Saprotrophic invasion by the soil-borne fungal plant pathogen Rhizoctonia solani and percolation thresholds

Transcription

1 RESEARCH New Phytol. (),, Saprotrophic invasion by the soil-borne fungal plant pathogen Rhizoctonia solani and percolation thresholds D. J. BAILEY*, W. OTTEN, AND C. A. GILLIGAN Department of Plant Sciences, University of Cambridge, Downing Street, Cambridge CB EA, UK Received 9 August 999; accepted February SUMMARY In this paper we distinguish between invasive and noninvasive (finite) saprotrophic spread of the soil-borne fungal plant pathogen, Rhizoctonia solani amongst discrete sites of nutrient resource. Using simple concepts of percolation theory, we predict the critical threshold distance, associated with a threshold probability, between donor (colonized) and recipient (uncolonized) nutrient sites at which R. solani can spread invasively by mycelial growth through a population of nutrient sites on a lattice. The critical distance for invasive spread is estimated from colonization profiles derived from placement experiments that summarize the probability of colonization with distance between replicated pairs of colonized and uncolonized sites. Colonization profiles were highly nonlinear, decaying sigmoidally with distance. Thresholds for invasive spread were predicted at inter-site distances of 8. mm and.8 mm for sites of low and high nutrient agar, respectively. In population experiments with inter-site distances below the predicted thresholds, the spread of the fungus was invasive in all replicates. At large distances ( mm for low, and mm for high nutrient sites) the spread of the fungus was always finite, with the proportion of finite replicates decreasing sharply close to the percolation threshold. Invasive spread did not depend on the furthest extent of growth of the fungus but on distances predicted by the percolation thresholds. Invasive spread of the fungus is also examined in a more natural and variable, nonsterile system involving the growth and colonization of a lattice of poppy seeds over sand. The system is characterized by a decay in the probability of colonization between older poppy seeds, which effectively quenches saprotrophic spread. Hence in the population experiments with poppy seeds all growth was ultimately finite. The threshold distance, corresponding to the critical percolation probability for invasive growth changed from 8 mm to mm over d leading to a switch from invasive to finite growth. We conclude that percolation theory can be used to link the growth of individual mycelial colonies to the formation of patches that result from the colonization of particulate organic matter. The nonlinearity of the colonization profiles combined with the presence of a percolation threshold means that small changes in the distance between nutrient sites can result in large differences in final patch size. The rapid decay of particulate organic matter in a more natural system can have a profound effect on the dynamics of colonization, restricting saprotrophic invasion of the soil. The consequences of invasion thresholds for colony growth of saprotrophic and parasitic fungi in dynamical systems are briefly discussed. Key words: percolation, heterogeneity, Rhizoctonia solani, invasive spread, pathozone. INTRODUCTION One important feature in the invasion and persistence of many ecologically and economically important fungal parasites and saprotrophs in soil is the ability of the fungus to spread by mycelial growth and expansion of fungal colonies. The lateral spread of parasites depends on the endogenous supply and translocation of nutrients within the fungal colony, the growth-habit of the colony and the distances between susceptible host roots or other organs. If susceptible hosts are too far apart, local *Author for correspondence (tel 9; fax 9; djb cus.cam.ac.uk). invasion ceases as the fungus exhausts its nutrient supply before infecting a new host. Similar constraints apply to the transmission of mycorrhizal fungi between infected and uninfected roots (Smith & Read, 997) and to the transmission of saprotrophs between discrete fragments of colonized and uncolonized organic matter. In each case it is convenient, at least initially, to visualize spread occurring through a population of discrete sites on a lattice. The sites correspond to discrete nutrient sources, comprising susceptible roots or discrete fragments of organic matter, and the entire lattice represents the soil matrix in which spread occurs. The lattice might

