# Asymptotic Behavior of Discrete Time Fuzzy Single Species Model

Authors: Qianhong Zhang; Fubiao Lin; Xiaoying Zhong
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This work is concerned with the qualitative behavior of discrete time single species model with fuzzy environmentxn+1=xnexp⁡A-Bxn,n=0,1,2,…, where xn denotes the number of individuals of generation n, A is the intrinsic growth rate, and B is interpreted as the carrying capacity of the surrounding environment. xn is a sequence of positive fuzzy number. A,B and the initial value x0 are positive fuzzy numbers. Applying difference of Hukuhara (H-difference), the existence, uniqueness of the positive solution, and global asymptotic behavior of all positive solution with the model are obtained. Moreover a numerical example is presented to show the effectiveness of theoretic results obtained.


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## Body

## 1. Introduction

Difference equations or discrete time dynamical systems have many applications in economics, biology, computer science, control engineering, etc. (see, for example, [1–10] and the references therein). In the recent years, many researchers pay a close attention to the study on qualitative behavior of difference equation in mathematical biology and population dynamics [11–13]. In theoretical ecology, the difference equation models are often described the interactions of species with nonoverlapping generations. For example, in 1954, Ricker [14] forecasted fish stock recruitment using the following discrete time single species model(1)xn+1=xnexp⁡r-xnK,n=0,1,2,….where xn denotes the number of individuals of generation n, r is the intrinsic growth rate, and K is interpreted as the carrying capacity of the surround environment. In fact, from a biological point of view, researchers focus on whether or not all species in a multispecies community can be permanent or bounded.As is all known, the parameters of the model are usually based on statistical method or on the choice of some method adapted to the identification. Therefore, these models are subjected to inaccuracies (fuzzy uncertainty) that can be caused by the nature of the state variables, by coefficients of the model and by initial conditions. In our real life, scientists are concerned with uncertainty and accept the fact that uncertainty is very important influencing factor of dynamical behavior of dynamical system. Fuzzy set introduced by Zadeh [15] and its development have been growing rapidly to various situation of theory and application including the theory of differential and difference equations with uncertainty. It is well known that a fuzzy difference equation is a difference equation where parameters or the initial values of systems are fuzzy numbers, and its solutions are sequences of fuzzy numbers.In fact, the dynamical behavior of fuzzy difference equation is different from the behavior of corresponding parametric ordinary difference equation. In recent decades, researchers have an increasing interest in studying fuzzy difference equation. Some results on fuzzy difference equations have been reported (see, for example, [16–31]). Barros, Bassanezi, and Tonelli [32] have investigated the dynamical behavior of population model with fuzzy uncertainty. However, to the best of our knowledge, few authors study discrete time single species model under fuzzy environment. This paper is the first to study the dynamical behavior of discrete time single species model using fuzzy sets theory.The main aim of this paper is to study the dynamical behaviors of the following discrete time single species model(2)xn+1=xnexp⁡A-Bxn,n=0,1,…,where xn denotes the number of individuals of generation n and xn is a sequence of positive fuzzy numbers. A, B are the intrinsic growth rate and the carrying capacity of the surround environment, respectively. Parameters A,B and the initial condition x0 are positive fuzzy numbers. This paper is, to some extent, a generalization of classic discrete time single species model, using the subjectivity which comes from “fuzziness” of the biological phenomenon.The rest of this paper is organized as follows. In the next section, we introduce some definitions and preliminaries. In Section3, the dynamical behaviors on the existence, uniqueness, and global asymptotic behaviors of the positive fuzzy solution to system (2) are studied. A numerical example is given to show effectiveness of results obtained in Section 4. Finally, a general conclusion is drawn in Section 5.


