International Journal of Nonlinear Analysis and Applications
On the food chain model with prey refuge and fear effect
Document Type : Research Paper
Author
Department of Mathematics, Vivekananda Mahavidyalaya, Purba Bardhaman-713103, West Bengal, India
Abstract
Of concern the present study deals with an updated food chain model in a natural environment with the inclusion of fear effect in the prey population through Holling type II functional response in presence of prey refuge effect. The present model is affluent with intra-specific competition among the hunter species having specific mortality. The model system emphasizes its characteristics in the proximity of the probable equilibrium position in the realm of biological dynamics. The response of the system is explored further for its stability analysis based on prerequisites and Hopf-bifurcation phenomena as well with respect to some significant model parameters. Extensive numerical simulation reveals the validity of the proposed model so as to indicate the ecological implications.
Keywords
[1] N. Ali and S. Chakravarty, Consequence of prey refuge in a tri-trophic prey-dependent food chain model with
intra-specific competition, J. Appl. Non. Dyn. 3 (2014), 1–6.[2] R. Arditi and L. Ginzburg, Coupling in predator-prey dynamics: ratiodependence, J. Theo. Bio. 139 (1989),
311–326.
[3] A.D. Bazykin, Nonlinear dynamics of interacting populations, World Scientific, 1998.
[4] J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Ani.
Ecol. 44 (1975), 331–340.
[5] G. Birkhoff and G.C. Rota, Ordinary differential equations, Ginn, Boston, 1989.
[6] C. Castellano, S. Fortunato and V. Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81 (2009),
591.
[7] P. Cong, M. Fan and X. and Zou, Dynamics of a three-species food chain model with fear effect, Commn. Non.
Sc. Num. Sim. 99 (2021), 105809.
[8] S. Creel and N.M. Creel, Communal hunting and pack size in African wild dogs, Lycaon pictus, Anim. Behav. 50
(1995), 1325–1339.
[9] A. Dejean, C. Leroy, B. Corbara, O. Roux, P. Cereghino, J. Orivel and R. Boulay, Arboreal ants use the “velcro
principle” to capture very large prey, Plos. One 5 (2010), e11331.
[10] J.P. Fran¸coise and J. Llibre, Analytical study of a triple Hopf bifurcation in a tritrophic food chain model, Appl.
Math. Comp. 217 (2011), 7146–7154.
[11] S. Gakkhar and R. Naji, Seasonally perturbed prey-predator system with predator-dependent functional response,
Chaos. Sol. Frac. 18 (2003), 1075–1083.
[12] M. Haque, Ratio-dependent predator-prey models of interacting populations, B. Math. Bio. 71 (2009), 430–452.
[13] M. Haque, N. Ali and S. Chakravarty, Study of a tri-trophic prey-dependent food chain model of interacting
populations, Math. Bio. 246 (2013), 55–71.
[14] A. Hastings and T. Powell, Chaos in a three-species food chain, Ecol. 72 (1991), 896–903.
[15] C.S. Holling, The components of predation as revealed by a study of small-mammal predation of the European
pine sawfly, Canad. Entomologist. 91 (1959), 293–329.
[16] E. Holmes, M. Lewis, J. Banks and R. Veit, Partial differential equations in ecology: Spatial interactions and
population dynamics, Ecol. 75 (1994), 17–29.
[17] A. Klebanoff and A. Hastings, Chaos in three species food chains, J. Math. Bio. 32 (1994) 427–451.
[18] M. Kot, Elements of mathematical ecology, Cambridge University Press, 2001.
[19] V. Kumar and N. Kumari, Controlling chaos in three species food chain model with fear effect, AIMS Math. 5
(2020), 828–842.
[20] S. Lima and L.M. Dill, Behavioral decisions made under the risk of predation: A review and prospectus, Canad.
J. Zoo. 68 (1990), 619–640.
[21] A.J. Lotka, Elements of mathematical biology, Dover Publications, 2011.
[22] S. Lv and M. Zhao, The dynamic complexity of a three species food chain model, Chaos Solitons Fractals 37 (2008)
1469–1480.
[23] R.M. May, Stability and complexity in model ecosystems, Princeton University Press, 2001.
[24] X. Meng, R. Liu and T. Zhang, Adaptive dynamics for a non-autonomous Lotka-Volterra model with size-selective
disturbance, Non. Anal.: Real. Wrold. Appl. 16 (2014), 202–213.
[25] H. Molla, M.S. Rahman and S. Sarwardi, Dynamics of a predator-prey model with Holling type II functional
response incorporating a prey refuge depending on both the species, Int. J. Nonl. Sc. Num. Simul. 20 (2019), no.
