Adaptive predator-prey dynamics

Antagonistic interactions between species, such as prey-predator and host-parasite system, have an inherent tendency of instability. When prey’s defence and predator’s offense change in an adaptive manner, either by genetic evolution, by phenotypic plasticity, or by behavioral choice, the population dynamics of a prey-predator system may stay in a stable equilibrium or fluctuate with a large amplitude. Stability of this adaptive dynamics critically depends on which species has the faster rate of adaptation, and on the predator's handling time and the prey’s carrying capacity. When the predator's handling time is long and the prey's carrying capacity is small, the stable equilibrium is more likely if the predator evolves faster than the prey. In such a situation, the prey’s population size can be nearly constant but the predator’s can have a very large amplitude, or two populations may oscillate in an anti-phase manner. These patterns are completely opposite to the results when the ecological dynamics (without adaptive changes) are of Lotka-Volterra type. We also observed that the amplitude of fluctuation is the largest when the speed of adaptation is an intermediate value, rather than very small or very large. These behaviors explain recent findings in the experimental studies with micro-organisms and provide several testable predictions.