Evolution towards oscillation or stability in a predator-prey system.

We studied a prey-predator system in which both species evolve. We discuss here the conditions that result in coevolution towards a stable equilibrium or towards oscillations. First, we show that a stable equilibrium or population oscillations with small amplitude is likely to occur if the prey's (host's) defence is effective when compared with the predator's (parasite's) attacking ability at equilibrium, whereas large-amplitude oscillations are likely if the predator's (parasite's) attacking ability exceeds the prey's (host's) defensive ability. Second, a stable equilibrium is more likely if the prey's defensive trait evolves faster than the predator's attack trait, whereas population oscillations are likely if the predator's trait evolves faster than that of the prey. Third, when the adaptation rates of both species are similar, the amplitude of the fluctuations in their abundances is small when the adaptation rate is either very slow or very fast, but at an intermediate rate of adaptation the fluctuations have a large amplitude. We also show the case in which the prey's abundance and trait fluctuate greatly, while those of the predator remain almost unchanged. Our results predict that populations and traits in host-parasite systems are more likely than those in prey-predator systems to show large-amplitude oscillations.