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Global analysis of Ivlev’s type predator-prey dynamic systems. (English) Zbl 1231.37054
Summary: Consider a class of Ivlev’s type predator-prey dynamic systems with prey and predator both having linear density restricts. By using the qualitative methods of ODE, the global stability of positive equilibrium and existence and uniqueness of non-small amplitude stable limit cycle are obtained. Especially under certain conditions, it shows that existence and uniqueness of non-small amplitude stable limit cycle is equivalent to the local un-stability of positive equilibrium and the local stability of positive equilibrium implies its global stability. That is to say, the global dynamic of the system is entirely determined by the local stability of the positive equilibrium.
MSC:
37N25Dynamical systems in biology
92B05General biology and biomathematics
References:
[1]Jitsuro Sugie. Two-parameter bifurcation in a predator-prey system of Ivelv type[J]. Journal of Mathematical Analysis and Application, 1998, 217(2):349–371. · Zbl 0894.34025 · doi:10.1006/jmaa.1997.5700
[2]Kooij R E, Zegeling A. A predator-prey model with Ivlev’s functional response[J]. Journal of Mathematical Analysis and Application, 1996, 198(2):473–489. · Zbl 0851.34030 · doi:10.1006/jmaa.1996.0093
[3]DeAngelis D L, Goldstein R A, O’Neill R V. A model for tropic interaction[J]. Ecology, 1975, 56:881–892. · doi:10.2307/1936298
[4]Tang Qiulin. A Predator-prey System with Ivlev’s Functional Response[J]. Journal of Beihua University(Natural Science), 2002, 3(5):381–384 (in Chinese).
[5]Feng Jianwen, Chen Shihua. Global asympotic behavior for the competing predators of the Ivlev types[J]. Mathematica Applicata, 2000, 13(4):85–88.