RESEARCH New Phytol. (2000), 146, 535–544

Saprotrophic invasion by the soil-borne

fungal plant pathogen Rhizoctonia solani

and percolation thresholds

D. J. BAILEY*, W. OTTEN, C. A. GILLIGAN

Department of Plant Sciences, University of Cambridge, Downing Street,

Cambridge CB2 3EA, UK

Received 9 August 1999 ; accepted 16 February 2000

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 8.1 mm and 11.8 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 ("10 mm for low, and "14 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 18 mm to 4 mm over 21d

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.

One important feature in the invasion and per-

sistence 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 ­44 1223 333 900; fax ­44 1223

333 953; e-mail djb21!cus.cam.ac.uk).

invasion ceases as the fungus exhausts its nutrient

supply before infecting a new host. Similar con-

straints apply to the transmission of mycorrhizal

fungi between infected and uninfected roots (Smith

& Read, 1997) 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

536 RESEARCH D. J. Bailey et al.

Donor Recipient

1.0

0.8

0.6

0.4

0.2

0.0

0 5 10 15 20Distance between donor

and recipient sites (r, mm)

Pro

bab

ility

of

colo

niz

atio

n

Pc

rc

r >rc

r <rc

(b) (c)(a)

r

Fig. 1. 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, rc, which corresponds to the critical probability, P

cfor percolation. (c) For

distances ! rcand probabilities " P

c, invasive spread of the pathogen amongst a population of nutrient sites

might occur whereas for distances " rcand 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 frag-

ments, 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, 1991)

under certain assumptions identified by Grassberger

& Scheunert (1980), Grassberger (1983) and Cardy

& Grassberger (1985) 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. 1). If the probability of colonizing a neigh-

bouring 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. 1c). In this paper,

we investigate the effects of site density and nutrient

status on the probability of spread between neigh-

bouring 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. 1b) from

placement experiments (Gilligan & Simons, 1987) to

summarize changes in the probability of colonization

with distance between donor and recipient sites

(colonization efficiency, Fig. 1a). These curves are

used to estimate the critical inter-site distance that

corresponds with a percolation probability (Fig. 1b)

that defines the threshold relating to spread amongst

a population of sites organized on a two-dimensional,

triangular lattice (Fig. 1c). 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 (1995) and White et al.

(1998), 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 prob-

ability 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

RESEARCH Saprotrophic invasion by Rhizoctonia solani 537

1.0

0.8

0.6

0.4

0.2

0.0

0 2 10 18 20

Distance (r, mm)

Pro

bab

ility

of

colo

niz

atio

n (

P)

Pc

1614124 6 8

Fig. 2. Change in the probability of colonization, P, of

Rhizoctonia solani with distance, r, between donor and

recipient sites with low (1% potato dextrose agar (PDA),

open circles) and high (10% PDA, crosses) nutrient agar.

Data are fitted with logistic functions P ¯ 1}(1­exp(0.51(r ®6.85))) and P ¯ 1}(1 ­ exp(0.61(r ®10.83))),

respectively, to provide profiles for the probability of

colonization with distance. Dotted vertical lines represent

95% confidence intervals about the estimated threshold

distance for invasion, rc.

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 equi-

valent to a ‘quenching’ term on the dynamics of

spread (Kleczkowski et al., 1996). 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 sapro-

trophic and parasitic fungi are briefly discussed.

System I: percolation with no decay of substrate

(agar sites)

Changes in the probability of colonization with dis-

tance. 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 Pc

¯ 2 sin (π}18) ¯ 0.35 for bond

percolation (where Pc

is the critical probability for

percolation; Stauffer & Ahorney, 1994). To estimate

the critical distance, rc, for P

c¯ 0.35, we constructed

probability profiles (Gilligan & Bailey, 1997)

describing the change in probability of colonization

with distance between a donor (colonized) and

a recipient (uncolonized) site. Small spots of agar

(20 mg, 3 mm diameter) of 1% (low nutrient) or 10%

(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

4, 6, 8, 10, 12, 14, 16, 18, 20, or 22 mm apart (from

centre to centre) in Petri plates (90-mm diameter).

There were 15 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. 1 mm in length, removed from the growing

edge of a 4-d-old colony of R. solani Ku$ hn (AG2–1)

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 23°C. Plates were assessed daily for 20 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 (3-mm

diameter) of either low or high nutrient agar (20

mg per site) were spotted onto a triangular lattice in

large Petri plates (140-mm diameter) at 6, 8, 10, 12,

14, 16, 18, 20 or 22 mm apart. The centre agar site

of each plate was inoculated with a single hyphal

strand (1 mm in length) removed from the growing

edge of a 4-d-old colony of R. solani grown on water

agar. Each experiment involved six replicates for all

distances except 6 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 23°C. Plates were

assessed daily for 21 d using a dissecting microscope

(magnification ¬40) and the number and locations

of colonized sites recorded.

