# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2020288

## Single species population dynamics in seasonal environment with short reproduction period

 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.

Citation: Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, 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 [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

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
The function $f(t,x)$ for $x\in\{5,10,100\}$ and $\hat\alpha = 1000$
Solutions of (1.1) with periodic Ricker-type birth function for different values of parameter $\hat\alpha$
Solutions of (1.1) with periodic Beverton–Holt-type birth function for different values of parameter $\hat\alpha$
 [1] Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042 [2] Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 [3] Kengo Nakai, Yoshitaka Saiki. Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1079-1092. doi: 10.3934/dcdss.2020352 [4] Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021005 [5] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [6] Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 [7] Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113 [8] Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464 [9] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454 [10] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316 [11] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [12] Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032 [13] Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020291 [14] Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 [15] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107 [16] John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044 [17] Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003 [18] Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 [19] Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021010 [20] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

2019 Impact Factor: 1.105

## Tools

Article outline

Figures and Tables