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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

Julián López-Gómez 1, , Eduardo Muñoz-Hernández 1, and  Fabio Zanolin 2,,

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.

Keywords: Periodic predator-prey model of Volterra type, subharmonic coexistence states, Poincaré–Birkhoff twist theorem, degenerate versus non-degenerate models, point-wise behavior of the low order subharmonics as the model degenerates.
Mathematics Subject Classification: 34C25, 37B55, 37E40.
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 and Continuous Dynamical Systems, 2020, 40 (4) : 2393-2419. doi: 10.3934/dcds.2020119
References:
[1]

M. Begon, C. R. Townsend and J. L. Harper, Ecology: From Individuals to Ecosystems, 4th Edition, Blackwell Scientific Publications, United Kingdom, 2006.

[2]

A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: A rotation number approach, Adv. Nonlinear Stud., 11 (2011), 77-103. 

[3]

A. Boscaggin and F. Zanolin, Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions, Discrete & Continuous Dynamical Systems - A, 33 (2013), 89-110.  doi: 10.3934/dcds.2013.33.89.

[4]

M. Braun, Differential Equations and Their Applications: An Introduction to Applied Mathematics, Third edition, Applied Mathematical Sciences, 15. Springer-Verlag, New York-Berlin, 1983.

[5]

G. J. Butler and H. I. Freedman, Periodic solutions of a predator-prey system with periodic coefficients, Math Biosci., 55 (1981), 27-38.  doi: 10.1016/0025-5564(81)90011-0.

[6]

A. Casal, J. C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Diff. Int. Eqns., 7 (1994), 411-439. 

[7]

J. M. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math., 32 (1977), 82-95.  doi: 10.1137/0132006.

[8]

F. Dalbono and C. Rebelo, Poincaré-Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar hamiltonian systems, Rend. Sem. Mat. Univ. Pol. Torino, 60 (2003), 233-263. 

[9]

E. N. Dancer, J. López-Gómez and R. Ortega, On the spectrum of some linear noncooperative weakly coupled elliptic systems, Diff. Int. Eqns., 8 (1995), 515-523. 

[10]

T. R. Ding and F. Zanolin, Harmonic solutions and subharmonic solutions for periodic Lotka-Volterra systems, Dynamical Systems (Tianjin, 1990/1991), Nankai Ser. Pure Appl. Math. Theoret. Phys., World Sci. Publ., River Edge, NJ, 4 (1993), 55-65. 

[11]

T. R. Ding and F. Zanolin, Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra type, World Congress of Nonlinear Analysts '92, de Gruyter, Berlin, 1-4 (1996), 395-406. 

[12]

W. Y. Ding, Fixed points of twist mappings and periodic solutions of ordinary differential equations, Acta Math. Sinica, 25 (1982), 227-235. 

[13]

T. Dondè and F. Zanolin, Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach, preprint, (2019), arXiv: 1901.09406 [math.DS].

[14]

A. Fonda, Playing Around Resonance. An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations, Birkhäuser Advanced Texts, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-47090-0.

[15]

A. Fonda, M. Sabatini and F. Zanolin, Periodic solutions of perturbed hamiltonian systems in the plane by the use of Poincaré-Birkhoff theorem, Topol. Meth. Nonlin. Anal., 40 (2012), 29-52. 

[16]

A. Fonda and R. Toader, Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal., 8 (2019), 583-602.  doi: 10.1515/anona-2017-0040.

[17]

A. Fonda and A. J. Ureña, A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 679-698.  doi: 10.1016/j.anihpc.2016.04.002.

[18]

A. R. Hausrath and R. F. Manásevich, Periodic solutions of a periodically perturbed Lotka-Volterra equation using the Poincaré-Birkhoff theorem, J. Math. Anal. Appl., 157 (1991), 1-9.  doi: 10.1016/0022-247X(91)90132-J.

[19]

J. López-Gómez, A bridge between operator theory and mathematical biology, Operator Theory and its Applications, Fields Inst. Comm. Amer. Math. Soc., Providence, RI, 25 (2000), 383-397. 

[20]

J. López-Gómez and E. Muñoz-Hernández, Global structure of subharmonics in a class of periodic predator-prey models, Nonlinearity, 33 (2020), 34-71. 

[21]

J. López-Gómez, R. Ortega and A. Tineo, The periodic predator-prey Lotka-Volterra model, Adv. Diff. Eqns., 1 (1996), 403-423. 

[22]

J. López-Gómez and R. M. Pardo, The existence and the uniqueness for the predator-prey model with diffusion, Diff. Int. Eqns., 6 (1993), 1025-1031. 

[23]

J. López-Gómez and R. M. Pardo, Invertibility of linear noncooperative elliptic systems, Nonlin. Anal., 31 (1998), 687-699.  doi: 10.1016/S0362-546X(97)00640-8.

[24]

A. Margheri, C. Rebelo and F. Zanolin, Maslov index, Poincaré-Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, J. Differential Equations, 183 (2002), 342-367.  doi: 10.1006/jdeq.2001.4122.

[25]

J. D. Murray, Mathematical Biology. I. An Introduction, Third edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.

[26]

C. Rebelo, A note on the Poincaré-Birkhoff fixed point theorem and periodic solutions of planar systems, Nonlin. Anal., 29 (1997), 291-311.  doi: 10.1016/S0362-546X(96)00065-X.

show all references

References:
[1]

M. Begon, C. R. Townsend and J. L. Harper, Ecology: From Individuals to Ecosystems, 4th Edition, Blackwell Scientific Publications, United Kingdom, 2006.

[2]

A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: A rotation number approach, Adv. Nonlinear Stud., 11 (2011), 77-103. 