2 RESEARCH D. J. Bailey et al. (a) (b) (c) Donor r Recipient Probability of colonization r c Distance between donor and recipient sites (r, mm) P c r >r c r <r c Fig.. Use of colonization profiles between donor and recipient sites to predict the threshold distances for invasive or finite colony expansion of Rhizoctonia solani from a point source in a population of nutrient sites. (a) Colonization between a donor and recipient site over a distance r apart can be used to characterize (b) the infection efficiency (probability of colonization) between individual donor and recipient sites. This curve is used to identify a threshold distance, r c, which corresponds to the critical probability, P c for percolation. (c) For distances r c and probabilities P c, invasive spread of the pathogen amongst a population of nutrient sites might occur whereas for distances r c and probabilities P c, spread of the pathogen is finite. be three-dimensional for discrete nutrient fragments such as comminuted straw, leaf or root tissue. However, for lateral spread of disease, it might be restricted to a horizontal plane passing through a population of seeds, roots, hypocotyls or other organs. Local spread, and hence progressive invasion by colony expansion, then depends on the spatial distribution of uncolonized sites. The fungus will continue to spread as long as it makes contact with these sites, creating an expanding patch, the size of which is limited only by the size of the system in which the fungus is growing. However, the fungus will stop spreading if it fails to make contact with new sources of uncolonized hosts or organic fragments, resulting in a patch of finite size. The outcome of such a stochastic, spatial process can be described by the theory of percolation (Bunde & Havlin, 99) under certain assumptions identified by Grassberger & Scheunert (98), Grassberger (98) and Cardy & Grassberger (98) whereby there is a critical probability for transmission between sites below which invasion ceases. By applying percolation theory to the saprotrophic growth of a soil-borne fungal plant pathogen, Rhizoctonia solani, spreading in a finite system amongst a population of nutrient sites organised on a triangular lattice we predict the existence of a threshold probability of colonization between sites (Fig. ). If the probability of colonizing a neighbouring site is above this threshold the fungus might spread invasively creating large patches but below the threshold growth is finite and restricted to comparatively small patches (Fig. c). In this paper, we investigate the effects of site density and nutrient status on the probability of spread between neighbouring sites and hence on finite and invasive spread within a population of sites. Specifically, we ask; is there an invasion threshold? Is it possible to predict invasive spread from a percolation threshold? How are the dynamics affected by the nutrient status and by declining susceptibility to colonization? We use curves (colonization profiles, Fig. b) from placement experiments (Gilligan & Simons, 987) to summarize changes in the probability of colonization with distance between donor and recipient sites (colonization efficiency, Fig. a). These curves are used to estimate the critical inter-site distance that corresponds with a percolation probability (Fig. b) that defines the threshold relating to spread amongst a population of sites organized on a two-dimensional, triangular lattice (Fig. c). These predictions are tested empirically in a simple system for populations of agar sites that vary in density and nutritional status. Other authors have used agar to study colonization, notably Ritz (99) and White et al. (998), who used a tessellation of agar tiles differing in nutrient availability to examine the growth response of R. solani and other fungi. That work was focused on the exploitation of heterogeneous nutrient sources, with quantitative assessment of hyphal growth on individual tiles and small gaps between the tiles to prevent nutrient diffusion between tiles. Here we are concerned with the role of the gaps between nutrient sources in determining the probability of colonization and, in particular, the threshold probability that is important for scaling from individual behaviour to invasive spread within a population. The predictions of invasive and noninvasive spread for the simple agar system depend on constant colonization efficiency with time. In more natural (nonsterile) systems patch size is limited, not only by

