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We propose a fuzzy system that simulates dispersion of individuals whose movements are described by diffusion. We will use only the position of the population as an input variable for describing the process. We emphasize that the classical diffusion equation along with its analytical solution in no time was used for obtaining our solution.

The differential equations and deterministic differences are a powerful tool for modeling phenomena whose state variables are subject to changes over time. However, for the deterministic modeling be efficient is necessary to have a deep well of the relationship between the variables and their variations knowledge. It is the knowledge of the phenomenon that makes it possible to choose the functions that determine the variations with respect to the state (value) of the variable. In many cases, however, this relationship between variables and variations is only partially known, which make the deterministic model is less applicable [

On the other hand, models of fuzzy variational equations though behaving subjectivities are also not applicable modeling phenomena with partially known relationships. This comes from the fact that these models are derived from deterministic models. Subjectivity supported by fuzzy equations refers to uncertainties as the initial states of

The p-fuzzy systems incorporate subjective information in both variables as the variations and their relationships with the variables and are therefore a very useful tool for modeling phenomena whose behavior is partially known.

The fuzzy systems are generally the result of a generalization of the classical systems; that is, in this approach the uncertain concepts are incorporated into these systems. A central feature of fuzzy systems is that they are based on the concept of fuzzy partition information. The use of fuzzy sets allows a generalization of information that is associated with the introduction of imprecision ignoring the phenomena. In essence, the representation of information in fuzzy systems tries to imitate the process of human reasoning, considering heuristic knowledge and information across the disconnected principle [

In this work, we describe a diffusive process without the use of their analytical solution, using dynamical systems p-fuzzy and given a rule base. It is worth noting that the results obtained in terms of solution are very similar to the deterministic.

A subset (classic)

Thus, the membership function describes completely the set

Allowing a kind of relaxation in the image set of the membership function of a set, Zadeh mathematically formulated the concept of fuzzy set [

A fuzzy rule-based system has basically four components: an input processor (or fuzzificator), a set of linguistic rules, a model of fuzzy inference, and a processor output (or defuzzificator), generating a real number as output. Figure

Structure of the fuzzy controller.

The fuzzification is the process by which the input values of the system are converted to fuzzy sets with respective ranges of values where they are defined. It is a mapping of the field of real numbers to fuzzy field.

We can define the fuzzy roles by structures of the form If

A set of rules (or rule base) can describe a system in its various possibilities, fulfilling the role of translating mathematically the information of basis of knowledge of the fuzzy system. The rule base systems fuzzy (RBSF), in this case called fuzzy controllers, has four modules: the fuzzification module, the module based on linguistic rules, fuzzy inference, and defuzzification module. These modules are connected as shown in Figure

Structure of operation of a p-fuzzy system.

The

It is the definition of the rule base the information of the phenomenon under study are used. For each state defined by the linguistic terms of the input variable is a rule base. Thus, the more linguistic terms more details are incorporated in the model.

The relationship between the linguistic variable is characterized MIN by the operator; that is, each rule is considered a fuzzy relation

The relationship between each rule is characterized by the maximum operator, that is, fuzzy relation

Now we want to find each entry a corresponding action; that is, for a

If the entry is a classic unitary set, then

Mamdani inference engine with two linguistic input variables and one output.

The role of the

One of the main methods of defuzzification is the center of mass, for continuous variables, which is given by

This defuzzification method will be used throughout this paper. Note that the fuzzy controller can be seen as a function

In this section, we are interested in developing a base of rules that enable us to find a solution to a graphical problem involving diffusion. For this, we use only the position population, initial condition, and population growth. By these ways we may estimate the population density in an instant

Classical models of population dynamics and/or epidemiology, in overall, are given by a system of differential equations. In this case, the parameters of the models are often taken as mean values obtained from one set of data such that the model is to be deterministically known. However, admitting uncertainty due to partial knowledge, which is common in biological phenomena, an alternative is to model such knowledge from a set of rules of the form if-then.

It is common to adopt an equation

Thus, consider how linguistic variables to position of the population (distance to origin):

Linguistic variables for population position.

Likewise, as output variable, consider the variation population and linguistic variables:

Linguistic variables for population growth.

Considering the known results about diffusion process, consider the following of fuzzy rules:

if the position is of the individuals

if the position is of the individuals

if the position is of the individuals

if the position is of the individuals

if the position is of the individuals

if the position is of the individuals

if the position is of the individuals

if the position is of the individuals

Figure

Curve generated from the rules of the previous figure using Mamdani.

So with this rule base, the Mamdani controller and defuzzification given by the center of mass, the p-fuzzy system in leads to the path illustrated by the sequence of Figures

Solution to

Solution to

Solution to

Solution to

Solution to

Solution to

Solution to

Solution to

Another important observation is that time in our problem means the number of iterations in simulated MATLAB. This information is of great importance because it reduces the amount of data needed to describe the problem. We can thus write p-fuzzy dynamical system in the following form:

As for each iteration

Union of p-fuzzy solutions.

The most interesting in this process is not possible to know what the best model, it is the deterministic model or the p-fuzzy model, since the results are very similar as we can see in Figure

From the educational point of view the best model is secondary because you can always do better than the previous one and you can always imagine different situations for the same phenomenon.

We show that it is possible to use a FRBS to model the behavior of the population density of a species when you want to take into account the diffusion of individuals.

The authors declare that there is no conflict of interests regarding the publication of this paper.