Skip to main content
SpringerLink
Log in

Exploring the cascading effect of fear on the foraging activities of prey in a three species Agroecosystem

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

Fear of top predators may be as or more important than direct killing in causing trophic cascades. The mere presence of top predators gives rise to a “landscape of fear”, buffering lower trophic levels from overconsumption by herbivores. In the present work, we have proposed a three-species food chain model of wolf spiders (top predator), insect pests (herbivores), and plant community interactions incorporating the cost of fear of wolf spiders in the predation rate of insect pests. It is assumed that some of the insect pests migrate due to the fear of wolf spiders. Positivity, boundedness, and parameter regimes for an ecologically realistic parameter set is identified. We have derived the local and global stability conditions for the proposed system. The existence of Hopf-bifurcation, direction and stability conditions of the bifurcating periodic solutions have been discussed. It is observed that the fear of wolf spiders has a stabilizing impact on system dynamics. It contributes positively to the role of wolf spiders as a biocontrol agent. One and two-parameter bifurcation diagrams have been plotted. It is observed that fear level decreases with an increase in the migration rate and for relatively large values of conversion rate, insect pests become less sensitive toward the perceived predation risk.

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

Data availability

The data that support the findings of this study are available within the article.

References

  1. R.P. Prasad, W.E. Snyder, Diverse trait-mediated interactions in a multi-predator, multi-prey community. Ecology 87(5), 1131–1137 (2006)

    Article  Google Scholar 

  2. A. Sih, K.J. Mathot, M. Moirón, P.O. Montiglio, M. Wolf, N.J. Dingemanse, Animal personality and state-behaviour feedbacks: a review and guide for empiricists. Trends Ecol. Evol. 30(1), 50–60 (2015)

    Article  Google Scholar 

  3. A. Lang, J. Filser, J.R. Henschel, Predation by ground beetles and wolf spiders on herbivorous insects in a maize crop. Agric. Ecosyst. Environ. 72(2), 189–199 (1999)

    Article  Google Scholar 

  4. W.F. Fagan, A.L. Hakim, H. Ariawan, S. Yuliyantiningsih, Interactions between biological control efforts and insecticide applications in tropical rice agroecosystems: the potential role of intraguild predation. Biol. Control 13(2), 121–126 (1998)

    Article  Google Scholar 

  5. K. Sunderland, Mechanisms underlying the effects of spiders on pest populations. Journal of Arachnology , 308–316 (1999)

  6. C.W. Agnew, J.W. Smith Jr., Ecology of spiders (Araneae) in a peanut agroecosystem. Environ. Entomol. 18(1), 30–42 (1989)

    Article  Google Scholar 

  7. P. Marc, A. Canard, F. Ysnel, Spiders (Araneae) useful for pest limitation and bioindication. Agric. Ecosyst. Environ. 74(1–3), 229–273 (1999)

    Article  Google Scholar 

  8. A.L. Rypstra, C.M. Buddle, Spider silk reduces insect herbivory. Biol. Lett. 9(1), 20120948 (2013)

    Article  Google Scholar 

  9. M.C. Barnes, M.H. Persons, A.L. Rypstra, The effect of predator chemical cue age on antipredator behavior in the wolf spider pardosa milvina (Araneae: Lycosidae). J. Insect Behav. 15(2), 269–281 (2002)

    Article  Google Scholar 

  10. S.L. Lima, L.M. Dill, Behavioral decisions made under the risk of predation: a review and prospectus. Can. J. Zool. 68(4), 619–640 (1990)

    Article  Google Scholar 

  11. J.P. Suraci, M. Clinchy, L.M. Dill, D. Roberts, L.Y. Zanette, Fear of large carnivores causes a trophic cascade. Nat. Commun. 7, 10698 (2016)

    Article  ADS  Google Scholar 

  12. C.J. Krebs, S. Boutin, R. Boonstra, A.R.E. Sinclair, J.N.M. Smith, M.R. Dale, K. Martin, R. Turkington, Impact of food and predation on the snowshoe hare cycle. Science 269(5227), 1112–1115 (1995)

    Article  ADS  Google Scholar 

  13. B.L. Peckarsky, C.A. Cowan, M.A. Penton, C. Anderson, Sublethal consequences of stream-dwelling predatory stoneflies on mayfly growth and fecundity. Ecology 74(6), 1836–1846 (1993)

    Article  Google Scholar 

  14. O.J. Schmitz, A.P. Beckerman, K.M. O’Brien, Behaviorally mediated trophic cascades: effects of predation risk on food web interactions. Ecology 78(5), 1388–1399 (1997)

