Skip to main content

Advertisement

SpringerLink
Log in

Dynamics of One-prey and Two-predator System Highlighting the Significance of Additional Food for Predators with Beddington–DeAngelis Functional Response

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

Role of additional food to predators is significant to retain biological balance for an improved ecosystem. Much attempts have been performed from the aspects of biological control and its consequences on global warming is investigated. Work has been done exhibiting the implication of mutual interference in stabilising the prey predator system which has a phenomenal impact on the dynamics of the system. In the proposed model, dynamics of additional food to predators on one-prey and two-predator system with Beddington–DeAngelis functional response is investigated. The proposed system also throws light on the role of mutual interference in predators and it differentiates the predators on the basis of the characteristic of consuming the additional food or to be solely dependent on preys for survival. Both local and global stability analysis of the system has been performed and at the end, numerical simulation is carried out which signifies the effect of changing the additional food parameters on the dynamics.

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

Similar content being viewed by others

References

  1. Wackers, F.L., van Rijn, P.C.J.: Food for protection: an introduction. In: Wackers, F.L., van Rijn, P.C.J., Bruin, J. (eds.) Plant-provided food for carnivorous insects: A protective mutualism and its applications, pp. 1–14. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  2. Perdikis, D., Lykouressis, D.: Effects of various items, host plants, and temperature on the development and survival of Macrolophus pygmaeus Rambur (Hemiptera: Miridae). Biol. Control 17, 5560 (2000)

    Google Scholar 

  3. Perdikis, D., Lykouressis, D.: Macrolophus pygmaeus (Hemiptera: Miridae) population parameters and biological characteristics when feeding on eggplant and tomato without prey. J. Econ. Entomol. 97, 12911298 (2004)

    Google Scholar 

  4. Perdikis, D., Lykouressis, D., Economou, L.: The influence of temperature, photo period and plant type on the predation rate of Macrolophus pygmaeus on Myzus persicae. BioControl 44, 281289 (1999)

    Google Scholar 

  5. Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331–340 (1975)

    Google Scholar 

  6. Koss, A.M., Snyder, W.E.: Alternative prey disrupt biocontrol by a guild of generalist predators. Biol. Control 32, 243–251 (2005)

    Google Scholar 

  7. Lorenzon, M., Pozzebon, A., Duso, C.: Effects of potential food sources on biological and demographic parameters of the predatory mites Kampimodromus aberrans, Typhlodromus pyri and Amblyseius andersoni. Exp. Appl. Acarol. 58, 259–278 (2012)

    Google Scholar 

  8. Put, K., Bollens, T., Wackers, F.L., Pekas, A.: Type and spatial distribution of food supplements impact population development and dispersal of the omnivore predator Macrolophus pygmaeus (Rambur) (Hemiptera: Miridae). Biol. Control 63, 172–180 (2012)

    Google Scholar 

  9. Cai, L., Yu, J., Zhu, G.: A stage-structured predator–prey model with Beddington–DeAngelis functional repsonse. J. Appl. Math. Comput. 26, 85–103 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Chattopadhayay, J., Sarkar, R., Hoballah, M.E.F., Turlings, T.C.J., Bersier, L.F.: Parasitoids may determine plant fitness—mathematical model based on experimental data. J. Theor. Biol. 212, 295–302 (2001)

    Google Scholar 

  11. Harwood, J.D., Obrycki, J.J.: The role of alternative prey in sustaining predator populations. In: Hoddle, M.S. (ed.) Proceedings of second international symposium on biological control of arthropods, Vol. II, pp. 453-462 (2005)

  12. Srinivasu, P.D.N., Prasad, B.S.R.V., Venkatesulu, M.: Biological control through provision of additional food to predators: a theoretical study. Theor. Popul. Biol. 72, 111–120 (2007)

    MATH  Google Scholar 

  13. Srinivasu, P.D.N., Prasad, B.S.R.V.: Time optimal control of an additional food provided predator–prey system with applications to pest management and biological conservation. J. Math. Biol. 60, 591–613 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Srinivasu, P.D.N., Prasad, B.S.R.V.: Erratum to: Time optimal control of an additional food provided predator–prey system with applications to pest management and biological conservation. J. Math. Biol. 61, 591–613 (2010a)

