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Some global dynamics of a Lotka-Volterra competition-diffusion-advection system
Stable periodic solutions for Nazarenko's equation
1. | Bolyai Institute, University of Szeged, 1 Aradi vértanúk tere, Szeged, Hungary |
2. | MTA-SZTE Analysis and Stochastics Research Group, , Bolyai Institute, University of Szeged, 1 Aradi vértanúk tere, Szeged, Hungary |
$ \begin{equation*} \dot{y}(t) = -py(t)+\dfrac{qy(t)}{r+y^{n}(t-\tau)},\qquad t>0, \end{equation*} $ |
$ p,q,r,\tau\in\left(0,\infty\right) $ |
$ n\in\mathbb{N} = \left\{ 1,2,\ldots\right\} $ |
$ q/p>r $ |
$ \tau $ |
$ n $ |
$ K = \left(q/p-r\right)^{1/n} $ |
$ n\rightarrow\infty $ |
References:
[1] |
Y. Cao,
Multiexistence of slowly oscillating periodic solutions for differential delay equations, SIAM J. Math. Anal., 26 (1995), 436-445.
doi: 10.1137/0526022. |
[2] |
Y. Cao,
Uniqueness of periodic solution for differential delay equations, J. Differ. Equ., 128 (1996), 46-57.
doi: 10.1006/jdeq.1996.0088. |
[3] |
J. L. Kaplan and J. A. Yorke,
On the stability of a periodic solution of a differential delay equation, SIAM J. Math. Anal., 6 (1975), 268-282.
doi: 10.1137/0506028. |
[4] |
B. Kennedy and E. Stumpf,
Multiple slowly oscillating periodic solutions for $x' (t) = f(x(t- 1))$ with negative feedback, Ann. Polon. Math., 118 (2016), 113-140.
doi: 10.4064/ap3899-10-2016. |
[5] |
I. Kubiaczyk and S. H. Saker,
Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Model., 35 (2002), 295-301.
doi: 10.1016/S0895-7177(01)00166-2. |
[6] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.
![]() ![]() |
[7] |
E. Liz and G. Röst,
Dichotomy results for delay differential equations with negative Schwarzian derivative, Nonlinear Anal. Real World Appl., 11 (2010), 1422-1430.
doi: 10.1016/j.nonrwa.2009.02.030. |
[8] |
J. Mallet-Paret and G. R. Sell,
The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differ. Equ., 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[9] |
V. G. Nazarenko,
Influence of delay on auto-oscillations in cell populations, Biofisika, 21 (1976), 352-356.
|
[10] |
R. D. Nussbaum,
A global bifurcation theorem with applications to functional differential equations, J. Funct. Anal., 19 (1975), 319-338.
doi: 10.1016/0022-1236(75)90061-0. |
[11] |
R. D. Nussbaum,
Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl., 101 (1974), 263-306.
doi: 10.1007/BF02417109. |
[12] |
R. D. Nussbaum,
The range of periods of periodic solutions of $x' (t) = - \alpha f(x(t- 1))$, J. Math. Anal. Appl., 58 (1977), 280-292.
doi: 10.1016/0022-247X(77)90206-2. |
[13] |
R. D. Nussbaum,
Uniqueness and nonuniqueness for periodic solutions of $x'(t) = -g(x(t -1))$, J. Differ. Equ., 34 (1979), 25-54.
doi: 10.1016/0022-0396(79)90016-0. |
[14] |
S. Ruan, Delay differential equations in single species dynamics, in Delay differential equations and applications (eds. O. Arino, M. L. Hbid and E. Ait Dads), NATO Sci. Ser. Ⅱ Math. Phys. Chem., Vol. 205, Springer, Dordrecht, (2006), 477–517.
doi: 10.1007/1-4020-3647-7_11. |
[15] |
S. H. Saker and J. O. Alzabut,
Existence of periodic solutions, global attractivity and oscillation of impulsive delay population model, Nonlinear Anal. Real World Appl., 8 (2007), 1029-1039.
doi: 10.1016/j.nonrwa.2006.06.001. |
[16] |
Y. Song and Y. Peng,
Periodic solutions of a nonautonomous periodic model of population with continuous and discrete time, J. Comput. Appl. Math., 188 (2006), 256-264.
doi: 10.1016/j.cam.2005.04.017. |
[17] |
Y. Song, J. Wei and M. Han,
Local and global Hopf bifurcation in a delayed hematopoiesis model, Int. J. Bifurcation Chaos Appl. Sci. Eng., 14 (2004), 3909-3919.
doi: 10.1142/S0218127404011697. |
[18] |
H. O. Walther, Contracting return maps for some delay differential equations, in Topics in Functional Differential and Difference Equations (Lisbon, 1999), Fields Inst. Commun., Vol. 29, American Mathematical Society, Providence, RI, (2001), 349–360. |
[19] |
Q. Wang, J. Wen, S. Qiu and C. Guo, Numerical oscillations for first-order nonlinear delay differential equations in a hematopoiesis model, Adv. Differ. Equ., (2013), 17.
doi: 10.1186/1687-1847-2013-163. |
[20] |
J. Wu,
Symmetric functional-differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.
doi: 10.1090/S0002-9947-98-02083-2. |
show all references
References:
[1] |
Y. Cao,
Multiexistence of slowly oscillating periodic solutions for differential delay equations, SIAM J. Math. Anal., 26 (1995), 436-445.
doi: 10.1137/0526022. |
[2] |
Y. Cao,
Uniqueness of periodic solution for differential delay equations, J. Differ. Equ., 128 (1996), 46-57.
doi: 10.1006/jdeq.1996.0088. |
[3] |
J. L. Kaplan and J. A. Yorke,
On the stability of a periodic solution of a differential delay equation, SIAM J. Math. Anal., 6 (1975), 268-282.
doi: 10.1137/0506028. |
[4] |
B. Kennedy and E. Stumpf,
Multiple slowly oscillating periodic solutions for $x' (t) = f(x(t- 1))$ with negative feedback, Ann. Polon. Math., 118 (2016), 113-140.
doi: 10.4064/ap3899-10-2016. |
[5] |
I. Kubiaczyk and S. H. Saker,
Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Model., 35 (2002), 295-301.
doi: 10.1016/S0895-7177(01)00166-2. |
[6] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.
