Research article

Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials

  • Received: 23 March 2023 Revised: 24 April 2023 Accepted: 27 April 2023 Published: 05 May 2023
  • 35K58, 35K65

  • In this article, we investigate the initial-boundary value problem for a class of finitely degenerate semilinear parabolic equations with singular potential term. By applying the Galerkin method and Banach fixed theorem we establish the local existence and uniqueness of the weak solution. On the other hand, by constructing a family of potential wells, we prove the global existence, the decay estimate and the finite time blow-up of solutions with subcritical or critical initial energy.

    Citation: Huiyang Xu. Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials[J]. Communications in Analysis and Mechanics, 2023, 15(2): 132-161. doi: 10.3934/cam.2023008

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  • In this article, we investigate the initial-boundary value problem for a class of finitely degenerate semilinear parabolic equations with singular potential term. By applying the Galerkin method and Banach fixed theorem we establish the local existence and uniqueness of the weak solution. On the other hand, by constructing a family of potential wells, we prove the global existence, the decay estimate and the finite time blow-up of solutions with subcritical or critical initial energy.



    Based on the experimental data which were observed and summarized by Nicholson [1], Gurney et al. [2] presented a classic biological dynamical system model

    N(t)=δN(t)+pN(tτ)eaN(tτ). (1.1)

    Here, N(t) is the size of the population at time t, p is the maximum per capita daily egg production, 1a is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time. The research on the Nicholson's blowfly model and its modifications has realized a remarkable progress in the past fifty years and an abundance of results on the existence of positive solutions, persistence, oscillation, stability, periodic solutions, almost periodic solutions, pseudo almost periodic solutions, etc. (see [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]) have been obtained. Furthermore, Berezansky et al. [22] systematically collected and compared the results in the above-mentioned studies and put forward several open problems that have been partially answered in recent works [23,24,25,26,27,28,29,30], such as Nicholson's blowfly model with impulsive perturbation, harvesting term and nonlinear density-dependent mortality term. On the other hand, Liu [31] considered the following cooperative Nicholson's blowfly equation with a patch structure:

    xi(t)=nj=1aijxj(t)+βxi(tri)exi(tri)dixi(t),i=1,2,,n. (1.2)

    Also, Berezansky et al. [32] studied the following cooperative Nicholson-type delay differential system

    {x1(t)=a1x1(t)+b1x2(t)+c1x1(tτ)ex1(tτ),x2(t)=a2x2(t)+b2x1(t)+c2x2(tτ)ex2(tτ). (1.3)

    In this paper, we will discuss the following competitive and cooperative Nicholson's blowfly system

    {x1(t)=δ1(t)x1(t)+a1(t)x2(t)+nj=1c1j(t)x1(tτ1j(t))eb1j(t)x1(tτ1j(t))k1(t)x1(t)x2(t),x2(t)=δ2(t)x2(t)+a2(t)x1(t)+nj=1c2j(t)x2(tτ2j(t))eb2j(t)x2(tτ2j(t))k2(t)x1(t)x2(t) (1.4)

    where δi,ai,bij,cij,τij,ki:R1[0,+) are almost periodic functions i=1,2,j=1,2,,n. x1(t),x2(t) denote the sizes of the different populations at time t, cij denotes the maximum per capita daily egg production of xi, 1bij represents the size at which the population xi reproduces at its maximum rate, δi is the per capita daily adult death rate of xi, and τij denotes the generation time of xi. a1(t) represents the rate at which x2 contributes to x1 and a2(t) represents the rate at which x1 contributes to x2 at time t. ki(t) denotes the death rate of xi due to the competition between x1 and x2 at time t. The cooperative terms a1(t)x2(t) and a2(t)x1(t) and the competitive terms k1(t)x1(t)x2(t) and k1(t)x1(t)x2(t) reflect the degree at which they cooperate and compete with each other, respectively.

