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Bifurcation analysis of predator–prey model with time delay and harvesting efforts using interval parameter

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Abstract

This paper presents a prey–predator harvesting model with time delay for bifurcation analysis. We consider the parameters of the proposed model with imprecise data as form of interval in nature, due to the lack of precise numerical information of the biological parameters such as prey population growth rate and predator population decay rate. The proposed prey–predator harvesting model is presented with Holling type of predation and time delay under impreciseness of parameters by introducing parametric functional form of interval number. Our study reveals that along with delay and harvesting efforts role on the stability of the system, interval parameters also play a significant role. Computer simulations of numerical examples are given to explain our proposed imprecise model and for observing of chaotic behaviors.

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References

  1. Hoekstra J, Bergh JCJMV (2005) Harvesting and conservation in a predator–prey system. J Econ Dyn Control 29:1097–1120

    Article  Google Scholar 

  2. Palma A, Olivares E (2012) Optimal harvesting in a predator–prey model with Allee effect and sigmoid functional response. Appl Math Comput 36:1864–1874

    Google Scholar 

  3. Gupta RP, Chandra P (2013) Bifurcation analysis of modified Leslie-Gower predator–prey model with Michaelis–Menten type prey harvesting. J Math Anal Appl 338:278–295

    Article  MathSciNet  Google Scholar 

  4. Pal D, Mahapatra GS, Samanta GP (2012) A proportional harvesting dynamical model with fuzzy intrinsic growth rate and harvesting quantity. Pac Asian J Math 6:199–213

    Google Scholar 

  5. Duncan S, Hepburn C, Papachristodoulou A (2011) Optimal harvesting of fish stocks under a time-varying discount rate. J Theor Biol 269:166–173

    Article  MathSciNet  Google Scholar 

  6. Anita L, Anita S, Arnautu V (2009) Optimal harvesting for periodic age-dependent population dynamics with logistic term. Appl Math Comput 215:2701–2715

    Article  MathSciNet  Google Scholar 

  7. Chen F, Ma Z, Zhang H (2012) Globala symptotical stability of the positive equilibrium of the Lotka-Volterra prey–predator model incorporating a constant number of prey refuges. Nonlinear Anal Real World Appl 13:2790–2793

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen L, Chen F, Wang Y (2013) Influence of predator mutual interference and prey refuge on Lotka–Volterra predator–prey dynamics. Commun Nonlinear Sci Numer Simul 18:3174–3180

    Article  MathSciNet  Google Scholar 

  9. Wang H, Morrison W, Sing A, Weiss H (2009) Modeling inverted biomass pyramids and refuges in ecosystems. Ecol Model 220:1376–1382

    Article  MATH  Google Scholar 

  10. Rebaza J (2012) Dynamics of prey threshold harvesting and refuge. J Comput Appl Math 236:1743–1752

    Article  MathSciNet  MATH  Google Scholar 

  11. Das KP, Roy S, Chattopadhyay J (2009) Effect of disease-selective predation on prey infected by contact and external sources. Biosystems 95:188–199

    Article  MATH  Google Scholar 

  12. Bairagi N, Chaudhuri S, Chattopadhyay J (2009) Harvesting as a disease control measure in an eco-epidemiological system: a theoretical study. Math Biosci 217:134–144

    Article  MathSciNet  MATH  Google Scholar 

  13. Hethcote HW, Wang W, Han L, Ma Z (2004) A predator–prey model with infected prey. Theor Popul Biol 66:259–268

    Article  Google Scholar 

  14. Pal AK, Samanta GP (2010) Stability analysis of an eco-epidemiological model incorporating a Prey Refuge. Nonlinear Anal Model Control 15:473–491

    MathSciNet  MATH  Google Scholar 

  15. Samanta GP (2011) Analysis of a nonautonomous dynamical model of diseases through droplet infection and direct contact. Appl Math Comput 217:5870–5888

    Article  MathSciNet  Google Scholar 

  16. Gopalsamy K (1983) Harmless delay in model systems. Bull Math Biol 45:295–309

    Article  MathSciNet  Google Scholar 

  17. Kar TK (2003) Selective harvesting in a prey–predator fishery with time delay. Math Comput Model 38:449–458

    Article  MathSciNet  MATH  Google Scholar 

  18. Yongzhen P, Shuping L, Changguo L (2011) Effect of delay on a predator–prey model with parasite infection. Nonlinear Dyn 63:311–321

    Article  MathSciNet  Google Scholar 

  19. Qu Y, Wei J (2007) Bifurcation analysis in a time-delay model for prey–predator growth with stage-structure. Nonlinear Dyn 49:285–294

    Article  MathSciNet  MATH  Google Scholar 

  20. Shao Y (2010) Analysis of a delayed predator–prey system with impulsive diffusion between two patches. Math Comput Model 52:120–127

