Dataset for model testing
For model testing we used data from experiments on the development of resistance in powdery mildew (Blumeria graminis f. sp. hordei) on spring barley (Hordeum vulgare) in response to different treatments of the quinone outside inhibitor (QoI) fungicide azoxystrobin (Fraaije et al., 2006). To increase the likelihood of mildew infection the susceptible barley cv. Golden Promise was sown in replicated plots. Experiments were conducted in the UK at sites near ADAS Terrington, Norfolk (52º45′N), Edinburgh (55º57′N) and Inverness (57º30′N) in 2002 and repeated at sites near Terrington and Edinburgh in 2003. At these five site/year combinations, total doses of either 1, 2 or 3 L ha^{−1} of the commercial product Amistar (suspension concentrate containing 250 g azoxystrobin L^{−1}; Syngenta) were applied in one, two or three sprays (Table 1). For QoI fungicides, the substitution of glycine by alanine at codon 143 of the mitochondrial cytochrome b gene (G143A) correlates with resistance development in cereal mildews (Fraaije et al., 2002; Baumler et al., 2003). The frequency of the G143A mutation was determined before the application of the first spray and at the end of the growing season using quantitative allelespecific PCR assays (Fraaije et al., 2006). For each site/year, treatment and treatment replicate, we calculated the ‘selection ratio’, defined here according to the following equation:
 (1)
Table 1. Azoxystrobin treatments (Amistar; suspension concentrate of 250 g azoxystrobin L^{−1}) applied at each site/year experiment included in the dataset for model testing to determine the development of resistance to azoxystrobin in powdery mildew (Blumeria graminis f. sp. hordei) on spring barley (Hordeum vulgare). The threespray programme was not applied at Edinburgh in 2002 Treatment number  Number of sprays  Dose per spray (L ha^{−1})  Total applied dose (L ha^{−1}) 

1  0  0  0 
2  1  1  1 
3  1  2  2 
4  1  3  3 
5  2  0·5  1 
6  2  1  2 
7  2  1·5  3 
8  3  0·33  1 
9  3  0·66  2 
10  3  1  3 
The selection ratio was thus the factor by which the frequency of the resistant powdery mildew strain changed over one growing season.
Data on the threespray programme at Edinburgh 2002 were missing. Data on the threespray programme at Terrington 2002 were excluded from the model testing dataset because the selection ratio increased according to an exponential curve with the total applied dose. This was inconsistent with the data from other site/years and incompatible with the asymptotic shape of doseresponse curves of fungicides in general (Lockley & Clark, 2005; Oxley & Hunter, 2005).
When the effects of the different azoxystrobin treatments on the selection ratio were analysed statistically, the selection ratio was shown to increase significantly (P < 0·05) with increasing total applied dose at all site/ year combinations. Given a total applied dose, the selection ratio (P < 0·05) increased significantly with spray number at all site/year combinations, except Edinburgh 2002 (PHF, S. Powers, BF, JL, unpublished data).
From the five site/year combinations, data collected from Edinburgh 2003 were randomly selected to be used for parameter estimation, leaving four site/year combinations for model testing.
Model structure
An ordinary differential equation (ODE) model was constructed to describe the development of resistance against azoxystrobin in a powdery mildew population growing on the leaves of a spring barley crop in response to applications of fungicide during one growing season. We assumed that resistance develops in a similar way for all plants in the crop. Definitions and dimensions of state variables and parameters in the model are given in Tables 2 and 3, respectively.
Table 2. Definitions and dimensions of the state variables in the model of azoxystrobin fungicide resistance in Blumeria graminis f. sp. hordei derived in this paper State variable  Definition  Dimension 


A  Total leaf area^{a}  cm^{2} 
H  Healthy leaf area  
L_{s}  Leaf area occupied by latent lesions of the sensitive pathogen strain  cm^{2} 
I_{s}  Leaf area occupied by infectious lesions of the sensitive pathogen strain  cm^{2} 
L_{r}  Leaf area occupied by latent lesions of the resistant pathogen strain  cm^{2} 
I_{r}  Leaf area occupied by infectious lesions of the resistant pathogen strain  cm^{2} 
F  Area of lower leaves^{b} occupied by infectious lesions of both the sensitive and resistant pathogen strain  cm^{2} 
C  Azoxystrobin concentration  L ha^{−1} 
Table 3. Definitions, values and dimensions of the parameters in the fungicide resistance model derived in this paper Parameter  Definition  Value  Dimension  Reference^{e} 