    Recently, there have been wide-ranging results obtained on competitive and cooperative systems added the literatures [33,34,35,36,37,38] due to its extensive applicability. However, to the best of our knowledge, few results are presented in literatures about the existence of positive almost periodic solutions for competitive and cooperative Nicholson's blowfly system. In the real world, since the competition is inevitable and there exists an almost periodically changing environment, it is worth studying the positive almost periodic solution for competitive and cooperative Nicholson's blowfly system. Based on the above idea, we shall consider the existence and exponential convergence of positive almost periodic solutions of system (1.4) which possesses obvious dynamics significance.

    For convenience, we introduce some notations. Throughout this paper, given a bounded continuous function g defined on R1, let g+ and g be defined as follows:

    g=inftR1g(t),g+=suptR1g(t).

    It will be assumed that

    δi>0,bij>0,ri=max1jn{τ+ij}>0i=1,2,j=1,2,,n. (1.5)

    Let Rn(Rn+) be the set of all (nonnegative) real vectors; we will use x=(x1,x2,,xn)TRn to denote a column vector, in which the symbol (T) denotes the transpose of a vector. we let |x| denote the absolute-value vector given by |x|=(|x1|,|x2|,,|xn|)T and define ||x||=max1in|xi|. For a matrix A=(aij)n×n, AT, A1, |A| and ρ(A) denote the transpose, inverse, the absolute-value matrix and the spectral radius of A respectively. A matrix or vector A0 means that all entries of A are greater than or equal to zero. A>0 can be defined similarly. For matrices or vectors A and B, AB (resp. A>B) means that AB0 (resp. AB>0). Denote C=2i=1C([ri,0],R1)  and  C+=2i=1C([ri,0],R1+) as Banach spaces equipped with the supremum norm defined by

    ||φ||=suprit0max1i2|φi(t)|   for all  φ(t)=(φ1(t),φ2(t))TC(orC+).

    If xi(t) is defined on [t0ri,ν) with t0,νR1 and i=1,2, then we define xtC as xt=(x1t,x2t)T where xit(θ)=xi(t+θ) for all θ[ri,0] and i=1,2.

    The initial conditions associated with system (1.4) are of the following form:

    xt0=φ, φ=(φ1,φ2)TC+. (1.6)

    We write xt(t0,φ)(x(t;t0,φ)) for a solution of the initial value problems (1.4) and (1.6). Also, let [t0,η(φ)) be the maximal right-interval of the existence of xt(t0,φ).

    The remaining part of this paper is organized as follows. In Section 2, we shall give some definitions and preliminary results. In Section 3, we shall derive sufficient conditions for checking the existence, uniqueness and exponential convergence of the positive almost periodic solution of (1.4). In Section 4, we shall give an example and numerical simulation to illustrate the results obtained in the previous section.

    In this section, some lemmas and definitions will be presented, which are of importance in proving our main results in Section 3.

    Definition 2.1. [39,40] Let u(t):R1Rn be continuous in t. u(t) is said to be almost periodic on R1, if for any ε>0, the set T(u,ε)={δ:|u(t+δ)u(t)|<ε  for all  tR1} is relatively dense, i.e., for any ε>0, it is possible to find a real number l=l(ε)>0, such that for any interval with length l(ε), there exists a number δ=δ(ε) in this interval such that |u(t+δ)u(t)|<ε,  for all  tR1.

    Definition 2.2. [39,40] Let xRn and Q(t) be an n×n continuous matrix defined on R1. The linear system

    x(t)=Q(t)x(t) (2.1)

    is said to admit an exponential dichotomy on R1 if there exist positive constants k,α, projection P and the fundamental solution matrix X(t) of (2.1) satisfying

    X(t)PX1(s)keα(ts)for allts,X(t)(IP)X1(s)keα(st)for all ts.

    Definition 2.3. A real n×n matrix K=(kij) is said to be an M-matrix if kij0,i,j=1,,n,ij and K10.

