Abstract A tumor–immune model with time-delay and diffusion is considered. Firstly, the local stability of equilibria and the existence of Hopf bifurcation are studied. Secondly, the direction and stability of Hopf bifurcation are discussed. Finally, the numerical simulations are used to verify the effectiveness of the theoretical results. It is found that the time-delay can destroy the stability of positive equilibrium and then affect the occurrence of Hopf branch. Specifically, the equilibrium is stable if the model is without delay or with small delay, and so there is no bifurcation; Conversely, when the delay is large, it induces the instability of equilibrium and the Hopf bifurcation occurs, the model then exhibits rich spatiotemporal dynamics.

中文翻译：

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具有时滞和扩散的肿瘤免疫模型的分叉和模式形成

摘要 考虑了具有时间延迟和扩散的肿瘤免疫模型。首先研究了平衡点的局部稳定性和Hopf分岔的存在性。其次，讨论了Hopf分岔的方向和稳定性。最后,通过数值模拟验证了理论结果的有效性。发现时滞会破坏正平衡的稳定性，进而影响Hopf分支的发生。具体来说，如果模型没有延迟或延迟很小，则平衡是稳定的，因此不存在分叉；反之，当延迟较大时，会引起平衡不稳定，出现Hopf分岔，模型表现出丰富的时空动态。