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Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media
Phase portraits of the Higgins–Selkov system
1. | Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain |
2. | Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran |
$ \begin{equation*} \dot{x} = 1-xy^\gamma, \quad\dot{y} = \alpha y(xy^{\gamma -1}-1), \end{equation*} $ |
$ \alpha $ |
$ \gamma>1 $ |
$ \gamma = 3, 4, 5, 6, $ |
$ \alpha $ |
$ \alpha $ |
References:
[1] |
M. J. Álvarez, A. Ferragut and X. Jarque,
A survey on the blow up technique, Int. J. Bifurcation and Chaos, 21 (2011), 3103-3118.
doi: 10.1142/S0218127411030416. |
[2] |
J. C. Arteś, J. Llibre and C. Valls,
Dynamics of the Higgins–Selkov and Selkov system, Chaos. Sol. Frac., 114 (2018), 145-150.
doi: 10.1016/j.chaos.2018.07.007. |
[3] |
P. Brechmann and A. D. Rendall,
Dynamics of the Selkov oscillator, Mathematical Biosciences, 306 (2018), 152-159.
doi: 10.1016/j.mbs.2018.09.012. |
[4] |
P. Brechmann and A. D. Rendall, Unbounded solutions of models for glycolysis, preprint, arXiv: 2003.07140. Google Scholar |
[5] |
H. Chen, J. Llibre and Y. Tang,
Global dynamics of a SD oscillator, Nonlinear Dynamics, 91 (2018), 1755-1777.
doi: 10.1007/s11071-017-3979-y. |
[6] |
H. Chen and Y. Tang,
Proof of Arté-Llibre-Valls's conjectures for the Higgins–Selkov and Selkov systems, J. Differential Equations, 266 (2019), 7638-7657.
doi: 10.1016/j.jde.2018.12.011. |
[7] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, Berlin, 2006. |
[8] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, in: Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[9] |
J. Higgins,
A chemical mechanism for oscillation of glycolytic intermediates in yeast cells, Proc. Natl. Acad. Sci. (USA), 51 (1964), 989-994.
doi: 10.1073/pnas.51.6.989. |
[10] |
T.-W. Hwang and H.-J. Tsai,
Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions, J. Phys. A, 38 (2005), 8211-8223.
doi: 10.1088/0305-4470/38/38/003. |
[11] |
L. Perko, Differential Equations and Dynamical Systems, 3rd Ed., Springer, 2001.
doi: 10.1007/978-1-4613-0003-8. |
[12] |
E. E. Sel'kov,
Self-oscillations in glycolysis. I. A simple kinetic model, Eur. J. Biochem., 4 (1968), 79-86.
doi: 10.1111/j.1432-1033.1968.tb00175.x. |
[13] |
L. Yang,
Recent advances on determining the number of real roots of parametric polynomials, J. Symb. Comput., 28 (1999), 225-242.
doi: 10.1006/jsco.1998.0274. |
[14] |
Z. Zhang, T. Ding and W. Huang, Qualitative Theory of Differential Equations, Transl. Math. Monogr. Amer. Soc., Providence, RI, 1992. Google Scholar |
show all references
References:
[1] |
M. J. Álvarez, A. Ferragut and X. Jarque,
A survey on the blow up technique, Int. J. Bifurcation and Chaos, 21 (2011), 3103-3118.
doi: 10.1142/S0218127411030416. |
[2] |
J. C. Arteś, J. Llibre and C. Valls,
Dynamics of the Higgins–Selkov and Selkov system, Chaos. Sol. Frac., 114 (2018), 145-150.
doi: 10.1016/j.chaos.2018.07.007. |
[3] |
P. Brechmann and A. D. Rendall,
Dynamics of the Selkov oscillator, Mathematical Biosciences, 306 (2018), 152-159.
doi: 10.1016/j.mbs.2018.09.012. |
[4] |
P. Brechmann and A. D. Rendall, Unbounded solutions of models for glycolysis, preprint, arXiv: 2003.07140. Google Scholar |
[5] |
H. Chen, J. Llibre and Y. Tang,
Global dynamics of a SD oscillator, Nonlinear Dynamics, 91 (2018), 1755-1777.
doi: 10.1007/s11071-017-3979-y. |
[6] |
H. Chen and Y. Tang,
Proof of Arté-Llibre-Valls's conjectures for the Higgins–Selkov and Selkov systems, J. Differential Equations, 266 (2019), 7638-7657.
doi: 10.1016/j.jde.2018.12.011. |
[7] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, Berlin, 2006. |
[8] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, in: Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[9] |
J. Higgins,
A chemical mechanism for oscillation of glycolytic intermediates in yeast cells, Proc. Natl. Acad. Sci. (USA), 51 (1964), 989-994.
doi: 10.1073/pnas.51.6.989. |
[10] |
T.-W. Hwang and H.-J. Tsai,
Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions, J. Phys. A, 38 (2005), 8211-8223.
doi: 10.1088/0305-4470/38/38/003. |
[11] |
L. Perko, Differential Equations and Dynamical Systems, 3rd Ed., Springer, 2001.
doi: 10.1007/978-1-4613-0003-8. |
[12] |
E. E. Sel'kov,
Self-oscillations in glycolysis. I. A simple kinetic model, Eur. J. Biochem., 4 (1968), 79-86.
doi: 10.1111/j.1432-1033.1968.tb00175.x. |
[13] |
L. Yang,
Recent advances on determining the number of real roots of parametric polynomials, J. Symb. Comput., 28 (1999), 225-242.
doi: 10.1006/jsco.1998.0274. |
[14] |
Z. Zhang, T. Ding and W. Huang, Qualitative Theory of Differential Equations, Transl. Math. Monogr. Amer. Soc., Providence, RI, 1992. Google Scholar |


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