
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+β1e−xn−1γ1+yn, yn+1=α2+β2e−yn−1γ2+xn,xn+1=α1+β1e−yn−1γ1+xn, yn+1=α2+β2e−xn−1γ2+yn,
where the parameters αi, βi, γi for i∈{1,2} and the initial conditions x−1,x0,y−1,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
[1] | Tran Hong Thai, Nguyen Anh Dai, Pham Tuan Anh . Global dynamics of some system of second-order difference equations. 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+β1e−xn−1γ1+yn, yn+1=α2+β2e−yn−1γ2+xn,xn+1=α1+β1e−yn−1γ1+xn, yn+1=α2+β2e−xn−1γ2+yn,
where the parameters αi, βi, γi for i∈{1,2} and the initial conditions x−1,x0,y−1,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+βxn−1e−xn, | (1) |
where the parameters
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+βe−ynγ+yn−1, | (2) |
where the parameters
Papaschinopoulos et al. [16] have studied the following systems of two difference equations of exponential form:
xn+1=α+βe−ynγ+yn−1, yn+1=δ+ϵe−xnη+xn−1,xn+1=α+βe−ynγ+xn−1, yn+1=δ+ϵe−xnη+yn−1,xn+1=α+βe−xnγ+yn−1, yn+1=δ+ϵe−ynη+xn−1, |
where
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+βxne−xn−1, |
where the parameters
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+β1e−xn−1γ1+yn, yn+1=α2+β2e−yn−1γ2+xn, | (3) |
xn+1=α1+β1e−yn−1γ1+xn, yn+1=α2+β2e−xn−1γ2+yn, | (4) |
where the parameters
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,xn−1,yn,yn−1),yn+1=g(xn,xn−1,yn,yn−1),n=0,1,.... | (5) |
where
¯x=f(¯x,¯x,¯y,¯y),¯y=g(¯x,¯x,¯y,¯y). |
The point
Definition 2.1. Let
(ⅰ) An equilibrium point
(ⅱ) An equilibrium point
(ⅲ) An equilibrium point
(ⅳ) An equilibrium point
(ⅴ) An equilibrium point
Definition 2.2. Let
Xn+1=JFXn, |
where
Lemma 2.3. (see [19]) Assume that
Definition 2.4. A positive solution
m≤xn,yn≤M,n≥N. |
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
(i)
(ii) Let
m1=f(M1,M2),M1=f(m1,m2),m2=g(M1,M2),M2=g(m1,m2) | (6) |
then
Then the systems of difference equations
xn+1=f(xn−1,yn),yn+1=g(xn,yn−1), | (7) |
xn+1=f(xn,yn−1),yn+1=g(xn−1,yn) | (8) |
have a unique equilibrium point
The following results give the rate of convergence of solutions of a system of difference equations
Xn+1=[A+B(n)]Xn | (9) |
where
‖B(n)‖→0 when n→ ∞, | (10) |
where
‖(x, y)‖=√x2+y2. |
Proposition 2.6 (Perron's theorem [18]). Assume that condition 10 holds. If
ρ=limn→∞n√‖Xn‖ | (11) |
exists and is equal to the modulus of one of the eigenvalues of matrix
Proposition 2.7 (See [18]). Assume that condition 10 holds. If
ρ=limn→∞‖Xn+1‖‖Xn‖ | (12) |
exists and is equal to the modulus of one of the eigenvalues of matrix
In this section, we show the boundedness and persistence of the positive solutions of system 3.
Lemma 3.1. Every positive solution
Proof. For any positive solution
xn+1≤α1+β1γ1=b1,yn+1≤α2+β2γ2=d1,n=0,1,2,… | (13) |
Furthermore, from system 3 and 13, we obtain that
xn+1≥α1+β1e−b1γ1+d1=a1,yn+1≥α2+β2e−d1γ2+b1=c1,n=2,3,4,… | (14) |
From 13 and 14, it follows that
a1≤xn≤b1,c1≤yn≤d1,n=3,4,5,… |
So the proof is complete.
