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Research article Special Issues

Transmission dynamics and optimal control of stage-structured HLB model

  • Received: 25 March 2019 Accepted: 30 May 2019 Published: 06 June 2019
  • Citrus Huanglongbing (HLB) is one of severe quarantine diseases affecting citrus production both in abroad and domestic. Based on the mechanism and characteristics of citrus HLB transmission, we establish a vector-borne model with stage structure and integrated strategy and investigate the effect of the strategy in controlling the spread of HLB. By calculating, we obtain the basic reproductive number R0, and prove that the disease can be eradicated if R0<1, whereas the disease will persist if R0>1. Meanwhile, we apply the optimal control theory to obtain an optimal integrated strategy. Finally, we use our model to simulate the data of the numbers of inspected and infected citrus trees in "Yuan Orchard", located in Ganzhou City, Jiangxi Province in the southeast of P.R China. We also give some numerical simulations for our theoretical findings.

    Citation: Yunbo Tu, Shujing Gao, Yujiang Liu, Di Chen, Yan Xu. Transmission dynamics and optimal control of stage-structured HLB model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5180-5205. doi: 10.3934/mbe.2019259

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  • Citrus Huanglongbing (HLB) is one of severe quarantine diseases affecting citrus production both in abroad and domestic. Based on the mechanism and characteristics of citrus HLB transmission, we establish a vector-borne model with stage structure and integrated strategy and investigate the effect of the strategy in controlling the spread of HLB. By calculating, we obtain the basic reproductive number R0, and prove that the disease can be eradicated if R0<1, whereas the disease will persist if R0>1. Meanwhile, we apply the optimal control theory to obtain an optimal integrated strategy. Finally, we use our model to simulate the data of the numbers of inspected and infected citrus trees in "Yuan Orchard", located in Ganzhou City, Jiangxi Province in the southeast of P.R China. We also give some numerical simulations for our theoretical findings.


    Difference equations are the essentials required to understand even the simplest epidemiological model: the SIR-susceptible, infected, recovered-model. This model is a compartmental model, which results in the basic difference equation used to measure the actual reproduction number. It is this basic model that helps us determine whether a pathogen is going to die out or whether we end up having an epidemic. This is also the basis for more complex models, including the SVIR, which requires a vaccinated state, which helps us to estimate the probability of herd immunity.

    There has been some recent interest in studying the qualitative analysis of difference equations and system of difference equations. Since the beginning of nineties there has be considerable interest in studying systems of difference equations composed by two or three rational difference equations (see, e.g., [4,5,6,2,8,9,11,10,14,15,17,19,20] and the references therein). However, given the multiplicity of factors involved in any epidemic, it will be important to study systems of difference equations composed by many rational difference equations, which is what we will do in this paper.

    In [2], Devault et al. studied the boundedness, global stability and periodic character of solutions of the difference equation

    xn+1=p+xnmxn (1)

    where m{2,3,}, p is positive and the initial conditions are positive numbers.

    In [20], Zhang et al. investigated the behavior of the following symmetrical system of difference equations

    xn+1=A+ynmyn,yn+1=A+xnmxn (2)

    where the parameter A is positive, the initial conditions xi,yi are arbitrary positive numbers for i=m,m+1,,0 and mN. While this study is good, we note that the authors did not investigate various device properties, such as the stability nature, the rate of convergence and the asymptotic behavior.

    Complement of the work above, in [8], Gümüş studied the global asymptotic stability of positive equilibrium, the rate of convergence of positive solutions and he presented some results about the general behavior of solutions of system (2). Our aim in this paper is to generalize the results concerning equation (1) and system (2) to the system of p nonlinear difference equations

    x(1)n+1=A+x(2)nmx(2)n,x(2)n+1=A+x(3)nmx(3)n,,x(p)n+1=A+x(1)nmx(1)n,n,m,pN0 (3)

    where A is a nonnegative constant and x(j)m,x(j)m+1,,x(j)1,x(j)0,j=1,2,,p are positive real numbers.

    The remainder of the paper is organized as follows. In Section (2), we introduce some definitions and notations that will be needed in the sequel. Moreover, we present, in Theorem (2.4), a result concerning the linearized stability that will be useful in the main part of the paper. Section (3) discuses the behavior of positive solutions of system (3) via semi-cycle analysis method. Furthermore, Section (4) is devoted to study the local stability of the equilibrium points and the asymptotic behavior of the solutions when 0A<1,A=1 and A>1. In Section (5), we turn our attention to estimate the rate of convergence of a solution that converges to the equilibrium point of the system (3) in the region of parameters described by A>1. Some numerical examples are carried out to support the analysis results in Section (6). Section (7) summarizes the results of this work, draws conclusions and give some interesting open problems for difference equations theory researchers.

