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.

• Kai Wang
• Yanling Zu

• Predator-prey model
• Mutual interference
• Delays
• Permanence
• Global attractivity

<p>[1] L.S. Chen. Mathematical Models and Methods in Ecology. Science Press,
Beijing, 1988 (in Chinese).
[2] M.P. Hassell. Mutual Interference between Searching Insect Parasites. J.
Animal Ecology 40 (1971) 473-486.
[3] M.P. Hassell. Density dependence in single-species population. J.Animal
Ecology 44 (1975) 283-295.
[4] 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.
[5] 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.
[6] L.H. Erbe, H.I. Freedman. Modeling persistence and mutual interference
among subpopulations of ecological communities. Bull. Math. Biol. 47
(2) (1985) 295-304.
[7] 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.
[8] K. Wang, Y.L. Zhu. Global attractivity of positive periodic solution for a
Volterra model. Appl. Math. Comput. 203 (2008) 493-501.
[9] 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.
[10] K. Wang. Permanence and global asymptotical stability of a predatorprey
model with mutual interference. Nonlinear Anal. Real World Appl.
12 (2011) 1062-1071.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] Y. Kuang. Delay Differential Equations with Applications in Population
Dynamics. Academic Press, New York, 1993.
[17] N. Macdonald. Biological Delay Systems: Linear Stability Theory.
Cambridge University Press, Cambridge, 1989.
[18] K. Gopalsamy. Stability and Oscillations in Delay Differential Equations
of Population Dynamics. Kluwer Academic Publisher, Boston, 1992.
[19] 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.</p>