# American Institute of Mathematical Sciences

June  2017, 22(4): 1565-1573. doi: 10.3934/dcdsb.2017075

## Stability of equilibriums of stochastically perturbed delay differential neoclassical growth model

 School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel

Received  May 2016 Revised  October 2016 Published  February 2017

The known nonlinear delay differential neoclassical growth model is considered. It is assumed that this model is influenced by stochastic perturbations of the white noise type and these perturbations are directly proportional to the deviation of the system state from the zero or a positive equilibrium. Sufficient conditions for stability in probability of the positive equilibrium and for exponential mean square stability of the zero equilibrium are obtained. Numerical calculations and figures illustrate the obtained stability regions and behavior of stable and unstable solutions of the considered model. The proposed investigation procedure can be applied for arbitrary nonlinear stochastic delay differential equations with the order of nonlinearity higher than one.

Citation: Leonid Shaikhet. Stability of equilibriums of stochastically perturbed delay differential neoclassical growth model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1565-1573. doi: 10.3934/dcdsb.2017075
##### References:

show all references

##### References:
Stability regions for equation (1.3) (red and green) and equation (3.3) (yellow), $\gamma=3$ , $b=1$ , $h=0.02$ , $p=20$
Stability regions for equation (1.3) (red and green) and equation (3.3) (yellow), $\gamma=2$ , $b=2$ , $h=0.02$ , $p=20$
Trajectories of solution of equation (1.3) in unstable equilibrium for different initial functions: $x_0=1.19$ (green), $x_0=1.1805$ (red), $x_0=1.17$ (yellow), $A(700,300)$ , $\gamma=3$ , $b=1$ , $h=0.02$ , $p=20$ , $x_1.*=1.1805$
Trajectories of solution of equation (1.3) in stable equilibrium for different initial functions: $x_0=4.6$ (green), $x_0=1.65$ (red), $x_0=1.1$ (yellow), $A(700,300)$ , $\gamma=3$ , $b=1$ , $h=0.02$ , $p=20$ , $x_2.*=3.1270$
Trajectories of solution of equation (1.3) in unstable equilibrium for different initial functions: $x_0=0.6536$ (yellow) and $x_0=4.5215$ (red), $B(900,200)$ , $\gamma=3$ , $b=1$ , $h=0.02$ , $p=20$ , $x_1.*=0.6527$ and $x_2.*=4.5215$
Trajectories of solution of equation (3.3) in stable zero equilibrium for different initial functions: $b_0=0.8$ (green), $b_0=1.55$ (red), $b_0=100$ (yellow), $C(600,400)$ , $\gamma=3$ , $b=1$ , $h=0.02$ , $p=20$ , $x.*=0$
 [1] Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 [2] Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 [3] Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362 [4] 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 [5] 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 [6] Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial & Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133 [7] Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108 [8] Jingjing Wang, Zaiyun Peng, Zhi Lin, Daqiong Zhou. On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. Journal of Industrial & Management Optimization, 2021, 17 (2) : 869-887. doi: 10.3934/jimo.2020002 [9] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [10] 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 [11] Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 867-884. doi: 10.3934/cpaa.2020294 [12] Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017 [13] 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 [14] 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 [15] 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 [16] 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 [17] Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028 [18] Theresa Lange, Wilhelm Stannat. Mean field limit of ensemble square root filters - discrete and continuous time. Foundations of Data Science, 2021  doi: 10.3934/fods.2021003 [19] A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441 [20] Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275

2019 Impact Factor: 1.27