Dynamics of some simple host-parasite models with more than two genotypes in each species

A two-species genetic model of host-parasite interaction is used to study the dynamical consequences of varying the number of genotypes in each species, and the recombination rate in the host. With two genotypes in each species, the model's behaviour is very simple; there is either a stable interior equilibrium, a stable cycle or a smooth outward spiral toward the boundaries. But with three or more genotypes, complex cycles and apparently chaotic behaviour may arise over wide ranges of parameter values. Increasing the number of genotypes also tends to slow the rate of gene-frequency change. Recombination in the host does not affect the stability of the interior fixed point, but intermediate rates of recombination may give dynamic stability to an otherwise dynamically unstable pattern of cycling. Intermediate rates of recombination also tend to decrease the amplitudes of gene-frequency cycles in the host, which implies that they could promote the accumulation of genetic variation involved in complementary, antagonistic interactions with parasites.