Stability analysis of a Filippov Gause predator-prey model with or without hunting cooperation among predators with respect to prey density

Figure 1.  General dynamics of the nullclines (6) of model (2) for the case where $ c-b\gamma>0 $. The purple and dark red nullclines are given by the functions (7) and (8), respectively. In particular, note that, if $ \hat{x}<K $ or $ K<\hat{x} $, then $ E_{0}^{+} $ is saddle point or stable node, respectively, with $ \hat{x} $ as shown in (8)

Figure 2.  Bifurcation diagram in the plane $ (c, \gamma) $, together with phase portraits describing each bifurcation region, of model (2), with fixed parameters $ a = 0.8 $, $ b = 0.4 $, $ K = 5 $ and $ r = 2.5 $. The magenta curve shows the existence of an asymptotically stable limit cycle $ \Gamma_{G_{0}}^{s} $ in $ \Omega $. The red and blue points represent the boundary equilibria $ E_{0}^{0} $ and $ E_{0}^{+} $, respectively. The black point represents the positive inner equilibrium $ E_{0}^{1} $, which is unstable in region Ⅲ and globally asymptotically stable in region Ⅱ. In region Ⅰ, model (2) has no positive inner equilibria, so $ E_{0}^{+} $ is a globally asymptotically stable node, unlike in regions Ⅱ and Ⅲ where $ E_{0}^{+} $ is a saddle point. The dashed and normal lines in panel (a) represent a transcritical and a Hopf bifurcation, respectively

Figure 3.  General dynamics of the nullclines (9) of model (3) for $ c-b\gamma>0 $. Note that the function (12), represented by the orange curve, is decreasing, which intersects with the nullcline $ \dot{x} = 0 $, given by the dark blue curve, at one or two points if $ E_{\nu}^{+} $ is a saddle point or a stable node, represented in panels (a, b), respectively. In particular, note that if $ \hat{x}<K $ or $ K<\hat{x} $, then $ E_{\nu}^{+} $ is a saddle point or stable node, respectively, with $ \hat{x} $ as shown in (8)

Figure 4.  Bifurcation diagram in the plane $ (\nu, \gamma) $, together with phase portraits describing each bifurcation region, of model (3), with fixed parameters $ a = 0.8 $, $ b = 0.4 $, $ c = 0.05 $, $ K = 5 $, and $ r = 2.5 $. The magenta curve shows the existence of an asymptotically stable limit cycle $ \Gamma_{G_{\nu}}^{s} $ in $ \Omega $. The dark red and turquoise points represent the boundary equilibria $ E_{\nu}^{0} $ and $ E_{\nu}^{+} $, respectively. The grey, magenta, and orange points represent the positive inner equilibria $ E_{\nu}^{i} $, $ i = 1, 2, 3 $, respectively. In region Ⅰ, we have that model (3) has no positive inner equilibria, so $ E_{\nu}^{+} $ is a globally asymptotically stable node. In regions Ⅱ and Ⅲ, we have that $ E_{\nu}^{+} $ is saddle point, and $ E_{\nu}^{1} $ is the positive inner equilibrium in $ \Omega $, which is unstable in region Ⅲ and globally asymptotically stable in region Ⅱ. For regions Ⅳ and Ⅴ, we have that $ E_{\nu}^{+} $ is a locally asymptotically stable node, and $ E_{\nu}^{k} $, $ k = 2, 3 $, are positive inner equilibria in $ \Omega $, where $ E_{\nu}^{3} $ is saddle point, and $ E_{\nu}^{2} $ is unstable or locally asymptotically stable in regions Ⅴ or Ⅳ, respectively. The dashed, normal and dotted lines shown in panel (a) represent transcritical, Hopf, and saddle-node bifurcations, respectively

