# American Institute of Mathematical Sciences

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

 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.
Citation: Haiyin Li, Yasuhiro Takeuchi. Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response. Discrete and 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. [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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##### 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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