## 2. Mathematical Preliminaries

For convenience, we give some definitions used in the sequel.Definition 1 (see [33]).
A is said to be a fuzzy number if A:R→[0,1] satisfies the below (i)-(iv)
(i)A is normal; i.e., there exists an x∈R such that A(x)=1.
(ii)A is fuzzy convex, i.e., for all t∈[0,1] and x1,x2∈R such that(3)Atx1+1-tx2≥min⁡Ax1,Ax2;
(iii)A is upper semicontinuous.
(iv) The support ofA, supp⁡A=⋃α∈(0,1]Aα¯=x:Ax>0¯ is compact, where A¯ denotes the closure of A.LetE1 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) [34]).
A fuzzy numberu in a parametric form is a pair (u_,u¯) of function u_(r),u¯(r),0≤r≤1, 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),0≤r≤1. A crisp (real) number x is simply represented by (u_(r),u2(r))=(x,x),0≤r≤1. The fuzzy number space u_r,u¯r becomes a convex cone E1 which could be embedded isomorphically and isometrically into a Banach space. [26]Definition 3.
Letu=(u_(r),u¯(r)),v=(v_(r),v¯(r))∈E1,0≤r≤1, and arbitrary k∈R. 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)(4)ku=ku_r,ku¯r,k≥0;ku¯r,ku_r,k<0,
(iv)uv = min⁡u_rv_r,u_rv¯r,u¯rv_r,u¯rv¯r, max⁡u_rv_r,u_rv¯r,u¯rv_r,u¯rv¯r.Definition 4.
Letu=(u_(r),u¯(r)),v=(v_(r),v¯(r))∈E1,0≤r≤1, and if there exists w=(w_(r),w¯(r))∈E1 such that u=v+w, then w is called the H-difference of u and v and it is denoted by w=u-v=(u_(r)-v_(r),u¯(r)-v¯(r))In this paper the “−” sign stands always for H-difference and let us remark thatu-v≠u+(-1)v.Definition 5 (triangular fuzzy number [34]).
A triangular fuzzy number (TFN) denoted byA is defined as (a,b,c) where the membership function(5)Ax=0,x≤a;x-ab-a,a≤x≤b;1,x=b;c-xc-b,b≤x≤c;0,x≥c.Theα-cuts of A=(a,b,c) are denoted by Aα=x∈R:Ax≥α = [a+α(b-a),c-α(c-b)] = [Al,α,Ar,α], α∈[0,1], and it is clear that the Aα are closed interval. A fuzzy number is positive if supp⁡A⊂(0,∞).The following proposition is fundamental since it characterizes a fuzzy set through theα-levels.Proposition 6 (see [33]).
IfAα:α∈0,1 is a compact, convex and not empty subset family of Rn such that
(i)⋃Aα¯⊂A0,
(ii)Aα2⊂Aα1 if α1≤α2,
(iii)Aα=⋂k≥1Aαk if αk↑α>0,
then there isu∈En such that uα=Aα for all α∈0,1 and u0=⋃0<α≤1Aα¯⊂A0.Definition 7 (see [27]).
A sequence of positive fuzzy numbersxn persists (resp., is bounded) if there exists a positive real number M (resp., N) such that(6)supp⁡xn⊂M,∞resp.supp⁡xn⊂0,N,n=1,2,….A sequence of positive fuzzy numbers xn is bounded and persists if there exist positive real numbers M,N>0 such that(7)supp⁡xn⊂M,N,n=1,2,….Definition 8.
x n is a positive solution of (2), if xn is a sequence of positive fuzzy numbers which satisfies (2). The equilibrium of (2) is the solution of the equation x=xexp⁡(A-Bx).Definition 9 (see [20]).
LetA,B be fuzzy numbers with Aα=[Al,α,Ar,α], Bα=[Bl,α,Br,α]. Then the metric of A and B is defined as(8)DA,B=supα∈0,1⁡max⁡Al,α-Bl,α,Ar,α-Br,α,and (E1,D) is a complete metric space.Definition 10 (see [23]).
Letxn be a sequence of positive fuzzy numbers and x is a positive fuzzy number. Suppose that(9)xnα=Ln,α,Rn,α,α∈0,1,n=0,1,…,and(10)xα=Lα,Rα,α∈0,1.The sequence {xn} converges to x with respect to D as n→∞ if limn→∞D(xn,x)=0.Definition 11.
Letx be a positive equilibrium of (2). The positive equilibrium x is stable, if for every ε>0, there exists a δ=δ(ε)>0 such that for every positive solution xn of (2), which satisfies D(x0,x)≤δ, we have D(xn,x)≤ε for all n≥0. The positive equilibrium x is asymptotically stable, if it is stable and every positive solution of (2) converges to the positive equilibrium of (2) with respect to D as n→∞.