    Set

    B={φ|φ=(φ1(t),φ2(t))T  is an almost periodic vector function on R1}.

    For any φB, we define an induced module φB=suptR1max1i2|φi(t)|; then, B is a Banach space.

    Lemma 2.1. [39,40] If the linear system (2.1) admits an exponential dichotomy, then the almost periodic system

    x(t)=Q(t)x+g(t) (2.2)

    has a unique almost periodic solution x(t), and

    x(t)=tX(t)PX1(s)g(s)ds+tX(t)(IP)X1(s)g(s)ds. (2.3)

    Lemma 2.2. [39,40] Let ci(t) be an almost periodic function on R1 and

    M[ci]=limT+1Tt+Ttci(s)ds>0,i=1,2,,n.

    Then the linear system

    x(t)=diag(c1(t),c2(t),,cn(t))x(t)

    admits an exponential dichotomy on R1.

    Lemma 2.3. [41,42] Let A0 be an n×n matrix and ρ(A)<1; then, (InA)10, where In denotes the identity matrix of size n.

    Lemma 2.4. Suppose that there exist positive constants Ei1 and Ei2 such that

    Ei1>Ei2,nj=1c+1jδ1b1je+a+1E21δ1<E11,nj=1c+2jδ2b2je+a+2E11δ2<E21, (2.4)
    a1δ+1E22+nj=1c1jδ+1E11eb+1jE11k+1δ+1E11E21>E121min1jmb1j, (2.5)
    a2δ+2E12+nj=1c2jδ+2E21eb+2jE21k+2δ+2E11E21>E221min1jmb2j, (2.6)

    where i=1,2. Let

    C0:={φ|φC,Ei2<φi(t)<Ei1,  for all  t[ri,0], i=1,2}.

    Moreover, assume that x(t;t0,φ) is the solution of (1.4) with φC0. Then,

    Ei2<xi(t;t0,φ)<Ei1,  for all   t[t0,η(φ)),i=1,2, (2.7)

    and η(φ)=+.

    Proof. We rewrite the system (1.4) as

    x(t)=f(t,xt),

    where x(t)=(x1(t),x2(t))T, f(t,φ)=(f1(t,φ),f2(t,φ))T, f1(t,φ)=δ1(t)φ1(0)+a1(t)φ2(0)+nj=1c1j(t)φ1(τ1j(t))eb1j(t)φ1(τ1j(t))k1(t)φ1(0)φ2(0), f2(t,φ)=δ2(t)φ2(0)+a2(t)φ1(0)+nj=1c2j(t)φ2(τ2j(t))eb2j(t)φ2(τ2j(t))k2(t)φ1(0)φ2(0), φ(t)=(φ1(t),φ2(t))TC0. It is obvious that f:R1×C0R2 is continuous and C0C is open. Let ϕ,ψC0; then, considering that supu0|1ueu|=1 and the inequality

    |xexyey|=|1(x+θ(yx))ex+θ(yx)||xy|       |xy|  where  x,y[0,+),0<θ<1, (2.8)

    we obtain

    |f1(t,ϕ)f1(t,ψ)|δ1(t)|ϕ1(0)ψ1(0)|+a1(t)|ϕ2(0)ψ2(0)|+nj=1c1j(t)×|ϕ1(τ1j(t))eb1j(t)ϕ1(τ1j(t))ψ1(τ1j(t))eb1j(t)ψ1(τ1j(t))|+k1(t)|ϕ1(0)ϕ2(0)ψ1(0)ψ2(0)|(δ+1+a+1)||ϕψ||+nj=1c1j(t)b1j(t)×|b1j(t)ϕ1(τ1j(t))eb1j(t)ϕ1(τ1j(t))b1j(t)ψ1(τ1j(t))eb1j(t)ψ1(τ1j(t))|+k+1(|ϕ1(0)||ϕ2(0)ψ2(0)|+|ψ2(0)||ϕ1(0)ψ1(0)|)(δ+1+a+1)||ϕψ||+nj=1c+1jb1j(t)×|b1j(t)ϕ1(τ1j(t))b1j(t)ψ1(τ1j(t))|+k+1(E11||ϕψ||+E21||ϕψ||)(δ+1+a+1+nj=1c+1j+k+1E11+k+1E21)||ϕψ||. (2.9)

