Small propagules like pollen or fungal spores may be dispersed by the wind over distances of hundreds or thousands of kilometres, even though the median dispersal may be only a few metres. Such long-distance dispersal is a stochastic event which may be exceptionally important in shaping a population. It has been found repeatedly in field studies that subpopulations of wind-dispersed fungal pathogens virulent on cultivars with newly introduced, effective resistance genes are dominated by one or very few genotypes. The role of propagule dispersal distributions with distinct behaviour at long distances in generating this characteristic population structure was studied by computer simulation of dispersal of clonal organisms in a heterogeneous environment with fields of unselective and selective hosts. Power-law distributions generated founder events in which new, virulent genotypes rapidly colonized fields of resistant crop varieties and subsequently dominated the pathogen population on both selective and unselective varieties, in agreement with data on rust and powdery mildew fungi. An exponential dispersal function, with extremely rare dispersal over long distances, resulted in slower colonization of resistant varieties by virulent pathogens or even no colonization if the distance between susceptible source and resistant target fields was sufficiently large. The founder events resulting from long-distance dispersal were highly stochastic and exact quantitative prediction of genotype frequencies will therefore always be difficult.
Population genetic processes are usually dominated by short-distance dispersal because rates of dispersal generally decline steeply with distance. Individual long-distance dispersal (LDD) events are therefore rare and sporadic, and often remain unobserved, but can be exceptionally important in shaping a population. In natural populations, LDD plays a significant role in the colonization of formerly uninhabitated niches like islands (Cain et al., 2000) and in recolonization of niches after drastic environmental change, such as post-glacial range expansion of plants (Cain et al., 1998; Clark, 1998). Long-distance dispersal of small propagules by the wind is a general mechanism by which organisms may be widely dispersed; aerosolized desert dusts, for example, may transport microbiota across continents (Kellogg & Griffin, 2006). In plant pathology, LDD by wind is especially relevant to fungal spores, such as those of powdery mildew and rust pathogens. Small, wind-dispersed conidia or urediospores can be displaced from their origin over hundreds to thousands of kilometres (Brown & Hovmøller, 2002; Hovmøller et al., 2002, 2011; Viljanen-Rollinson et al., 2007) and thus become a major threat to crop production (Anderson et al., 2004).
The potential severity of this threat is exacerbated by the low degree of biological variation in modern agriculture, which increases the risk of epidemics and major crop losses. This is caused by the limited diversity in crop species compared to most wild plants, particularly the small number of important varieties of those crops. Within this scenario, LDD increases the risk that a disease outbreak might achieve a worldwide distribution (Brown & Hovmøller, 2002; Kellogg & Griffin, 2006), as occurred recently with the outbreak of genotypes of Puccinia striiformis f.sp. striiformis (yellow or stripe rust of wheat) adapted to high temperatures (Hovmøller et al., 2008, 2010, 2011) and the spread of the Ug99 group of pathotypes of Puccinia graminis f.sp. tritici (stem or black rust of wheat) across eastern Africa and western Asia (Singh et al., 2008, 2011).
Growing resistant varieties is generally an effective and economic method of disease control (Chen, 2005). In response to the threat posed by pathogens, plant breeders have often introduced single resistance genes into cultivars with useful agronomic traits, whether deliberately or unintentionally. As a long-term strategy, this has not been successful because new genotypes of the pathogen have emerged (Brown et al., 1991; Brown, 1994). In a gene-for-gene interaction, a single mutation may be sufficient for a pathogen to become virulent on varieties with the corresponding resistance gene.
Despite the large numerical size of the populations of newly emerged virulent genotypes of rust and mildew fungi, their diversity has often been very low. Repeatedly, only one or a few virulent genotypes have become especially abundant, forming a considerable fraction of populations sampled at widely separated locations (Brown, 1994; Hovmøller et al., 2002). In 1988, for example, only about 5% of the spring barley acreage in the UK was planted with new varieties carrying a recently introduced resistance gene, Mla13, against barley powdery mildew (Blumeria graminis f.sp. hordei). The resistance of Mla13 became ineffective that summer, owing to the appearance of two B. graminis clones lacking avirulence to Mla13, which dominated populations of the pathogen sampled from Mla13 fields and trial sites throughout the British Isles (Brown et al., 1991). The ability of a few genotypes to dominate a huge agricultural area was confirmed by the even more striking example of P. striiformis f.sp. tritici, in which two genotypes, virulent on recently introduced wheat varieties with the Yr17 resistance gene, were found in four European countries, up to 1700 km apart (Hovmøller et al., 2002). An explanation for such findings begins by noting that the first genotypes virulent to a new resistance gene and arriving in fields planted with crops containing the gene have a very strong selective advantage over all other genotypes in those fields. These genotypes can therefore produce many spores, which, as a result of the LDD characteristics of transport by wind, can be dispersed to other fields in which they have a similar selective advantage, whence they will spread to fields without the resistance gene. In this way, the first colonizing genotypes may dominate the total pathogen population until mutation or recombination restores diversity (Brown et al., 1993).
