Rare events in a stochastic vegetation-water dynamical system based on machine learning

TL;DR

A machine learning-based method for analyzing rare events in stochastic vegetation-water dynamical systems with multiplicative Gaussian noise can effectively predict early warnings of vegetation degradation, providing new theoretical foundations and mathematical tools for ecological management and conservation.

Abstract

Stochastic vegetation-water dynamical systems play a pivotal role in ecological stability, biodiversity, water resource management, and adaptation to climate change. This research proposes a machine learning-based method for analyzing rare events in stochastic vegetation-water dynamical systems with multiplicative Gaussian noise. Utilizing the Freidlin-Wentzell large deviation theory, we derive the asymptotic expressions for the quasipotential and the mean first exit time. Based on the decomposition of vector field, we design a neural network architecture to compute the most probable transition paths and the mean first exit time for both non-characteristic and characteristic boundary scenarios. The results indicate that this method can effectively predict early warnings of vegetation degradation, providing new theoretical foundations and mathematical tools for ecological management and conservation. Moreover, the method offers new possibilities for exploring more complex and higher-dimensional stochastic dynamical systems.

Figures (11)

• Figure 1: Bifurcation digram of the vegetation system about the parameter $R$. It exhibits two branches, the blue curve representing the stable equilibria, while the red dashed branch representing the unstable equilibria. Upon varying the control parameter $R$, the two branches approach each other until they meet at the critical value of $R_c=1.4278$. At this point, they annihilate each other in a saddle-node bifurcation.
• Figure 2: For the selected value of $R=1.55$, the system \ref{['dvw']} possesses two stable fixed points $SN1=(0,1.55)$ and $SN2=(4.6366,0.9959)$, which are separated by the stable manifold of the saddle point US=(1.6667,1.2917). This stable manifold is delineated by a purple curve. Additionally, the unstable manifold of the saddle point US is denoted by a green curve.
• Figure 3: A representative transition trajectory of the stochastic vegetation-water model \ref{['vw']} depicted through Monte Carlo simulation.
• Figure 4: The loss function is decreasing with respect to the increasing number of epochs during the training of the neural network.
• Figure 5: The quasipotential function $V_\theta(x)$ and the two rotational components $l_{1\theta}(x)$, $l_{2\theta}(x)$ are learned by machine learning.