# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021200
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## Effects of fear and anti-predator response in a discrete system with delay

 1 Indian Institute of Engineering Science and Technology, Shibpur, Howrah -711103, India 2 Vivekananda College, Thakurpukur, Kolkata - 700063, India

* Corresponding author

Received  April 2021 Revised  May 2021 Early access August 2021

In this paper a discrete-time two prey one predator model is considered with delay and Holling Type-Ⅲ functional response. The cost of fear of predation and the effect of anti-predator behavior of the prey is incorporated in the model, coupled with inter-specific competition among the prey species and intra-specific competition within the predator. The conditions for existence of the equilibrium points are obtained. We further derive the sufficient conditions for permanence and global stability of the co-existence equilibrium point. It is observed that the effect of fear induces stability in the system by eliminating the periodic solutions. On the other hand the effect of anti-predator behavior plays a major role in de-stabilizing the system by giving rise to predator-prey oscillations. Finally, several numerical simulations are performed which support our analytical findings.

Citation: Ritwick Banerjee, Pritha Das, Debasis Mukherjee. Effects of fear and anti-predator response in a discrete system with delay. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021200
##### References:
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##### References:
 [1] S. Aoki, U. Kurosu and S. Usuba, First instar larvae of the sugar-cane wooly aphid, ceratovacuna lanigera (homotera, pemphigidae), attack its predators, Kontyu, 52 (1984), 458-460.   Google Scholar [2] R. Banerjee, P. Das and D. Mukherjee, Stability and permanence of a discrete-time two-prey one-predator system with Holling Type-Ⅲ functional response, Chaos, Solitons & Fractals, 117 (2018), 240-248.  doi: 10.1016/j.chaos.2018.10.032.  Google Scholar [3] R. Banerjee, P. Das and D. Mukherjee, Global dynamics of a Holling Type-Ⅲ two prey–one predator discrete model with optimal harvest strategy, Nonlinear Dynamics, 99 (2020), 3285-3300.  doi: 10.1007/s11071-020-05490-0.  Google Scholar [4] M. C. and A. Barkai, Predator-prey role reversal in a marine benthic ecosystem, Science, (1988), 62–64. Google Scholar [5] M. Clinchy, M. J. Sheriff and L. Y. Zanette, Predator-induced stress and the ecology of fear, Functional Ecology, 27 (2013), 56-65.  doi: 10.1111/1365-2435.12007.  Google Scholar [6] R. Kaushik and S. Banerjee, Predator-prey system: Prey's counter-attack on juvenile predators shows opposite side of the same ecological coin, Applied Mathematics and Computation, 388 (2021), 125530. doi: 10.1016/j.amc.2020.125530.  Google Scholar [7] R. H. MacArthur and E. R. Pianka, On optimal use of a patchy environment, The American Naturalist, 100 (1966), 603-609.  doi: 10.1086/282454.  Google Scholar [8] R. J. Mrowicki and N. E. O'Connor, Wave action modifies the effects of consumer diversity and warming on algal assemblages, Ecology, 96 (2015), 1020-1029.  doi: 10.1890/14-0577.1.  Google Scholar [9] P. Panday, N. Pal, S. Samanta and J. Chattopadhyay, A three species food chain model with fear induced trophic cascade, Int. J. Appl. Comput. Math., 5 (2019), 26 pp. doi: 10.1007/s40819-019-0688-x.  Google Scholar [10] P. Panja, S. Jana and S. k. Mondal, Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey, Numerical Algebra, Control & Optimization, 11 (2021), 391-405.  doi: 10.3934/naco.2020033.  Google Scholar [11] W. Ripple, L. Painter, R. Beschta and C. Gates, Wolves, elk, bison, and secondary trophic cascades in Yellowstone National Park, The Open Ecology Journal, 3 (2010). Google Scholar [12] W. J. Ripple and R. L. Beschta, Wolves and the ecology of fear: Can predation risk structure ecosystems?, BioScience, 54 (2004), 755-766.   