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On Dynamical Behavior of Discrete Time Fuzzy Logistic Equation.

1. 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].

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, [3-18]).

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:

[x.sub.n+1] = A[x.sub.n] ([??] - [x.sub.n]), n = 0, 1, ..., (1)

where parameter A, 1 and the initial condition [x.sub.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 (1). Some numerical examples are given to show effectiveness of results obtained in Section 4. A general conclusion ii drawn in Section 5.

2. Mathematical Preliminaries

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

Definition 1 (see [3]). A is said to be a fuzzy number if A : R [right arrow] [0,1] satisfies (i)-(iv).

(i) A is normal; i.e., there exists an x [member of] R such that A(x) = 1.

(ii) A is fuzzy convex; i.e., for all t [member of] [0,1] and [x.sub.1], [x.sub.2] [member of] R such that

A(t[x.sub.1] + (1 - t)[x.sub.2]) [greater than or equal to] min{A([x.sub.1]),A([x.sub.2])}. (2)

(iii) A is upper semicontinuous.

(iv) The support of A, supp A = [bar.[[union].sub.[alpha][member of](0,1)][[A].sup.[alpha]]] = [bar.{x : A(x) > 0}] is compact, where [bar.A] denotes the closure of A.

Let [E.sup.1] be the set of all real fuzzy numbers which are normal, upper semicontinuous, convex, and compactly supported fuzzy sets.

Definition 2 (fuzzy number (parametric form) [19]). A fuzzy number m in a parametric form is a pair ([u.bar], [bar.u]) of functions [u.bar](r), [bar.u](r),0 [less than or equal to] r [less than or equal to] 1, which satisfies the following requirements:

(1) [u.bar](r) is a bounded monotonie increasing left continuous function.

(2) [bar.u](r) is a bounded monotonie decreasing left continuous function.

(3) [u.bar](r) [less than or equal to] [bar.u](r),0 [less than or equal to] r [less than or equal to] 1.

A crisp (real) number x is simply represented by ([u.bar](r),[u.sub.2](r)) = (x,x),0 [less than or equal to] r [less than or equal to] 1. The fuzzy number space {([u.bar](r), [bar.u](r))} becomes a convex cone [E.sup.1] which could be embedded isomorphically and isometrically into a Banach space [17].

Definition 3 (see [19]). Let m = ([u.bar](r), [bar.u](r)),v = ([v.bar](r),[bar.v](r)) [member of] [E.sup.1], 0 [less than or equal to] r [less than or equal to] 1, and arbitrary k [member of] R. Then

(i) u = v iff [u.bar](r) = [v.bar](r), [bar.u](r) = [bar.v](r)

(ii) u + v = ([u.bar](r) + [v.bar](r), [bar.u](r) + [bar.v](r))

(iii) u - v = ([u.bar](r) - [bar.v](r), [bar.u](r) - [v.bar](r))

(iv)

[mathematical expression not reproducible], (3)

(v) [mathematical expression not reproducible]

Definition 4 (triangular fuzzy number [19]). A triangular fuzzy number (TFN) denoted by A is defined as (b, c, d) where the membership function

[mathematical expression not reproducible]. (4)

The a-cuts of A = (b,c,d) are denoted by [[A].sup.[alpha]] = {x [member of] R : A(x) [greater than or equal to] [alpha]} = [b + [alpha](c -b),d - [alpha](d - c)] = [[A.sub.l,[alpha]],[A.sub.r,[alpha]]], [alpha] [member of] [0,1]; it is clear that [[A].sup.[alpha]] is a closed interval. A fuzzy number is positive if supp A [subset] (0, [infinity]).

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

Proposition 5 (see [19]). If {[A.sup.[alpha]] : [alpha] [member of] [0,1]} is a compact, convex, and not empty subset family of [R.sup.n] such that

(i) [[bar.[union][A.sup.[alpha]]] [subset] [A.sup.0]

(ii) [mathematical expression not reproducible]

(iii) [mathematical expression not reproducible]

then there is u [member of] [E.sup.n] such that [[u].sup.[alpha]] = [A.sup.[alpha]] for all [alpha] [member of] (0,1] and [[u].sup.0]] = [[bar.[[union].sub.0<[alpha][less than or equal to]1][A.sup.[alpha]]] [subset] [A.sup.0].