  15. S. Creel, J. Winnie Jr., B. Maxwell, K. Hamlin, M. Creel, Elk alter habitat selection as an antipredator response to wolves. Ecology 86(12), 3387–3397 (2005)

    Article  Google Scholar 

  16. S.L. Lima, P.A. Bednekoff, Temporal variation in danger drives antipredator behavior: the predation risk allocation hypothesis. Am. Naturalist 153(6), 649–659 (1999)

    Article  Google Scholar 

  17. J. Winnie Jr., S. Creel, Sex-specific behavioural responses of elk to spatial and temporal variation in the threat of wolf predation. Anim. Behav. 73(1), 215–225 (2007)

    Article  Google Scholar 

  18. M.J. Childress, M.A. Lung, Predation risk, gender and the group size effect: does elk vigilance depend upon the behaviour of conspecifics? Anim. Behav. 66(2), 389–398 (2003)

    Article  Google Scholar 

  19. Z. Barta, A. Liker, F. Mónus, The effects of predation risk on the use of social foraging tactics. Anim. Behav. 67(2), 301–308 (2004)

    Article  Google Scholar 

  20. C. Boesch, The effects of leopard predation on grouping patterns in forest chimpanzees. Behaviour, 220–242 (1991)

  21. D. Fortin, H.L. Beyer, M.S. Boyce, D.W. Smith, T. Duchesne, J.S. Mao, Wolves influence elk movements: behavior shapes a trophic cascade in yellowstone national park. Ecology 86(5), 1320–1330 (2005)

    Article  Google Scholar 

  22. L.Y. Zanette, A.F. White, M.C. Allen, M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year. Science 334(6061), 1398–1401 (2011)

    Article  ADS  Google Scholar 

  23. X. Wang, L. Zanette, X. Zou, Modelling the fear effect in predator-prey interactions. J. Math. Biol. 73(5), 1179–1204 (2016)

    Article  MathSciNet  Google Scholar 

  24. X. Wang, X. Zou, Modeling the fear effect in predator-prey interactions with adaptive avoidance of predators. Bull. Math. Biol. 79(6), 1325–1359 (2017)

    Article  MathSciNet  Google Scholar 

  25. P. Panday, N. Pal, S. Samanta, J. Chattopadhyay, Stability and bifurcation analysis of a three-species food chain model with fear. Int. J. Bifurcat. Chaos 28(01), 1850009 (2018)

    Article  MathSciNet  Google Scholar 

  26. J. Wang, Y. Cai, S. Fu, W. Wang, The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge. Chaos: An Interdisciplinary. J. Nonlinear Sci. 29(8), 083109 (2019)

    MATH  Google Scholar 

  27. H. Zhang, Y. Cai, S. Fu, W. Wang, Impact of the fear effect in a prey-predator model incorporating a prey refuge. Appl. Math. Comput. 356, 328–337 (2019)

    Article  MathSciNet  Google Scholar 

  28. S.K. Sasmal, Y. Takeuchi, Dynamics of a predator-prey system with fear and group defense. J. Math. Anal. Appl 481(1), 123471 (2020)

    Article  MathSciNet  Google Scholar 

  29. X. Wang, X. Zou, Pattern formation of a predator-prey model with the cost of anti-predator behaviors. Math. Biosci. Eng. 15(3), 775 (2018)

    Article  MathSciNet  Google Scholar 

  30. R.K. Upadhyay, S. Mishra, Population dynamic consequences of fearful prey in a spatiotemporal predator-prey system. Math. Biosci. Eng. 16(1), 338–372 (2018)

    Article  MathSciNet  Google Scholar 

  31. S. Mishra, R.K. Upadhyay, Strategies for the existence of spatial patterns in predator-prey communities generated by cross-diffusion. Nonlinear Anal. Real World Appl. 51, 103018 (2020)

    Article  MathSciNet  Google Scholar 

  32. D. Jana, S. Batabyal, M. Lakshmanan, Self-diffusion-driven pattern formation in prey-predator system with complex habitat under fear effect. Eur. Phys. J. Plus 135(11), 1–42 (2020)

    Article  Google Scholar 

  33. F. Souna, S. Djilali, A. Lakmeche, Spatiotemporal behavior in a predator-prey model with herd behavior and cross-diffusion and fear effect. Eur. Phys. J. Plus 136(5), 1–21 (2021)

    Article  Google Scholar 

  34. R.K. Upadhyay, Multiple attractors and crisis route to chaos in a model food-chain. Chaos, Solitons & Fractals 16(5), 737–747 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  35. O.P. Young, G.B. Edwards, Spiders in united states field crops and their potential effect on crop pests. Journal of Arachnology , 1–27 (1990)

