Article Contents
Article Contents

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

• *Corresponding author: Jianhe Shen

The research was supported by the Natural Science Foundation of China (NO. 11771082) and Guangxi College Enhancing Youths Capacity Project (2021KY0653)

• This paper studies bifurcations of canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect. Based on geometric singular perturbation theory (GSPT) and canard theory, canard explosion is observed and the associated bifurcation curve is determined. Due to the canard point, a homoclinic orbit with slow and fast segments and homoclinic to a saddle can also exist, in which, the stable and unstable manifolds of the saddle are connected under certain parameter value. By analyzing the slow divergence integral, it is proved that the cyclicity of canard cycles in this model is at most four. Finally, by calculating the entry-exit function explicitly, a unique, orbitally stable canard relaxation oscillation passing through a transcritical bifurcation point is detected. All these theoretical predictions on the birth of canard explosion, canard limit cycles and homoclinic orbits are verified by numerical simulations.

Mathematics Subject Classification: 34D23, 92B05, 34D40.

 Citation:

• 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

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