Definition 6 (see [4]). A sequence {[x.sub.n]} of positive fuzzy numbers persists (resp., is bounded) if there exists a positive real number M (resp., N) such that

supp [x.sub.n] [subset] [M, [infinity]) (resp.supp[x.sub.n] [subset] (0,N]), (5) N = 1,2,.....

A sequence {[x.sub.n]} of positive fuzzy numbers is bounded and persists if there exist positive real numbers M,N > 0 such that

supp [x.sub.n] [subset] [M, N], n= 1, 2,.... (6)

Definition 7. [x.sub.n] is called a positive solution of (1), if {[x.sub.n]} is a sequence of positive fuzzy numbers which satisfies (1). The equilibrium of (1) is the solution of the following equation: x = A x ([??] - x).

Definition 8 (see [3]). Let A,B be fuzzy numbers with [[A].sup.[alpha]] = [[A.sub.l,[alpha]], [A.sub.r,[alpha]]], [[B].sup.[alpha]] = [[B.sub.l,[alpha]], [B.sub.r,[alpha]]]; then the metric of A and B is defined as

[mathematical expression not reproducible]. (7)

Definition 9. Let {[x.sub.n]} be a sequence of positive fuzzy numbers and x is a positive fuzzy number. Suppose that

[[[x.sub.n]].sup.[alpha]] =[[Ln,[alpha]],[R.sub.n,[alpha]], [alpha] [member of] (0,1], n=0,1, ..., (8)

and

[[x].sup.[alpha]] = [[L.sub.[alpha]],[R.sub.[alpha]]], [alpha] [member of] (0,1], (9)

The sequence {[x.sub.n]} converges to x with respect to D as n [right arrow] [infinity] if [lim.sub.n [right arrow] [infinity]]D([x.sub.n],x) = 0.

Definition 10. Let x be a positive equilibrium of (1). The positive equilibrium x is stable, if, for every [epsilon] > 0, there exists a [delta] = [delta]([epsilon]) such that for every positive solution [x.sub.n] of (1), which satisfies D([x.sub.0], x) [less than or equal to] [delta], we have D([x.sub.n], x) [less than or equal to] [epsilon] for all n [greater than or equal to] 0. The positive equilibrium x is asymptotically stable, if it is stable and every positive solution of (1) converges to the positive equilibrium of (1) with respect to D as n [right arrow] [infinity].

3. Main Results

First we study the existence of the positive solutions of (1). We need the following lemma.

Lemma 11 (see [20]). Let f : [R.sup.+] x [R.sup.+] [right arrow] [R.sup.+] be continuous; A and B are fuzzy numbers. Then

[[f(A,B)].sup.[alpha]] =f([[A].sup.[alpha]], [[B].sup.[alpha]]), [alpha] [member of] (0,1] (10)

Theorem 12. Consider (1) where A, [??] are positive fuzzy numbers. Then, for any positive fuzzy numbers [x.sub.0], there exists a unique positive solution [x.sub.n] of(1).

Proof. The proof is similar to Proposition 2.1 [5]. Suppose that there exists a sequence {[x.sub.n]} of fuzzy numbers satisfying (1) with initial condition [x.sub.0]. Consider [alpha]-cuts, [alpha] [member of] (0,1], n = 0,1,2, ..., applying Lemma 11; we have

[mathematical expression not reproducible]. (11)

From (11), we can get the following system of ordinary difference equation with parameter [alpha] [member of] (0,1], for n = 0,1,2, ...,

[mathematical expression not reproducible]. (12)

Then, for any initial condition ([L.sub.0,[alpha]], [R.sub.0,[alpha]]), [alpha] [member of] (0,1], there exists a unique solution ([L.sub.n,[alpha]], [R.sub.n,[alpha]]).