    MATH  Google Scholar 

  15. Srinivasu, P.D.N., Prasad, B.S.R.V.: Role of quantity of additional food to predators as a control in predatorprey systems with relevance to pest management and biological conservation. Bull. Math. Biol. 73, 2249–2276 (2011)

    MathSciNet  MATH  Google Scholar 

  16. Coll, M.: Parasitioids in diversified intercropped systems. In: Pickett, C.H., Bugg, R. (eds.) Enhancing biological control: Habitat management to promote natural enemies of argicultural pets, pp. 85–120. University of California Press, Berkely (1998)

    Google Scholar 

  17. Coll, M.: (2009) Feeding on non-prey ressources by natural enemies. In: Lundgren, J.G., Relationships of natural enemies and non-prey foods, Springer, Berlin, pp. ix–xxiii

  18. Hagen, K.S.: Ecosystem analysis: plant cultivars (HPR), entomophagous species and food supplements. In: (Boethel, D.J., Eikenbary, R.D. (eds.), Interactions of plant resistance and parasitoids and predators of insects, Halsted, New York, pp. 151-197 (1986)

  19. Lundgren, J.G.: Relationships of natural enemies and non-prey foods. Springer, New York (2009)

    Google Scholar 

  20. Holt, R.D., Lawton, J.H.: The ecological consequences of shared natural enemies. Annu. Rev. Ecol. Syst. 25, 495–520 (1994)

    Google Scholar 

  21. Holt, R.D.: Predation, apparent competition, and the the structure of prey communities. Theor. Popul. Biol. 12, 197–229 (1977)

    MathSciNet  Google Scholar 

  22. Murdoch, W.W., Chesson, J., Chesson, P.L.: Biological control in thoery and practice. Am. Nat. 125, 344–366 (1985)

    Google Scholar 

  23. Hoddle, M., van Driesche, R., Sanderson, J.: Biological control of Bemisia argentifolii (Homoptera:Aleyrodidae) on Poinsettia with inundative releases of Encarsia formosa (Hymenoptera: Aphelinidae): Are higher release rates necessarily better? Biol. Control 10, 166–179 (1997)

    Google Scholar 

  24. Prasad, B.S.R.V.: Dynamics of additional food provided predator–prey system with mutually interfering predators. J. Math. Biosci. 246, 176–190 (2013)

    MathSciNet  MATH  Google Scholar 

  25. DeLong, J.P., Vasseur, D.A.: Mutual interference is common and mostly intermediate in magnitude. BMC Ecol. 11(1), 1–8 (2011)

    Google Scholar 

  26. de Villemereuil, P.B., Lopez-Sepulcre, A.: Consumer functional responses under intra- and inter-specific interference competition. Ecol. Model. 222, 419–426 (2011)

    Google Scholar 

  27. Ginzburg, L.R., Jensen, C.X.J.: From controversy to consensus: the indirect interference functional response. Verh. Internat. Verein. Limnol. 30(2), 297–301 (2008)

    Google Scholar 

  28. Kidd, N.A.C., Jervis, M.A.: Population dynamics. In: Jervis, M.A. (ed.) Insects as natural enemies: A practical perspective, pp. 435–524. SpringerVerlag, The Netherlands (2007)

    Google Scholar 

  29. Reigadaa, C., Araujob, S.B.L., de Aguiar, M.A.M.: Patch exploitation strategies of parasitoids: The role of sex ratio and foragers interference in structuring metapopulations. Ecol. Model. 230, 11–21 (2012)

    Google Scholar 

  30. DeAngelis, D.L., Goldstein, R.A., ONeill, R.V.: A model for trophic interaction. Ecology 56, 881–892 (1975)

    Google Scholar 

  31. Cantrell, R.S., Cosner, C.: On the dynamics of predator–prey models with the Beddington–DeAngelis funcational response. J. Math. Anal. Appl. 257, 206–222 (2001)

