Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Population dynamic consequences of fearful prey in a spatiotemporal predator-prey system

Department of Applied Mathematics, Indian Institute of Technology (ISM) Dhanbad, Jharkhand, 826004, India

Fear can influence the overall population size of an ecosystem and an important drive for change in nature. It evokes a vast array of responses spanning the physiology, morphology, ontogeny and the behavior of scared organisms. To explore the effect of fear and its dynamic consequences, we have formulated a predator-prey model with the cost of fear in prey reproduction term. Spatial movement of species in one and two dimensions have been considered for the better understanding of the model system dynamics. Stability analysis, Hopf-bifurcation, direction and stability of bifurcating periodic solutions have been studied. Conditions for Turing pattern formation have been established through diffusion-driven instability. The existence of both supercritical and subcritical Hopfbifurcations have been investigated by numerical simulations. Various Turing patterns are presented and found that the change in the level of fear and diffusion coe cients alter these structures significantly. Holes and holes-stripes mixed type of ecologically realistic patterns are observed for small values of fear and relative increase in the level of fear may reduce the overall population size.
  Article Metrics

Keywords predator-prey interactions; fear effect; anti-predator response; stability; Hopf-bifurcation; pattern formation

Citation: Ranjit Kumar Upadhyay, Swati Mishra. Population dynamic consequences of fearful prey in a spatiotemporal predator-prey system. Mathematical Biosciences and Engineering, 2019, 16(1): 338-372. doi: 10.3934/mbe.2019017


