competition version of the model, this combination yields the minimum host fitness. In this situation, the host bears the cost of resistance but the pathogen suffers no cost of virulence because there is no competition from the avirulent pathogen phenotype.
Neither host fitness nor pathogen fitness is maximized at equilibrium. The reason for this is that maximum fitness in the host is attained by minimizing fitness of the pathogen, and conversely, host fitness is minimized as a consequence when pathogen fitness is maximized. The host population has maximum fitness when all the pathogen population is avirulent and the host is resistant (Table 9. 1). Maximum pathogen fitness occurs when all the host plants are susceptible and the pathogen is avirulent; in the case of the competition version of the model, it may also occur when the hosts are all resistant and the pathogen population is completely virulent (Fig. 9.3). At equilibrium, both host and pathogen fitness are balanced somewhere between the minimum and maximum levels that they can reach.
Both the hard selection and competition versions of the host-pathogen coevolution model indicate that at equilibrium, resistance should occur at relatively low frequencies in wild plant populations while the frequency of virulence should be high. Wild host-pathogen systems that have been studied thoroughly exhibit these characteristics. Clarke et al. ( 1990) found that resistance genes in groundsel occur in relatively low frequencies over a wide geographical area, although individual resistance genes can occur at high frequency in small plots within the area. In contrast, virulence occurs at high frequency, and most mildew isolates have complex genotypes with many virulence genes. This pattern of moderate to low frequencies of resistance genes and high frequencies of virulence genes has been demonstrated also for wild oats, Avena spp., and the crown rust fungus, Puccinia coronata, in both Israel ( Wahl 1970) and Australia ( Burdon 1987). Wahl ( 1970) and Burdon ( 1987) showed that wild oats in areas with environments unfavorable to crown rust have less resistance to the disease. Burdon ( 1987) showed that both frequency and diversity of virulence in P. coronata populations is reduced in environments unfavorable to the disease.
Both the hard selection and competition versions of the model have unstable limit cycles. The position of the limit cycle varies depending upon the values assigned to the key parameters in the model. Because gene frequencies inside the unstable limit cycle spiral in toward the equilibrium point, we can use the size of the limit cycle as a measure of the relative evolutionary stability of the equilibrium for each set of parameter values. With both versions of the model, there is a tendency for the equilibria to be more evolutionarily stable for genes that have high fitness costs associated with them, but this is less pronounced with the competition version than with the hard selection version. In general, equilibria are more evolutionarily stable in the competition version, particularly for resistance genes with low associated fitness costs. Also, in the competition version, the evolutionary stability of