The transitional functions

A relevant issue to be considered with regard to landscape simulation models, especially the mosaic structured models, refers to the neighborhood influence in the transition probabilities and, consequently, in the landscape patches dynamics. This can be explained by the fact that most of the transitions occur along the patch border. One way in which the DINAMICA software deals with this matter consists splitting the cell selection mechanism into two processes. The first process is dedicated only to the expansion and contraction of previous landscape patches and it is called Expander Function. By its turn, the second process is designed to generate or form new patches through a seedling mechanism and it is called Pacther Function.

The combination of the two process is shown in the equation bellow:

Where Qij means the total amount of transitions of type ij specified for one step and, r and s respectively represent the percentage of transitions executed by each function, being r + s = 1.

Varying the proportions of transition executed by the two functions, the simulation outputs can be made to approximate the structure of a landscape.

Both of the above mentioned functions employ a stochastic selecting mechanism. The applied algorithm consists of firstly reading the landscape map to pick up the cells with higher probabilities and order them in a data array, known as dynamic heap. In sequence, the selection of cells takes place randomly from top to bottom (the internal stochastic choosing mechanism can be loosened or tightened depending on the degree of randomization desired), storing at the end the locations of the selected cells. In a second step, the landscape map is again scanned to execute the selected transitions. In this way, it is guaranteed that there will be no bias from the reading of the landscape map, which it is always done sequentially from the top left of the raster. The above described procedures are used in both transitional functions. For each function, the number iterations needed to accomplish the amount of specified transitions is computed. In the event of the Expander function does not realize the amount of desired transitions after a limited number of iterations, it passes on to the Patcher Function a residual number of transitions, so that the total number of transitions always reaches an expected value. As a result, the final landscape map will only need to be evaluated with regard to its spatial configuration, because the distributional model will be coincident to that of the reference landscape.

The expander algorithm is realized by the following equation:

Where nj corresponds to the amount of cells of type j occurring in a window 3*3. This method guarantees that the maximum Pij will be the original Pij, whenever a cell type i is surrounded by at least 50% of neighbor cells of type j.

Despite the fact that DINAMICA is a mosaic structured model, it can work using this function, with the concept of landscape patches, which are either contracted or expanded depending on the type of transition.

The patcher function aims at reproducing the actual landscape structure, impeding the formation of tiny single cell patches, which probably would occur if only a simple allocation process were used.

This function employs a device which looks for cells around a chosen location for a combined transition. This is done, firstly by electing the core cell of the new patch and then selecting a specific number of cells around the core cell for the combined transition according to their Pij transition probabilities. As an example of the application of this function, it can be cited that the new deforested patches produced by the simulation are set to be equal to an area equivalent to the yearly forest cleared by a typical Amazonian colonist, about 5 hectares.

For each phase of the simulation, the percentage of the amount of transitions is set for each of the above functions. Yet to avoid infinite loops, a maximum number of iterations for the Expander Function is specified. The number of successful transited cells is subtracted from the total number desired and the rest is passed on to the following function.

Both described functions also incorporate an allocation device, which is responsible for locating on the map the cells with the highest transition probabilities for each ij desired transition. This function picks up those cells and orders them for subsequent selection. In this process, each newly selected cell will form a core for a new patch or an expansion fringe, which will still need to be developed by the use of the transitional functions.

The size of the new patches and the expansion fringes are set up according to a lognormal probability distribution, which implies that as input it is necessary to specify the parameters of this distribution, which are represented by the mean size and the variance of each type of new patch and expansion fringe to be formed.

By discovering the optimum combination of the two developed functions together with their input parameters, the simulation can be fine-tuned to make the output map have a similar structure of a particular landscape. This can be done by running several simulation tests with varying input parameters in a systematic way. Landscape structure indices, such as Contagion and Fractal dimension, can be used to evaluate the results. Those indices can be assessed through FRAGSTATS, a software developed by Mccarigall (1992). In addition to these methods, multiple fitting procedure can be used to evaluate the spatial configuration of the simulated landscape maps.

Click here to visit the CSR site and to get new versions of the DINAMICA software together with its documentation and the database of the two Amazon areas, where the software was tested – Terra Nova and Guarantã. If you are interested in further readings, get also the paper: Landscape analysis, in PDF and Winzip formats.