Global property in a delayed periodic predator-prey model with stage-structure in prey and density-independence in predator. (English) Zbl 1470.34224

Summary: We study the global property in a delayed periodic predator-prey model with stage-structure in prey and density-independence in predator. The sufficient conditions on the ultimate boundedness of all positive solutions are obtained, and the sufficient conditions of the integrable form for the permanence and extinction are further established, respectively. Some well-known results on the predator density-dependency are improved and extended to the predator density-independent cases. The theoretical results are confirmed by the special examples and the numerical simulations.


34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K25 Asymptotic theory of functional-differential equations
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[1] Sokol, W.; Howell, J. A., Kinetics of phenol oxidation by washed cells, Biotechnology and Bioengineering, 23, 9, 2039-2049 (1980) · doi:10.1002/bit.260230909
[2] Cui, J.; Song, X., Permanence of predator-prey system with stage structure, Discrete and Continuous Dynamical Systems B, 4, 3, 547-554 (2004) · Zbl 1100.92062 · doi:10.3934/dcdsb.2004.4.547
[3] Cui, J.; Takeuchi, Y., A predator-prey system with a stage structure for the prey, Mathematical and Computer Modelling, 44, 11-12, 1126-1132 (2006) · Zbl 1132.92340 · doi:10.1016/j.mcm.2006.04.001
[4] Cui, J.; Sun, Y., Permanence of predator-prey system with infinite delay, Electronic Journal of Differential Equations, 2004, 81, 1-12 (2004) · Zbl 1056.92050
[5] Teng, Z.; Chen, L., Permanence and extinction of periodic predator-prey systems in a patchy environment with delay, Nonlinear Analysis: Real World Applications, 4, 2, 335-364 (2003) · Zbl 1018.92033 · doi:10.1016/S1468-1218(02)00026-3
[6] Hale, J. K., Theory of Functional Differential Equations (1977), New York, NY, USA: Springer, New York, NY, USA · Zbl 0352.34001
[7] Hale, J. K.; Kato, J., Phase space for retarded equations with infinite delay, Funkcialaj Ekvacioj, 21, 1, 11-41 (1978) · Zbl 0383.34055
[8] Kuang, Y., Delay Differential Equation with Applications in Population Dynamics (1993), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0777.34002
[9] Cui, J.; Chen, L.; Wang, W., The effect of dispersal on population growth with stage-structure, Computers & Mathematics with Applications, 39, 1-2, 91-102 (2000) · Zbl 0968.92018 · doi:10.1016/S0898-1221(99)00316-8
[10] Zhao, X., Dynamical Systems in Population Biology (2003), New York, NY, USA: Springer, New York, NY, USA · Zbl 1023.37047
[11] Teng, Z.; Chen, L., The positive periodic solutions of periodic Kolmogorov type systems with delays, Acta Mathematicae Applicatae Sinica, 22, 3, 446-454 (1999) · Zbl 0976.34063
[12] Smith, H. L., Cooperative systems of differential equations with concave nonlinearities, Nonlinear Analysis: Theory, Methods & Applications, 10, 10, 1037-1052 (1986) · Zbl 0612.34035 · doi:10.1016/0362-546X(86)90087-8
[13] Zhao, X., The qualitative analysis of \(n\)-species Lotka-Volterra periodic competition systems, Mathematical and Computer Modelling, 15, 11, 3-8 (1991) · Zbl 0756.34048 · doi:10.1016/0895-7177(91)90100-L
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