## Permanence, extinction and periodic solution of the predator-prey system with Beddington-DeAngelis functional response and stage structure for prey.(English)Zbl 1142.34051

Summary: We study permanence, extinction and periodic solutions of a periodic predator-prey system with Beddington-DeAngelis functional response and stage structure for prey. A set of sufficient and necessary conditions which guarantee the predator and prey species to be permanent are obtained. In addition, sufficient conditions are derived for the existence of positive periodic solutions to the system. Numeric simulations show the feasibility of the main results.

### MSC:

 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K13 Periodic solutions to functional-differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general) 34K25 Asymptotic theory of functional-differential equations
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### References:

 [1] Aiello, W.G.; Freedman, H.I.; Wu, J., Analysis of a model representing stage- structure population growth with state-dependent time delay, SIAM J. appl. math., 52, 855-869, (1992) · Zbl 0760.92018 [2] Beddington, J.R., Mutual interference between parasites or predators and its effect on searching efficiency, J. animal. ecol., 44, 331-340, (1975) [3] Bence, J.R.; Nisbet, R.M., Space-limited recruitment in open system: the importance of time delays, Ecology, 70, 1434-1441, (1989) [4] Bernard, O.; Souissi, S., Qualitative behavior of stage-structure populations: application to structural validation, J. math. biol., 37, 291-308, (1998) · Zbl 0919.92035 [5] Berryman, A.A., The origins and evolution of predator – prey theory, Ecology, 75, 1530-1535, (1992) [6] Cantrell, R.S.; Cosner, C., On the dynamics of predator – prey models with the beddington – deangelis functional response, J. math. anal. appl., 25, 1, 206-222, (2001) · Zbl 0991.34046 [7] Chen, F.D., Periodic solutions of a delayed predator – prey model with stage structure for predator, J. appl. math., 2005, 2, 153-169, (2005) [8] Chen, F.D., On a nonlinear non-autonomous predator – prey model with diffusion and distributed delay, J. comput. appl. math., 180, 1, 33-49, (2005) · Zbl 1061.92058 [9] Chen, F.D., On a periodic multi-species ecological model, Appl. math. comput., 171, 1, 492-510, (2005) · Zbl 1080.92059 [10] Chen, F.D., Positive periodic solutions of neutral lotka – volterra system with feedback control, Appl. math. comput., 162, 3, 1279-1302, (2005) · Zbl 1125.93031 [11] Chen, F.D., Some new results on the permanence and extinction of nonautonomous gilpin – ayala type competition model with delays, Nonlinear anal. real world appl., 7, 5, 1205-1222, (2006) · Zbl 1120.34062 [12] Chen, F.D.; Chen, X.X.; Shi, J.L., Dynamic behavior of a nonlinear single species diffusive system, Adv. complex systems, 8, 4, 399-417, (2005) · Zbl 1163.34360 [13] Chen, F.D.; Shi, J.L., Periodicity in a logistic type system with several delays, Comput. math. appl., 48, 1-2, 35-44, (2004) · Zbl 1061.34050 [14] Chen, F.D.; Sun, D.X.; Shi, J.L., Periodicity in a food-limited population model with toxicants and state dependent delays, J. math. anal. appl., 288, 1, 136-146, (2003) · Zbl 1087.34045 [15] Chen, W.; Wang, M.X., Qualitative analysis of predator – prey models with beddington – deangelis functional response and diffusion, Math. comput. modelling, 42, 1-2, 31-44, (2005) · Zbl 1087.35053 [16] Cui, J.A., Dispersal permanence of a periodic predator – prey system with beddington – deangelis functional response, Nonlinear anal., 64, 3, 440-456, (2006) · Zbl 1101.34035 [17] Cui, J.; Chen, L.; Wang, W., The effect of dispersal on population growth with stage-structure, Computers math. appl., 39, 91-102, (2000) · Zbl 0968.92018 [18] Cui, J.A.; Song, X.Y., Permanence of a predator – prey system with stage structure, Discrete continuous dynam. systems ser. B, 4, 3, 547-554, (2004) · Zbl 1100.92062 [19] Cui, J.A.; Takeuchi, Y., Permanence, extinction and periodic solution of predator – prey system with beddington – deangelis functional response, J. math. anal. appl., 317, 2, 464-474, (2006) · Zbl 1102.34033 [20] DeAngelis, D.L.; Goldstein, R.A.; O’neil, R.V., A model for tropic interaction, Ecology, 56, 881-892, (1975) [21] Dimitrov, D.T.; Kojouharov, H.V., Complete mathematical analysis of predator – prey models with linear prey growth and beddington – deangelis functional response, Appl. math. comput., 162, 2, 523-538, (2005) · Zbl 1057.92050 [22] Fan, M.; Kuang, Y., Dynamics of a nonautonomous