  We identify a new type of pattern formation in spatially distributed active
systems. We simulate one-dimensional two-component systems with predator-prey
local interaction and pursuit-evasion taxis between the components. In a
sufficiently large domain, spatially uniform oscillations in such systems are
unstable with respect to small perturbations. This instability, through a
transient regime appearing as spontanous focal sources, leads to establishment
of periodic traveling waves. The traveling waves regime is established even if
boundary conditions do not favor such solutions. The stable wavelength are
within a range bounded both from above and from below, and this range does not
coincide with instability bands of the spatially uniform oscillations.
