Stability of equilibria in a gene-for-gene coevolution model of host-parasite interactions

Stability of resistance/susceptibility and virulence/avirulence polymorphisms in a gene-for-gene host-parasite coevolution model was tested by numerical analysis. Computer simulations were run for 752 different combinations of parameter values in the model. Repeated simulations with different initial frequencies of resistance and virulence alleles revealed the presence of an unstable limit cycle for each combination of parameter values. Represented in a phase plane, unstable limit cycles repel gene frequencies (i.e., gene frequencies starting inside the limit cycle spiral inward toward an internal equilibrium point; those starting outside the limit cycle spiral outward toward fixation or extinction). Depending on their initial frequencies in the model, alleles for virulence and susceptibility either spiraled toward equilibrium or they became fixed. Likewise, alleles for avirulence and resistance moved either toward equilibrium or extinction. Thus, the position of the unstable limit cycle and the initial gene frequencies determined whether the system went toward a stable equilibrium or fixation of virulence and susceptibility. The position of the unstable limit cycle depended on the values of key parameters in the model. For some combinations of parameter values, the unstable limit cycles extended so far from the equilibrium point that new genes for virulence could not possibly enter the parasite population at frequencies outside the limit cycle. In those cases, the polymorphisms were regarded as stable in biological terms. Two versions of the coevolution model were compared. In the hard-selection version, virulence alleles carry an associated fitness cost of reduced inherent rate of reproduction on either susceptible or resistant hosts. In the competition version, only unnecessary virulence carries a fitness cost, because the cost of virulence is expressed as reduced competitive ability on susceptible hosts. Polymorphisms were stable for moderate costs of unnecessary virulence in the competition version of the model but usually were not stable for the hard-selection version. In the competition version, polymorphisms were stable even when there was no cost of resistance, provided that the cost of unnecessary virulence was moderately high.