DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi 10.1155/2018/8742397 8742397 Research Article On Dynamical Behavior of Discrete Time Fuzzy Logistic Equation http://orcid.org/0000-0001-9553-5443 Zhang Qianhong 1 Lin Fubiao 1 Anderson Douglas R. School of Mathematics and Statistics Guizhou University of Finance and Economics Guiyang Guizhou 550025 China gufe.edu.cn 2018 11122018 2018 15 08 2018 29 11 2018 11122018 2018 Copyright © 2018 Qianhong Zhang and Fubiao Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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. xn+1=Axn(1~-xn),n=0,1,2,, where {xn} is a sequence of positive fuzzy numbers. A,1~ and the initial value x0 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.

National Natural Science Foundation of China 11761018 Guizhou University of Finance and Economics 2018XZD02
1. Introduction

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

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:(1)xn+1=Axn1~-xn,n=0,1,,where parameter A,1~ and the initial condition x0 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 is drawn in Section 5.

2. Mathematical Preliminaries

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

Definition 1 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

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

(i) A is normal; i.e., there exists an xR such that A(x)=1.

(ii) A is fuzzy convex; i.e., for all t[0,1] and x1,x2R such that(2)Atx1+1-tx2minAx1,Ax2. (iii) A is upper semicontinuous.

(iv) The support of A, suppA=α(0,1][A]α¯={x:A(x)>0}¯ is compact, where A¯ denotes the closure of A.

Let E1 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) [<xref ref-type="bibr" rid="B20">19</xref>]).

A fuzzy number u in a parametric form is a pair (u_,u¯) of functions u_(r),u¯(r),0r1, which satisfies the following requirements:

(1) u_(r) is a bounded monotonic increasing left continuous function.

(2) u¯(r) is a bounded monotonic decreasing left continuous function.

(3) u_(r)u¯(r),0r1.

A crisp (real) number x is simply represented by (u_(r),u2(r))=(x,x),0r1. The fuzzy number space {(u_(r),u¯(r))} becomes a convex cone E1 which could be embedded isomorphically and isometrically into a Banach space .

Definition 3 (see [<xref ref-type="bibr" rid="B20">19</xref>]).

Let u=(u_(r),u¯(r)),v=(v_(r),v¯(r))E1,0r1, and arbitrary kR. Then

(i) u=v iff u_(r)=v_(r),u¯(r)=v¯(r)

(ii) u+v=(u_(r)+v_(r),u¯(r)+v¯(r))

(iii) u-v=(u_(r)-v¯(r),u¯(r)-v_(r))

(iv) (3)ku=ku_r,ku¯r,k0;ku¯r,ku_r,k<0,

(v) uv=(min{u_(r)v_(r),u_(r)v¯(r),u¯(r)v_(r),u¯(r)v¯(r)}, max{u_(r)v_(r),u_(r)v¯(r),u¯(r)v_(r),u¯(r)v¯(r)})

Definition 4 (triangular fuzzy number [<xref ref-type="bibr" rid="B20">19</xref>]).

A triangular fuzzy number (TFN) denoted by A is defined as (b,c,d) where the membership function(4)Ax=0,xb;x-bc-b,bxc;1,x=c;d-xd-c,cxd;0,xd.

The α-cuts of A=(b,c,d) are denoted by [A]α={xR:A(x)α}=[b+α(c-b),d-α(d-c)]=[Al,α,Ar,α], α[0,1]; it is clear that [A]α is a closed interval. A fuzzy number is positive if suppA(0,).

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

Proposition 5 (see [<xref ref-type="bibr" rid="B20">19</xref>]).

If {Aα:α[0,1]} is a compact, convex, and not empty subset family of Rn such that

(i) Aα¯A0

(ii) Aα2Aα1 if α1α2

(iii) Aα=k1Aαk if αkα>0

then there is uEn such that [u]α=Aα for all α(0,1] and [u]0=0<α1Aα¯A0.

Definition 6 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

A sequence {xn} of positive fuzzy numbers persists (resp., is bounded) if there exists a positive real number M (resp., N) such that(5)suppxnM,resp.suppxn0,N,n=1,2,.A sequence {xn} of positive fuzzy numbers is bounded and persists if there exist positive real numbers M,N>0 such that(6)suppxnM,N,n=1,2,.