3 RESEARCH Saprotrophic invasion by Rhizoctonia solani 7 Probability of colonization (P)..8.. P c Distance (r, mm) Fig.. Change in the probability of colonization, P, of Rhizoctonia solani with distance, r, between donor and recipient sites with low (% potato dextrose agar (PDA), open circles) and high (% PDA, crosses) nutrient agar. Data are fitted with logistic functions P ( exp(.(r.8))) and P ( exp(.(r.8))), respectively, to provide profiles for the probability of colonization with distance. Dotted vertical lines represent 9% confidence intervals about the estimated threshold distance for invasion, r c. the stochastic process associated with the distance between sites, but also by a time-dependent decay of soil organic material as it becomes colonized by other soil microbes. This has the effect of reducing the colonization efficiency of the pathogen and is equivalent to a quenching term on the dynamics of spread (Kleczkowski et al., 99). To test this quenching effect, we compare the properties of patch formation in the agar system with a more natural system involving the spread of R. solani amongst poppy seeds (Papaver somniferum), again with a fixed distance between sites on a lattice. In particular, we examine the dynamics of colony expansion through the lattice to identify a change in the rate of colonization when the fungus is forced to switch from invasive to noninvasive spread as the nutritional status of the poppy seed decays. Some consequences of invasion thresholds for colony growth of saprotrophic and parasitic fungi are briefly discussed. MATERIALS AND METHODS System I: percolation with no decay of substrate (agar sites) Changes in the probability of colonization with distance. For sites distributed on a triangular lattice, percolation theory predicts that the threshold for transition between production of a small patch and the growth of a large patch will occur when the probability of colonization between individual sites exceeds P c sin (π 8). for bond percolation (where P c is the critical probability for percolation; Stauffer & Ahorney, 99). To estimate the critical distance, r c, for P c., we constructed probability profiles (Gilligan & Bailey, 997) describing the change in probability of colonization with distance between a donor (colonized) and a recipient (uncolonized) site. Small spots of agar ( mg, mm diameter) of % (low nutrient) or % (high nutrient) (w v) potato dextrose agar (PDA) were used to provide reproducible substrates at each site, with no significant decay (see later) during the course of the experiment. Pairs of agar sites, comprising a donor and recipient, were positioned at,, 8,,,,, 8,, or mm apart (from centre to centre) in Petri plates (9-mm diameter). There were replicates of each distance in a fully randomized design and the experiment was repeated for low and high nutrient sites. The donor site of each agar pair was inoculated with a single hyphal strand, c. mm in length, removed from the growing edge of a -d-old colony of R. solani Ku hn (AG ) grown on water agar. Moist filter paper was placed in the lid of each Petri plate to avoid desiccation of the agar and the plates were sealed and incubated in the dark at C. Plates were assessed daily for d for colonization of the recipient sites. The effect of inter-site distance on finite and invasive spread in populations. To demonstrate the existence of a threshold distance between sites for invasive spread of R. solani, the dynamics of patch formation were measured for growth of the fungus amongst replicate populations of agar sites. Sites (-mm diameter) of either low or high nutrient agar ( mg per site) were spotted onto a triangular lattice in large Petri plates (-mm diameter) at, 8,,,,, 8, or mm apart. The centre agar site of each plate was inoculated with a single hyphal strand ( mm in length) removed from the growing edge of a -d-old colony of R. solani grown on water agar. Each experiment involved six replicates for all distances except mm for which there were two replicates. The experiment was fully randomized. To avoid desiccation of the agar, a moist filter paper was placed into the lid of each plate. The plates were sealed and incubated in the dark at C. Plates were assessed daily for d using a dissecting microscope (magnification ) and the number and locations of colonized sites recorded. Fungal growth on each replicate plate was scored as invasive or finite after d, by which time most patches had either reached the edge of the system or had stopped growing and before nutrient sites showed visible signs of desiccation. Invasive patches were identified as patches which had reached the outer edge of the experimental system whereas finite patches had not.

