On Dynamical Behavior of Discrete Time Fuzzy Logistic Equation

Theaimofthispaperistoinvestigatethedynamicalbehaviorofthefollowingmodelwhichdescribesthelogisticdifferenceequationtakingintoaccountthesubjectivityinthestatevariablesandintheparameters. 𝑥 𝑛+1 = 𝐴𝑥 𝑛 (̃1 − 𝑥 𝑛 ), 𝑛 = 0, 1, 2,⋅ ⋅⋅ , where {𝑥 𝑛 } is a sequence of positive fuzzy numbers. 𝐴, ̃1 and the initial value 𝑥 0 are positive fuzzy numbers. The existence and uniqueness of the positive solution and global asymptotic behavior of all positive solution of the fuzzy logistic difference equation are obtained. Moreover, some numerical examples are presented to show the effectiveness of results obtained.


Introduction
The first models for growing population were the classical Malthus and Verhulst (or logistic) models which deal with populations with one species [1]. In these models, the identification of the parameters is usually based on statistical methods, starting from data experimentally obtained and on the choice of some method adapted to the identification. These models are often subjected to inaccuracies (fuzzy uncertainty) that can be caused by the nature of the state variables, by parameters as coefficients of the model and by initial conditions.
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 [2] is one of suitable tools and its development has been growing rapidly to various situations of theory and application including the theory of differential equations and difference equations with uncertainty. The latter is known as fuzzy difference equation whose parameters or the initial values are fuzzy numbers, and its solution is a sequence of fuzzy numbers. It has been used to model a dynamical system under possibility uncertainty [3].
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: where parameter ,1 and the initial condition 0 are positive fuzzy numbers. The rest of this paper is organized as follows. In Section 2, we introduce some definitions and preliminaries. In Section 3, we study the existence, uniqueness, and global asymptotic behavior of the positive fuzzy solutions to system 2 Discrete Dynamics in Nature and Society (1). Some numerical examples are given to show effectiveness of results obtained in Section 4. A general conclusion is drawn in Section 5.

Mathematical Preliminaries
To be convenience, we give some definitions used in the sequel.
Let 1 be the set of all real fuzzy numbers which are normal, upper semicontinuous, convex, and compactly supported fuzzy sets.
( ( ) , ( )) , < 0, Definition 4 (triangular fuzzy number [19]). A triangular fuzzy number (TFN) denoted by is defined as ( , , ) where the membership function The The following proposition is fundamental since it characterizes a fuzzy set through the -levels.
Proposition 5 (see [19]). If { : ∈ [0, 1]} is a compact, convex, and not empty subset family of such that Definition 6 (see [4]). A sequence { } of positive fuzzy numbers persists (resp., is bounded) if there exists a positive real number (resp., ) such that supp ⊂ [ , ∞) (resp.supp ⊂ (0, ]) , A sequence { } of positive fuzzy numbers is bounded and persists if there exist positive real numbers , > 0 such that is called a positive solution of (1), if { } is a sequence of positive fuzzy numbers which satisfies (1). The equilibrium of (1) is the solution of the following equation: Definition 8 (see [3] The sequence { } converges to with respect to as → ∞ if lim →∞ ( , ) = 0.
Definition 10. Let be a positive equilibrium of (1). The positive equilibrium is stable, if, for every > 0, there exists a = ( ) such that for every positive solution of (1), which satisfies ( 0 , ) ≤ , we have ( , ) ≤ for all ≥ 0. The positive equilibrium is asymptotically stable, if it is stable and every positive solution of (1) converges to the positive equilibrium of (1) with respect to as → ∞.

Main Results
First we study the existence of the positive solutions of (1). We need the following lemma.
Proof. (i) It is clear that (0, 0) is always equilibrium. We can easily obtain that the linearized system of (24) about the positive equilibrium (0, 0) is from which we can easily obtain that the eigenvalue = , = . Since < 1 and < 1, so all of eigenvalue lie inside the unit disk. This implies that the equilibrium (0, 0) is local asymptotically stable.
(ii) We can obtain that the linearized system of (24) about the positive equilibrium (0, ) is where Φ = ( ) , The character equation of the linearized system of (28) about equilibrium (0, ) is It is clear that the characteristic root of (30) has a root = = 1. Hence the equilibrium (0, ) is unstable.
(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 (28) about equilibrium ( , ) is The character root of (31) is = 1 ± √( − 1)( − 1). It is clear that there exists a character root larger than 1. Hence the equilibrium ( , ) is unstable.
Remark 17. From Theorem 16, we can conclude that the dynamical behavior of (1) is correlated with the initial value 0 . When the initial value 0 is a positive fuzzy number which is smaller than one, no matter how long the support of 0 is, the solution of fuzzy difference equation (1) eventually converges to 0. From a biological point of view, if the population initial value is too small, even if the growth rate of the population is high (equal one), the population eventually becomes extinct.

Some Illustrative Examples
In order to illustrate our results obtained, we give some numerical examples to show effectiveness of results.
It results in a coupled system of difference equations with parameter ∈ (0, 1].   Clearly, all conditions of Theorem 16 are satisfied. Therefore, the solution of (50) converges to 0 (see Figures 4-6).

Conclusion
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 , < ,1 , < , < 1, then the positive solution of (1) eventually converges to 0 no matter how much the number of population initial values are. 8 Discrete Dynamics in Nature and Society (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 = 1, the system has unique positive equilibrium .
Finally, some examples are presented to show effectiveness of results.

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

Conflicts of Interest
The authors declare that they have no conflicts of interest.