  • 1. B.R. Anholt and E.E.Werner, Density-dependent consequences of induced behavior, in The ecology and evolution of inducible defenses (eds. R. Tollrian and C.D. Harvell), Princeton University Press, USA, 1999, 218–230.
  • 2. Z. Barta, A. Liker and F. Mónus, The effects of predation risk on the use of social foraging tactics, Anim. Behav., 67 (2004), 301–308.
  • 3. B.A. Belgrad and B.D. Griffen, Predator–prey interactions mediated by prey personality and predator hunting mode, Proc. R. Soc. B, 283 (2016), 20160408.
  • 4. C. Boesch, The effects of leopard predation on grouping patterns in forest chimpanzees, Behaviour, 117 (1991), 220–241.
  • 5. M.J. Childress and M.A. Lung, Predation risk, gender and the group size effect: does elk vigilance depend upon the behaviour of conspecifics?, Anim. Behav., 66 (2003), 389–398.
  • 6. S.N. Chow and J.K. Hale, Methods of Bifurcation Theory, vol. 251, Springer-Verlag, New York, 1982.
  • 7. S. Creel and D. Christianson, Relationships between direct predation and risk effects, Trends Ecol. Evol., 23 (2008), 194–201.
  • 8. S. Creel, D. Christianson, S. Liley and J.A. Winnie Jr., Predation risk affects reproductive physiology and demography of elk, Science, 315 (2007), 960–960.
  • 9. S. Creel, J. Winnie Jr., B. Maxwell, K. Hamlin and M. Creel, Elk alter habitat selection as an antipredator response to wolves, Ecology, 86 (2005), 3387–3397.
  • 10. D. Fortin, H.L. Beyer, M.S. Boyce, D.W. Smith, T. Duchesne and J.S. Mao, Wolves influence elk movements: behavior shapes a trophic cascade in Yellowstone National Park, Ecology, 86 (2005), 1320–1330.
  • 11. M.R. Garvie, Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator– Prey Interactions in MATLAB, Bull. Math. Biol., 69 (2007), 931–956.
  • 12. B.D. Hassard, N.D. Kazarinoff and Y.H. Wan, Theory and applications of Hopf bifurcation, vol. 41, CUP Archive, 1981.
  • 13. M. Hebblewhite, C.A. White, C.G. Nietvelt, J.A. McKenzie, T.E. Hurd, J.M. Fryxell, S.E. Bayley and P.C. Paquet, Human activity mediates a trophic cascade caused by wolves, Ecology, 86 (2005), 2135–2144.
  • 14. D.S. Hik, Does risk of predation influence population dynamics? Evidence from the cyclic decline of snowshoe hares, Wildl. Res., 22 (1995), 115–129.
  • 15. S.B. Hsu and T.W. Huang, Global stability for a class of predator–prey systems, SIAM J. Appl. Math., 55 (1995), 763–783.
  • 16. M.S. Khudr, O.Y. Buzhdygan, J.S. Petermann and S.Wurst, Fear of predation alters clone-specific performance in phloem-feeding prey, Sci. Rep., 7 (2017), 7695.
  • 17. C.J. Krebs, Two complementary paradigms for analysing population dynamics, Philos. Trans. R. Soc. Lond. B. Biol. Sci., 357 (2002), 1211–1219.
  • 18. S.L. Lima, Nonlethal effects in the ecology of predator-prey interactions, Bioscience, 48 (1998), 25–34.
  • 19. S.L. Lima, Predators and the breeding bird: behavioral and reproductive flexibility under the risk of predation, Biol. Rev., 84 (2009), 485–513.
  • 20. S.L. Lima and P.A. Bednekoff, Temporal variation in danger drives antipredator behavior: the predation risk allocation hypothesis, Am. Nat., 153 (1999), 649–659.
  • 21. S.J. McCauley, L. Rowe and M.J. Fortin, The deadly effects of
  • 22. J.P. Michaud, P.R.R. Barbosa, C.L. Bain and J.B. Torres, Extending the
  • 23. E.H. Nelson, C.E. Matthews and J.A. Rosenheim, Predators reduce prey population growth by inducing changes in prey behavior, Ecology, 85 (2004), 1853–1858.
  • 24. K.L. Pangle, S.D. Peacor and O.E. Johannsson, Large nonlethal effects of an invasive invertebrate predator on zooplankton population growth rate, Ecology, 88 (2007), 402–412.
  • 25. M. Pascual, Diffusion-induced chaos in a spatial predator–prey system, in Proc. R. Soc. Lond. B, vol. 251, 1993, 1–7.
  • 26. B.L. Peckarsky, C.A. Cowan, M.A. Penton and C. Anderson, Sublethal consequences of streamdwelling predatory stoneflies on mayfly growth and fecundity, Ecology, 74 (1993), 1836–1846.
  • 27. E.L. Preisser, D.I. Bolnick and M.F. Benard, Scared to death? The effects of intimidation and consumption in predator–prey interactions, Ecology, 86 (2005), 501–509.
  • 28. R.M. Sapolsky, Why Zebras Don't Get Ulcers, 3rd edn., Henry Holt and Company, New York, NY, 2004.
  • 29. O.J. Schmitz, A.P. Beckerman and K.M. O'Brien, Behaviorally mediated trophic cascades: effects of predation risk on food web interactions, Ecology, 78 (1997), 1388–1399.
  • 30. G. Sun, S. Sarwardi, P.J. Pal and M.S. Rahman, The spatial patterns through diffusion-driven instability in modified Leslie-Gower and Holling-type II predator-prey model, J. Biol. System., 18 (2010), 593–603.
  • 31. J.T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855–867.
  • 32. M. Travers, M. Clinchy, L. Zanette, R. Boonstra and T.D. Williams, Indirect predator effects on clutch size and the cost of egg production, Ecol. Lett., 13 (2010), 980–988.
  • 33. R.K. Upadhyay, V. Volpert and N.K. Thakur, Propagation of Turing patterns in a plankton model, J. Biol. Dynam., 6 (2012), 524–538.
  • 34. R.K. Upadhyay and P. Roy, Disease spread and its effect on population dynamics in heterogeneous environment, Int. J. Bifurc. Chaos, 26 (2016), 1650004.
  • 35. R.K. Upadhyay, P. Roy and J. Datta, Complex dynamics of ecological systems under nonlinear harvesting: Hopf bifurcation and Turing instability, Nonlinear Dyn., 79 (2015), 2251–2270.
  • 36. C.Wang, L. Chang and H. Liu, Spatial patterns of a predator-prey system of Leslie type with time delay, PloS one, 11 (2016), e0150503.
  • 37. X. Wang, L. Zanette and X. Zou, Modelling the fear effect in predator–prey interactions, J. Math. Biol., 73 (2016), 1179–1204.
  • 38. X. Wang and X. Zou, Modeling the Fear Effect in Predator–Prey Interactions with Adaptive Avoidance of Predators, Bull. Math. Biol., 79 (2017), 1325–1359.
  • 39. X. Wang and X. Zou, Pattern formation of a predator-prey model with the cost of anti–predator behaviors, Math. Biosci. Engineer., 15 (2018), 775–805.
  • 40. J. Winnie Jr., D. Christianson, S. Creel and B. Maxwell, Elk decision-making rules are simplified in the presence of wolves, Behav. Ecol. Sociobiol., 61 (2006), 277–289.
  • 41. J. Winnie Jr. and S. Creel, Sex-specific behavioural responses of elk to spatial and temporal variation in the threat of wolf predation, Anim. Behav., 73 (2007), 215–225.
  • 42. L.Y. Zanette, A.F. White, M.C. Allen and M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398–1401.
  • 43. J.F. Zhang, W.T. Li and X.P. Yan, Hopf bifurcations in a predator-prey diffusion system with Beddington-DeAngelis response, Acta Appl. Math., 115 (2011), 91–104.