    In this section we recall some definitions and results that will be useful in our investigation, for more details see [3,7,14,13].

    Definition 2.1. (see, [14]) A 'string' of sequential terms {x(j)μ,,x(j)ν}, μ1, ν+ is said to be a positive semi-cycle if x(j)i¯x(j), i{μ,,ν}, x(j)μ1<¯x(j) and x(j)ν+1<¯x(j), j{1,2,,p}.

    A 'string' of sequential terms {x(j)μ,,x(j)ν}, μ1, ν+ is said to be a negative semi-cycle if x(j)i<¯x(j), i{μ,,ν}, x(j)μ1¯x(j) and x(j)ν+1¯x(j), j{1,2,,p}.

    A 'string' of sequential terms {(x(1)μ,x(2)μ,,x(p)μ),,(x(1)ν,x(2)ν,,x(p)ν)}, μ1, ν+ is said to be a positive semi-cycle (resp. negative semi-cycle) if if {x(1)μ,,x(1)ν},,{x(p)μ,,x(p)ν} are positive semi-cycles (resp. negative semi-cycles).

    A 'string' of sequential terms {(x(1)μ,x(2)μ,,x(p)μ),,(x(1)ν,x(2)ν,,x(p)ν)}, μ1, ν+ is said to be a positive semi-cycle (resp. negative semi-cycle) with respect to x(q)n and negative semi-cycle (resp. positive semi-cycle) with respect to x(s)n if {x(q)μ,,x(q)ν} is a positive semi-cycle (resp. negative semi-cycle) and {x(s)μ,,x(s)ν} is a negative semi-cycle (resp. positive semi-cycle).

    Definition 2.2. (see, [14]) A function x(i)n oscillates about ¯x(i) if for every ξN there exist μ,νN, μξ, νξ such that

    (x(i)μ¯x(i))(x(i)μ¯x(i))0,i=1,2,,p.

    We say that a solution {x(1)n,x(2)n,,x(p)n}nm of system (3) oscillates about (¯x(1),¯x(2),,¯x(p)) if x(q)n oscillates about ¯x(q), q{1,2,,p}.

    Let f(1),f(2),,f(p) be p continuously differentiable functions:

    f(i):Ik+11×Ik+12××Ik+1pIk+1i,i=1,2,,p,

    where Ii,i=1,2,,p are some intervals of real numbers. Consider the system of difference equations

    {x(1)n+1=f(1)(x(1)n,x(1)n1,,x(1)nk,x(2)n,x(2)n1,,x(2)nk,,x(p)n,x(p)n1,,x(p)nk)x(2)n+1=f(2)(x(1)n,x(1)n1,,x(1)nk,x(2)n,x(2)n1,,x(2)nk,,x(p)n,x(p)n1,,x(p)nk)x(p)n+1=f(p)(x(1)n,x(1)n1,,x(1)nk,x(2)n,x(2)n1,,x(2)nk,,x(p)n,x(p)n1,,x(p)nk) (4)

    where n,kN0, (x(i)k,x(i)k+1,,x(i)0)Ik+1i,i=1,2,,p.

    Define the map

    F:I(k+1)1×I(k+1)2××I(k+1)pI(k+1)1×I(k+1)2××I(k+1)p

    by

    F(W)=(f(1)0(W),f(1)1(W),,f(1)k(W),f(2)0(W),f(2)1(W),,
    ,f(2)k(W),,f(p)0(W),f(p)1(W),,f(p)k(W)),

    where

    W=(u(1)0,u(1)1,,u(1)k,u(2)0,u(2)1,,u(2)k,,u(p)0,u(p)1,,u(p)k)T,
    f(i)0(W)=f(i)(W),f(i)1(W)=u(i)0,,f(i)k(W)=u(i)k1,i=1,2,,p.

    Let

    Wn=(x(1)n,x(1)n1,,x(1)nk,x(2)n,x(2)n1,,x(2)nk,,x(p)n,x(p)n1,,x(p)nk)T.

    Then, we can easily see that system (4) is equivalent to the following system written in vector form

    Wn+1=F(Wn),nN0. (5)

    Definition 2.3. (see, [13]) Let (¯x(1),¯x(2),,¯x(p)) be an equilibrium point of the map F where f(i), i=1,2,,p are continuously differentiable functions at (¯x(1),¯x(2),,¯x(p)). The linearized system of (3) about the equilibrium point (¯x(1),¯x(2),,¯x(p)) is

    Xn+1=F(Xn)=BXn

    where Xn=(x(1)n,x(1)n1,,x(1)nk,x(2)n,x(2)n1,,x(2)nk,,x(p)n,x(p)n1,,x(p)nk)T and B is a Jacobian matrix of the system (3) about the equilibrium point (¯x(1),¯x(2),,¯x(p)).

