Example of allele frequency dynamics with and without Lotka-Volterra population size fluctuations. Top: Lines show the deterministic Lotka-Volterra dynamics, as often considered in theoretical studies, cf. Eqs. (4). Middle: When stochasticity is included (thin lines show the results of 50 individual stochastic Gillespie simulations), then simulations may initially produce allele oscillations as above and below. However, alleles usually spread to fixation (or go extinct) at a much faster rate. Bottom: Dynamics without Lotka-Volterra cycles, fixing the average population size of both species to Navg=1000 by resetting it after every Navg reactions, while maintaining the ratio between the alleles. The 50 individual stochastic simulations now only rarely reach fixation. The figure illustrates the scenario where the rare host allele (H1) is more likely to reach fixation than the frequent host allele (see Figure 3). This fixation probability decreases with increasing initial frequency (cf. Figure 3). The simulation parameters are a=5, c=2.5, b=10/Navg=0.01 with H1=5%, H2=95%, P1=20%, P2=80% as initial condition.