The complexity of spatial interactions in nature often gives rise to emergent patterns that cannot be explained by local dynamics alone. An investigation is conducted into how cross-diffusion and cross-advection govern the stability of general reaction-advection-diffusion system on an infinite spatial domain. The analysis is formulated for nonlinear reaction terms using the reaction Jacobian, diffusion matrix, and advection matrix. By applying normal-mode decomposition, the Routh-Hurwitz criterion for complex characteristic polynomials, and the critical neutral curve method, we derive stability conditions and identify instability boundaries in parameter space. The results show that a constant equilibrium stable under the local reaction dynamics remains stable under self-diffusion and self-advection. In the baseline kinetic setting, cross-advection alone also preserves stability when cross-diffusion is absent. In contrast, cross-diffusion can destabilize the equilibrium and interact with cross-advection to reshape the unstable region. Under different kinetic-sign conditions, asymmetric cross-advection may induce instability even in the absence of cross-diffusion. The predator-prey application supports these findings, as weak transport interactions preserve homogeneous coexistence while sufficiently strong cross-transport generates unstable spatial modes and visible spatiotemporal patterns. Cross-diffusion initiates spatial organization through density-gradient responses between prey and predator. In contrast, cross-advection modifies the direction, intensity, and propagation of the resulting patterns through directed movement such as prey avoidance and predator pursuit. These results provide a systematic framework for detecting cross-transport-induced instability and explaining the transition from homogeneous coexistence to pattern-forming dynamics.
Citation: Ririn Setiyowati, Lina Aryati, Sumardi, Nanang Susyanto. Stability transitions induced by cross-transport mechanisms in a coupled reaction-advection-diffusion system[J]. AIMS Mathematics, 2026, 11(9): 27791-27825. doi: 10.3934/math.20261111
The complexity of spatial interactions in nature often gives rise to emergent patterns that cannot be explained by local dynamics alone. An investigation is conducted into how cross-diffusion and cross-advection govern the stability of general reaction-advection-diffusion system on an infinite spatial domain. The analysis is formulated for nonlinear reaction terms using the reaction Jacobian, diffusion matrix, and advection matrix. By applying normal-mode decomposition, the Routh-Hurwitz criterion for complex characteristic polynomials, and the critical neutral curve method, we derive stability conditions and identify instability boundaries in parameter space. The results show that a constant equilibrium stable under the local reaction dynamics remains stable under self-diffusion and self-advection. In the baseline kinetic setting, cross-advection alone also preserves stability when cross-diffusion is absent. In contrast, cross-diffusion can destabilize the equilibrium and interact with cross-advection to reshape the unstable region. Under different kinetic-sign conditions, asymmetric cross-advection may induce instability even in the absence of cross-diffusion. The predator-prey application supports these findings, as weak transport interactions preserve homogeneous coexistence while sufficiently strong cross-transport generates unstable spatial modes and visible spatiotemporal patterns. Cross-diffusion initiates spatial organization through density-gradient responses between prey and predator. In contrast, cross-advection modifies the direction, intensity, and propagation of the resulting patterns through directed movement such as prey avoidance and predator pursuit. These results provide a systematic framework for detecting cross-transport-induced instability and explaining the transition from homogeneous coexistence to pattern-forming dynamics.
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Ririn Setiyowati, Lina Aryati, Sumardi, Nanang Susyanto. Stability transitions induced by cross-transport mechanisms in a coupled reaction-advection-diffusion system[J]. AIMS Mathematics, 2026, 11(9): 27791-27825. doi: 10.3934/math.20261111
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