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

  • * Corresponding author: F. Zanolin

    * Corresponding author: F. Zanolin

This paper has been written under the auspices of the Ministry of Science, Technology and Universities of Spain, under Research Grant PGC2018-097104-B-100, and of the IMI of Complutense University. The second author, ORCID 0000-0003-1184-6231, has been also supported by contract CT42/18-CT43/18 of Complutense University of Madrid

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  • This paper studies the existence of subharmonics of arbitrary order in a generalized class of non-autonomous predator-prey systems of Volterra type with periodic coefficients. When the model is non-degenerate it is shown that the Poincaré–Birkhoff twist theorem can be applied to get the existence of subharmonics of arbitrary order. However, in the degenerate models, whether or not the twist theorem can be applied to get subharmonics of a given order might depend on the particular nodal behavior of the several weight function-coefficients involved in the setting of the model. Finally, in order to analyze how the subharmonics might be lost as the model degenerates, the exact point-wise behavior of the $ T $-periodic solutions of a non-degenerate model is ascertained as a perturbation parameter makes it degenerate.

    Mathematics Subject Classification: 34C25, 37B55, 37E40.


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  • 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 $

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