4 8 RESEARCH D. J. Bailey et al. mm % PDA % PDA mm 8 mm 8 mm No. of sites colonized 8 7 mm mm 8 mm mm 8 mm mm mm Time (d) Fig.. Change in the number of agar sites colonized by Rhizoctonia solani with time for replicate populations with different distances between sites (given in the top left of each plot) arranged in a triangular lattice for high nutrient (% potato dextrose agar (PDA)) and low nutrient (% PDA) agar. Solid lines indicate replicates exhibiting invasive spread (patches had reached the edge of the system after d) and dotted lines, replicates showing finite spread (patches which had not spread to the edge of the system). System II: percolation with substrate decay (poppy seed) Changes in the probability of colonization with distance and time. To examine the effect of seed (inoculum) decay on the threshold distance for invasive spread, r c, probability profiles were constructed using seeds that had been incubated for three different times (, 7 and d) on nonsterile sand. Poppy seeds (Papaver somniferum L.) measuring approx. mm in diameter were initially sterilized by autoclaving for h at kpa. Large Petri plates (-cm diameter) were filled with g of sand (Hepworth Minerals & Chemicals Ltd, Redhill, UK; Grade, with % gravimetric water content). Donor seeds were positioned on the sand surface at,,, 9,,, 8,, and 7 mm from uncolonized seeds. Donor seeds were inoculated with a single hyphal strand of R. solani removed from the growing edge of a -d-old colony grown on water agar either immedi-

5 RESEARCH Saprotrophic invasion by Rhizoctonia solani 9 ately or after 7 or d incubation of the seeds on sand in the dark at C. There were replicates per treatment and the plates were assessed daily for colonization until no change in the number of seeds colonized was detected. Time after innoculation (d) Invasive spread Finite spread Patch formation and seed density. The dynamics of patch formation were examined for R. solani growing through a population of poppy seeds. Replicate seed trays ( mm deep) were filled to a depth of cm with sand (Hepworth Minerals & Chemicals Ltd). Sterile poppy seeds were placed on the surface of the sand in a triangular lattice. Nine seed densities were prepared with distances between neighbouring seeds of,,, 8,,,,, and 8 mm; each density was replicated five times. Spread and colonization of the seeds was initiated from the central poppy seed, colonized by R. solani. The trays were sealed in plastic bags to reduce evaporation and incubated in the dark at C. The bags were opened daily and the poppy seeds were assessed for colonization using a binocular microscope ( ) for 7 d. RESULTS System I: percolation with no decay of substrate (agar spots) Predicting the threshold distance for invasion from profiles describing changes in the probability of colonization with distance. The probability of colonization, P, decayed sigmoidally as the distance, r, between donor and recipient sites increased (Fig. ). Threshold distances, r c, for bond percolation on a lattice (corresponding to P P c.) were obtained by inverse prediction from a logistic model, P θ ( exp (θ (r θ ))), where θ i are parameters. The model was fitted to the experimental data by maximum likelihood under the assumption of binomial errors using Genstat (Numerical Algorithms Group Ltd, Oxford, UK). Increasing the nutrient status of the agar resulted in a steeper profile shifted to the right and an increase in the threshold distance between sites from r c 8..9 mm for low nutrient sites to r c.8.9 mm for sites with a high nutrient status (Fig. ). Effect of inter-site distance and nutrient status on finite and invasive spread in populations. Increasing the distance between sites and reducing the nutrient status of sites created finite patches (Figs, ). For high nutrient sites, all replicates with inter-site distances mm displayed invasive growth. At mm, five out of six replicates produced finite patches and at and mm all replicates were finite. The threshold distance between sites for invasive spread was estimated from the population data (Fig. ) at approx. 