Dryland ecosystems: The coupled stochastic dynamics of soil water and vegetation and the role of rainfall seasonality

In drylands the soil water availability is a key factor ruling the architecture of the ecosystem. The soil water reflects the exchanges of water among soil, vegetation, and atmosphere. Here, a dryland ecosystem is investigated through the analysis of the local interactions between soil water and vegetation forced by rainfall having seasonal and stochastic occurrence. The evolution of dryland ecosystems is represented by a system of two differential equations, having two steady states, one vegetated and the other unvegetated. The rainfall forcing is described by a diffusion process with monthly parameters. In each of the two possible steady states, the probability density functions of soil water and vegetation are derived analytically in terms of the rainfall distribution. The results show how the seasonality of rainfall influences the oscillation of the ecosystem between its vegetated steady state during the wet season and its unvegetated steady state during the dry season.

Figure 1

Comparison between observed (●) and calculated (○) monthly means of daily rainfall with the relative standard deviation (error bars) at Skukuza, South Africa. The gray line represents the threshold value $P=ϵ\mu w_1/\left(\gamma -\mu \right)$. All values are in m/yr.

Figure 2

Monthly estimates of the parameters of the $P$ distribution, for Skukuza data set. Estimates of (a) $\mathbb{P}\mathbf{\left(}P\left(\phantom{|}t|i\right)=0\mathbf{\right)}$, (b) ${\theta }_i$, (c) ${\lambda }_i$, and (d) ${\zeta }_i$. The standard deviation of estimate of each parameter is given as error bars.

Figure 3

Steady density function for the variables $P$, $S$, and $N$ and for each month (black lines). (●) indicates the atom probability. (a) $f_{P,i,st}$, (b) $f_{S,i,st}$ for the bare soil case, (c) $f_{S,i,st}$, and (d) $f_{N,i,st}$ for the vegetated state. For a clear comparison among the 12 monthly densities, these are projected in the same plane (gray lines).

Figure 4

Probability density function of the stable state solution for each month, for $S$ in (a) and $N$ in (b). The black line indicates that the stable solution is vegetated, and the gray line that the stable solution is unvegetated. (●) indicates the atom probability.