Discrete-time models for gene frequencies and population densities in plant pathosystems

Discrete-time models for the gene frequencies and population densities of plants and pathogens are formulated and analyzed. The gene-frequency models are based on the gene-for-gene hypothesis; a resistant gene in the plant has a corresponding virulent gene in the pathogen. Some basic population genetics models with selection are summarized. Then, gene-frequency models for two populations are formulated. A gene-frequency model for plant pathosystems developed in the 1970's is discussed. For the general model, conditions are derived for local stability of the equilibria. The polymorphic equilibrium E_P is shown to be nonhyperbolic but numerical simulations indicate that cycling occurs about E_P. In addition, population models for the host and pathogen are formulated, where the genetics of the pathogen are modeled. It is shown that the asymptotic dynamics depend on the basic reproductive number R₀, which in turn depends on the fitness values of the host and pathogen.