具有三次源项的二元细胞趋化系统的稳定性及动态跃迁

Stability and Dynamic Transitions of a Binary Cell Chemotaxis System with a Cubic Source Term

摘要: 本文研究一类带有三次多项式生长源的二元细胞趋化系统在齐次Neumann边界条件下的稳定性与动态跃迁问题。其中考虑了趋化系数不为零的一般情形。利用线性全连续场谱理论和中心流形约化方法，给出了系统两个常数稳态解——零平衡点和正平衡点的特征值序列，建立了特征值实部发生正负穿越的PES条件，并求得了相应的临界扩散系数。分析表明，以细胞扩散系数为控制参数时，当该参数穿过临界值，系统在零平衡点处会发生连续型动态跃迁；而在正平衡点处，趋化项的引入使得线性化矩阵产生非对角耦合项，即使生长源项在该点的一阶导数为负，若趋化强度足够大，系统仍可能发生扩散驱动的失稳现象，从而丰富了系统的动力学行为。

Abstract: This paper investigates the stability and dynamic transitions of a two-component chemotaxis system with a cubic polynomial growth source under homogeneous Neumann boundary conditions, taking into account the general case with nonzero chemotaxis coefficient. By employing the linear completely continuous field spectral theory and the center manifold reduction method, we derive the eigenvalue sequences for the two constant steady-state solutions, namely the zero equilibrium and the positive equilibrium, establish the PES conditions under which the real parts of eigenvalues change sign, and obtain the corresponding critical diffusion coefficients. The analysis reveals that when the cell diffusion coefficient is taken as the control parameter and crosses the critical value, a continuous-type dynamic transition occurs at the zero equilibrium. For the positive equilibrium, the introduction of the chemotaxis term yields off-diagonal coupling in the linearization matrix; consequently, even if the first derivative of the growth source term at that point is negative, the system may still undergo diffusion-driven instability provided that the chemotactic intensity is sufficiently large, thereby enriching the dynamical behaviors of the system.