# American Institute of Mathematical Sciences

April  2020, 40(4): 2393-2419. doi: 10.3934/dcds.2020119

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

 1 Universidad Complutense de Madrid, Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Plaza de las Ciencias 3, 28040 Madrid, Spain 2 Università degli Studi di Udine, Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Via delle Scienze 2016, 33100 Udine, Italy

* Corresponding author: F. Zanolin

Received  July 2019 Revised  October 2019 Published  January 2020

Fund Project: 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.

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.

Citation: Julián López-Gómez, Eduardo Muñoz-Hernández, Fabio Zanolin. On the applicability of the poincaré–Birkhoff twist theorem to a class of planar periodic predator-prey models. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2393-2419. doi: 10.3934/dcds.2020119
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##### References:
A genuine case when $\alpha\beta = 0$ in ${\mathbb R}$
Two weights such that $\alpha\beta \gneq 0$
The weight functions $\alpha_ \varepsilon(t)$ and $\beta(t)$
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]
Behavior of $u(t, \varepsilon_n)$ and $v(t, \varepsilon_n)$ in Case 1.A for small $n$
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 $n$
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 $n$
Admissible $u(t, \varepsilon_n)$ and $v(t, \varepsilon_n)$ in Case 2 for large $n$
Admissible $u(t, \varepsilon_n)$ and $v(t, \varepsilon_n)$ in Case 2 for large $n$
Admissible components with $u_0( \varepsilon_n)>1$ for sufficiently large $n\geq 1$
Admissible components in the Subcase 4.B for sufficiently large $n$
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