Now we show that [[L.sub.n,[alpha]], [R.sub.n,[alpha]]], [alpha] [member of] (0,1], determines the solution of (1) with initial value [x.sub.0], where ([L.sub.n,[alpha]], [R.sub.n,[alpha]]) is the solution of system (12) with initial conditions ([L.sub.0,[alpha]], [R.sub.0,[alpha]]), satisfying

[[[x.sub.n]].sup.[alpha]] = [[L.sub.n,[alpha]], [R.sub.n,[alpha]]], n=0,1,2, ..., [alpha] [member of] (0,1]. (13)

Since A, [??], [x.sub.0] are positive fuzzy numbers, for [[alpha].sub.1], [[alpha].sub.2] [member of] (0,1], [[alpha].sub.1] [less than or equal to] [[alpha].sub.2], and, from Definition 2, then we have

[mathematical expression not reproducible] (14)

We claim that, for n = 0,1, ...,

[mathematical expression not reproducible]. (15)

Inductively, it is clear that (15) is true for n = 0. Suppose that (15) holds true for n [less than or equal to] k, k [member of] {1,2, ...}. Then, from (12), (14), and (15) for n [less than or equal to] k, it follows that

[mathematical expression not reproducible]. (16)

Therefore, (15) is satisfied.

Moreover, for [for all][alpha] [member of] (0,1], it follows from (12) that

[L.sub.1,[alpha]] = [A.sub.l,[alpha]][L.sub.0,[alpha]] ([[??].sub.l,[alpha]] - [R.sub.0,[alpha]]), [R.sub.1,[alpha]] = [A.sub.r,[alpha]][R.sub.0,[alpha]] ([[??].sub.r,[alpha]] - [L.sub.0,[alpha]]). (17)

Since A, [x.sub.0] are positive fuzzy numbers, by Definition 2, then we have that [A.sub.l,[alpha]], [A.sub.r,[alpha]], [L.sub.0,[alpha]], [R.sub.0,[alpha]] are left continuous. From (17) we have that [L.sub.1[alpha]], [R.sub.1[alpha]] are left continuous. By induction we can get that [L.sub.n,[alpha]], [R.aub.n,[alpha]] are left continuous.

Next we prove that the support of [x.sub.n], supp [x.sub.n] = [[bar.[[union].sub.[alpha] [member of] (0,1]][[L.sub.n,[alpha]], [R.sub.n,[alpha]]]] is compact. It is sufficient to prove that [[bar.[[union].sub.[alpha] [member of] (0,1]][[L.sub.n,[alpha]], [R.sub.n,[alpha]]]] is bounded.

Let n = 1; since A, [??] and [x.sub.0] are positive fuzzy numbers, there exist constants [M.sub.A] > 0, [N.sub.A] >0, P > 0, Q > 0, [M.sub.0] > 0, [N.sub.0] >0 such that, for [alpha] [member of] (0,1],

[mathematical expression not reproducible]. (18)

Hence, from (17) and (18), for [alpha] [member of] (0,1],we get

[[L.sub.1,[alpha]], [R.sub.1,[alpha]] [subset] [[M.sub.A][M.sub.0] (P - [N.sub.0]), [N.sub.A][N.sub.0] (Q - [M.sub.0])]. (19)

It is clear that

[mathematical expression not reproducible]. 20)

Therefore, (20) implies [[bar.[[union].sub.[alpha][member of](0,1)][[L.sub.1,[alpha]].[R.sub.1,[alpha]]]] is compact, and [[bar.[[union].sub.[alpha][member of](0,1)][[L.sub.1,[alpha]],[R.sub.1,[alpha]]] [subset] (0, [infinity]). Deducing inductively it can follow easily that

[mathematical expression not reproducible]. (21)

Therefore, from (15), (21), and [L.sub.n,[alpha]], [R.sub.n,[alpha]] being left continuous, it can be concluded that [[L.sub.n,[alpha]],[R.sub.n,[alpha]]] determines a sequence {[x.sub.n]} of positive fuzzy numbers satisfied with (13).

We prove that [x.sub.n] is a solution of (11) with initial value [x.sub.0], since, for [for all][alpha] [member of] (0,1],

[mathematical expression not reproducible]. (22)

Namely, [x.sub.n] is a solution of (11) with initial value [x.sub.0].

