具有单扩散系数的S-K-T生物竞争系统的动力学行为分析

Dynamical Behavior Analysis of the S-K-T Biological Competition System with a Single Diffusion Coefficient

摘要: 本文利用线性全连续场谱理论、动态跃迁理论和线性稳定性理论研究了单扩散系数S-K-T生物竞争模型在正平衡点处的动态跃迁。首先利用线性全连续场谱理论，得到线性化系统的特征方程，从而计算出系统的特征值序列以及相应条件下临界控制参数${\lambda }_c$ 发生正负穿越的PES条件。借助动态跃迁理论，得到系统发生连续型动态跃迁并且分歧出一个渐近稳定的奇点。最后利用数值模拟对理论进行验证。

Abstract: In this paper, by employing the linear completely continuous field spectral theory, the dynamic transition theory, and the linear stability theory, we investigate the dynamic transition of the S-K-T biological competition system with a single diffusion coefficient at its positive equilibrium point. First, by means of the linear completely continuous field spectral theory, we derive the characteristic equation of the linearized system, and thereby compute the sequence of eigenvalues. Furthermore, under the corresponding conditions, we establish the Positive Eigenvalue Switching condition under which the critical control parameter crosses from negative to positive. Then, with the aid of the dynamic transition theory, we prove that the system undergoes a continuous dynamic transition and bifurcates an asymptotically stable singular point. Finally, numerical simulations are carried out to verify the theoretical results.