Lemma 3.2. Let
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=α1+β1e−¯xγ1+¯y,¯y=α2+β2e−¯yγ2+¯x. |
The linearized form of system 3 about the equilibrium point
Xn+1=JF(¯x,¯y)Xn, |
where
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
Proof. Consider the following functions:
f(x,y)=α1+β1e−xγ1+y,g(x,y)=α2+β2e−yγ2+x, |
where
m=f(M,R),M=f(m,r),r=g(M,R),R=g(m,r). |
Then, one has
m=α1+β1e−Mγ1+R,M=α1+β1e−mγ1+r,r=α2+β2e−Rγ2+M,R=α2+β2e−rγ2+m. | (16) |
Moreover arguing as in the proof of Theorem 1.16 of [6], it suffices to assume that
m≤M,r≤R. | (17) |
From 16, we get
β1e−m=M(γ1+r)−α1,β1e−M=m(γ1+R)−α1, |
β2e−r=R(γ2+m)−α2,β2e−R=r(γ2+M)−α2, |
which imply that
β1(e−m−e−M)=γ1(M−m)+Mr−mR,β2(e−r−e−R)=γ2(R−r)+mR−Mr. | (18) |
Then by adding the two relations 18, we obtain
β1(e−m−e−M)+β2(e−r−e−R)=γ1(M−m)+γ2(R−r). |
Moreover, we get
eR−er=eξ(R−r), min{R,r}≤ξ≤max{R,r},eM−em=eθ(M−m), min{M,m}≤θ≤max{M,m}. | (19) |
Then from 19, imply that
β1e−m−M+θ(M−m)+β2e−r−R+ξ(R−r)=γ1(M−m)+γ2(R−r). | (20) |
Hence from 20, we have
γ1(M−m)(1−β1γ1e−m−M+θ)+γ2(R−r)(1−β2γ2e−r−R+ξ)=0. | (21) |
Finally, from 15, 17 and 21, it follows that
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
β1e−a1γ1+c1+β2e−c1γ2+a1+b1d1+β1β2e−a1−c1(γ1+c1)(γ2+a1)<1. | (22) |
Then the unique positive equilibrium point
Proof. First we will prove that
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+¯x≤β1e−a1γ1+c1+β2e−c1γ2+a1+b1d1+β1β2e−a1−c1(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
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+1−¯x=α1+β1e−xn−1γ1+yn−α1+β1e−¯xγ1+¯y=(α1+β1e−xn−1)(γ1+¯y)−(α1+β1e−¯x)(γ1+yn)(γ1+yn)(γ1+¯y)=−α1(yn−¯y)+β1γ1(e−xn−1−e−¯x)+β1(e−xn−1¯y−e−¯xyn)(γ1+yn)(γ1+¯y)=−α1(yn−¯y)−β1γ1e−xn−1(exn−1−¯x−1)(γ1+yn)(γ1+¯y)+β1(e−xn−1¯y−e−xn−1yn+e−xn−1yn−e−¯xyn)(γ1+yn)(γ1+¯y)=−β1e−xn−1(exn−1−¯x−1)(γ1+¯y)(xn−1−¯x)(xn−1−¯x)−α1+β1e−xn−1(γ1+yn)(γ1+¯y)(yn−¯y), |
and
yn+1−¯y=α2+β2e−yn−1γ2+xn−α2+β2e−¯yγ2+¯x=(α2+β2e−yn−1)(γ2+¯x)−(α2+β2e−¯y)(γ2+xn)(γ2+xn)(γ2+¯x)=−α2(xn−¯x)+β2γ2(e−yn−1−e−¯y)+β2(e−yn−1¯x−e−¯yxn)(γ2+xn)(γ2+¯x)=−α2(xn−¯x)−β2γ2e−yn−1(eyn−1−¯y−1)(γ2+xn)(γ2+¯x)+β2(e−yn−1¯x−e−yn−1xn+e−yn−1xn−e−¯yxn)(γ2+xn)(γ2+¯x)=−α2+β2e−yn−1(γ2+xn)(γ2+¯x)(xn−¯x)−β2e−yn−1(eyn−1−¯y−1)(γ2+¯x)(yn−1−¯y)(yn−1−¯y). |
Let
e1n+1=ane1n−1+bne2n,e2n+1=cne1n+dne2n−1, |
where
