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January  2014, 19(1): 173-187. doi: 10.3934/dcdsb.2014.19.173

Global stability of a predator-prey system with stage structure and mutual interference

1. 

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

2. 

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350116, China

Received  September 2012 Revised  August 2013 Published  December 2013

In this paper, we consider a predator-prey system with stage structure and mutual interference. By analyzing the characteristic equations, we study the local stability of the interior equilibrium of the system. Using an iterative method, we investigate the global stability of this equilibrium.
Citation: Zhong Li, Maoan Han, Fengde Chen. Global stability of a predator-prey system with stage structure and mutual interference. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 173-187. doi: 10.3934/dcdsb.2014.19.173
References:
[1]

S. Ahmad, On the nonautonomous Volterra-Lotka competition equations, Proc. Amer. Math. Soc., 117 (1993), 199-204. doi: 10.1090/S0002-9939-1993-1143013-3.

[2]

S. Ahmad, Extinction of species in nonautonomous Lotka-Volterra systems, Proc. Amer. Math. Soc., 127 (1999), 2905-2910. doi: 10.1090/S0002-9939-99-05083-2.

[3]

S. Ahmad and A. C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Nonlinear Anal., 40 (2000), 37-49. doi: 10.1016/S0362-546X(00)85003-8.

[4]

W. G. Aiello and H. I. Freedman, A time delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.

[5]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086.

[6]

F. D. Chen, Z. Li and X. D. Xie, Permanence of a nonlinear integro-differential prey-competition model with infinite delays, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 2290-2297. doi: 10.1016/j.cnsns.2007.05.022.

[7]

F. D. Chen, X. D. Xie and J. L. Shi, Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays, J. Comput. Appl. Math., 194 (2006), 368-387. doi: 10.1016/j.cam.2005.08.005.

[8]

Z. J. Du and Y. S. Lv, Permanence and Almost Periodic Solution of a Lotka-Volterra Model with mutual interference and time delays, Appl. Math. Model., 3 (2013), 1054-1068. doi: 10.1016/j.apm.2012.03.022.

[9]

H. I. Freedman, Stability analysis of a predator-prey system with mutual interference and density-dependent death rates, Bull. Math. Biol., 41 (1979), 67-78. doi: 10.1016/S0092-8240(79)80054-3.

[10]

H. I. Freedman and V. S. H. Rao, The trade-off between mutual interference and time lags in predator-prey system, Bull. Math. Biol., 45 (1983), 991-1004. doi: 10.1016/S0092-8240(83)80073-1.

[11]

H. J. Guo and X. X. Chen, Existence and global attractivity of positive periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response, Appl. Math. Comput., 217 (2011), 5830-5837. doi: 10.1016/j.amc.2010.12.065.

[12]

G. H. Guo and J. H. Wu, The effect of mutual interference between predators on a predator-prey model with diffusion, J. Math. Anal. Appl., 389 (2012), 179-194. doi: 10.1016/j.jmaa.2011.11.044.

[13]

M. P. Hassell, Density dependence in single-species population, J. Anim. Ecol., 44 (1975), 283-295. doi: 10.2307/3863.

[14]

S. B. Hsu, T. W. Hwang and Y. Kuang, Global dynamics of a predator-prey model with Hassell-Varley type functional response, Disc. Cont. Dyn. Sys. B, 10 (2008), 857-871. doi: 10.3934/dcdsb.2008.10.857.

[15]

H. J. Hu and L. H. Huang, Stability and Hopf bifurcation in a delayed predator-prey system with stage structure for prey, Nonlinear Analysis RWA, 11 (2010), 2757-2769. doi: 10.1016/j.nonrwa.2009.10.001.

[16]

Z. Li, F. D. Chen and M. X. He, Permanence and global attractivity of a periodic predator-prey system with mutual interference and impulses, Commun Nonlinear Sci Numer Simulat, 17 (2012), 444-453. doi: 10.1016/j.cnsns.2011.05.026.

[17]

X. Lin and 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. doi: 10.1016/j.amc.2009.04.028.

[18]

S. Q. Liu, L. S. Chen, G. L. Luo and Y. L. Jiang, Asymptotic behaviors of competitive Lotka-Volterra system with stage structure, J. Math. Anal. Appl., 271 (2002), 124-138. doi: 10.1016/S0022-247X(02)00103-8.

[19]

S. Q. Liu, L. S. Chen and Z. J. Liu, Extinction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl., 274 (2002), 667-684. doi: 10.1016/S0022-247X(02)00329-3.

[20]

Y. S. Lv and Z. J. Du, Existence and global attractivity of a positive periodic solution to a Lotka-Volterra model with mutual interference and Holling III type functional response, Nonlinear Analysis RWA, 12 (2011), 3654-3664. doi: 10.1016/j.nonrwa.2011.06.022.

