The Bifurcation Theory For The Network-organized Reaction-diffusion Systems

Reaction-diffusion processes exist universally in nature and human society.Fruitful achievements have been made in the research of partial differential reaction-diffusion systems in continuous space.However,the exploration of basic theories and their applications for the reaction-diffusion systems defined on complex networks are still severely limited due to their ultra-high dimension and strong nonlinearity.Therefore,there is urgently imperative necessity for a breakthrough.In this thesis,we carry out a series of fundamental theoretical and application-oriented investigation on Turing bifurcation,steady state bifurcation,Hopf bifurcation and even general bifurcation of the network-organized reaction-diffusion systems.The main research contents and innovation points are as follows:(1)Turing bifurcation of the network-organized reaction-diffusion systems and its application.Mathematically,we strictly establish the general analysis framework for Turing bifurcation in the network-organized reaction-diffusion systems,derive the necessary and sufficient conditions for the occurrence of Turing bifurcation,and reveal the influence of the eigenvalues of network Laplacian matrix on Turing bifurcation.To expand the way for investigating patterning,we put forward a novel concept of ‘spatial network pattern',constructe two kinds of complex spatial networks,and then study the formation and transitions of epidemiological patterns in such spatial networks.Through a series of simulations,the effects of some fators including heterogeneous and random connectivity of the network on the qualitative characteristics of epidemiological patterns are studied.It is found that the strong heterogeneous connectivity does not lead to qualitative changes of patterns,and random connectivity can be viewed as a new mechanism to promote the emergence of irregular patterns.(2)Steady state bifurcation of the network-organized reaction-diffusion systems and its application.We give out the calculation method for the steady state bifurcation normal form of the network-organized reaction-diffusion systems,and analyze in detail all possible 36 types of bifurcation diagrams of the third-order truncated steady state bifurcation normal form,which provides some theoretical basis for revealing the robustness of complex networks.The obtained theoretical results are applied to a specific network-organized epidemic reaction-diffusion systems,and we analytically determine its steady state bifurcation points,calculate the corresponding steady state bifurcation normal forms.Combined with simulations,we study the influence of complex networks' structure on the steady state bifurcation,and find some new multi-stability and strong hysteresis phenomena in irregular small-world networks and scale-free networks.(3)Hopf bifurcation of the network-organized reaction-diffusion systems and its application.We give out the calculation method for the Hopf bifurcation normal form of the network-organized reaction-diffusion systems,which provides a mathematical tool for studying the periodic oscillation behaviors of complex networks.The results show that the calculation process fot the Hopf bifurcation normal form of the network-organized reaction-diffusion systems is more intricate than its corresponding process of partial differential reaction-diffusion systems.The obtained theoretical results are applied to a specific network-organized predator-prey reaction-diffusion system with Holling II functional response function,and we analytically determine its Hopf bifurcation points,judge the types of the corresponding Hopf bifurcations.Combined with simulations,we explore the influence of complex networks' structure on Hopf bifurcation,and find that the reduction of the largest non-zero eigenvalue of the Laplacian matrix of the network can significantly reduce the number of its spatially heterogeneous Hopf bifurcation points.(4)General bifurcation of the network-organized reaction-diffusion systems and its application.Further deepen the above two theoretical results,we strictly derive the method of directly calculating the general bifurcation normal forms of the network-organized reaction-diffusion systems,and so provide a more universal theory for studying their complex dynamic behaviors.We particularly derive another series of calculation formulas for the steady state bifurcation and Hopf bifurcation normal forms,and prove the equivalence of those two kinds of formulas.The obtained theoretical results are applied to a network-organized predator-prey reaction-diffusion system with Michaelis-Menten functional response function,and we conduct a strict analysis for Turing bifurcation,steady state bifurcation,Hopf and Turing-Hopf bifurcation of this system,and calculate the corresponding steady state and Hopf bifurcation normal forms.Combined with simulations,we dig out the effects of the structure of complex network on steady state bifurcation and Hopf bifurcation,and further cover the complexity of dynamic behaviors of the network-organized reaction-diffusion systems.