Advertisement

Bulletin of Mathematical Biology

, Volume 80, Issue 3, pp 626–656 | Cite as

Modeling the Effect of Prey Refuge on a Ratio-Dependent Predator–Prey System with the Allee Effect

  • Maitri Verma
  • A. K. Misra
Original Article

Abstract

The extinction of species is a major threat to the biodiversity. The species exhibiting a strong Allee effect are vulnerable to extinction due to predation. The refuge used by species having a strong Allee effect may affect their predation and hence extinction risk. A mathematical study of such behavioral phenomenon may aid in management of many endangered species. However, a little attention has been paid in this direction. In this paper, we have studied the impact of a constant prey refuge on the dynamics of a ratio-dependent predator–prey system with strong Allee effect in prey growth. The stability analysis of the model has been carried out, and a comprehensive bifurcation analysis is presented. It is found that if prey refuge is less than the Allee threshold, the incorporation of prey refuge increases the threshold values of the predation rate and conversion efficiency at which unconditional extinction occurs. Moreover, if the prey refuge is greater than the Allee threshold, situation of unconditional extinction may not occur. It is found that at a critical value of prey refuge, which is greater than the Allee threshold but less than the carrying capacity of prey population, system undergoes cusp bifurcation and the rich spectrum of dynamics exhibited by the system disappears if the prey refuge is increased further.

