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Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity
Single species population dynamics in seasonal environment with short reproduction period
Bolyai Institute, University of Szeged, H-6720 Szeged, Hungary |
We present a periodic nonlinear scalar delay differential equation model for a population with short reproduction period. By transforming the equation to a discrete dynamical system, we reduce the infinite dimensional problem to one dimension. We determine the basic reproduction number not merely as the spectral radius of an operator, but as an explicit formula and show that is serves as a threshold parameter for the stability of the trivial equilibrium and for permanence.
References:
[1] |
M. Gyllenberg, I. Hanksi and T. Lindström, Continuous versus discrete single species population models with adjustable reproduction strategies, Bull. Math. Biol., 59 (1997), 679–705.
doi: 10.1007/BF02458425. |
[2] |
E. Liz, Clark's equation: a useful difference equation for population models, predictive control, and numerical approximations, Qual. Theory Dyn. Syst., 19 (2020), 11 pp.
doi: 10.1007/s12346-020-00405-1. |
[3] |
K. Nah and G. Röst, Stability threshold for scalar linear periodic delay differential equations, Canad. Math. Bull., 59 (2016), 849–857.
doi: 10.4153/CMB-2016-043-0. |
[4] |
R. Qesmi, A short survey on delay differential systems with periodic coefficients, J. Appl. Anal. Comput., 8 (2018), 296–330.
doi: 10.11948/2018.296. |
[5] |
G. Röst, Neimark–Sacker bifurcation for periodic delay differential equations, Nonlinear Anal., 60(2005), 1025–1044.
doi: 10.1016/j.na.2004.08.043. |
[6] |
H. L. Smith, An introduction to delay differential equations with applications to the life sciences, Texts in Applied Mathematics, Springer, New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[7] |
H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.
![]() |
[8] |
X. Q. Zhao,
Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Differ. Equ., 29 (2017), 67-82.
doi: 10.1007/s10884-015-9425-2. |
show all references
References:
[1] |
M. Gyllenberg, I. Hanksi and T. Lindström, Continuous versus discrete single species population models with adjustable reproduction strategies, Bull. Math. Biol., 59 (1997), 679–705.
doi: 10.1007/BF02458425. |
[2] |
E. Liz, Clark's equation: a useful difference equation for population models, predictive control, and numerical approximations, Qual. Theory Dyn. Syst., 19 (2020), 11 pp.
doi: 10.1007/s12346-020-00405-1. |
[3] |
K. Nah and G. Röst, Stability threshold for scalar linear periodic delay differential equations, Canad. Math. Bull., 59 (2016), 849–857.
doi: 10.4153/CMB-2016-043-0. |
[4] |
R. Qesmi, A short survey on delay differential systems with periodic coefficients, J. Appl. Anal. Comput., 8 (2018), 296–330.
doi: 10.11948/2018.296. |
[5] |
G. Röst, Neimark–Sacker bifurcation for periodic delay differential equations, Nonlinear Anal., 60(2005), 1025–1044.
doi: 10.1016/j.na.2004.08.043. |
[6] |
H. L. Smith, An introduction to delay differential equations with applications to the life sciences, Texts in Applied Mathematics, Springer, New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[7] |
H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.
![]() |
[8] |
X. Q. Zhao,
Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Differ. Equ., 29 (2017), 67-82.
doi: 10.1007/s10884-015-9425-2. |


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