Lienard Equation Infinite Boundary Value Problems Existence And Rm Prey Ecological Model

The dissertation consists of three chapters. Chapter one is the summary.In chapter two, the existence of monotone and non-monotone solutions of boundary value problem on the real line for Lienard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector fields defined by Lienard system and the system given by symmetric transformation or quasi-symmetric transformation, the invariant regions of the system are constructed. The existence of connecting orbits can be proved. A lot of sufficient conditions to guarantee the existence of solutions of the boundary value problem are obtained.The key words in the chapter: reaction-diffusion equation; Lienard system; travelling wave solutions; connecting orbits.In chapter three, R.osenzweing-MacArthur predator-prey biology models is studied. At first, by taking Jfc, the environmental carrying capacity of the prey, as the bifurcation parameter, the existence of the small amplitude stable limit cycle created by Hopf bifurcation is obtained. Then by using qualitative analysis methods, it follows that the existence and uniqueness of non-small amplitude stable limit cycle. The global stability of the positive equilibrium in the first quadrant is also discussed. Some relevant results in some references are generalized.