    MathSciNet  MATH  Google Scholar 

  32. Huang, G., Ma, W., Takeuchi, Y.: Global properties for virus dynamics models with Beddington–DeAngelis functional response. Appl. Math. Lett. 22, 1690–1693 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Hwang, T.-W.: Global analysis of the predator–prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 281, 395–401 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Hwang, T.-W.: Uniqueness of limit cycles of the predator-prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 290, 113–122 (2004)

    MathSciNet  MATH  Google Scholar 

  35. Kratina, P., Vos, M., Bateman, A., Anhold, B.R.: Functional responses modified by predator density. Oecologia 159, 425–433 (2009)

    Google Scholar 

  36. Naji, R.K., Balasim, A.T.: Dynamical behavior of a three species food chain model with Beddington–DeAngelis functional response. Choas Solitons Fractals 32, 1853–1866 (2007)

    MathSciNet  MATH  Google Scholar 

  37. Negi, K., Gakkhar, S.: Dynamics in a Beddington–DeAngelis preypredator system with impulsive harvesting. Ecol. Model. 206, 421–430 (2007)

    Google Scholar 

  38. Skalski, G.T., Gilliam, J.F.: Functional responses with predator interference: Viable alternatives to the Holling type II model. Ecology 82, 3083–3092 (2001)

    Google Scholar 

  39. Dimitrov, D.T., Kojouharov, H.V.: Complete mathematical analysis of predator–prey models with linear prey growth and Beddington–DeAngelis functional response. Appl. Math. Comput. 162, 523–538 (2005)

    MathSciNet  MATH  Google Scholar 

  40. Nagumo, M.: Uber die Lage der Integralkurven gew onlicher Differentialgeichungen. Proc. Phys. Math. Soc. Jpn. 24, 555 (1942)

    Google Scholar 

  41. Birkhoff, G., Rota, G.C.: Ordinary Differential Equations. Ginn Boston (1982)

  42. Kar, T.K., Ghosh, Bapan: Sustainability and optimal control of an exploited prey predator system through provision of alternative food to predator. Biosystems 109(2), 220–232 (2012)

    Google Scholar 

  43. Sahoo, Banshidhar, Poria, Swarup: Effects of supplying alternative food in a predator prey model with harvesting. Appl. Math. Comput. 234, 150–166 (2014)

    MathSciNet  MATH  Google Scholar 

  44. Sahoo, Banshidhar, Poria, Swarup: Disease control in a food chain model supplying alternative food. Appl. Math. Model. 37(8), 5653–5663 (2013)

    MathSciNet  MATH  Google Scholar 

  45. Skalaki, Garrick T., Gilliam, James F.: Functional responses with predator interferenve: Viable alternatives to the Holling type II model. Ecol. Soc. Am. 82(11), 3083–3092 (2001)

    Google Scholar 

  46. Harrison, Gary W.: Global stability of predator–prey interactions. J. Math. Biol. 8(2), 159–171 (1979)

    MathSciNet  MATH  Google Scholar 

  47. Chen, Fengde: On a nonlinear nonautonomous predator–prey model with diffusion and distributed delay. J. Comput. Appl. Math. 180, 33–49 (2005)

    MathSciNet  MATH  Google Scholar 

  48. Hsu, Sze-Bi, Huang, Tzy-Wei: Global stability for a class of predator–prey systems. SIAM J. Appl. Math 55(3), 763–783 (1995)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Authors are grateful to Professor Balram Dubey, BITS Pilani for the valuable suggestions and constant inspiration while preparing this paper.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Delhi Technological University, Delhi, 110042, India

    Charu Arora & Vivek Kumar

Authors
  1. Charu Arora
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vivek Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vivek Kumar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arora, C., Kumar, V. Dynamics of One-prey and Two-predator System Highlighting the Significance of Additional Food for Predators with Beddington–DeAngelis Functional Response. Differ Equ Dyn Syst 30, 411–431 (2022). https://doi.org/10.1007/s12591-018-0442-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-018-0442-6

Keywords

Search

Navigation