**I| Choice of the system**

Define what a patch is. In our case of Mediterranean ecosystems, we defined a patch as an area totally covered by vegetation.

**A) In case of a real system**

The line-intercept method is an appropriate sampling method to measure vegetation cover in patchy systems8. This method consists in measuring the vertical projection of vegetation on the ground. It provides a one-dimensional measure, but it is proven to be a good estimation for patch size8. This method assumes isotropy of the system. We selected areas with gentle slope, and to avoid anisotropy we established transects perpendicular to the slope. If the terrain would have been flat, we would have taken random orientation for the transects.

- Choice of different sites corresponding to the different conditions to be compared. Except for the condition, the sites should be as similar as possible. In our case, the studied condition was the level of grazing pressure. We chose sites under different grazing pressures that had equivalent climate, vegetation type, soil type, slope aspect, and slope angle.
- The condition must then be evaluated for each site. In our case, three grazing pressures were identified: low, medium and high. Effective stocking rates (animals ha-1 year-1) were calculated by animals observation. Animals (goats and sheep) were followed in the field. Their movement (position located by GPS and transferred to a map in a GIS) and the time spent in each site were recorded. Effective stocking rate was calculated as the average stocking rate multiplied by the percentage of time each grazing site was used
9. - Decide the time of the year to collect the data. In our case, Mediterranean climate presents a strong seasonality. Therefore, the data were collected at the peak vegetation growth period, which is from April to June.
- Choose the size of the transects. This depends on the size of the patches. A transect should be big enough to include all the spatial variability of interest. For the areas in Greece (shrubland) and Spain (scrubland), we took transects of 32 meters long. In Morocco however, we estimated a priori that 16 m was long enough, because the vegetation type was different (mountain grassland).
- Choose the number of transects (number of repetitions). We recorded 30 transects per grazing pressure (low, medium and high) and per area (Spain, Greece, Morocco).
- Choose a random location in the sites for all the transects so that the estimates are accurate.
- Put a nail at each of the ends of the transects.
- Put the measuring tape in between the two nails of each transect on top of the elements (e.g. above the shrubs in shrublands and herbaceous species in grasslands).
- Recording the location of the transects by GPS can be useful to follow the transects in time or to plot the transects on a map (this step is not necessary)
- It is easier to be two persons to record a transect. One person goes along the transect, holding a stick vertically in his hand. The stick follows the measuring tape. When the sticks touches an element of the patch (e.g. a plant), the second person records the beginning point of the patch as a distance on the transect from the beginning nail. The first person keeps going along the transect, still holding the stick vertically. The end point of the patch (again as a distance from the beginning nail) is recorded as soon as the stick does not touch any element of the patch anymore (Fig. 2).

**B) In case of a model system**

Let us assume that there is a spatial lattice-structured model that describes the system of interest. At the end of the simulation of the system dynamics, the model provides the spatial repartition of the elements that form the patches (e.g. vegetation). The lattice is then a two-phases mosaic: the patches of elements (e.g. patches of vegetation) and the rest of the lattice (e.g. sites without any vegetation).

- Define the type of neighbourhood. In our case, we decided that the four nearest neighbours of a lattice cell (top, bottom, left, right) constitute the neighbourhood of this cell. This is called the von Neumann neighbourhood
10. - Based on this definition, the patches can be delimited: two elements are part of the same patch if they are neighbours according to the definition.

**2| Extraction of the number and size of the patches**

Define sizes-classes. We took classes of 5 cells for the model analyses, and of 10 cm for the system analyses. The choice of the size-classes was motivated by the final number of classes obtained (and was therefore determined by the maximum patch size), but it does not affect the results.

**A) In case of a real system**

- Transform the data into patch size. The length of a patch along the transect is calculated as: end point minus beginning point (in cm). For each transect, record the size of all the patches, and the number of patches.
- Evaluate the number of patches in each size-class (e.g. number of patches whose size is between 0 and 10 cm, between 10 and 20 cm, etc).

**B) In case of a model system**

- Evaluate the size of the patches (number of cells belonging to the same patch).
- Calculate the number of patches in each class (e.g. number of patches that have a size between 1 and 5 cells, between 5 and 10 cells, etc)

*a) If the model is stochastic:*

(i) Along a simulation, once the steady state reached (e.g. the vegetation cover does not vary anymore), record the number and size of the patches at each time step during several time steps11. We did this for 2000 time step after steady state.

(ii) Average these distributions (e.g. total number of patches between 0 and 5 cells divided by the number of time steps).

Note that these latest points are different than what is done in real systems. Indeed in a stochastic model, simulations outputs differ at each time step even if the steady state is reached (e.g. the vegetation cover does not vary anymore). Having a high number of repetitions allows having a much better estimation of the law describing the patch size distribution.

*b) If the model is deterministic:*

(i) Once the steady state reached, evaluate the number and size of the patches in the same way as in the real system.

**III| Analysis of the patch size distribution**

- On a logarithmic scale, plot the number of patches in a given size-class as a function of the size-class.
- Compare different model fits statistically (see next point).

**IV| Comparison of different models for the description of the patch size distribution**

In our case of arid Mediterranean ecosystems, both in the data and in the model, the patch size distribution of the vegetation on a logarithm scale was either linear, meaning that the distribution follows a power law, or bended, what we call truncated power law in the following. Let *N(S)* be the number of patches of size *S*. A power law can be described by: *N(S)=C S*-γ, and a truncated power law by *N(S)=C S*-γ e-S/Sx (*C*, *γ* and *S*x are constants). These two models are nested. The truncated power law is the full model, and the power law the reduced model.

- Fit all the patch size distributions to the possible models (in our case power law and truncated power law).
- For each patch size distribution, perform a sum of square reduction test
12 to decide which model describes best the data. The *F* value of the sum of square reduction test can be obtained as following *F*obs=(SSRr - SSRf)/(q MSRf), with *SSR*f and *SSR*r the residuals sum of squares in the full and reduced model, respectively, *MSR*f the residual mean square in the full model, and *q* is the difference in the number of parameters between the full and the reduced model (in our case *q*=1, the only different parameter is *S*x).

Under the null hypothesis, the statistics *F*obs has an *F* distribution with *q* numerator and *dfR*f denominator degrees of freedom. *F*obs is compared to *F*α,q,dfRf. If the resulting *p*-value that is larger than *α*, the reduced model describes best the data (In our case, we took *α*=0.05).

Doing this for patch size distributions corresponding to different conditions allows checking whether a different model describes the patch size distribution under different external conditions (Fig. 3).

In the case of arid Mediterranean ecosystems, at low level of external stress (i.e. low grazing pressure), the patch size distribution was linear on a logarithm scale, meaning that the patch size distribution followed a power law. For low grazing pressure, a power law described best the model. For high grazing pressure the patch size distribution was bended on a logarithm scale, and a truncated power law described best the model.