    In the same way, we also get

    |f2(t,ϕ)f2(t,ψ)|(δ+2+a+2+nj=1c+2j+k+2E11+k+2E21))||ϕψ||. (2.10)

    Then (2.9) and (2.10) imply that f satisfies the Lipschitz condition in its second argument on each compact subset of R1×C0. Moreover, since φC+, it is easy to get that xt(t0,φ)C+forallt[t0,η(φ)) by using Theorem 5.2.1 from [43, p. 81]. Set x(t)=x(t;t0,φ) for all  t[t0,η(φ)).

    We claim that

    0xi(t)<Ei1  for all  t[t0,η(φ)),i=1,2. (2.11)

    By contradiction, assume that (2.11) does not hold. Then, there exists t1(t0,η(φ)) such that one of the following two cases must occur:

    (1)x1(t1)=E11,0xi(t)<Ei1for allt[t0ri,t1),i=1,2; (2.12)
    (2)x2(t1)=E21,0xi(t)<Ei1for allt[t0ri,t1),i=1,2. (2.13)

    In the sequel, we consider two cases.

    Case (ⅰ). Suppose that (2.12) holds. Considering the derivative of x1(t), together with (2.4) and the fact that supu0ueu=1e, we have

    0x1(t1)=δ1(t1)x1(t1)+a1(t1)x2(t1)+nj=1c1j(t1)x1(t1τ1j(t1))eb1j(t1)x1(t1τ1j(t1))k1(t1)x1(t1)x2(t1)δ1x1(t1)+a+1x2(t1)+nj=1c+1jb1j1eδ1E11+a+1E21+nj=1c+1jb1j1e=δ1(E11+nj=1c+1jδ1b1je+a+1E21δ1)<0,

    which is a contradiction.

    Case (ⅱ). Suppose that (2.13) holds. Considering the derivative of x2(t), together with (2.4) and the fact that supu0ueu=1e, we have

    0x2(t1)=δ2(t1)x2(t1)+a2(t1)x1(t1)+nj=1c2j(t1)x2(t1τ1j(t1))eb2j(t1)x2(t1τ1j(t1))k2(t1)x1(t1)x2(t1)δ2x2(t1)+a+2x1(t1)+nj=1c+2jb2j1eδ2E21+a+2E11+nj=1c+2jb2j1e=δ2(E21+nj=1c+2jδ2b2je+a+2E11δ2)<0,

    which is a contradiction. Together with Cases (ⅰ) and (ⅱ), (2.11) holds for t[t0,η(φ)).

    We next show that

    xi(t)>Ei2,  for all  t(t0,η(φ)),i=1,2. (2.14)

    Suppose, for the sake of contradiction, that (2.14) does not hold. Then, there exists t2(t0,η(φ)) such that one of the following two cases must occur:

    (1)x1(t2)=E12,Ei2<xi(t)<Ei1for allt[t0ri,t2),i=1,2; (2.15)
    (2)x2(t2)=E22,Ei2<xi(t)<Ei1for allt[t0ri,t2),i=1,2. (2.16)

    If (2.15) holds, from (2.5), (2.6), (2.11) and (2.15), we get

    Ei2<xi(t)<Ei1,  b+ijxi(t)b+ijEi2b+ij1min1jnbij1, (2.17)

    for all t[t0ri,t2),i=1,2, j=1,2,n. Calculating the derivative of x1(t), together with (2.5) and the fact that min1uκueu=κeκ, (1.4), (2.15) and (2.17) imply that

