Emergence of fractional chemotactic Turing patterns and spatiotemporal chaos in memory-dependent Keller–Segel cross-diffusion systems

This paper develops a novel fractional-order Keller–Segel cross-diffusion framework for investigating memory-dependent chemotactic aggregation, biological pattern formation, and anomalous transport in microbial populations and cellular systems. The proposed model extends the classical Keller–Segel system by incorporating a Caputo fractional time derivative and a nonlinear cross-diffusion mechanism to account for hereditary transport and density-dependent migration. A rigorous mathematical analysis is presented, including positivity and boundedness of solutions, existence of weak solutions, linear stability analysis, and explicit conditions for fractional Turing instability. The critical wave number and instability interval are also determined to characterize the onset of diffusion-driven pattern formation. Numerical simulations validate the theoretical predictions and demonstrate the emergence of spot-like, stripe-like, and mixed spot–stripe patterns. Furthermore, comparison with the corresponding integer-order model shows that fractional memory delays the onset of instability, prolongs transient dynamics, and enhances pattern persistence by slowing the temporal growth of unstable modes. These results demonstrate that fractional memory plays a fundamental role in regulating chemotactic self-organization and provide a unified mathematical framework for studying memory-dependent pattern formation in biological systems.