Stability of a Gene-for-Gene Coevolution System Under Constant Perturbations

Abstract

in which ni and mi (ni + mi = 1) are the frequencies of genes for virulence and avirulence in the parasite population at the end of the ith host generation, respectively; pi and qi (pi + qi = 1) are the frequencies of genes for resistance and susceptibility in the host population, respectively; relative fitness of the avirulent parasite on the resistant host is 1 – t, where t represents the effectiveness of the resistance; relative fitness of the virulent parasite on the susceptible host is 1 – k, where k is the cost of virulence; c is the fitness cost of resistance; s is the severity of disease; and 1 – k + a is the relative fitness of the virulent parasite on the resistant host, where a determines the type of selection in the parasite population. When a = 0, there is a cost of virulence even when it is necessary to parasitize the resistant host. This is considered a hard selection, in which the presence of the virulence gene causes a reduction in the intrinsic rate of reproduction by the parasite (7,10). When a = k, virulence has a fitness cost when it is unnecessary, as it is when a virulent parasite infects a susceptible host. This can be considered a soft selection, in which the relative fitness of the virulent parasite is reduced in competition with the avirulent parasite on a susceptible host (7,10). The model has a single internal equilibrium point (EP; ne, pe), at which selection pressures are balanced and there is no change in the frequency of either resistance or virulence. The internal EP can be calculated by the following equation set: