Bifurcation Analysis Of The Holling-? Type Model With Diffusion

Bifurcation theory plays a very important role in nonlinear science and many application fields.It has been a central topic in the field of dynamical systems and differential equation.In general,the classical bifurcation theory consists of static bifurcation and dynamic bifurcation,in which the dynamic bifurcation focuses on the mutation at the balance point.For example,to produce periodic rails,homecords,and other or invariant sets.The most common of dynamic bifurcation is the Hopf bifurcation.The bifurcation theory of autonomous system has been fully developed,and it has been widely used in real world,such as the predation model and population growth model established by the cross spread of population.In this paper,we choose the Holling-? type of the diffusion system,which is a bifurcation argument with beta.Including the Turing bifurcation and Hopf bifurcation,the main discussion is the Hopf bifurcation.Firstly,we can solve the static equation and get the constant steady-state solution of this model;And then we linearize the original system in constant state solution,and find the eigenvalues and eigenvectors of the linear system;Secondly,verify that the desired eigenvalue satisfies the condition of the Hopf bifurcation.When Hopf bifurcation occurs,the system at the critical point has a pair of conjugate pure virtual characteristic value,while other eigenvalue real part is not zero,then there is the center manifold,next according to the theory of type specification we calculate center manifold,the original system is reduced to the center manifold,so as to convert an infinite dimensional differential equation to finite dimensional differential equation,finally calculated for determining when the subcritical and supercritical parameter conditions.