    Article  MATH  Google Scholar 

  21. Jiao J, Chen I, Yang X, Cai S (2009) Dynamical analysis of a delayed predator–prey model with impulsive diffusion between two patches. Math Comput Simul 80:522–532

    Google Scholar 

  22. MacDonald M (1989) Biological delay systems: linear stability theory. Cambridge University Press, Cambridge

  23. Misra AK, Dubey B (2010) A ratio-dependent predator–prey model with delay and harvesting. J Biol Syst 18:437–453

    Article  MathSciNet  Google Scholar 

  24. Zhang J (2012) Bifurcation analysis of a modified Holling–Tanner predator–prey model with time delay. Appl Math Model 36:1219–1231

    Article  MathSciNet  MATH  Google Scholar 

  25. Bandyopadhyay M, Banerjee S (2006) A stage-structured prey–predator model with discrete time delay. Appl Math Comput 182:1385–1398

    Article  MathSciNet  MATH  Google Scholar 

  26. Bassanezi RC, Barros LC, Tonelli A (2000) Attractors and asymptotic stability for fuzzy dynamical systems. Fuzzy Sets Syst. 113:473–483

    Article  MATH  Google Scholar 

  27. Barros LC, Bassanezi RC, Tonelli PA (2000) Fuzzy modelling in population dynamics. Ecol Model 128:27–33

    Article  Google Scholar 

  28. Peixoto M, Barros LC, Bassanezi RC (2008) Predator–prey fuzzy model. Ecol Model 214:39–44

    Article  MATH  Google Scholar 

  29. Tuyako MM, Barros LC, Bassanezi RC (2009) Stability of fuzzy dynamic systems. Int J Uncertain Fuzzyness Knowl Based Syst 17:69–83

    Article  Google Scholar 

  30. Pal D, Mahapatra GS, Samanta GP (2013) Quota harvesting model for a single species population under fuzziness. Int J Math Sci 12:33–46

    Google Scholar 

  31. Abundo M (1991) Stochastic model for predator–prey systems: basic properties, stability and computer simulation. J Math Biol 29:495–511

    Article  MathSciNet  MATH  Google Scholar 

  32. Rudnicki R (2003) Long-time behaviour of a stochastic prey–predator model. Stoch Process Appl 108:93–107

    Article  MathSciNet  MATH  Google Scholar 

  33. Liu M, Wang K (2012) Extinction and global asymptotical stability of a nonautonomous predator–prey model with random perturbation. Appl Math Model 36:5344–5353

    Article  MathSciNet  MATH  Google Scholar 

  34. Vasilova M (2013) Asymptotic behavior of a stochastic Gilpin–Ayala predator–prey system with time-dependent delay. Math Comput Model 57:764–781

    Article  MathSciNet  Google Scholar 

  35. Aguirre P, Olivares EG, Torres S (2013) Stochastic predator–prey model with Allee effect on prey. Nonlinear Anal Real World Appl 14:768–779

    Article  MathSciNet  Google Scholar 

  36. Mahapatra GS, Mandal TK (2012) Posynomial parametric geometric programming with interval valued coefficient. J Optim Theory Appl 154:120–132

    Article  MathSciNet  MATH  Google Scholar 

  37. Pal D, Mahapatra GS, Samanta GP (2013) Optimal harvesting of prey–predator system with interval biological parameters: a bioeconomic model. Math Biosci 24:181–187

    Article  Google Scholar 

  38. Kuang Y, Freedman HI (1988) Uniqueness of limit cycles in Gause-type models of predator–prey systems. Math Biosci 88:67–84

    Article  MathSciNet  MATH  Google Scholar 

  39. Freedman HI, Hari Rao V Sree (1983) The trade-of between mutual interference and time lags in predator–prey systems. Bull Math Biol 45:991–1003

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to express their gratitude to the Editor and Referees for their encouragement and constructive comments in revising the paper.

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Authors and Affiliations

  1. Chandrahati Dilip Kumar High School (H.S.), Chandrahati , 712504, West Bengal, India

    D. Pal

  2. Department of Mathematics, National Institute of Technology-Puducherry, Karaikal , 609605, India

    G. S. Mahapatra

  3. Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah , 711103, India

    G. P. Samanta

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  1. D. Pal
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  2. G. S. Mahapatra
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  3. G. P. Samanta
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Correspondence to G. S. Mahapatra.

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Pal, D., Mahapatra, G.S. & Samanta, G.P. Bifurcation analysis of predator–prey model with time delay and harvesting efforts using interval parameter. Int. J. Dynam. Control 3, 199–209 (2015). https://doi.org/10.1007/s40435-014-0083-8

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  • DOI: https://doi.org/10.1007/s40435-014-0083-8

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