  36. S.E. Riechert, T. Lockley, Spiders as biological control agents. Annu. Rev. Entomol. 29(1), 299–320 (1984)

    Article  Google Scholar 

  37. M.A. Aziz-Alaoui, M.D. Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Appl. Math. Lett. 16(7), 1069–1075 (2003)

    Article  MathSciNet  Google Scholar 

  38. M. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population model. Chaos, Solitons & Fractals 14(8), 1275–1293 (2002)

  39. R.K. Upadhyay, S. Iyengar, V. Rai, Chaos: an ecological reality? Int. J. Bifurc. Chaos 8(06), 1325–1333 (1998)

    Article  MathSciNet  Google Scholar 

  40. B.D. Hassard, B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Y.W. Wan, Theory and Applications of Hopf Bifurcation, vol. 41 (CUP Archive, 1981)

  41. R.K. Upadhyay, S. Kumari, Bifurcation analysis of an e-epidemic model in wireless sensor network. Int. J. Comput. Math. 95(9), 1775–1805 (2018)

    Article  MathSciNet  Google Scholar 

  42. Y. Lin, X. Zeng, Z. Jing, Dynamical behaviors for a three-dimensional differential equation in chemical system. Acta Math. Appl. Sin. 12(2), 144–154 (1996)

    Article  MathSciNet  Google Scholar 

  43. S.L. Lima, Nonlethal effects in the ecology of predator-prey interactions. Bioscience 48(1), 25–34 (1998)

    Article  Google Scholar 

  44. O.J. Schmitz, J.H. Grabowski, B.L. Peckarsky, E.L. Preisser, G.C. Trussell, J.R. Vonesh, From individuals to ecosystem function: toward an integration of evolutionary and ecosystem ecology. Ecology 89(9), 2436–2445 (2008)

    Article  Google Scholar 

  45. O.J. Schmitz, K.B. Suttle, Effects of top predator species on direct and indirect interactions in a food web. Ecology 82(7), 2072–2081 (2001)

    Article  Google Scholar 

  46. J.S. Thaler, C.A.M. Griffin, Relative importance of consumptive and non-consumptive effects of predators on prey and plant damage: the influence of herbivore ontogeny. Entomol. Exp. Appl. 128(1), 34–40 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics & Computing, Indian Institute of Technology (ISM), Dhanbad, Jharkhand, 826004, India

    Swati Mishra & Ranjit Kumar Upadhyay

Authors
  1. Swati Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ranjit Kumar Upadhyay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ranjit Kumar Upadhyay.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Appendix

