  A discrete chemotactic predator-prey model is proposed in which the prey
secrets a diffusing chemical which is sensed by the predator and vice versa.
Two dynamical states corresponding to catching and escaping are identified and
it is shown that steady hunting is unstable. For the escape process, the
predator-prey distance is diffusive for short times but exhibits a transient
subdiffusive behavior which scales as a power law $t^{1/3}$ with time $t$ and
ultimately crosses over to diffusion again. This allows to classify the
motility and dynamics of various predatory bacteria and phagocytes. In
particular, there is a distinct region in the parameter space where they prove
to be infallible predators.