8. mm and mm for low and Fig.. Examples of replicate microcosms showing patches with invasive and finite growth of Rhizoctonia solani between sites of low (% potato dextrose agar) nutrient agar. Note that gaps occur in invasive patches. Distance between sites: mm, invasive spread; 8 mm, finite spread. high nutrient sites, respectively. These estimates were consistent with the predicted thresholds of 8. and.8 mm obtained from the colonization profiles for pairs of nutrient sites (Fig. ). However, maps of invasive spread (Fig. ) show that coverage is not complete and gaps occur even within invasive patches. Effect of the probability of colonization on finite and invasive spread. The proportion of patches exhibiting invasive spread was plotted against inter-site distance (Fig. a) and the corresponding probability (derived from Fig. ; Fig. b). The relationships were highly nonlinear with a very marked change in the proportion of invasive patches over short distances. Whereas invasive spread occurred at different inter-site distances for high and low nutrient status (Fig. a), there was no difference in the empirical estimate of the critical probability (Fig. b). The latter probability was estimated as..7 by inverse prediction after fitting a logistic function to the data for the combined profiles (Fig. b). System II: percolation with decay of substrate (poppy seed) Changes in the probability of colonization with distance and time. The probability of colonization of recipient poppy seeds declined sigmoidally with distance from the donor (Fig. ). Compared with the more artificial

6 RESEARCH D. J. Bailey et al. Proportion of patches showing invasive spread (Pi) (a) (b) 8 8 Inter-site distance (r, mm) Probability of colonization (P) between donor and recipient Fig.. Change in the proportion of patches of Rhizoctonia solani exhibiting invasive spread (P i ) with (a) distance between sites and (b) probability of colonization between donor and recipient sites (P), for growth amongst a population of sites of low (% potato dextrose agar (PDA), open circles) and high (% PDA, crosses) nutrient agar. The probability of colonization (P) was estimated from the relationship in Fig., given the inter-site distance, r. The fitted curves relating the proportion of invasive patches to inter-site distance are given by the logistic functions P i ( exp(.(r.8) and P i ( exp(.(r 7.) for high and low nutrient agar, respectively (Fig. a), and by P i ( exp( (P.))) relating the proportion of invasive patches to the probability of colonization between individual sites (Fig. b). Note that, whilst invasive spread occurs at different inter-site distances for high and low nutrient sites, the probability at which invasive spread occurs is not affected.. Probability of colonization (P) Distance (r, mm) Fig.. Change in the probability of colonization, P, byrhizoctonia solani with distance, r, between pairs of poppy seeds (circles), 7 (squares) or (triangles) d old. Data are fitted with the logistic functions P ( exp( (r.8))), P ( exp(.7(r 7.))) and P ( exp(.9(r.))) for, 7 and - d-old poppy seed, respectively, to provide profiles for the probability of colonization with distance. agar spots, the poppy seed profile was shifted further to the right but dropped less steeply. Ageing of inoculum before exposure of recipients suppressed colonization, shifting the curve to the left (Fig. ). Threshold probabilities, r c, for invasion (P c., for bond percolation on a triangular lattice) corresponded with a maximum distance of 8 mm between neighbouring seeds for young inoculum but declined to 8. mm for seeds incubated on sand for 7 d and to. mm for seed incubated for d (Fig. ). Dynamics of fungal spread. We summarize the dynamics of fungal spread for individual replicates by describing changes in the number of colonized seeds over time for each replicate population (Fig. 7). Although none of the patches had reached the edge of the system after 8 d, the rate of colonization had slowed to almost zero in all replicates, with the exception of a late and meagre start of one replicate at the maximum distance. There was a marked difference in the extent and rate of spread of the fungus above and below the

7 RESEARCH Saprotrophic invasion by Rhizoctonia solani 8 7 mm mm mm 8 mm mm mm No. of sites colonized 8 7 mm 7 mm 8 mm Time (d) Fig. 7. Change in the number of poppy seeds colonized by Rhizoctonia solani with time for replicate populations of poppy seeds with different distances between sites (given in top left of each plot) arranged in a triangular lattice. Data are summarized by sigmoidal (continuous) or monomolecular (dotted) curves. Note that the vertical scales for the numbers of colonized seeds differ amongst the plots. No. of seed colonized 7 (a) 8 8 Inter-seed distance (mm) Furthest colony extent (mm) (b) 8 8 Inter-seed distance (mm) Fig. 8. Change in the average patch size after 8 d given as (a) numbers of colonized poppy seeds (described by an exponential function) and (b) furthest extent of fungal colonization (described by a logistic function), with inter-seed distance for the fungal pathogen Rhizoctonia solani.

