Skip to main content
SpringerLink
Log in

Positive Effect of Predator’s Mortality in Predator-Prey System via Turing Patterns

  • Statistical
  • Published:
Brazilian Journal of Physics Aims and scope Submit manuscript

Abstract

Hydra effect is the positive effect which defines that the population of species increases with an increase in mortality. Hence, this paper encapsulates the positive effect of predator mortality in a predator-prey system via Turing patterns. First, the existence of equilibrium points is obtained and the condition for system stability is derived. Hopf-bifurcation analysis has been carried out for the feasible equilibrium point and the necessary condition for hydra effect in predators is derived. Further, in the presence of diffusion, the random movement of the species is studied to establish conditions for the system’s stability, and derive the Turing instability condition. Numerical simulation with Neumann boundary condition revealed that the system experienced the Turing instability and gave various Turing patterns such as spots, stripes, and mixed spots-stripes. Our investigation shows the patterns are sensitive to mortality rate and self-diffusion.

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

Similar content being viewed by others

References

  1. P.A. Abrams, When does greater mortality increase population size? The long history and diverse mechanisms underlying the hydra effect. Ecol. Lett. 12, 462–474 (2009)

    Article  Google Scholar 

  2. P.A. Abrams, On classifying interactions between populations. Oecologia 73, 272–281 (1987)

    Article  ADS  Google Scholar 

  3. E.A. Bender, T.J. Case, M.E. Gilpin, Perturbation experiments in community ecology: theory and practice. Ecology 65, 1–13 (1984)

    Article  Google Scholar 

  4. J.M. Fryxell, I.M. Smith, D.H. Lynn, Evaluation of alternate harvesting strategies using experimental microcosms. Oikos 111, 143–149 (2005)

    Article  Google Scholar 

  5. A.J. Nicholson, Compensatory reactions of populations to stresses, and their evolutionary significance. Aust. J. Zool. 2, 1–8 (1954)

    Article  Google Scholar 

  6. A.J. Nicholson, The self-adjustment of populations to change. Cold Spring Harbor Laboratory Press 22, 153–173 (1957)

    Article  Google Scholar 

  7. A. Schroder, A. Leeuwen, T.C. Cameron, When less is more: positive population-level effects of mortality. Trends Ecol. Evol. 29, 614–624 (2014)

    Article  Google Scholar 

  8. A. Schroder, A. Leeuwen, T.C. Cameron, Empirical support for different types of positive mortality effects. A reply to Abrams. Trends Ecol. Evol. 30, 180–181 (2015)

  9. E.F. Zipkin, P.J. Sullivan, E.G. Cooch, C.E. Kraft, B.J. Shuter, B.C. Weidel, Overcompensatory response of a smallmouth bass (Micropterus dolomieu) population to harvest: Release from competition? Can. J. Fish. Aquat. 65, 2279–2292 (2008)

    Article  Google Scholar 

  10. M. Sieber, F.M. Hilker, The hydra effect in predator-prey models. J. Math. Biol. 64, 341–360 (2012)

    Article  MathSciNet  Google Scholar 

  11. P.A. Abrams, M.H. Cortez, The many potential indirect interactions between predators that share competing prey. Eco. Monogr. 85, 625–641 (2015)

  12. P.A. Abrams, C. Quince, The impact of mortality on predator population size and stability in systems with stage-structured prey. Theor. Popul. Biol. 68, 253–266 (2005)

    Article  Google Scholar 

  13. M.I.S. Costa, L. Anjos, Multiple hydra effect in a predator-prey model with Allee effect and mutual interference in the predator. Ecol. Modell. 373, 22–24 (2018)

    Article  Google Scholar 

  14. H.M. Cortez, P.A. Abrams, Hydra effects in stable communities and their implications for system dynamics. Ecology 97, 1135–1145 (2016)

    Article  Google Scholar 

  15. P.D. Adhikary, S. Mukherjee, B. Ghosh, Bifurcations and hydra effects in Bazykins predator-prey model. Theor. Popul. Biol. 140, 44–53 (2021)

