Based on the study of the plankton population system, a diffusive toxic plankton model with Tissiet type functional response function and predation delay is proposed. Firstly, the sufficient conditions for locally asymptotic stability of the diffusion system without delay at the positive equilibrium are given, the existence conditions of Hopf bifurcation caused by diffusion are given, and the conditions under which diffusion makes spatially homogeneous and nonhomogeneous periodic solutions bifurcate from the positive constant equilibrium are given. Secondly, the time delay effect on the plankton reaction–diffusion system is studied, the existence of Hopf bifurcation at the positive equilibrium induced by delay is discussed. By applying the central manifold theory and normal form method of partial functional differential equations, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied. Finally, the reliability of theoretical research is verified by numerical simulation.

在对浮游生物种群系统研究的基础上，提出了一种具有Tissiet型功能响应函数和捕食延迟的扩散有毒浮游生物模型。首先，给出了扩散系统在正平衡时无延迟局部渐近稳定的充分条件，给出了扩散引起的Hopf分岔的存在条件，以及扩散使空间齐次和非齐次周期解分岔的条件。给出了正常数平衡。其次，研究了时滞对浮游生物反应-扩散系统的影响，讨论了时滞引起的正平衡时Hopf分岔的存在。应用偏函数微分方程的中心流形理论和范式方法，研究了Hopf分岔的方向和分岔周期解的稳定性。最后通过数值模拟验证了理论研究的可靠性。