Definition 7.

x n is called a positive solution of (1), if {xn} is a sequence of positive fuzzy numbers which satisfies (1). The equilibrium of (1) is the solution of the following equation: x=Ax(1~-x).

Definition 8 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Let A,B be fuzzy numbers with [A]α=[Al,α,Ar,α],[B]α=Bl,α,Br,α; then the metric of A and B is defined as(7)DA,B=supα0,1maxAl,α-Bl,α,Ar,α-Br,α.

Definition 9.

Let {xn} be a sequence of positive fuzzy numbers and x is a positive fuzzy number. Suppose that(8)xnα=Ln,α,Rn,α,α0,1,n=0,1,,and(9)xα=Lα,Rα,α0,1.The sequence {xn} converges to x with respect to D as n if limnD(xn,x)=0.

Definition 10.

Let x be a positive equilibrium of (1). The positive equilibrium x is stable, if, for every ε>0, there exists a δ=δ(ε) such that for every positive solution xn of (1), which satisfies D(x0,x)δ, we have D(xn,x)ε for all n0. 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.

3. Main Results

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

Lemma 11 (see [<xref ref-type="bibr" rid="B19">20</xref>]).

Let f:R+×R+R+ be continuous; A and B are fuzzy numbers. Then(10)fA,Bα=fAα,Bα,α0,1

Theorem 12.

Consider (1) where A,1~ are positive fuzzy numbers. Then, for any positive fuzzy numbers x0, there exists a unique positive solution xn of (1).

Proof.

The proof is similar to Proposition 2.1 . Suppose that there exists a sequence {xn} of fuzzy numbers satisfying (1) with initial condition x0. Consider α-cuts, α(0,1],n=0,1,2,, applying Lemma 11; we have(11)xn+1α=Ln+1,α,Rn+1,α=Axn1~-xnα=Aαxnα1~α-xnα=Al,α,Ar,α×Ln,α,Rn,α×1~l,α,1~r,α-Ln,α,Rn,α=Al,αLn,α1~l,α-Rn,α,Ar,αRn,α1~r,α-Ln,α.From (11), we can get the following system of ordinary difference equation with parameter α(0,1], for n=0,1,2,,(12)Ln+1,α=Al,αLn,α1~l,α-Rn,α,Rn+1,α=Ar,αRn,α1~r,α-Ln,α.Then, for any initial condition (L0,α,R0,α),α(0,1], there exists a unique solution (Ln,α,Rn,α).

Now we show that [Ln,α,Rn,α],α(0,1], determines the solution of (1) with initial value x0, where (Ln,α,Rn,α) is the solution of system (12) with initial conditions (L0,α,R0,α), satisfying(13)xnα=Ln,α,Rn,α,n=0,1,2,,α0,1.Since A,1~,x0 are positive fuzzy numbers, for α1,α2(0,1],α1α2, and, from Definition 2, then we have(14)0<Al,α1Al,α2Ar,α2Ar,α10<1~l,α11~l,α21~r,α21~r,α10<L0,α1L0,α2R0,α2R0,α1We claim that, for n=0,1,,(15)0<Ln,α1Ln,α2Rn,α2Rn,α1.Inductively, it is clear that (15) is true for n=0. Suppose that (15) holds true for nk,k{1,2,}. Then, from (12), (14), and (15) for nk, it follows that (16)Lk+1,α1=Al,α1Lk,α11~l,α1-Rk,α1Al,α2Lk,α21~l,α2-Rk,α2=Lk+1,α2Ar,α2Rk,α21~r,α2-Lk,α2=Rk+1,α2Ar,α1Rk,α11~r,α1-Lk,α1=Rk+1,α1. Therefore, (15) is satisfied.

Moreover, for α(0,1], it follows from (12) that(17)L1,α=Al,αL0,α1~l,α-R0,α,R1,α=Ar,αR0,α1~r,α-L0,α. Since A,x0 are positive fuzzy numbers, by Definition 2, then we have that Al,α,Ar,α,L0,α,R0,α are left continuous. From (17) we have that L1,α,R1,α are left continuous. By induction we can get that Ln,α,Rn,α are left continuous.

