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The role of vigilance on a discrete-time predator-prey model

  • *Corresponding author: Nikhil Pal

    *Corresponding author: Nikhil Pal 

The first author is supported by AUS grant FRG19-S-S141

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  • The change of behaviors of prey in the form of vigilance significantly affects the dynamics of a predator-prey system. In this paper, we consider a discrete-time predator-prey model, where the vigilance of prey acts as a trade-off between the safety and growth rate of the prey. Mathematical properties such as stability, permanence, both flip and Neimark-Sacker bifurcations of the model are investigated. Numerical simulations are carried out to illustrate the analytical findings and to explore the impact of prey vigilance on the dynamics of the system.

    Mathematics Subject Classification: 92D25, 39A28, 39A33.


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  • Figure 1.  Figure (a) shows the region $ \Omega_1 $ when $ \bar x_2\geq M_1 $. Figure (b) shows the region $ \Omega_\infty $ when $ \bar x_2<x^* $

    Figure 2.  This figure shows the geometric illustration for establishing a positively invariant region. The top blue line segment represents $ L(\gamma_{c_1}(t)), $ while the two other blue line segments represent $ L(\gamma_{c}(t)) $ for two random choices of $ c $

    Figure 3.  This figure illustrates the stability scenarios of the coexistence equilibrium $ (\bar x_2,\bar y_2) $ based on Lemma 3.1 and the values of $ X: = \frac{a}{1+v}\bar x_2, $ $ Y: = \frac{mp}{k+v}\bar y_2. $ The shaded dashed-region reflects the local stability region of the coexistence equilibrium

    Figure 4.  The figure shows a bifurcation diagram (left) and maximum Lyapunov exponent (right) of the system (1.3) with respect to the vigilance parameter $ v $, where other parameters are same as equation (5.1)

    Figure 5.  The figure shows the region of survival of the populations in $ v-r $ bi-parameter space. In region $ I $, populations show chaotic and quasiperiodic oscillations. In region $ II $, both prey and predator show stable coexistence. In region $ III $, the prey population shows stable behavior, but predator populations extinct from the system. In region $ IV $, both prey and predator extinct from the system

    Figure 6.  The figure shows variation of the densities of prey and predator populations in $ v-r $ bi-parameter space

    Figure 7.  (A) Phase portrait of the system showing period-$ 12 $ and period-$ 13 $ attractors with initial conditions $ (0.5, 0.2) $, and $ (0.1, 0.2) $, (B) phase portrait of the system showing period-$ 13 $ and period-$ 14 $ attractors with initial conditions $ (0.1, 0.2) $, and $ (0.7, 0.2) $, (C) basins of attraction for the coexisting period-$ 12 $ and period-$ 13 $ attractors, (D) basins of attraction for the coexisting period-$ 13 $ and period-$ 14 $ attractors

    Figure 8.  This figure shows bifurcation diagram (left) and maximum Lyapunov exponent (right) of the system (1.3) with respect to the vigilance parameter $ v $, where other parameters are same as equation (5.2)

    Figure 9.  The figure shows the region of survival of the populations in $ v-r $ bi-parameter space. The meanings of the regions are the same as in figure 4

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