## 3. Main Results

### 3.1. Existence and Uniqueness of Positive Solution

First we study the existence and uniqueness of the positive solutions of (2). We need the following lemma.Lemma 12 (see [33]).
Letf:R+×R+→R+ be continuous; A,B are fuzzy numbers. Then(11)fA,Bα=fAα,Bα,α∈0,1Theorem 13.
Consider (2) where A,B are positive fuzzy numbers, if there exists fuzzy number C such that A=Bxn+C for n≥0. Then for any positive fuzzy numbers x0, there exists a unique positive solution xn of (2).Proof.
The proof is similar toTheorem 2.3 [24]. Suppose that there exists a sequence of fuzzy numbers xn satisfying (2) with initial conditions x0. Consider α-cuts, α∈(0,1],n=0,1,2,…, and applying Lemma 12, and A=Bxn+C for n≥0, we have(12)xn+1α=Ln+1,α,Rn+1,α=xnexp⁡A-Bxnα=xnαexp⁡Aα-Bαxnα=Ln,α,Rn,α×exp⁡Al,α,Ar,α-Bl,α,Br,α×Ln,α,Rn,α=Ln,α,Rn,α×exp⁡Al,α-Bl,αLn,α,Ar,α-Br,αRn,α=Ln,αexp⁡Al,α-Bl,αLn,α,Rn,αexp⁡Ar,α-Br,αRn,αIt follows from (12) and H-difference of fuzzy numbers that, for n=0,1,2,…,α∈(0,1],(13)Ln+1,α=Ln,αexp⁡Al,α-Bl,αLn,α,Rn+1,α=Rn,αexp⁡Ar,α-Br,αRn,α.Then it is obvious that, for any initial condition (L0,α,R0,α),α∈(0,1], there exists a unique solution (Ln,α,Rn,α).
Now we prove that[Ln,α,Rn,α],α∈(0,1], where (Ln,α,Rn,α) is the solution of system (13) with initial conditions (L0,α,R0,α), determines the solution of (2) with initial value x0, such that,(14)xnα=Ln,α,Rn,α,n=0,1,2,…,α∈0,1.From Definition 2 and since A,B,x0 are positive fuzzy numbers, for any α1,α2∈(0,1],α1≤α2, we have(15)0<Al,α1≤Al,α2≤Ar,α2≤Ar,α10<Bl,α1≤Bl,α2≤Br,α2≤Br,α10<L0,α1≤L0,α2≤R0,α2≤R0,α1We claim that, for n=0,1,…,(16)0<Ln,α1≤Ln,α2≤Rn,α2≤Rn,α1.We prove it by induction. It is clear that (16) is true for n=0. Suppose that (16) holds true for n≤k,k∈1,2,…,. Then, from (13), (15), and (16) for n≤k, it follows that (17)Lk+1,α1=Lk,α1exp⁡Al,α1-Bl,α1Lk,α1≤Lk,α2exp⁡Al,α2-Bl,α2Lk,α2=Lk+1,α2≤Rk,α2exp⁡Ar,α2-Br,α2Rk,α2=Rk+1,α2≤Rk,α1exp⁡Ar,α1-Br,α1Rk,α1=Rk+1,α1.Therefore (16) is satisfied. Moreover, for ∀α∈(0,1], it follows from (13) that(18)L1,α=L0,αexpAl,α-Bl,αL0,α,R1,α=R0,αexpAr,α-Br,αR0,α.Since A,B,x0 are positive fuzzy numbers, from Definition 2, then Al,α, Ar,α, Bl,α, Br,α, L0,α, R0,α are left continuous. From (18) we have 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 ofxn, supp⁡xn=⋃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, and since A,B and x0 are positive fuzzy numbers, there exist constants MA>0, NA>0, MB>0, NB>0, M0>0, N0>0 such that, for α∈(0,1],(19)Al,α,Ar,α⊂⋃α∈0,1Al,α,Ar,α¯⊂MA,NA,Bl,α,Br,α⊂⋃α∈0,1Bl,α,Br,α¯⊂MB,NB,L0,α,L0,α⊂⋃α∈0,1L0,α,R0,α¯⊂M0,N0,Hence, from (18) and (19), for α∈(0,1], we get(20)L1,α,R1,α⊂M0exp⁡MA-NBN0,N0exp⁡NA-MBM0,from which it is clear that(21)⋃α∈0,1L1,α,R1,α⊂M0exp⁡MA-NBN0,N0exp⁡NA-MBM0,Therefore (21) 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(22)⋃α∈0,1Ln,α,Rn,α¯is  compact,  and⋃α∈0,1Ln,α,Rn,α¯⊂0,∞.Therefore, from (16), (22) and Ln,α, Rn,α are left continuous. It can conclude that [Ln,α,Rn,α] determines a sequence of positive fuzzy numbers xn such that (14) holds.
We prove thatxn is a solution of (12) with initial value x0. Since ∀α∈(0,1], (23)xn+1α=Ln+1,α,Rn+1,α=Ln,αexp⁡Al,α-Bl,αLn,α,Rn,αexp⁡Ar,α-Br,αRn,α=xnexp⁡A-Bxnα.Namely, xn is a solution of (12) with initial value x0.
Suppose that there exists another solutionx¯n of (12) with initial value x0. Then from arguing as above we can easily get that, for n=0,1,2,…,(24)x¯nα=Ln,α,Rn,α,α∈0,1.Then from (14) and (24), we have xnα=x¯nα,α∈(0,1],n=0,1,2,…, and hence xn=x¯n,n=0,1,2,…, and this completes the proof of Theorem 13.