Suppose that there exists another solution [[bar.x].sub.n] of (11) with initial value [x.sub.0]. Then from arguing as above we can easily get that, for n = 0,1,2, ...,

[[[[bar.x].sub.n]].sup.[alpha]] = [[L.sub.n,[alpha]],[R.sub.n,[alpha]]], [alpha] [member of] (0,1], (23)

Then from (13) and (23), we have [[x.sub.n]].sup.[alpha] = [[[[bar.x].sub.n]].sup.[alpha]], [alpha] [member of] (0,1], n = 0,1,2, ..., and hence [x.sub.n] = [x.sub.n], n = 0,1,2,.... This completes the proof of Theorem 12.

In order to study the dynamical behavior of the solution [x.sub.n] to (1), we first consider the following system of difference equations

[y.sub.n+1] = p[y.sub.n](a - [z.sub.n]), [z.sub.n+1] = q[z.sub.n] (b - [y.sub.n]), n = 0, 1,.... (24)

It is clear that the equilibrium points of (24) include the following four cases:

(i) (0,0), (ii) (0, z), (iii) (y, 0), (iv) ([qb - 1]/q, [pa - 1]/p). (25)

Lemma 13. Consider the system of difference equations (24), where p, q, a, and b are positive real constants, and the initial values [y.sub.0], [z.sub.0] are positive real numbers; then the following statements are true.

(i) The equilibrium (0,0) is local asymptotically stable if pa < 1, qb < 1.

(ii) System (24) has infinite numbers equilibrium (0, z) which is unstable if qb = 1.

(iii) System (24) has infinite numbers equilibrium (y, 0) which is unstable if pa = 1.

(iv) Suppose that pa > 1, qb > 1; then system (24) has a unique positive equilibrium

([bar.y], [bar.z]) = ([qb - 1]/q, [pa - 1]/p), (26)

which is unstable.

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

[y.sub.n+1] = [pay].sub.n], [z.sub.n+1] = [qbz.sub.n], (27) N = 0, 1, ...,

from which we can easily obtain that the eigenvalue [lambda] = pa, [lambda] = qb. Since pa < 1 and qb < 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, z) is

[[PHI].sub.n+1] = D[[PHI].sub.n], (28)

where

[mathematical expression not reproducible] (29)

The character equation of the linearized system of (28) about equilibrium (0, z) is

[[lambda] - p(a - z)][[lambda] - qb] = 0. (30)

It is clear that the characteristic root of (30) has a root [lambda] = qb = 1. Hence the equilibrium (0, z) 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 ([bar.y], [bar.z]) is

[[lambda].sup.2] - 2[lambda] + 1 - (pa - 1) (qb - 1) = 0 (31)

The character root of (31) is [lambda] = 1 [+ or -] [square root of ((pa - 1)(qb - 1))]. It is clear that there exists a character root larger than 1. Hence the equilibrium ([bar.y], [bar.z]) is unstable.

Theorem 14. Consider the fuzzy difference equation (1). Suppose that there exists positive constants [N.sub.A], Q, for all [alpha] [member of] (0,1] such that

[A.sub.r,[alpha] < [N.sub.A], [[??].sub.r,[alpha]] < Q, [N.sub.A]Q < 1. (32)

Then

[mathematical expression not reproducible]. (33)

Proof. Let [x.sub.n] be a positive solution of (1) with initial conditions [x.sub.0]; applying (i) of Lemma 13, we get that ([L.sub.n,[alpha]], [R.sub.n,[alpha]]), n = 0,1, ..., satisfies the following family of systems of parametric ordinary difference equations

[mathematical expression not reproducible], (34)

From (34), we have that for all [alpha] [member of] (0,1]

[mathematical expression not reproducible] (35)

Since 0 < [N.sub.A]Q < 1, it follows from (35) that for all [alpha] [member of] (0,1]

[mathematical expression not reproducible] (36)

Therefore, [lim.sub.n[right arrow][infinity]]D([x.sub.n], 0) = 0.

