Research on Pattern Dynamics Behavior of a Fractional Vegetation-Water Model in Arid Flat Environment

Abstract

In order to stop and reverse land degradation and curb the loss of biodiversity, the United Nations 2030 Agenda for Sustainable Development proposes to combat desertification. In this paper, a fractional vegetation–water model in an arid flat environment is studied. The pattern behavior of the fractional model is much more complex than that of the integer order. We study the stability and Turing instability of the system, as well as the Hopf bifurcation of fractional order α, and obtain the Turing region in the parameter space. According to the amplitude equation, different types of stationary mode discoveries can be obtained, including point patterns and strip patterns. Finally, the results of the numerical simulation and theoretical analysis are consistent. We find some novel fractal patterns of the fractional vegetation–water model in an arid flat environment. When the diffusion coefficient, d, changes and other parameters remain unchanged, the pattern structure changes from stripes to spots. When the fractional order parameter, β, changes, and other parameters remain unchanged, the pattern structure becomes more stable and is not easy to destroy. The research results can provide new ideas for the prevention and control of desertification vegetation patterns.

Keywords

• vegetation pattern
• fractional vegetation–water model
• weakly nonlinear analysis
• Hopf bifurcation