### 3.2. Dynamical Behaviour of Positive Solution

In order to study the dynamical behavior of the solutionxn to (2), we first consider the following system of difference equations(25)yn+1=ynexp⁡a-pyn,zn+1=znexp⁡b-qzn,n=0,1,…,It is clear that the equilibrium points (y¯,z¯) of (25) include the following four cases:(26)i0,0,ii0,bq,iiiap,0,ivap,bq.Lemma 14.
Consider difference equation(27)yn+1=ynexp⁡a-pyn,n=0,1,2,….If a>0,p>0, then 0<yn≤1/pexp⁡(a-1).Proof.
Consider functionfy=yexp⁡(a-py), and we have f′(y)=1-pyexp⁡(a-py). It follows that(28)f′y>0ify<1p,f′y<0ify>1p.It is clear that the maximum of f is equal to 1/pexp⁡(a-1). Therefore we have 0<yn≤1/pexp⁡(a-1).Lemma 15 (see [2]).
Suppose the vector difference equation(29)Xn+1=HXn,n=0,1,2,…,where Xn∈Rk+1,H∈C1[Rk+1,Rk+1]. Let X¯ be an equilibrium point (29), and DH(X¯) denote the Jacobian Matrix of function H at X¯. Then
(i)X¯ is called a hyperbolic equilibrium if DH(X¯) has no eigenvalues with absolute value equal to 1.
(ii)X¯ is called a sink or an attracting equilibrium if every eigenvalue of DH(X¯) has absolute value less than 1.
(iii)X¯ is called a source or a repelling equilibrium if every eigenvalue of DH(X¯) has absolute value greater than 1.
(iv)X¯ is called a saddle point if some of the eigenvalues of DH(X¯) are greater and some are less than 1 in absolute value.Theorem 16.
Consider the system of difference equations (25), where a,b,p, and q 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 a source (a repelling equilibrium).
(ii) The equilibrium0,b/q is a saddle point if 0<b<2; the equilibrium 0,b/q is a source (a repelling equilibrium) if b>2.
(iii) The equilibriuma/p,0 is a saddle point if 0<a<2; the equilibrium a/p,0 is a source (a repelling equilibrium) if a>2.Proof.
(i) It is clear that(0,0) is always an equilibrium. We can easily obtain that the linearized system of (25) about the positive equilibrium (0,0) is(30)Xn+1=D1Xn,n=0,1,…,where(31)Xn=ynzn,D1=ea00eb.We can easily obtain that the eigenvalue of Jacobian matrix D1, λ1=ea, λ2=eb. It is clear that λ>1. This implies that the equilibrium (0,0) is a source (a repelling equilibrium).
(ii) We can obtain that the linearized system of (25) about the equilibrium 0,a/p is(32)Xn+1=D2Xn,where(33)Xn=ynzn,D2=ea001-bIt is easy to obtain the following: if 0<b<2, the eigenvalue of D2, λ1>1, λ2<1, then 0,a/p is a saddle point. if b>2, all the eigenvalue of D2, λ>1, then 0,a/p is a source (a repelling equilibrium).
(iii) The proof of (iii) is similar to the proof of (ii). So we omit it.Theorem 17.
Consider the system of difference equations (25), where the initial values y0,z0 are positive real numbers. If 0<a<2,0<b<2, then (25) has a unique positive equilibrium point a/p,b/q which is globally asymptotically stable.Proof.
It follows easily from (25) that a/p,b/q is a unique positive equilibrium point.
We can obtain that the linearized system of (25) about the positive equilibrium a/p,b/q is(34)Xn+1=D3Xn,where(35)Xn=ynzn,D3=1-a001-b
We can easily obtain that the eigenvalue of Jacobian matrixD3, λ1=1-a,λ2=1-b. It is clear that λ<1. Hence the equilibrium a/p,b/q is a sink (an attracting equilibrium).
Noting Lemma14, we have(36)limn→∞⁡sup⁡yn=L1,limn→∞⁡inf⁡yn=l1,limn→∞⁡sup⁡zn=L2,limn→∞⁡inf⁡zn=l2.where Li,li∈(0,∞),i=1,2. Then from (25) we have(37)L1≤L1exp⁡a-pl1,l1≥l1exp⁡a-pL1,L2≤L2exp⁡b-ql2,l2≥l2exp⁡b-qL2.From this, we have(38)1≤exp⁡a-pl1,1≥exp⁡a-pL1,1≤exp⁡b-ql2,1≥exp⁡b-qL2.The relation (38) implies l1≤L1, l2≤L2. We claim that(39)l1=L1,l2=L2.Suppose on the contrary that l1<L1,l2<L2. From (38), we have(40)l1≤ap<L1≤1pexp⁡a-1,l2≤bq<L2≤1qexp⁡b-1,from which we have(41)a<exp⁡a-1,b<exp⁡b-1,It follows from (41) that a≠1 and b≠1, which is contradicting with 0<a<2,0<b<2. So Li=li,i=1,2.. Hence we have limn→∞yn=a/p, limn→∞zn=b/q. Therefore the unique positive equilibrium a/p,b/q is globally asymptotically stable.Theorem 18.
Consider the fuzzy difference equation (2), where A,B are positive fuzzy numbers. Consider that there exists positive constants NA, for all α∈(0,1] such that(42)Ar,α<NA<2.Then the following statements are true.
(i) Every positive solutionxn of (2) is bounded.
(ii) Equation (2) has a unique positive equilibrium x.
(iii) Every positive solutionxn of (2) converges the unique equilibrium x with respect to D as n→∞.
Moreover the unique positive equilibriumx is asymptotically stable.Proof.
(i) Letxn be a positive solution of (2) with initial conditions x0. Suppose that (19) holds; from (25) and using Lemma 14, we get that(43)0<Ln,α≤1Bl,αexp⁡Al,α-1,0<Rn,α≤1Br,αexp⁡Ar,α-1,α∈0,1,
From (19) and (43), we have that for all α∈(0,1](44)Ln,α,Rn,α⊂0,N,n≥1,where N=1/MBexp⁡(NA-1). From (44), we have for n≥1, ⋃α∈(0,1][Ln,α,Rn,α]⊂(0,N], so ⋃α∈(0,1][Ln,α,Rn,α]¯⊂(0,N]. Thus the proof of Part (i) is completed.
(ii) We consider the system(45)Lα=LαexpAl,α-Bl,αLα,Rα=RαexpAr,α-Br,αRα.Then the positive solution (Lα,Rα) of (45) is given by(46)Lα=Al,αBl,α,Rα=Ar,αBr,α,α∈0,1.Let xn be a positive solution of (2) such that xnα=[Ln,α,Rn,α], α∈(0,1], n=0,1,2,⋯. Then applying Theorem 17 to system (13) we have(47)limn→∞⁡Ln,α=Lα,limn→∞⁡Rn,α=RαFrom (44) and (47) we have for 0<α1<α2<1,(48)0<Lα1≤Lα2≤Rα2≤Rα1.Since Al,α, Ar,α, Bl,α, Br,α are left continuous, from (45), it follows that Lα,Rα are also left continuous. From (46) and (19), we have(49)c=MANB≤Lα≤Rα≤NAMB=d.Therefore (49) implies that [Lα,Rα]⊂[c,d], and so ⋃α∈(0,1][Lα,Rα]⊂[c,d]. It is clear that(50)⋃α∈0,1Lα,Rαis  compact  and⋃α∈0,1Lα,Rα⊂0,∞.So from Definition 2, (45), (48), (50) and Lα,Rα,α∈(0,1] determine a fuzzy number x such that(51)x=xexp⁡A-Bx,xα=Lα,Rα,α∈0,1.
Provided that there exists another positive equilibriumx¯ of (2), then there exist functions L¯α,R¯α:(0,1)→(0,∞) such that(52)x¯=x¯exp⁡A-Bx¯,x¯α=L¯α,R¯α,α∈0,1.From this we get(53)L¯α=L¯αexp⁡Al,α-Bl,αL¯α,R¯α=R¯αexp⁡Ar,α-Br,αR¯α.So L¯α=Lα,R¯α=Rα,α∈(0,1]. Hence x=x¯. This completes the proof of Part (ii).
(iii) From (47) we have(54)limn→∞⁡Dxn,x=limn→∞⁡supα∈0,1⁡max⁡Ln,α-Lα,Rn,α-Rα=0.Namely, every positive solution xn of (2) converges the unique equilibrium x with respect to D as n→∞.
Letε be an arbitrary positive real number; we consider the positive real number δ as follows(55)0<δ<ε1-MBγexp⁡NA-MBγ.where 0<γ=min⁡L0,α,infn≥1Ln,α,α∈(0,1].
Letxn be a positive solution of (2) such that D(x0,x)≤δ<ε. From this we have(56)L0,α-Lα≤δ,R0,α-Rα≤δ,α∈0,1.From (18), (19), and (45), we have(57)L1,α-Lα=L0,αexp⁡Al,α-Bl,αL0,α-Lαexp⁡Al,α-Bl,αLα≤1-Bl,αξ_exp⁡Al,α-Bl,αξ_L0,α-Lα=δ1-Bl,αξ_exp⁡Al,α-Bl,αξ_<δ1-MBξ_exp⁡NA-MBξ_(58)R1,α-Rα=R0,αexp⁡Ar,α-Br,αR0,α-Rαexp⁡Ar,α-Br,αRα≤1-Br,αη_exp⁡Ar,α-Br,αη_R0,α-Rα=δ1-Br,αη_exp⁡Ar,α-Br,αη_<δ1-MBη_exp⁡NA-MBη_where ξ_=min⁡L0,α,Lα, η_=min⁡R0,α,Rα, α∈(0,1].
From (55), it is obvious that(59)L1,α-Lα<ε,R1,α-Rα<ε.From this and working inductively we can easily prove that(60)Ln,α-Lα<ε,Rn,α-Rα<ε,α∈0,1,n=0,1,2,⋯.And so D(xn,x)<ε. Therefore the positive equilibrium x is stable. Moreover relation (54) holds. So the equilibrium x is asymptotically stable. The proof of Theorem 18 is completed.Remark 19.
From Theorem18, we can know that the dynamical behavior of (2) is relevant to the intrinsic growth rate A of population. No matter what the initial population quantity x0 is small and no matter what the carrying capacity of the surround environment B is large. As long as the intrinsic growth rate of population A satisfies Ar,α<2,0<Al,α-Bl,αLn,α≤Ar,α-Br,αRn,α, for all α∈(0,1), the solution of fuzzy difference equation (2) is bounded and eventually converges to the unique positive equilibrium.


