The impact of alternative resources and fear on the dynamics of the food chain

Document Type : Research Paper


Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq


In this paper, a three-species food chain model is proposed and studied. It is assumed that there are fear costs in the first two-level due to predation risk and there exists an alternative source of food for a top predator. Holling’s disc function is adopted to describe the food transition throughout the chain. All the solution properties are discussed. The conditions of local stability and persistence of the model are established. The Lyapunov function is used to specify the basin of attraction for each equilibrium. Local bifurcation analyses are studied. The global dynamics of the model are investigated numerically. Different bifurcation diagrams and attractors are obtained. It is obtained that the system is rich in dynamics that include chaos.


[1] W. Cresswell,Predation in bird populations, J. Ornithol. 152 (2010) 251—263.[2] N.H. Fakhry and R.K. Naji, The dynamics of a square root prey-predator model with fear, Iraqi J. Sci. 61 (2020)
[3] S. Gakkhar and R.K. Naji, On a food web consisting of a specialist and a generalist predator, J. Biol. Syst. 11
(2003a) 365–376.
[4] K. Garain, U. Kumar and P.S. Mandal,Global Dynamics in a Beddington–DeAngelis Prey–Predator Model with
Density Dependent Death Rate of Predator, Differ. Equ. Dyn. Syst. 29 (2021) 265–283.
[5] M. Haque, Ratio-Dependent Predator-Prey Models of Interacting Populations, Bull. Math. Biol. 71 (2008)430-
[6] A. Hastings and T. Powell, Chaos in the three-species food chain, Ecology 72 (1991) 896–903.
[7] C.S. Holling, The components of predation as revealed by a study of small-mammal predation of the European
pine sawfly, Can. Entomol. 91 (1959) 293–320.
[8] F. Hua, K. E. Sieving, R. J. Fletcher and C. A. Wright, Increased perception of predation risk to adults and
offspring alters avian reproductive strategy and performance, Behav. Ecol. 25 (2014) 509-–519.
[9] J. Liu, P. Lv, B. Liu and T. Zhang, Dynamics of a Predator-Prey Model with Fear Effect and Time Delay,
Complex. 2021 (2021) 16 pages.
[10] X. Liu and Y. Lou, Global dynamics of a predator–prey model, J. MATH. ANAL. APPL. 371 (2010) 323–340.
[11] J. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edition, Springer-Verlag,
NY, 2003.
[12] R.K. Naji and A.T. Balasim, Dynamical behavior of a three-species food chain model with Beddington–DeAngelis
functional response, Chaos Solitons Fract. 32 (2007) 1853–1866.
[13] P. Panday, N, Pal, S. Samanta and J. Chattopadhyay, Stability and bifurcation analysis of a three-species food
chain model with fear, IJBC. 28 (2018) 1850009-1-1850009-20.
[14] S.D. Peacor, B.L. Peckarsky, G.C. Trussell and J.R. Vonesh, Costs of predator-induced phenotypic plasticity: a
graphical model for predicting the contribution of nonconsumptive and consumptive effects of predators on prey,
Oecologia 171 (2012) 1–10.
[15] M.L. Rosenzweig and R.H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat. 97 (1963) 209–223.
[16] S.K. Sasmal, Population dynamics with multiple Allee effects induced by fear factors – A mathematical study on
prey-predator interactions, Appl. Math. Model. 64 (2018) 1-–14.
[17] J.P. Suraci, M. Clinchy, L.M. Dill, D. Roberts and L.Y. Zanette, Fear of large carnivores causes a trophic cascade,
Nature Commun. 7 (2016) 1–7.
[18] R.K. Upadhyay and R.K. Naji, Dynamics of a three-species food chain model with Crowley–Martin type functional
response, Chaos Solitons Fract. 42, (2009) 1337—1346.
[19] X. Wang, L. Zanette and X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol. 73 (2016)
[20] 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.
[21] L.Y. Zanette, A. F. White, M. C. Allen and M. Clinchy, Perceived predation risk reduces the 609 number of
offspring songbirds produce per year, Science, 334 (2011) 1398–1401.
[22] H. Zhang, Y. Cai, S. Fu and W. Wang, Impact of the fear effect in a prey-predator model incorporating a prey
refuge, Appl. Math. Comput. 356 (2019) 328-–337.
Volume 12, Issue 2
November 2021
Pages 2207-2234
  • Receive Date: 03 April 2021
  • Revise Date: 08 June 2021
  • Accept Date: 16 August 2021