A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator

Figure 1.  In the left panel, we show the per capita growth rate of the logistic function (blue line), the strong Allee effect with $ m = 0.1 $ (red curve), the weak Allee effect with $ m = -0.1 $ (orange curve), multiple Allee effects with $ m = 0.1 $ and $ b = 0.15 $ (grey curve) and multiple Allee effects with $ m = 0.1 $ and $ b = 0.05 $ (green curve). In the right panel, we show the size of the depensation region for the strong Allee effect (6) (red curve) and for the multiple Allee effects (5) (grey curve) as function of the non-fertile prey population $ b $. We observe that the depensation region for the multiple Allee effects is always smaller than the depensation region for the strong Allee effect

Figure 2.  The intersections of the functions $ p(u) $ (red line) and $ d(u) $ (blue lines) for three different possible cases: (a) If $ \Delta<0 $ (10) then $ p(u) $ and $ d(u) $ do not intersect, and (8) does not have positive equilibrium points; (b) If $ \Delta = 0 $ then $ p(u) $ and $ d(u) $ intersect in one point, and (8) has a unique positive equilibrium point; (c) If $ \Delta>0 $ then $ p(u) $ and $ d(u) $ intersect in two points, and (8) has two distinct positive equilibrium points

Figure 3.  Phase plane of system (8) and its invariant regions $ \Phi $ and $ \Gamma\backslash\Phi $

Figure 4.  For $ M = 0.05 $, $ B = 0.05 $, $ C = 0.5 $, $ Q = 0.8 $, and $ S = 0.175 $, such that $ \Delta<0 $ (10), the equilibrium point $ (0,C) $ is a global attractor for trajectories starting in the first quadrant. The blue (red) curve represents the prey (predator) nullcline

Figure 5.  Let the system parameter $ (M,B,C,Q) = (0.07,0.0645,0.32,0.736) $ be such that $ \Delta>0 $ (10). (a) If $ S = 0.15 $ such that $ C<C_{H} $, then the equilibrium point $ P_2 $ is stable. (b) If $ S = 0.05 $ such that $ C>C_{H} $, then the equilibrium point $ P_2 $ is unstable. The blue (red) curve represents the prey (predator) nullcline. The orange (light blue) region represents the basin of attraction of the equilibrium point $ (0,C) $ ($ P_2 $). Note that the same color conventions are used in the upcoming figures

Figure 6.  If $ M = 0.05 $, $ B = 0.05 $, $ S = 0.125 $ and $ Q = 0.60821818 $, then $ \Delta = 0 $. Therefore, the equilibrium point $ P_3 $ is (a) a saddle-node repeller if $ C>C_{SN} $ and (b) a saddle-node attractor if $ C<C_{SN} $

Figure 7.  For $ M = 0.05 $, $ B = 0.05 $, $ C = 0.58951256 $, $ S = 0.125 $ and $ Q = 0.60821818 $, such that $ \Delta = 0 $ and $ f(u_3) = C_{SN} $, the point $ (0,C) $ is an attractor and the equilibrium point $ P_3 $ is a cusp point

Figure 8.  The bifurcation diagram of system (8) for $ M = 0.05 $ and $ S = 0.071080895 $ fixed and created with the numerical bifurcation package MATCONT [17]. In the left panel $ B = 0.1 $ fixed and varying $ Q $ and $ C $ and in the right panel $ Q = 0.608 $ fixed and varying $ B $ and $ C $. The curve $ C_H $ represents the Hopf curve, $ C_{HOM} $ represents the homoclinic curve, $ C_{SN} $ represents the saddle-node curve, and $ BT $ represents the Bogdanov-Takens bifurcation.The corresponding phase planes for the different regions are shown in Figure 9

Figure 9.  The phase planes of system (8) for $ B = 0.1 $, $ M = 0.05 $, $ Q = 0.75 $ and $ S = 0.071080895 $ fixed and varying $ C $. This last parameter impacts the number of equilibrium points of system (8). The light blue area in the phase plane represent the basins of attraction of the equilibrium points $ P_2 $, while the orange area in the phase plane represent the basins of attraction of the equilibrium points $ (0,C) $

Figure 10.  The size of the basin of attraction of $ p_2 $, in units$ ^2 $, of the stable equilibrium point $ p_2 $ of system (7) considering strong Allee effect (red line) and multiple Allee effect (blue line) for varying the non-fertile population $ b $ and with other system parameters $ r = 14 $, $ K = 150 $, $ m = 15 $, $ q = 1.08 $, $ s = 1.25 $, $ n = 0.05 $ and $ c = 0.75 $ fixed. The blue dotted-dashed line represents the region where the stable manifold of the saddle equilibrium point $ p_1 $ connects with (K, 0) and the blue dashed line represent the region where the equilibrium point $ p_2 $ is surrounded by an unstable limit cycle