Periodic solutions and spatial patterns induced by mixed delays in a diffusive spruce budworm model with Holling II predation function

In present article, under homogeneous Neumann boundary condition, we put forward a diffusive spruce budworm model with mixed delays and Holling II predation function firstly. Then, choosing delay (discrete delay or distributed delay) as bifurcating parameter together with characteristic equation, we derive that not only can discrete delay induce the appearance of Hopf bifurcations for this model, but also distributed delay can do it. However, to our knowledge, in the known literatures, Hopf bifurcation can only be deduced by discrete delay or distributed delay. So, the obtained results in present article are new. At last, by carrying out numerical simulations, we obtain periodic solutions and spatial patterns deduced by discrete delay or distributed delay, which illustrates the results in this article.