1, 89–104.
[26] N. Mukherjee, S. Ghorai and M. Banerjee, Detection of turing patterns in a three species food chain model via
amplitude equation, Commn. Non. Sc. Num. Sim. 69 (2019), 219–236.[27] J.D. Murray, Mathematical biology II: Spatial models and biomedical applications, Springer-Verlag, New York,
2001.
[28] S. Pal, N. Pal, S. Samanta and J. Chattopadhyay, Effect of hunting cooperation and fear in a predator-prey model,
Ecol. Comp. 39 (2019), 100770.
[29] P. Pandey, N. Pal, S. Samanta J. and Chattopadhyay, Stability and bifurcation analysis of a three-species food
chain model with fear, Int. J. Bif. Chaos. 28 (2018), 1850009.
[30] E.C. Pielou, An introduction to mathematical ecology, John Wiley & Sons, 1969.
[31] M.L. Rosenzweig, and R.H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Amer. Nat. 97 (1963), 209–223.
[32] K. Sarkar and S. Khajanchi, Impact of fear effect on the growth of prey in a predator-prey interaction model, Ecol.
Comp. 42 (2020), 100826.
[33] S.K. Sasmal, Population dynamics with multiple Allee effects induced by fear factors-A mathematical study on
prey-predator interactions, Appl. Math. Modl. 64 (2018), 1–14.
[34] G. Seo and D.L. DeAngelis A predator-prey model with a Holling type I functional response including a predator
mutual interference, J. Nonlinear Sci. 21 (2011), 811–833.
[35] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theo. Biol. 79 (1979),
83–99.
[36] N. Sk, P.K. Tiwari, Y. Kang and S. Pal, A nonautonomous model for the interactive effects of fear, refuge and
additional food in a prey-predator system, J. Bio. Syst. 29 (2021), 107–145.
[37] N. Sk, P.K. Tiwari and S. Pal, A delay nonautonomous model for the impacts of fear and refuge in a three species
food chain model with hunting cooperation, Math. Comp. Simul. 192 (2022), 136–166.
[38] A. Szolnoki, M. Mobilia, L.-L. Jiang, B. Szczesny, A.M. Rucklidge and M. Perc, Cyclic dominance in evolutionary
games: A review, J. Royl. Soc. Intrf. 11 (2014), 20140735.
[39] R.J. Taylor, Predation, Chapman & Hall, 1984.
[40] X. Wang, L. Zanette and X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol. 73 (2016),
1179–1204.
[41] D. Xiao and S. Ruan, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J.
Appl. Math. 61 (2001), 1445–1472.
intra-specific competition, J. Appl. Non. Dyn. 3 (2014), 1–6.[2] R. Arditi and L. Ginzburg, Coupling in predator-prey dynamics: ratiodependence, J. Theo. Bio. 139 (1989),
311–326.
[3] A.D. Bazykin, Nonlinear dynamics of interacting populations, World Scientific, 1998.
[4] J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Ani.
Ecol. 44 (1975), 331–340.
[5] G. Birkhoff and G.C. Rota, Ordinary differential equations, Ginn, Boston, 1989.
[6] C. Castellano, S. Fortunato and V. Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81 (2009),
591.
[7] P. Cong, M. Fan and X. and Zou, Dynamics of a three-species food chain model with fear effect, Commn. Non.
Sc. Num. Sim. 99 (2021), 105809.
[8] S. Creel and N.M. Creel, Communal hunting and pack size in African wild dogs, Lycaon pictus, Anim. Behav. 50
(1995), 1325–1339.
[9] A. Dejean, C. Leroy, B. Corbara, O. Roux, P. Cereghino, J. Orivel and R. Boulay, Arboreal ants use the “velcro
principle” to capture very large prey, Plos. One 5 (2010), e11331.
[10] J.P. Fran¸coise and J. Llibre, Analytical study of a triple Hopf bifurcation in a tritrophic food chain model, Appl.
Math. Comp. 217 (2011), 7146–7154.
[11] S. Gakkhar and R. Naji, Seasonally perturbed prey-predator system with predator-dependent functional response,
Chaos. Sol. Frac. 18 (2003), 1075–1083.
[12] M. Haque, Ratio-dependent predator-prey models of interacting populations, B. Math. Bio. 71 (2009), 430–452.
[13] M. Haque, N. Ali and S. Chakravarty, Study of a tri-trophic prey-dependent food chain model of interacting
populations, Math. Bio. 246 (2013), 55–71.