Next we prove that the support of xn, suppxn=α(0,1][Ln,α,Rn,α]¯ is compact. It is sufficient to prove that α(0,1][Ln,α,Rn,α] is bounded.

Let n=1; since A,1~ and x0 are positive fuzzy numbers, there exist constants MA>0,NA>0,P>0,Q>0,M0>0,N0>0 such that, for α(0,1], (18)Al,α,Ar,αMA,NA,1~l,α,1~r,αP,Q,L0,α,L0,αM0,N0.Hence, from (17) and (18), for α(0,1], we get(19)[L1,α,R1,αMAM0P-N0,NAN0Q-M0. It is clear that(20)α0,1[L1,α,R1,αMAM0P-N0,NAN0Q-M0.Therefore, (20) implies α(0,1][L1,α,R1,α]¯ is compact, and α(0,1][L1,α,R1,α]¯(0,). Deducing inductively it can follow easily that(21)α0,1Ln,α,Rn,α¯iscompact,andα0,1Ln,α,Rn,α¯0,.Therefore, from (15), (21), and Ln,α,Rn,α being left continuous, it can be concluded that [Ln,α,Rn,α] determines a sequence {xn} of positive fuzzy numbers satisfied with (13).

We prove that xn is a solution of (11) with initial value x0, since, for α(0,1],(22)xn+1α=Ln+1,α,Rn+1,α=Al,αLn,α1~l,α-Rn,α,Ar,αRn,α1~r,α-Ln,α=Axn1~-xnα. Namely, xn is a solution of (11) with initial value x0.

Suppose that there exists another solution x¯n of (11) with initial value x0. Then from arguing as above we can easily get that, for n=0,1,2,,(23)x¯nα=Ln,α,Rn,α,α0,1.Then from (13) and (23), we have [xn]α=[x¯n]α,α(0,1],n=0,1,2,, and hence xn=x¯n,n=0,1,2,. This completes the proof of Theorem 12.

In order to study the dynamical behavior of the solution xn to (1), we first consider the following system of difference equations(24)yn+1=pyna-zn,zn+1=qznb-yn,n=0,1,. It is clear that the equilibrium points of (24) include the following four cases:(25)i0,0,ii0,z,iiiy,0,ivqb-1q,pa-1p.

Lemma 13.

Consider the system of difference equations (24), where p,q,a, and b are positive real constants, and the initial values y0,z0 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(26)y¯,z¯=qb-1q,pa-1p, 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(27)yn+1=payn,zn+1=qbzn,n=0,1,,from which we can easily obtain that the eigenvalue λ=pa,λ=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(28)Φn+1=DΦn,where(29)Φn=ynzn,D=pa-z0-qzqbThe character equation of the linearized system of (28) about equilibrium (0,z) is(30)λ-pa-zλ-qb=0. It is clear that the characteristic root of (30) has a root λ=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 (y¯,z¯) is(31)λ2-2λ+1-pa-1qb-1=0 The character root of (31) is λ=1±(pa-1)(qb-1). It is clear that there exists a character root larger than 1. Hence the equilibrium (y¯,z¯) is unstable.

Theorem 14.

Consider the fuzzy difference equation (1). Suppose that there exists positive constants NA,Q, for all α(0,1] such that(32)Ar,α<NA,1~r,α<Q,NAQ<1.Then(33)xnα=Ln,α,Rn,α,limnDxn,0=0.

Proof.

Let xn be a positive solution of (1) with initial conditions x0; applying (i) of Lemma 13, we get that (Ln,α,Rn,α),n=0,1,, satisfies the following family of systems of parametric ordinary difference equations(34)Ln+1,α=Al,αLn,α1~l,α-Rn,α,Rn+1,α=Ar,αRn,α1~r,α-Ln,α,α0,1,From (34), we have that for all α(0,1](35)0<Ln,α=Al,αLn-1,α1~l,α-Rn-1,α<NAQnL0,α,0<Rn,α=Ar,αRn-1,α1~r,α-Ln-1,α<NAQnR0,αSince 0<NAQ<1, it follows from (35) that for all α(0,1](36)limnLn,α=0,limnRn,α=0.Therefore, limnD(xn,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(37)AM. Then (1) has a unique positive equilibrium x such that(38)xα=Lα,Rα,Lα=Rα=A-1A,α0,1.

