and were adequately described by a monomolecular function (Fig. 2). The third parameter, d^ decreased approximately linearly with time, when all parameters were estimated separately at each time. However, formal testing for common parameters across time and within treatments suggested that 6^ could be assumed to be common over time, whilst 6-^ and X" varied with time. The outlier for 6.^ in Figure 2 reflects a difference in the peak probability for infection associated with only one day (see day 6, Fig. la). In the absence of additional data between days 5 and 7 with which to define this trend, it was not identified as being of either statistical or biological importance. The model for the change in the pathozone profile over time then yields (using the reparameterised form in eqn (3)), ' P{r,t) = {d^{t) + d^r)k{ty\ where in the absence of T. viride. (16a) Now if there are P propagules within an area A and A'" plants, the number of infected plants at time t is given by. The parameters for the population dynamics given in eqn (15) are all derived from pathozone behaviour studied around single plants. Equation (15) was used to predict the progress of infection and disease, together with 95 % confidence intervals (based on binomial expectations), in the presence and absence of T. viride. These predictions were compared with the observed data from the population experiment. Equation (15) is therefore a prediction of population behaviour scaled up from individual behaviour. RESULTS Placement experiments Pathozone profiles describing the changes in the probability of disease with distance increased to a and d, = 0-22, and in the presence of T. viride, (166) ^,(0 = 0-26(1-exp(-0-41(i~5-09))). p( (16c) and (92 = The corresponding response surfaces for P{r, t) are given in Figure 3 in which the inhibitory effect of T. viride on the probability of disease is shown over distance and time. The models accounted for 98-5 % and 91-5% respectively of the variance. Pairs of pathozone profiles with and without Trichoderma were tested for common parameters at each time, using the reparameterised form of P{r) given in (2). The models failed to converge when testing for common parameters at days 5 and 6 after infestation. Thereafter, addition of T. viride was associated with a reduction only in the intercept parameter, d^. It was not possible with the current data-sets to refine the models in (16) to a simpler

6 364 D. y. Bailey and C. A. Gilligan (a) Trichoderma absent (b) Trichoderma present Distance (mm) Distance (mm) Figure 3. Response surfaces of the form, P{r, t) = {6^{t) + d^r)k{ty {a) in the absence of T. viride, with 6^{t) = 0-61(1-exp(-1-05(^-5-02))), K(0 = 0-81 (1-exp(-2-22(f-3-99))), ^^ = 0-22 and (b) in the presence of T. viride, ^1 (0 = (1 - exp (-0-41 (f-5-09))), K(0 = 0-74(1-exp (-0-75 (i-3-99))), ^^ = Solid points represent the data. 20 Figure 4. Change in the extent (i?) of pathozone influence, beyond which ^ 5 % of propagules succeed in causing disease, for Rhizoctonia solani and radish over time, in the presence (#) and absence (O) of Trichoderma viride placed at the host surface. The fitted curves are given by eqns (17a, b). function, P(r,t;Tv) form in which fewer of the parameters (in 16fe, c) are affected by T. viride (Tv). Changes in pathozone width, computed at each time using eqn (4), were plotted against time (Figure 4) and a smooth monomolecular function fitted to curves, giving, in the absence of T. viride. i? = 22-6(1-exp (-0-65(^-4-64))), and in the presence of T. viride, R= 13-8(1-exp(-0-38(f-4-63))). 10 (17a) (176) It follows that the asymptotic extent of the pathozone is reduced from 22-6 mm for R. solani and radish when T. viride is absent, to 13-8 mm when the biocontrol agent is present. Population experiment The observed and predicted average disease progress curves (DPCs) for the unprotected and protected 0-0- Figure 5. Comparison of observed (circles) and predicted (lines) disease progress curves for Rhizoctonia solani in populations of radish plants in the presence (#) and absence (O) of Trichoderma viride, placed in contact with host plants. Data represent the means of three replicates. The dotted lines represent 95 % confidence intervals about the predicted curve, based on binomial sampling. crops are shown in Figure 5. Trichoderma reduced disease from a maximum of 42 %, in the absence of biocontrol to 13%. The average DPC for the