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Global dynamics of some system of second-order difference equations

  • Received: 01 June 2021 Published: 08 October 2021
  • Primary: 39A10, 39A30; Secondary: 40A05

  • In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations

    xn+1=α1+β1exn1γ1+yn, yn+1=α2+β2eyn1γ2+xn,xn+1=α1+β1eyn1γ1+xn, yn+1=α2+β2exn1γ2+yn,

    where the parameters αi, βi, γi for i{1,2} and the initial conditions x1,x0,y1,y0 are positive real numbers. Some numerical example are given to illustrate our theoretical results.

    Citation: Tran Hong Thai, Nguyen Anh Dai, Pham Tuan Anh. Global dynamics of some system of second-order difference equations[J]. Electronic Research Archive, 2021, 29(6): 4159-4175. doi: 10.3934/era.2021077

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  • In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations

    xn+1=α1+β1exn1γ1+yn, yn+1=α2+β2eyn1γ2+xn,xn+1=α1+β1eyn1γ1+xn, yn+1=α2+β2exn1γ2+yn,

    where the parameters αi, βi, γi for i{1,2} and the initial conditions x1,x0,y1,y0 are positive real numbers. Some numerical example are given to illustrate our theoretical results.



    Mathematical models of population dynamics are often described by difference equations and systems of difference equations. In particular, the population models involving exponential difference equations are quite popular, although their stability analysis can be complicated. In recent years, the global asymptotic behavior of the difference equations of exponential form has been one of the main topics in the theory of difference equations (see [2,3,4,5,7,8,12,13,14,15,16,17,20] and reference cited therein).

    In [4], El-Metwally et al. investigated the following population model:

    xn+1=α+βxn1exn, (1)

    where the parameters α and β are positive numbers and the initial conditions x1 and x0 are arbitrary non-negative numbers. Later in [5], Fotiades et al. studied the existence, uniqueness and attractivity of prime period two solution of this equation.

    Papaschinopoulos et al. [15] and Papaschinopoulos and Schinas [17] investigated the dynamical properties of two-species model described by systems of difference equations, which is natural extension of single-species population model depicted in 1.

    Ozturk et al. [12] have investigated the following difference equation:

    yn+1=α+βeynγ+yn1, (2)

    where the parameters α,β,γ are positive numbers and the initial conditions are arbitrary non-negative numbers.

    Papaschinopoulos et al. [16] have studied the following systems of two difference equations of exponential form:

    xn+1=α+βeynγ+yn1, yn+1=δ+ϵexnη+xn1,xn+1=α+βeynγ+xn1, yn+1=δ+ϵexnη+yn1,xn+1=α+βexnγ+yn1, yn+1=δ+ϵeynη+xn1,

    where α,β,γ,δ,ϵ,η are positive constants and the initial values x1,x0,y1,y0 are positive numbers.

    In 2016, Wang and Feng [20] have investigated the dynamics of positive solution of the following difference equation which is naturally a new form of single-species model described in 1:

    xn+1=α+βxnexn1,

    where the parameters α and β are positive numbers and the initial conditions x1 and x0 are arbitrary non-negative numbers.

    Motivated by the aforementioned study, our goal in this paper is to investigate the qualitative behavior of positive solutions of some systems of exponential difference equations

    xn+1=α1+β1exn1γ1+yn, yn+1=α2+β2eyn1γ2+xn, (3)
    xn+1=α1+β1eyn1γ1+xn, yn+1=α2+β2exn1γ2+yn, (4)

    where the parameters αi, βi, γi for i{1,2} and the initial conditions x1,x0,y1,y0 are positive real numbers.

    More precisely, we investigate the boundedness character, persistence, existence of invariant rectangle, local asymptotic stability and global behavior of unique positive equilibrium point, and rate of convergence of positive solutions of system 3 and 4 which converges to its unique positive equilibrium point. For applications and basic theory of difference equations we refer to [1,6,10,11,19].

    In this section, we present some definitions and theorems which are used throughout this study.

    Let us consider fourth-dimensional discrete dynamical system of the following form:

    xn+1=f(xn,xn1,yn,yn1),yn+1=g(xn,xn1,yn,yn1),n=0,1,.... (5)

    where f:I2×J2I and g:I2×J2J are continuously differentiable functions and I,J are some intervals of real numbers. Furthermore, a solution {xn,yn}n=1 of system 5 is uniquely determined by initial conditions (xi,yi)I×J for i{1,0}. Along with system 5, we consider the corresponding vector map F=(f,xn,g,yn). An equilibrium point of 5 is a point (¯x,¯y) that sitisfies

    ¯x=f(¯x,¯x,¯y,¯y),¯y=g(¯x,¯x,¯y,¯y).