It is unethical, and perhaps even impossible, to conduct experiments on the ‘breakdown’ of an effective, useful resistance gene through the evolution of virulent genotypes of a parasite capable of escape and widespread dispersal. Here, the results of computer simulations designed to test these theoretical predictions about the impact of LDD on adaptation to host resistance are reported. In these models, the movement of defined clonal lineages in a heterogeneous landscape represents agricultural fields and the dispersal of fungal spores between them.
Description of model
This paper focuses on one step in an evolutionary process, in which one or very few genotypes of a plant pathogen become very abundant over a large agricultural area. An individual-based, spatially explicit model of haploid, asexual individuals was used for this purpose, developing a model published previously (Wingen et al., 2007). Simulated individuals represented plant pathogens on crop plants, the latter represented as homogeneous habitats of specific types: permissive, selective or non-permissive. Individuals were allowed to multiply on only a few patches, on habitat of either the permissive or selective type. These patches stand for agricultural fields of a suitable crop surrounded by non-permissive habitat representing fields of an unsuitable crop or other unsuitable habitat. This has been described as resembling oases in a desert, with separate pockets of favourable habitat surrounded by expanses of hostile environment (Brown et al., 2002).
Rules for individuals
The model operated in discrete time steps. Individuals gave birth to offspring once, after a short latent period of one time step, and then died. The duration of one generation was thus one time step. Offspring were immediately dispersed according to a chosen dispersal function (see below). The mean number of offspring was 1·5 per birth process following a Poisson distribution. Individuals had a genotype of 32 biallelic loci, coded as ‘0’ or ‘1’. The loci could represent any genes but the first locus is here considered to be an avirulence (effector) gene, which has an incompatible gene-for-gene interaction with the corresponding plant resistance. The other loci could be other avirulences, molecular markers or any other biallelic trait. Simulations were initiated with one individual of a random genotype, but with the first locus being in state ‘0’, indicating inability to grow in the selective habitat, specifically avirulence on a plant genotype with a new resistance gene. The genotype of a new individual was either the same as that of its parent or was mutated by conversion of one random bit of the 32 biallelic loci. Mutation took place at random with a set frequency. Offspring were not placed closer to other offspring than the minimal interaction distance of one unit. One unit could, for example, be equivalent to about 10 cm for B. graminis (O’Hara & Brown, 1997). Furthermore, offspring did not survive outside the habitat patches; if the simulation generated such an event, the offspring was lost.
Dispersal of the spores was modelled as described previously (Wingen et al., 2007), by an inverse power-law probability density function with the spore concentration t(r|θ) at distance r from the source along a given bearing θ given by
The parameters b =1·5 and b =2·0 were used (PLr−1·5 or PLr−2·0 dispersal functions) because theoretical and experimental results suggest that this range of b covers relevant values for wind dispersal of small particles like fungal spores and pollen (see references in Wingen et al., 2007).
For comparison, simulations were also done with the commonly used dispersal model of an exponential decline of spore concentration with a characteristic scale of k:
The median dispersal distance was set to 30 units for the exponentially bounded function and the power-law functions were adjusted to have approximately the same median value. This equals a median distance of about 3 m in the real life examples of B. graminis and P. striiformis urediospores (see references in Wingen et al., 2007).
A simple spatial layout with two patches was used for the main model to investigate the processes that shape pathogen population structure within a simple geographic situation. The two patches were filled homogeneously with plants, representing fields of crops that were natural hosts for the pathogen. The patch size was 256 × 256 square units, so that one patch would be a small field in real life. Results from the simulations are scalable and using small patches was a compromise between computational feasibility and a realistic scenario.
