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

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