predator – prey system with the beddington – deangelis functional response, J. math. anal. appl., 295, 1, 15-39, (2004) · Zbl 1051.34033 [23] H.F. Huo, W.T. Li, J.J. Nieto, Periodic solutions of delayed predator – prey model with the Beddington-DeAngelis functional response, Chaos, Solitons and Fractals, in press. · Zbl 1155.34361 [24] Hwang, T.W., Global analysis of the predator – prey system with beddington – deangelis functional response, J. math. anal. appl., 281, 1, 395-401, (2003) · Zbl 1033.34052 [25] Hwang, T.W., Uniqueness of limit cycles of the predator – prey system with beddington – deangelis functional response, J. math. anal. appl., 290, 1, 113-122, (2004) · Zbl 1086.34028 [26] T.K. Kar, U.K. Pahari, Modelling and analysis of a prey – predator system with stage-structure and harvesting, Nonlinear Analysis: Real World Appl., in press, doi:10.1016/j.nonrwa.2006.01.004. · Zbl 1152.34374 [27] Li, Z.Q.; Wang, W.M.; Wang, H.L., The dynamics of a beddington-type system with impulsive control strategy, Chaos, solitons and fractals, 29, 5, 1229-1239, (2006) · Zbl 1142.34305 [28] Liu, S.Q.; Chen, L.S.; Liu, Z.J., Extinction and permanence in nonautonomous competitive system with stage structure, J. math. anal. appl., 274, 2, 667-684, (2002) · Zbl 1039.34068 [29] Liu, S.Q.; Chen, L.S.; Luo, G.L., Extinction and permanence in competitive stage structured system with time-delays, Nonlinear anal., 51, 8, 1347-1361, (2002) · Zbl 1021.34065 [30] Liu, Z.H.; Yuan, R., Stability and bifurcation in a delayed predator – prey system with beddington – deangelis functional response, J. math. anal. appl., 296, 2, 521-537, (2004) · Zbl 1051.34060 [31] R.K. Naji, A.T. Balasim, Dynamical behavior of a three species food chain model with Beddington-DeAngelis functional response, Chaos, Solitons and Fractals, in press. · Zbl 1195.92061 [32] Qui, Z.P.; Yu, J.; Zou, Y., The asymptotic behavior of a chemostat model with the beddington – deangelis functional response, Math. biosci., 187, 2, 175-187, (2004) · Zbl 1049.92039 [33] Teng, Z.; Chen, L., The positive periodic solutions in periodic Kolmogorov type systems with delays (in Chinese), Acta math. appl. sinica,, 22, 446-456, (1999) · Zbl 0976.34063 [34] Wang, W.; Mulone, G.; Salemi, F.; Salone, V., Permanence and stability of a stage-structured predator – prey model, J. math. anal. appl., 262, 2, 499-528, (2001) · Zbl 0997.34069 [35] Wang, H.L.; Zhong, S.M., Asymptotic behavior of solutions in nonautonomous predator – prey patchy system with beddington-type functional response, Appl. math. comput., 172, 1, 122-140, (2006) · Zbl 1102.34030 [36] Y. Xiao, Study on the eco-epidemiology dynamical system, Ph.D. Thesis, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, 2001. [37] Xu, R.; Chaplain, M.A.J.; Davidson, F.A., Permanence and periodicity of a delayed ratio-dependent predator – prey model with stage structure, J. math. anal. appl., 303, 2, 602-621, (2005) · Zbl 1073.34090 [38] Xu, R.; Chaplain, M.A.J.; Davidson, F.A., A Lotka-Volterra type food chain model with stage structure and time delays, J. math. anal. appl., 315, 1, 90-105, (2006) · Zbl 1096.34055 [39] Zhang, Z.Q., Periodic solutions of a predator – prey system with stage-structures for predator and prey, J. math. anal. appl., 302, 2, 291-305, (2005) · Zbl 1098.34035 [40] Zhang, S.W.; Chen, L.S., A study of predator – prey models with the beddington – deanglis functional response and impulsive effect, Chaos, solitons and fractals, 27, 1, 237-248, (2006) · Zbl 1102.34032 [41] Zhang, X.; Chen, L.; Neumann, A.U., The stage-structured predator – prey model and optimal harvesting policy, Math. biosci., 101, 139-153, (2000) [42] Zhang, S.W.; Tan, D.J.; Chen, L.S., Chaotic behavior of a chemostat model with beddington – deangelis functional response and periodically impulsive invasion, Chaos, solitons and fractals, 29, 2, 474-482, (2006) · Zbl 1121.92070 [43] Zhang, S.W.; Tan, D.J.; Chen, L.S., Dynamic complexities of a food chain model with impulsive perturbations and beddington – deangelis functional response, Chaos, solitons and fractals, 27, 3, 768-777, (2006) · Zbl 1094.34031 [44] Zhang, Z.Q.; Wu, J.; Wang, Z.C., Periodic solutions of nonautonomous stage-structured cooperative system, Comput. math. appl., 47, 4-5, 699-706, (2004) · Zbl 1069.34066 [45] Zhao, X.Q., The qualitative analysis of $$N$$-species lotka – volterra periodic competition systems, Math. comput. modeling, 15, 3-8, (1991) · Zbl 0756.34048
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