Permanence and Exponential Stability of a Predator-prey Model with HV-Holling Functional Response
In this paper, a delayed predator-prey system with Hassell-Varley-Holling type functional response is studied. A sufficient criterion for the permanence of the system is presented, and further some sufficient conditions for the global attractivity and exponential stability of the system are established. And an example is to show the feasibility of the results by simulation.
[1] A. Lotka, The Elements of Physical Biology, Williams and Wilkins,
Baltimore, 1925.
[2] V. Volterra, Variazioni e fluttuazioni del numero di individui in specie
animali conviventi, Mem. Accd. Lincei. 2 (1926) 31-113.
[3] Y. Kuang, Delay Differential Equations with Applications in Population
Dynamics, Academic Press, New York, 1993.
[4] C. Chen and 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.
[5] X.Z. He, Stability and delays in a predator-prey system. J. Math. Anal.
Appl. 198 (1996) 355-370.
[6] F.D. Chen, Permanence and global stability of nonautonomous Lotka-
Volterra system with predator-prey and deviating arguments. Appl. Math.
Comput. 173 (2006) 1082-1100.
[7] M. Fan, Global existence of positive periodic solution of predator-prey
system with deviating arguments. Acta Math. Appl. Sin. 23 (2000) 557-
561.
[8] J.D. Zhao and W.C. Chen, Global asymptotic stability of a periodic
ecological model. Appl. Math. Comput. 147 (2004) 881-892.
[9] F.D. Chen, The permanence and global attractivity of Lotka-Volterra
competition system with feedback controls. Nonlinear Anal. Real World
Appl. 7 (2006) 133-143.
[10] Y.H. Xia, J.D. Cao and S.S. Cheng, Periodic solutions for a Lotka-
Volterra mutualism system with several delays. Appl. Math. Modelling
31 (2007) 1960-1969.
[11] C. Cosner, D.L. DeAngelis, J.S. Ault and D.B. Olson, Effect of spatial
grouping on the functional response of predators, Theor. Pop. Biol. 56
(1999) 65-75.
[12] C.S. Holling, The components of predation as revealed by a study of
small mammal predation of the European pine sawfly, Canad. Entomological
Soc. 91 (1959) 293-320.
[13] C.S. Holling, Some characteristics of simple types of predation and
parasitism, Canad. Entomologist 91 (1959) 385-395.
[14] J.R. Beddington, Mutual interference between parasites or predators and
its effect on searching efficiency, J. Animal Ecology 44 (1975) 331-341.
[15] D.L. DeAngelis, Goldsten RA and Neill R, A model for trophic
interaction, Ecology 56 (1975) 881-892.
[16] P.H. Crowley and E.K. Martin, Functional response and interference
within and between year classes of a dragonfly population. J. North
American Benthological 8 (1989) 211-21.
[17] M.P. Hassell and G.C. Varley, New inductive population model for insect
parasites and its bearing on biological control, Nature 223 (1969) 1133-
1137.
[18] S.-B. Hsu, T.-W. Hwang and Y. Kuang, Global dynamics of a predatorCprey
model with Hassell-Varley type functional response, Discrete
Contin. Dyn. Syst. Ser. B 10 (2008) 857-871.
[19] K. Wang, Permanence and global asymptotic stability of a delayed
predator-prey model with Hassell-Varley type functional response, Bullet.
Iranian Math. Soci. 37 (2011) 203-220.
[20] D. Schenk, L. Bersier and S. Bacher, An experimental test of the nature
of predation: neither prey- nor ratio-dependent, J. Animal Ecology 74
(2005) 86-91.
[21] X.X. Liu and Y.J. Lou, Global dynamics of a predator-prey model, J.
Math. Anal. Appl. 371 (2010) 323-340.
[1] A. Lotka, The Elements of Physical Biology, Williams and Wilkins,
Baltimore, 1925.
[2] V. Volterra, Variazioni e fluttuazioni del numero di individui in specie
animali conviventi, Mem. Accd. Lincei. 2 (1926) 31-113.
[3] Y. Kuang, Delay Differential Equations with Applications in Population
Dynamics, Academic Press, New York, 1993.
[4] C. Chen and 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.
[5] X.Z. He, Stability and delays in a predator-prey system. J. Math. Anal.
Appl. 198 (1996) 355-370.
[6] F.D. Chen, Permanence and global stability of nonautonomous Lotka-
Volterra system with predator-prey and deviating arguments. Appl. Math.
Comput. 173 (2006) 1082-1100.
[7] M. Fan, Global existence of positive periodic solution of predator-prey
system with deviating arguments. Acta Math. Appl. Sin. 23 (2000) 557-
561.
[8] J.D. Zhao and W.C. Chen, Global asymptotic stability of a periodic
ecological model. Appl. Math. Comput. 147 (2004) 881-892.
[9] F.D. Chen, The permanence and global attractivity of Lotka-Volterra
competition system with feedback controls. Nonlinear Anal. Real World
Appl. 7 (2006) 133-143.
[10] Y.H. Xia, J.D. Cao and S.S. Cheng, Periodic solutions for a Lotka-
Volterra mutualism system with several delays. Appl. Math. Modelling
31 (2007) 1960-1969.
[11] C. Cosner, D.L. DeAngelis, J.S. Ault and D.B. Olson, Effect of spatial
grouping on the functional response of predators, Theor. Pop. Biol. 56
(1999) 65-75.
[12] C.S. Holling, The components of predation as revealed by a study of
small mammal predation of the European pine sawfly, Canad. Entomological
Soc. 91 (1959) 293-320.
[13] C.S. Holling, Some characteristics of simple types of predation and
parasitism, Canad. Entomologist 91 (1959) 385-395.
[14] J.R. Beddington, Mutual interference between parasites or predators and
its effect on searching efficiency, J. Animal Ecology 44 (1975) 331-341.
[15] D.L. DeAngelis, Goldsten RA and Neill R, A model for trophic
interaction, Ecology 56 (1975) 881-892.
[16] P.H. Crowley and E.K. Martin, Functional response and interference
within and between year classes of a dragonfly population. J. North
American Benthological 8 (1989) 211-21.
[17] M.P. Hassell and G.C. Varley, New inductive population model for insect
parasites and its bearing on biological control, Nature 223 (1969) 1133-
1137.
[18] S.-B. Hsu, T.-W. Hwang and Y. Kuang, Global dynamics of a predatorCprey
model with Hassell-Varley type functional response, Discrete
Contin. Dyn. Syst. Ser. B 10 (2008) 857-871.
[19] K. Wang, Permanence and global asymptotic stability of a delayed
predator-prey model with Hassell-Varley type functional response, Bullet.
Iranian Math. Soci. 37 (2011) 203-220.
[20] D. Schenk, L. Bersier and S. Bacher, An experimental test of the nature
of predation: neither prey- nor ratio-dependent, J. Animal Ecology 74
(2005) 86-91.
[21] X.X. Liu and Y.J. Lou, Global dynamics of a predator-prey model, J.
Math. Anal. Appl. 371 (2010) 323-340.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:52305", author = "Kai Wang and Yanling Zu", title = "Permanence and Exponential Stability of a Predator-prey Model with HV-Holling Functional Response", abstract = "In this paper, a delayed predator-prey system with Hassell-Varley-Holling type functional response is studied. A sufficient criterion for the permanence of the system is presented, and further some sufficient conditions for the global attractivity and exponential stability of the system are established. And an example is to show the feasibility of the results by simulation.
", keywords = "Predator-prey system, Hassell-Varley-Holling, delay, permanence, exponential stability.", volume = "5", number = "12", pages = "1851-5", }