    0x1(t2)=δ1(t2)x1(t2)+a1(t2)x2(t2)+nj=1c1j(t2)x1(t2τ1j(t2))eb1j(t2)x1(t2τ1j(t2))k1(t2)x1(t2)x2(t2)δ+1x1(t2)+a1x2(t2)+nj=1c1j(t2)b+1jb+1jx1(t2τ1j(t2))eb+1jx1(t2τ1j(t2))k+1x1(t2)x2(t2)>δ+1E12+a1E22+nj=1c1jE11eb+1jE11k+1E11E21=δ+1(E12+a1δ+1E22+nj=1c1jδ+1E11eb+1jE11k+1δ+1E11E21)>0,

    which is absurd and implies that (2.14) holds. If (2.16) holds, we can prove that (2.14) also holds in a similar way.

    It follows from (2.11) and (2.14) that (2.7) is true. From Theorem 2.3.1 in [44], we easily obtain η(φ)=+. This completes the proof.

    Theorem 3.1. Let (2.4) – (2.6) hold. Moreover, suppose that

    ρ(A1B)<1. (3.1)

    where

    A=(δ100δ2),B=(nj=1c+1je2+k+1E21a+1+k+1E11a+2+k+2E21nj=1c+2je2+k+2E11).

    Then, there exists a unique positive almost periodic solution of system (1.4) in the region B={φ|φB,Ei2φi(t)Ei1,  for all  tR1,i=1,2,,n}.

    Proof. For any ϕB, we consider the following auxiliary system

    {x1(t)=δ1(t)x1(t)+a1(t)ϕ2(t)+nj=1c1j(t)ϕ1(tτ1j(t))eb1j(t)ϕ1(tτ1j(t))k1(t)ϕ1(t)ϕ2(t)x2(t)=δ2(t)x2(t)+a2(t)ϕ1(t)+nj=1c2j(t)ϕ2(tτ2j(t))eb2j(t)ϕ2(tτ2j(t))k2(t)ϕ1(t)ϕ2(t). (3.2)

    Since M[δi]>0 (i=1,2), it follows from Lemma 2.2 that the linear system

    xi(t)=δi(t)xi(t),i=1,2 (3.3)

    admits an exponential dichotomy on R1. Thus, by Lemma 2.1, we obtain that the system (3.2) has exactly one almost periodic solution xϕ(t)=(xϕ1(t),xϕ2(t))T:

    {xϕ1(t)=tetsδ1(u)du[a1(s)ϕ2(s)+nj=1c1j(s)ϕ1(sτ1j(s))eb1j(s)ϕ1(sτ1j(s))k1(s)ϕ1(s)ϕ2(s)]dsxϕ2(t)=tetsδ2(u)du[a2(s)ϕ1(s)+nj=1c2j(s)ϕ2(sτ2j(s))eb2j(s)ϕ2(sτ2j(s))k2(s)ϕ1(s)ϕ2(s)]ds. (3.4)

    Define a mapping T:BB by setting

    T(ϕ(t))=xϕ(t),  ϕB.

    Since B={φ|φB,Ei2φi(t)Ei1,  for all  tR1,i=1,2}, it is obvious that B is a closed subset of B. For i=1,2 and any ϕB, from (2.4), (3.4) and the fact that supu0ueu=1e, we have

          xϕ1(t)tetsδ1(u)du[a+1E21+nj=1c1j(s)1b1j(s)e]ds1δ1[a+1E21+nj=1c+1jb1je]=nj=1c+1jδ1b1je+a+1E21δ1<E11      for all  tR1,                               (3.5)

    and

          xϕ2(t)tetsδ2(u)du[a+2E11+nj=1c2j(s)1b2j(s)e]ds1δ2[a+2E11+nj=1c+2jb2je]=nj=1c+2jδ2b2je+a+2E11δ2<E21      for all  tR1.                               (3.6)