    The point (¯x,¯x,¯y,¯y) is also called a fixed point of the vector map F.

    Definition 2.1. Let (¯x,¯y) be an equilibrium point of system 5.

    (ⅰ) An equilibrium point (¯x,¯y) is called stable if for any ε>0 there is δ>0 such that for every initial conditions (x1,y1) and (x0,y0), if ||(x1,y1)(¯x,¯y)||+||(x0,y0)(¯x,¯y)||<δ implies that ||(xn,yn)(¯x,¯y)||<ε for all n>0, where ||.|| is usual Euclidean norm in R2.

    (ⅱ) An equilibrium point (¯x,¯y) is called unstable if it is not stable.

    (ⅲ) An equilibrium point (¯x,¯y) is called locally asymptotically stable if it stable and if, in addition, there exists r>0 such that (xn,yn)(¯x,¯y) as n for all (x1,y1) and (x0,y0) that satisfy ||(x1,y1)(¯x,¯y)||+||(x0,y0)(¯x,¯y)||<r.

    (ⅳ) An equilibrium point (¯x,¯y) is called global attractor if (xn,yn)(¯x,¯y) as n.

    (ⅴ) An equilibrium point (¯x,¯y) is called globally asymptotically stable if it stable and a global attractor.

    Definition 2.2. Let (¯x,¯x,¯y,¯y) be a fixed point of a map F=(f,xn,g,yn) where f and g are continuously differentiable functions at (¯x,¯y). The linearized system of 5 about the equilibrium point (¯x,¯y) is

    Xn+1=JFXn,

    where Xn=(xnxn1ynyn1) and JF is the Jacobian matrix of system 5 about the equilibrium point (¯x,¯y).

    Lemma 2.3. (see [19]) Assume that Xn+1=F(Xn),n=0,1,..., is a system of difference equations such that ¯X is a fixed point of F. If all eigenvalues of the Jacobian matrix JF about ¯X lie inside the open unit disk |λ|<1, then ¯X is locally asymptotically stable. If one of them has a modulus greater than one, then ¯X is unstable.

    Definition 2.4. A positive solution {xn,yn}n=1 of system 5 is bounded and persists if there exist positive constants m,M and an interger N1 such that

    mxn,ynM,nN.

    In order to study the asymptotic behavior of positive equilibrium, we state the following lemma which is a slight modification of Theorem 1.16 of [6] and for readers convenience we state it without its proof.

    Lemma 2.5. Assume that f:(0,)×(0,)(0,) and g:(0,)×(0,)(0,) be continuous functions and a,b,c,d are positive real numbers with a<b,c<d. Moreover, suppose that f:[a,b]×[c,d][a,b] and g:[a,b]×[c,d][c,d] such that following conditions are satisfied:

    (i) f(x,y),g(x,y) are decreasing with respect to x (resp. y) for all y (resp. x);

    (ii) Let m1,M1,m2,M2 are real numbers such that

    m1=f(M1,M2),M1=f(m1,m2),m2=g(M1,M2),M2=g(m1,m2) (6)

    then m1=M1 and m2=M2.

    Then the systems of difference equations

    xn+1=f(xn1,yn),yn+1=g(xn,yn1), (7)
    xn+1=f(xn,yn1),yn+1=g(xn1,yn) (8)

    have a unique equilibrium point (¯x,¯y) and every solution (xn,yn) of the system 7 (resp. 8) with x1,x0[a,b],y1,y0[c,d] converges to the unique equilibrium (ˉx,ˉy).

    The following results give the rate of convergence of solutions of a system of difference equations

    Xn+1=[A+B(n)]Xn (9)

    where Xn is a m-dimensional vector, ACm×m is a constant matrix, and B: Z+Cm×m is a matrix function satisfying

    B(n)0 when n , (10)

    where ||.|| denotes any matrix norm which is associated with the vector norm

    (x, y)=x2+y2.

    Proposition 2.6 (Perron's theorem [18]). Assume that condition 10 holds. If Xn is a solution of system 9, then either Xn=0 for all large n or

    ρ=limnnXn (11)

    exists and is equal to the modulus of one of the eigenvalues of matrix A.

