Turing–Hopf bifurcation in the reaction–diffusion equations and its applications

Turing–Hopf bifurcation analysis of a predator–prey model with herd behavior and cross-diffusion

“…But, as far as we know, the work on the rich dynamics deduced by the Turing-Hopf bifurcation in predatorprey type reaction-diffusion systems [53][54][55] near the bifurcation point was all reported numerically, except paper [56,57]. In [56,57], the authors have discussed the classification of the spatiotemporal dynamics in a neighborhood of the bifurcation point in detail, which can be figured out in the framework of the normal forms.…”

“…Based on the paper [58], we shall continue to explore the other dynamics of model (2) such as the existence and priori bound of a solution for the model without cross-diffusion, and complex spatiotemporal dynamics near the Turing-Hopf bifurcation point of the model with cross-diffusion in the framework of normal form, which are different from the results in [58]; for instance, the stable spatially inhomogeneous periodic solutions are found. Here, we have to point out the fact that, although the method of computing the normal form in this paper is motivated by the one in [56,57], because of the existence of crossdiffusion, the procedure of computing normal form in this paper need be deduced again. This implies that our results generalize the application scope of references [56,57].…”

Dynamical analysis of a competition and cooperation system with multiple delays

“…Furthermore, we do not consider systems with a spatial variable, which is the diffusive model subject to a suitable boundary condition. We intend to make the comparison between the two models, and find what is the influence on the dynamical behaviour with different delays and diffusive terms [32][33][34][35][36][37][38], and then illustrate with the theoretical predictions. Lastly, our problem is only restricted to the theoretical analysis of such economical phenomena.…”

Stability and Hopf Bifurcation Analysis of a Mathematical Model in Tumor Angiogenesis

“…They observe that Turing-Hopf and spatial resonance bifurcation can occur. Similarly, in [4,6] the authors consider the Turing-Hopf bifurcation arising from the reaction-diffusion equations, and in [3,5] the authors derive a necessary and sufficient condition for Turing instabilities to occur in two-component systems of reaction-diffusion equations with Neumann boundary conditions. In [7] the authors study the stability analysis of the steady-state solution of a mathematical model in tumor angiogenesis whereas in [10] Hopf bifurcation of a ratio -dependent predator-prey model involving two discrete maturation time delays is considered.…”