    Theorem 2.4. (see, [13])

    1. If all the eigenvalues of the Jacobian matrix B lie in the open unit disk |λ|<1, then the equilibrium point ¯X of system (3) is asymptotically stable.

    2. If at least one eigenvalue of the Jacobian matrix B has absolute value greater than one, then the equilibrium point ¯X of system (3) is unstable.

    In this section, we discuss the behavior of positive solutions of system (3) via semi-cycle analysis method. It is easy to see that system (3) has a unique positive equilibrium point (¯x(1),¯x(2),,¯x(p))=(A+1,A+1,,A+1).

    Lemma 3.1. Let {(x(1)n,x(2)n,,x(p)n)}nm be a solution to system (3). Then, either {(x(1)n,x(2)n,,x(p)n)}nm consists of a single semi-cycle or {(x(1)n,x(2)n,,x(p)n)}nm oscillates about the equilibrium (¯x(1),¯x(2),,¯x(p))=(A+1,A+1,,A+1) with semi-cycles having at most m terms.

    Proof. Suppose that {(x(1)n,x(2)n,,x(p)n)}nm has at least two semi-cycles. Then, there exists n0m such that either

    x(j)n0<A+1x(j)n0+1 or x(j)n0+1<A+1x(j)n0,j=1,2,,p.

    We suppose the first case, that is, x(j)n0<A+1x(j)n0+1. The other case is similar and will be omitted. Assume that the positive semi-cycle beginning with the term (x(1)n0+1,x(2)n0+1,,x(p)n0+1) have m terms. In this case we have

    x(j)n0<A+1x(j)n0+m,j=1,2,,p.

    So, we get from system (3)

    x(j)n0+m+1=A+x(j+1)mod(p)n0x(j+1)mod(p)n0+m<A+1,j=1,2,,p.

    The Lemma is proved.

    Lemma 3.2. Let {(x(1)n,x(2)n,,x(p)n)}nm be a solution to system (3) which has m1 sequential semi-cycles of length one. Then, every semi-cycle after this point is of length one.

    Proof. Assume that there exists n0m such that either

    x(j)n0,x(j)n0+2,,x(j)n0+m1<A+1x(j)n0+1,x(j)n0+3,,x(j)n0+m,j=1,2,,p, (6)

    or

    x(j)n0+1,x(j)n0+3,,x(j)n0+m<A+1x(j)n0,x(j)n0+2,,x(j)n0+m1,j=1,2,,p. (7)

    We will prove the case (6). The case (7) Is identical and will not be included. According to system (3) we obtain

    x(j)n0+m+1=A+x(j+1)mod(p)n0x(j+1)mod(p)n0+m<A+1,j=1,2,,p,

    and

    x(j)n0+m+2=A+x(j+1)mod(p)n0+1x(j+1)mod(p)n0+m+1>A+1,j=1,2,,p,

    The result proceeds by induction. Thus, the proof is completed.

    Lemma 3.3. System (3) has no nontrivial periodic solutions of (not necessarily prime) period m.

    Proof. Suppose that

    (α(1)1,α(2)1,,α(p)1),(α(1)2,α(2)2,,α(p)2),,(α(1)m,α(2)m,,α(p)m),(α(1)1,α(2)1,,α(p)1),

    is a m-periodic solution of system (3). It is obvious then that for this solution,

    (x(1)nm,x(2)nm,,x(p)nm)=(x(1)n,x(2)n,,x(p)n),n0.

    So, the equilibrium solution (¯x(1),¯x(2),,¯x(p))=(A+1,A+1,,A+1) must be this solution. Thus, the proof is completed.

    Lemma 3.4. All non-oscillatory solutions of system (3) converge to the equilibrium (¯x(1),¯x(2),,¯x(p))=(A+1,A+1,,A+1).

    Proof. We assume there exists non-oscillatory solutions of system (3). We will prove this lemma for the case of a single positive semi-cycle, the situation is identical for a single negative semi-cycle, so it will be omitted. Assume that (x(1)n,x(2)n,,x(p)n)(¯x(1),¯x(2),,¯x(p)) for all nm. From system (3) we have

    x(j)n+1=A+x(j+1)mod(p)nmx(j+1)mod(p)nA+1,j=1,2,,p,

    So, we get

    A+1x(j)nx(j)nm,n0,j=1,2,,p (8)

    From (8), there exists δ(j)i fori=0,1,,m1 such that

    limn+x(j)nm+i=δ(j)i.

