Bifurcation Analysis of a Single‐Species Model With a General Advection Term and Two Delays

ABSTRACT This paper is concerned with a single‐species population model that incorporates a general advection term and a reaction term with two distinct time delays. We first obtain the existence of a positive steady state of the system via the implicit function theorem. Then a critical prior estimate is developed, and based on this, we investigate the stability of positive steady state and the direction in which the purely imaginary eigenvalues pass through the imaginary axis of the complex plane when delays cross stability crossing curves. Finally, numerical simulations are presented with an illustrative example, demonstrating the emergence of Hopf bifurcation as the delays vary, which is consistent with the theoretical results.