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February  2021, 20(2): 755-762. doi: 10.3934/cpaa.2020288

Single species population dynamics in seasonal environment with short reproduction period

Attila Dénes , and  Gergely Röst

Bolyai Institute, University of Szeged, H-6720 Szeged, Hungary

* Corresponding author

Received  July 2020 Revised  September 2020 Published  December 2020

Fund Project: A. Dénes was supported by the Hungarian National Research, Development and Innovation Office grant NKFIH PD_128363 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. G. Röst was supported by EFOP-3.6.1-16-2016-00008 and by the Hungarian National Research, Development and Innovation Office the grant NKFIH KKP_129877 and TUDFO/47138-1/2019-ITM

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.

Keywords: Scalar population model, delay differential equation, threshold parameter, stability.
Mathematics Subject Classification: Primary:34K05, 34K20, 92D25.
Citation: Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, 2021, 20 (2) : 755-762. doi: 10.3934/cpaa.2020288
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[7] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.   Google Scholar
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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.   Google Scholar
[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.  Google Scholar

Figure 1.  The function $ f(t,x) $ for $ x\in\{5,10,100\} $ and $ \hat\alpha = 1000 $
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Figure 2.  Solutions of (1.1) with periodic Ricker-type birth function for different values of parameter $ \hat\alpha $
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Figure 3.  Solutions of (1.1) with periodic Beverton–Holt-type birth function for different values of parameter $ \hat\alpha $
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