# Instability Induced by Random Background Noise in a Delay Model of Landslide Dynamics
## 4 Discussion and Conclusions  
In the present paper, we propose a landslide model that includes delayed interaction and the effect of random background noise. In the first phase of the paper, the introduction of background seismic noise is justified by the results of the analysis of the monitored  
Figure 9: Effect of noise intensity on dynamics of the observed system (3): (**a**) for high values of time delay \(\tau\) and low values of spring stiffness \(k\) and (**b**) for low values of time delay \(\tau\) and high values of spring stiffness \(k\). The constant values of other parameters are the same as in Figure 7.  
ambient noise within the location of the existing landslide in Serbia. The analysis is conducted using surrogate data testing and results in a classification of the analyzed monitoring results as linear stationary stochastic processes with Gaussian inputs. This justified the introduction of background random noise in the dynamical system proposed in the present paper. The model is suggested in the form of stochastic delay-differential equations, with included effect of delayed interaction and background seismic noise. Friction is assumed to be rate-dependent which is very well aligned with the results of previous studies. The dynamics of the proposed dynamical system are examined using mean-filed approximation: the derived mean-field model is shown to exhibit qualitatively the same dynamics as the starting stochastic model. The justification for using a single mean-field model instead of n globally coupled stochastic units lies in the very nature of the landslides formed in weathered rock mass: dynamics of such landslides could be observed as the average dynamics of the whole slope. Results of the conducted numerical analyses indicated that with the increase of spring stiffness and time delay, a direct Hopf supercritical bifurcation occurs, with the transition from a creeping state (constant displacement of low intensity) to regular periodic oscillations (unstable regime). The effect of noise on the dynamics of the suggested model is examined further numerically. It is shown that background noise has a positive effect on the stability of the landslide dynamics. In particular, noise has the same effect for highly weathered rock masses, with rather low values of spring stiffness and high values of time delay, and for slightly weathered rock masses, with high values of spring stiffness and low values of time delay. In both cases, an increase of \(D\) \"stabilizes\" the landslide, and vice versa--reduction of noise leads to the onset of slope instability. Such a stabilizing effect of the background noise confirms the significance of its effect, particularly near the bifurcation point, which is valid for the creeping landslides. From the geodynamical viewpoint, the obtained results could be interpreted in the following way:  
* Creeping slopes composed of highly weathered rocks, where individual unstable parts have loose interaction with high delay, are sensitive to the change of the background noise intensity. In this case, the interaction of time delay and noise intensity is particularly emphasized: an increase of noise intensity for two units of order leads to the significant stabilization of landslide activity, with the increase of time delay for 1. The interaction of noise and spring stiffness is not so significant, and approximately one order of units is smaller compared to the interaction of noise and time delay. In particular, an increase of noise intensity for two orders of units leads to the change in critical values of spring stiffness for approximately 0.1. * Creeping landslides composed of slightly weathered rock masses, where individual unstable parts have strong interactions, with low values of time delay, are more sensitive to the interaction of delay and spring stiffness compared to the interaction of \(D\) and \(\tau\).