Stability and bifurcation in a stage-structured predator–prey system with Holling-II functional response and multiple delays

References

• W.G. Aiello and H.I. Freedman, A time-delay model of single species growth with stage structure, Math. Biosci. 101 (1990), pp. 139–153. doi: 10.1016/0025-5564(90)90019-U
   PubMed Web of Science ®Google Scholar
• M.A. Aziz-Alaoui, Study of a Leslie¨CGower-type tritrophic population, Chaos Solitons Fractals. 14 (2002), pp. 1275–1293. doi: 10.1016/S0960-0779(02)00079-6
   Web of Science ®Google Scholar
• S. Gakkhar and A. Singh, Complex dynamics in a prey predator system with multiple delays, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), pp. 914–929. doi: 10.1016/j.cnsns.2011.05.047
   Web of Science ®Google Scholar
• Q.T. Gan, R. Xu, and P.H. Yang, Bifurcation and chaos in a ratio-dependent predator–prey system with time delay, Chaos Solitons Fractals. 39 (2009), pp. 1883–1895. doi: 10.1016/j.chaos.2007.06.122
   Web of Science ®Google Scholar
• S.J. Gao, L.S. Chen, and Z.D. Teng, Hopf bifurcation and global stability for a delayed predator–prey system with stage structure for predator, Appl. Math. Comput. 202 (2008), pp. 721–729. doi: 10.1016/j.amc.2008.03.011
   Web of Science ®Google Scholar
• 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), pp. 5830–5837. doi: 10.1016/j.amc.2010.12.065
   Web of Science ®Google Scholar
• B.D. Hassard, N.D. Kazarinoff, and Y.H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
   Google Scholar
• H.J. Hu and L.H. Huang, Stability and Hopf bifurcation in a delayed predator–prey system with stage structure for prey, Nonlinear Anal. RWA. 11 (2010), pp. 2757–2769. doi: 10.1016/j.nonrwa.2009.10.001
   Web of Science ®Google Scholar
• S. Jana and T.K. Kar, Modeling and analysis of a prey–predator system with disease in the prey, Chaos Solitons Fractals. 47 (2013), pp. 42–53. doi: 10.1016/j.chaos.2012.12.002
   Web of Science ®Google Scholar
• Y.F. Jia, A sufficient and necessary condition for the existence of positive solutions for a prey–predator system with Ivlev-type functional response, Appl. Math. Lett. 24 (2011), pp. 1084–1088. doi: 10.1016/j.aml.2011.01.027
   Web of Science ®Google Scholar
• Y.F. Jia, H.K. Xu, and R.P. Agarwal, Existence of positive solutions for a prey–predator model with refuge and diffusion, Appl. Math. Comput. 217 (2011), pp. 8264–8276. doi: 10.1016/j.amc.2011.02.111
   Web of Science ®Google Scholar
• J.J. Jiao, L.S. Chen, and J.J. Nieto, Permanence and global attractivity of stage-structured predator–prey model with continuous harvesting on predator and impulsive stocking on prey, Appl. Math. Mech. 29 (2008), pp. 653–663. doi: 10.1007/s10483-008-0509-x
   Web of Science ®Google Scholar
• T.K. Kar and A. Ghorai, Dynamic behaviour of a delayed predator–prey model with harvesting, Appl. Math. Comput. 217 (2011), pp. 9085–9104. doi: 10.1016/j.amc.2011.03.126
   Web of Science ®Google Scholar
• Y. Kuang and J.W.H. So, Analysis of a delayed two-stage population with space-limited recruitment, SIAM J. Appl. Math. 55 (1995), pp. 1675–1695. doi: 10.1137/S0036139993252839
   Web of Science ®Google Scholar
• K.Q. Lan and C.R. Zhu, Phase portraits of predator–prey systems with harvesting rates, Discrete Contin. Dyn. Syst. Ser. A. 32 (2012), pp. 901–933. doi: 10.3934/dcds.2012.32.901
   Web of Science ®Google Scholar
• F. Li and H.W. Li, Hopf bifurcation of a predator–prey model with time delay and stage structure for the prey, Math. Comput. Model. 55 (2012), pp. 672–679. doi: 10.1016/j.mcm.2011.08.041
   Web of Science ®Google Scholar
• W. Lin, Description of complex dynamics in a class of impulsive differential equations, Chaos Solitons Fract. 25 (2005), pp. 1007–1017. doi: 10.1016/j.chaos.2005.01.043
   Web of Science ®Google Scholar
• J. Liu, Z.Z. Zhang, and M. Fu, Stability and bifurcation in a delayed Holling-Tanner predator–prey system with ratio-dependent functional response, J. Appl. Math. 2012 (2012), Article ID 384293, 19 pages.