At the start of the simulation, only one patch, the source patch, was present and was filled with a permissive cultivar susceptible to all pathogen genotypes. The pathogen population was initiated with one individual in the source patch, which was avirulent on the cultivar in the target patch and therefore unable to reproduce there. It was allowed to reproduce and spread within the source patch for 500 generations, during which a dense population of about 18 000 individuals developed. This source patch population was not in equilibrium, just as most natural populations are not, but some genetic diversity was present (see Results). After this burn-in period, a second patch, the target, was created. The habitat of the target patch was selective, representing the introduction of a cultivar carrying an effective resistance gene. Colonization of the target patch was only possible for individuals with a state of ‘1’ at the first genotype locus. Individual pathogens born to a mother with a state of ‘0’ at the first locus could mutate to ‘1’ by chance at birth, representing the mutation of an avirulence allele to a virulence allele, typical of a gene-for-gene relationship. If colonization of the target patch took place within 6000 generations, simulations were run for a further 6000 generations after the first colonization event; if not, simulations were discarded.
The two-patch simulations were conducted with three separation distances between patches (d =100, 300 or 1000 units) and two mutation rates (μ= 0·001 or 0·0001 mutations per genome per generation), for each dispersal function. All two-patch simulations were repeated until 50 successful runs with colonization were produced, with the exception of exponential dispersal and d =1000 units, where no colonization of the second patch took place in 100 replicates.
To test if the results from the simple two-patch layout were valid in a more complex spatial setting, simulations with a five- by five-patch layout were also conducted. Patches were arranged in a regular lattice of equidistant patches. Most patches consisted of the permissive cultivar. In alternative designs, either only one patch in the centre of the layout or two patches, immediately northeast and southwest of the central patch, had a selective cultivar. This equals a percentage of 4% or 8%, respectively, of area sown with a selective cultivar, typical of the percentage area of cereal cultivars with a new resistance gene in the early years of its use (Brown et al., 1991).
For both kinds of five-by-five layout, five replicate simulations were conducted with PLr−1·5 dispersal and d =100 and 1000 units, and with exponential dispersal using d =100 units. The mutation rate used in these simulations was μ= 0·001.
Characterization of populations
Statistical characterization of populations was done for two-patch simulations only. Simulations in which no colonization took place were excluded from statistical analysis, including all those with exponential dispersal and d =1000 units. Standard population genetic statistics were calculated using all individuals present and regarding each patch as a subpopulation. Mean local genetic diversity within subpopulations, HS= 1−IS, was calculated from the weighted mean of gene identities, , where wi was the relative size of the ith subpopulation (Chakraborty, 1974), m was the number of subpopulations (i.e. two in this case), pij was the frequency of the jth genotype in the ith subpopulation and n was the total number of genotypes present. Total genetic diversity was calculated as where pj was the frequency of the jth genotype in the entire population, including all patches. Genetic differentiation between subpopulations was calculated as GST = HT − HS/HT. The frequency of the first genotype colonizing the selective patch was also recorded. If colonization was started by more than one individual of different genotypes simultaneously, one genotype was picked at random as the first colonizing genotype when this was required for the purposes of statistical analysis.
Calculation of migration rates
The dispersal functions compared here specify the frequency of point-to-point movements. To convert these to movements between large square patches, immigrant arrival rates (MP) per source individual were calculated numerically by integrating migration from a square patch of size w to another square patch of the same size at distance d away, with dispersal governed by function f:
where x and y were the two-dimensional spatial coordinates in the first patch and u and z the coordinates in the second patch. One of the three dispersal functions t(r|θ) (Eqns 1 and 2) was used as function f with a median dispersal rate of 30 units, as described above. The actual movement between patches in the simulation was then determined by generating a Poisson random variate with the mean set by MP.
To compare results of simulations with analytical expectations, the expected logarithm of waiting time to migration from the source patch to the target was calculated using the integral in Eqn 3 divided by the area of the source patch as a measure of the rate of migration per virulent individual. This rate was used to calculate expected waiting times with the average numbers of virulent individuals when the new patch was made available. Waiting times were calculated from migration rates using a geometric series. The mean of the logarithm of waiting time was calculated analytically from the migration rate and transformed back to a natural scale.