biocontrol treatment was sigmoidal with an asymmetrical point of inflection approximately 7 d after sowing. The average DPC in the unprotected crop rose smoothly without inflection towards an asymptote (Figure 5). The models for disease progress based upon pathozone behaviour eqns ((12-15, 16b, c)) closely predicted the average DPCs in both the unprotected and in the protected crops (Figure 5). DISCUSSION We have shown that the biological control agent, T. viride, reduces the extent of the pathozone influence and the infection efficiency of R. solani infecting radish and that this change in pathozone behaviour can be used to predict changes in disease progress in 10

7 Pathozone dynamics and biocontrol CD CO a> CA T3 s O XJ CO O ic) e-^3=0-70 e"^3=o.8o n-n " " max Distance (rmm) Figure 6. Effects of changing {a) 6.^, {b) d^_ and {c) exp {-(93) on the shape of the profile described by P{r) = {d^ + d^r) exp{ 6^r). {d) Relationships amongst parameters and characteristic features (intercept, maximum ''max' ^^^ inflection t^^^) of the function. Default values for the parameters were 6^ = 0'6, 6^ 0-2, exp ( (Jg) = 0-8. a population of plants. This is an important example of interpreting population behaviour of an epidemic from the infection and disease of single plants. The shape of the pathozone profile rises and falls with distance and is accurately described by the critical exponential function. It is the result of two interacting processes involving the density of mycelium that reaches the host, which decreases with distance of inoculum from the host, and the net infectivity at the host surface, which has been shown to increase during the period of inoculum challenge (Gilligan & Bailey, 1997). When inoculum is placed close to the host surface, the highmyeelial density is disrupted by the mechanical disturbance of the germinating seedling. Gilligan & Bailey (1997) present quantitative evidence for this change in the net infectivity at the surface of the host. This reduces the probability of infection for inoculum close to the host. When inoculum is placed a little further away, contact with the host is delayed, avoiding the period of disturbance and thereby increasing the probability of infection. When placed further away still, the reduction in the density of mycelium making contact with the host results in a concomitant decay in infectivity. We used a critical exponential model (1) as a convenient, parsimonious non-linear description of the shape of the pathozone profile. The critical exponential model has three parameters. Certain features of the model are summarized in Figure 6. The pathozone profile characterizes the infection efficiency of inoculum which is defined as the change in the probability of infection with distance of inoculum from the host (Gilligan, 1985; Gilligan & Simons, 1987; Gilligan & Bailey, 1997). The pathozone profile is also used to define the extent of pathozone influence theoretically, beyond which the probability of infection is < 5%. Two parameters, (9i and K. = exp ( ^3) increase monotonically over time, towards limiting values (Fig. 2, eqns 166, c). 6^ defines the probability^ of infection for inoculum located at the host surface (Figure 6). The change in 6-^ is caused by a build up in fungal mycelium as hyphae grow out of the inoculum in contact with the host. As the density of mycelium increases on the hypocotyl, so the probability of infection increases (Gilligan & Bailey, 1997). Formally, 1/6^ defines the distance between the maximum and the point of inflection. In practice, 6^ has marked effects on the furthest extent of the pathozone (Fig. 6). Hence the change in K over time reflects the expansion of the pathozone as mycelium reaches and infects the host from more distant inoculum. Placement of T. viride in contact with a germinating radish seed reduced 6^ but had no significant effect on 6^. This is consistent with the main antagonistic eftect being local to the site of placement. We note, however, that there is empirical evidence for a difference in trend of 6^ with time due to T. viride (Fig. 2), which would indicate an effect extending out into the pathozone (cf. Fig. 6 c), though the difference was not statistically significant. Prediction of the course of disease progress from pathozone behaviour (Fig. 5) matched not only the shapes but also the magnitudes of independent disease progress curves with and without T. viride.

