The model (Gilad et al., 2004) consists of three dynamical variables: (1) a biomass density, B(r,t), representing the plant's biomass above ground level and has units of [kg/m2], (2) the soil-water density variable, W(r,t), describing the amount of soil water available to the plants per unit area of ground surface in units of [kg/m2], and (3) the surface water variable, H(r,t), describing the height of a thin water layer above ground level in units of [mm]. The model equations are:
where
| GB[1/yr] represents biomass growth rate, |
| K[kg/m2] is the maximum standing biomass, |
| GW[1/yr] represents the soil water consumption rate, |
| I[1/yr] represents the infiltration rate of surface water into the soil, |
| P[mm/yr] is the precipitation rate, |
| N[1/yr] represents soil water evaporation rate, |
| R describes the reduction in soil-water evaporation rate due to shading, |
| M[1/yr] describes the rate of biomass loss due to mortality and disturbances, |
| Z(r) is the surface height, |
| DH is a friction coefficient between the surface water and the ground surface, |
and the Laplacians of B and W represent, respectively, seed dispersal and soil water transport in non-saturated soil.
The explicit forms of the rates of biomass growth, water consumption and water infiltration are given by
The integral forms model the effects of the plants' roots. The biomass growth rate depends not only on the amount of soil water at the plant location, but also on the amount of soil water in the neighborhood which the plant's roots extend to. Similarly the soil water consumption rate at a given point is due to all plants whose roots extend to this point.
The forms for the biomass growth rate and the water infiltration model two important positive feedbacks between biomass and water:
Water uptake by plants' roots: The bigger the plant the longer the roots (E>0), the larger the water uptake by the roots and the higher the biomass growth rate.
Increased infiltration at vegetation patches: The larger the biomass the higher the water infiltration rate (for f<<1).