    In view of the fact that min1uκueu=κeκ, from (2.5)–(2.7) and (3.4), we obtain

    xϕ1(t)tetsδ1(u)du[a1E22+nj=1c1j(s)1b+1jb+1jϕ1(sτ1j(s))eb+1jϕ1(sτ1j(s))k+1ϕ1(s)ϕ2(s)]ds1δ+1[a1E22+nj=1c1jE11eb+1jE11k+1E11E21]=a1δ+1E22+nj=1c1jδ+1E11eb+1jE11k+1δ+1E11E21>E12  for all  tR1, (3.7)

    and

    xϕ2(t)tetsδ2(u)du[a2E12+nj=1c2j(s)1b+2jb+2jϕ2(sτ2j(s))eb+2jϕ1(sτ2j(s))k+2ϕ1(s)ϕ2(s)]ds1δ+2[a2E12+nj=1c2jE21eb+2jE21k+2E11E21]=a2δ+2E12+nj=1c2jδ+2E21eb+2jE21k+2δ+2E11E21>E22  for all  tR1. (3.8)

    Therefore, (3.5)–(3.8) show that the mapping T is a self-mapping from B to B.

    Let φ,ψB; for i=1,2, we get

    suptR1|(T(φ(t))T(ψ(t)))1|=suptR1|tetsδ1(u)du[a1(s)(φ2(s)ψ2(s))+nj=1c1j(s)(φ1(sτ1j(s))eb1j(s)φ1(sτ1j(s))ψ1(sτ1j(s))eb1j(s)ψ1(sτ1j(s)))k1(s)(φ1(s)φ2(s)ψ1(s)ψ2(s))]ds|=suptR1|tetsδ1(u)du[a1(s)(φ2(s)ψ2(s))+nj=1c1j(s)b1j(s)×(b1j(s)φ1(sτ1j(s))eb1j(s)φ1(sτ1j(s))b1j(s)ψ1(sτ1j(s))eb1j(s)ψ1(sτ1j(s)))k1(s)(φ1(s)φ2(s)φ1(s)ψ2(s)+φ1(s)ψ2(s)ψ1(s)ψ2(s))]ds|, (3.9)

    and

    suptR1|(T(φ(t))T(ψ(t)))2|=suptR1|tetsδ2(u)du[a2(s)(φ1(s)ψ1(s))+nj=1c2j(s)(φ2(sτ2j(s))eb2j(s)φ2(sτ2j(s))ψ2(sτ2j(s))eb2j(s)ψ2(sτ2j(s)))k2(s)(φ1(s)φ2(s)ψ1(s)ψ2(s))]ds|=suptR1|tetsδ2(u)du[a2(s)(φ1(s)ψ1(s))+nj=1c2j(s)b2j(s)×(b2j(s)φ2(sτ2j(s))eb2j(s)φ2(sτ2j(s))b2j(s)ψ2(sτ2j(s))eb2j(s)ψ2(sτ2j(s)))k2(s)(φ1(s)φ2(s)φ1(s)ψ2(s)+φ1(s)ψ2(s)ψ1(s)ψ2(s))]ds|. (3.10)

    Since

    bij(s)φi(sτij(s))bijEi2bij1min1jnbij1,  for all  sR1,i=1,2,j=1,2,,n,

    and

     bij(s)ψi(sτij(s))bijEi2bij1min1jnbij1,  for all  sR1,i=1,2,j=1,2,,n.

    According to (1.4), (2.5), (3.5), (3.7) and (3.9), together with supu1|1ueu|=1e2 and the inequality

    |xexyey|=|1(x+θ(yx))ex+θ(yx)||xy|         1e2|xy|  where  x,y[1,+),0<θ<1, (3.11)

