Discontinuity, Nonlinearity, and Complexity
Modeling and Analysis of a Onepredator Twoprey Ecological System with Fear Effect
Discontinuity, Nonlinearity, and Complexity 10(4) (2021) 585604  DOI:10.5890/DNC.2021.12.001
Anindita Bhattacharyya$^1$, Sanghita Bose$^1$, Ashok Mondal$^2$, A. K. Pal$^3$
$^{1}$ Department of Mathematics, Amity University, Kolkata700 135, India
$^2$ Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah
711 103, India
$^3$ Department of Mathematics, Seth Anandram Jaipuria College, Kolkata700 005, India
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Abstract
The present study deals with the dynamical response of a twoprey onepredator model inculcating the antipredator fear effect. The proposed model considers a Holling type II response function and it is intended to investigate the effect of the presence of fear among preys due to a predator. It is first shown that the system is bounded and the conditions of existence and stability of the equilibria of the proposed model have been furnished. Next the presence of Hopf bifurcation and limit cycles have been shown to explain the transition of the model from a stable to an unstable one. The study reveals that along with fear the interaction between the preys and predator can also be effectively stated as a control factor in determining dynamics of the model. The effect of antipredator fear and mutual interaction between the preys and predator has been numerically simulated in order to potray the dynamics of the model and the occurrence of limit cycles.
References

[1] 
Jones, D.S and Sleeman, B.D. (1983), Differential Equations and Mathematical Biology, George Allen and Unwin, Boston.


[2] 
May, R.M. (1974), Stability and Complexity in Model Ecosystem, Princeton University Press.


[3] 
Smith, J.M. (1975), Models in Ecology, Cambridge University Press, Cambridge.


[4] 
Kesh, D., Sarkar, A., and Roy, A.B. (2000), Persistence of two prey onepredator system with ratio dependent predator influence, Math. Appl. Sci., 23, 347356.


[5] 
Hsu, S.B, Hwang, T.W., and Kuang, Y. (2001), Rich dynamics of a ratiodependent one prey two predator model, J. Math. Biol., 43, 377396.


[6] 
Clinchy, M, Sheriff, M.J., and Zanette, L.Y. (2013), Predator induced stress and the ecology of fear, Functional Ecology, 27, 5665.


[7] 
Preisser, E.L. and Bolnick, D.I. (2008), The many faces of fear: Comparing the pathways and impacts of Nonconsumptive predator effects on prey populations, PLoS One, 3, e2465.


[8] 
Sasmal, S. (2018), Population dynamics with multiple Allee effects induced by fear factors induced by fear factorsa mathematical study on preypredator, Appl. Math. Model, 64, 114.


[9] 
Upadhyay, R. and Mishra, S. (2018), Population dynamic consequences of fearful prey in a spatiotemporal predatorprey system, Math. Biosci. Eng., 16(1), 338372.


[10] 
Wang, X., Zanette, L., and Zou, X. (2016), Modelling the fear effect in predator prey interactions, J. Math. Biol., 73(5), 11791204.


[11] 
Wang, X. and Zou, X. (2017), Modelling the fear effect in predatorprey interactions with adaptive avoidance of predators, Bull. Math. Biol., 79(6), 13251359.


[12] 
Das, A. and Samanta, G.P. (2018), Stochastic preypredator model with additional food for predator, J. Phys. A. Math. Theor, 51, 465601.


[13] 
Sahoo, B. and Poria, S. (2015), Effects of additional food in a delayed prey predator model, Math. Biosci, 261, 62.


[14] 
Sahoo, B. and Poria, S. (2011), Effects of additional food in a susceptile exposedinfected prey predator model, Int. J. Ecosystem, 1(1), 10.


[15] 
Suraci, J.P., Clinchy, M., Dill, L.M., Roberts, D., and Zanette, L.Y. (2016), Fear of large carnivores causes a trophic cascade, Nature Communications, 7, 10698.


[16] 
Holling, C.S. (1959), The components of predation as revealed by a study of smallmammal predation of the european pine sawfly, Canadian Entomol., 91(5), 293320.


[17] 
Holling, C.S (1959), Some characteristic of simple types of predation and parasitism, Canadian Entomol., 91(7), 385398.


[18] 
Holling, C.S. (1965), The functional response ofpredators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97(S45), 560.


[19] 
Huang, J., Ruan, S., and Song, J. (2014), Bifurcations in a predatorprey system of Leslie type with generalized Holling type III functional response, J. Diff. Equ., 257(6), 17211752.


[20] 
May R.M. (1972), Limit cycles in predatorprey communities, Science, 177, 900902.


[21] 
Mondal, A., Pal, A.K., and Samanta, G.P. (2019), Analysis of a Delayed EcoEpidemiological PestPlant Model with Infected Pest, Biophysi. Rev. Lett., 14(3), 141170.


[22] 
Seo G. and DeAngelis D. L. (2011), A predatorprey model with a Holling type I functional response including a predator mutual interference, J Nonlinear Sci, 21(6), 811833.


[23] 
Mondal, A., Pal, A.K., and Samanta, G.P. (2018), Stability and bifurcation analysis of a delayed three species food chain model with crowleymartin, Appl. Applied Maths.: An Int. J., 13(2), 709749.


[24] 
Freedman, H.I. and Wolkowicz, G.S.K. (2018), Predatorprey systems with group defence: the paradox of enrichment revisited, Bull Math Biol, 48(5/6), 493508.


[25] 
Ruan, S. and Xiao, D. (2001), Global analysis in a predatorprey system with nonmonotonic functional response, SIAM J. Appl Math, 61(4), 14451472.


[26] 
Wolkowicz, G.S.K. (1988), Bifurcation analysis of a predatorprey system involving group defence, SIAM J. Appl Math, 48(3), 592606.


[27] 
Zhu, H., Campbell, S.A., and Wolkowicz, G.S.K. (2003), Bifurcation analysis of a predatorprey system with nonmonotonic functional response, SIAM J. Appl Math, 63(2), 636682.


[28] 
Cai, Y., Gui, Z., Zhang, X., Shi, H., and Wang, W.M (2018), Bifurcations and pattern formation in a predatorprey model, Inter. J. Bifur. Chaos, 28(11), 1850140.


[29] 
Murray, J.D. (1989), Mathematical Biology, SpringerVerleg, Berlin.
