In semi-arid ecosystems, successful use of the limited water resources is of central importance in determining the evolutionary trends of the vegetation. The competition between different species and individuals for this resource is driven by variations in physiology and metabolic regulation strategies, expressed by such parameters as rooting depth, wilting point or stomatal opening, among others. It is typically not practical to simulate the full evolutionary dynamics of every plant individual in the landscape because of the difficulties introduced by the spatial heterogeneity, as well as the many timescales involved, ranging from hourly up to intergenerational. Instead, the amount of biomass of a given species assimilated per unit area of the landscape may serve as a proxy for its competitiveness and evolutionary success. It is the behaviour of the biomass, which must be described probabilistically due to the stochasticity of the rainfall, which is the subject here. This paper develops a new analytical description of the stationary and transient joint behaviour of plant biomass and soil moisture. Additionally, the effects of climatic fluctuations are considered, including the important case of a bi-seasonal climate regime consisting of alternating wet and dry seasons, which is characteristic of many ecosystems of interest.