   Google Scholar
• Y.S. Lv and Z.J. Du, Existence and global attractivity of a positive periodic solution to a Lotk-Volterra model with mutual interference and Holling III type functional response, Nonlinear Anal. Real. World. Appl. 12 (2011), pp. 3654–3664. doi: 10.1016/j.nonrwa.2011.06.022
   Web of Science ®Google Scholar
• J.L. Lv and K. Wang, Asymptotic properties of a stochastic predator–prey system with Holling II functional response, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), pp. 4037–4048. doi: 10.1016/j.cnsns.2011.01.015
   Web of Science ®Google Scholar
• X.Y. Meng, H.F. Huo, and X.B. Zhang, Stability and Hopf bifurcation in a three-species system with feedback delays, Nonlinear Dyn. 64 (2011), pp. 349–364. doi: 10.1007/s11071-010-9866-4
   Web of Science ®Google Scholar
• Y. Xiao and L. Chen, Global stability of a predator–prey system with stage structure for the predator, Acta Math. Appl. Sin. Engl. Ser. 20 (2004), pp. 63–70. doi: 10.1007/s10114-002-0234-2
   Web of Science ®Google Scholar
• R. Xu, Global dynamics of a predator–prey model with time delay and stage structure for the prey, Nonlinear Anal. RWA. 12 (2011), pp. 2151–2162. doi: 10.1016/j.nonrwa.2010.12.029
   Web of Science ®Google Scholar
• R. Xu, Global stability and Hopf bifurcation of a predator–prey model with stage structure and delayed predator response, Nonlinear Dyn. 67 (2012), pp. 1683–1693. doi: 10.1007/s11071-011-0096-1
   Web of Science ®Google Scholar
• R. Xu and Z. Ma, Stability and Hopf bifurcation in a ratio-dependent predator prey system with stage-structure, Chaos Solitons Fractals. 38 (2008), pp. 669–684. doi: 10.1016/j.chaos.2007.01.019
   Web of Science ®Google Scholar
• Y. Yang, Hopf bifurcation in a two-competitor, one-prey system with time delay, Appl. Math. Comput. 214 (2009), pp. 228–235. doi: 10.1016/j.amc.2009.03.078
   Web of Science ®Google Scholar
• J.F. Zhang, Bifurcation analysis of a modified Holling–Tanner predator–prey model with time delay, Appl. Math. Model. 36 (2012), pp. 1219–1231. doi: 10.1016/j.apm.2011.07.071
   Web of Science ®Google Scholar
• H. Zhang, L. Chen, and R. Zhu, Permanence and extinction of a periodic predator–prey delay system with functional response and stage structure for prey, Appl. Math. Comput. 184 (2007), pp. 931–944. doi: 10.1016/j.amc.2006.05.195
   Web of Science ®Google Scholar
• F.Q. Zhang and C.W. Zheng, Positive periodic solutions for the neutral ratio-dependent predator–prey model, Comput. Math. Appl. 61 (2011), pp. 2221–2226. doi: 10.1016/j.camwa.2010.09.021
   Web of Science ®Google Scholar
• T. Zhao, Y. Kuang, and H.L. Smith, Global existence of periodic solutions in a class of delayed Gauss-type predator–prey systems, Nonlinear Anal. 28 (1997), pp. 1373–1394. doi: 10.1016/0362-546X(95)00230-S
   Web of Science ®Google Scholar
• J. Zhou and C.L. Mu, Positive solutions for a three-trophic food chain model with diffusion and Beddington-Deangelis functional response, Nonlinear Anal. RWA. 12 (2011), pp. 902–917. doi: 10.1016/j.nonrwa.2010.08.015
   Web of Science ®Google Scholar
• Y.L. Zhu and K. Wang, Existence and global attractivity of positive periodic solutions for a predator–prey model with modified Leslie-Gower Holling-type II schemes, J. Math. Anal. Appl. 384 (2011), pp. 400–408. doi: 10.1016/j.jmaa.2011.05.081
   Web of Science ®Google Scholar