Abstract: The vegetative cover in semi-arid lands typically occurs as patches of individual species more or less separated from one another by bare ground. Klausmeier [1999. Regular and irregular patterns in semiarid vegetation. Science 284 (5421), 1826-1828] reported that the vegetation striped patterns can grow lying along the contours of gentle slopes. He has proposed a model of vegetation stripes based on competition for water. In this paper, our main aim is to study the positive feedback effects between the water and biomass on the vegetation spatial pattern formation within a nonsaturated soil, which arises from the suction of water by the roots and processes of water resource redistribution. According to the dispersion relation formula, we discuss the changes of the wavelength, wave speed, as well as the conditions of the spatial pattern formation. Our numerical results show that trees are more sensitive than grasses to the positive feedback function to format the spatial heterogeneous pattern, and the stronger positive feedback increases the parameters region where vegetation bands occur, which indicates that the positive feedback raises the possibility of shift from green to desert states in semi-arid areas for the long term. Our numerical results also show that the positive feedback can increase the migration velocity of the vegetation stripes.

Abstract: We investigate epidemic models with spatial structure based on the cellular automata method. The construction of the cellular automata is from the study by Weimar and Boon about the reaction-diffusion equations [Phys. Rev. E 49, 1749 (1994)]. Our results show that the spatial epidemic models exhibit the spontaneous formation of irregular spiral waves at large scales within the domain of chaos. Moreover, the irregular spiral waves grow stably. The system also shows a spatial period-2 structure at one dimension outside the domain of chaos. It is interesting that the spatial period-2 structure will break and transform into a spatial synchronous configuration in the domain of chaos. Our results confirm that populations embed and disperse more stably in space than they do in nonspatial counterparts.