    we have

    suptR1|(T(φ(t))T(ψ(t)))1|a+1δ1suptR1|φ2(t)ψ2(t)|+suptR1tetsδ1(u)dunj=1c+1j1e2|φ1(sτ1j(s))ψ1(sτ1j(s))|ds+suptR1tetsδ1(u)duk1(s)(|φ1(s)||φ2(s)ψ2(s)|+|ψ2(s)||φ1(s)ψ1(s)|)dsa+1δ1suptR1|φ2(t)ψ2(t)|+nj=1c+1jδ1e2suptR1|φ1(t)ψ1(t)|+k+1δ1E11suptR1|φ2(t)ψ2(t)|+k+1δ1E21suptR1|φ1(t)ψ1(t)|,=(nj=1c+1jδ1e2+k+1δ1E21)suptR1|φ1(t)ψ1(t)|+(a+1δ1+k+1δ1E11)suptR1|φ2(t)ψ2(t)|. (3.12)

    Similarly, we also get

    suptR1|(T(φ(t))T(ψ(t)))2|(nj=1c+2jδ2e2+k+2δ2E11)suptR1|φ2(t)ψ2(t)|+(a+2δ2+k+2δ2E21)suptR1|φ1(t)ψ1(t)|. (3.13)

    Hence

    (suptR1|(T(φ(t))T(ψ(t)))1|,suptR1|(T(φ(t))T(ψ(t)))2|)T((nj=1c+1jδ1e2+k+1δ1E21)suptR1|φ1(t)ψ1(t)|+(a+1δ1+k+1δ1E11)suptR1|φ2(t)ψ2(t)|,(nj=1c+2jδ2e2+k+2δ2E11)suptR1|φ2(t)ψ2(t)|+(a+2δ2+k+2δ2E21)suptR1|φ1(t)ψ1(t)|)T=F(suptR1|φ1(t)ψ1(t)|,suptR1|φ2(t)ψ2(t)|)T=F(suptR1|(φ(t)ψ(t))1|,suptR1|(φ(t)ψ(t))2|)T, (3.14)

    where F=A1B. Let μ be a positive integer. Then, from (3.14) we get

    (suptR1|(Tμ(φ(t))Tμ(ψ(t)))1|,suptR1|(Tμ(φ(t))Tμ(ψ(t)))2|)T=(suptR1|(T(Tμ1(φ(t)))T(Tμ1(ψ(t))))1|, suptR1|(T(Tμ1(φ(t)))T(Tμ1(ψ(t))))2|)TF(suptR1|(Tμ1(φ(t))Tμ1(ψ(t)))1|,suptR1|(Tμ1(φ(t))Tμ1(ψ(t)))2|)T Fμ(suptR1|(φ(t)ψ(t))1|,suptR1|(φ(t)ψ(t))2|)T=Fμ(suptR1|φ1(t)ψ1(t)|,suptR1|φ2(t)ψ2(t)|)T. (3.15)

    Since ρ(F)<1, we obtain

    limμ+Fμ=0,

    which implies that there exist a positive integer N and a positive constant r<1 such that

    FN=(A1B)N=(gij)2×2and2j=1gijr,i=1,2. (3.16)

    In view of (3.15) and (3.16), we have

    |(TN(φ(t))TN(ψ(t)))i|suptR1|(TN(φ(t))TN(ψ(t)))i|2j=1gijsuptR1|φj(t)ψj(t)|suptR1max1j2|φj(t)ψj(t)|2j=1gijrφ(t)ψ(t)B,

    for all tR,i=1,2. It follows that

    TN(φ(t))TN(ψ(t))B=suptR1max1i2|(TN(φ(t))TN(ψ(t)))i|rφ(t)ψ(t)B. (3.17)

    This implies that the mapping TN:BB is a contraction mapping.

    By the fixed point theorem for Banach space, T possesses a unique fixed point φB such that Tφ=φ. By (3.2), φ satisfies (1.4). So φ is an almost periodic solution of (1.4) in B. The proof of Theorem 3.1 is now completed.

    Theorem 3.2. Let x(t) be the positive almost periodic solution of system (1.4) in the region B. Suppose that (2.4)–(2.6) and (3.1) hold. Then, the solution x(t;t0,φ) of (1.4) with φC0 converges exponentially to x(t) as t+.

