Keywords: cellular automata, forest fire model, simulation, ecological modelling, environmental systems, discrete dynamical systems.

Forest fires is one of the major ecological agents in forests. Each year, fire burns between six and fourteen million hectares of forest; consequently they alter the structure and composition of forests and adversely affect human health and the supply of goods and services on which communities depend.

As a consequence, fires have received increased public attention worldwide. In this way the scientific community is responding to the challenge addresses forest fires as interesting and complex phenomena requiring a multi-disciplinary approach. Specifically, several mathematical approaches to the study of the spreading of fire have been appeared in the literature (see, for example [1,2,4]).

The main goal of this work is to introduce a new model for predicting forest fire spreading. It is based on a particular type of discrete dynamical system called two-dimensional cellular automata (2D-CA for short). Roughly speaking, 2D-CA are dynamical systems for which time and space are discrete. They consist of a collection of a finite two-dimensional array of simple objects, called cells, interacting locally with each other. Each cell can assume a state such that it changes in every time step according to a local rule whose variables are the states of some cells (its neighborhood) at previous time steps.

The proposed model is a modification of the model introduced by Karafyllidis and Thanailakis in [3]. In that model, the forest is divided into a matrix of identical square cells and it is represented by a 2D-CA, where each square cell of the forest is considered as a 2D-CA cell. Moreover, the local state of each cell at time is defined as the ratio of the burned out cell area to the total cell area. The state of an unburned cell is zero, whereas the state of a fully burned out cell is 1. Furthermore, each cell is endowed with a rate of fire spread , which is given by some other model. It determines the time needed for this cell to be fully burned out. The state of the -th cell at time step , , is affected by the states of all eight cells in its neighbourhood at time step and by its own state at time step as follows:

where and are specific values which are assigned to the state of the -th cell in order to incorporate the effect of the wind and the height, respectively. The time step in the model is taken to be equal to the time needed for the cells with the larger rate of fire spread to be fully burned out, when only one cell in its neighborhood is fully burned out at the previous time step.

The modified model introduced in this work proposes to change the parameter and the value of the time step. Specifically, we show that one must take in order to obtain a more realistic situation.

1

M.I. Asensio, L. Ferragut, J. Simon, "Modelling of convective phenomena in forest fire", RACSAM, Rev. Real Academia de Ciencias, Serie A. Mat. Vol. 96 (3), 299-313, 2002.

2

M. Dimopoulou, I. Giannikos, "Towards an integrated framework for forest fire control", European Journal of Operational Research, 152, 476-486, 2004. doi:10.1016/S0377-2217(03)00038-9

3

I. Karafyllidis, A. Thanailakis, "A model for predicting forest fire spreading using cellular automata", Ecological Modelling, 99, 87-97, 1997. doi:10.1016/S0304-3800(96)01942-4

4

R.C. Rothermel, "A mathematical model for predicting fire spread in windland fuels", Ogden Utah: USDA Forest Service Research Paper INT-115, 1972.

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