Software and hardware
Simulation software was written in Kylix (Delphi for Linux, v. 14.5, Open Edition) using the Free Pascal Compiler (v. 1.9.8). Simulations were run on a GNU/Linux platform with 3·6 GHz dual core processors. Statistical analysis of populations and graphical output were done using the free R software suite (v. 2.4.0) (R Development Core Team, 2004). Numerical integration was done with mathematica v. 4.0 (Wolfram Research).
Colonization start times and fill-up times
Two things must happen for colonization of the target patch to begin: first the source patch must produce individuals of the right genotype, then one of these individuals must be dispersed from the source patch over the intervening space to the target patch. Colonization was initiated by the arrival of the first virulent individual that reproduced successfully on the selective host genotype. The simulated pathogen genotype had ‘1’ as the value of the first of 32 bits. The mean waiting times until colonization started differed greatly between simulations with different parameter values, from 1·9 to over 2000 time steps (Fig. 1; Table 1). The distribution of colonization start times (Fig. 1) reflects stochastic variation both in the composition of the source, non-selective population and in spore dispersal events. In none of the runs with the least LDD (the exponential distribution) and the widest spacing of patches (d =1000) was the target patch colonized after 6000 generations of the simulated population (Fig. 1). In approximately 70% of runs with exponential dispersal, d =300 and μ= 10−4, the target population was not colonized, because the frequency of virulent individuals that moved from the source patch to the target patch was very low.
Table 1. Geometric mean and variance of start times of pathogen colonization of the target patch for different dispersal functions, patch separation distances (d units) and mutation rates (μ) per locus per genome per generation
PLr−1·5 and PLr−2·0: power-law dispersal with exponents −1·5 and −2·0, respectively; exp.: exponential dispersal; NA: information not available because colonization did not take place.
Geometric means were calculated from 50 replicates; theoretical means were calculated as described in the text.
5·0 × 1013
Variance (using geometric mean)
1·2 × 104
3·4 × 105
2·3 × 106
2·7 × 104
4·2 × 104
3·4 × 106
3·0 × 105
8·4 × 105
1·0 × 107
2·8 × 107
1·7 × 104
6·7 × 104
Before the start of colonization, the frequency of virulent individuals in the source patch was similar at the same point in time in different simulations with the same mutation rate, regardless of the patch spacing and dispersal kernel. When colonization began, 1–4% of individuals had a virulent genotype with μ= 10−4 and 0·2–2% with μ= 10−3. The lower values within those ranges applied to simulations in which colonization was quick, because longer waiting times provided more opportunity for mutation. Between 4% and 9% of pathogen genotypes at the start of colonization were virulent on the selective cultivar.
The probability of a virulent genotype being dispersed to the target patch depended on the dispersal function used and the distance (d) separating the patches. As expected, in general, colonization started later if d was larger, given the same dispersal function. It also began later with a less fat-tailed dispersal function. Successful colonization was initiated later with exponential dispersal than with the power-law functions because LDD events were rarer. With longer d, the PLr−1·5 function generally initiated colonization quicker (Fig. 1), but start times were similar for both power-law dispersal functions for short d.
The ratio between the migration rate, measured by the time until the start of colonization, and the time the colonizing population took to fill the newly colonized patch, was an important parameter in the population process as it determined the amount of stochastic variation in the colonization process, particularly in the frequency of the first pathogen genotype to colonize the target patch (Fig. 2). The mean time to fill the target patch was similar for all simulations, as it mainly depended on the birth rate and patch size, which were the same for all simulations. There was minor variation as exponential dispersal resulted in a somewhat denser population than PLr−1·5 dispersal (average population size 19 700 and 15 200 in a patch, respectively; note later time points in Figure 3). It took about 20 time steps to fill a patch to 80–90% or 30 time steps to fill it completely. This agrees with the expectation of 24 time steps, calculated as the time it takes an exponentially growing population with a multiplication rate of 1·5 per time step to reach a population size of 18 000.
