Skip to main content
SpringerLink
Log in

Dynamic behaviour of a reaction–diffusion predator–prey model with both refuge and harvesting

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

An appropriate mathematical structure to describe the population dynamics is given by the partial differential equations of reaction–diffusion type. The spatiotemporal dynamics and bifurcations of a ratio-dependent Holling type II predator–prey model system with both the effect of linear prey harvesting and constant proportion of prey refuge are investigated. The existence of all ecologically feasible equilibria for the non-spatial model is determined, and the dynamical classifications of these equilibria are developed. The model system representing initial boundary value problem under study is subjected to zero flux boundary conditions. The conditions of diffusion-driven instability and the Turing bifurcation region in two parameter space are explored. The consequences of spatial pattern analysis in two-dimensional domain by means of numerical simulations reveal that the typical dynamics of population density variation is the formation of isolated groups, i.e. spotted or stripe-like patterns or coexistence of both the patterns or labyrinthine patterns and so on. The results around the unique interior feasible equilibrium solution indicate that the effect of refuge and harvesting plays a significant role on the control of spatial pattern formation of the species. Finally, the paper ends with a comprehensive discussion of biological implications of our findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Akçakaya, H.R., Arditi, R., Ginzburg, L.R.: Ratio-dependent predation: an abstraction that works. Ecology 76(3), 995–1004 (1995)

    Article  Google Scholar 

  2. Arditi, R., Ginzburg, L.R.: Coupling in predator–prey dynamics: ratio-dependence. J. Theor. Biol. 139(3), 311–326 (1989)

  3. Arditi, R., Perrin, N., Saïah, H.: Functional responses and heterogeneities: an experimental test with cladocerans. Oikos 60(1), 69–75 (1991)

    Article  Google Scholar 

  4. Brauer, F., Soudack, A.: Stability regions in predator–prey systems with constant-rate prey harvesting. J. Math. Biol. 8(1), 55–71 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. Clark, C.W.: Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, New York (1990)

    MATH  Google Scholar 

  6. Cosner, C., DeAngelis, D.L., Ault, J.S., Olson, D.B.: Effects of spatial grouping on the functional response of predators. Theor. Popul. Biol. 56(1), 65–75 (1999)

    Article  MATH  Google Scholar 

  7. Dai, G., Tang, M.: Coexistence region and global dynamics of a harvested predator–prey system. SIAM J. Appl. Math. 58(1), 193–210 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dubey, B., Das, B., Hussain, J.: A predator–prey interaction model with self and cross-diffusion. Ecol. Model. 141(1), 67–76 (2001)

    Article  Google Scholar 

  9. Freedman, H.: Deterministic Mathematical Method in Population Ecology. Dekker, New York (1980)

  10. Garvie, M.R.: Finite-difference schemes for reaction–diffusion equations modeling predator–prey interactions in MATLAB. Bull. Math. Biol. 69(3), 931–956 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gatto, M.: Some remarks on models of plankton densities in lakes. Am. Nat. 137(2), 264–267 (1991)

    Article  Google Scholar 

  12. Guin, L.N.: Existence of spatial patterns in a predator–prey model with self- and cross-diffusion. Appl. Math. Comput. 226, 320–335 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Guin, L.N., Haque, M., Mandal, P.K.: The spatial patterns through diffusion-driven instability in a predator–prey model. Appl. Math. Model. 36(5), 1825–1841 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guin, L.N., Mandal, P.K.: Effect of prey refuge on spatiotemporal dynamics of reaction–diffusion system. Comput. Math. Appl. 68(10), 1325–1340 (2014)

    Article  MathSciNet  Google Scholar 

  15. Gutierrez, A.: Physiological basis of ratio-dependent predator–prey theory: the metabolic pool model as a paradigm. Ecology 73(5), 1552–1563 (1992)

    Article  Google Scholar 

  16. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1971)

    Book  MATH  Google Scholar 

  17. Hanski, I.: The functional response of predators: worries about scale. Trends Ecol. Evol. 6(5), 141–142 (1991)

    Article  Google Scholar 

  18. Haque, M.: Existence of complex patterns in the Beddington-DeAngelis predator–prey model. Math. Biosci. 239(2), 179–190 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hassell, M.P., May, R.M.: Stability in insect host-parasite models. J. Anim. Ecol. 42, 693–726 (1973)