    Hence,

    (δ(1)0,δ(2)0,,δ(p)0),(δ(1)1,δ(2)1,,δ(p)1),,(δ(1)m1,δ(2)m1,,δ(p)m1)

    is a periodic solution of (not necessarily prime period) period m. But, from Lemma (3.3), we saw system (3) has no nontrivial periodic solutions of (not necessarily prime period) period m. Thus, the solution must be the equilibrium solution. So, the proof is over.

    Theorem 4.1. Suppose 0<A<1 and {(x(1)n,x(2)n,,x(p)n)}nm be a positive solution to system (3). Then the following statements hold.

    ⅰ): If m is odd, and 0<x(j)2k1<1, x(j)2k>11A for k=1m2,3m2,,0, then

    limn+x(j)2n=+,limn+x(j)2n+1=A.

    ⅱ): If m is odd, and 0<x(j)2k<1, x(j)2k1>11A for k=1m2,3m2,,0, then

    limn+x(j)2n=A,limn+x(j)2n+1=+.

    Proof. (ⅰ): From (3), for i=1,2,,p, we get

    x(i)1=A+x(i+1)mod(p)mx(i+1)mod(p)0<A+1x(i+1)mod(p)0<A+(1A)=1,x(i)2=A+x(i+1)mod(p)1mx(i+1)mod(p)1>A+x(i+1)mod(p)1m>x(i+1)mod(p)1m>11A.

    By induction, for n =0,1,2, and i=1,2,,p, we obtain

    x(i)2n1<1,x(i)2n>11A. (9)

    So, from (3) and (9), we have

    x(i)2n=A+x(i+1)mod(p)2n1mx(i+1)mod(p)2n1>A+x(i+1)mod(p)2n1m>2A+x(i+1)mod(p)2n3m>3A+x(i+1)mod(p)2n5m>

    So

    x(i)2n>nA+x(i+1)mod(p)0. (10)

    By limiting the inequality (10), we get

    limnx(i)2n=. (11)

    On the other hand, from(3), (9) and (11), we get

    limnx(i)2n+1=limn(A+x(i+1)mod(p)2nmx(i+1)mod(p)2n)=A.

    (ⅱ): The proof is similar to the proof of (ⅰ).

    Open Problem. Investigate the asymptotic behavior of the system (3) when m is even.

    Lemma 4.2. Suppose A=1. Then every positive solution of the system (3) is bounded and persists.

    Proof. Let {(x(1)n,x(2)n,,x(p)n)}nm be a positive solution to system (3). Then, it is clear that for n1, x(j)n>A=1,j=1,2,,p. So, we get

    x(j)i[L,LL1],i=1,2,,m+1,j=1,2,,p,

    where

    L=min{α,ββ1}>1,α=min1jm+1{x(1)j,x(2)j,,x(p)j},
    β=max1jm+1{x(1)j,x(2)j,,x(p)ji}.

    So, we get

    L=1+LL/(L1)x(j)m+2=1+x(j+1)mod(p)1x(j+1)mod(p)m+1LL1,

    thus, the following is obtained

    Lx(j)mLL1.

    By induction, we get

    x(j)i[L,LL1],j=1,2,,p,i=1,2, (12)

    Theorem 4.3. Suppose A=1 and {(x(1)n,x(2)n,,x(p)n)}nm be a positive solution to system (3). Then

    lim infn+x(i)n=lim infn+x(j)n,i,j=1,2,,p,lim supn+x(i)n=lim supn+x(j)n,i,j=1,2,,p.

    Proof. From (12), we can set

    Li=limnsupx(i)n,mi=limninfx(i)n,i=1,2,,p. (13)

    We first prove the theorem for p=2. From system (3), we have

    L11+L2m2,L21+L1m1,m11+m2L2,m21+m2L2,

    which implies

    L1m2m2+L2m1L2m1+L1m2L1

    thus, the following equalities are obtained

    m2+L2=m1+L1,L1m2=m1L2.

    So, we get that m1=m2 and L1=L2. Now we suppose that

    Li=Lj,mi=mj,i,j=1,2,,p1,

    From system (3), we have

    Lp11+Lpmp,Lp1+Lp1mp1,mp11+mpLp,mp1+mpLp,

    hence, we get

    Lp1mpmp+Lpmp1Lpmp1+Lp1mpLp1,

    consequently, the following equalities are obtained

    mp+Lp=mp1+Lp1,Lp1mp=mp1Lp.

    So, we get that mp=mp1 and Lp=Lp1. Thus, the proof completes.

    Theorem 4.4. Assume that A>1. Then, the unique positive equilibrium (¯x(1),¯x(2),,¯x(p))=(A+1,A+1,,A+1) of system (3) is locally asymptotically stable.

