American Institute of Mathematical Sciences

November  2019, 12(7): 2195-2209. doi: 10.3934/dcdss.2019141

Bifurcation analysis of a stage-structured predator-prey model with prey refuge

 a. School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China b. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China c. School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

* Corresponding author: Huaqin Peng

Received  July 2017 Revised  July 2018 Published  December 2018

Fund Project: The research is supported by NNSF of China grant 11301102, 11771104, 11701113, Guangzhou Education Bureau 1201431215 and Key Laboratory of Mathematics and Statistical Model of Guangxi Colleges Open Foundation 2017GXKLMS007.

A stage-structured predator-prey model with prey refuge is considered. Using the geometric stability switch criteria, we establish stability of the positive equilibrium. Stability and direction of periodic solutions arising from Hopf bifurcations are obtained by using the normal form theory and center manifold argument. Numerical simulations confirm the above theoretical results.

Citation: Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141
References:
 [1] W. G. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.  doi: 10.1016/0025-5564(90)90019-U. [2] W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.  doi: 10.1137/0152048. [3] J. F. M. Al-Omari, The Effect of State Dependent Delay and Harvesting on a Stage-structured Predator-prey Model, Elsevier Science Inc., (2015).  doi: 10.1016/j.amc.2015.08.119. [4] Y. Z. Bai and X. P. Zhang, Stability and Hopf bifurcation in a diffusive predator-prey system with Beddington-DeAngelis functional response, Abstr. Appl. Anal., 2011 (2011), Art. ID 463721, 22 pp. doi: 10.1155/2011/463721. [5] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependant parameters, SIAM J. Math. Anal., 33 (2001), 1144-1165.  doi: 10.1137/S0036141000376086. [6] F. Chen, L. Chen and X. Xie, On a Leslie-Gower predator-prey model incorporaring a prey refuge, Nonlinear Anal. Real World Appl., 10 (2009), 2905-2908.  doi: 10.1016/j.nonrwa.2008.09.009. [7] X. Y. Chen and L. H. Huang, A Filippov system describing the effect of prey refuge use on a ratio-dependent predator-prey model, J. Math. Anal. Appl., 428 (2015), 817-837.  doi: 10.1016/j.jmaa.2015.03.045. [8] R. Cressmana and J. Garay, A predator-prey refuge system: Evolutionary stability in ecological systems, Theor. Popul. Bio., 76 (2009), 248-257. [9] S. Devi, Effects of prey refuge on a ratio-dependent predator-prey model with stage-structure of prey population, Appl. Math. Model., 37 (2013), 4337-4349.  doi: 10.1016/j.apm.2012.09.045. [10] E. Gonzalez-Olivares and R. Ramos-Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecol. Model., 166 (2003), 135-146. [11] Z. Gui and W. Ge, The effect of harvesting on a predator-prey system with stage structure, Ecol. Model., 187 (2005), 329-340. [12] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, 1981. [13] Y. Huang, F. Chen and L. Zhong, Stability analysis of a predator-prey model with Honing type Ⅲ response function incorporating a prey refuge, Appl. Math. Comput., 182 (2006), 672-683.  doi: 10.1016/j.amc.2006.04.030. [14] T. K. Kar and U. K. Pahari, Modelling and analysis of a prey-predator system with stage-structure and harvesting, Nonlinear Anal. Real World Appl., 8 (2007), 601-609.  doi: 10.1016/j.nonrwa.2006.01.004. [15] Z. Ma, W. Li, Y. Zhao, W. Wang, H. Zhang and Z. Li, Effects of prey refuges on a predator-prey model with a class of functional response: The role of refuges, Math. Biosci., 218 (2009), 73-79.  doi: 10.1016/j.mbs.2008.12.008. [16] J. M. McNair, The effects of refuges on predator-prey interactions: A reconsideration, Theor. Popul. Biol., 29 (1986), 38-63.  doi: 10.1016/0040-5809(86)90004-3. [17] Y. Qu and J. J. Wei, Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure, Nonlinear Dyn., 49 (2007), 285-294.  doi: 10.1007/s11071-006-9133-x. [18] Y. Qu and J. J. Wei, Bifurcation analysis in a predator-prey system with stage-structure and harvesting, J. Franklin Inst., 347 (2010), 1097-1113.  doi: 10.1016/j.jfranklin.2010.03.017. [19] X. Y. Song and L. S. Chen, Optimal harvesting and stability for a predator-prey system model with age structure, Acta Math. Appl. Sin., 18 (2002), 423-430.  doi: 10.1007/s102550200042. [20] J. J. Wei, H. Wang and W. Jiang, The Bifurcations Theory and Applications of Differential Equations, Science Press, Beijing, 2012. [21] A. A. Yakubu, Prey dominance in discrete predator-prey systems with a prey refuge, Math. Biosci., 144 (1997), 155-178.  doi: 10.1016/S0025-5564(97)00026-6. [22] R. Z. Yang and C. R. Zhang, Dynamics in a diffusive predator-prey system with a constant prey refuge and delay, Nonlinear Anal. Real World Appl., 31 (2016), 1-22.  doi: 10.1016/j.nonrwa.2016.01.005. [23] F. Zhang and Y. Li, Stability and Hopf bifurcation of a delayed-diffusive predator-prey model with hyperbolic mortality and nonlinear prey harvesting, Nonlinear Dynam., 88 (2017), 1397-1412.  doi: 10.1007/s11071-016-3318-8.

