    doi: 10.3934/dcdsb.2021286
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Stability of positive steady-state solutions to a time-delayed system with some applications

 1 School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China 2 School of Navigation and Shipping, Shandong Jiaotong University, Weihai, Shandong 264200, China

*Corresponding author: Shihe Xu

Received  April 2021 Revised  October 2021 Early access December 2021

Fund Project: The first author and the third author are supported by NSF of Guangdong Province (2018A030313536), Foundation of Science and Technology Innovation Project of Zhaoqing (202004031510) and Foundation of Characteristic Innovation Project of Universities in Guangdong, China (2021KTSCX142). The second author is supported by Shanghai Pujiang Program (2019PJC062)

In this paper, we study a general nonlinear retarded system:
 $\begin{equation} y'(t) = a(t)F(y(t),y(t-\tau)), \; \; t\geq 0, \end{equation}$
where
 $\tau>0$
is a constant,
 $a(t)$
is a positive value function defined on
 $[0,\infty)$
,
 $F(y,z)$
is continuous in
 $\mathscr{D} = \mathbb{R}_+^2$
, where
 $\mathbb{R_+} = (0,+\infty)$
. Sufficient conditions for stability of the unique positive equilibrium are established. Our results show that if
 $F_z(y,z)>0$
for
 $y,z\in \mathbb{R_+}$
, then the unique positive equilibrium of (1) which denoted by
 $\bar{y}$
is globally stable for any positive initial value and all
 $\tau>0$
; if
 $F(y,z)$
is decreasing in
 $y$
, then
 $\bar{y}$
is globally stable for small
 $\tau$
. Some applications are given.
Citation: Shihe Xu, Fangwei Zhang, Meng Bai. Stability of positive steady-state solutions to a time-delayed system with some applications. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021286
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References:
  J. Belair and M. C. Mackey, Consumor memory and price fluctuations in commodity markets:An intergrodifferential model, J. Dynam. Diff. Equations, 1 (1989), 299-325.  doi: 10.1007/BF01053930.  Google Scholar  S. Cui and S. Xu, Analysis of mathematical models for the growth of tumors with time delays in cell proliferation, J. Math. Anal. Appl., 336 (2007), 523-541.  doi: 10.1016/j.jmaa.2007.02.047.  Google Scholar  U. Foryś and M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Modelling, 37 (2003), 1201-1209.  doi: 10.1016/S0895-7177(03)80019-5. Google Scholar  G. Huang, A. Liu and U. Foryś, Global stability analysis of some nonlinear delay differential equations in populaton dynamics, J. Nonlinear Sci, 26 (2016), 27-41.  doi: 10.1007/s00332-015-9267-4.  Google Scholar  Y. Kuang, Global attractivity and periodic solutions in delay differential equations related to models of physiology and population biology, Japan J. Indust. Appl. Math, 9 (1992), 205-238.  doi: 10.1007/BF03167566.  Google Scholar  Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. Google Scholar  M. Mackey, Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors, J. Econom. Theory, 48 (1989), 497-509.  doi: 10.1016/0022-0531(89)90039-2.  Google Scholar  M. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.  doi: 10.1126/science.267326. Google Scholar  C. Qian, Global attractivity in a nonlinear delay differential equtuion with applications, Nonlinear Anal., 71 (2009), 1893-1900.  doi: 10.1016/j.na.2009.01.024.  Google Scholar  C. Qian, Global attractivity in a delay differential equation with application in a commodity model, Appl. Math. Lett., 24 (2011), 116-121.  doi: 10.1016/j.aml.2010.08.029.  Google Scholar  C. Qian, Global attractivity of solutions of nonliinear delay differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 13B (2006), 25-37. Google Scholar  S. Xu, Analysis of a free boundary problem for tumor growth in a periodic external environment, Bound Value Probl., 2015 (2015), 1-12.  doi: 10.1186/s13661-015-0399-0.  Google Scholar  S. Xu, M. Bai and X. Zhao, Analysis of a solid avascular tumor growth model with time delays in proliferation process, J. Math. Anal. Appl., 391 (2012), 38-47.  doi: 10.1016/j.jmaa.2012.02.034.  Google Scholar  S. Xu, Analysis of a free boundary problem for tumor growth with angiogenesis and time delays in proliferation, Nonlinear Anal. Real World Appl., 51 (2020), 103005.  doi: 10.1016/j.nonrwa.2019.103005.  Google Scholar
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