    Proof. The linearized equation of system (3) about the equilibrium point (¯x(1),¯x(2),,¯x(p)) is

    Xn+1=BXn

    where Xn=(x(1)n,x(1)n1,,x(1)nm,x(2)n,x(2)n1,,x(2)nm,,x(p)n,x(p)n1,,,x(p)nm)t, and B=(bij), 1i,jpm+p is an (pm+p)×(pm+p) matrix such that

    B=(JAOOOOOJAOOOOOJAOOOOOOJAAOOOOJ)

    where A,J and O are (m+1)×(m+1) matrix defined as follows

    J=(000010000010),O=(000000000000), (14)
    A=(1A+1001A+100000000). (15)

    Let λ1,λ2,,λpm+p denote the eigenvalues of matrix B and let

    D=diag(d1,d2,,dpm+p)

    be a diagonal matrix where d1=dm+2=d2m+3==d(p1)m+p=1, dk=dm+1+k=1kε for k{1,2,,p2(m+1)}. Since A>1, we can take a positive number ε such that

    0<ε<A1(m+1)(A+1). (16)

    It is obvious that D is an invertible matrix. Computing matrix DBD1, we get

    DBD1=(J(1)A(1)OOOOOJ(2)A(2)OOOOOJ(3)A(3)OOOOOOJ(p1)A(p1)A(p)OOOOJ(p)),

    where

    J(j)=(0000d(j1)m+j+1d(j1)m+j00000d(j1)m+m+jd(j1)m+m+j10),j=0,1,,p,
    A(j)=(1A+1djdjm+j+1001A+1djdjm+j+100000000),j=0,1,,p1,

    and

    A(p)=(1A+1d(p1)m+pd1001A+1d(p1)m+pdm+100000000).

    From d1>d2>>dp2(m+1) and dp2(m+1)+1>dp2(m+1)+2>>dpm+p we can get that

    A(p)=(1A+1d(p1)m+pd1001A+1d(p1)m+pdm+100000000).

    Moreover, from A>1 and (16) we have

    1A+1+1(1(m+1)ε)(A+1)<1(1(m+1)ε)(A+1)+1(1(m+1)ε)(A+1)<2(1(m+1)ε)(A+1)<1.

    It is common knowledge that B has the same eigenvalues as DBD1, we have that

    1A+1+1(1(m+1)ε)(A+1)<1(1(m+1)ε)(A+1)+1(1(m+1)ε)(A+1)<2(1(m+1)ε)(A+1)<1.

    We have that all eigenvalues of B lie inside the unit disk. According to Theorem (2.4) we obtain that the unique positive equilibrium (¯x(1),¯x(2),,¯x(p))=(A+1,A+1,,A+1) is locally asymptotically stable. Thus, the proof is completed.

    To prove the global stability of the positive equilibrium, we need the following lemma.

    Lemma 4.5. Suppose A>1. Then every positive solution of the system (3) is bounded and persists.

    Proof. Let {(x(1)n,x(2)n,,x(p)n)}nm be a positive solution to system (3). Then, it is clear that for n1, x(j)n>A>1,j=1,2,,p. So, we get

    x(j)i[L,LLA],i=1,2,,m+1,j=1,2,,p,

    where

    L=min{α,ββ1}>1,α=min1jm+1{x(1)j,x(2)j,,x(p)j},
    β=max1jm+1{x(1)j,x(2)j,,x(p)ji}.

    So, we get

    L=A+LL/(LA)x(j)m+2=A+x(j+1)mod(p)1x(j+1)mod(p)m+1LL1,

    thus, the following is obtained

    Lx(j)mLL1.

    By induction, we get

    x(j)i[L,LL1],j=1,2,,p,i=1,2, (17)

    Theorem 4.6. Assume that A>1. Then the positive equilibrium of system (3) is globally asymptotically stable.

    Proof. Let {(x(1)n,x(2)n,,x(p)n)}nm be a solution of system (3). By Theorem (4.4) we need only to prove that the equilibrium point (A+1,A+1,,A+1) is global attractor, that is

    limn(x(1)n,x(2)n,,x(p)n)=(A+1,A+1,,A+1).

    To do this, we prove that for i=1,2,,p, we have

    limnx(i)n=A+1.

    From Lemma (4.5), we can set

    Li=limnsupx(i)n,mi=limninfx(i)n,i=1,2,,p. (18)

    So, from (3) and (13), we have

    LiA+L(i+1)mod(p)m(i+1)mod(p),miA+m(i+1)mod(p)L(i+1)mod(p). (19)

    We first prove the theorem for p=2. From (19), we get

    AL1+m1L1m2Am2+L2,AL2+m2L2m1Am1+L1.

