The present paper is concerned with a diffusive population model of Logistic type with an instantaneous density-dependent term and two delayed density-dependent terms and subject to the zero-Dirichlet boundary condition. By regarding the delay as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation for the sufficiently small positive steady state solution are shown. It is found that under the suitable condition, the positive steady state solution of the model will become ultimately unstable after a single stability switch (or change) at a certain critical value of delay through a Hopf bifurcation. However, under the other condition, the positive steady state solution of the model will become ultimately unstable after multiple stability switches at some certain critical values of delay through Hopf bifurcations. In addition, the direction of the above Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed by means of the center manifold theory and normal form method for partial functional differential equations. Finally, in order to illustrate the correction of the obtained theoretical results, some numerical simulations are also carried out.

本论文涉及具有瞬时密度相关项和两个延迟密度相关项并服从零狄利克雷边界条件的 Logistic 类型的扩散种群模型。通过将时延作为分岔参数并详细分析相关的特征值问题，给出了足够小的正稳态解的局部渐近稳定性和Hopf分岔的存在性。发现在合适的条件下，模型的正稳态解经过Hopf分岔在某个时延临界值处进行单次稳定性切换（或变化）后，最终会变得不稳定。但是，在其他条件下，模型的正稳态解将在通过 Hopf 分岔的某些特定延迟临界值处进行多次稳定性切换后最终变得不稳定。此外，利用偏函数微分方程的中心流形理论和范式方法分析了上述Hopf分岔的方向和分岔周期解的稳定性。最后，为了说明所得理论结果的修正，还进行了一些数值模拟。