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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:
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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