Proof.

Suppose that there exists a fuzzy number x such that(39)x=Ax1-x,xα=Lα,Rα,α0,1, where Lα,Rα0. Then from (39) we can obtain that(40)Lα=ALα1-Rα,Rα=ARα1-Lα. Hence we can easily obtain that (Lα,Rα) is a unique positive solution of (40).

Conversely, we prove that [Lα,Rα],α(0,1], where (Lα,Rα) is the unique positive solution of (40) satisfying (38), determining a fuzzy number x which satisfies (39). From (38) we get, for α(0,1],(41)Lα=AA-1A1-A-1ARα=AA-1A1-A-1AThen from (41) for all α1,α2,0α1α21 we have(42)0<Lα1Lα2Rα2Rα1.

Moreover, since A is a fuzzy number, we have that Al,α,Ar,α are left continuous. Then from (37) and (38) we have that Lα,Rα are left continuous.

Furthermore since (37) holds and A is a positive fuzzy number, there exists a constant N such that [Al,α,Ar,α][M,N],α(0,1]. Then from (37) and (38) we obtain that(43)Lα,RαM-1N,N-1M, from which we have that(44)α0,1Lα,Rα¯iscompact,α0,1Lα,Rα¯0,,n=1,2,,Therefore, from Theorem 2.1 of , (42), (44), and Lα,Rα 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 x0 is a positive fuzzy number, suppose that there exists a positive constant P,Q,P<Q<1, such that(45)α0,1L0,α,R0,αP,Q. Then the solution of system (1) converges to 0.

Proof.

By Theorem 12, we have that a sequence of a positive fuzzy number {xn} with [xn]α=[Ln,α,Rn,α] is the solution of (1), satisfying(46)Ln+1,α=Ln,α1-Rn,α,Rn+1,α=Rn,α1-L0,α,n=0,1,. From (45) and (46), it follows that(47)L1,α=L0,α1-R0,α<1-PL0,α,R1,α=R0,α1-L0,α<1-PR0,α,α0,1. We can prove inductively that(48)Ln,α<1-PnL0,α,Rn,α<1-PnR0,α,n1,α0,1. From (46), it is easy to get(49)limnLn,α=0,limnRn,α=0. Namely, limnD(xn,0)=0.

Remark 17.

From Theorem 16, we can conclude that the dynamical behavior of (1) is correlated with the initial value x0. When the initial value x0 is a positive fuzzy number which is smaller than one, no matter how long the support of x0 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 x0:(50)xn+1=Axn1~-xn,n=0,1,, where A,1~ and the initial value x0 are positive fuzzy numbers such that(51)A=0.4,0.5,0.6,1~=0.5,1,1.5,x0=0.3,0.4,0.5 From that, we get, for α(0,1],(52)Aα=0.4+0.1α,0.6-0.1α,1~α=0.5+0.5α,1.5-0.5α,x0α=0.3+110α,5-110α. From (50) and (52), it results in a coupled system of difference equation with parameter α(0,1],(53)Ln+1,α=0.4+0.1αLn,α0.5+0.5α-Rn,α,Rn+1,α=0.6-0.1αRn,α1.5-0.5α-Ln,α. It is clear that (32) of Theorem 14 is satisfied. Therefore, the solution of (50) converges to 0. (see Figures 13)

The dynamics of system (50) with initial value x0=(0.3,0.4,0.5).

The solution of system (53) at α=0 and α=0.25.

The solution of system (53) at α=0.75 and α=1.

Example 19.

Consider discrete fuzzy logistic system (50) with initial value x0 where A=1~=1 and initial value x0=(0.7,0.8,0.9).

It results in a coupled system of difference equations with parameter α(0,1].(54)Ln+1,α=Ln,α1-Rn,α,Rn+1,α=Rn,α1-Ln,α,n=0,1,2,.Clearly, all conditions of Theorem 16 are satisfied. Therefore, the solution of (50) converges to 0 (see Figures 46).

The dynamics of system (54) with initial value x0=(0.7,0.8,0.9).

The solution of system (54) at α=0 and α=0.25.

The solution of system (54) at α=0.75 and α=1.

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 Ar,α<NA,1~r,α<Q,NAQ<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.

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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