Early warning for spatial ecological system: Fractal dimension and deep learning

Ecological dynamic systems often undergo catastrophic regime shifts and have tipping points. Due to the complex interactions and feedbacks among different components of the systems, predicting such transition is a challenging task. This paper investigates the transition patterns of vegetation collapse in semiarid grazing systems. We propose the fractal dimension as a spatial early warning signal to detect this transition. The fractal dimension considers the spatial evolution from the perspective of self-similarity between vegetation. We show that the fractal dimension always decreases to a minimum when the system approaches the critical region, indicating a loss of resilience. We also assess the sensitivity of the fractal dimension under different scenarios of diffusion coefficients and noise levels, which affect the spatial patterns of the vegetation. We compare and analyze the fractal dimension with two commonly used metrics, spatial variance and skewness, and a novel deep learning method in the current research. We also investigate how well the fractal dimension performs with lower-resolution spatial data. Results indicate that the fractal dimension successfully predicts impending critical transition. It turns out that the fractal dimension is a reliable indicator and has significant implications for preventing desertification.