Appendix

$$\begin{aligned} Q_1=&\frac{1}{\varDelta }\left[ q_{31}\left\{ -\frac{p_{11}^2r_1}{K_1} +\frac{\omega }{D^2}(1-kS^*)(P^*-D)p_{11}p_{21}\right\} \right. \\&\left. +q_{32} \Bigg \lbrace -\frac{\omega _1}{D^2}(1-kS^*)(P^*-D)p_{11}p_{21}\right. \\&\left. +\frac{2I^*\omega _2}{D_1^4}(I^*{^2}-D_1^2)p_{21}p_{31}+\frac{\omega _2S^*}{D_1^4}(I^*{^2}-D_1^2)p_{21}^2\Bigg \rbrace \right. \\&\left. + q_{33}\left\{ -\frac{p_{31}^2r_2}{K_2}-\frac{2I^*(I^*{^2}-D_1^2) \omega _3p_{21}p_{31}}{D_1^4}\right. \right. \\&\left. \left. -\frac{\omega _3S^*(I^*{^2}-D_1^2)p_{21}^2}{D_1^4}\right\} \right] ,\\ Q_2=&\frac{1}{\varDelta }\left[ q_{31}\left\{ -\frac{2p_{11}p_{12}r_1}{K_1} +\frac{\omega }{D^2}(1-kS^*)(P^*-D)(p_{12}p_{21}+p_{11}p_{22})\right\} \right. \\&\left. +q_{32}\bigg \lbrace -\frac{\omega _1}{D^2}(1-kS^*)(P^*-D)\times (p_{12}p_{21}+p_{11}p_{22}) \right. \\&\left. +\frac{2I^*\omega _2}{D_1^4}(I^*{^2}-D_1^2)(p_{22}p_{31}+p_{21}p_{32}) +\frac{2\omega _2S^*}{D_1^4}(I^*{^2}-D_1^2)p_{21}p_{22}\bigg \rbrace \right. \\&\left. + q_{33}\Bigg \lbrace -\frac{2p_{31}p_{32}r_2}{K_2}-\frac{2I^*(I^*{^2}-D_1^2) \omega _3(p_{22}p_{31}+p_{21}p_{32})}{D_1^4}\right. \\&\left. -\frac{2\omega _3S^*(I^*{^2}-D_1^2)p_{21}p_{22}}{D_1^4}\Bigg \rbrace \right] ,\\ Q_3=&\frac{1}{\varDelta }\left[ q_{31}\left\{ -\frac{p_{12}^2r_1}{K_1}+\frac{\omega }{D^2}(1-kS^*)(P^*-D)p_{12}p_{22}\right\} \right. \\&\left. +q_{32} \bigg \lbrace -\frac{\omega _1}{D^2}(1-kS^*)(P^*-D)p_{12}p_{22} \right. \\&\left. +\frac{2I^*\omega _2}{D_1^4}(I^*{^2}-D_1^2)p_{22}p_{32}+\frac{\omega _2S^*}{D_1^4}(I^*{^2}-D_1^2)p_{22}^2\bigg \rbrace \right. \\&\left. + q_{33}\left\{ -\frac{p_{32}^2r_2}{K_2}-\frac{2I^*(I^*{^2}-D_1^2) \omega _3p_{22}p_{32}}{D_1^4}\right. \right. \\&\left. \left. -\frac{\omega _3S^*(I^*{^2}-D_1^2)p_{22}^2}{D_1^4}\right\} \right] . \end{aligned}$$
$$\begin{aligned} m_1=&-\frac{r_1}{K_1}\left\{ p_{11}y_1+p_{12}y_2+\frac{p_{13}}{2}(b_{11}y_{1}^2+2b_{12}y_1y_2 +b_{22}y_2^2)\right\} ^2\\&+\frac{\omega }{D^2}(1-kS^*)(P^*-D)\times \left\{ p_{11}y_1+p_{12}y_2 \right. \\&\left. +\frac{p_{13}}{2}(b_{11}y_1^2+2 b_{12}y_1y_2+b_{22}y_2^2) \right\} \\&\left\{ p_{21} y_1+ p_{22} y_2+\frac{p_{23}}{2}(b_{11}y_1^2+2 b_{12}y_1 y_2+b_{22}y_2^2)\right\} +\text {h.o.t},\\ m_2=&\frac{\omega _2 S^*(I^*{^2}-D_1^2)}{D_1^4}\left\{ p_{21}y_1+p_{22}y_2+\frac{p_{23}}{2}(b_{11}y_1^2+2 b_{12}y_1y_2+b_{22}y_2^2) \right\} ^2\\&+\frac{2\omega _2I^*}{D_1^4}(I^*{^2}-D_1^2)\times \left\{ p_{21}y_1 \right. \\&\left. +p_{22}y_2+\frac{p_{23}}{2}(b_{11}y_1^2+2 b_{12}y_1y_2+b_{22}y_2^2)\right\} \times \\&\left\{ p_{31}y_1+p_{32}y_2+\frac{p_{33}}{2}(b_{11}y_1^2+2 