Hopf bifurcation for general network-organized reaction-diffusion systems and its application in a multi-patch predator-prey system

Hopf Bifurcation and Its Normal Form of Reaction Diffusion Systems Defined on Directed Networks

“…On the other hand, since the undirected network can be regarded as a special balanced directed network too, the derivation process for Hopf bifurcation normal form of the reaction-diffusion systems defined on directed networks is also applicable to the ones defined on undirected networks. In this sense, our current results extend the previous results in [9].…”

“…And, this special limitation also exists in other works, for example in [4,7]. Since the undirected network can also be regarded as a special kind of balanced directed networks, this work extended the results of Hopf bifurcation theory for the general two-species reaction-diffusion systems on undirected networks in [9]. Because the Laplacian matrix of the underlying directed networks is asymmetrical, it is found that there involves some important different details in calculating the Hopf bifurcation nor-mal form of the general two-species reaction-diffusion systems defined on the balanced directed networks comparing the calculating the Hopf bifurcation normal form of the general two-species reaction-diffusion systems defined on undirected networks, especially the ways of determining the vector q and q * .…”

“…For one specific system, it is more reasonable to find a feasible way to derive the Hopf bifurcation normal form according to its own characteristics, rather than just apply the general theory mechanically. For system (2.1), as we did in our work [9], we particularly introduced the tensor product of the matrix to represent the Jacobian matrix (2.4), and then determine the eigenvalues and corresponding eigenvectors of the matrix in a straightforward way based on the fundamental Theorem 2.1, which allows us to successfully derive the final Hopf bifurcation normal form. We believe that this is not applicable to any high-dimensional ODEs.…”

“…2. There exist some noteworthy differences and relations between the derivation processes for Hopf bifurcation normal form of the reaction-diffusion systems defined on directed networks and the ones defined on undirected networks, the later one is provided in detail in our previous work [9]. The foremost difference is that the analysis for the eigenvalues given by characteristic equation (2.6) need to be considered in complex space when the Laplacian eigenvalues of the underlying directed networks are complex.…”

“…To fill this gap, we are to strictly derive the Hopf normal form of the reaction-diffusion systems defined on directed networks in this paper, with adopting the framework in [14,Chapter 5]. Compared with the derivation for Hopf normal form of the reaction-diffusion systems defined in undirected networks [9], as well for parabolic partial differential equations defined in continuous space, the complex eigenvalue of underlying directed networks does bring about some important difference in this process. We also apply directly the obtained theoretical results in a specifical SI epidemic system defined on the balanced directed ring networks or small-word directed NW networks.…”