Google Scholar [13] W. J. Ripple and R. L. Beschta, Trophic cascades in Yellowstone: The first 15 years after wolf reintroduction, Biological Conservation, 145 (2012), 205-213.   Google Scholar [14] Y. Saito, Prey kills predator: Counter-attack success of a spider mite against its specific phytoseiid predator, Experimental & Applied Acarology, 2 (1986), 47-62.  doi: 10.1007/BF01193354.  Google Scholar [15] 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.   Google Scholar [16] Y. N. P. Service, \em 2019 Late Winter Survey of Northern Yellowstone Elk, 2019. Available from: https://www.nps.gov/yell/learn/news/2019-late-winter-survey-of-northern-yellowstone-elk.htm. Google Scholar [17] Y. N. P. Service, \em Questions & Answers About Bison Management, 2021. Available from: https://www.nps.gov/yell/learn/news/2019-late-winter-survey-of-northern-yellowstone-elk.htm. Google Scholar [18] D. W. Smith, L. D. Mech, M. Meagher, W. E. Clark, R. Jaffe, M. K. Phillips and J. A. Mack, Wolf–bison interactions in Yellowstone National Park, Journal of Mammalogy, 81 (2000), 1128-1135.   Google Scholar [19] J. P. Suraci, M. Clinchy, L. M. Dill, D. Roberts and L. Y. Zanette, Fear of large carnivores causes a trophic cascade, Nat Commun., 7 (2016), 10698. Google Scholar [20] V. Tiwari, J. P. Tripathi, S. Mishra and R. K. Upadhyay, Modeling the fear effect and stability of non-equilibrium patterns in mutually interfering predator–prey systems, Applied Mathematics and Computation, 371 (2020), 124948. doi: 10.1016/j.amc.2019.124948.  Google Scholar [21] H. Zhang, Y. Cai, S. Fu and W. Wang, Impact of the fear effect in a prey-predator model incorporating a prey refuge, Applied Mathematics and Computation, 356 (2019), 328-337.  doi: 10.1016/j.amc.2019.03.034.  Google Scholar
Existence of the equilibrium point $E_5(0, x_2', z')$ of system (2) with the parameter values $r_2 = 2, b_2 = 1 , \beta = 1.5, g_2 = 0.03 , d = 1 , d_1 = 0.1 , \alpha = 5, m = 0.025, c_2 = 1$. For the chosen parameters the conditions of Theorem 1 are verified with $Q = 8.73724>0$, $S = -2.89871<0$
Time series of (2) with the parameter values $r_1 = 1, h = 1, p = 1, q = 0.01, b_1 = 0.6, b_2 = 0.6, g_1 = 0.02, g_2 = 0.03, d = 1, d_1 = 0.02, \alpha = 5, \beta = 1.5, m = 0.01, c_1 = 0.3, c_2 = 0.3, \tau_1 = 2, \tau_2 = 2$
Phase portraits and time series of (2) with the parameter values $r_1 = 2 , h = 3.8, p = 0.3 , q = 0.6, , b_1 = 1.5, b_2 = 1 , \beta = 1.5 , g_1 = 0.08 , g_2 = 0.03 , d = 1 , d_1 = 0.1 , \alpha = 5, m = 0.025, c_2 = 1, c_1 = 1.4, \tau_1 = 2, \tau_2 = 2$
Bifurcation diagram and chaos 0-1 test of system (2) with the parameter values $r_1 = 2 , h = 3.8, p = 0.3 , q = 0.6, , b_1 = 1.5, b_2 = 1 , \beta = 1.5 , g_1 = 0.08 , g_2 = 0.03 , d = 1 , d_1 = 0.1 , \alpha = 5, m = 0.025, c_2 = 1, c_1 = 1.4, \tau_1 = 2, \tau_2 = 2$, showing the existence of chaos with increase in $r_2$
Stability region of system (2) for the varying parameters $r_2$ and $m$. The system shows stable dynamics in region A, period-2 oscillations in region B, period-4 oscillations in region C, period-3 oscillations in region D and chaotic dynamics in region E
Bifurcation diagram and chaos 0-1 test of system (2) with the parameter values $r_1 = 2 , p = 0.3 , q = 0.6, b_1 = 1.5, b_2 = 1 , \beta = 1.5 , g_1 = 0.08 , g_2 = 0.03 , d = 1 , d_1 = 0.1 , \alpha = 5, c_2 = 1, c_1 = 1.4, \tau_1 = 2, \tau_2 = 2$
$(p, q)$ plots for varying values of $\tau_{1, 2}$ with the parameter values $r_1 = 2, r_2 = 3.7, p = 0.3 , q = 0.6 , b_2 = 1 , \beta = 1.5 , g_1 = 0.08 , g_2 = 0.03 , d = 1 , d_1 = 0.1 , \alpha = 1, c_2 = 1, c_1 = 1.4, b_1 = 1.5, h = 0.5, m = 3.265$
$(p, q)$ plots for varying values of $r_2$ with the parameter values $r_1 = 2, p = 0.3 , q = 0.6 , b_2 = 1 , \beta = 1.5 , g_1 = 0.08 , g_2 = 0.03 , d = 1 , d_1 = 0.1 , \alpha = 1, c_2 = 1, c_1 = 1.4, b_1 = 1.5, h = 1, m = 2, \tau_{1} = 2, \tau_2 = 2$
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