The colonization start time was much shorter than the fill-up time only for the two power-law functions with the higher mutation rate (μ= 0·001) and shortest patch separation (d =100), as follows. With high μ, colonization of the target patch was often initiated by more than one individual (mean 1·6 individuals). For the same d and μ, the mean start time of colonization with exponential dispersal was after 2·8 time steps, with populations in target patches initiated by 1·1 individuals on average. For larger d, the colonization times for the two power-law functions were mostly still shorter than the fill-up time. At the lower mutation rate (μ= 0·0001), the time taken to fill up the target patch was shorter than the time that elapsed before colonization was initiated. The large amount of variation in the frequency of the first colonizing genotype generated with PLr−1·5 dispersal, μ= 0·001 and d =100 is evident in the top left panel of Figure 2.
As a check on the simulations, expected log waiting times were calculated analytically. The observed means were mostly a little longer than the theoretical values but broadly consistent with them (Table 1). With low μ, there were some large discrepancies, especially at d =100 units. However, the variance in estimated waiting time was approximately of the same magnitude as the mean, so these discrepancies probably reflect sampling variation in the simulations.
Spatial population structure
With all combinations of parameters, newly colonized target patch subpopulations had low diversity shortly after the start of colonization (Figs 3 and 4; time steps 15 and 50, first colonizing genotype plotted in red). Patches were colonized by one or only a few genotypes and were quickly filled by the descendants of the founder individuals. Later, the predominant genotype of the target patch also became abundant in the source patch owing to back-colonization. This was more pronounced for shorter d and for power-law dispersal (Figs 3 and 4; time step 500), so homogenization was faster under these conditions. Genotypic diversity in both patches then increased through mutation. At μ= 0·001 the initial colonizing genotypes were frequently completely lost from the population (e.g. Fig. 3, generation 6000, where the ‘red’ genotype has mainly disappeared), but the mutation conferring the ability to live on the resistant host was of course retained. With lower μ, however, the first colonizing genotype remained predominant until the end of the simulation (Fig. 4, generation 6000, where the ‘red’ genotypes are still abundant).
Mean genotype numbers in the source patches at time of colonization were 58 or 13 on average with μ= 0·001 or μ= 0·0001, respectively. Owing to colonization of the target patch, the total population size doubled and genotype number increased to up to 250 with high μ and 28 with low μ by 6000 generations after colonization.
Genetic differentiation (GST)
GST followed a similar course in all populations. It rose sharply within the first 30 generations after colonization of the target patch from near zero to a maximum, then decreased more slowly back to near zero (Fig. 5). The rise of GST was related to population growth, being largest when the target patch was newly filled with one or a very few genotypes and the two subpopulations were thus differentiated. The later decrease of GST occurred when population numbers were stable and genetic differentiation was lost as a result of further migration between the two patches. There was an inverse correlation between the duration of decrease of GST and the migration rate because GST decreased fastest for PLr−1·5 dispersal, which produced more migrants to long distances than the other dispersal functions. Mutation influenced GST too, as lower μ resulted in stronger genetic differentiation of patches and a slower decrease in GST as that differentiation was eroded more slowly.
Total genetic diversity (HT)
HT initially increased in all simulations as the target patch filled up with genotypes that were generally rare in the source patch. It then decreased in PLr−1·5 dispersed populations with low d (100 and 300 units) because LDD enabled the genotype that had colonized the target patch to back-migrate to the source patch. As it then formed a large proportion of the population there, total genetic diversity across the population decreased [Fig. 5, log10(time) = 1·5 and later]. With other parameter combinations, HT did not fall, but the rate of diversification slowed following colonization of the target patch and rapid population growth there. HT increased in all populations at later times as a result of mutation.
Local genetic diversity (HS1 and HS2) and genotype frequencies
The population process can be understood further by study of genetic diversity in the separate subpopulations (Fig. 6). With power-law dispersal and low d, diversity in the source patch, HS1, decreased after the target patch was filled up, owing to back-colonization from the uniform population in the target patch. This effect was also observed to a less marked extent with exponential dispersal and d =100. By contrast, diversity increased in the target patch (HS2) in all cases because of further mutation following the initial colonization (Fig. 6, right-hand columns). HS2 varied considerably in simulations with power-law dispersal and d =100 units (results for PLr−1·5 shown), reflecting stochastic variation in the number of simultaneous colonizing individuals of different genotypes. It varied less, however, in other simulations in which descendants of the first colonizing individual formed most of the population. There was a high variance of the frequency of the first colonizing genotype for power-law dispersal functions and d =100 units (Fig. 2, top left, PLr−1·5 dispersal). For most other parameter values, the variance was much lower for the same reason as before. In most cases, the frequency of the first colonizing genotype increased from near zero to about 40% as the target patch filled up. Diversity was especially low with PLr−1·5 dispersal and d =100 units, when this frequency increased further, up to 80% with high mutation and 96% with low mutation.
