Predator-Prey Model with Prey Group Defense and Non-linear Predator Harvesting

  • Rajat KaushikEmail author
  • Sandip Banerjee
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 308)


This paper is concerned with a predator-prey system with a prey group defense and non-linear harvesting of the predator incorporating deterrence hypothesis for predators. Inclusion of predator deterrence rate makes the modelling approach more practicable and exhibits significant impact on the net predation. Taking all possible interactions into account, model equations are formulated. In brief qualitative analysis, existence of interior equilibrium and stabilities of all equilibrium points of the system are discussed to investigate the dynamical behavior of the ecosystem. Hopf, transcritical and saddle-node bifurcations are illustrated for various parameters. Numerical simulations are ecologically justified and supportive of theoretical results.


Predator-prey Co-existence Local stability Hopf bifurcation Stability switches 


  1. 1.
    Ajraldi, V., Pittavino, M., Venturino, E.: Modelling herd behavior in population system. Nonlinear Anal. RWA 12, 2319–2338 (2011)CrossRefGoogle Scholar
  2. 2.
    Bancala, F.: Function of mucus secretion by lamellose ormer, haliotis tuberculata lamellosa, in response to starfish predation. Anim. Behav. 78, 1189–1194 (2009)CrossRefGoogle Scholar
  3. 3.
    Belvisi, S., Venturino, E.: An ecoepidemic model with diseased predators and prey group defense. Simulat. Model. Pract. Theor. 34, 144–155 (2013)CrossRefGoogle Scholar
  4. 4.
    Bera, S.P., Maiti, A., Samanta, G.P.: Modelling herd behavior of prey: analysis of a prey-predator model. World J. Model. Simul. 11, 3–14 (2015)Google Scholar
  5. 5.
    Birkhoff, G., Rota, G.C.: Ordinary Differential Equations. Needham Heights, Ginn (1982)zbMATHGoogle Scholar
  6. 6.
    Braza, P.A.: Predator-prey dynamics with square root functional responses. Nonlinear Anal. RWA 13(1837), 2012 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Clark, R.W.: Pursuit-deterrent communication between prey animals and timber rattlesnakes (crotalus horridus): the response of snakes to harassment displays. Behav. Ecol. Sociobiol. 59, 258–261 (2005). Scholar
  8. 8.
    Eisner, T., Grant, R.P.: Toxicity, odor aversion, and ‘olfactory aposematism. Science 213(4506), 476 (1981). Scholar
  9. 9.
    Gimmelli, G., Kooi, B.W., Venturino, E.: Ecoepidemic models with prey group defense and feeding saturation. Ecol. Complex. 22, 50–58 (2015)CrossRefGoogle Scholar
  10. 10.
    Ma, X., Shao, Y., Wang, Z., Luo, M., Fang, X., Ju, Z.: An impulsive two-stage predator-prey model with stage structure and square root functional responses. Math. Comput. Simulat. 119, 91–107 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Maan, M.E., Cummings, M.E.: Poison frog colors are honest signals of toxicity, particularly for bird predators. Am. Nat. 179(1), E1–E2 (2012). Scholar
  12. 12.
    Matia, S.N., Alam, S.: Prey-predator dynamics under herd behavior of prey. Univ. J. Appl. Math. 1, 251–257 (2013)Google Scholar
  13. 13.
    Rosenzweing, M.: The paradox of enrichment. Science 171, 385–387 (1971)CrossRefGoogle Scholar
  14. 14.
    Tang, X., Song, Y.: Bifurcation analysis and turing instability in a diffusive predator-prey model with herd behavior and hyperbolic mortality. Chaos Solitons Fractals 81, 303–314 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Tang, X., Song, Y.: Stability, hopf bifurcation and spatial patterns in a delayed diffusive predator-prey model with herd behavior. Appl. Math. Comput. 254, 375–391 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Thomas, C., Candy, R.: Avian predators change their foraging strategy on defended prey when undefended prey are hard to find. Anim. Behav. 93, 97–103 (2014)CrossRefGoogle Scholar
  17. 17.
    Wang, X., Pan, Q., Kang, Y., He, M.: Predator group size distributions in predator-prey system. Ecol. Complex. 26, 117–127 (2016)CrossRefGoogle Scholar
  18. 18.
    Xu, C., Yuan, X., Zhang, T.: Global dynamics of a predator-prey model with defense mechanism for prey. Appl. Math. Lett. 62, 42–48 (2016)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Indian Institute of Technology RoorkeeRoorkeeIndia

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