## 3.1. Existence and Uniqueness of Positive Solution

First we study the existence and uniqueness of the positive solutions of (2). We need the following lemma.Lemma 12 (see [33]).
Letf:R+×R+→R+ be continuous; A,B are fuzzy numbers. Then(11)fA,Bα=fAα,Bα,α∈0,1Theorem 13.
Consider (2) where A,B are positive fuzzy numbers, if there exists fuzzy number C such that A=Bxn+C for n≥0. Then for any positive fuzzy numbers x0, there exists a unique positive solution xn of (2).Proof.
The proof is similar toTheorem 2.3 [24]. Suppose that there exists a sequence of fuzzy numbers xn satisfying (2) with initial conditions x0. Consider α-cuts, α∈(0,1],n=0,1,2,…, and applying Lemma 12, and A=Bxn+C for n≥0, we have(12)xn+1α=Ln+1,α,Rn+1,α=xnexp⁡A-Bxnα=xnαexp⁡Aα-Bαxnα=Ln,α,Rn,α×exp⁡Al,α,Ar,α-Bl,α,Br,α×Ln,α,Rn,α=Ln,α,Rn,α×exp⁡Al,α-Bl,αLn,α,Ar,α-Br,αRn,α=Ln,αexp⁡Al,α-Bl,αLn,α,Rn,αexp⁡Ar,α-Br,αRn,αIt follows from (12) and H-difference of fuzzy numbers that, for n=0,1,2,…,α∈(0,1],(13)Ln+1,α=Ln,αexp⁡Al,α-Bl,αLn,α,Rn+1,α=Rn,αexp⁡Ar,α-Br,αRn,α.Then it is obvious that, for any initial condition (L0,α,R0,α),α∈(0,1], there exists a unique solution (Ln,α,Rn,α).
Now we prove that[Ln,α,Rn,α],α∈(0,1], where (Ln,α,Rn,α) is the solution of system (13) with initial conditions (L0,α,R0,α), determines the solution of (2) with initial value x0, such that,(14)xnα=Ln,α,Rn,α,n=0,1,2,…,α∈0,1.From Definition 2 and since A,B,x0 are positive fuzzy numbers, for any α1,α2∈(0,1],α1≤α2, we have(15)0<Al,α1≤Al,α2≤Ar,α2≤Ar,α10<Bl,α1≤Bl,α2≤Br,α2≤Br,α10<L0,α1≤L0,α2≤R0,α2≤R0,α1We claim that, for n=0,1,…,(16)0<Ln,α1≤Ln,α2≤Rn,α2≤Rn,α1.We prove it by induction. It is clear that (16) is true for n=0. Suppose that (16) holds true for n≤k,k∈1,2,…,. Then, from (13), (15), and (16) for n≤k, it follows that (17)Lk+1,α1=Lk,α1exp⁡Al,α1-Bl,α1Lk,α1≤Lk,α2exp⁡Al,α2-Bl,α2Lk,α2=Lk+1,α2≤Rk,α2exp⁡Ar,α2-Br,α2Rk,α2=Rk+1,α2≤Rk,α1exp⁡Ar,α1-Br,α1Rk,α1=Rk+1,α1.Therefore (16) is satisfied. Moreover, for ∀α∈(0,1], it follows from (13) that(18)L1,α=L0,αexpAl,α-Bl,αL0,α,R1,α=R0,αexpAr,α-Br,αR0,α.Since A,B,x0 are positive fuzzy numbers, from Definition 2, then Al,α, Ar,α, Bl,α, Br,α, L0,α, R0,α are left continuous. From (18) we have 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 ofxn, supp⁡xn=⋃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, and since A,B and x0 are positive fuzzy numbers, there exist constants MA>0, NA>0, MB>0, NB>0, M0>0, N0>0 such that, for α∈(0,1],(19)Al,α,Ar,α⊂⋃α∈0,1Al,α,Ar,α¯⊂MA,NA,Bl,α,Br,α⊂⋃α∈0,1Bl,α,Br,α¯⊂MB,NB,L0,α,L0,α⊂⋃α∈0,1L0,α,R0,α¯⊂M0,N0,Hence, from (18) and (19), for α∈(0,1], we get(20)L1,α,R1,α⊂M0exp⁡MA-NBN0,N0exp⁡NA-MBM0,from which it is clear that(21)⋃α∈0,1L1,α,R1,α⊂M0exp⁡MA-NBN0,N0exp⁡NA-MBM0,Therefore (21) 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(22)⋃α∈0,1Ln,α,Rn,α¯is  compact,  and⋃α∈0,1Ln,α,Rn,α¯⊂0,∞.Therefore, from (16), (22) and Ln,α, Rn,α are left continuous. It can conclude that [Ln,α,Rn,α] determines a sequence of positive fuzzy numbers xn such that (14) holds.
We prove thatxn is a solution of (12) with initial value x0. Since ∀α∈(0,1], (23)xn+1α=Ln+1,α,Rn+1,α=Ln,αexp⁡Al,α-Bl,αLn,α,Rn,αexp⁡Ar,α-Br,αRn,α=xnexp⁡A-Bxnα.Namely, xn is a solution of (12) with initial value x0.
Suppose that there exists another solutionx¯n of (12) with initial value x0. Then from arguing as above we can easily get that, for n=0,1,2,…,(24)x¯nα=Ln,α,Rn,α,α∈0,1.Then from (14) and (24), we have xnα=x¯nα,α∈(0,1],n=0,1,2,…, and hence xn=x¯n,n=0,1,2,…, and this completes the proof of Theorem 13.