[3]

A. Boscaggin and F. Zanolin, Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions, Discrete & Continuous Dynamical Systems - A, 33 (2013), 89-110.  doi: 10.3934/dcds.2013.33.89.

[4]

M. Braun, Differential Equations and Their Applications: An Introduction to Applied Mathematics, Third edition, Applied Mathematical Sciences, 15. Springer-Verlag, New York-Berlin, 1983.

[5]

G. J. Butler and H. I. Freedman, Periodic solutions of a predator-prey system with periodic coefficients, Math Biosci., 55 (1981), 27-38.  doi: 10.1016/0025-5564(81)90011-0.

[6]

A. Casal, J. C. Eilbeck and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Diff. Int. Eqns., 7 (1994), 411-439. 

[7]

J. M. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math., 32 (1977), 82-95.  doi: 10.1137/0132006.

[8]

F. Dalbono and C. Rebelo, Poincaré-Birkhoff fixed point theorem and periodic solutions of asymptotically linear planar hamiltonian systems, Rend. Sem. Mat. Univ. Pol. Torino, 60 (2003), 233-263. 

[9]

E. N. Dancer, J. López-Gómez and R. Ortega, On the spectrum of some linear noncooperative weakly coupled elliptic systems, Diff. Int. Eqns., 8 (1995), 515-523. 

[10]

T. R. Ding and F. Zanolin, Harmonic solutions and subharmonic solutions for periodic Lotka-Volterra systems, Dynamical Systems (Tianjin, 1990/1991), Nankai Ser. Pure Appl. Math. Theoret. Phys., World Sci. Publ., River Edge, NJ, 4 (1993), 55-65. 

[11]

T. R. Ding and F. Zanolin, Periodic solutions and subharmonic solutions for a class of planar systems of Lotka-Volterra type, World Congress of Nonlinear Analysts '92, de Gruyter, Berlin, 1-4 (1996), 395-406. 

[12]

W. Y. Ding, Fixed points of twist mappings and periodic solutions of ordinary differential equations, Acta Math. Sinica, 25 (1982), 227-235. 

[13]

T. Dondè and F. Zanolin, Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach, preprint, (2019), arXiv: 1901.09406 [math.DS].

[14]

A. Fonda, Playing Around Resonance. An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations, Birkhäuser Advanced Texts, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-47090-0.

[15]

A. Fonda, M. Sabatini and F. Zanolin, Periodic solutions of perturbed hamiltonian systems in the plane by the use of Poincaré-Birkhoff theorem, Topol. Meth. Nonlin. Anal., 40 (2012), 29-52. 

[16]

A. Fonda and R. Toader, Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal., 8 (2019), 583-602.  doi: 10.1515/anona-2017-0040.

[17]

A. Fonda and A. J. Ureña, A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 679-698.  doi: 10.1016/j.anihpc.2016.04.002.

[18]

A. R. Hausrath and R. F. Manásevich, Periodic solutions of a periodically perturbed Lotka-Volterra equation using the Poincaré-Birkhoff theorem, J. Math. Anal. Appl., 157 (1991), 1-9.  doi: 10.1016/0022-247X(91)90132-J.

[19]

J. López-Gómez, A bridge between operator theory and mathematical biology, Operator Theory and its Applications, Fields Inst. Comm. Amer. Math. Soc., Providence, RI, 25 (2000), 383-397. 

[20]

J. López-Gómez and E. Muñoz-Hernández, Global structure of subharmonics in a class of periodic predator-prey models, Nonlinearity, 33 (2020), 34-71. 

[21]

J. López-Gómez, R. Ortega and A. Tineo, The periodic predator-prey Lotka-Volterra model, Adv. Diff. Eqns., 1 (1996), 403-423. 

[22]

J. López-Gómez and R. M. Pardo, The existence and the uniqueness for the predator-prey model with diffusion, Diff. Int. Eqns., 6 (1993), 1025-1031. 

[23]

J. López-Gómez and R. M. Pardo, Invertibility of linear noncooperative elliptic systems, Nonlin. Anal., 31 (1998), 687-699.  doi: 10.1016/S0362-546X(97)00640-8.

[24]

A. Margheri, C. Rebelo and F. Zanolin, Maslov index, Poincaré-Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, J. Differential Equations, 183 (2002), 342-367.  doi: 10.1006/jdeq.2001.4122.

[25]

J. D. Murray, Mathematical Biology. I. An Introduction, Third edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.

[26]

C. Rebelo, A note on the Poincaré-Birkhoff fixed point theorem and periodic solutions of planar systems, Nonlin. Anal., 29 (1997), 291-311.  doi: 10.1016/S0362-546X(96)00065-X.

Figure 1.  A genuine case when $ \alpha\beta = 0 $ in $ {\mathbb R} $
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Figure 2.  Two weights such that $ \alpha\beta \gneq 0 $
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Figure 3.  The weight functions $ \alpha_ \varepsilon(t) $ and $ \beta(t) $
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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]
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Figure 5.  Behavior of $ u(t, \varepsilon_n) $ and $ v(t, \varepsilon_n) $ in Case 1.A for small $ n $
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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 $
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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 $
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Figure 8.  Admissible $ u(t, \varepsilon_n) $ and $ v(t, \varepsilon_n) $ in Case 2 for large $ n $
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Figure 9.  Admissible $ u(t, \varepsilon_n) $ and $ v(t, \varepsilon_n) $ in Case 2 for large $ n $
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Figure 10.  Admissible components with $ u_0( \varepsilon_n)>1 $ for sufficiently large $ n\geq 1 $
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Figure 11.  Admissible components in the Subcase 4.B for sufficiently large $ n $
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