an=−β1e−xn−1(exn−1−¯x−1)(γ1+¯y)(xn−1−¯x),bn=−α1+β1e−xn−1(γ1+yn)(γ1+¯y),cn=−α2+β2e−yn−1(γ2+xn)(γ2+¯x),dn=−β2e−yn−1(eyn−1−¯y−1)(γ2+¯x)(yn−1−¯y). |
Moreover,
limn→∞an=−β1e−¯xγ1+¯y,limn→∞bn=−¯xγ1+¯y,limn→∞cn=−¯yγ2+¯x,limn→∞dn=−β2e−¯yγ2+¯x. |
So, the limiting system of the error terms can be written as
(e1n+1e1ne2n+1e2n)=(0−β1e−¯xγ1+¯y−¯xγ1+¯y01000−¯yγ2+¯x00−β2e−¯yγ2+¯x0010)(e1ne1n−1e2ne2n−1) |
which similar to the linearized system of 3 about the equilibrium point
Theorem 3.5. Assume that
limn→∞(||en||)1n=|λi|,limn→∞||en+1||||en||=|λi|,i=1,2,3,4, |
where
In the following lemma, we study the boundedness and persistence of the positive solutions of system 4.
Lemma 3.6. Every positive solution
Proof. Let
xn∈[a2,b2],yn∈[c2,d2], |
where
a2=α1+β1e−α2+β2γ2γ1+α1+β1γ1,b2=α1+β1γ1, |
c2=α2+β2e−α1+β1γ1γ2+α2+β2γ2,d2=α2+β2γ2. |
So the proof is complete.
Corollary 3.7. Let
In this section, we shall investigate the asymptotic behavior of system 4. Let
¯x=α1+β1e−¯yγ1+¯x,¯y=α2+β2e−¯xγ2+¯y. |
The linearized form of system 4 about the equilibrium point
Xn+1=JF(¯x,¯y)Xn, |
where
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
Proof. Consider the following functions:
f(x,y)=α1+β1e−yγ1+x,g(x,y)=α2+β2e−xγ2+y, |
where
m=f(M,R),M=f(m,r),r=g(M,R),R=g(m,r). |
Then, one has
m=α1+β1e−Rγ1+M,M=α1+β1e−rγ1+m,r=α2+β2e−Mγ2+R,R=α2+β2e−mγ2+r. | (25) |
From 25, we get
β1e−r=M(γ1+m)−α1,β1e−R=m(γ1+M)−α1, |
β2e−m=R(γ2+r)−α2,β2e−M=r(γ2+R)−α2, |
which imply that
β1(e−r−e−R)=γ1(M−m),β2(e−m−e−M)=γ2(R−r). | (26) |
Moreover, we get
eR−er=eξ(R−r), min{R,r}≤ξ≤max{R,r},eM−em=eθ(M−m), min{M,m}≤θ≤max{M,m}. | (27) |
Then from 26 and 27, we have
M−m=β1γ1(e−r−e−R)=β1γ1e−r−R(eR−er)=β1γ1e−r−R+ξ(R−r),R−r=β2γ2(e−m−e−M)=β2γ2e−m−M(eM−em)=β2γ2e−m−M+θ(M−m), | (28) |
and so
|M−m|≤β1γ1|R−r|,|R−r|≤β2γ2|M−m|. |
Therefore, we get
(1−β1β2γ1γ2)|M−m|≤0,(1−β1β2γ1γ2)|R−r|≤0. | (29) |
Finally, from 24 and 29, it follows that
Theorem 3.9. Consider system 4 where 24 holds true. Also suppose that
b2γ1+a2+d2γ2+c2+b2d2+β1β2e−a2−c2(γ1+a2)(γ2+c2)<1. | (30) |
Then the unique positive equilibrium point
Proof. First we will prove that
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+¯y≤b2γ1+a2+d2γ2+c2+b2d2+β1β2e−a2−c2(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
In this section, we give the rate of convergence of a solution that converges to the equilibrium of the systems 4.
