Qualitative structures of a degenerate fixed point of a Ricker competition model

Jun, Zhang; Zhong, Jiyu

doi:10.1080/10236198.2019.1581181

Cited by 6 publications

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References 12 publications

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“…In the cases that (k, b) lies at one of the points (1,4), (2,1) and (3,1), the qualitative properties of E are not given because the proofs involve computing the normal forms of some strong resonances, the Takens theorem, which shows that the analysis of dynamics for a map can be reduced to the discussion of a differential equation, and the blowing-up techniques. The specific processes can be referred to the references [24,26]. Furthermore, the system may produce 1:2 resonance, 1:4 resonance and 1:3 resonance respectively.…”

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Qualitative properties and bifurcations of a leaf-eating herbivores model

Zhong¹

2021

DCDS-B

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In this paper, we discuss the dynamics of a discrete-time leafeating herbivores model. First of all, to investigate the bifurcations of the model, we study the qualitative properties of a fixed point, including hyperbolic and non-hyperbolic. Secondly, applying the center manifold theorem, we give the conditions that the model produces a supercritical flip bifurcation and a subcritical flip bifurcation respectively, from which we find a generalized flip bifurcation point. And then, we prove rigorously that the model undergoes a generalized flip bifurcation and give three parameter regions that the model possesses two period-two cycles, one period-two cycles and none respectively. Next, computing the normal form, we prove that the model undergoes a subcritical Neimark-Sacker bifurcation and produces a unique unstable invariant circle near the fixed point. Finally, by numerical simulations, we not only verify our results but also show a saddle period-five cycle and a saddle period-six cycle on the invariant circle.

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