Stable Polymorphisms in a Two-Locus Gene-for-Gene System

Abstract

A two-locus gene-for-gene model is presented to analyze coevolutionary dynamics in interactions between host plants and their pathogens. Using both analytical and simulation approximations, we show that the behavior of the model is very simple with one locus. In the reciprocal genetic feedback version, there is a smooth outward spiral toward the boundaries. In the delayed feedback version, there is an infinite family of closed curves corresponding to different initial conditions. Both versions of the model are stabilized by the addition of recurrent mutation. Either a stable interior equilibrium or a stable limit cycle appears. But with the two-locus model, different coevolutionary outcomes are predicted according to the parameter values. For a wide range of small and medium values of virulence and resistance costs, complex fluctuations arise. The number of virulence alleles per isolate and the number of resistance alleles per plant cycle indefinitely. If the costs of both virulence and resistance are above a threshold, the final state of the coevolutionary dynamics is a stable single-resistance static polymorphism in the host and avirulence in the parasite. An equivalent threshold to maintain a disease-free host population was obtained analytically for a multilocus system. These expressions can be used to determine the number of single-resistance host genotypes that would have to be present in a mixture to prevent the spread of any virulent race of pathogen. The model demonstrates that it is preferable to use mixtures of single-resistant genotypes rather than using multiple resistance alleles in the same cultivar.