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References:
 [1] W. G. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.  doi: 10.1016/0025-5564(90)90019-U. [2] W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.  doi: 10.1137/0152048. [3] J. F. M. Al-Omari, The Effect of State Dependent Delay and Harvesting on a Stage-structured Predator-prey Model, Elsevier Science Inc., (2015).  doi: 10.1016/j.amc.2015.08.119. [4] Y. Z. Bai and X. P. Zhang, Stability and Hopf bifurcation in a diffusive predator-prey system with Beddington-DeAngelis functional response, Abstr. Appl. Anal., 2011 (2011), Art. ID 463721, 22 pp. doi: 10.1155/2011/463721. [5] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependant parameters, SIAM J. Math. Anal., 33 (2001), 1144-1165.  doi: 10.1137/S0036141000376086. [6] F. Chen, L. Chen and X. Xie, On a Leslie-Gower predator-prey model incorporaring a prey refuge, Nonlinear Anal. Real World Appl., 10 (2009), 2905-2908.  doi: 10.1016/j.nonrwa.2008.09.009. [7] X. Y. Chen and L. H. Huang, A Filippov system describing the effect of prey refuge use on a ratio-dependent predator-prey model, J. Math. Anal. Appl., 428 (2015), 817-837.  doi: 10.1016/j.jmaa.2015.03.045. [8] R. Cressmana and J. Garay, A predator-prey refuge system: Evolutionary stability in ecological systems, Theor. Popul. Bio., 76 (2009), 248-257. [9] S. Devi, Effects of prey refuge on a ratio-dependent predator-prey model with stage-structure of prey population, Appl. Math. Model., 37 (2013), 4337-4349.  doi: 10.1016/j.apm.2012.09.045. [10] E. Gonzalez-Olivares and R. Ramos-Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecol. Model., 166 (2003), 135-146. [11] Z. Gui and W. Ge, The effect of harvesting on a predator-prey system with stage structure, Ecol. Model., 187 (2005), 329-340. [12] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, 1981. [13] Y. Huang, F. Chen and L. Zhong, Stability analysis of a predator-prey model with Honing type Ⅲ response function incorporating a prey refuge, Appl. Math. Comput., 182 (2006), 672-683.  doi: 10.1016/j.amc.2006.04.030. [14] T. K. Kar and U. K. Pahari, Modelling and analysis of a prey-predator system with stage-structure and harvesting, Nonlinear Anal. Real World Appl., 8 (2007), 601-609.  doi: 10.1016/j.nonrwa.2006.01.004. [15] Z. Ma, W. Li, Y. Zhao, W. Wang, H. Zhang and Z. Li, Effects of prey refuges on a predator-prey model with a class of functional response: The role of refuges, Math. Biosci., 218 (2009), 73-79.  doi: 10.1016/j.mbs.2008.12.008. [16] J. M. McNair, The effects of refuges on predator-prey interactions: A reconsideration, Theor. Popul. Biol., 29 (1986), 38-63.  doi: 10.1016/0040-5809(86)90004-3. [17] Y. Qu and J. J. Wei, Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure, Nonlinear Dyn., 49 (2007), 285-294.  doi: 10.1007/s11071-006-9133-x. [18] Y. Qu and J. J. Wei, Bifurcation analysis in a predator-prey system with stage-structure and harvesting, J. Franklin Inst., 347 (2010), 1097-1113.  doi: 10.1016/j.jfranklin.2010.03.017. [19] X. Y. Song and L. S. Chen, Optimal harvesting and stability for a predator-prey system model with age structure, Acta Math. Appl. Sin., 18 (2002), 423-430.  doi: 10.1007/s102550200042. [20] J. J. Wei, H. Wang and W. Jiang, The Bifurcations Theory and Applications of Differential Equations, Science Press, Beijing, 2012. [21] A. A. Yakubu, Prey dominance in discrete predator-prey systems with a prey refuge, Math. Biosci., 144 (1997), 155-178.  doi: 10.1016/S0025-5564(97)00026-6. [22] R. Z. Yang and C. R. Zhang, Dynamics in a diffusive predator-prey system with a constant prey refuge and delay, Nonlinear Anal. Real World Appl., 31 (2016), 1-22.  doi: 10.1016/j.nonrwa.2016.01.005. [23] F. Zhang and Y. Li, Stability and Hopf bifurcation of a delayed-diffusive predator-prey model with hyperbolic mortality and nonlinear prey harvesting, Nonlinear Dynam., 88 (2017), 1397-1412.  doi: 10.1007/s11071-016-3318-8.
For $\tau\in[0,\bar{\tau})$, the graph of $S_0(\tau)$ and $S_1(\tau)$
For $\tau = 0.6<\tau^*$ and the initial value $" 1.0 , 1.0 "$, the positive equilibrium point of system (49) is stable
For τ = 22 > τ* and the initial value " 0.81, 0.61 ", system (49) exhibits a periodic solution
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