    Proposition 2.7 (See [18]). Assume that condition 10 holds. If Xn is a solution of system 9, then either Xn=0 for all large n or

    ρ=limnXn+1Xn (12)

    exists and is equal to the modulus of one of the eigenvalues of matrix A.

    In this section, we show the boundedness and persistence of the positive solutions of system 3.

    Lemma 3.1. Every positive solution {(xn,yn)} of system 3 is bounded and persists.

    Proof. For any positive solution {(xn,yn)} of system 3, one has

    xn+1α1+β1γ1=b1,yn+1α2+β2γ2=d1,n=0,1,2, (13)

    Furthermore, from system 3 and 13, we obtain that

    xn+1α1+β1eb1γ1+d1=a1,yn+1α2+β2ed1γ2+b1=c1,n=2,3,4, (14)

    From 13 and 14, it follows that

    a1xnb1,c1ynd1,n=3,4,5,

    So the proof is complete.

    Lemma 3.2. Let {(xn,yn)} be a positive solution of system 3. Then [a1,b1]×[c1,d1] is an invariant set for system 3.

    Proof. The proof follows by induction.

    In this section, we shall investigate the asymptotic behavior of system 3. Similar method can be found in [9].

    Let (¯x,¯y) be the equilibrium point of system 3 then

    ¯x=α1+β1e¯xγ1+¯y,¯y=α2+β2e¯yγ2+¯x.

    The linearized form of system 3 about the equilibrium point (¯x,¯y) is given by

    Xn+1=JF(¯x,¯y)Xn,

    where Xn=(xnxn1ynyn1) and JF(¯x,¯y)=(0β1e¯xγ1+¯y¯xγ1+¯y01000¯yγ2+¯x00β2e¯yγ2+¯x0010).

    In the following theorem, we show the asymptotic behavior of the positive solutions of system 3.

    Theorem 3.3. Suppose that the following relation holds true:

    β1<γ1,β2<γ2. (15)

    Then system 3 has a unique positive equilibrium (¯x,¯y) and every positive solution of system 3 tends to the unique positive equilibrium as n.

    Proof. Consider the following functions:

    f(x,y)=α1+β1exγ1+y,g(x,y)=α2+β2eyγ2+x,

    where xI1=[a1,b1],yI2=[c1,d1] which implies that f(x,y)I1,g(x,y)I2 and so that f:I1×I2I1,g:I1×I2I2. Then, it is easy to see that f(x,y),g(x,y) are decreasing with respect to x (resp. y) for all y (resp. x). Let (m,M,r,R) be a solution of the system

    m=f(M,R),M=f(m,r),r=g(M,R),R=g(m,r).

    Then, one has

    m=α1+β1eMγ1+R,M=α1+β1emγ1+r,r=α2+β2eRγ2+M,R=α2+β2erγ2+m. (16)

    Moreover arguing as in the proof of Theorem 1.16 of [6], it suffices to assume that

    mM,rR. (17)

    From 16, we get

    β1em=M(γ1+r)α1,β1eM=m(γ1+R)α1,
    β2er=R(γ2+m)α2,β2eR=r(γ2+M)α2,

    which imply that

    β1(emeM)=γ1(Mm)+MrmR,β2(ereR)=γ2(Rr)+mRMr. (18)

    Then by adding the two relations 18, we obtain

    β1(emeM)+β2(ereR)=γ1(Mm)+γ2(Rr).

    Moreover, we get

    eRer=eξ(Rr), min{R,r}ξmax{R,r},eMem=eθ(Mm), min{M,m}θmax{M,m}. (19)

    Then from 19, imply that

    β1emM+θ(Mm)+β2erR+ξ(Rr)=γ1(Mm)+γ2(Rr). (20)

    Hence from 20, we have

    γ1(Mm)(1β1γ1emM+θ)+γ2(Rr)(1β2γ2erR+ξ)=0. (21)

    Finally, from 15, 17 and 21, it follows that M=m and R=r. Therefore, from Lemma 2.5, it follows that system 3 has a unique positive equilibrium (¯x,¯y) and every positive solution of system 3 tends to the unique positive equilibrium as n. This completes the proof of the theorem.

    In the next theorem of this section, we will study the global asymptotic stability of the positive equilibrium of system 3.