b_{12}y_1y_2+b_{22}y_2^2)\right\} \\&+\frac{\omega _2}{D_1^4} (I^*{^2}-D_1^2)\left\{ p_{21}y_1+p_{22}y_2+\frac{p_{23}}{2}(b_{11}y_1^2+2 b_{12}y_1y_2+b_{22}y_2^2)\right\} ^2 \\&\times \bigg \lbrace p_{31}y_1+p_{32}y_2+\frac{p_{33}}{2}(b_{11}y_1^2 \\&+2 b_{12}y_1y_2+b_{22}y_2^2)\bigg \rbrace -\frac{\omega _1}{D}(1-kS^*)(P^*-D)\\&\left\{ p_{11}y_1+p_{12}y_2+\frac{p_{13}}{2}(b_{11}y_1^2+2 b_{12}y_1 y_2+b_{22}y_2^2)\right\} \\&\times \left\{ p_{21}y_1+p_{22}y_2+\frac{p_{23}}{2}(b_{11}y_1^2+2 b_{12}y_1 y_2+b_{22}y_2^2)\right\} +\text {h.o.t},\\ m_3=&-\frac{r_2}{K_2}\left\{ p_{31}y_1+p_{32}y_2+\frac{p_{33}}{2}(b_{11}y_{1}^2+2b_{12}y_1y_2 +b_{22}y_2^2)\right\} ^2\\&-\frac{\omega _3 S^* (I^*{^2}-D_1^2)}{D_1^4}\left\{ p_{21}y_1+p_{22}y_2\right. \\&\left. +\frac{p_{23}}{2}(b_{11}y_1^2+2 b_{12}y_1 y_2+b_{22}y_2^2)\right\} ^2-\frac{2\omega _3I^*(I^*{^2}-D_1^2)}{D_1^4}\\&\bigg \lbrace p_{21}y_1+p_{22}y_2+\frac{p_{23}}{2}(b_{11}y_1^2+2 b_{12}y_1 y_2\\&+b_{22}y_2^2)\bigg \rbrace \times \left\{ p_{31}y_1+p_{32}y_2+\frac{p_{33}}{2}(b_{11}y_1^2+2 b_{12}y_1 y_2+b_{22}y_2^2)\right\} \\&-\frac{\omega _3 (I^*{^2}-D_1^2) }{D_1^4}\left\{ p_{21}y_1+p_{22}y_2+\frac{p_{23}}{2}(b_{11}y_1^2+2 b_{12}y_1 y_2+b_{22}y_2^2) \right\} ^2 \\&\left\{ p_{31}y_1+p_{32}y_2+\frac{p_{33}}{2}(b_{11}y_1^2+2 b_{12}y_1 y_2+b_{22}y_2^2)\right\} \\&+\text {h.o.t}. \end{aligned}$$
$$\begin{aligned} f_{11}^1=&\frac{2}{\varDelta }\left[ q_{11}\left\{ -\frac{p_{11}^2r_1}{K_1}+\frac{\omega }{D^2}(1-kS^*)(P^*-D)p_{11}p_{21} \right\} \right. \\&\left. +q_{12}\left\{ -\frac{\omega _1}{D^2}(1-kS^*)(P^*-D)p_{11}p_{21} \right. \right. \\&\left. \left. +\frac{2I^*\omega _2}{D_1^4}(I^*{^2}-D_1^2) p_{21}p_{31}+\frac{\omega _2 S^*}{D_1^4}(I^*{^2}-D_1^2)p_{21}^2\right\} \right. \\&\left. -q_{13}\left\{ \frac{p_{31}^2r_2}{K_2}+\frac{2I^*(I^*{^2} -D_1^2)\omega _3}{D_1^4}p_{21}p_{31}\right. \right. \\&\left. \left. +\frac{\omega _3S^*(I^*{^2}-D_1^2)}{D_1^4}p_{21}^2 \right\} \right] ,\\ f_{11}^2=&\frac{2}{\varDelta }\left[ q_{21}\left\{ -\frac{p_{11}^2r_1}{K_1}+\frac{\omega }{D^2}(1-kS^*)(P^*-D)p_{11}p_{21}\right\} \right. \\&\left. +q_{22} \left\{ -\frac{\omega _1}{D^2}(1-kS^*)(P^*-D)p_{11}p_{21}\right. \right. \\&\left. \left. +\frac{2I^*\omega _2}{D_1^4}(I^*{^2}-D_1^2)p_{21}p_{31} +\frac{\omega _2 S^*}{D_1^4}(I^*{^2}-D_1^2)p_{21}^2\right\} \right. \\&\left. -q_{23}\left\{ \frac{p_{31}^2r_2}{K_2}+\frac{2I^*(I^*{^2}-D_1^2)\omega _3}{D_1^4} p_{21}p_{31}\right. \right. \\&\left. \left. +\frac{\omega _3S^*(I^*{^2}-D_1^2)}{D_1^4}p_{21}^2 \right\} \right] ,\\ f_{22}^1=&\frac{2}{\varDelta }\left[ q_{11}\left\{ -\frac{p_{12}^2r_1}{K_1} +\frac{\omega }{D^2}(1-kS^*)(P^*-D)p_{12}p_{22}\right\} \right. \\&\left. +q_{12} \bigg \lbrace -\frac{\omega _1}{D^2}(1-kS^*)(P^*-D)p_{12}p_{22}\right. \\&\left. +\frac{2I^*\omega _2}{D_1^4}(I^*{^2}-D_1^2)p_{22}p_{32} +\frac{\omega _2 S^*}{D_1^4}(I^*{^2}-D_1^2)p_{22}^2\bigg \rbrace \right. \\&\left. -q_{13} \left\{ \frac{p_{32}^2r_2}{K_2}+\frac{2I^*(I^*{^2}-D_1^2) \omega _3}{D_1^4}p_{22}p_{32}\right. \right. \\&\left. \left. +\frac{\omega _3S^*(I^*{^2}-D_1^2)}{D_1^4}p_{22}^2 \right\} \right] ,\\ \end{aligned}$$
$$\begin{aligned} f_{22}^2=&\frac{2}{\varDelta }\left[ q_{21}\left\{ -\frac{p_{12}^2r_1}{K_1} +\frac{\omega }{D^2}(1-kS^*)(P^*-D)p_{12}p_{22}\right\} \right. \\&\left. +q_{22}\bigg \lbrace -\frac{\omega _1}{D^2}(1-kS^*)(P^*-D)p_{12}p_{22}\right. \\&\left. +\frac{2I^*\omega _2}{D_1^4}(I^*{^2}-D_1^2)p_{22}p_{32} +\frac{\omega _2 S^*}{D_1^4}(I^*{^2}-D_1^2)p_{22}^2\bigg \rbrace \right. \\&\left. -q_{23}\left\{ \frac{p_{32}^2r_2}{K_2}+\frac{2I^*(I^*{^2}-D_1^2) \omega _3}{D_1^4}p_{22}p_{32}\right. \right. \\&\left. \left. +\frac{\omega _3S^*(I^*{^2}-D_1^2)}{D_1^4}p_{22}^2 \right\} \right] ,\\ f_{12}^1=&\frac{1}{\varDelta }\left[ q_{11}\left\{ -\frac{2p_{11} p_{12}r_1}{K_1}+\frac{\omega }{D^2}(1-kS^*)(P^*-D)(p_{12}p_{21}+p_{11}p_{22})\right\} \right. \\&\left. +q_{12}\bigg \lbrace -\frac{\omega _1}{D^2}(1-kS^*)\right. \\&\left. \left. \times (P^*-D)(p_{12}p_{21}+p_{11}p_{22}) +\frac{2I^*\omega _2}{D_1^4}(I^*{^2}-D_1^2)(p_{22}p_{31}+p_{21}p_{32})\right. \right. \\&\left. +\frac{2\omega _2 S^*}{D_1^4}(I^*{^2}-D_1^2)p_{21}p_{22}\bigg \rbrace \right. \\&\left. -q_{13}\Bigg \lbrace \frac{2p_{31}p_{32}r_2}{K_2} +\frac{2I^*(I^*{^2}-D_1^2)\omega _3}{D_1^4}(p_{22}p_{31}+p_{21}p_{32})\right. \\&\left. +\frac{2\omega _3S^*(I^*{^2}-D_1^2)}{D_1^4}p_{21}p_{22}\Bigg \rbrace \right] ,\\ f_{12}^2=&\frac{1}{\varDelta }\left[ q_{21}\left\{ -\frac{2p_{11} p_{12}r_1}{K_1}+\frac{\omega }{D^2}(1-kS^*)(P^*-D) (p_{12}p_{21}+p_{11}p_{22})\right\} \right. \\&\left. +q_{22}\left\{ -\frac{\omega _1}{D^2}(1-kS^*)\right. \right. \\&\left. \left. \times (P^*-D)(p_{12}p_{21}+p_{11}p_{22}) +\frac{2I^*\omega _2}{D_1^4}(I^*{^2}-D_1^2)(p_{22}p_{31}+p_{21}p_{32})\right. \right. \\&\left. \left. +\frac{2\omega _2 S^*}{D_1^4}(I^*{^2}-D_1^2)p_{21}p_{22}\right\} \right. \\&\left. -q_{23}\Bigg \lbrace \frac{2p_{31}p_{32}r_2}{K_2} +\frac{2I^*(I^*{^2}-D_1^2)\omega _3}{D_1^4}(p_{22}p_{31}+p_{21}p_{32})\right. \\&\left. +\frac{2\omega _3S^*(I^*{^2}-D_1^2)}{D_1^4}p_{21}p_{22}\Bigg \rbrace \right] ,\\ f_{111}^1=&\frac{3}{\varDelta }\left[ q_{11}\left\{ -\frac{2b_{11}p_{11} p_{13}r_1}{K_1}+\frac{\omega b_{11}}{D^2}(1-kS^*) (P^*-D)(p_{13}p_{21}+p_{11}p_{23})\right\} \right. \\&\left. +q_{12}\left\{ -\frac{\omega _1 