Figure 5.  General dynamics of the nullclines (6) and (9) of model (15) for the case where $ c-b\gamma>0 $, where the color of the curves in each panel are described in Figures 1 and 3. The blue line represents $ \Sigma_{s}^{\epsilon} $. On the other hand, in panel (a) we have that $ E_{\nu}^{1} \notin \Sigma_{\nu}^\epsilon $ and $ E_{0}^{1} \in \Sigma_{0}^\epsilon $. In panel (b), we have that $ E_{\nu}^{i} \in \Sigma_{\nu}^\epsilon $, $ i = 2, 3 $

Figure 6.  Example of regular quadratic tangency point $ T_{0} $ in model (3). The blue normal and black dotted curves represent $ \Sigma_{s}^{\epsilon} $ and $ \Sigma_{2c}^{\epsilon} $, respectively

Figure 7.  Examples of a limit cycle $ \Gamma_{G_{\epsilon}} $ and cycle $ \Upsilon_{G_{\epsilon}} $ in model (15) represented by the purple curve. The black curves are trajectories $ \varphi_{G_{\epsilon}} $ that are not periodic

Figure 8.  Bifurcation diagram of model (3) in the plane $ (P, \gamma) $ with fixed parameters $ a = 0.8 $, $ b = 0.4 $, $ c = 0.05 $ and $ r = 2.5 $. Note that the phase portraits showing each bifurcation region are shown in Figures 9, 10, and 11. The dotted, dashed, and normal line shows the existence of a transcritical, saddle-node, and Hopf bifurcation of model (3) at $ \Omega $. On the other hand, the dash-dotted line represents the $ \Sigma $-equilibrium bifurcation. The dark red and black dash dotted lines show the existence of crossing bifurcation and touching bifurcation, respectively, unlike the green dash-dotted line which shows the intersection of limit cycle $ \Gamma_{G_{\epsilon}} $ with $ \Sigma $. The blue dash-dotted line shows the intersection of $ W_{u}^{+}(E_{\nu}^{+}) $ with the regular tangent point $ T_{0} $

Figure 9.  Phase portraits in bifurcation regions Ⅰ to Ⅵ of model (15) shown in Figure 8. The light red and light green trajectories are described by the vector fields $ G_{i} $, $ i = 0, \nu $, respectively. The blue line represents the sliding segment $ \Sigma_{s}^{\epsilon} $. The asymptotically stable limit cycle $ \Gamma_{G_{\epsilon}}^{s} $ or cycle $ \Upsilon_{G_{\epsilon}}^{s} $ are described by the closed magenta curve. The boundary equilibria $ E_{0}^{0} \in \Sigma_{0}^{\epsilon} $ and $ E_{\nu}^{+} \in \Sigma_{\nu}^{\epsilon} $ are described by the red and turquoise points, respectively, where $ E_{0}^{+} $ is a stable node in regions Ⅰ to Ⅳ, or a saddle point in regions Ⅴ to Ⅵ. Furthermore, the gray, magenta, and orange points describe the positive inner equilibria $ E_{\nu}^{i} \in \Sigma_{\nu}^{\epsilon} $, $ i = 1, 2, 3 $, respectively. In region Ⅰ, we have that model (15) has no positive inner equilibria or pseudo-equilibria. In regions Ⅱ to Ⅳ, we have that $ E_{\nu}^{i} \in \Sigma_{\nu}^{\epsilon} $, $ i = 2, 3 $, where $ E_{\nu}^{2} $ is stable in region Ⅱ, and unstable in regions Ⅲ and Ⅳ. In regions Ⅴ to Ⅵ we have that $ E_{\nu}^{1} \in \Sigma_{\nu}^{\epsilon} $, stable in region Ⅴ, and unstable in region Ⅵ

