American Institute of Mathematical Sciences

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June  2015, 20(4): 1117-1134. doi: 10.3934/dcdsb.2015.20.1117

Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response

Haiyin Li 1, and  Yasuhiro Takeuchi 2,

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

Received  March 2014 Revised  June 2014 Published  February 2015

We investigate the dynamics of a non-autonomous and density dependent predator-prey system with Beddington-DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered. First, we derive a sufficient condition of permanence by comparison theorem, at the same time we propose a weaker condition ensuring some positive bounded set to be positive invariant. Next, we obtain two existence conditions for positive periodic solution by Brouwer fixed-point theorem and by continuation theorem, where the second condition is weaker than the first and gives the existence range of periodic solution. Further we show the global attractivity of the bounded positive solution by constructing Lyapunov function. Similarly, we have sufficient condition of global attractivity of boundary periodic solution.
Keywords: Density dependent predator, periodic solution., Beddington-DeAngelis functional response, permanence.
Mathematics Subject Classification: Primary: 34D23; Secondary: 92D2.
Citation: Haiyin Li, Yasuhiro Takeuchi. Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1117-1134. doi: 10.3934/dcdsb.2015.20.1117
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.  Google Scholar

[2]

D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: Stability Theory and Applications, Ellis Horwood Limited, Chichester, 1989.  Google Scholar

[3]

D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, New York, 1993.  Google Scholar

[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.  Google Scholar

[5]

L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann., 71 (1911), 97-115.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. L. DeAngelis, R. A. Goldstein and R. V. Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. E. Gaines and R. M. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Canad. Entomologist, 91 (1959), 382-384. doi: 10.4039/Ent91385-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, New York, 2011. doi: 10.1090/surv/176.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: Stability Theory and Applications, Ellis Horwood Limited, Chichester, 1989.  Google Scholar

[3]

D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, New York, 1993.  Google Scholar

[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.  Google Scholar

[5]

L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann., 71 (1911), 97-115.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. L. DeAngelis, R. A. Goldstein and R. V. Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

R. E. Gaines and R. M. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

C. S. Holling, Some characteristics of simple types of predation and parasitism, Canad. Entomologist, 91 (1959), 382-384. doi: 10.4039/Ent91385-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, New York, 2011. doi: 10.1090/surv/176.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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