STABILITY OF EQUILIBRIA IN A GENE-FOR-GENE COEVOLUTION MODEL OF HOST-PARASITE INTERACTIONS

Stability of resistance/susceptibility and virulence/avirulence polymo rphisms 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 w ith 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 cycl es 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 ext inction). Depending on their initial frequencies in the model, alleles for virulence and susceptibility either spiraled toward equilibrium o r they became fixed. Likewise, alleles for avirulence and resistance m oved either toward equilibrium or extinction. Thus, the position of th e unstable limit cycle and the initial gene frequencies determined whe ther the system went toward a stable equilibrium or fixation of virule nce and susceptibility. The position of the unstable limit cycle depen ded on the values of key parameters in the model. For some combination s 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 biologica l terms. Two versions of the coevolution model were compared. In the h ard-selection version, virulence alleles carry an associated fitness c ost of reduced inherent rate of reproduction on either susceptible or resistant hosts. In the competition version, only unneccessary virulen ce carries a fitness cost, because the cost of virulence is expressed as reduced competitive ability on susceptible hosts. Polymorphisms wer e stable for moderate costs of unnecessary virulence in the competitio n version of the model but usually were not stable for the hard-select ion version. In the competition version, polymorphisms were stable eve n when there was no cost of resistance, provided that the cost of unne cessary virulence was moderately high.