## 3.2. Dynamical Behaviour of Positive Solution

In order to study the dynamical behavior of the solutionxn to (2), we first consider the following system of difference equations(25)yn+1=ynexp⁡a-pyn,zn+1=znexp⁡b-qzn,n=0,1,…,It is clear that the equilibrium points (y¯,z¯) of (25) include the following four cases:(26)i0,0,ii0,bq,iiiap,0,ivap,bq.Lemma 14.
Consider difference equation(27)yn+1=ynexp⁡a-pyn,n=0,1,2,….If a>0,p>0, then 0<yn≤1/pexp⁡(a-1).Proof.
Consider functionfy=yexp⁡(a-py), and we have f′(y)=1-pyexp⁡(a-py). It follows that(28)f′y>0ify<1p,f′y<0ify>1p.It is clear that the maximum of f is equal to 1/pexp⁡(a-1). Therefore we have 0<yn≤1/pexp⁡(a-1).Lemma 15 (see [2]).
Suppose the vector difference equation(29)Xn+1=HXn,n=0,1,2,…,where Xn∈Rk+1,H∈C1[Rk+1,Rk+1]. Let X¯ be an equilibrium point (29), and DH(X¯) denote the Jacobian Matrix of function H at X¯. Then
(i)X¯ is called a hyperbolic equilibrium if DH(X¯) has no eigenvalues with absolute value equal to 1.
(ii)X¯ is called a sink or an attracting equilibrium if every eigenvalue of DH(X¯) has absolute value less than 1.
(iii)X¯ is called a source or a repelling equilibrium if every eigenvalue of DH(X¯) has absolute value greater than 1.
(iv)X¯ is called a saddle point if some of the eigenvalues of DH(X¯) are greater and some are less than 1 in absolute value.Theorem 16.
Consider the system of difference equations (25), where a,b,p, and q 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 a source (a repelling equilibrium).
(ii) The equilibrium0,b/q is a saddle point if 0<b<2; the equilibrium 0,b/q is a source (a repelling equilibrium) if b>2.
(iii) The equilibriuma/p,0 is a saddle point if 0<a<2; the equilibrium a/p,0 is a source (a repelling equilibrium) if a>2.Proof.
(i) It is clear that(0,0) is always an equilibrium. We can easily obtain that the linearized system of (25) about the positive equilibrium (0,0) is(30)Xn+1=D1Xn,n=0,1,…,where(31)Xn=ynzn,D1=ea00eb.We can easily obtain that the eigenvalue of Jacobian matrix D1, λ1=ea, λ2=eb. It is clear that λ>1. This implies that the equilibrium (0,0) is a source (a repelling equilibrium).
(ii) We can obtain that the linearized system of (25) about the equilibrium 0,a/p is(32)Xn+1=D2Xn,where(33)Xn=ynzn,D2=ea001-bIt is easy to obtain the following: if 0<b<2, the eigenvalue of D2, λ1>1, λ2<1, then 0,a/p is a saddle point. if b>2, all the eigenvalue of D2, λ>1, then 0,a/p is a source (a repelling equilibrium).
(iii) The proof of (iii) is similar to the proof of (ii). So we omit it.Theorem 17.
Consider the system of difference equations (25), where the initial values y0,z0 are positive real numbers. If 0<a<2,0<b<2, then (25) has a unique positive equilibrium point a/p,b/q which is globally asymptotically stable.Proof.
It follows easily from (25) that a/p,b/q is a unique positive equilibrium point.
We can obtain that the linearized system of (25) about the positive equilibrium a/p,b/q is(34)Xn+1=D3Xn,where(35)Xn=ynzn,D3=1-a001-b
We can easily obtain that the eigenvalue of Jacobian matrixD3, λ1=1-a,λ2=1-b. It is clear that λ<1. Hence the equilibrium a/p,b/q is a sink (an attracting equilibrium).
Noting Lemma14, we have(36)limn→∞⁡sup⁡yn=L1,limn→∞⁡inf⁡yn=l1,limn→∞⁡sup⁡zn=L2,limn→∞⁡inf⁡zn=l2.where Li,li∈(0,∞),i=1,2. Then from (25) we have(37)L1≤L1exp⁡a-pl1,l1≥l1exp⁡a-pL1,L2≤L2exp⁡b-ql2,l2≥l2exp⁡b-qL2.From this, we have(38)1≤exp⁡a-pl1,1≥exp⁡a-pL1,1≤exp⁡b-ql2,1≥exp⁡b-qL2.The relation (38) implies l1≤L1, l2≤L2. We claim that(39)l1=L1,l2=L2.Suppose on the contrary that l1<L1,l2<L2. From (38), we have(40)l1≤ap<L1≤1pexp⁡a-1,l2≤bq<L2≤1qexp⁡b-1,from which we have(41)a<exp⁡a-1,b<exp⁡b-1,It follows from (41) that a≠1 and b≠1, which is contradicting with 0<a<2,0<b<2. So Li=li,i=1,2.. Hence we have limn→∞yn=a/p, limn→∞zn=b/q. Therefore the unique positive equilibrium a/p,b/q is globally asymptotically stable.Theorem 18.
