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Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response
1. | Department of Mathematics and Information, Henan University of, Economics and Law, Zhengzhou, China |
2. | College of Science and Engineering, Aoyama Gakuin University, Sagamihara, 2525258 |
References:
[1] |
R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326.
doi: 10.1016/S0022-5193(89)80211-5. |
[2] |
D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: Stability Theory and Applications, Ellis Horwood Limited, Chichester, 1989. |
[3] |
D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, New York, 1993. |
[4] |
J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.
doi: 10.2307/3866. |
[5] |
L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann., 71 (1911), 97-115. |
[6] |
R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.
doi: 10.1006/jmaa.2000.7343. |
[7] |
R. S. Cantrell, C. Cosner and S. G. Ruan, Intraspecific interference and consumer-resource dynamics, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 527-546.
doi: 10.3934/dcdsb.2004.4.527. |
[8] |
F. Chen, Y. Chen and J. Shi, Stability of the boundary solution of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 344 (2008), 1057-1067.
doi: 10.1016/j.jmaa.2008.03.050. |
[9] |
P. H. Crowley and E. K. Martin, Functional response and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 8 (1989), 211-221.
doi: 10.2307/1467324. |
[10] |
J. Cui and Y. Takeuchi, Permanence, extinction and periodic solution of predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 317 (2006), 464-474.
doi: 10.1016/j.jmaa.2005.10.011. |
[11] |
D. L. DeAngelis, R. A. Goldstein and R. V. Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. |
[12] |
D. T. Dimitrov and H. V. Kojouharov, Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response, Appl. math. Comput., 162 (2005), 523-538.
doi: 10.1016/j.amc.2003.12.106. |
[13] |
M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15-39.
doi: 10.1016/j.jmaa.2004.02.038. |
[14] |
R. E. Gaines and R. M. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977. |
[15] |
M. P. Hassell and C. C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature, 223 (1969), 1133-1137.
doi: 10.1038/2231133a0. |
[16] |
C. S. Holling, The components of predation as revealed by a study of small mammal predation of European pine sawfly, Canad. Entomologist, 91 (1959), 291-292.
doi: 10.4039/Ent91293-5. |
[17] |
C. S. Holling, Some characteristics of simple types of predation and parasitism, Canad. Entomologist, 91 (1959), 382-384.
doi: 10.4039/Ent91385-7. |
[18] |
T. W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401.
doi: 10.1016/S0022-247X(02)00395-5. |
[19] |
T. W. Hwang, Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 290 (2004), 113-122.
doi: 10.1016/j.jmaa.2003.09.073. |
[20] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, New York, 2011.
doi: 10.1090/surv/176. |
[21] |
P. Kratina, M. Vos, A. Bateman and B. R. Anholt, Functional response modified by predator density, Oecologia, 159 (2009), 425-423.
doi: 10.1007/s00442-008-1225-5. |
[22] |
Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.
doi: 10.1007/s002850050105. |
[23] |
H. Li and Y. Takeuchi, Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 374 (2011), 644-654.
doi: 10.1016/j.jmaa.2010.08.029. |
[24] |
S. Liu and E. Beretta, A stage-structured predator-prey model of Beddington-DeAngelis type, SIAM J. Appl. Math., 66 (2006), 1101-1129.
doi: 10.1137/050630003. |
[25] |
Z. Liu and R. Yuan, Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 296 (2004), 521-537.
doi: 10.1016/j.jmaa.2004.04.051. |
[26] |
Z. P. Qiu, J. Yu and Y. Zou, The asymptotic behavior of a chemostat model with the Beddington-DeAngelis functional response, Math. Biosci., 187 (2004), 175-187.
doi: 10.1016/j.mbs.2003.10.001. |
show all references
References:
[1] |
R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326.
doi: 10.1016/S0022-5193(89)80211-5. |
[2] |
D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: Stability Theory and Applications, Ellis Horwood Limited, Chichester, 1989. |
[3] |
D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, New York, 1993. |
[4] |
J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.
doi: 10.2307/3866. |
[5] |
L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann., 71 (1911), 97-115. |
[6] |
R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.
doi: 10.1006/jmaa.2000.7343. |
[7] |
R. S. Cantrell, C. Cosner and S. G. Ruan, Intraspecific interference and consumer-resource dynamics, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 527-546.
doi: 10.3934/dcdsb.2004.4.527. |
[8] |
F. Chen, Y. Chen and J. Shi, Stability of the boundary solution of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 344 (2008), 1057-1067.
doi: 10.1016/j.jmaa.2008.03.050. |
[9] |
P. H. Crowley and E. K. Martin, Functional response and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 8 (1989), 211-221.
doi: 10.2307/1467324. |
[10] |
J. Cui and Y. Takeuchi, Permanence, extinction and periodic solution of predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 317 (2006), 464-474.
doi: 10.1016/j.jmaa.2005.10.011. |
[11] |
D. L. DeAngelis, R. A. Goldstein and R. V. Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. |
[12] |
D. T. Dimitrov and H. V. Kojouharov, Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response, Appl. math. Comput., 162 (2005), 523-538.
doi: 10.1016/j.amc.2003.12.106. |
[13] |
M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 295 (2004), 15-39.
doi: 10.1016/j.jmaa.2004.02.038. |
[14] |
R. E. Gaines and R. M. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977. |
[15] |
M. P. Hassell and C. C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature, 223 (1969), 1133-1137.
doi: 10.1038/2231133a0. |
[16] |
C. S. Holling, The components of predation as revealed by a study of small mammal predation of European pine sawfly, Canad. Entomologist, 91 (1959), 291-292.
doi: 10.4039/Ent91293-5. |
[17] |
C. S. Holling, Some characteristics of simple types of predation and parasitism, Canad. Entomologist, 91 (1959), 382-384.
doi: 10.4039/Ent91385-7. |
[18] |
T. W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401.
doi: 10.1016/S0022-247X(02)00395-5. |
[19] |
T. W. Hwang, Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 290 (2004), 113-122.
doi: 10.1016/j.jmaa.2003.09.073. |
[20] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, New York, 2011.
doi: 10.1090/surv/176. |
[21] |
P. Kratina, M. Vos, A. Bateman and B. R. Anholt, Functional response modified by predator density, Oecologia, 159 (2009), 425-423.
doi: 10.1007/s00442-008-1225-5. |
[22] |
Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.
doi: 10.1007/s002850050105. |
[23] |
H. Li and Y. Takeuchi, Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 374 (2011), 644-654.
doi: 10.1016/j.jmaa.2010.08.029. |
[24] |
S. Liu and E. Beretta, A stage-structured predator-prey model of Beddington-DeAngelis type, SIAM J. Appl. Math., 66 (2006), 1101-1129.
doi: 10.1137/050630003. |
[25] |
Z. Liu and R. Yuan, Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 296 (2004), 521-537.
doi: 10.1016/j.jmaa.2004.04.051. |
[26] |
Z. P. Qiu, J. Yu and Y. Zou, The asymptotic behavior of a chemostat model with the Beddington-DeAngelis functional response, Math. Biosci., 187 (2004), 175-187.
doi: 10.1016/j.mbs.2003.10.001. |
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