scholarly journals Stability of positive steady-state solutions to a time-delayed system with some applications

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Shihe Xu ◽  
Fangwei Zhang ◽  
Meng Bai
Keyword(s):  
Steady State ◽  
Value Function ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Initial Value ◽  
Positive Steady State ◽  
Delayed System ◽  
T Tau ◽  
Steady State Solutions ◽  
Globally Stable

<p style='text-indent:20px;'>In this paper, we study a general nonlinear retarded system:</p><p style='text-indent:20px;'><disp-formula> <label>1</label> <tex-math id="E1"> \begin{document}$ \begin{equation} y'(t) = a(t)F(y(t),y(t-\tau)), \; \; t\geq 0, \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \tau&gt;0 $\end{document}</tex-math></inline-formula> is a constant, <inline-formula><tex-math id="M2">\begin{document}$ a(t) $\end{document}</tex-math></inline-formula> is a positive value function defined on <inline-formula><tex-math id="M3">\begin{document}$ [0,\infty) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ F(y,z) $\end{document}</tex-math></inline-formula> is continuous in <inline-formula><tex-math id="M5">\begin{document}$ \mathscr{D} = \mathbb{R}_+^2 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R_+} = (0,+\infty) $\end{document}</tex-math></inline-formula>. Sufficient conditions for stability of the unique positive equilibrium are established. Our results show that if <inline-formula><tex-math id="M7">\begin{document}$ F_z(y,z)&gt;0 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ y,z\in \mathbb{R_+} $\end{document}</tex-math></inline-formula>, then the unique positive equilibrium of (1) which denoted by <inline-formula><tex-math id="M9">\begin{document}$ \bar{y} $\end{document}</tex-math></inline-formula> is globally stable for any positive initial value and all <inline-formula><tex-math id="M10">\begin{document}$ \tau&gt;0 $\end{document}</tex-math></inline-formula>; if <inline-formula><tex-math id="M11">\begin{document}$ F(y,z) $\end{document}</tex-math></inline-formula> is decreasing in <inline-formula><tex-math id="M12">\begin{document}$ y $\end{document}</tex-math></inline-formula>, then <inline-formula><tex-math id="M13">\begin{document}$ \bar{y} $\end{document}</tex-math></inline-formula> is globally stable for small <inline-formula><tex-math id="M14">\begin{document}$ \tau $\end{document}</tex-math></inline-formula>. Some applications are given.</p>

scholarly journals Dynamics of a Diffusive Predator-Prey Model with General Nonlinear Functional Response

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Wensheng Yang
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Nonlinear Functional ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Parameter Configuration ◽  
Sufficient And Necessary Conditions ◽  
Steady State Solutions

We study a diffusive predator-prey model with nonconstant death rate and general nonlinear functional response. Firstly, stability analysis of the equilibrium for reduced ODE system is discussed. Secondly, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. Furthermore, sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the system are derived by using the method of Lyapunov function. Finally, we show that there are no nontrivial steady state solutions for certain parameter configuration.

Positive Solutions of a Strongly Coupled Prey-Predator System

2013 ◽  
Vol 748 ◽  
pp. 432-436
Author(s):  
Xiao Zhou Feng ◽  
Mei Hua Wei ◽  
Yan Ling Li
Keyword(s):  
Steady State ◽  
Fixed Point Index ◽  
Equation System ◽  
Point Index ◽  
Differential Equation System ◽  
Strongly Coupled ◽  
Positive Steady State ◽  
Holling Ii Functional Response ◽  
Steady State Solutions ◽  
Predator System

In this paper, the positive steady-state solutions of a strongly coupled partial differential equation system with Holling II functional response is studied. The existence for positive steady-state solutions of system is established by calculating the fixed point index in cone.

scholarly journals Stability of the positive steady-state solutions of systems of nonlinear Volterra difference equations of population models with diffusion and infinite delay

2000 ◽  
Vol 23 (4) ◽  
pp. 261-270 ◽  
Author(s):  
B. Shi
Keyword(s):  
Steady State ◽  
Difference Equations ◽  
Infinite Delay ◽  
Steady State Solution ◽  
Population Models ◽  
Monotone Iterative ◽  
Positive Steady State ◽  
Infinite Delays ◽  
Steady State Solutions ◽  
Volterra Difference Equations

An open problem given by Kocic and Ladas in 1993 is generalized and considered. A sufficient condition is obtained for each solution to tend to the positive steady-state solution of the systems of nonlinear Volterra difference equations of population models with diffusion and infinite delays by using the method of lower and upper solutions and monotone iterative techniques.

GLOBAL STABILITY OF A CHEMOSTAT MODEL WITH DELAYED RESPONSE IN GROWTH AND A LETHAL EXTERNAL INHIBITOR

2008 ◽  
Vol 01 (04) ◽  
pp. 503-520 ◽  
Author(s):  
ZHIQI LU ◽  
JINGJING WU
Keyword(s):  
Global Stability ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Delayed Response ◽  
Qualitative Properties ◽  
Chemostat Model ◽  
Stability And Instability ◽  
Discrete Delays ◽  
Globally Stable

A competition model between two species with a lethal inhibitor in a chemostat is analyzed. Discrete delays are used to describe the nutrient conversion process. The proved qualitative properties of the solution are positivity, boundedness. By analyzing the local stability of equilibria, it is found that the conditions for stability and instability of the boundary equilibria are similar to those in [9]. In addition, the global asymptotic behavior of the system is discussed and the sufficient conditions for the global stability of the boundary equilibria are obtained. Moreover, by numerical simulation, it is interesting to find that the positive equilibrium may be globally stable.

scholarly journals Feedback regulation of logistic growth

1993 ◽  
Vol 16 (1) ◽  
pp. 177-192 ◽  
Author(s):  
K. Gopalsamy ◽  
Pei-Xuan Weng
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
State Feedback ◽  
Control Variable ◽  
Sufficient Conditions ◽  
Feedback Regulation ◽  
Logistic Growth ◽  
Positive Equilibrium ◽  
Indirect Control ◽  
T Tau

Sufficient conditions are obtained for the global asymptotic stability of the positive equilibrium of a regulated logistic growth with a delay in the state feedback of the control modelled bydn(t)dt=rn(t)[1−(a1n(t)+a2n(t−τ)K)−cu(t)]dn(t)dt=−au(t)+bn(t−τ)whereudenotes an indirect control variable,r,a2,τ,a,b,c∈(0,∞)anda1∈[0,∞).

Sign in / Sign up

Export Citation Format

Share Document