Host – Spring barley 
γ  Growth rate of leaf area  1·22E−02  t^{−1a}  1, 2 
A_{max}  Maximum leaf area  88·8  cm^{2}  2 
σ  Senescence rate  Eqn 14 in text  t^{−1}  1, 2 
Disease – Powdery mildew 
F_{0}  Initial area of infectious lesions on lower leaves^{b}  0·82  cm^{2}  3 
λ  Rate of decrease of area of infectious lesions on lower leaves^{b}  5·5E−03  t^{−1}  2 
θ  Initial frequency of resistant mildew strain  Variable, see text  ^{c}  3 
ρ  Transmission rate^{d}  1·17E−02  t^{−1}  3 
1/δ  Length of latent stage  99  t  4 
1/μ  Length of infectious stage  262  t  5 
Fungicide – Azoxystrobin 
v  Decay rate of azoxystrobin  Variable, see text  t^{−1}  3 
α  Proportional reduction of infection efficiency by azoxystrobin  Eqn 12 in text  –  – 
α_{max}  Maximum proportional reduction of infection efficiency by azoxystrobin  1  –  6 
β  Shape parameter of doseresponse curve  9·5  –  3 
The model describes the seasonal development of the spring barley canopy in order to account for the effect of the availability of susceptible host tissue on the growth of the powdery mildew population. The model describes the development of the combined area of leaves one to four during a growing season, because we assume that most of the sprayed fungicides will be intercepted by leaves one to four (leaf one being the flag leaf) and because pathogen samples for mutation analysis were taken from upper leaves. The total area of leaves one to four, hereafter called total leaf area (A), is assumed to increase according to the monomolecular equation (Thornley & Johnson, 1990) and reaches its maximum value (A_{max}) at growth stage (GS) 39 on Zadoks’ scale (Zadoks et al., 1974):
 (2)
We used the monomolecular equation since it predicts an approximately constant growth of the total leaf area during the emergence of leaves two to four on a time scale in degreedays. This is in agreement with the approximately constant length of phyllochrons of leaves two to four of spring barley in degreedays (Anonymous, 2006) and the approximately similar size of these leaves (NP, unpublished data). The growth of the total leaf area is not affected by disease and consists of the sum of healthy, dead and infected leaf tissue.
The development of the healthy leaf area (H) consists of a growth stage, a plateau with no leaf growth or senescence followed by a senescence stage. The end of the growth, plateau and senescence stages correspond to GS 39, 61 and 87 (Zadoks et al., 1974), respectively. In the absence of disease, the equation for healthy leaf area is:
 (3)
In this equation, parameter σ represents the senescence rate.
The powdery mildew population on leaves one to four of spring barley consists of strains that are sensitive to azoxystrobin and those that are assumed to be completely resistant to this fungicide (with the range of doses applied) as a result of the presence of the G143A mutation. The life cycle of each powdery mildew strain is divided into a latent (L) and subsequently an infectious stage (I). Leaf tissue occupied by latent lesions of powdery mildew stays green and is still capable of photosynthesis. The length of the latent period is 1/δ. The length of the infectious period is 1/μ. Because powdery mildew is a biotroph, leaf senescence also kills the pathogen and thus senescence decreases the density of both latent and infectious lesions (Carver & Griffiths, 1981; Asher & Thomas, 1984). Subscripts s and r are used to distinguish between lesions of the sensitive and resistant powdery mildew strains.
At the start of the growing season, healthy leaf area becomes infected with powdery mildew as a result of deposition of spores produced by infectious lesions on lower leaves. The size (F) and therefore spore production rate of these lesions is assumed to decline according to an exponential function:
 (4)
In this equation, λ represents the loss rate of infectious lesions on lower leaves as a result of both senescence and reaching the end of the infectious period. For the reasons outlined herein, we also modelled the size of infectious lesions on lower leaves for site/year Edinburgh 2002 according to the function:
 (5)
We assume that a fraction θ of the infectious lesions at lower leaves (F) consists of the resistant powdery mildew strain. Parameter θ is kept constant during the growing season, because higher leaves intercept most of the sprayed azoxystrobin.
The rate at which an infectious lesion generates new infections, the transmission rate, is determined by the product of (i) the sporulation rate of infectious lesions, (ii) the probability that spores land on leaves one to four, (iii) the probability that a spore lands on healthy leaf tissue, given that it lands on these leaves and (iv) the infection efficiency of spores. Points (i), (ii) and (iv) are combined in the compound parameter ρ. We account for point (iii) by multiplying parameter ρ with the fraction of the total area of leaves that consists of healthy leaf tissue, ^{H}/_{A}. This makes the growth of the sensitive and resistant powdery mildew strains dependent on the availability of healthy host tissue.
The decay of the azoxystrobin concentration is modelled as
 (11)
with decay rate v. Azoxystrobin reduces the infection efficiency of sensitive powdery mildew strains (Bartlett et al., 2002). We model the dependence of the infection efficiency on the azoxystrobin concentration (C) by multiplying this parameter with a factor (1 −α(C)), in which α(C) is the fraction by which the infection efficiency is reduced at fungicide dose C. This fraction depends on the azoxystrobin concentration according to the asymptotic function:
 (12)
In this equation, parameter α_{max} is the maximum reduction of infection efficiency and β determines the curvature of the doseresponse curve.
All state variables are expressed in cm^{2} leaf area per degreeday. A degreeday scale was used to incorporate temperature effects on the development of spring barley and powdery mildew. The number of degreedays (t_{stage}) that it takes to complete a developmental stage was calculated as
 (13)
in which l is the length of a developmental stage in days, T_{av} is the average temperature and T_{threshold} is the minimum temperature required for development (taken as approximating to 0°C for all developmental processes). Hereafter, t represents the number of degreedays accumulated since the start of a model simulation.
Comparison of the predicted selection ratio and observed selection ratio
To quantify how well the model described the observed data for each site/year combination, the percentage of variance in the observed data that was accounted for by the model predictions was calculated as:
 (17)
To quantify how well the model described the data over all site/year combinations, we pooled the predicted and mean observed selection ratios for all treatments and site/year combinations. When the predicted selection ratios exactly match the observed ones, the relationship between these two variables is described by the 1:1 line. We calculated the percentage of the variance in the observed selection ratios that was explained by this 1:1 line as a quantitative measure of the predictive power of the model.