Single species population dynamics in seasonal environment with short reproduction period

Attila Dénes; Gergely Röst

doi:10.3934/cpaa.2020288

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2021, Volume 20, Issue 2: 755-762. Doi: 10.3934/cpaa.2020288

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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
^* Corresponding author

Received: July 2020

Revised: September 2020

Early access: December 2020

Published: February 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary:34K05, 34K20, 92D25.

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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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References

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