## Fuzzy Sets for Continuous Inputs

In the definition of fuzzy sets the inputs were given by enumeration. Spatial modeling of ecological processes often involves inputs which are continuous variables such as distances or density functions. This continuous values must be 'fuzzified', that is, the modeler must specify membership function values for the fuzzy sets. Membership functions with triangular or trapezoidal shape are frequently used. A triangular membership function is defined as:

The lower-left edge point is x1, x2 the point where the triangle is 1, and x3 the right-hand point of the triangle. This membership function describes fuzzy values, for example, 'about 10' (meaning numbers around 10). In this example x1 = 8 and x3 = 12. The 10 belongs to this set with MA(10) = 1, and 9 belongs to this set with a degree Ma(9) = (9 8)/(10 8) = 1/2.

Another commonly used membership function type has a trapezoidal shape, and is used to describe ranges. A range is a generalization of classical sets. In the middle of a range, between [x2, x3] all values belong to the set with a membership of muA(x) = 1. But there are some values between (x1, x2) and (x3, x4) with a strict fuzziness. This type of membership function is often used in habitat modeling. If an effect of disturbance on a habitat has to be modeled, then this range is often helpful. In the center of the range the animal can tolerate the disturbance, but if the disturbance exceeds a determined level, the reaction may be strong. In a formal way this range can be defined as ma(x) = <

The triangular and the trapezoidal membership functions are simple and useful in modeling environmental processes. More sophisticated membership function types like bell-shaped functions are possible but often overrated. A trapezoidal membership can approximate a bell-shaped function quite well, but the bell-shaped function is itself an approximation. The ultimate objective is to use fuzzy models to transform expert knowledge into formal knowledge that can be used to perform simulations. 