A predator-prey coevolution: instability but stabilization caused by evolutionary instability

Coevolution of predator and prey has been analyzed by numerous theoretical studies mainly in two ways. The most of studies assume that the interaction is based on the degree of phenotype matching among species (e.g., size specific predation), however few assumes that the interaction is based on the degree that the phenotype of one species exceeds that of the other (e.g., speed-speed interaction), in spite of the fact that the latter case also occurs widely in nature. Here we develop a simple mathematical model for analyzing the coevolutionary dynamics of predator and prey whose interaction is based on the degree that the phenotype of one species exceeds that of the other. The ecological dynamic is assumed to be Lotka-Volterra system and the evolutionary dynamics is described by a quantitative genetic model. We analytically show the conditions under which the coevolutioary cycles can occur through a local stability analysis and demonstrate the cycles occur under broad conditions. Furthermore, we find an interesting pattern of non-equilibrium eco-evolutionary dynamics. The local stability of slower evolution is destroyed by relatively faster evolution, however, the amplitudes of population cycles decrease even if those of trait cycles increase with increasing the evolutionary speed.