Permanence and Global Attractivity of a Delayed Predator-Prey Model with Mutual Interference
By utilizing the comparison theorem and Lyapunov second method, some sufficient conditions for the permanence and global attractivity of positive periodic solution for a predator-prey model with mutual interference m ∈ (0, 1) and delays τi are obtained. It is the first time that such a model is considered with delays. The significant is that the results presented are related to the delays and the mutual interference constant m. Several examples are illustrated to verify the feasibility of the results by simulation in the last part.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1088188Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2050
 L.S. Chen. Mathematical Models and Methods in Ecology. Science Press, Beijing, 1988 (in Chinese).
 M.P. Hassell. Mutual Interference between Searching Insect Parasites. J. Animal Ecology 40 (1971) 473-486.
 M.P. Hassell. Density dependence in single-species population. J.Animal Ecology 44 (1975) 283-295.
 H.I. Freedman. Stability analysis of a predator-prey system with mutual interference and density-dependent death rates. Bull. Math. Biol. 41 (1) (1979) 4267-78.
 H.I. Freedman, V.S. Rao. The trade-off between mutual interference and time lags in predator-prey systems. Bull. Math. Biol. 45 (6) (1983) 991- 1004.
 L.H. Erbe, H.I. Freedman. Modeling persistence and mutual interference among subpopulations of ecological communities. Bull. Math. Biol. 47 (2) (1985) 295-304.
 L.H. Erbe, H.I. Freedman, V.S. Rao, Three-species food-chain models with mutual interference and time delays. Math. Biosci. 80 (1) (1986) 57-80.
 K. Wang, Y.L. Zhu. Global attractivity of positive periodic solution for a Volterra model. Appl. Math. Comput. 203 (2008) 493-501.
 K. Wang. Existence and global asymptotic stability of positive periodic solution for a Predator-Prey system with mutual interference. Nonlinear Anal. Real World Appl. 10 (2009) 2774-2783.
 K. Wang. Permanence and global asymptotical stability of a predatorprey model with mutual interference. Nonlinear Anal. Real World Appl. 12 (2011) 1062-1071.
 X. Lin, F.D. Chen. Almost periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response. Appl. Math. Comput. 214 (2009) 548-556.
 X.L. Wang, Z.J. Du, J. Liang. Existence and global attractivity of positive periodic solution to a Lotka-Volterra model. Nonlinear Anal. Real World Appl. 11 (2010) 4054-4061.
 L.J. Chen, L.J. Chen. Permanence of a discrete periodic volterra model with mutual interference. Discrete Dyn. Nat. Soc. (2009), doi:10.1155/2009/205481.
 R. Wu. Permanence of a discrete periodic Volterra model with mutual interference and Beddington-Deangelis functional response. Disc. Dyna. Natu. Soci. (2010), doi:10.1155/2010/246783.
 Permanence and almost periodic solution of a Lotka-Volterra model with mutual interference and time delays. Appl. Math. Modelling (2012), http://dx.doi.org/10.1016/j.apm.2012.03.022.
 Y. Kuang. Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York, 1993.
 N. Macdonald. Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge, 1989.
 K. Gopalsamy. Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publisher, Boston, 1992.
 C. Chen, F.D. Chen. Conditions for global attractivity of multispecies ecological competition-predator system with Holling III type functional response. J. Biomath. 19 (2004) 136-140.