The qualitative properties of a predatorprey system with Holling-（n ＋ 1） functional response and a fairly general growth rate are completely investigated. The necessary and sufficient condition to guarantee the uniqueness of limit cycles is given. Our work extends the previous relevant results in the reference.
Bifurcations of a degenerate homoclinic orbit with orbit flip in high dimensional system are studied. By establishing a local coordinate system and a Poincare map near the homoclinic orbit, the existence and uniqueness of 1-homoclinic orbit and 1-periodic orbit are given. Also considered is the existence of 2-homoclinic orbit and 2-periodic orbit. In additon, the corresponding bifurcation surfaces are given.
By using the continuation theorem of coincidence degree theory, the sufficient conditions to guarantee the existence of positive periodic solutions are established for nonautonomous predator-prey systems with discrete and continuously distributed delays.
A class of Beddington-DeAngelis＇ type predator-prey dynamic system with prey and predator both having linear density restriction is considered. By using the qualitative methods of ODE, the existence and uniqueness of positive equilibrium and its global asymptotic stability are analyzed. The direct criterions for local stability of positive equilibrium and existence of limit cycle are also established when inference parameter of predator is small.
Xiao Haibin Department of Mathematics, Ningbo University, Zhejiang 315211,China
In this paper we study the singularity at the origin with three–fold zeroeigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C~∞–equivalent toy(partial deriv)/(partial deriv) + z(partial deriv)/(partial deriv)y + ax^2y (partialderiv)/(partial deriv)/z with a ≠ 0. We first obtain several subfamilies of the symmetric versalunfoldings of this singularity by using the normal form and blow–up methods under some conditions,and derive the local and global bifurcation behavior, then prove analytically the existence of theSilnikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of thissingularity, by using the generalized Melnikov methods of a homoclinic orbit to a hyperbolic ornon–hyperbolic equilibrium in a highdimensional space.