Fungal growth on each replicate plate was scored

as invasive or finite after 21 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.

538 RESEARCH D. J. Bailey et al.

500400300200100

0

6 mm

250200150100500

8 mm

1401201008060

0

10 mm

4020

0

12 mm

10203040506070

0

14 mm

81012

642

0

16 mm45

23

1

0 5 10 15 20 25

Time (d)

14 mm

0 5 10 15 20 25

0123456

12 mm

0123456

10 mm

4

6

8

2

0

8 mm

05

101520253035

6 mm400

300

200

100

0

10% PDA 1% PDA

No

. of

site

s co

lon

ized

Fig. 3. 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 (10% potato dextrose agar (PDA)) and low nutrient (1% PDA) agar. Solid lines indicate replicates

exhibiting invasive spread (patches had reached the edge of the system after 21 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,

rc, probability profiles were constructed using seeds

that had been incubated for three different times (0,

7 and 21 d) on nonsterile sand. Poppy seeds (Papaver

somniferum L.) measuring approx. 1 mm in diameter

were initially sterilized by autoclaving for 1 h at

121 kPa. Large Petri plates (14-cm diameter) were

filled with 200 g of sand (Hepworth Minerals &

Chemicals Ltd, Redhill, UK; Grade 16}30, with

10% gravimetric water content). Donor seeds were

positioned on the sand surface at 0, 3, 6, 9, 12, 15, 18,

21, 25 and 27 mm from uncolonized seeds. Donor

seeds were inoculated with a single hyphal strand

of R. solani removed from the growing edge of a

4-d-old colony grown on water agar either immedi-

RESEARCH Saprotrophic invasion by Rhizoctonia solani 539

ately or after 7 or 21 d incubation of the seeds on

sand in the dark at 23°C. There were 15 replicates

per treatment and the plates were assessed daily for

colonization until no change in the number of seeds

colonized was detected.

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 (240¬360 ¬50 mm deep) were filled to a depth

of 4 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 12, 14, 16, 18, 20, 22, 24, 26, and 28 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 23°C. The bags were opened

daily and the poppy seeds were assessed for

colonization using a binocular microscope (¬40)

for 27 d.

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.

2). Threshold distances, rc, for bond percolation on

a lattice (corresponding to P ¯ Pc

¯ 0.35) were

obtained by inverse prediction from a logistic model,

P ¯ θ"}(1­exp (θ

#(r®θ

$))), where θ

iare parameters.

The model was fitted to the experimental data by

maximum likelihood under the assumption of bi-

nomial 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 rc¯ 8.1³0.49 mm for

low nutrient sites to rc

¯ 11.8³0.93 mm for sites

with a high nutrient status (Fig. 2).

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 3, 4). For

high nutrient sites, all replicates with inter-site

distances % 10 mm displayed invasive growth. At

12 mm, five out of six replicates produced finite

patches and at 14 and 16 mm all replicates were finite.

The threshold distance between sites for invasive

spread was estimated from the population data

(Fig. 3) at approx. 8.0 mm and 12 mm for low and

Invasive spread Finite spreadTime after

innoculation(d)

4

12

20

Fig. 4. Examples of replicate microcosms showing patches

with invasive and finite growth of Rhizoctonia solanibetween sites of low (1% potato dextrose agar) nutrient

agar. Note that gaps occur in invasive patches. Distance

between sites : 6 mm, invasive spread; 8 mm, finite spread.

high nutrient sites, respectively. These estimates

were consistent with the predicted thresholds of 8.1

and 11.8 mm obtained from the colonization profiles

for pairs of nutrient sites (Fig. 2). However, maps

of invasive spread (Fig. 4) 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 dis-

tance (Fig. 5a) and the corresponding probability

(derived from Fig. 2; Fig. 5b). 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. 5a), there was no difference in

the empirical estimate of the critical probability (Fig.

5b). The latter probability was estimated as

0.365³0.07 by inverse prediction after fitting a

logistic function to the data for the combined profiles

(Fig. 5b).

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. 6). Compared with the more artificial

540 RESEARCH D. J. Bailey et al.

1.0

0.8

0.6

0.4

0.2

0.0

0 2 4 6 8 10 12 14 16 18

Inter-site distance (r, mm)

0.0 0.2 0.4 0.6 0.8 1.0

Probability of colonization (P) betweendonor and recipient

1.0

0.8

0.6

0.4

0.2

0.0

(a) (b)

Pro

po

rtio

n o

f p

atch

es s

ho

win

gin

vasi

ve s

pre

ad (

Pi)

Fig. 5. Change in the proportion of patches of Rhizoctonia solani exhibiting invasive spread (Pi) 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 (1% potato dextrose agar (PDA), open circles) and high (10% PDA, crosses) nutrient

agar. The probability of colonization (P) was estimated from the relationship in Fig. 2, 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 Pi¯ 1}(1­exp(5.0(r® 11.18) and P

i¯ 1}(1­exp(5.0(r®7.5) for high and low nutrient agar,

respectively (Fig. 5a), and by Pi¯ 1}(1­exp(®25(P®0.45))) relating the proportion of invasive patches to the

probability of colonization between individual sites (Fig. 5b). 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.