    Article  Google Scholar 

  16. A.M. Turing, The chemical basis of morphogenesis. Bull. Math. Biol. 52, 153–197 (1990)

    Article  Google Scholar 

  17. L. Szili, J. Toit, Necessary condition of the Turing instability. Phys. Rev. E 48, 183 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  18. J.D. Murray, A pre-pattern formation mechanism for animal coat markings. J. Theor. Biol. 88, 161–199 (1981)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  20. L.A. Segal, J.L. Jackson, Dissipative structure: an explanation and an ecological example. J. Theor. Biol. 37, 545–559 (1972)

    Article  ADS  Google Scholar 

  21. T. Zhang, Y. Xing, H. Zang, M. Han, Spatio-of a reaction-diffusion system for a predator-prey model with hyperbolic mortality. Nonlinear Dyn. 78, 265–277 (2014)

    Article  MathSciNet  Google Scholar 

  22. T. Singh, S. Banerjee, Spatial aspect of hunting cooperation in predators with Holling type II functional response. J. Biol. Syst. 26, 511–531 (2018)

    Article  MathSciNet  Google Scholar 

  23. H. Jiang, Turing bifurcation in a diffusive predator-prey model with schooling behavior. Appl. Math. Lett. 96, 230–235 (2019)

    Article  MathSciNet  Google Scholar 

  24. W. Hao, C. Xue, Spatial pattern formation in reaction-diffusion models: a computational approach. J. Math. Biol. 80, 521–543 (2020)

    Article  MathSciNet  Google Scholar 

  25. S. Yan, D. Jia, T. Zhang, S. Yuan, Pattern dynamics in a diffusive predator-prey model with hunting cooperations. Chaos Solit. Fractals 130, 109428 (2020)

    Article  MathSciNet  Google Scholar 

  26. T. Singh, R. Dubey, V.N. Mishar, M. Abdel-Aty, Modeling of diffusive patterns in predator-prey system using Turing instability and amplitude equations. Info. Sci. Lett. 10, 2 (2021)

    Google Scholar 

  27. T. Singh, R. Dubey, Spatial patterns dynamics of a diffusive predator-prey system with cooperative behavior in predators. Fractals 29, 2150085 (2021)

    Article  ADS  Google Scholar 

  28. J.E. Satulovsky, T. Tome, Stochastic lattice gas model for a predator-prey system. Phy. Rev. E 49, 5073 (1994)

    Article  ADS  Google Scholar 

  29. S. Sinha, O.P. Misra, J. Dhar, Modelling a predator-prey system with infected prey in polluted environment. Appl. Math. Model. 34, 1861–1872 (2010)

    Article  MathSciNet  Google Scholar 

  30. S. Sinha, O.P. Misra, J. Dhar, A two species competition model under the simultaneous effect of toxicant and disease. Nonlinear Anal. Real World Appl. 11, 1131–1142 (2010)

    Article  MathSciNet  Google Scholar 

  31. Shivam, T. Singh, M. Kumar, Spatiotemporal dynamical analysis of a predator-prey system with fear and group defense in prey. J. Biol. Syst. 30, 1–36 (2022)

  32. Shivam, T. Singh, M. Kumar, Hopf-bifurcation and pattern selections in a three trophic level food web system. Int. J. Bifurc. Chaos 32, 1–36 (2022)

  33. A.D. Bazykin, Structural and dynamic stability of model predator-prey systems. IIASA Research Memorandum (1976)

  34. Y.M. Aponin, A.D. Bazykin, Model of eutrophication in predator-prey systems. IIASA Research Memorandum (1977)

Download references

Author information

Author notes

  1. All authors contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Graphic Era (Deemed to be) University, Dehradun, 248002, Uttarakhand, India

    Shivam & Mukesh Kumar

  2. School of Computer Science, University of Petroleum and Energy Studies, Dehradun, 248007, Uttarakhand, India

    Teekam Singh

  3. Amity Institute of Applied Science, Amity University, Noida, 201301, Uttar Pradesh, India

    Sudipa Chauhan

Authors
  1. Shivam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mukesh Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Teekam Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sudipa Chauhan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Teekam Singh or Sudipa Chauhan.

Ethics declarations

Conflict of Interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shivam, Kumar, M., Singh, T. et al. Positive Effect of Predator’s Mortality in Predator-Prey System via Turing Patterns. Braz J Phys 52, 159 (2022). https://doi.org/10.1007/s13538-022-01154-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13538-022-01154-z

Keywords

Search

Navigation