# American Institute of Mathematical Sciences

April  2022, 27(4): 2427-2440. doi: 10.3934/dcdsb.2021138

## Global population dynamics of a single species structured with distinctive time-varying maturation and self-limitation delays

 1 School of Mathematics and Statistics, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha, 410114, China 2 Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, M3J 1P3, Canada

* Corresponding author: Jianhong Wu

Received  September 2020 Revised  February 2021 Published  April 2022 Early access  May 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 11971076), the Natural Sciences and Engineering Research Council of Canada, and the Canada Research Chairs program

We consider the classical Nicholson's blowflies model incorporating two distinctive time-varying delays. One of the delays corresponds to the length of the individual's life cycle, and another corresponds to the specific physiological stage when self-limitation feedback takes place. Unlike the classical formulation of Nicholson's blowflies equation where self-regulation appears due to the competition of the productive adults for resources, the self-limitation of our considered model can occur at any developmental stage of an individual during the entire life cycle. We aim to find sharp conditions for the global asymptotic stability of a positive equilibrium. This is a significant challenge even when both delays are held at constant values. Here, we develop an approach to obtain a sharp and explicit criterion in an important situation where the two delays are asymptotically apart. Our approach can be also applied to the non-autonomous Mackey-Glass equation to provide a partial solution to an open problem about the global dynamics.

Citation: Chuangxia Huang, Lihong Huang, Jianhong Wu. Global population dynamics of a single species structured with distinctive time-varying maturation and self-limitation delays. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2427-2440. doi: 10.3934/dcdsb.2021138
##### References:
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show all references

##### References:
 [1] L. Berezansky and E. Braverman, A note on stability of Mackey-Glass equations with two delays, J. Math. Anal. Appl., 450 (2017), 1208-1228.  doi: 10.1016/j.jmaa.2017.01.050. [2] L. Berezansky and E. Braverman, Boundedness and persistence of delay differential equations with mixed nonlinearity, Appl. Math. Comput., 279 (2016), 154-169.  doi: 10.1016/j.amc.2016.01.015. [3] L. Berezansky, J. Bastinec, J. Diblík and Z. Smarda, On a delay population model with quadratic nonlinearity, Adv. Difference Equ., 2012 (2012), 1-9.  doi: 10.1186/1687-1847-2012-230. [4] L. Berezansky and E. Braverman, Global linearized stability theory for delay differential equations, Nonlinear Anal., 71 (2009), 2614-2624.  doi: 10.1016/j.na.2009.01.147. [5] L. Berezansky, E. Braverman and E. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417.  doi: 10.1016/j.apm.2009.08.027. [6] Q. Cao, G. Wang, H. Zhang and S. Gong, New results on global asymptotic stability for a nonlinear density-dependent mortality Nicholson's blowflies model with multiple pairs of time-varying delays, J. Inequal. Appl., 7 (2020), 1-12.  doi: 10.1186/s13660-019-2277-2. [7] Y. Chen, Periodic solutions of delayed periodic Nicholson's blowflies models, Canad. Appl. Math. Quart., 11 (2003), 23-28. [8] H. El-Morshedy and A. Ruiz-Herrera, Global convergence to equilibria in non-monotone delay differential equations, Proc. Amer. Math. Soc., 147 (2019), 2095-2105.  doi: 10.1090/proc/14360. [9] I. Győri, F. Hartung and N. Mohamady, Permanence in a class of delay differential equations with mixed monotonicitys, Electronic J. Qual. Theory Differ. Equ., 53 (2018), 1-21.  doi: 10.14232/ejqtde.2018.1.53. [10] C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Diff. Equ., 256 (2014), 2101-2114.  doi: 10.1016/j.jde.2013.12.015. [11] C. Huang, X. Zhao, J. Cao and F. Alsaadi, Global dynamics of neoclassical growth model with multiple pairs of variable delays, Nonlinearity, 33 (2020), 6819-6834.  doi: 10.1088/1361-6544/abab4e. [12] C. Huang and Y. Tan, Global behavior of a reaction-diffusion model with time delay and Dirichlet condition, J. Differ. Equ., 271 (2021), 186-215.  doi: 10.1016/j.jde.2020.08.008. [13] C. Huang, Y. Tan, B. Sun and T. Wang, Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition, J. Math. Anal. Appl., 458 (2018), 1115-1130.  doi: 10.1016/j.jmaa.2017.09.045. [14] X. Long, Novel stability criteria on a patch structure nicholson's blowflies model with multiple pairs of time-varying delays, AIMS Mathematics, 5 (2020), 7387-7401.  doi: 10.3934/math.2020473. [15] Y. Muroya, Global stability for separable nonlinear delay differential equations, Comput. Math. Appl., 49 (2005), 1913-1927.  doi: 10.1016/j.camwa.2004.02.013. [16] C. Qian and Y. Hu, Novel stability criteria on nonlinear density-dependent mortality nicholson's blowflies systems in asymptotically almost periodic environments, J. Inequal. Appl., 2020 (2020), 1-18.  doi: 10.1186/s13660-019-2275-4. [17] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [18] J. So and J. Yu, Global attractivity and uniform persistence in Nicholson's blowflies, Diff. Equ. & Dyn. Sys., 2 (1994), 11-18. [19] D. Son, L. HienB and T. Tuan Anh, Global attractivity of positive periodic solution of a delayed Nicholson model with nonlinear density-dependent mortality term, J. Qual. Theory Differ. Equ., 2019 (2019), 1-21.  doi: 10.14232/ejqtde.2019.1.8. [20] Y. Xu, New stability theorem for periodic Nicholson's model with mortality term, Appl. Math. Lett., 94 (2019), 59-65.  doi: 10.1016/j.aml.2019.02.021. [21] Y. Xu, Q. Cao and X. Guo, Stability on a patch structure nicholson's blowflies system involving distinctive delays, Appl. Math. Lett., 105 (2020), 106340, 7pp. doi: 10.1016/j.aml.2020.106340. [22] L. Yao, Dynamics of Nicholson's blowflies models with a nonlinear density-dependent mortality, Appl. Math. Model., 64 (2018), 185-195.  doi: 10.1016/j.apm.2018.07.007. [23] X. Zhang and J. Wu, Implication of vector attachment and host grooming behaviours for vector population dynamics and vector-on-host distribution patterns, Appl. Math. Model., 81 (2020), 1-15.  doi: 10.1016/j.apm.2019.12.012. [24] C. Zhao, L. Debnath and K. Wang, Positive periodic solutions of a delayed model in population, Appl. Math. Lett., 16 (2003), 561-565.  doi: 10.1016/S0893-9659(03)00037-5.
Numerical solutions $x(t)$ to (17) with the two delays $h(t)$ and $g(t)$ given by (18) and different initial values $6$, $15$, $20$
Numerical solutions $x(t)$ to (17) with delays given in (19) and different initial values $5$, $17$, $30$
Numerical solutions $x(t)$ to (17) with delays given in (20) and different initial values $6$, $15$, $20$
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