On the applicability of the poincaré–Birkhoff twist theorem to a class of planar periodic predator-prey models

Figure 1.  A genuine case when $ \alpha\beta = 0 $ in $ {\mathbb R} $

Figure 2.  Two weights such that $ \alpha\beta \gneq 0 $

Figure 3.  The weight functions $ \alpha_ \varepsilon(t) $ and $ \beta(t) $

Figure 4.  Subharmonics of (17) under condition (21). The figure represents an ideal global bifurcation diagram for subharmonics with the parameter $ A = B $ (in the abscissa) versus the value of the initial point $ x = u_0 = v_0 $ of the periodic solution (in the ordinate). Each bifurcation curve is labelled with the period of the corresponding subharmonic solution. For a detailed analysis of the real bifurcation diagrams, we refer to [20]

Figure 5.  Behavior of $ u(t, \varepsilon_n) $ and $ v(t, \varepsilon_n) $ in Case 1.A for small $ n $

Figure 6.  Behavior of $ u(t, \varepsilon_n) $ and $ v(t, \varepsilon_n) $ in Case 1.A for large $ n $. Notice that $ v $ is constant on $ [0,T/2] $ while, on the same interval, $ u $ is near to a constant for large $ n $

Figure 7.  Admissible $ u(t, \varepsilon_n) $ and $ v(t, \varepsilon_n) $ in Subcase 1.B for large $ n $. As in Figure 6, $ v $ is constant on $ [0,T/2] $ while, on the same interval, $ u $ is near to a constant for large $ n $

Figure 8.  Admissible $ u(t, \varepsilon_n) $ and $ v(t, \varepsilon_n) $ in Case 2 for large $ n $

Figure 9.  Admissible $ u(t, \varepsilon_n) $ and $ v(t, \varepsilon_n) $ in Case 2 for large $ n $

Figure 10.  Admissible components with $ u_0( \varepsilon_n)>1 $ for sufficiently large $ n\geq 1 $

Figure 11.  Admissible components in the Subcase 4.B for sufficiently large $ n $