Let
xn+1−¯x=α1+β1e−yn−1γ1+xn−α1+β1e−¯yγ1+¯x=(α1+β1e−yn−1)(γ1+¯x)−(α1+β1e−¯y)(γ1+xn)(γ1+xn)(γ1+¯x)=−α1(xn−¯x)+β1γ1(e−yn−1−e−¯y)+β1(e−yn−1¯x−e−¯yxn)(γ1+xn)(γ1+¯x)=−α1(xn−¯x)−β1γ1e−yn−1(eyn−1−¯y−1)(γ1+xn)(γ1+¯x)+β1(e−yn−1¯x−e−yn−1xn+e−yn−1xn−e−¯yxn)(γ1+xn)(γ1+¯x)=−α1+β1e−yn−1(γ1+xn)(γ1+¯x)(xn−¯x)−β1e−yn−1(eyn−1−¯y−1)(γ1+¯x)(yn−1−¯y)(yn−1−¯y), |
and
yn+1−¯y=α2+β2e−xn−1γ2+yn−α2+β2e−¯xγ2+¯y=(α2+β2e−xn−1)(γ2+¯y)−(α2+β2e−¯x)(γ2+yn)(γ2+yn)(γ2+¯y)=−α2(yn−¯y)+β2γ2(e−xn−1−e−¯x)+β2(e−xn−1¯y−e−¯xyn)(γ2+yn)(γ2+¯y)=−α2(yn−¯y)−β2γ2e−xn−1(exn−1−¯x−1)(γ2+yn)(γ2+¯y)+β2(e−xn−1¯y−e−xn−1yn+e−xn−1yn−e−¯xyn)(γ2+yn)(γ2+¯y)=−β2e−xn−1(exn−1−¯x−1)(γ2+¯y)(xn−1−¯x)(xn−1−¯x)−α2+β2e−xn−1(γ2+yn)(γ2+¯y)(yn−¯y). |
Let
e1n+1=fne1n+gne2n−1,e2n+1=hne1n−1+kne2n, |
where
fn=−α1+β1e−yn−1(γ1+xn)(γ1+¯x),gn=−β1e−yn−1(eyn−1−¯y−1)(γ1+¯x)(yn−1−¯y),hn=−β2e−xn−1(exn−1−¯x−1)(γ2+¯y)(xn−1−¯x),kn=−α2+β2e−xn−1(γ2+yn)(γ2+¯y). |
Moreover,
limn→∞fn=−¯xγ1+¯x,limn→∞gn=−β1e−¯yγ1+¯x, |
limn→∞hn=−β2e−¯xγ2+¯y,limn→∞kn=−¯yγ2+¯y. |
So, the limiting system of the error terms can be written as
(e1n+1e1ne2n+1e2n)=(−¯xγ1+¯x00−β1e−¯yγ1+¯x10000−β2e−¯xγ2+¯y−¯yγ2+¯y00010)(e1ne1n−1e2ne2n−1) |
which similar to the linearized system of 4 about the equilibrium point
Theorem 3.10. Assume that
limn→∞(||en||)1n=|λi|,limn→∞||en+1||||en||=|λi|,i=1,2,3,4, |
where
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
xn+1=30+1.4e−xn−11.5+yn, yn+1=45+2.5e−yn−12.8+xn, | (32) |
with initial conditions
In this case, the unique positive equilibrium point of the system 32 is given by
Example 4.2. Let
xn+1=0.2+19e−xn−14+yn, yn+1=0.3+20e−yn−12+xn, | (33) |
with initial conditions
In this case, the unique positive equilibrium point of the system 33 is unstable. Moreover, the plot of
Example 4.3. Let
xn+1=401+1e−yn−11.75+xn, yn+1=395+1.5e−xn−11+yn, | (34) |
with initial conditions
In this case, the unique positive equilibrium point of the system 34 is given by
Example 4.4. Let
xn+1=4+10e−yn−11.6+xn, yn+1=5.5+10.8e−xn−11+yn, | (35) |
with initial conditions
In this case, the unique positive equilibrium point of the system (35) is unstable. Moreover, the plot of
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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