8 366 D.J. Bailey and C.A. Gilligan Time (d) Distance (rmm) Distance (mm) Figure 7. (a) Effects of changing K = exp( ^3) on disease progress curves; the disease progress curves were computed from eqns (12-16), with the asymptotic value of K (eqn 16) changing, (b) Relationship between/(r) and radial distance, r. (c) Relationship between the product/(r).p(r) and radial distance, r, measured from the host surface. The three curves correspond to i? = 0-80, upper curve, K 0-75, middle curve, K = 0-70 lower curve. See text for interpretation of regions I, II and III. 20 We conclude that the effects of T. viride on pathozone dynamics were sufficient to cause the substantial reduction in disease progress observed in the experiments. Much has been written on possible mechanisms for biological control (see e.g. Baker & Cook, 1974; Cook & Baker, 1983; Harman & Lumsden, 1990; Hornby, 1990) but, to the best of our knowledge, this is the first analysis of biological control that links the dynamics of disease progress to detailed behaviour of the biological control agent on the pathogen in the pathozone. Although the system was restricted to primary infection, the disease progress curves, for the case with T. viride, are characteristically sigmoidal. Moreover the asymptotic levels of disease were < 100%, even in the absence of biological control. Both features reflect the temporal change in pathozone behaviour. This can be conveniently shown by considering the differential equation associated with eqn (15). Consider, then, d(n-j) which after further substitution and minor rearrangement gives, d(n-i) dt Since N is constant dl/dt = ~d(n l)/dt, yielding a differential equation for the proportion of infected plants, of the form. dt where <p(t) (18) This is analogous to a monomolecular function with a time varying rate (cf. Jeger, 1987). The solution depends on the form of ^(t) but generic solutions include the Gompertz function (with its characteristic point of inflection) when j3(t) is a simple exponential function. Our function for ^{t) is a more complicated expression involving ratios of exponentials but it can be shown that the point of inflection is given by the solution of ^'(t) ^(t) = 0, which depends on (f){t). Strictly, the model in eqns (15, 18) predicts 100% infection (I = N; t-^oo). In practice this equilibrium is not reached because the growth of the fungal colony is finite and incapable of reaching all host plants. The epidemic, therefore, ceases at a level below 100% infection. The interruption of transient disease progress curves for R. solani by decreasing susceptibility of the host and exhaustion of inoculum is discussed in Kleczkowski et al. (1996). Following the arguments of Kleczkowski et al. (1996), we note that the exponential terms within fi{t) in eqn (18) reflect these processes. Some practical inciplications for the optimisation of biocontrol come from analysis of the sensitivity of disease progress curves to parameters that characterize the shape of the pathozone. The asymptotic

9 Pathozone dynamics and biocontrol 367 level of infection and disease is particularly sensitive to K{ = exp ( /^g) (Fig. 7 a). Even a small decrease in K (equivalent to an increase in 6*3), due, for example, to the presence of a biological control agent, can result in a comparatively large reduction in the asymptotic level of disease (Fig. la). This, in turn, reflects the changes in spatial distribution of propagules within the pathozone and the efficiency of infection. The probability of inoculum occurring in the pathozone/(r) increases with r (Fig. Ib), while infection efficiency {Pir)) increases slightly with distance and then declines exponentially (cf. Fig. 6c). The product, f{r).p{r), is a measure of the contribution of inoculum that falls within concentric zones to infection (cf. eqn (6)). We distinguish subjectively three regions (Fig. 7 c): an inner region (I) dominated by low values of/(r), an outer region (III) dominated by low values of P{r) and an intermediate region (II), between 3 12 mm from which most