    Theorem 3.4. Consider system 3 where 15 holds true. Also suppose that

    β1ea1γ1+c1+β2ec1γ2+a1+b1d1+β1β2ea1c1(γ1+c1)(γ2+a1)<1. (22)

    Then the unique positive equilibrium point (¯x,¯y) of system 3 is globally asymptotically stable.

    Proof. First we will prove that (¯x,¯y) is locally asymptotically stable. The charateristic equation of the Jacobian matrix JF(¯x,¯y) about (¯x,¯y) is given by

    λ4+p2λ2+p4=0, (23)

    where

    p2=β1e¯xγ1+¯y+β2e¯yγ2+¯x¯x.¯y(γ1+¯y)(γ2+¯x),
    p4=β1e¯xγ1+¯y.β2e¯yγ2+¯x.

    From condition 22, we get

    |p2|+|p4|=β1e¯xγ1+¯y+β2e¯yγ2+¯x+¯x.¯y(γ1+¯y)(γ2+¯x)+β1e¯xγ1+¯y.β2e¯yγ2+¯xβ1ea1γ1+c1+β2ec1γ2+a1+b1d1+β1β2ea1c1(γ1+c1)(γ2+a1)<1.

    Therefore, follows Remark 1.3.1 of reference [10], all the roots of equation 23 are of modulus less than 1, and it follows from Lemma 2.3 that the unique positive equilibrium point (¯x,¯y) of system 3 is locally asymptotically stable. Using Theorem 3.3, we obtain that (¯x,¯y) is globally asymptotically stable. This completes the proof of the theorem.

    In this section, we give the rate of convergence of a solution that converges to the equilibrium of the systems 3. Similar method can be found in [8,9].

    Let {(xn,yn)} be an arbitrary solution of system 3 such that limnxn=¯x, and limnyn=¯y, where ¯x[a1,b1], and ¯y[c1,d1]. To find the error terms, one has from the system 3

    xn+1¯x=α1+β1exn1γ1+ynα1+β1e¯xγ1+¯y=(α1+β1exn1)(γ1+¯y)(α1+β1e¯x)(γ1+yn)(γ1+yn)(γ1+¯y)=α1(yn¯y)+β1γ1(exn1e¯x)+β1(exn1¯ye¯xyn)(γ1+yn)(γ1+¯y)=α1(yn¯y)β1γ1exn1(exn1¯x1)(γ1+yn)(γ1+¯y)+β1(exn1¯yexn1yn+exn1yne¯xyn)(γ1+yn)(γ1+¯y)=β1exn1(exn1¯x1)(γ1+¯y)(xn1¯x)(xn1¯x)α1+β1exn1(γ1+yn)(γ1+¯y)(yn¯y),

    and

    yn+1¯y=α2+β2eyn1γ2+xnα2+β2e¯yγ2+¯x=(α2+β2eyn1)(γ2+¯x)(α2+β2e¯y)(γ2+xn)(γ2+xn)(γ2+¯x)=α2(xn¯x)+β2γ2(eyn1e¯y)+β2(eyn1¯xe¯yxn)(γ2+xn)(γ2+¯x)=α2(xn¯x)β2γ2eyn1(eyn1¯y1)(γ2+xn)(γ2+¯x)+β2(eyn1¯xeyn1xn+eyn1xne¯yxn)(γ2+xn)(γ2+¯x)=α2+β2eyn1(γ2+xn)(γ2+¯x)(xn¯x)β2eyn1(eyn1¯y1)(γ2+¯x)(yn1¯y)(yn1¯y).

    Let e1n=xn¯x, and e2n=yn¯y, then one has

    e1n+1=ane1n1+bne2n,e2n+1=cne1n+dne2n1,

    where

    an=β1exn1(exn1¯x1)(γ1+¯y)(xn1¯x),bn=α1+β1exn1(γ1+yn)(γ1+¯y),cn=α2+β2eyn1(γ2+xn)(γ2+¯x),dn=β2eyn1(eyn1¯y1)(γ2+¯x)(yn1¯y).

    Moreover,

    limnan=β1e¯xγ1+¯y,limnbn=¯xγ1+¯y,limncn=¯yγ2+¯x,limndn=β2e¯yγ2+¯x.

    So, the limiting system of the error terms can be written as

    (e1n+1e1ne2n+1e2n)=(0β1e¯xγ1+¯y¯xγ1+¯y01000¯yγ2+¯x00β2e¯yγ2+¯x0010)(e1ne1n1e2ne2n1)

    which similar to the linearized system of 3 about the equilibrium point (¯x,¯y). Using Proposition 2.6 and 2.7, one has the following result.