Figure 10.  Phase portraits in bifurcation regions Ⅶ to Ⅻ of model (15) shown in Figure 8. The light red and light green trajectories are described by the vector fields $ G_{i} $, $ i = 0, \nu $, respectively. The blue line represents the sliding segment $ \Sigma_{s}^{\epsilon} $. The asymptotically stable limit cycle $ \Gamma_{G_{\epsilon}}^{s} $ or cycle $ \Upsilon_{G_{\epsilon}}^{s} $ are described by the closed magenta curve. The boundary equilibria $ E_{0}^{0} \in \Sigma_{0}^{\epsilon} $ and $ E_{\nu}^{+} \in \Sigma_{\nu}^{\epsilon} $ are described by the red and turquoise points, respectively, where $ E_{0}^{+} $ is a saddle point. In addition, the black, gray, orange, and green points describe the positive inner equilibria $ E_{0}^{1} \in \Sigma_{i}^{\epsilon} $, $ i = 0, \nu $, $ E_{\nu}^{3} \in \Sigma_{\nu}^{\epsilon} $, and the pseudo-equilibrium $ PN \in \Sigma_{s}^{\epsilon} $, respectively. In regions Ⅶ and Ⅷ, we have that $ E_{\nu}^{1} \in \Sigma_{\nu}^{\epsilon} $ is unstable. In regions Ⅸ and Ⅺ, we have that $ PN \in \Sigma_{s}^{\epsilon} $, where $ E_{\nu}^{3} \in \Sigma_{\nu}^{\epsilon} $ in region Ⅸ. In region Ⅻ, we have that $ E_{0}^{1} \in \Sigma_{0}^{\epsilon} $, which is stable

Figure 11.  Phase portraits in bifurcation regions ⅫⅠ to XV of model (15) shown in Figure 8. The light red and light green trajectories are described by the vector fields $ G_{i} $, $ i = 0, \nu $, respectively. The blue line represents the sliding segment $ \Sigma_{s}^{\epsilon} $. The asymptotically stable limit cycle $ \Gamma_{G_{\epsilon}}^{s} $ or cycle $ \Upsilon_{G_{\epsilon}}^{s} $ are described by the closed magenta curve. The boundary equilibria $ E_{0}^{0} \in \Sigma_{0}^{\epsilon} $ and $ E_{\nu}^{+} \in \Sigma_{\nu}^{\epsilon} $ are described by the red and turquoise points, respectively, where $ E_{0}^{+} $ is a saddle point. In addition, the black point describes the positive inner equilibrium $ E_{0}^{1} \in \Sigma_{0}^{\epsilon} $, which is unstable

Figure 12.  General dynamics of the nullclines (6) and (9) of model (19) for $ c-b\gamma>0 $, where the color of the curves in each panel are described in Figures 1 and 3. The red line represents $ \Sigma_{e}^{\tau} $. On the other hand, in panel (a) we have that $ E_{\nu}^{1} \in \Sigma_{\nu}^\tau $ and $ E_{0}^{1} \notin \Sigma_{0}^\tau $. In panel (b), we have that $ E_{\nu}^{i} \notin \Sigma_{\nu}^\tau $, $ i = 2, 3 $

Figure 13.  Bifurcation diagram of model (19) in the plane $ (P, \gamma) $ with fixed parameters $ a = 0.8 $, $ b = 0.4 $, $ c = 0.05 $, $ r = 2.5 $. Note that the phase portraits showing each bifurcation region are shown in Figures 14, 15, and 16. The dotted, dashed, and normal line shows the existence of a transcritical, saddle-node, and Hopf bifurcation of model (19) at $ \Omega $. On the other hand, the dash-dotted line represents the boundary bifurcation. The green dash-dotted line shows the intersection of the limit cycle $ \Gamma_{G_{\epsilon}} $ with $ \Sigma $