successful infections occur. We conclude that successful biological control must be able to inhibit potential infections that arise from inoculum in this intermediate region, where there is optimization of the inoculum location and infection efficiency. While there is no direct evidence for synergism amongst propagules of R. solani in pathozones, the incorporation of synergism into pathozone models is discussed elsewhere (Gilligan & Simons, 1987; Brassett & Gilligan, 1988). The experiments and models have been restricted here to epidemics with primary infection. Further work is underway to predict epidemics involving dual sources of primary and secondary infection from pathozone behaviour (Kleczkowski et al, 1997). ACKNOWLEDGEMENTS This work was funded by the award of a Research Grant from the Biotechnology and Biological Sciences Research Council which we gratefully acknowledge. We thank Dr A. Kleczkowski and Dr W. Otten for helpful discussion of the manuscript and B. V. D. Goddard for assistance with the figures. REFERENCES Anonymous Genstat 5 : Reference manual. Oxford: Oxford University Press. Baker KF, Cook RJ Biological control of plant pathogens. San Francisco: Freeman. Benson DM, Baker R Epidemiology of Rhizoctonia solani pre-emergence damping-off of radish: influence of pentachloronitrobenzene. Phytopathology 64: 38^0. Brassett PR, Gilligan CA A discrete probability model for polycyclic infection by soil-borne plant parasites. New Phytologist 109: Brown ME, Hornby D Behaviour of Ophiobolus graminis on slides buried in soil in the presence and absence of wheat seedlings. Transactions of the British Mycological Society 56: Cook RJ, Baker KF The nature and practice of biological control of plant pathogens. St. Paul: American Phytopathological Society. Gilligan CA Zone of potential infection between hosts roots and inoculum units of the take-all fungus, Gaeumannomyces graminis var. triciti. Soil Biology & Biochemistry 12: Gilligan CA Probability models for host infections by soilborne plant pathogens. Phytopathology 75: Gilligan CA Comparison of disease progress curves. New Phytologist 115: Gilligan CA Temporal aspects of the development of root disease epidemics. In: Campbell CL, Benson DM, eds. Epidemiology and Management of Root Diseases. Heidelberg: Springer-Verlag, Gilligan CA, Bailey DJ Components of pathozone behaviour. New Phytologist 136: Gilligan CA, Simons SA Inoculum efficiency and pathozone width for two host-parasite systems. New Phytologist 107: Harman GE, Lumsden RD Biological disease control. In: Lynch J, ed. The Rhizosphere. Chichester, UK: Wilev, Henis Y, Ben-Yephet Y Effect of propagule size of Rhizoctonia solani on saprophytic growth, infectivity and virulence of bean seedlings. Phytopathology 60: Hornby D Biological control of soil-borne plant pathogens. Wallingford, UK: CAB International. Jeger MJ The influence of root growth and inoculum density on the dynamics of root disease epidemics: Theoretical analysis. New Phytologist 107: Kleczkowski A, Bailey DJ, Gilligan CA Dynamically generated variability in plant pathogen systems with biological control. Proceedings of the Royal Society Series B 263: Kleczkowski A, Gilligan CA, Bailey, DJ Scaling and spatial dynamics in plant pathogen systems: from individuals to populations. Proceedings of the Royal Society Series B. (in press). Lynch JM The rhizosphere. Chichester, UK: Wiley. Punja ZK, Grogan RG Myeelial growth and infection without a food base by eruptiveh' germinating sclerotia of Sclerotium rolfsii. Phytopathology 7i: Reynolds KM, Benson DM, Bruck RI Epidemiology of Phytophthora root rot of Fraser fir: rhizosphere width and inoculum efliciency. Phytopathology 75: Ross GJS Maximum likelihood program. Oxford: Numerical Algorithms Group. Ross GJS Nonlitiear estimation. New York: Springer Verlag. Wildermuth GB, Warcup JH, Rovira AD, Growth of Gaeumannomyces graminis var. tritici in soil in the presence and absence of wheat roots. Transactions of the British Mycological Society 82:

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