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Global analysis in Bazykin’s model with Holling II functional response and predator competition. (English) Zbl 1464.34070
Summary: In this paper, we study the well-known Bazykin’s model with Holling II functional response and predator competition. A detailed bifurcation analysis, depending on all four parameters, reveals a rich bifurcation structure, including supercritical and subcritical Bogdanov-Takens bifurcation, degenerate Hopf bifurcation of codimension at most 2, and a focus type degenerate Bogdanov-Takens bifurcation of codimension 3, originating from a nilpotent focus of codimension 3 which acts as the organizing center for the bifurcation set. Moreover, some sufficient conditions to guarantee the global asymptotical stability of the semi-trivial equilibrium or the unique positive equilibrium are also given. Our analysis indicates that we can classify the long-time dynamics of the model with a threshold value \(c_0\) for the natural mortality rate \(c\) of predators, in detail, the following are true. (i) When \(c \geq c_0\), the prey will persist and predators will eventually go extinct for all positive initial populations. (ii) When \(c < c_0\), the prey and predators will coexist, for all positive initial populations, in the form of multiple positive equilibria or multiple periodic orbits. Our results can be seen as a complement to the work by Bazykin et al., Hainzl, Kuznetsov.
MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
34C23 Bifurcation theory for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
92D25 Population dynamics (general)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
Software:
AUTO2000; HomCont; Matlab
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