Bifurcation analysis of a predator-prey model with strong Allee effect and Beddington-DeAngelis functional response

Figure 1.  Real roots of the polynomial $ p(x) $. (A) Three positive real roots $ x_1, x_2 $ and $ x_3 $. (B) and (C) Two positive real roots $ x_- $ and $ x_+ $. (D) Unique positive real root $ x_- = x_+ $

Figure 2.  Number of interior equilibria. (A) Three equilibria at $ M = 0.05, \alpha_1 = 0.1, \alpha_2 = 0.4, A = 0.3, \beta = 0.3, \gamma = 0.1 $. (B) Two equilibria at $ M = 0.1, \alpha_1 = 0.1, \alpha_2 = 0.4, A = 0.3, \beta = 0.3, \gamma = 0.1 $. (C) Unique equilibrium at $ M = 0.02, \alpha_1 = 0.174293, \alpha_2 = 0.4, A = 0.3, \beta = 0.4, \gamma = 0.1 $

Figure 3.  Phase portrait of three interior equilibria $ E_i^* $ ($ i = 1, 2, 3 $) and three boundary equilibria $ E_j $ ($ j = 0, 1, 2 $) at $ M = 0.07, \alpha_1 = 0.1, \alpha_2 = 0.4, A = 0.3, \beta = 0.3, \gamma = 0.1 $. $ E_3^* $ is stable hyperbolic focus, $ E_2^* $, $ E_2 $ and $ E_1 $ are saddles nodes, $ E_1^* $ is an unstable node and $ E_0 $ is stable node

Figure 4.  (A) Transcritical bifurcation around $ E_1(M, 0) $ and $ E_2( 1, 0) $, and saddle-node bifurcation around $ E^*(0.568240, 0.881665) $ with respect to $ \alpha_2 $ under condition (19). (B) Hopf bifurcation around $ E^*(0.842105, 0.0701758) $ with respect to $ M $ under condition (19)

Figure 5.  Phase portrait of the limit cycle and corresponding time diagram of Hopf bifurcation in Figure 4(B). (A) Stable limit cycle generated by supercritical Hopf bifurcation at $ M = 0.6666 $. (B) Time diagram of stable limit cycle

Figure 6.  Codimension two bifurcation diagrams of system (3). (A) In $ \alpha_1-\alpha_2 $ plane under condition (17). (B) In $ \alpha_1-M $ plane under condition (17). (C) In $ \alpha_2-M $ plane under condition (18). (D) In $ \alpha_1-M $ plane under condition (19)

Figure 7.  Neighborhood of $ BT $ and $ GH $ points in Figure 6. (A) and (C) The neighborhood of $ BT $ point in Figure 6 (A) and (B). (B) and (D) The neighborhood of $ GH $ point in Figure 6 (A) and (D)

Figure 8.  Phase portraits in the neighborhood of Bogdanov-Takens bifurcation point $ BT $ in Figure 7(A). (A) No interior equilibria. (B) Unstable hyperbolic focus. (C) Unstable limit cycle enclosing a stable hyperbolic focus

Figure 9.  Phase portraits in the neighborhood of $ GH $ point ($ l_2>0 $) in Figure 7(B). (A) Single stable hyperbolic focus. (B) Unique and stable limit cycle enclosing unstable hyperbolic focus. (C) Two limit cycles (the inner one is unstable and the outer is stable) enclosing stable hyperbolic focus. (D) Unstable limit cycle enclosing stable hyperbolic focus

Figure 10.  Phase portraits in the neighborhood of Bogdanov-Takens bifurcation point $ BT $ in Figure 7(A). (A) No interior equilibria. (B) Unstable hyperbolic focus. (C) Unstable limit cycle enclosing a stable hyperbolic focus

Figure 11.  Phase portraits for the neighborhood of generalized Hopf bifurcation $ GH $ ($ l_2<0 $) in Figure 7(D). (A) Single stable equilibrium. (B) Unique and stable limit cycle enclosing unstable hyperbolic focus. (C) Two limit cycles (the inner one is unstable and the outer is stable) enclosing an stable hyperbolic focus. (D) Unstable limit cycle enclosing stable hyperbolic focus