    So,

    AL1+m1(Am1+L1)Am2+L2(AL2+m2),

    hence

    (A1)(L1m1+L2m2)0,

    since A>1, It follows that

    L1m1+L2m2=0,

    we know that L1m10 and L2m20, so we obtain L1=m1 and L2=m2. Now we assume that the theorem holds for p1, that is Li=mi,i=1,2,,p1 and prove the theorem for p. From (19), we have

    ALp+mpLpm1Am1+L1,AL1+m1L1mpAmp+Lp.

    So,

    ALp+mp(Amp+Lp)Am1+L1(AL1+m1),

    Thus, the following inequality is obtained

    (A1)(Lpmp+L1m1)0,

    since A>1, L1m10 and Lpmp0, we obtain Lp=mp, it signify that

    Li=mi,=1,2,,p.

    Therefore every positive solution {(x(1)n,x(2)n,,x(p)n)}n1 of system (3) tends to (A+1,A+1,,A+1) as n+.

    In this section, we estimate the rate of convergence of a solution that converges to the equilibrium point (¯x(1),¯x(2),,¯x(p))=(A+1,A+1,,A+1) of the system (3) in the region of parameters described by A>1. We give precise results about the rate of convergence of the solutions that converge to the equilibrium point by using Perron's theorems. The following result gives the rate of convergence of solutions of a system of difference equations

    Xn+1=(A+Bn)Xn (20)

    where Xn is a (pm+p)-dimensional vector, AC(pm+p)×(pm+p) is a constant matrix and B:Z+C(pm+p)×(pm+p) is a matrix function satisfying

    Bn0, when n (21)

    where . indicates any matrix norm which is associated with the vector norm ..

    Theorem 5.1. (Perron's first Theorem, see [16]) Suppose that condition (21) holds. If Xn is a solution of (20), then either Xn=0 for all largen or

    ρ=limn+Xn+1Xn

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

    Theorem 5.2. (Perron's second Theorem, see [16]) Suppose that condition (21) holds. If Xn is a solution of (20), then either Xn=0 for all largen or

    ρ=limn+(Xn)1n

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

    Theorem 5.3. Assume that a solution {(x(1)n,x(2)n,,x(p)n)}nm of system (3) converges to the equilibrium (¯x(1),¯x(2),,¯x(p)) which is globally asymptotically stable. Then, the error vector

    en=(e(1)ne(1)n1e(1)nme(p)ne(p)n1e(p)nm)=(x(1)n¯x(1)x(1)n1¯x(1)x(1)nm¯x(1)x(p)n¯x(p)x(p)n1¯x(p)x(p)nm¯x(p))

    of every solution of system (3) satisfies both of the following asymptotic relations:

    limn+en+1en=|λiJF((¯x(1),¯x(2),,¯x(p)))|,i=1,2,,m
    limn+(en)1n=|λiJF((¯x(1),¯x(2),,¯x(p)))|,i=1,2,,m

    where |λiJF((¯x(1),¯x(2),,¯x(p)))| is equal to the modulus of one the eigenvalues of the Jacobian matrix evaluated at the equilibrium point (¯x(1),¯x(2),,¯x(p)).

    Proof. First, we will find a system that satisfies the error terms. The error terms are given as

    x(j)n+1¯x(j)=mi=0(j)A(1)i(x(1)ni¯x(1))+mi=0(j)A(2)i(x(2)ni¯x(2))++mi=0(j)A(1)i(x(p)ni¯x(p)), (22)

    for i=1,2,,m,j=1,2,,p. Set

    e(j)n=x(j)n¯x(j),j=1,2,,p

    Then, system (22) can be written as

    e(j)n+1=mi=0(j)A(1)ie(1)ni+mi=0(j)A(2)ie(2)ni++mi=0(j)A(1)ie(p)ni

    where

    e(j)n+1=mi=0(j)A(1)ie(1)ni+mi=0(j)A(2)ie(2)ni++mi=0(j)A(1)ie(p)ni

    and the others parameters (k)A(j)i are equal zero.

    If we consider the limiting case, It is obvious then that

    e(j)n+1=mi=0(j)A(1)ie(1)ni+mi=0(j)A(2)ie(2)ni++mi=0(j)A(1)ie(p)ni

    That is

    e(j)n+1=mi=0(j)A(1)ie(1)ni+mi=0(j)A(2)ie(2)ni++mi=0(j)A(1)ie(p)ni

    where α(i)n,β(i)n0 when n. Now we have the following system of the form (20)

    en+1=(A+Bn)en

    where en=(e(1)n,e(1)n1,,e(1)nm,e(2)n,e(2)n1,,e(2)nm,,e(p)n,e(p)n1,,e(p)nm)t and

    A=JF((¯x(1),¯x(2),,¯x(p)))=(JA(1)nOOOOOJA(2)nOOOOOJA(3)nOOOOOOJA(p1)nA(p)nOOOOJ)
    Bn=(JAOOOOOJAOOOOOJAOOOOOOJAAOOOOJ)

    where

    A(j)n=(α(j)n00β(j)n00000000),j=1,2,,p.

    and A,J and O are the (m+1)×(m+1) matrix defined in (14) and (15).

