Host-parasite coevolution in populations of variable size

Many theoretical and empirical studies in host-parasite coevolution have been motivated by the hypothesis that pathogens are responsible for the evolution of sex or high genetic diversity in their hosts. The matching- allele (MA) and gene-for-gene (GfG) models are popular models to describe host-parasite interactions. While in the MA models the parasite infection is strictly type specific, in the GfG models a parasite can infect multiple hosts. Agrawal and Lively (Evolutionary Ecology Research 4:79-90, 2002) captured these two models in a single framework and numeri- cally explored its dynamics. As in most of the theoretical studies, they assume that the population sizes are constant. Here, we take account of changes in host and parasite population size via the Lotka-Volterra equations, a popular framework for modeling population dynamics in predator and prey. To keep the model simple, we consider a system of two hosts and two parasites. The population dynamics become complex under changing population size when the model moves away from pure matching-allele and becomes more gene-for-gene-like. In a gene-for-gene-like model with changing population sizes, we are dealing with a system of four nonlinear differen- tial equations. While the populations oscillate with a single oscillation frequency in the pure MA model, a second oscillation frequency arises under GfG-like conditions. We have calculated the interior fixed point of the four-dimensional system and the oscillation frequency at this point. A constant of motion can be found on a two-dimensional hyperplane. This basic understanding of the model system is important, in particular when stochasticity is included, which can have a strong impact on such systems (Gokhale et al., BMC Evolutionary Biology 13:254, 2013).