Consider the fuzzy difference equation (2), where A,B are positive fuzzy numbers. Consider that there exists positive constants NA, for all α∈(0,1] such that(42)Ar,α<NA<2.Then the following statements are true.
(i) Every positive solutionxn of (2) is bounded.
(ii) Equation (2) has a unique positive equilibrium x.
(iii) Every positive solutionxn of (2) converges the unique equilibrium x with respect to D as n→∞.
Moreover the unique positive equilibriumx is asymptotically stable.Proof.
(i) Letxn be a positive solution of (2) with initial conditions x0. Suppose that (19) holds; from (25) and using Lemma 14, we get that(43)0<Ln,α≤1Bl,αexp⁡Al,α-1,0<Rn,α≤1Br,αexp⁡Ar,α-1,α∈0,1,
From (19) and (43), we have that for all α∈(0,1](44)Ln,α,Rn,α⊂0,N,n≥1,where N=1/MBexp⁡(NA-1). From (44), we have for n≥1, ⋃α∈(0,1][Ln,α,Rn,α]⊂(0,N], so ⋃α∈(0,1][Ln,α,Rn,α]¯⊂(0,N]. Thus the proof of Part (i) is completed.
(ii) We consider the system(45)Lα=LαexpAl,α-Bl,αLα,Rα=RαexpAr,α-Br,αRα.Then the positive solution (Lα,Rα) of (45) is given by(46)Lα=Al,αBl,α,Rα=Ar,αBr,α,α∈0,1.Let xn be a positive solution of (2) such that xnα=[Ln,α,Rn,α], α∈(0,1], n=0,1,2,⋯. Then applying Theorem 17 to system (13) we have(47)limn→∞⁡Ln,α=Lα,limn→∞⁡Rn,α=RαFrom (44) and (47) we have for 0<α1<α2<1,(48)0<Lα1≤Lα2≤Rα2≤Rα1.Since Al,α, Ar,α, Bl,α, Br,α are left continuous, from (45), it follows that Lα,Rα are also left continuous. From (46) and (19), we have(49)c=MANB≤Lα≤Rα≤NAMB=d.Therefore (49) implies that [Lα,Rα]⊂[c,d], and so ⋃α∈(0,1][Lα,Rα]⊂[c,d]. It is clear that(50)⋃α∈0,1Lα,Rαis  compact  and⋃α∈0,1Lα,Rα⊂0,∞.So from Definition 2, (45), (48), (50) and Lα,Rα,α∈(0,1] determine a fuzzy number x such that(51)x=xexp⁡A-Bx,xα=Lα,Rα,α∈0,1.
Provided that there exists another positive equilibriumx¯ of (2), then there exist functions L¯α,R¯α:(0,1)→(0,∞) such that(52)x¯=x¯exp⁡A-Bx¯,x¯α=L¯α,R¯α,α∈0,1.From this we get(53)L¯α=L¯αexp⁡Al,α-Bl,αL¯α,R¯α=R¯αexp⁡Ar,α-Br,αR¯α.So L¯α=Lα,R¯α=Rα,α∈(0,1]. Hence x=x¯. This completes the proof of Part (ii).
(iii) From (47) we have(54)limn→∞⁡Dxn,x=limn→∞⁡supα∈0,1⁡max⁡Ln,α-Lα,Rn,α-Rα=0.Namely, every positive solution xn of (2) converges the unique equilibrium x with respect to D as n→∞.
Letε be an arbitrary positive real number; we consider the positive real number δ as follows(55)0<δ<ε1-MBγexp⁡NA-MBγ.where 0<γ=min⁡L0,α,infn≥1Ln,α,α∈(0,1].
Letxn be a positive solution of (2) such that D(x0,x)≤δ<ε. From this we have(56)L0,α-Lα≤δ,R0,α-Rα≤δ,α∈0,1.From (18), (19), and (45), we have(57)L1,α-Lα=L0,αexp⁡Al,α-Bl,αL0,α-Lαexp⁡Al,α-Bl,αLα≤1-Bl,αξ_exp⁡Al,α-Bl,αξ_L0,α-Lα=δ1-Bl,αξ_exp⁡Al,α-Bl,αξ_<δ1-MBξ_exp⁡NA-MBξ_(58)R1,α-Rα=R0,αexp⁡Ar,α-Br,αR0,α-Rαexp⁡Ar,α-Br,αRα≤1-Br,αη_exp⁡Ar,α-Br,αη_R0,α-Rα=δ1-Br,αη_exp⁡Ar,α-Br,αη_<δ1-MBη_exp⁡NA-MBη_where ξ_=min⁡L0,α,Lα, η_=min⁡R0,α,Rα, α∈(0,1].
From (55), it is obvious that(59)L1,α-Lα<ε,R1,α-Rα<ε.From this and working inductively we can easily prove that(60)Ln,α-Lα<ε,Rn,α-Rα<ε,α∈0,1,n=0,1,2,⋯.And so D(xn,x)<ε. Therefore the positive equilibrium x is stable. Moreover relation (54) holds. So the equilibrium x is asymptotically stable. The proof of Theorem 18 is completed.Remark 19.
From Theorem18, we can know that the dynamical behavior of (2) is relevant to the intrinsic growth rate A of population. No matter what the initial population quantity x0 is small and no matter what the carrying capacity of the surround environment B is large. As long as the intrinsic growth rate of population A satisfies Ar,α<2,0<Al,α-Bl,αLn,α≤Ar,α-Br,αRn,α, for all α∈(0,1), the solution of fuzzy difference equation (2) is bounded and eventually converges to the unique positive equilibrium.