This article has been cited by

  • 1. Huisen Zhang, Yongli Cai, Shengmao Fu, Weiming Wang, Impact of the fear effect in a prey-predator model incorporating a prey refuge, Applied Mathematics and Computation, 2019, 356, 328, 10.1016/j.amc.2019.03.034
  • 2. Jing Wang, Yongli Cai, Shengmao Fu, Weiming Wang, The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge, Chaos: An Interdisciplinary Journal of Nonlinear Science, 2019, 29, 8, 083109, 10.1063/1.5111121
  • 3. Swati Mishra, Ranjit Kumar Upadhyay, Strategies for the existence of spatial patterns in predator–prey communities generated by cross-diffusion, Nonlinear Analysis: Real World Applications, 2020, 51, 103018, 10.1016/j.nonrwa.2019.103018
  • 4. Ting Qiao, Yongli Cai, Shengmao Fu, Weiming wang, Stability and Hopf Bifurcation in a Predator–Prey Model with the Cost of Anti-Predator Behaviors, International Journal of Bifurcation and Chaos, 2019, 29, 13, 1950185, 10.1142/S0218127419501852
  • 5. Vandana Tiwari, Jai Prakash Tripathi, Swati Mishra, Ranjit Kumar Upadhyay, Modeling the fear effect and stability of non-equilibrium patterns in mutually interfering predator–prey systems, Applied Mathematics and Computation, 2020, 371, 124948, 10.1016/j.amc.2019.124948
  • 6. Mainul Hossain, Nikhil Pal, Sudip Samanta, Impact of fear on an eco-epidemiological model, Chaos, Solitons & Fractals, 2020, 134, 109718, 10.1016/j.chaos.2020.109718
  • 7. Mainul Hossain, Nikhil Pal, Sudip Samanta, Joydev Chattopadhyay, Fear Induced Stabilization in an Intraguild Predation Model, International Journal of Bifurcation and Chaos, 2020, 30, 04, 2050053, 10.1142/S0218127420500534
  • 8. Debasis Mukherjee, Role of fear in predator–prey system with intraspecific competition, Mathematics and Computers in Simulation, 2020, 10.1016/j.matcom.2020.04.025
  • 9. Pijush Panday, Nikhil Pal, Sudip Samanta, Joydev Chattopadhyay, A Three Species Food Chain Model with Fear Induced Trophic Cascade, International Journal of Applied and Computational Mathematics, 2019, 5, 4, 10.1007/s40819-019-0688-x
  • 10. Shilpa Samaddar, Mausumi Dhar, Paritosh Bhattacharya, Effect of fear on prey–predator dynamics: Exploring the role of prey refuge and additional food, Chaos: An Interdisciplinary Journal of Nonlinear Science, 2020, 30, 6, 063129, 10.1063/5.0006968

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved