Transition and bifurcation analysis for chemotactic systems with double eigenvalue crossings

Our main objective of this research is to study the dynamic transition for diffusive chemotactic systems modeled by Keller-Segel equations in a rectangular domain. The main tool used is the recently developed dynamic transition theory. Through a reduction analysis and focusing on systems with certain symmetry where double eigenvalue crossing occurs during the instability process, it is shown that the chemotactic system can undergo both continuous and jump type transitions from the steady states, depending on non-dimensional parameters $\alpha$, $\mu$ and the side length $L_1$ and $L_2$ of the container. Detailed dynamic structures during transition, including metastable and stable states and orbital connections between them, are rigorously obtained. This result extends the previous work with only one eigenvalue crossing at critical parameters and offers more complex insights given the symmetry of our settings.