Search

This paper, as a complement to the works by Hsu et al [SIAM. J. Appl. Math. 55 (1995)] and Huang et al [J. Differential Equations 257 (2014)], aims to examine the Hopf bifurcation and global dynamics of a predator–prey system of Leslie type with generalized Holling type III functional response for the two cases: (A) system has a unique anti-saddle positive equilibrium, which is not semi-hyperbolic or nilpotent; (B) system has three distinct positive equilibria. For each case, the type and stability of each equilibrium, Hopf bifurcation at each weak focus, and the number and distribution of limit cycles in the first quadrant are studied. It is shown that every equilibrium is not a center for all parameter values. For the case (A), i limit cycle(s) can appear near the unique positive equilibrium, i = 1 , ... , 4. For i = 3 or 4, system has two stable limit cycles, which gives a positive answer to the open problem proposed by Coleman [Differential equations model,1983]: finding at least two ecologically stable cycles in any natural (or laboratory) predator–prey or other interacting system. For the case (B), one positive equilibrium is a saddle and the others are both anti-saddle. If one of the two anti-saddles is a weak focus and the other is not, then the order of the weak focus is at most 3. If both anti-saddles are weak foci, then they are unstable weak foci of order one. Moreover, one limit cycle can bifurcate from each of them simultaneously. Numerical simulations demonstrate that there is also a big stable limit cycle enclosing these two limit cycles. Our results indicate that the maximum number of limit cycles in the model of this kind is at least 4, which improves the preceding results that this number is at least 2. [ABSTRACT FROM AUTHOR]

Litvak-Hinenzon et al. developed the phenomenological model to simulate the horizontal motion of particles in the atmosphere, and a series of their papers (see e.g., Litvak-Hinenzon et al. (Phys. D 164:213-250, 2002; Nonlinearity 15:1149-1177, 2002; SIAM J. Appl. Dyn. Syst. 3:525-573, 2004)) focused on studying the fate of parabolic resonance lower dimensional tori (as part of a quasi-periodic Hamiltonian pitchfork bifurcation (HPB) scenario in the unperturbed phenomenological model) under perturbations. However, the structural stability of quasi-periodic HPB involved therein has not been fully exposed theoretically (Litvak-Hinenzon et al. have only given some numerical explanations). Based on BCKV singularity theory established by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993), we consider a more general quasi-periodic HPB triggered by the Z 2 -invariant universal unfolding N b c k v = a y 2 2 − (λ + b I 1) x 2 2 + c x 4 4 with respect to Z 2 -equivariant BCKV-restricted morphisms of the planar singularity a 2 y 2 + c 4 x 4 (the coefficients a , b , c ≠ 0 , the I 1 is regarded as distinguished parameter with respect to the external parameter λ). We prove a KAM (Kolmogorov–Arnold–Moser) theorem concerning parabolic tori in such quasi-periodic HPB, by which Diophantine parabolic tori (and the whole corresponding Diophantine HPB scenario) survive non-integrable and Z 2 -invariant Hamiltonian perturbations, parametrized by pertinent large Cantor sets. In the context of Z 2 -symmetry, our results can be seen as rigorous proof of the structural stability problem of bifurcations of Floquet-tori triggered by the universal unfolding N b c k v with distinguished parameters which is proposed by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993). Ultimately, we similarly obtain the structural stability result of quasi-periodic HPB in the phenomenological model mentioned above, which can be utilized as a starting point for a deeper understanding of the various resonances and chaotic dynamics (in the gaps of the Cantor sets), just as Litvak-Hinenzon et al. did for normally parabolic tori undergoing a HPB. • We prove a KAM theorem in a quasi-periodic Hamiltonian pitchfork bifurcation. • We obtain the structural stability of a bifurcation in a phenomenological model. • We verify some of the previous numerical phenomena of Litvak-Hinenzon et al. [ABSTRACT FROM AUTHOR]