Global analysis in Bazykin's model with Holling II functional response and predator competition

By Min Lu, Jicai Huang. 2021

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Abstract 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 ≥ 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. [2] , [3] , [4] , [5] , Hainzl [22] , [23] , Kuznetsov [30] .

Subfields: Ecology, Genetics, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/efc1c9430efb0e35ad5704945a83d001854fac79