More complex spatial scenario
The simple two-patch layout investigated so far is not very realistic. Additional simulations were conducted with a more complicated spatial layout of five-by-five equidistant patches including one or two selective patches. This represents a very simplified model of an agricultural environment in industrialized countries, where 4–8% coverage of the acreage might be typical of a newly introduced crop variety with a resistance gene that is still effective. The first genotype that colonized the selective patches then quickly began to dominate neighbouring patches with PLr−1·5 dispersal and eventually dominated the whole area (Fig. 7). An exponential dispersal function, representing short-range dispersal only, did not have such an extreme effect. Exponential dispersal was not in fact strong enough to counteract local diversification driven by high mutation and therefore resulted in a patchy population structure (Fig. 7). The PLr−1·5 dispersal is very fat-tailed, so the first colonizing genotype reached all patches even with d =1000 units (data not shown). Its maximum frequency remained lower with higher values of d, however. These conclusions are essentially similar to those from the two-patch model analysed in the remainder of this paper, implying that the simple model captures important features of more complex scenarios.
Spatial heterogeneity has a major influence on pathogen adaptation to host genotypes (Laine et al., 2011). Here, the adaptation of wind-dispersed fungal plant pathogens to crops was modelled by computer simulations of haploid individuals dispersing in a heterogeneous arena. The aim was to understand processes in a pathogen population that cause a resistance gene to become ineffective because of the emergence of virulent pathogen genotypes. Fields of resistant crops, modelled as patches selective for specific pathogen genotypes, can be regarded as empty niches, in which the genotype that first colonizes them has a strong selective advantage (Brown et al., 1991; Brown, 1994; Hovmøller et al., 2002). This confers an overall selective advantage because the mutant genotypes have a broadened niche and hence an increased maximum population size.
The model supported observations from population genetic studies of two pathogens, the wind-dispersed fungi that cause yellow rust of wheat or powdery mildew of barley (Brown & Hovmøller, 2002). In both species, one or very few genotypes with novel virulences have been observed to reach very high proportions of the total. Moreover, because the emergence of those particular genotypes owed a great deal to chance, their alleles at loci other than that of the new virulence were not necessarily typical of the population as a whole (Brown & Wolfe, 1990; Hovmøller et al., 2002). The simulations reported here show that these features of a founder event in a pathogen population are most likely to occur if there is LDD with a fat-tailed dispersal function and are much less likely if dispersal follows an exponentially bounded function with a similar median value. The highest frequencies of the first colonizing genotypes were found in simulations with PLr−1·5 or PLr−2·0 dispersal functions (Fig. 2), which are appropriate for modelling LDD of small airborne propagules (Shaw et al., 2006), and these genotypes were most widespread over the arena (Figs 3, 4 and 7). Moreover, the frequencies themselves were highly variable as the process was strongly stochastic and depended on the number of different genotypes involved in the colonization of the empty niche. The magnitude of stochastic variation depended largely on the ratio between the time taken to fill up the selective niche and the average time taken for a new virulent genotype to reach this niche. It was greatest for PLr−1·5 dispersal with d =100, because this had the shortest mean time to the start of colonization (Fig. 2). The results reported here provide quantitative and qualitative insights into the process by which a very small number of virulent genotypes of a pathogen with a fat-tailed dispersal mechanism can initiate rapid population change – and potentially vast epidemics – on a host population into which a new resistance gene has been introduced (Brown & Hovmøller, 2002).
This also applied in a more complex and realistic simulation. A few genotypes could still become dominant even if the selective patch only occupied 4–8% suitable habitat (Fig. 7). Here, LDD was of even greater importance than in the two-patch case because the fields of susceptible varieties were scattered across an area of otherwise hostile habitat, in the manner of ‘oases in a desert’ (Brown et al., 2002).
The simulations presented here show that the founder event process (Brown & Wolfe, 1990; Brown, 1994; Brown & Hovmøller, 2002) is most likely to occur when: (i) the distance between fields is sufficiently great that dispersal between them is rare, (ii) the dispersal function is fat-tailed, so there is a finite probability of dispersal over long distances, (iii) the rate of colonization of a formerly uninfected field is rapid and (iv) mutation rates are low (Figs 3, 4 and 7).