    Article  Google Scholar 

  20. Hsu, S.B., Hwang, T.W., Kuang, Y.: Global analysis of the Michaelis–Menten-type ratio-dependent predator–prey system. J. Math. Biol. 42(6), 489–506 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hsu, S.B., Hwang, T.W., Kuang, Y.: A ratio-dependent food chain model and its applications to biological control. Math. Biosci. 181(1), 55–83 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hu, J.H., Xue, Y.K., Sun, G.Q., Jin, Z., Zhang, J.: Global dynamics of a predator–prey system modeling by metaphysiological approach. Appl. Math. Comput. 283, 369–384 (2016)

    MathSciNet  Google Scholar 

  23. Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  24. Kuang, Y., Beretta, E.: Global qualitative analysis of a ratio-dependent predator–prey system. J. Math. Biol. 36(4), 389–406 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, L., Jin, Z., Li, J.: Periodic solutions in a herbivore-plant system with time delay and spatial diffusion. Appl. Math. Model. 40(7), 4765–4777 (2016)

    Article  MathSciNet  Google Scholar 

  26. Liu, C., Zhang, Q., Zhang, X., Duan, X.: Dynamical behavior in a stage-structured differential-algebraic prey–predator model with discrete time delay and harvesting. J. Comput. Appl. Math. 231(2), 612–625 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, C., Zhang, Q., Zhang, Y., Duan, X.: Bifurcation and control in a differential-algebraic harvested prey–predator model with stage structure for predator. Int. J. Bifurc. Chaos 18(10), 3159–3168 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lotka, A.J.: Elements of Mathematical Biology. Dover Publications, Mineola (1956)

    MATH  Google Scholar 

  29. Lv, Y., Yuan, R., Pei, Y.: Effect of harvesting, delay and diffusion in a generalist predator–prey model. Appl. Math. Comput. 226, 348–366 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Ma, J., Qin, H., Song, X., Chu, R.: Pattern selection in neuronal network driven by electric autapses with diversity in time delays. Int. J. Mod. Phys. B 29(1), 1450239 (2015)

    Article  Google Scholar 

  31. Ma, J., Tang, J.: A review for dynamics of collective behaviors of network of neurons. Sci. China Technol. Sci. 58(12), 2038–2045 (2015)

    Article  Google Scholar 

  32. Ma, J., Xu, Y., Ren, G., Wang, C.: Prediction for breakup of spiral wave in a regular neuronal network. Nonlinear Dyn. 84(2), 497–509 (2016)

    Article  MathSciNet  Google Scholar 

  33. May, R.M.: Stability and Complexity in Model Ecosystems, vol. 6. Princeton University Press, Princeton (1973)

    Google Scholar 

  34. Medvinsky, A.B., Petrovskii, S.V., Tikhonova, I.A., Malchow, H., Li, B.L.: Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44(3), 311–370 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Meyer, J.J., Byers, J.E.: As good as dead? Sublethal predation facilitates lethal predation on an intertidal clam. Ecol. Lett. 8(2), 160–166 (2005)

    Article  Google Scholar 

  36. Murray, J.D.: Mathematical Biology II. Springer, Heidelberg (2002)

  37. Myerscough, M., Gray, B., Hogarth, W., Norbury, J.: An analysis of an ordinary differential equation model for a two-species predator–prey system with harvesting and stocking. J. Math. Biol. 30(4), 389–411 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ruxton, G.: Short term refuge use and stability of predator–prey models. Theor. Popul. Biol. 47(1), 1–17 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  39. Sambath, M., Balachandran, K.: Spatiotemporal dynamics of a predator–prey model incorporating a prey refuge. J. Appl. Anal. Comput. 3(1), 71–80 (2013)

    MathSciNet  MATH  Google Scholar 

  40. Schreiber, S.J.: Generalist and specialist predators that mediate permanence in ecological communities. J. Math. Biol. 36(2), 133–148 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sotomayor, J.: Generic bifurcations of dynamical systems. In: Peixoto, M.M. (ed.) Dynamical Systems, Proceedings of a Symposium at the University of Bahia, Salvador, pp. 549–560. Academic Press, New York (1973)

  42. Sun, G.Q.: Mathematical modeling of population dynamics with Allee effect. Nonlinear Dyn. 85, 1–12 (2016)

    Article  MathSciNet  Google Scholar 

  43. Sun, G.Q., Chakraborty, A., Liu, Q.X., Jin, Z., Anderson, K.E., Li, B.L.: Influence of time delay and nonlinear diffusion on herbivore outbreak. Commun. Nonlinear Sci. Numer. Simul. 19(5), 1507–1518 (2014)

