Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect

Figure 1.  The intersections between the curve $ v = V(u) $ and the line $ v = u+w $. In the figure, the intersection points are the equilibriums of model (1.2)

Figure 2.  The slow-fast limiting dynamics associated with (1.2). In the figure, the orbits with double arrow indicates the fast fibers while the ones marked with single arrow are the slow orbits

Figure 3.  (a)-(b): Canard cycles of system (1.2); (c): Singular homoclinic orbit of system (1.2). In the figure, the orbits with double arrow indicates the fast fibers while the ones marked with single arrow are the slow orbits

Figure 4.  (a) The canard cycle of system (1.2) when $ h\in(0,v_0+a/p\beta) $; (b) The canard cycle of system (1.2) when $ h\in(0,v_0-v^*_1) $

Figure 5.  The integral on the function $ \overline{\Phi}(v) $, i.e., the slow divergence integral $ I(h,\mu_0) $

Figure 6.  (a) A homoclinic orbit of system (1.2); (b) The birth of the homoclinic orbit in system (1.2)

Figure 7.  An orbit of system (4.1) entering at $ (\bar{u}_0,\bar{v}_0) $ and exiting at $ (\bar{u}_0, p_{\varepsilon}(\bar{v}_0)) $, where $ 0<\varepsilon\ll1 $

Figure 8.  A singular slow-fast cycle $ \Gamma_0: = PLRMP $ of system (1.2)

Figure 9.  Singular Hopf bifurcation curve and canard curve of (1.2) when $ a = -0.4,\,\,p = 0.4,\,\,\beta = 2,\,\,\varepsilon = 0.001 $

Figure 10.  The birth and bifurcation of canard limit cycle when $ a = -0.4,\,\,p = 0.4,\,\,\beta = 2,\,\,\varepsilon = 0.001 $. (a): The small amplitude Hopf cycle when $ w = 0.362 $, where the initial value is $ (0.2, 0.54159) $; (b): Local zooming of the small amplitude Hopf cycle in Fig. 10 (a); (c) The time history of the small amplitude Hopf cycle when $ w = 0.362 $; (d): The canard limit cycle when $ w = 0.375 $, where the initial value is $ (0.2, 0.5048) $; (e): Local zooming of the canard limit cycle displayed in Fig. (d); (f) The time history of the canard cycle when $ w = 0.375 $; (g): The canard relaxation oscillation passing through the transcritical bifurcation point when $ w = 0.381 $, where the initial value is $ (0.381, 0.51) $; (h): Local zooming of the canard relaxation oscillation displayed in Fig. (i); The time history of the canard relaxation oscillation when $ w = 0.381 $

Figure 11.  The whole bifurcation process of canard explosion

Figure 12.  Singular Hopf bifurcation curve and canard curve of (1.2) when $ a = -0.3,\,\,p = 0.15,\,\,\beta = 2.13,\,\,\varepsilon = 0.0055685 $

Figure 13.  (a) and (b): The homoclinic cycle of system (1.2) when $ a = -0.3,\,\,p = 0.15,\,\,\beta = 2.13,\,\,\varepsilon = 0.0055685 $ and $ w = 0.9364 $; (c): The local zooming of the homoclinic cycle near the saddle (0.001387109324, 0.9413871091); (d): The time history of the homoclinic orbit under the present parameter value