b_{11}}{D^2}(1-kS^*)\right. \right. \\&\left. \left. \times (P^*-D)(p_{13}p_{21}+p_{11}p_{23}) +\frac{2\omega _2}{D_1^4}(I^*{^2}-D_1^2)p_{21}^2p_{31}\right. \right. \\&\left. \left. +\frac{2b_{11} I^*\omega _2 }{D_1^4}(I^*{^2}-D_1^2)(p_{23}p_{31}+p_{21}p_{33})\right. \right. \\&\left. \left. +\frac{2b_{11}(I^*{^2}-D_1^2)S^*\omega _2}{D_1^4}p_{21}p_{23}\right\} -q_{13}\left\{ \frac{2b_{11}p_{31}p_{33}r_2}{K_2}+\frac{2(I^*{^2}-D_1^2)\omega _3}{D_1^4}p_{21}^2p_{31}\right. \right. \\&\left. \left. +\frac{2b_{11}I^*(I^*{^2}-D_1^2)\omega _3}{D_1^4}(p_{23}p_{31}+p_{21}p_{33})+\frac{2b_{11}\omega _3S^*(I^*{^2}-D_1^2)}{D_1^4}p_{21}p_{23}\right\} \right] ,\\ f_{222}^2=&\frac{3}{\varDelta }\left[ q_{21}\left\{ -\frac{2b_{22}p_{12} p_{13}r_1}{K_1}+\frac{\omega b_{22}}{D^2}(1-kS^*)(P^*-D)(p_{13}p_{22} +p_{12}p_{23})\right\} +q_{22}\left\{ -\frac{\omega _1 b_{22}}{D^2}(1-kS^*)\right. \right. \\&\left. \left. \times (P^*-D) (p_{13}p_{22}+p_{12}p_{23})+\frac{2\omega _2}{D_1^4}(I^*{^2}-D_1^2) p_{22}^2p_{32}+\frac{2b_{22} I^*\omega _2 }{D_1^4}(I^*{^2}-D_1^2) (p_{23}p_{32}+p_{22}p_{33})\right. \right. \\&\left. \left. +\frac{2b_{22}(I^*{^2}-D_1^2)S^*\omega _2}{D_1^4}p_{22}p_{23}\right\} -q_{23}\left\{ \frac{2b_{22}p_{32}p_{33}r_2}{K_2} +\frac{2(I^*{^2}-D_1^2)\omega _3}{D_1^4}p_{22}^2p_{32}\right. \right. \\&\left. \left. +\frac{2b_{22}I^*(I^*{^2}-D_1^2)\omega _3}{D_1^4} (p_{23}p_{32}+p_{22}p_{33})+\frac{2b_{22}\omega _3S^*(I^*{^2}-D_1^2)}{D_1^4}p_{22}p_{23}\right\} \right] ,\\ \end{aligned}$$
$$\begin{aligned} f_{122}^1=&\frac{1}{\varDelta }\left[ q_{11}\left\{ -\frac{2b_{22}p_{11} p_{13}r_1}{K_1}-\frac{4b_{12}p_{12}p_{13}r_1}{K_1} +\frac{\omega b_{22}}{D^2}(1-kS^*)(P^*-D)(p_{13}p_{21}+p_{11}p_{23}) \right. \right. \\&\left. \left. +\,\,\frac{2b_{12}(1-kS^*)(P^*-D) \omega }{D^2}(p_{12}p_{23}+p_{13}p_{22})\right\} +q_{12}\left\{ -\frac{\omega _1 b_{22}}{D^2}(1-kS^*)(P^*-D)\right. \right. \\&\left. \left. \times \,(p_{13}p_{21}+p_{11}p_{23}) -\frac{2b_{12}(1-kS^*)(P^*-D)\omega _1}{D^2}(p_{13}p_{22}+p_{12}p_{23})\right. \right. \\&\left. \left. +\frac{2\omega _2(I^*{^2}-D_1^2)p_{22}^2p_{31}}{D_1^4}\right. \right. \\&\left. \left. +\,\frac{2\omega _2b_{22}I^*}{D_1^4}(I^*{^2}-D_1^2) (p_{23}p_{31}+p_{21}p_{33})+\frac{4\omega _2(I^*{^2}-D_1^2)p_{21} p_{22}p_{32}}{D_1^4}\right. \right. \\&\left. \left. +\,\frac{4b_{12} I^*\omega _2 }{D_1^4}(I^*{^2}-D_1^2) \right. \right. \\&\left. \left. \times (p_{23}p_{32}+p_{22}p_{33}) +\frac{2b_{22}(I^*{^2}-D_1^2)S^*\omega _2}{D_1^4}p_{21}p_{23} +\frac{4b_{12}(I^*{^2}-D_1^2)S^*\omega _2}{D_1^4}p_{22}p_{23}\right\} \right. \\&\left. +\,q_{13}\left\{ -\frac{b_{22}(1-kS^*)(P^*-D)\omega _1}{D^2}(p_{13}p_{21}+p_{11}p_{23}) -\frac{2b_{12}(1-kS^*)(P^*-D)\omega _1}{D^2}\right. \right. \\&\left. \left. \times (p_{13}p_{22}+p_{12}p_{23}) +\frac{2(I^*{^2}-D_1^2)\omega _2}{D_1^4}p_{22}^2p_{31} +\frac{2b_{22}I^*(I^*{^2}-D_1^2)\omega _2}{D_1^4}(p_{23}p_{31}+p_{21}p_{33}) \right. \right. \\&\left. \left. +\,\frac{4(I^*{^2}-D_1^2)p_{21}p_{22}p_{32}\omega _2}{D_1^4} +\frac{4b_{12}I^*(I^*{^2}-D_1^2)\omega _2}{D_1^4}(p_{23}p_{32} +p_{22}p_{33})\right. \right. \\&\left. \left. +\,\frac{2b_{22}\omega _2S^*(I^*{^2}-D_1^2)}{D_1^4}\right. \right. \\&\left. \left. \times p_{21}p_{23}+\frac{4b_{12} \omega _2S^*(I^*{^2}-D_1^2)}{D_1^4}p_{22}p_{23}\right\} \right] ,\\ f_{112}^2=&\frac{1}{\varDelta }\left[ q_{21}\left\{ -\frac{4b_{12}p_{11} p_{13}r_1}{K_1}-\frac{2b_{11}p_{12}p_{13}r_1}{K_1} +\frac{2\omega b_{12}(1-kS^*)(P^*-D)}{D^2}(p_{13}p_{21}+p_{11}p_{23}) \right. \right. \\&\left. \left. +\,\frac{b_{11}(1-kS^*)(P^*-D) \omega }{D^2}(p_{13}p_{22}+p_{12}p_{23})\right\} +q_{22}\left\{ -\frac{2\omega _1 b_{12}(1-kS^*)(P^*-D)}{D^2}\right. \right. \\&\left. \left. \times (p_{13}p_{21}+p_{11}p_{23}) -\frac{b_{11}(1-kS^*)(P^*-D)\omega _1}{D^2}(p_{13}p_{22}+p_{12}p_{23}) +\frac{4\omega _2(I^*{^2}-D_1^2)}{D_1^4}\right. \right. \\&\left. \left. \times p_{21}p_{22}p_{31}+\frac{4\omega _2b_{12}I^*}{D_1^4}(I^*{^2}-D_1^2) (p_{23}p_{31}+p_{21}p_{33})+\frac{2\omega _2(I^*{^2}-D_1^2)p_{21}^2p_{32}}{D_1^4}\right. \right. \\&\left. \left. +\,\frac{2b_{11} I^*\omega _2 }{D_1^4}(I^*{^2}-D_1^2)(p_{23}p_{32}+p_{22}p_{33})+\frac{4b_{12} (I^*{^2}-D_1^2)S^*\omega _2}{D_1^4}p_{21}p_{23}\right. \right. \\&\left. \left. +\,\frac{2b_{11}(I^*{^2}-D_1^2)S^*\omega _2}{D_1^4}p_{22}p_{23}\right\} \right. \\&\left. -\,q_{23}\left\{ \frac{4b_{12}r_2}{K_2}p_{31}p_{33}+\frac{2b_{11}p_{32}p_{33}r_2}{K_2}+\frac{4(I^*{^2}-D_1^2)\omega _3}{D_1^4}\right. \right. \\&\left. \left. \times p_{21}p_{22}p_{31}+\frac{4b_{12}I^*(I^*{^2}-D_1^2)\omega _3}{D_1^4} (p_{23}p_{31}+p_{21}p_{33})+\frac{2(I^*{^2}-D_1^2)\omega _3}{D_1^4}p_{21}^2p_{32}\right. \right. \\&\left. \left. +\,\frac{2b_{11}I^*(I^*{^2}-D_1^2)\omega _3}{D_1^4}(p_{23}p_{32}+p_{22}p_{33}) +\frac{4b_{12}(I^*{^2}-D_1^2)p_{21}p_{23}S^*\omega _3}{D_1^4}\right. \right. \\&\left. \left. +\,\frac{2b_{11}(I^*{^2}-D_1^2)S^*\omega _3}{D_1^4}p_{22}p_{23}\right\} \right] . \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mishra, S., Upadhyay, R.K. Exploring the cascading effect of fear on the foraging activities of prey in a three species Agroecosystem. Eur. Phys. J. Plus 136, 974 (2021). https://doi.org/10.1140/epjp/s13360-021-01936-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-021-01936-5

Search

Navigation