    Theorem 3.5. Assume that {(xn,yn)} be a positive solution of system 3 such that limnxn=¯x, and limnyn=¯y, where ¯x[a1,b1] and ¯y[c1,d1]. Then the error vector en=(e1ne1n1e2ne2n1) of every solution of (3) satisfies both of the following asymptotic relations:

    limn(||en||)1n=|λi|,limn||en+1||||en||=|λi|,i=1,2,3,4,

    where λi is one of the charateristic roots of Jacobian matrix JF(¯x,¯y).

    In the following lemma, we study the boundedness and persistence of the positive solutions of system 4.

    Lemma 3.6. Every positive solution {(xn,yn)} of system 4 is bounded and persists.

    Proof. Let {(xn,yn)} be a positive solution of system 4. Similarly as Lemma 3.1, for n=3,4,5, by induction, we get

    xn[a2,b2],yn[c2,d2],

    where

    a2=α1+β1eα2+β2γ2γ1+α1+β1γ1,b2=α1+β1γ1,
    c2=α2+β2eα1+β1γ1γ2+α2+β2γ2,d2=α2+β2γ2.

    So the proof is complete.

    Corollary 3.7. Let {(xn,yn)} be a positive solution of system 4. Then [a2,b2]×[c2,d2] is an invariant set for system 4.

    In this section, we shall investigate the asymptotic behavior of system 4. Let (¯x,¯y) be the equilibrium point of system 4 then

    ¯x=α1+β1e¯yγ1+¯x,¯y=α2+β2e¯xγ2+¯y.

    The linearized form of system 4 about the equilibrium point (¯x,¯y) is given by

    Xn+1=JF(¯x,¯y)Xn,

    where Xn=(xnxn1ynyn1) and JF(¯x,¯y)=(¯xγ1+¯x00β1e¯yγ1+¯x10000β2e¯xγ2+¯y¯yγ2+¯y00010).

    In the following theorem, we show the asymptotic behavior of the positive solutions of system 4.

    Theorem 3.8. Suppose that the following relation holds true:

    β1β2<γ1γ2. (24)

    Then system 4 has a unique positive equilibrium (¯x,¯y) and every positive solution of system 4 tends to the unique positive equilibrium as n.

    Proof. Consider the following functions:

    f(x,y)=α1+β1eyγ1+x,g(x,y)=α2+β2exγ2+y,

    where xI3=[a2,b2],yI4=[c2,d2] which implies that f(x,y)I3,g(x,y)I4 and so that f:I3×I4I3,g:I3×I4I4. Then, it is easy to see that f(x,y),g(x,y) are decreasing with respect to x (resp. y) for all y (resp. x). Let (m,M,r,R) be a solution of the system

    m=f(M,R),M=f(m,r),r=g(M,R),R=g(m,r).

    Then, one has

    m=α1+β1eRγ1+M,M=α1+β1erγ1+m,r=α2+β2eMγ2+R,R=α2+β2emγ2+r. (25)

    From 25, we get

    β1er=M(γ1+m)α1,β1eR=m(γ1+M)α1,
    β2em=R(γ2+r)α2,β2eM=r(γ2+R)α2,

    which imply that

    β1(ereR)=γ1(Mm),β2(emeM)=γ2(Rr). (26)

    Moreover, we get

    eRer=eξ(Rr), min{R,r}ξmax{R,r},eMem=eθ(Mm), min{M,m}θmax{M,m}. (27)

    Then from 26 and 27, we have

    Mm=β1γ1(ereR)=β1γ1erR(eRer)=β1γ1erR+ξ(Rr),Rr=β2γ2(emeM)=β2γ2emM(eMem)=β2γ2emM+θ(Mm), (28)

    and so

    |Mm|β1γ1|Rr|,|Rr|β2γ2|Mm|.

    Therefore, we get

    (1β1β2γ1γ2)|Mm|0,(1β1β2γ1γ2)|Rr|0. (29)

    Finally, from 24 and 29, it follows that M=m and R=r. Therefore, from Lemma 2.5, it follows that system 4 has a unique positive equilibrium (¯x,¯y) and every positive solution of system 4 tends to the unique positive equilibrium as n. This completes the proof of the theorem. In the next theorem of this section, we will study the global asymptotic stability of the positive equilibrium of system 4.