[14] A. Hastings and T. Powell, Chaos in a three-species food chain, Ecol. 72 (1991), 896–903.
[15] C.S. Holling, The components of predation as revealed by a study of small-mammal predation of the European
pine sawfly, Canad. Entomologist. 91 (1959), 293–329.
[16] E. Holmes, M. Lewis, J. Banks and R. Veit, Partial differential equations in ecology: Spatial interactions and
population dynamics, Ecol. 75 (1994), 17–29.
[17] A. Klebanoff and A. Hastings, Chaos in three species food chains, J. Math. Bio. 32 (1994) 427–451.
[18] M. Kot, Elements of mathematical ecology, Cambridge University Press, 2001.
[19] V. Kumar and N. Kumari, Controlling chaos in three species food chain model with fear effect, AIMS Math. 5
(2020), 828–842.
[20] S. Lima and L.M. Dill, Behavioral decisions made under the risk of predation: A review and prospectus, Canad.
J. Zoo. 68 (1990), 619–640.
[21] A.J. Lotka, Elements of mathematical biology, Dover Publications, 2011.
[22] S. Lv and M. Zhao, The dynamic complexity of a three species food chain model, Chaos Solitons Fractals 37 (2008)
1469–1480.
[23] R.M. May, Stability and complexity in model ecosystems, Princeton University Press, 2001.
[24] X. Meng, R. Liu and T. Zhang, Adaptive dynamics for a non-autonomous Lotka-Volterra model with size-selective
disturbance, Non. Anal.: Real. Wrold. Appl. 16 (2014), 202–213.
[25] H. Molla, M.S. Rahman and S. Sarwardi, Dynamics of a predator-prey model with Holling type II functional
response incorporating a prey refuge depending on both the species, Int. J. Nonl. Sc. Num. Simul. 20 (2019), no.
1, 89–104.
[26] N. Mukherjee, S. Ghorai and M. Banerjee, Detection of turing patterns in a three species food chain model via
amplitude equation, Commn. Non. Sc. Num. Sim. 69 (2019), 219–236.[27] J.D. Murray, Mathematical biology II: Spatial models and biomedical applications, Springer-Verlag, New York,
2001.
[28] S. Pal, N. Pal, S. Samanta and J. Chattopadhyay, Effect of hunting cooperation and fear in a predator-prey model,
Ecol. Comp. 39 (2019), 100770.
[29] P. Pandey, N. Pal, S. Samanta J. and Chattopadhyay, Stability and bifurcation analysis of a three-species food
chain model with fear, Int. J. Bif. Chaos. 28 (2018), 1850009.
[30] E.C. Pielou, An introduction to mathematical ecology, John Wiley & Sons, 1969.
[31] M.L. Rosenzweig, and R.H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Amer. Nat. 97 (1963), 209–223.
[32] K. Sarkar and S. Khajanchi, Impact of fear effect on the growth of prey in a predator-prey interaction model, Ecol.
Comp. 42 (2020), 100826.
[33] S.K. Sasmal, Population dynamics with multiple Allee effects induced by fear factors-A mathematical study on
prey-predator interactions, Appl. Math. Modl. 64 (2018), 1–14.
[34] G. Seo and D.L. DeAngelis A predator-prey model with a Holling type I functional response including a predator
mutual interference, J. Nonlinear Sci. 21 (2011), 811–833.
[35] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theo. Biol. 79 (1979),
83–99.
[36] N. Sk, P.K. Tiwari, Y. Kang and S. Pal, A nonautonomous model for the interactive effects of fear, refuge and
additional food in a prey-predator system, J. Bio. Syst. 29 (2021), 107–145.
[37] N. Sk, P.K. Tiwari and S. Pal, A delay nonautonomous model for the impacts of fear and refuge in a three species
food chain model with hunting cooperation, Math. Comp. Simul. 192 (2022), 136–166.
[38] A. Szolnoki, M. Mobilia, L.-L. Jiang, B. Szczesny, A.M. Rucklidge and M. Perc, Cyclic dominance in evolutionary
games: A review, J. Royl. Soc. Intrf. 11 (2014), 20140735.
[39] R.J. Taylor, Predation, Chapman & Hall, 1984.
[40] X. Wang, L. Zanette and X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol. 73 (2016),
1179–1204.
[41] D. Xiao and S. Ruan, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J.
Appl. Math. 61 (2001), 1445–1472.
)
Volume 13, Issue 2
July 2022Pages 2071-2086
- Receive Date: 12 August 2021
- Revise Date: 13 April 2022
- Accept Date: 05 June 2022