## 4. An Illustrative Example

In order to illustrate our obtained results, we give a numerical example to show effectiveness of theoretic results.Example 1.
Consider discrete time fuzzy single species model(61)xn+1=xnexp⁡A-Bxn,n=0,1,…,where A,B and the initial value x0 are positive triangular fuzzy numbers such that(62)A=1.2,1.5,1.8,B=0.9,1,1.1,x0=1,1.1,1.2from which, we get, for α∈(0,1],(63)Aα=1.2+0.3α,1.8-0.3α,Bα=0.9+0.1α,1.1-0.1α,x0α=1+0.1α,1.2-0.1α.From (61) and (63), it results in a coupled system of difference equation with parameter α∈(0,1],(64)Ln+1,α=Ln,αexp⁡1.2+0.3α-0.9+0.1αLn,α,Rn+1,α=Rn,αexp⁡1.8-0.3α-1.1-0.1αRn,α.It is clear that (42) of Theorem 18 is satisfied. Therefore the solution of (61) converges to the unique positive equilibrium x=(4/3,1.5,18/11) (see Figures 1–5).Figure 1
The dynamics of system (64).Figure 2
The solution of system (64) at α=0.Figure 3
The solution of system (64) at α=0.25.Figure 4
The solution of system (64) at α=0.75.Figure 5
The solution of system (64) at α=1.


## 5. Conclusion

Difference equation is one of the most important models when it is applicable in various problems in different fields. It is also important if it is applied to study population dynamic in fuzzy environment. When it can be studied in fuzzy environment the behavior of it changes. In this work, we first consider the dynamical behavior of discrete time single speciesxn+1=xnexp⁡(A-Bxn) with fuzzy parameters and fuzzy initial conditions. Compared with crisp discrete time single species model, the dynamical behavior of system is different. The parameters of system A and B satisfy the condition Ar,α<2,0<Al,α-Bl,αLn,α≤Ar,α-Br,αRn,α, for all α∈(0,1],n=0,1,2,…, and the solution of (2) is bounded and converges to unique fuzzy positive equilibrium.



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