    Article  MathSciNet  Google Scholar 

  44. Sun, G.Q., Jin, Z., Li, L., Haque, M., Li, B.L.: Spatial patterns of a predator–prey model with cross diffusion. Nonlinear Dyn. 69(4), 1631–1638 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sun, G.Q., Jin, Z., Liu, Q.X., Li, L.: Pattern formation induced by cross-diffusion in a predator–prey system. Chin. Phys. B 17(11), 3936–3941 (2008)

    Article  Google Scholar 

  46. Sun, G.Q., Jin, Z., Zhao, Y.G., Liu, Q.X., Li, L.: Spatial pattern in a predator–prey system with both self-and cross-diffusion. Int. J. Mod. Phys. C 20(01), 71–84 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sun, G.Q., Li, L., Zhang, Z.K.: Spatial dynamics of a vegetation model in an arid flat environment. Nonlinear Dyn. 73(4), 2207–2219 (2013)

    Article  MathSciNet  Google Scholar 

  48. Sun, G.Q., Wang, S.L., Ren, Q., Jin, Z., Wu, Y.P.: Effects of time delay and space on herbivore dynamics: linking inducible defenses of plants to herbivore outbreak. Sci. Rep. 5(11), 246 (2015)

    Google Scholar 

  49. Sun, G.Q., Wu, Z.Y., Wang, Z., Jin, Z.: Influence of isolation degree of spatial patterns on persistence of populations. Nonlinear Dyn. 83(1–2), 811–819 (2016)

    Article  MathSciNet  Google Scholar 

  50. Sun, G.Q., Zhang, J., Song, L.P., Jin, Z., Li, B.L.: Pattern formation of a spatial predator–prey system. Appl. Math. Comput. 218(22), 11151–11162 (2012)

    MathSciNet  MATH  Google Scholar 

  51. Tripathi, J.P., Abbas, S., Thakur, M.: Dynamical analysis of a prey–predator model with beddington-deangelis type function response incorporating a prey refuge. Nonlinear Dyn. 80(1–2), 177–196 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  52. Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B Biol. Sci. 237(641), 37–72 (1952)

    Article  MathSciNet  Google Scholar 

  53. Wang, W., Liu, Q.X., Jin, Z.: Spatiotemporal complexity of a ratio-dependent predator–prey system. Phys. Rev. E 75(5), 051913 (2007)

    Article  MathSciNet  Google Scholar 

  54. Xiao, D., Jennings, L.S.: Bifurcations of a ratio-dependent predator–prey system with constant rate harvesting. SIAM J. Appl. Math. 65(3), 737–753 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  55. Xiao, D., Ruan, S.: Global dynamics of a ratio-dependent predator–prey system. J. Math. Biol. 43(3), 268–290 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  56. Xiao, M., Cao, J.: Hopf bifurcation and non-hyperbolic equilibrium in a ratio-dependent predator–prey model with linear harvesting rate: analysis and computation. Math. Comput. Model. 50(3), 360–379 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  57. Xiao, Y., Chen, L.: A ratio-dependent predator–prey model with disease in the prey. Appl. Math. Comput. 131(2), 397–414 (2002)

    MathSciNet  MATH  Google Scholar 

  58. Yodzis, P.: Predator–prey theory and management of multispecies fisheries. Ecol. Appl. 4(1), 51–58 (1994)

    Article  Google Scholar 

  59. Zhang, X., Zhang, Ql, Zhang, X.: Bifurcations of a class of singular biological economic models. Chaos Solitons Fractals 40(3), 1309–1318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees and the editors for their suggestions and constructive comments which greatly improved the presentation of this manuscript. Also, the authors gratefully acknowledge the financial support in part from Special Assistance Programme (SAP-III) sponsored by the University Grants Commission (UGC), New Delhi, India (Grant No. F.510 / 3 / DRS-III / 2015 (SAP-I)).

Author information

Authors and Affiliations

  1. Department of Mathematics, Visva-Bharati, Santiniketan, West Bengal, 731 235, India

    Lakshmi Narayan Guin & Sattwika Acharya

Authors
  1. Lakshmi Narayan Guin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sattwika Acharya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lakshmi Narayan Guin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guin, L.N., Acharya, S. Dynamic behaviour of a reaction–diffusion predator–prey model with both refuge and harvesting. Nonlinear Dyn 88, 1501–1533 (2017). https://doi.org/10.1007/s11071-016-3326-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-3326-8

Keywords

Mathematics Subject Classification

Search

Navigation