Figure 14.  Phase portraits in bifurcation regions Ⅰ to Ⅵ of model (19) shown in Figure 13. The light red and light green trajectories are described by the vector fields $ G_{i} $, $ i = 0, \nu $, respectively. The red line represents the escape region $ \Sigma_{e}^{\tau} $. The asymptotically stable limit cycle $ \Gamma_{G_{\tau}}^{s} $ is described by the closed magenta curve. The boundary equilibria $ E_{\nu}^{0} \in \Sigma_{\nu}^{\tau} $ and $ E_{0}^{+} \in \Sigma_{0}^{\tau} $ are described by the dark red and dark blue points, respectively, where $ E_{0}^{+} $ is a stable node. Furthermore, the green, magenta, and orange points describe the pseudo-equilibrium $ PN \in \Sigma_{e}^{\tau} $ and the positive inner equilibria $ E_{\nu}^{i} \in \Sigma_{\nu}^{\tau} $, $ i = 2, 3 $, respectively. In region Ⅰ, we have that model (15) has no positive inner equilibria and no pseudo-equilibria. In regions Ⅱ and Ⅲ, we have that $ E_{\nu}^{i} \in \Sigma_{\nu}^{\tau} $, $ i = 2, 3 $, where $ E_{\nu}^{2} \in \Sigma_{\nu}^{\tau} $ is globally asymptotically stable in region Ⅱ and unstable in region Ⅲ. In regions Ⅳ to Ⅵ, we have that $ E_{\nu}^{2} \in \Sigma_{\nu}^{\tau} $ and $ PN \in \Sigma_{e}^{\tau} $, where $ E_{\nu}^{2} \in \Sigma_{\nu}^{\tau} $ is globally asymptotically stable in region Ⅳ, and unstable in regions Ⅴ to Ⅵ

Figure 15.  Phase portraits in bifurcation regions Ⅶ to Ⅻ of model (19) shown in Figure 13. The light red and light green trajectories are described by the vector fields $ G_{i} $, $ i = 0, \nu $, respectively. The red line represents the escape region $ \Sigma_{e}^{\tau} $. The asymptotically stable limit cycle $ \Gamma_{G_{\tau}}^{s} $ is described by the closed magenta curve. The boundary equilibria $ E_{\nu}^{0} \in \Sigma_{\nu}^{\tau} $ and $ E_{0}^{+} \in \Sigma_{0}^{\tau} $ are described by the dark red and dark blue points, respectively, where $ E_{0}^{+} $ is a saddle point. In addition, the green, black, and gray points describe the pseudo-equilibrium $ PN \in \Sigma_{e}^{\tau} $ and the inner positive equilibria $ E_{i}^{1} \in \Sigma_{i}^{\tau} $, $ i = 0, \nu $, respectively. In regions Ⅶ to Ⅸ, we have that $ PN \in \Sigma_{e}^{\tau} $ and $ E_{i}^{1} \in \Sigma_{i}^{\tau} $, $ i = 0, \nu $, are positive inner equilibria of model (19) in $ \Omega $. Note that $ E_{i}^{1} \in \Sigma_{i}^{\tau} $, $ i = 0, \nu $, are locally asymptotically stable in region Ⅶ, $ E_{\nu}^{1} \in \Sigma_{\nu}^{\tau} $ is unstable and $ E_{0}^{1} \in \Sigma_{0}^{1} $ is globally asymptotically stable in region Ⅷ, and $ E_{i}^{1} \in \Sigma_{i}^{\tau} $, $ i = 0, \nu $, are unstable in region Ⅸ. In regions X to Ⅻ, we have that $ E_{0}^{1} \in \Sigma_{0}^{\tau} $, which is globally asymptotically stable in region X, and unstable in regions Ⅺ and Ⅻ

Figure 16.  Phase portraits in bifurcation regions ⅫⅠ to XV of model (19) shown in Figure 13. The light red and light green trajectories are described by the vector fields $ G_{i} $, $ i = 0, \nu $, respectively. The red line represents the escape region $ \Sigma_{e}^{\tau} $. The asymptotically stable limit cycle $ \Gamma_{G_{\tau}}^{s} $ is described by the closed magenta curve. The boundary equilibria $ E_{\nu}^{0} \in \Sigma_{\nu}^{\tau} $ and $ E_{0}^{+} \in \Sigma_{0}^{\tau} $ are described by the dark red and dark blue points, respectively, where $ E_{0}^{+} $ is a saddle point. In addition, the gray point describes the positive inner equilibrium $ E_{\nu}^{1} \in \Sigma_{\nu}^{\tau} $, which is globally asymptotically stable in region ⅫⅠ, and unstable in regions ⅪⅤ and ⅩⅤ