![]() ![]() |
[7] |
E. Liz and G. Röst,
Dichotomy results for delay differential equations with negative Schwarzian derivative, Nonlinear Anal. Real World Appl., 11 (2010), 1422-1430.
doi: 10.1016/j.nonrwa.2009.02.030. |
[8] |
J. Mallet-Paret and G. R. Sell,
The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differ. Equ., 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[9] |
V. G. Nazarenko,
Influence of delay on auto-oscillations in cell populations, Biofisika, 21 (1976), 352-356.
|
[10] |
R. D. Nussbaum,
A global bifurcation theorem with applications to functional differential equations, J. Funct. Anal., 19 (1975), 319-338.
doi: 10.1016/0022-1236(75)90061-0. |
[11] |
R. D. Nussbaum,
Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl., 101 (1974), 263-306.
doi: 10.1007/BF02417109. |
[12] |
R. D. Nussbaum,
The range of periods of periodic solutions of $x' (t) = - \alpha f(x(t- 1))$, J. Math. Anal. Appl., 58 (1977), 280-292.
doi: 10.1016/0022-247X(77)90206-2. |
[13] |
R. D. Nussbaum,
Uniqueness and nonuniqueness for periodic solutions of $x'(t) = -g(x(t -1))$, J. Differ. Equ., 34 (1979), 25-54.
doi: 10.1016/0022-0396(79)90016-0. |
[14] |
S. Ruan, Delay differential equations in single species dynamics, in Delay differential equations and applications (eds. O. Arino, M. L. Hbid and E. Ait Dads), NATO Sci. Ser. Ⅱ Math. Phys. Chem., Vol. 205, Springer, Dordrecht, (2006), 477–517.
doi: 10.1007/1-4020-3647-7_11. |
[15] |
S. H. Saker and J. O. Alzabut,
Existence of periodic solutions, global attractivity and oscillation of impulsive delay population model, Nonlinear Anal. Real World Appl., 8 (2007), 1029-1039.
doi: 10.1016/j.nonrwa.2006.06.001. |
[16] |
Y. Song and Y. Peng,
Periodic solutions of a nonautonomous periodic model of population with continuous and discrete time, J. Comput. Appl. Math., 188 (2006), 256-264.
doi: 10.1016/j.cam.2005.04.017. |
[17] |
Y. Song, J. Wei and M. Han,
Local and global Hopf bifurcation in a delayed hematopoiesis model, Int. J. Bifurcation Chaos Appl. Sci. Eng., 14 (2004), 3909-3919.
doi: 10.1142/S0218127404011697. |
[18] |
H. O. Walther, Contracting return maps for some delay differential equations, in Topics in Functional Differential and Difference Equations (Lisbon, 1999), Fields Inst. Commun., Vol. 29, American Mathematical Society, Providence, RI, (2001), 349–360. |
[19] |
Q. Wang, J. Wen, S. Qiu and C. Guo, Numerical oscillations for first-order nonlinear delay differential equations in a hematopoiesis model, Adv. Differ. Equ., (2013), 17.
doi: 10.1186/1687-1847-2013-163. |
[20] |
J. Wu,
Symmetric functional-differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.
doi: 10.1090/S0002-9947-98-02083-2. |


2.8 | 6 | 1.3 | 19 | 5.16 |
2.8 | 6.9 | 0.9 | 25 | 2.41 |
2.8 | 6.9 | 0.9 | 2 | 23.68 |
1.9 | 4.2 | 0.8 | 20 | 3.88 |
0.7 | 1.3 | 0.7 | 30 | 8.84 |
1.9 | 6.9 | 0.8 | 15 | 8.16 |
6.6 | 9.3 | 0.4 | 10 | 2.63 |
3 | 5.3 | 1.3 | 15 | 9.71 |
8.8 | 5.9 | 0.5 | 20 | 8.52 |
9 | 6.4 | 0.4 | 5 | 6.62 |
9 | 6.4 | 0.4 | 2 | 16.54 |
2.8 | 6 | 1.3 | 19 | 5.16 |
2.8 | 6.9 | 0.9 | 25 | 2.41 |
2.8 | 6.9 | 0.9 | 2 | 23.68 |
1.9 | 4.2 | 0.8 | 20 | 3.88 |
0.7 | 1.3 | 0.7 | 30 | 8.84 |
1.9 | 6.9 | 0.8 | 15 | 8.16 |
6.6 | 9.3 | 0.4 | 10 | 2.63 |
3 | 5.3 | 1.3 | 15 | 9.71 |
8.8 | 5.9 | 0.5 | 20 | 8.52 |
9 | 6.4 | 0.4 | 5 | 6.62 |
9 | 6.4 | 0.4 | 2 | 16.54 |
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