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On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem
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 |
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.
References:
[1] |
M. Begon, C. R. Townsend and J. L. Harper, Ecology: From Individuals to Ecosystems, 4th Edition, Blackwell Scientific Publications, United Kingdom, 2006. Google Scholar |
[2] |
A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: A rotation number approach, Adv. Nonlinear Stud., 11 (2011), 77-103. Google Scholar |
[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]. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[2] |
A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: A rotation number approach, Adv. Nonlinear Stud., 11 (2011), 77-103. Google Scholar |
[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]. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |





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