1.0

0.8

0.6

0.4

0.2

0.0

0 5 10 15 20 25 30

Distance (r, mm)

Pro

bab

ility

of

colo

niz

atio

n (

P)

Fig. 6. Change in the probability of colonization, P, by Rhizoctonia solani with distance, r, between pairs of

poppy seeds 0 (circles), 7 (squares) or 21 (triangles) d old. Data are fitted with the logistic functions P ¯1}(1­exp(0±25(r®15.83))), P ¯ 1}(1­exp(0.71(r®7.66))) and P ¯ 1}(1­exp(0.93(r®3.62))) for 0, 7 and 21-

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. 6).

Threshold probabilities, rc, for invasion (P

c¯ 0.35,

for bond percolation on a triangular lattice) corre-

sponded with a maximum distance of 18 mm

between neighbouring seeds for young inoculum

but declined to 8.0 mm for seeds incubated on sand

for 7 d and to 4.0 mm for seed incubated for 21 d

(Fig. 6).

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 28 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

RESEARCH Saprotrophic invasion by Rhizoctonia solani 541

100

80

60

40

20

00 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30

30

25

20

15

10

5

70

60

50

40

30

20

10

0

12 mm 14 mm 16 mm

5

0

18 mm

10

15

20

2020 mm

20

15

10

5

0

22 mm10

8

6

4

2

00 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30

0

24 mm

1

2

3

4

5

6

7

0 5 10 15 20 25 300

26 mm

1

2

3

4

5

6

7

0 5 10 15 20 25 300

28 mm

1

2

3

4

5

0 5 10 15 20 25 30Time (d)

No

. of

site

s co

lon

ized

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.

70

60

50

40

30

20

10

0

10 12 14 16 18 20 22 24 26 28 30

Inter-seed distance (mm) Inter-seed distance (mm)

10 12 14 16 18 20 22 24 26 28 30

100

90

80

70

60

50

40

30

Furt

hes

t co

lon

y ex

ten

t (m

m)

No

. of

seed

co

lon

ized

(a) (b)

Fig. 8. Change in the average patch size after 28 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.

542 RESEARCH D. J. Bailey et al.

initial threshold distance of 18 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 immedi-

ately 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 ar-

bitrarily 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).

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. 2). It

was, instead, associated with a threshold distance,

different for low and high nutrient sites, but

corresponding to a common threshold probability,

Pc

¯ 0.35. 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 experi-

mental data yields an estimate for P ¯ 0.36, 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., 1999). 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, 1990; Bailey & Gilligan,

1997; Gilligan & Bailey, 1997), the fungus grows

saprotrophically from a source of previously

colonized host material towards a susceptible host.

For secondary infection (Kleczkowski et al., 1997),

mycelial growth between infected and susceptible

hosts might be directly fuelled by translocated

nutrients from parasitic activity, though sapro-

trophic 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 sapro-

trophs spread through a population of discrete

nutrient sites. The analogy holds for saprotrophy as

long as the availability of exogenous nutrient be-

tween 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, 1993) 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 trans-

mission between sites evolves during the course of

the epidemic. Grassberger (1983) 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.Theprob-

ability of transmission is given by 1 ® exp(® βT),

where β is the rate of transmission (related to the

RESEARCH Saprotrophic invasion by Rhizoctonia solani 543

threshold probability) between donor and recipient

and T is a fixed period between colonization and

exhaustion. Grassberger (1983) 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,

1976; Grenfell et al., 1995) and dynamically

generated variability (Kleczkowski et al., 1996). 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, multi-

plicative 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 be-

tween 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 18 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. 6). 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 cau-

tiously 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 ( Pc, benefits

from an extended period of invasive growth. For

replicates with large inter-seed distances, P ! Pcand

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., 1996; Gilligan

et al., 1997; Gibson et al., 1999).

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 rc

(corresponding to a probability of

transmission " Pc) 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,

1999).

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, 1956; Burdon

& Chilvers, 1982). 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, 1993; Bolton & Boddy,

1993; Rayner & Coates, 1987; Ritz, 1995). We have

subsumed the detail of colony behaviour and for-

aging behaviour into the colonization profile. Un-

doubtedly, the radial density of hyphal growth, the

aggregation of hyphae into strands, the degree of

branching and anastomosis all influence the prob-

ability 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

(1982) and Edelstein-Keshet & Ermentrout (1989),

whereas theories of invasion through populations at

the larger scale are being developed (Shigesada &

Kiyosawa, 1997). It remains to link the microscopic

scales to invasion of multiple colonies through field

populations.

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

544 RESEARCH D. J. Bailey et al.

Botanical Epidemiology and Modelling Group for helpful

discussion of the work.

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