    Theorem 3.9. Consider system 4 where 24 holds true. Also suppose that

    b2γ1+a2+d2γ2+c2+b2d2+β1β2ea2c2(γ1+a2)(γ2+c2)<1. (30)

    Then the unique positive equilibrium point (¯x,¯y) of system 4 is globally asymptotically stable.

    Proof. First we will prove that (¯x,¯y) is locally asymptotically stable. The charateristic equation of the Jacobian matrix JF(¯x,¯y) about (¯x,¯y) is given by

    λ4+q1λ3+q2λ2+q4=0, (31)

    where

    q1=¯xγ1+¯x+¯yγ2+¯y,
    q2=¯x.¯y(γ1+¯x)(γ2+¯y),
    q4=β1e¯yγ1+¯x.β2e¯xγ2+¯y.

    From condition 30, we get

    |q1|+|q2|+|q4|=¯xγ1+¯x+¯yγ2+¯y+¯x.¯y(γ1+¯x)(γ2+¯y)+β1e¯yγ1+¯x.β2e¯xγ2+¯yb2γ1+a2+d2γ2+c2+b2d2+β1β2ea2c2(γ1+a2)(γ2+c2)<1.

    Therefore, follows Remark 1.3.1 of reference [10], all the roots of equation 31 are of modulus less than 1, and it follows from Lemma 2.3 that the unique positive equilibrium point (¯x,¯y) of system 4 is locally asymptotically stable. Using Theorem 3.8, we obtain that (¯x,¯y) is globally asymptotically stable. This completes the proof of the theorem.

    In this section, we give the rate of convergence of a solution that converges to the equilibrium of the systems 4.

    Let {(xn,yn)} be an arbitrary solution of system 4 such that limnxn=¯x, and limnyn=¯y, where ¯x[a2,b2], and ¯y[c2,d2]. To find the error terms, one has from the system 4

    xn+1¯x=α1+β1eyn1γ1+xnα1+β1e¯yγ1+¯x=(α1+β1eyn1)(γ1+¯x)(α1+β1e¯y)(γ1+xn)(γ1+xn)(γ1+¯x)=α1(xn¯x)+β1γ1(eyn1e¯y)+β1(eyn1¯xe¯yxn)(γ1+xn)(γ1+¯x)=α1(xn¯x)β1γ1eyn1(eyn1¯y1)(γ1+xn)(γ1+¯x)+β1(eyn1¯xeyn1xn+eyn1xne¯yxn)(γ1+xn)(γ1+¯x)=α1+β1eyn1(γ1+xn)(γ1+¯x)(xn¯x)β1eyn1(eyn1¯y1)(γ1+¯x)(yn1¯y)(yn1¯y),

    and

    yn+1¯y=α2+β2exn1γ2+ynα2+β2e¯xγ2+¯y=(α2+β2exn1)(γ2+¯y)(α2+β2e¯x)(γ2+yn)(γ2+yn)(γ2+¯y)=α2(yn¯y)+β2γ2(exn1e¯x)+β2(exn1¯ye¯xyn)(γ2+yn)(γ2+¯y)=α2(yn¯y)β2γ2exn1(exn1¯x1)(γ2+yn)(γ2+¯y)+β2(exn1¯yexn1yn+exn1yne¯xyn)(γ2+yn)(γ2+¯y)=β2exn1(exn1¯x1)(γ2+¯y)(xn1¯x)(xn1¯x)α2+β2exn1(γ2+yn)(γ2+¯y)(yn¯y).

    Let e1n=xn¯x, and e2n=yn¯y, then one has

    e1n+1=fne1n+gne2n1,e2n+1=hne1n1+kne2n,

    where

    fn=α1+β1eyn1(γ1+xn)(γ1+¯x),gn=β1eyn1(eyn1¯y1)(γ1+¯x)(yn1¯y),hn=β2exn1(exn1¯x1)(γ2+¯y)(xn1¯x),kn=α2+β2exn1(γ2+yn)(γ2+¯y).

    Moreover,

    limnfn=¯xγ1+¯x,limngn=β1e¯yγ1+¯x,
    limnhn=β2e¯xγ2+¯y,limnkn=¯yγ2+¯y.

    So, the limiting system of the error terms can be written as

    (e1n+1e1ne2n+1e2n)=(¯xγ1+¯x00β1e¯yγ1+¯x10000β2e¯xγ2+¯y¯yγ2+¯y00010)(e1ne1n1e2ne2n1)

    which similar to the linearized system of 4 about the equilibrium point (¯x,¯y). Using Proposition 2.6 and 2.7, one has the following result.