    Bn0 when n. Therefore, the limiting system of error terms can be written as

    en+1=(JAOOOOOJAOOOOOJAOOOOOOJAAOOOOJ)(e(1)ne(1)n1e(1)nme(p)ne(p)n1e(p)nm)

    and Bn0 when n. This system is exactly the linearized system of (3) evaluated at the equilibrium point (¯x(1),¯x(2),,¯x(p)). From Theorems (5.1) and (5.2), the result follows.

    In this section we will consider several interesting numerical examples to verify our theoretical results. These examples shows different types of qualitative behavior of solutions of the system (3). All plots in this section are drawn with Matlab.

    Exemple 6.1. Let m=1 and p=10 in system (3), then we obtain the system

    x(1)n+1=1.2+x(2)n1x(2)n,x(2)n+1=A+x(3)n1x(3)n,,x(10)n+1=1.2+x(1)n1x(1)n,nN0 (23)

    with A=1.2>1 and the initial values x(1)1=3.3,x(1)0=2,x(2)1=1.1,x(2)0=0.3,x(3)1=2.3,x(3)0=1.5,x(4)1=0.5,x(4)0=2,x(5)1=1.9,x(5)0=0.8,x(6)1=4,x(6)0=1.3,x(7)1=1.2,x(7)0=1.3,x(8)1=2.1,x(8)0=2.3,x(9)1=3.6,x(9)0=0.2,x(10)1=2.3,x(10)0=1.1. Then the positive equilibrium point (¯x(1),¯x(2),,¯x(10))= (2.2,2.2,,2.2) of system (23)) is globally asymptotically stable (see Figure (1), Theorem (4.4)).

    Figure 1.  The plot of system (23) with A=1.2>1.

    Exemple 6.2. Consider the system (23) with A=1 and the initial values x(1)1=0.3,x(1)0=1.1,x(2)1=1.3,x(2)0=0.3,x(3)1=1.4,x(3)0=1.5,x(4)1=0.5,x(4)0=2,x(5)1=1.9,x(5)0=0.8,x(6)1=4,x(6)0=1.3,x(7)1=1.4,x(7)0=1.3,x(8)1=0.1,x(8)0=1.1,x(9)1=1.6,x(9)0=1.7,x(10)1=1.9,x(10)0=1.1. Then the solution oscillates about the positive equilibrium point (¯x(1),¯x(2),,¯x(10))=(2,2,,2) of system (23) with semi-cycles having at most five terms. Also, the equilibrium is not globally asymptotically stable (see Figure (2), Theorem 4.2).

    Figure 2.  The plot of system (23) with A=1.

    Exemple 6.3. Consider the system (23) with A=0.9 and the initial values x(1)1=1.2,x(1)0=0.7,x(2)1=1.2,x(2)0=2.3,x(3)1=0.4,x(3)0=1.1,x(4)1=0.8,x(4)0=8,x(5)1=1.3,x(5)0=1.8,x(6)1=2.6,x(6)0=0.9,x(7)1=1.4,x(7)0=1.1,x(8)1=0.1,x(8)0=1.4,x(9)1=0.9,x(9)0=1.3,x(10)1=1.2,x(10)0=2.1. Then the positive equilibrium point (¯x(1),¯x(2),¯x(3),¯x(4))=(1.9,1.9,,1.9) of system (23) is not globally asymptotically stable. Also, this solution is unbounded solution see Figure (3), Theorem 4.2).

    Figure 3.  The plot of system (23) with A=0.9<1.

    Exemple 6.4. Let m=5 and p=4 in system (3), then we obtain the system

    x(1)n+1=A+x(2)n5x(2)n,x(2)n+1=A+x(3)n5x(3)n,x(3)n+1=A+x(4)n5x(4)n,x(4)n+1=A+x(1)n5x(1)n,nN0 (24)

    with A=1.4>1 and the initial values x(1)5=1.2,x(1)4=0.8,x(1)3=1.9,x(1)2=2.2,x(1)1=0.3,x(1)0=1.7,x(2)5=1.3,x(2)4=2.4,x(2)3=1.2,x(2)2=0.5,x(2)1=1.6,x(2)0=2.3,x(3)5=0.4,x(3)4=1.1,x(3)3=1.4,x(3)2=2.1,x(3)1=0.3,x(3)0=1.1,x(4)5=0.8,x(4)4=1.2,x(4)3=1.8,x(4)2=3.1,x(4)1=0.7,x(4)0=1.8,. Then the positive equilibrium point (¯x(1),¯x(2),¯x(3),¯x(4))=(2.4,2.4,2.4,2.4) of system (24)) is globally asymptotically stable (see Figure (4), Theorem (4.4)).

