Turing instabilities in prey-predator systems with dormancy of predators

In this paper, we study the stationary and oscillatory Turing instabilities of a homogeneous equilibrium in prey-predator reaction-diffusion systems with dormant phase of predators. We propose a simple criterion which is useful in classifying these Turing instabilities. Moreover, numerical simulations reveal transient spatio-temporal complex patterns which are a mixture of spatially periodic steady states and traveling/standing waves. In this mixture, the steady part is the stable Turing pattern bifurcated primarily from the homogeneous equilibrium, while wave parts are unstable oscillatory solutions bifurcated secondarily from the same homogeneous equilibrium. Although our criterion does not exclude the occurrence of oscillatory Turing instability, we have not yet found stable traveling/standing waves due to oscillatory Turing instability in our simulations. These results suggest that dormancy of predators is not a generator but an enhancer of spatio-temporal Turing patterns in prey-predator reaction-diffusion systems.