[21]

F. Montes De Oca and M. L. Zeeman, Extinction in nonautonomous competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 124 (1996), 3677-3687. doi: 10.1090/S0002-9939-96-03355-2.

[22]

F. Montes De Oca and L. Pérez, Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay, Nonlinear Anal., 75 (2012), 758-768. doi: 10.1016/j.na.2011.09.009.

[23]

S. G. Ruan and H. I. Freedman, Persistence in three-species food chain models with group defense, Math. Biosci., 107 (1991), 111-125. doi: 10.1016/0025-5564(91)90074-S.

[24]

Z. D. Teng, Uniform persistence of the periodic predator-prey Lotka-Volterra systems, Appl. Anal., 72 (1998), 339-352. doi: 10.1080/00036819908840745.

[25]

K. Wang, Permanence and global asymptotical stability of a predator-prey model with mutual interference, Nonlinear Analysis RWA, 12 (2011), 1062-1071. doi: 10.1016/j.nonrwa.2010.08.028.

[26]

K. Wang, Existence and global asymptotic stability of positive periodic solution for a predator-prey system with mutual interference, Nonlinear Analysis RWA, 10 (2009), 2774-2783. doi: 10.1016/j.nonrwa.2008.08.015.

[27]

X. L. Wang, Z. J. Du and J. Liang, Existence and global attractivity of positive periodic solution to a Lotka-Volterra model, Nonlinear Analysis RWA, 11 (2010), 4054-4061. doi: 10.1016/j.nonrwa.2010.03.011.

[28]

K. Wang and Y. L. Zhu, Global attractivity of positive periodic solution for a Volterra model, Appl. Math. Comput., 203 (2008), 493-501. doi: 10.1016/j.amc.2008.04.005.

[29]

R. Xu, Global dynamics of a predator-prey model with time delay and stage structure for the prey, Nonlinear Analysis RWA, 12 (2011), 2151-2162. doi: 10.1016/j.nonrwa.2010.12.029.

[30]

R. Xu, M. A. J. Chaplain and F. A. Davidson, Global stability of a Lotka-Volterra type predator-prey model with stage structure and time delay, Appl. Math. Comput., 159 (2004), 863-880. doi: 10.1016/j.amc.2003.11.008.

[31]

G. H. Zhu, X. Z. Meng and L. S. Chen, The dynamics of a mutual interference age structured predator-prey model with time delay and impulsive perturbations on predators, Appl. Math. Comput., 216 (2010), 308-316. doi: 10.1016/j.amc.2010.01.064.

show all references

References:
[1]

S. Ahmad, On the nonautonomous Volterra-Lotka competition equations, Proc. Amer. Math. Soc., 117 (1993), 199-204. doi: 10.1090/S0002-9939-1993-1143013-3.

[2]

S. Ahmad, Extinction of species in nonautonomous Lotka-Volterra systems, Proc. Amer. Math. Soc., 127 (1999), 2905-2910. doi: 10.1090/S0002-9939-99-05083-2.

[3]

S. Ahmad and A. C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Nonlinear Anal., 40 (2000), 37-49. doi: 10.1016/S0362-546X(00)85003-8.

[4]

W. G. Aiello and H. I. Freedman, A time delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.

[5]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086.

[6]

F. D. Chen, Z. Li and X. D. Xie, Permanence of a nonlinear integro-differential prey-competition model with infinite delays, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 2290-2297. doi: 10.1016/j.cnsns.2007.05.022.

[7]

F. D. Chen, X. D. Xie and J. L. Shi, Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays, J. Comput. Appl. Math., 194 (2006), 368-387. doi: 10.1016/j.cam.2005.08.005.

[8]

Z. J. Du and Y. S. Lv, Permanence and Almost Periodic Solution of a Lotka-Volterra Model with mutual interference and time delays, Appl. Math. Model., 3 (2013), 1054-1068. doi: 10.1016/j.apm.2012.03.022.

[9]

H. I. Freedman, Stability analysis of a predator-prey system with mutual interference and density-dependent death rates, Bull. Math. Biol., 41 (1979), 67-78. doi: 10.1016/S0092-8240(79)80054-3.

[10]

H. I. Freedman and V. S. H. Rao, The trade-off between mutual interference and time lags in predator-prey system, Bull. Math. Biol., 45 (1983), 991-1004. doi: 10.1016/S0092-8240(83)80073-1.

[11]

H. J. Guo and X. X. Chen, Existence and global attractivity of positive periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response, Appl. Math. Comput., 217 (2011), 5830-5837. doi: 10.1016/j.amc.2010.12.065.