Theorem 15. Suppose that A and 1 are positive real numbers (trivial fuzzy numbers) and there exists a constant M > 1 such that

A [greater than or equal to] M. (37)

Then (1) has a unique positive equilibrium x such that

[[x].sup.[alpha]] = [[L.sub.[alpha]],[R.sub.[alpha]]], [L.sub.[alpha]] = [R.sub.[alpha]] = [A - 1]/A, [alpha] [member of] (0,1]. (38)

Proof. Suppose that there exists a fuzzy number x such that

x = Ax (1 - x), [[x].sup.[alpha]] = [[L.sub.[alpha]],[R.sub.[alpha]]], [alpha] [member of] (0,1], (39)

where [L.sub.[alpha]], [R.sub.[alpha]] [greater than or equal to] 0. Then from (39) we can obtain that

[L.sub.[alpha]] = A[L.sub.[alpha]] (1 - [R.sub.[alpha]]), [R.sub.[alpha]] = A[R.sub.[alpha]] (1 - [L.sub.[alpha]]). (40)

Hence we can easily obtain that ([L.sub.[alpha]], [R.sub.[alpha]]) is a unique positive solution of (40).

Conversely, we prove that [[L.sub.[alpha]],[R.sub.[alpha]]], [alpha] [member of] (0,1], where ([L.sub.[alpha]], [R.sub.[alpha]]) is the unique positive solution of (40) satisfying (38), determining a fuzzy number x which satisfies (39). From (38) we get, for [alpha] [member of] (0,1],

[L.sub.[alpha]] = A[[A - 1]/A] (1 - [A - 1]/A) [R.sub.[alpha]] = A [[A - 1]/A] (1 - [A - 1]/A) (41)

Then from (41) for all [[alpha].sub.1], [[alpha].sub.2], 0 [less than or equal to] [[alpha].sub.1] [less than or equal to] [[alpha].sub.2] [less than or equal to] 1 we have

[mathematical expression not reproducible]. (42)

Moreover, since A is a fuzzy number, we have that [A.sub.l,[alpha]], [A.sub.r,[alpha]] are left continuous. Then from (37) and (38) we have that [L.sub.[alpha]], [R.sub.[alpha]] are left continuous.

Furthermore since (37) holds and A is a positive fuzzy number, there exists a constant N such that [[A.sub.l,[alpha]],[A.sub.r,[alpha]] [subset] [M, N], [alpha] [member of] (0,1]. Then from (37) and (38) we obtain that

[[L.sub.[alpha]], [R.sub.[alpha]]] [subset] [[M - 1]/N, [N - 1]/M], (43)

from which we have that

[mathematical expression not reproducible], (44)

Therefore, from Theorem 2.1 of [13], (42), (44), and [L.sub.[alpha]],[R.sub.[alpha]] being left continuous we have that x is a positive fuzzy number which satisfies (39). This completes the proof of Theorem 15.

Theorem 16. Consider fuzzy difference equation (1); if two positive fuzzy numbers A, 1 are all real number 1 (trivial fuzzy numbers) and the initial value [x.sub.0] is a positive fuzzy number, suppose that there exists a positive constant P, Q, P < Q < 1, such that

[mathematical expression not reproducible]. (45)

Then the solution of system (1) converges to 0.

Proof. By Theorem 12, we have that a sequence of a positive fuzzy number {[x.sub.n]} with [[[x.sub.n]].sup.[alpha]] = [[L.sub.n,[alpha]], [R.sub.n,[alpha]]] is the solution of (1), satisfying

[L.sub.n+1,[alpha]] = [L.sub.n,[alpha]] (1 - [R.sub.n,[alpha]]), [R.sub.n+1,[alpha]] = [R.sub.n,[alpha]] (1 - [L.sub.0,[alpha]]), n = 0,1,.... (46)

From (45) and (46), it follows that

[L.sub.1,[alpha] = [L.sub.0,[alpha]] (1 - [R.sub.0,[alpha]]) < (1 - P) [L.sub.0,[alpha]], [R.sub.1,[alpha]] = [R.sub.0,[alpha]] (1 - [L.sub.0,[alpha]]) < (1 - P)[R.sub.0,[alpha]], [alpha] [member of] (0,1]. (47)

We can prove inductively that

[mathematical expression not reproducible], (48)

From (46), it is easy to get

[mathematical expression not reproducible]. (49)

Namely, [lim.sub.n [right arrow] [infinity]] D([x.sub.n], 0) = 0.