8 RESEARCH D. J. Bailey et al. initial threshold distance of 8 mm. Two patterns of colonization were distinguished for curves with and without a point of inflection. Above the threshold distance, the rate of colonization declined immediately for all replicates, typified by a monomolecular increase in which the rate of colonization slowed progressively (Fig. 7, dotted lines). Below the threshold distance, most replicates revealed an initial increase in the rates of colonization followed by a decline giving a sigmoidal pattern. This was arbitrarily described by a logistic curve (Fig. 7, solid lines). The proportion of initially accelerating patches increased as the inter-seed distance decreased (Fig. 7). The relationship between final patch size or furthest extent of fungal colonization and inter-seed distance is highly nonlinear, decreasing as the distance between seeds increases (Fig. 8). DISCUSSION In this paper we have distinguished between invasive and noninvasive (finite) saprotrophic spread of R. solani and, using simple concepts of percolation theory, we have predicted how changing the distance between uncolonized sites can affect the colonization of organic matter. Colonization profiles that summarize the probability of transmission of the fungus with distance between donor (colonized) and recipient (uncolonized) sites identified a threshold distance for either invasive or finite spread of the fungus amongst a population of agar sites. Invasive spread did not depend on the furthest extent of mycelial growth evident in the tails of colonization profiles from the placement experiments (Fig. ). It was, instead, associated with a threshold distance, different for low and high nutrient sites, but corresponding to a common threshold probability, P c.. As the probability of colonization rose above the threshold, so the likelihood of invasive spread increased, leading to characteristically large patches. Conversely, below the threshold, finite growth was more common. We used the threshold probability successfully to predict critical distances above and below which invasive spread would occur. We have also shown that an a posteriori empirical estimate for percolation, derived from the experimental data yields an estimate for P., which corresponds very closely with the predicted value. In natural systems, the size, nutritional status and availability of organic matter is likely to be highly variable. Small differences in soil physical properties, in particular the continuity of air-filled pore space, can have profound effects on the saprotrophic growth and expansion of fungal colonies from a single nutrient site (Otten et al., 999). Here we have considered how to scale up from the behaviour around one site to predict invasion at the next larger scale through a population of sites. In the context of the current analyses, the effects of soil physical properties would be expressed in the colonization profile. This in turn affects the threshold distance for invasion. The colonization profile, devised here for saprotrophic growth, is analogous to the pathozone profile for the growth or movement of fungal parasites during primary and secondary infection. In primary infection (Gilligan, 99; Bailey & Gilligan, 997; Gilligan & Bailey, 997), the fungus grows saprotrophically from a source of previously colonized host material towards a susceptible host. For secondary infection (Kleczkowski et al., 997), mycelial growth between infected and susceptible hosts might be directly fuelled by translocated nutrients from parasitic activity, though saprotrophic growth might also be involved. The analogies between parasitic and saprotrophic invasion can be extended by considering saprotrophy as a special case of the general epidemic process in which colonization replaces infection. Hence nutrient sites might pass through a sequence from susceptible (i.e. uncolonized) to infected (i.e. colonized) and removed (i.e. exhausted) states, yielding the well known dynamics of SIR epidemics. This leads to a joint interpretation of the dynamics of invasion and lateral spread of saprotrophs and parasites, in which parasites are considered to spread horizontally through a population of, say, roots whereas saprotrophs spread through a population of discrete nutrient sites. The analogy holds for saprotrophy as long as the availability of exogenous nutrient between sites is small relative to availability within sites. The model experimental systems used here were restricted to two-dimensional lattices. Clearly most saprotrophic and parasitic activity occurs within three dimensions in soil but exceptions occur in considering colonization of seeds, colonization across the soil surface (Boddy, 99) and when parasitic spread up and down hosts is negligible relative to lateral spread within a soil layer. Strictly, percolating systems apply to static systems in which the transmission properties (which determine whether or not a substance at a certain site on a lattice can be passed on to a neighbour) are fixed at the beginning. This contrasts with dynamical systems, such as epidemics, in which the transmission between sites evolves during the course of the epidemic. Grassberger (98) considered these transitions for the general epidemic process in which susceptible sites become temporarily infected, after which they are permanently immune. He argued that since infection can only pass once between a donor and recipient (since the recipient becomes immune to further challenge) the dynamical process could be treated as a system involving bond percolation in which the probability that an infection can be passed on to a neighbour is fixed and independent. The probability of transmission is given by exp( βt), where β is the rate of transmission (related to the

