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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 & 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.  doi: 10.1090/S0002-9939-1993-1143013-3.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  doi: 10.1137/S0036141000376086.  Google Scholar

[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.  doi: 10.1016/j.cnsns.2007.05.022.  Google Scholar

[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.  doi: 10.1016/j.cam.2005.08.005.  Google Scholar

[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.  doi: 10.1016/j.apm.2012.03.022.  Google Scholar

[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.  doi: 10.1016/S0092-8240(79)80054-3.  Google Scholar

[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.  doi: 10.1016/S0092-8240(83)80073-1.  Google Scholar

[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.  doi: 10.1016/j.amc.2010.12.065.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2011.11.044.  Google Scholar

[13]

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

[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.  doi: 10.3934/dcdsb.2008.10.857.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2009.10.001.  Google Scholar

[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.  doi: 10.1016/j.cnsns.2011.05.026.  Google Scholar

[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.  doi: 10.1016/j.amc.2009.04.028.  Google Scholar

[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.  doi: 10.1016/S0022-247X(02)00103-8.  Google Scholar

[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.  doi: 10.1016/S0022-247X(02)00329-3.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2011.06.022.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.  doi: 10.1016/j.nonrwa.2008.08.015.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2010.03.011.  Google Scholar

[28]

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

[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.  doi: 10.1016/j.nonrwa.2010.12.029.  Google Scholar

[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.  doi: 10.1016/j.amc.2003.11.008.  Google Scholar

[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.  doi: 10.1016/j.amc.2010.01.064.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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.  doi: 10.1137/S0036141000376086.  Google Scholar

[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.  doi: 10.1016/j.cnsns.2007.05.022.  Google Scholar

[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.  doi: 10.1016/j.cam.2005.08.005.  Google Scholar

[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.  doi: 10.1016/j.apm.2012.03.022.  Google Scholar

[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.  doi: 10.1016/S0092-8240(79)80054-3.  Google Scholar

[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.  doi: 10.1016/S0092-8240(83)80073-1.  Google Scholar

[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.  doi: 10.1016/j.amc.2010.12.065.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2011.11.044.  Google Scholar

[13]

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

[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.  doi: 10.3934/dcdsb.2008.10.857.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2009.10.001.  Google Scholar

[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.  doi: 10.1016/j.cnsns.2011.05.026.  Google Scholar

[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.  doi: 10.1016/j.amc.2009.04.028.  Google Scholar

[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.  doi: 10.1016/S0022-247X(02)00103-8.  Google Scholar

[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.  doi: 10.1016/S0022-247X(02)00329-3.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2011.06.022.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.  doi: 10.1016/j.nonrwa.2008.08.015.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2010.03.011.  Google Scholar

[28]

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

[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.  doi: 10.1016/j.nonrwa.2010.12.029.  Google Scholar

[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.  doi: 10.1016/j.amc.2003.11.008.  Google Scholar

[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.  doi: 10.1016/j.amc.2010.01.064.  Google Scholar

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