Remark 17. From Theorem 16, we can conclude that the dynamical behavior of (1) is correlated with the initial value [x.sub.0]. When the initial value [x.sub.0] is a positive fuzzy number which is smaller than one, no matter how long the support of [x.sub.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.

4. Some Illustrative Examples

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

Example 18. Consider discrete fuzzy logistic system with initial value [x.sub.0]:

[x.sub.n+1] =A[x.sub.n] ([??] - [x.sub.n]), n = 0,1, ..., (50)

where A, [??] and the initial value [x.sub.0] are positive fuzzy numbers such that

A = (0.4, 0.5, 0.6), [??] = (0.5,1, 1.5), [x.sub.0] = (0.3,0.4,0.5) (51)

From that, we get, for [alpha] [member of] (0,1],

[[A].sup.[alpha]] = [0.4 +0.1[alpha], 0.6-0.1[alpha]], [[[??].sup.[alpha]] = [0.5 + 0.5[alpha], 1.5 - 0.5[alpha]], [[[x.sub.0]].sup.[alpha]] = [0.3 + [1/10][alpha], 5 - [1/10][alpha]]. (52)

From (50) and (52), it results in a coupled system of difference equation with parameter [alpha] [member of] (0,1],

[L.sub.n+1,[alpha]] = (0.4 + 0.1[alpha]) [L.sub.n,[alpha]] [0.5 + 0.5[alpha] - [R.sub.n,[alpha]]], [R.sub.n+1,[alpha]] = (0.6 - 0.1[alpha]) [R.sub.n,[alpha]] [1.5 - 0.5[alpha] - [L.sub.n,[alpha]]]. (53)

It is clear that (32) of Theorem 14 is satisfied. Therefore, the solution of (50) converges to 0. (see Figures 1-3)

Example 19. Consider discrete fuzzy logistic system (50) with initial value [x.sub.0] where A = [??] = 1 and initial value [x.sub.0] = (0.7,0.8,0.9).

It results in a coupled system of difference equations with parameter [alpha] [member of] (0,1].

[L.sub.n+1,[alpha]] = [L.sub.n,[alpha]] (1 - [R.sub.n,[alpha]]), [R.sub.n+1,[alpha]] = [R.sub.n,[alpha]] (1 - [L.sub.n,[alpha]]), n = 0,1,2,.... (54)

Clearly, all conditions of Theorem 16 are satisfied. Therefore, the solution of (50) converges to 0 (see Figures 4-6).

5. 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 [A.sub.r,[alpha]] < [N.sub.A], [[??].sub.r,[alpha]] < Q, [N.sub.A]Q < 1, then the positive solution of (1) eventually converges to 0 no matter how much the number of population initial values are.

(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 A = 1, the system has unique positive equilibrium x.

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

https://doi.org/10.1155/2018/8742397

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.

Acknowledgments

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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Qianhong Zhang (iD) and Fubiao Lin

School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, Guizhou 550025, China

Correspondence should be addressed to Qianhong Zhang; zqianhong68@163.com

Received 15 August 2018; Accepted 29 November 2018; Published 11 December 2018

Academic Editor: Douglas R. Anderson

Caption: Figure 1: The dynamics of system (50) with initial value [x.sub.0] = (0.3, 0.4, 0.5).

Caption: Figure 2: The solution of system (53) at [alpha] = 0 and [alpha] = 0.25.

Caption: Figure 3: The solution of system (53) at [alpha] = 0.75 and [alpha] = 1.

Caption: Figure 4: The dynamics of system (54) with initial value [x.sub.0] = (0.7, 0.8, 0.9).

Caption: Figure 5: The solution of system (54) at [alpha] = 0 and [alpha] = 0.25.

Caption: Figure 6: The solution of system (54) at [alpha] = 0.75 and [alpha] = 1.
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Title Annotation:Research Article
Author:Zhang, Qianhong; Lin, Fubiao
Publication:Discrete Dynamics in Nature and Society
Date:Jan 1, 2018
Words:4816
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