# American Institute of Mathematical Sciences

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:
 $$$y'(t) = a(t)F(y(t),y(t-\tau)), \; \; t\geq 0,$$$
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
##### References:

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