[12]

G. H. Guo and J. H. Wu, The effect of mutual interference between predators on a predator-prey model with diffusion, J. Math. Anal. Appl., 389 (2012), 179-194. doi: 10.1016/j.jmaa.2011.11.044.

[13]

M. P. Hassell, Density dependence in single-species population, J. Anim. Ecol., 44 (1975), 283-295. doi: 10.2307/3863.

[14]

S. B. Hsu, T. W. Hwang and Y. Kuang, Global dynamics of a predator-prey model with Hassell-Varley type functional response, Disc. Cont. Dyn. Sys. B, 10 (2008), 857-871. doi: 10.3934/dcdsb.2008.10.857.

[15]

H. J. Hu and L. H. Huang, Stability and Hopf bifurcation in a delayed predator-prey system with stage structure for prey, Nonlinear Analysis RWA, 11 (2010), 2757-2769. doi: 10.1016/j.nonrwa.2009.10.001.

[16]

Z. Li, F. D. Chen and M. X. He, Permanence and global attractivity of a periodic predator-prey system with mutual interference and impulses, Commun Nonlinear Sci Numer Simulat, 17 (2012), 444-453. doi: 10.1016/j.cnsns.2011.05.026.

[17]

X. Lin and 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. doi: 10.1016/j.amc.2009.04.028.

[18]

S. Q. Liu, L. S. Chen, G. L. Luo and Y. L. Jiang, Asymptotic behaviors of competitive Lotka-Volterra system with stage structure, J. Math. Anal. Appl., 271 (2002), 124-138. doi: 10.1016/S0022-247X(02)00103-8.

[19]

S. Q. Liu, L. S. Chen and Z. J. Liu, Extinction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl., 274 (2002), 667-684. doi: 10.1016/S0022-247X(02)00329-3.

[20]

Y. S. Lv and Z. J. Du, Existence and global attractivity of a positive periodic solution to a Lotka-Volterra model with mutual interference and Holling III type functional response, Nonlinear Analysis RWA, 12 (2011), 3654-3664. doi: 10.1016/j.nonrwa.2011.06.022.

[21]

F. Montes De Oca and M. L. Zeeman, Extinction in nonautonomous competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 124 (1996), 3677-3687. doi: 10.1090/S0002-9939-96-03355-2.

[22]

F. Montes De Oca and L. Pérez, Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay, Nonlinear Anal., 75 (2012), 758-768. doi: 10.1016/j.na.2011.09.009.

[23]

S. G. Ruan and H. I. Freedman, Persistence in three-species food chain models with group defense, Math. Biosci., 107 (1991), 111-125. doi: 10.1016/0025-5564(91)90074-S.

[24]

Z. D. Teng, Uniform persistence of the periodic predator-prey Lotka-Volterra systems, Appl. Anal., 72 (1998), 339-352. doi: 10.1080/00036819908840745.

[25]

K. Wang, Permanence and global asymptotical stability of a predator-prey model with mutual interference, Nonlinear Analysis RWA, 12 (2011), 1062-1071. doi: 10.1016/j.nonrwa.2010.08.028.

[26]

K. Wang, Existence and global asymptotic stability of positive periodic solution for a predator-prey system with mutual interference, Nonlinear Analysis RWA, 10 (2009), 2774-2783. doi: 10.1016/j.nonrwa.2008.08.015.

[27]

X. L. Wang, Z. J. Du and J. Liang, Existence and global attractivity of positive periodic solution to a Lotka-Volterra model, Nonlinear Analysis RWA, 11 (2010), 4054-4061. doi: 10.1016/j.nonrwa.2010.03.011.

[28]

K. Wang and Y. L. Zhu, Global attractivity of positive periodic solution for a Volterra model, Appl. Math. Comput., 203 (2008), 493-501. doi: 10.1016/j.amc.2008.04.005.

[29]

R. Xu, Global dynamics of a predator-prey model with time delay and stage structure for the prey, Nonlinear Analysis RWA, 12 (2011), 2151-2162. doi: 10.1016/j.nonrwa.2010.12.029.

[30]

R. Xu, M. A. J. Chaplain and F. A. Davidson, Global stability of a Lotka-Volterra type predator-prey model with stage structure and time delay, Appl. Math. Comput., 159 (2004), 863-880. doi: 10.1016/j.amc.2003.11.008.

[31]

G. H. Zhu, X. Z. Meng and L. S. Chen, The dynamics of a mutual interference age structured predator-prey model with time delay and impulsive perturbations on predators, Appl. Math. Comput., 216 (2010), 308-316. doi: 10.1016/j.amc.2010.01.064.

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