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

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... 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.