9 RESEARCH Saprotrophic invasion by Rhizoctonia solani threshold probability) between donor and recipient and T is a fixed period between colonization and exhaustion. Grassberger (98) derives scaling laws for the behaviour of epidemics around the critical threshold as well as velocities for the spread of the wave of infected sites on a square lattice, testing these by relatively large numbers of Monte Carlo simulations. We do not pursue this analysis here because of the shortage of replications around the phase transition from finite to invasive growth but future experiments might address this. The generation of large differences in final state caused by small initial differences in time dependent, dynamical, nonlinear systems has been known for some time and has been widely studied in biological systems in the context of chaotic behaviour (May, 97; Grenfell et al., 99) and dynamically generated variability (Kleczkowski et al., 99). In the latter example, large differences in the final amounts of disease (patches) between replicates were caused by small initial differences in the growth of the fungus combined with the nonlinear, multiplicative effects of secondary infection and an interruption of transients as the resistance of the host plant increased. We suggest that, for such intrisically spatial and dynamical systems such as the spread of fungus into soil, the presence of a percolation threshold can further enhance the nonlinear response and hence, dynamically generated variability between replicate populations. The growth of the fungal colony from poppy seeds was highly variable and colonization profiles were initially shallow with a threshold probability for invasive spread predicted at a distance of 8 mm between sites. Spread of the fungus in the population experiments was initially rapid in certain replicates but quenching, due to ageing of the inoculum, led to a decrease in distance over which the fungus could grow and colonize neighbours (Fig. ). Consequently all patches were finite, ceasing growth before reaching the edge of the experimental system although there were marked differences in the progress curves for colonization through populations of differing inter-seed distance (Fig. 7). We cautiously interpret the cause of these differences as a switch from invasive to noninvasive growth whereby the fungus, spreading between seeds in replicates with a small inter-seed distance for which, initially, the probability of transmission P P c, benefits from an extended period of invasive growth. For replicates with large inter-seed distances, P P c and growth is entirely restricted to noninvasive spread. Changes in infectivity, susceptibility or, as here, in the potential for growth from ageing sites are common phenomena in microbial growth dynamics but until recently, this form of quenching and its effect in interrupting growth dynamics has received little formal study (Kleczkowski et al., 99; Gilligan et al., 997; Gibson et al., 999). In contrast to the quenching effect on the saprotrophic spread of a fungus amongst a fixed population of ageing seeds, a fungal pathogen spreading parasitically in a growing root system might experience a switch from finite to invasive growth. At low root density the fungus is constrained by the availability of susceptible tissue but at root densities that provide an average inter-root distance less than r c (corresponding to a probability of transmission P c ) the fungus has the potential to spread invasively. This might, in part, explain the sudden increase in the rate of infection during the spread of take-all on wheat as the epidemic switches from primary to secondary infection in response to changes in the density of roots (Bailey & Gilligan, 999). The experiments presented here focus on a dynamical approach involving phase transitions in the analysis of saprotrophic growth of soil-borne fungi. It permits distinction between invasive and noninvasive growth between discrete nutrient sources and, by analogy, for growth between infected and susceptible hosts. It has long been known that the density of hosts affects the rate of mycelial spread of soil-borne plant pathogens (Gibson, 9; Burdon & Chilvers, 98). That work treated the effect of host density on the rate of epidemic development as a continuum. Here we infer that there are qualitative, stochastic differences around the percolation threshold distance, below which there is a high probability of invasion and above which invasion is unlikely to occur. Much work has also been done in the quantification and analysis of foraging strategies of soil-borne fungi (Boddy, 99; Bolton & Boddy, 99; Rayner & Coates, 987; Ritz, 99). We have subsumed the detail of colony behaviour and foraging behaviour into the colonization profile. Undoubtedly, the radial density of hyphal growth, the aggregation of hyphae into strands, the degree of branching and anastomosis all influence the probability of invasion. So too does the translocation ability of the fungus and the vital rates of birth and death of fungal hyphae. Some progress in scaling from hyphal to colony behaviour is given in Edelstein (98) and Edelstein-Keshet & Ermentrout (989), whereas theories of invasion through populations at the larger scale are being developed (Shigesada & Kiyosawa, 997). It remains to link the microscopic scales to invasion of multiple colonies through field populations. ACKNOWLEDGEMENTS This work was funded by the award of a research grant from the Biotechnology and Biological Sciences Research Council, which we gratefully acknowledge. Prof C. A. Gilligan also acknowledges the support of the Royal Society and the Leverhulme Trust. We are grateful to Dr Adam Kleczkowski and Dr Nico Stollenwerk of the

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