Volume 18, Issue 2, pp 154--162
Publication Date: 2018-01-26
- Department of Science Training, Fujian Institute of Education, Fuzhou, Fujian, 350025, P. R. China
Yiqin Wang - Department of Science Training, Fujian Institute of Education, Fuzhou, Fujian, 350025, P. R. China
In this paper, we consider a predator-prey model with square root functional response and prey refuge. The study reveals that the dynamical behavior near the origin of the model is subtle and interesting. We also show that the model undergoes Transcritical bifurcation and Hopf bifurcation. Numerical simulations not only illustrate our results, but also exhibit richer dynamical behaviors of the model than those with Holling II type functional response. Taking prey refuge as control variable, it is feasible to decrease predation rate and then control predator density properly so as to avoid two of population extinction and promote coexistence.
Square root functional response, prey refuges, limit cycle, global stability, transcritical bifurcation
 V. Ajraldi, M. Pittavino, E. Venturino, Modeling herd behavior in population systems, Nonlinear Anal. Real World Appl., 12 (2011), 2319–2338.
 P. A. Braza, Predator-prey dynamics with square root functional responses, Nonlinear Anal. RealWorld Appl., 13 (2012), 1837–1843.
 L.-J. Chen, F.-D. Chen, Global analysis of a harvested predator-prey model incorporating a constant prey refuge, Int. J. Biomath., 3 (2010), 205–223.
 L.-J. Chen, F.-D. Chen, L.-J. Chen, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear Anal. Real World Appl., 11 (2010), 246–252.
 F.-D. Chen, L.-J. Chen, X.-D. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. Real World Appl., 10 (2009), 2905–2908.
 E. González-Olivares, R. Ramos-Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecol. Modell., 166 (2003), 135–146.
 Y.-J. Huang, F.-D. Chen, Z. Li, Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Appl. Math. Comput., 182 (2006), 672–683.
 T.-W. Hwang, Uniqueness of the limit cycle for Gause-type predator-prey systems, J. Math. Anal. Appl., 238 (1999), 179–195.
 L.-L. Ji, C.-Q.Wu, Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating a constant prey refuge, Nonlinear Anal. Real World Appl., 11 (2010), 2285–2295.
 T. K. Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 681–691.
 T. K. Kar, Modelling and analysis of a harvested prey-predator system incorporating a prey refuge, J. Comput. Appl. Math., 185 (2006), 19–33.
 W.-Y. Ko, K.-M. Ryu, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge, J. Differential Equations, 231 (2006), 534–550.
 V. Křivan, Effects of optimal antipredator behavior of prey on predator-prey dynamics: the role of refuges, Theor. Popul. Biol., 53 (1998), 131–142.
 Z.-H. Ma, W.-L. Li, Y. Zhao, W.-T. Wang, H. Zhang, Z.-Z. Li, Effects of prey refuges on a predator-prey model with a class of functional responses: the role of refuges, Math. Biosci., 218 (2009), 73–79.
 J. N. McNair, The effects of refuges on predator-prey interactions: a reconsideration, Theoret. Population Biol., 29 (1986), 38–63.
 J. N. McNair, Stability effects of prey refuges with entry-exit dynamics, J. Theoret. Biol., 125 (1987), 449–464.