Asymptotical Analysis on Deterministic and Stochastic Climate Models of Vegetation-CO₂-Temperature

The surface temperature of the Earth is affected significantly by emission of (Formula presented.) and vegetation. To study this complex climate system, in this paper, we propose a deterministic and two stochastic mathematical models to describe the climate system of vegetation- (Formula presented.) -temperature and investigate the dynamical behavior of their solutions. Initially, a deterministic model (ODE) is developed by taking into account the relationship among temperature, (Formula presented.), and vegetation. The existence and linearized stability of vegetation-free and vegetation-cover equilibria for this model are discussed by applying Lyapunov functions method. Based on this, through introducing stochastic disturbances, we further establish two (periodic) stochastic models, and the asymptotic behavior of their solutions is studied as well, including existence of (periodic) positive solutions, linearized stability, and global attractiveness of the vegetation-free steady point. Finally, the obtained theoretical results are illustrated and discussed through numerical simulations.