The main parameters of the patch colonization process are: (i) the immigrant arrival rate (MP) at the target patch, which determines the waiting time between migrants, (ii) the mutation rate μ, which determines the waiting time λ = 1/(N1μ) until a new mutant is produced in the source patch with population size N1 and (iii) the population growth rate R that determines the fill-up time tF = log(N2)/log(R) of the empty patch (capacity N2). The relationships among these timescales determine the outcome of colonization.
The waiting time until the start of colonization Wc, which is closely related to Wm, depends on d, the patch separation distance, and the dispersal function (Table 1). On average, it is only a little lower for power-law dispersal than for exponential dispersal when d =100 units. Although Wc increased for all dispersal functions as d increased, it did so much more in the exponential case, to the point that no colonization was observed when d =1000. With lower μ, trends were similar but the number of migrants was much lower and waiting times longer. The waiting time between successive mutations in the source patch (λ) is complicated to calculate because the population is expanding exponentially, so N1 increases and space becomes occupied at the same time as mutants appear. However, λ is initially larger than tF until 40 or 400 individuals are present in the source patch (depending on μ), but then reduces and becomes, on average, of similar magnitude to tF.
The process by which the metapopulation was dominated by very few genotypes was thus driven by the balance between Wm and tF. For Wm<< tF many genotypes contributed to colonization of the target patch, but if the opposite was true (tF<< Wm) the patch was colonized by only one genotype. As patch size was constant and tF was thus almost constant, the ratio Wm/tF depended largely on Wm, which in turn depended on the dispersal function and d. The shortest values of Wm resulted in the contribution of a few individuals to the colonization of the target patch, as observed for d =100 units with power-law dispersal in more than half of the cases, resulting in large variation in the frequency of the first colonizing genotype (Fig. 2). This caused genetic diversity of the target to be initially low but variable (Fig. 6) owing to variation in the number of colonizing individuals. In other cases with higher Wm, there was only one colonizing individual.
These considerations support the hypothesis that the founder event process in populations of rusts, mildews and other aerially dispersed plant parasites results from rare LDD. If fields are far enough apart that dispersal from source fields to target fields of resistant varieties is rare, the new subpopulation of virulent parasites is predicted to have very little genetic diversity, as observed in agricultural populations (Brown & Wolfe, 1990; Hovmøller et al., 2002). Such rare, significant dispersal events are more likely with a fat-tailed dispersal function than with exponential dispersal, which has a similar median, a much thinner tail and an infinitesimal probability of very long-range dispersal.
Of further significance for population structure is unequal migration. As the number of migrants from the target patch into the source patch was initially higher than those moving in the reverse direction, owing to the selective nature of the target patch, the predominant clone or few clones in the target patch could back-colonize the source patch and thus dominate the whole population. In later time steps, the majority of individuals in the total population were virulent (i.e. able to grow in the selective, target patch), so the difference in migration rates was then of no further practical importance.
Sexual recombination in many fungi, including biotrophic parasites, occurs at the end of the host plant’s growing season, whereas asexual reproduction for the majority of the season allows the population to be dominated by a few genotypes. In the models presented here, a single allelic change confers virulence on the pathogen. As sex reassorts existing alleles, its main effect on this model would be to accelerate the increase of genotypic diversity in both source and target patches (Fig. 6). Sex is not predicted to alter the conclusion that LDD can lead to domination of the population in the target patch by a single pathogen genotype during the first season after colonization and possibly a little longer. In agreement with this, the genetic diversity in fungal pathogen populations with a sexual cycle increases in the years following the ‘founder event’, in which the first mutation to virulence to a new resistance becomes established (Brown et al., 1993; Brown, 1994, 1995).
Moreover, the model presented here does not consider changes in the plant variety in each field and could therefore be viewed as specifically relevant more to perennial than annual crops. To obtain greater quantitative precision of predictions about the evolution of virulence in pathogen populations, more realistic assumptions about pathogen life cycles and cropping systems would need to be incorporated in the model.
This research was supported by the Biotechnology and Biological Sciences Research Council and was part of the BioExploit project in the European Union Framework 6 Programme.