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The aim of this paper is to investigate the dynamical behavior of the following model which describes the logistic difference equation taking into account the subjectivity in the state variables and in the parameters.

The first models for growing population were the classical Malthus and Verhulst (or logistic) models which deal with populations with one species [

In our real life, scientists have accepted the fact that uncertainty is very important study in most applications and they also have learned how to deal with uncertainty. Modeling the real life problems in such cases usually involves vagueness or uncertainty in some of the parameters or initial conditions. It is well known that fuzzy set introduced by Zadeh [

To the best of our knowledge, the behavior of the parametric fuzzy difference equation is not always the same as the behavior of corresponding parametric ordinary difference equation. In recent decades, there is an increasing interest in studying fuzzy difference equation by many scholars. Some results concerning the study of fuzzy difference equations are included in these papers (see, for example, [

What we propose in this paper is, to some extent, a generalization of classical logistic discrete model, using the subjectivity which comes from “fuzziness” of the biological phenomenon. The main aim in this paper is to investigate the dynamical behavior of the following logistic discrete time system:

The rest of this paper is organized as follows. In Section

To be convenience, we give some definitions used in the sequel.

(i)

(ii)

(iv) The support of

Let

A fuzzy number

(1)

(2)

(3)

A crisp (real) number

Let

(i)

(ii)

(iii)

(iv)

(v)

A triangular fuzzy number (TFN) denoted by

The

The following proposition is fundamental since it characterizes a fuzzy set through the

If

(i)

(ii)

(iii)

then there is

A sequence

Let

Let

Let

First we study the existence of the positive solutions of (

Let

Consider (

The proof is similar to Proposition 2.1 [

Now we show that

Moreover, for

Next we prove that the support of

Let

We prove that

Suppose that there exists another solution

In order to study the dynamical behavior of the solution

Consider the system of difference equations (

(i) The equilibrium

(ii) System (

(iii) System (

(iv) Suppose that

(i) It is clear that

(ii) We can obtain that the linearized system of (

(iii) The proof of (iii) is similar to the proof of (ii). So it is omitted.

(iv) We can obtain that the characteristic equation of the linearized system of (

Consider the fuzzy difference equation (

Let

Suppose that

Suppose that there exists a fuzzy number

Conversely, we prove that

Moreover, since

Furthermore since (

Consider fuzzy difference equation (

By Theorem

From Theorem

In order to illustrate our results obtained, we give some numerical examples to show effectiveness of results.

Consider discrete fuzzy logistic system with initial value

The dynamics of system (

The solution of system (

The solution of system (

Consider discrete fuzzy logistic system (

It results in a coupled system of difference equations with parameter

The dynamics of system (

The solution of system (

The solution of system (

This paper deals with the dynamical behavior of single population logistic model under fuzzy environment. Firstly, the existence of positive fuzzy solution of this model is proved. Secondly, we obtained the following results

(i) If

(ii) If the number of population model is too small, even if the growth rate of population is 1, the population also converges to 0 (extinct).

(iii) When the parameter of model

Finally, some examples are presented to show effectiveness of results.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work was financially supported by the National Natural Science Foundation of China (Grant no. 11761018) and Key Research Project of Guizhou University of Finance and Economics (2018XZD02).

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