Keywords

Predator–prey system Allee effect Ratio dependent Prey refuge Bifurcation 

Mathematics Subject Classification

34C23 34C25 34D20 92B05 92D25 

References

  1. Ajraldi V, Pittavino M, Venturino E (2011) Modeling herd behavior in population systems. Nonlinear Anal-Real 12:2319–2338MathSciNetCrossRefMATHGoogle Scholar
  2. Aguirre P, Flores JD, Flores JD, González-Olivares E (2014) Bifurcations and global dynamics in a predator–prey model with a strong Allee effect on the prey and ratio-dependent functional response. Nonlinear Anal-Real 16:235–249MathSciNetCrossRefMATHGoogle Scholar
  3. Allee WC (1932) Animal aggregations: a study in general sociology. University of Chicago Press, ChicagoGoogle Scholar
  4. Arditi R, Ginzburg LR (1989) Coupling in predator-prey dynamics: ratio-dependence. J Theor Biol 139:311–326CrossRefGoogle Scholar
  5. Cai L, Chen G, Xiao D (2013) Multiparametric bifurcations of an epidemiological model with strong Allee effect. J Math Biol 67:185–215MathSciNetCrossRefMATHGoogle Scholar
  6. Chen L, Chen F, Chen L (2010) Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge. Nonlinear Anal-Real 11:246–252MathSciNetCrossRefMATHGoogle Scholar
  7. Fan RN (2015) A predator-prey model incorporating prey refuge and Allee effect. Appl Mech Mater 713–715:1534–1539CrossRefGoogle Scholar
  8. Fan M, Wu P, Feng Z, Swihar RK (2016) Dynamics of predator-prey metapopulations with Allee effects. Bull Math Biol 78:662–694MathSciNetCrossRefGoogle Scholar
  9. Flores JD, González-Olivares E (2014) Dynamics of a predator-prey model with Allee effect on prey and ratio-dependent functional response. Ecol Complex 18:59–66CrossRefMATHGoogle Scholar
  10. Gao Y, Li B (2013) Dynamics of a ratio-dependent predator-prey system with strong Allee effect. Disc Contin Dyn Syst Ser B 18(9):2283–2313MathSciNetCrossRefMATHGoogle Scholar
  11. González-Olivars E, Ramos-Jiliberto R (2003) Dynamics consequences of prey refuges in a simple model system: more prey, few predators and enhanced stability. Ecol Model 166:135–146CrossRefGoogle Scholar
  12. González-Olivars E, Rojas-Palma A (2011) Multiple limit cycles in a Gause type predator-prey model with Holling type III functional response and Allee effect on prey. Bull Math Biol 73:1378–1397MathSciNetCrossRefMATHGoogle Scholar
  13. Haque M, Rahman MS, Venturino E, Li BL (2014) Effect of a functional response-dependent prey refuge in a predator-prey model. Ecol Complex 20:248–256CrossRefGoogle Scholar
  14. Hassell MP (1978) The dynamics of arthropod predator-prey systems. Princeton University Press, PrincetonMATHGoogle Scholar
  15. Hsu SB, Hwang TW, Kuang Y (2001) Global analysis of the Michaelis–Menten type ratio-dependent predator-prey system. J Math Biol 42:489–506MathSciNetCrossRefMATHGoogle Scholar
  16. Holling CS (1965) The functional response of predators to prey density and its role in mimicry and population regulation. Mem Entomol Soc Can 45:3–60Google Scholar
  17. Huang J, Ruan S, Song J (2014) Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response. J Differ Equ 257:1721–1752MathSciNetCrossRefMATHGoogle Scholar
  18. Huang J, Xia X, Zhang X, Ruan S (2016) Bifurcation of codimension 3 in a predator-prey system of Leslie type with simplified Holling type IV functional response. Int J Bifurc Chaos 26(2):1650034MathSciNetCrossRefMATHGoogle Scholar
  19. Huang Y, Chen F, Zhong L (2006) Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge. Appl Math Comput 182:672–683MathSciNetMATHGoogle Scholar
  20. Kar TK (2005) Stability analysis of a prey-predator model incorporating a prey refuge. Commun Nonlinear Sci Numer Simul 10:681–691MathSciNetCrossRefMATHGoogle Scholar
  21. Kuang Y, Beretta E (1998) Global qualitative analysis of a ratio-dependent predator-prey system. J Math Biol 36:389–406MathSciNetCrossRefMATHGoogle Scholar
  22. Kuang Y (1999) Rich Dynamics of Gause-type ratio-dependent predator-prey systems. Fields Inst Commun 21:325–337MathSciNetMATHGoogle Scholar
  23. Kŕivan V (1998) Effects of optimal antipredator behvior of prey on predator-prey dynamics: the role of reufuges. Theor Popul Biol 53:131–142CrossRefMATHGoogle Scholar
  24. Kuznetsov YA (1998) Elements of applied bifurcation theory, 2nd edn. Springer, New YorkMATHGoogle Scholar
  25. Li B, Kuang Y (2007) Heteroclinic bifurcation in the Michaelis-Menten-type ratio-dependent predator-prey system. SIAM J Appl Math 67:1453–1464MathSciNetCrossRefMATHGoogle Scholar
  26. Ma Z, Li W, Zhao Y, Wang W, Zhang H, Li Z (2009a) Review effects of prey refuges on a predator-prey model with a class of functional responses: the role of refuges. Math Biosci 218:73–79MathSciNetCrossRefMATHGoogle Scholar
  27. Ma Z, Li W, Shu-fan W, Li Z (2009b) Dynamical analysis of prey refuges in a predator-prey system with lvlev functional response. Dyn Contin Discret Impuls Syst Ser B Appl Algorithms 16:741–748MATHGoogle Scholar
  28. Ma Z, Wang S, Li W, Li Z (2013) The effect of prey refuge in a patchy predator-prey system. Math Biosci 243:126–130MathSciNetCrossRefMATHGoogle Scholar
  29. McNair JN (1986) The effects of refuges on predator-prey interactions: a reconsideration. Theor Popul Biol 29(1):38–63MathSciNetCrossRefMATHGoogle Scholar
  30. Morozov A, Petrovskii S, Li BL (2004) Bifurcations and chaos in a predator-prey system with the Allee effect. Proc R Soc Lond B 271:1407–1414CrossRefGoogle Scholar
  31. Murray JD (1993) Mathematical biology. Springer, New YorkCrossRefMATHGoogle Scholar
  32. Perko L (2000) Differential equations and dynamical systems, 3rd edn. Springer, BerlinMATHGoogle Scholar
  33. Rana S, Bhowmick AR, Bhattacharya S (2014) Impact of prey refuge on a discrete time predator-prey system with Allee effect. Int J Bifurc Chaos 24:1450106MathSciNetCrossRefMATHGoogle Scholar
  34. Ruxton GD (1995) Short term refuge use and stability of predator-prey models. Theor Popul Biol 47:1–17CrossRefMATHGoogle Scholar
  35. Sih A (1987) Prey refuges and predator-prey stability. Theor Popul Biol 31:1–12MathSciNetCrossRefGoogle Scholar
  36. Tian X, Xu R (2011) Global dynamics of a predator-prey system with Holling type II functional response. Nonlinear Anal Model Control 16:242–253MathSciNetMATHGoogle Scholar
  37. Wang J, Shi J, Wei J (2011) Predator-prey system with strong Allee effect in prey. J Math Biol 62:291–331MathSciNetCrossRefMATHGoogle Scholar
  38. Wang Y, Wang J (2012) Influence of prey refuge on predator-prey dynamics. Nonlinear Dyn 67(1):191–201MathSciNetCrossRefGoogle Scholar
  39. Wang W, Zhu Y, Cai Y, Wang W (2014) Dynamical complexity induced by Allee effect in a predator-prey model. Nonlinear Anal-Real 16:103–119MathSciNetCrossRefMATHGoogle Scholar
  40. Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Springer, BerlinCrossRefMATHGoogle Scholar
  41. Xiao D, Ruan S (2001) Global dynamics of a ratio-dependent predator-prey system. J Math Biol 43:268–290MathSciNetCrossRefMATHGoogle Scholar
  42. Zhou X, Liu Y, Wang G (2005) The stability of predator-prey systems subject to the Allee effects. Theo Popul Biol 67:23–31CrossRefMATHGoogle Scholar
  43. Zua J, Mimurab M (2010) The impact of Allee effect on a predator-prey system with holling type II functional response. Appl Math Comput 217:3542–3556MathSciNetGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of Mathematics, School of Physical and Decision SciencesBabasaheb Bhimrao Ambedkar UniversityLucknowIndia
  2. 2.Department of Mathematics, Institute of ScienceBanaras Hindu UniversityVaranasiIndia

Personalised recommendations