    Figure 4.  The plot of system (24) with A=1.4>1.

    Exemple 6.5. Consider the system (24) with A=1 and the initial values x(1)5=0.4,x(1)4=1.3,x(1)3=2.9,x(1)2=1.2,x(1)1=0.8,x(1)0=1.2,x(2)5=0.3,x(2)4=1.4,x(2)3=1.3x(2)2=0.5,x(2)1=1.6,x(2)0=2.1,x(3)5=1.3,x(3)4=2.1,x(3)3=1.4,x(3)2=2.1,x(3)1=0.3,x(3)0=1.5,x(4)5=0.6,x(4)4=1.2,x(4)3=1.3,x(4)2=0.8,x(4)1=1.7,x(4)0=0.1,. Then the solution oscillates about the positive equilibrium point (¯x(1),¯x(2),¯x(3),¯x(4))=(2.4,2.4,2.4,2.4) of system (24) with semi-cycles having at most five terms. Also, the equilibrium is not globally asymptotically stable (see Figure (5), Theorem 4.2).

    Figure 5.  The plot of system (24) with A=1.

    Exemple 6.6. Consider the system (24) with A=0.7 and the initial values x(1)5=1.3,x(1)4=0.9,x(1)3=2.1,x(1)2=0.9,x(1)1=0.7,x(1)0=2.2,x(2)5=1.3,x(2)4=0.4,x(2)3=1.3x(2)2=1.5,x(2)1=1.2,x(2)0=1.1,x(3)5=1.7,x(3)4=1.6,x(3)3=1.5,x(3)2=2.3,x(3)1=0.9,x(3)0=1.5,x(4)5=0.6,x(4)4=1.4,x(4)3=2.3,x(4)2=3.1,x(4)1=2.7,x(4)0=1.9. Then the positive equilibrium point (¯x(1),¯x(2),¯x(3),¯x(4))=(1.7,1.7,1.7,1.7) of system (23) is not globally asymptotically stable. Also, this solution is unbounded solution see Figure (6), Theorem 4.2).

    Figure 6.  The plot of system (24) with A=0.7<1.

    In the paper, we studied the global behavior of solutions of system (3) composed by p rational difference equations. More exactly, we here study the global asymptotic stability of equilibrium, the rate of convergence of positive solutions. Also, we present some results about the general behavior of solutions of system (3) and some numerical examples are carried out to support the analysis results. Our system generalized the equations and systems studied in [2,8] and [20].

    The findings suggest that this approach could also be useful for extended to a system with arbitrary constant different parameters, or to a system with a nonautonomous parameter, or to a system with different parameters and arbitrary powers. So, we will give the following some important open problems for difference equations theory researchers.

    Open Problem 1. study the dynamical behaviors of the system of difference equations

    x(1)n+1=A1+x(2)nmx(2)n,x(2)n+1=A2+x(3)nmx(3)n,,x(p)n+1=Ap+x(1)nmx(1)n,n,m,pN0

    where Ai, i=1,2,,p are nonnegative constants and x(j)m,x(j)m+1,,x(j)1,x(j)0,j=1,2,,p are positive real numbers.

    Open Problem 2. study the dynamical behaviors of the system of difference equations

    x(1)n+1=αn+x(2)nmx(2)n,x(2)n+1=αn+x(3)nmx(3)n,,x(p)n+1=αn+x(1)nmx(1)n,n,m,pN0

    where αn is a sequence (this sequence can be chosen as convergent, periodic or bounded), and x(j)m,x(j)m+1,,x(j)1,x(j)0,j=1,2,,p are positive real numbers.

    Open Problem 3. study the dynamical behaviors of the system of difference equations

    x(1)n+1=A1+(x(2)nm)p1(x(2)n)q1,x(2)n+1=A2+(x(3)nm)p2(x(3)n)q2,,x(p)n+1=Ap+(x(1)nm)pp(x(1)n)qp,

    wheren,m,pN0, Ai, i=1,2,,p are nonnegative constants, the parameters pi,qi, i=1,2,,p are non-negative and x(j)m,x(j)m+1,,x(j)1,x(j)0,j=1,2,,p are positive real numbers.

    This work was supported by Directorate General for Scientific Research and Technological Development (DGRSDT), Algeria.



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