In this work we investigate a minimal reaction--diffusion system with nonlinear cross-diffusion, showing that this term alone is sufficient to trigger spontaneous pattern formation, even in a triangular configuration preserving the classical Lotka--Volterra competitive kinetics. Beyond the expected stationary Turing patterns, our analysis reveals the emergence of oscillatory non-homogeneous solutions, which is surprising, since no Hopf bifurcation, wave instability, or external periodic forcing occurs in the model.
A detailed spectral analysis reveals that these oscillations originate from spatial resonance mechanism, in which the fundamental mode interacts with its subharmonics and superharmonics. This resonance induces secondary instabilities and leads to the excitation of subharmonic oscillations. By deriving the normal form for the unstable amplitudes and applying a center--unstable manifold reduction, we characterize the amplitude dynamics on the resonant manifold and determine the critical threshold at which subharmonic excitation sets in.
These findings demonstrate that nonlinear cross-diffusion acts as the fundamental mechanism responsible for both spatial segregation and the onset of resonant oscillatory instabilities in minimal competitive systems.