    Theorem 3.10. Assume that {(xn,yn)} be a positive solution of system 4 such that limnxn=¯x, and limnyn=¯y, where ¯x[a2,b2] and ¯y[c2,d2]. Then the error vector en=(e1ne1n1e2ne2n1) of every solution of 4 satisfies both of the following asymptotic relations:

    limn(||en||)1n=|λi|,limn||en+1||||en||=|λi|,i=1,2,3,4,

    where λi is one of the charateristic roots of Jacobian matrix JF(¯x,¯y).

    In an effort to affirm our theoretical dialogue, we consider several numerical examples. These examples represent different types of qualitative behavior of solutions of the systems 3 and 4. All plots in this section are drawn with MATLAB.

    Example 4.1. Let α1=30,β1=1.4,γ1=1.5,α2=45,β2=2.5, and γ2=2.8. Then system 3 can be written as

    xn+1=30+1.4exn11.5+yn, yn+1=45+2.5eyn12.8+xn, (32)

    with initial conditions x1=0.59,x0=0.61,y1=0.96, and y0=0.94.

    In this case, the unique positive equilibrium point of the system 32 is given by (¯x,¯y)=(3.455959,7.193442). Moreover, the plot of xn is shown in Figure 1, the plot of yn is shown in Figure 2, and an attractor of the system 32 is shown in Figure 3.

    Figure 1.  Plot of xn for the system 32.
    Figure 2.  Plot of yn for the system 32.
    Figure 3.  An attractor of the system 32.

    Example 4.2. Let α1=0.2,β1=19,γ1=4,α2=0.3,β2=20, and γ2=2. Then system 3 can be written as

    xn+1=0.2+19exn14+yn, yn+1=0.3+20eyn12+xn, (33)

    with initial conditions x1=1,x0=1,y1=3, and y0=3.

    In this case, the unique positive equilibrium point of the system 33 is unstable. Moreover, the plot of xn is shown in Figure 4, the plot of yn is shown in Figure 5, and a phase portrait of system 33 is shown in Figure 6.

    Figure 4.  Plot of xn for the system 33.
    Figure 5.  Plot of yn for the system 33.
    Figure 6.  Phase portrait of system 33.

    Example 4.3. Let α1=401,β1=1,γ1=1.75,α2=395,β2=1.5, and γ2=1. Then system 4 can be written as

    xn+1=401+1eyn11.75+xn, yn+1=395+1.5exn11+yn, (34)

    with initial conditions x1=15,x0=24.5,y1=15, and y0=25.

    In this case, the unique positive equilibrium point of the system 34 is given by (¯x,¯y)=(19.169092,19.380895). Moreover, the plot of xn is shown in Figure 7, the plot of yn is shown in Figure 8, and an attractor of the system 34 is shown in Figure 9.

    Figure 7.  Plot of xn for the system 34.
    Figure 8.  Plot of yn for the system 34.
    Figure 9.  An attractor of the system 34.

    Example 4.4. Let α1=4,β1=10,γ1=1.6,α2=5.5,β2=10.8, and γ2=1. Then system 4 can be written as

    xn+1=4+10eyn11.6+xn, yn+1=5.5+10.8exn11+yn, (35)

    with initial conditions x1=1.3,x0=1.5,y1=2, and y0=1.7.

    In this case, the unique positive equilibrium point of the system (35) is unstable. Moreover, the plot of xn is shown in Figure 10, the plot of yn is shown in Figure 11, and a phase portrait of system 35 is shown in Figure 12.

    Figure 10.  Plot of xn for the system 35.
    Figure 11.  Plot of yn for the system 35.
    Figure 12.  Phase portrait of system 35.

    In this study, we investigate the qualitative behavior of some systems of exponential difference equations. We have proved the boundedness and persistence of positive solutions of system 3 and 4. Moreover, we have shown that unique positive equilibrium point of system 3 and 4 is locally as well as globally asymptotically stable under certain parametric conditions. Furthermore, the rate of convergence of positive solutions of 3 and 4 which converges to its unique positive equilibrium point is demonstracted. Finally, some illustrative numerical examples are provided.

    This research is funded by Hung Yen University of Technology and Education under grand number UTEHY.L.2020.11.



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