    Proof. Since ρ(A1B)<1, it follows from Theorem 3.1 that system (1.4) has a unique almost periodic solution x(t)=(x1(t),x2(t))T in the region B. Set x(t)=x(t;t0,φ), x(t)=x(t;t0,φ) and yi(t)=xi(t)xi(t), where φ,φC0, t[t0ri,+), i=1,2. Then, we have the following:

    {y1(t)=δ1(t)y1(t)+a1(t)y2(t)+nj=1c1j(t)(x1(tτ1j(t))eb1j(t)x1(tτ1j(t))x1(tτ1j(t))eb1j(t)x1(tτ1j(t)))k1(t)(x1(t)x2(t)x1(t)x2(t)),y2(t)=δ2(t)y2(t)+a2(t)y1(t)+nj=1c2j(t)(x2(tτ2j(t))eb2j(t)x2(tτ2j(t))x2(tτ2j(t))eb2j(t)x2(tτ2j(t)))k2(t)(x1(t)x2(t)x1(t)x2(t)). (3.18)

    Again from , it follows from Lemma 2.3 that is an M-matrix; we obtain that there exists a constant and a vector such that

    Therefore,

    which implies that

    (3.19)

    We can choose a positive constant such that

    (3.20)

    In the sequel, we consider the following Lyapunov function:

    (3.21)

    In view of (2.5)–(2.7), for and , we obtain

    and

    which, together with (3.11) and (3.18), imply that

    (3.22)

    and

    (3.23)

    Let denote an arbitrary real number such that

    It follows from (3.21) that

    We claim that

    (3.24)

    In contrast, there must exist and such that

    (3.25)

    Thus,

    (3.26)

    or

    (3.27)

    Together with (3.20), (3.22), (3.23), (3.26) and (3.27), we obtain

    (3.28)

    or

    (3.29)

    which are both contradictory. Hence, (3.24) holds. Let such that

    (3.30)

    In view of (3.24) and (3.30), we get

    This completes the proof.

    Corollary 3.1. Let (2.4)–(2.6) hold. Suppose that is an M-matrix. Then system (1.4) has exactly one almost periodic solution . Moreover, the solution of (1.4) with converges exponentially to as .

    Proof. Since is an M-matrix, it follows that there exists a vector such that

    (3.31)

    hence

    (3.32)

    For any matrix norm and nonsingular matrix , also defines a matrix norm. Let . Then (3.32) implies that the row norm of matrix is less than 1. Hence . Corollary 3.1 follows immediately from Theorems 3.1 and 3.2.

    In this section, we give an example and present a numerical simulation to demonstrate the results obtained in previous sections.

    Example 4.1. Consider the following competitive and cooperative Nicholson's blowfly system:

    (4.1)

    Obviously, and . Let and for ; we obtain

    (4.2)
    (4.3)
    (4.4)
    (4.5)
    (4.6)

    Then (4.2) (4.6) imply that the competitive and cooperative Nicholson's blowfly system (4.1) satisfies (2.4)–(2.6) and (3.1). Hence, from Theorems 3.1 and 3.2, system (4.1) has a positive almost periodic solution

    Moreover, if then converges exponentially to as . The fact is verified by the numerical simulation illustrated in Figure 1.

    Figure 1.  Numerical solution of system (4.1) for the initial value .

    Remark 4.1. To the best of our knowledge, few authors have considered the problems related to positive almost periodic solutions of competitive and cooperative Nicholson's blowfly systems. Therefore, the main results in [31,32] and the references therein can not be appled to prove that all solutions of (4.1) with initial the value converge exponentially to the positive almost periodic solution. This implies that the results in this paper are new and this complements previously obtained results.

    This article investigated a class of competitive and cooperative Nicholson's blowfly system. Unlike what has been done for some known cooperative Nicholson's blowfly systems [31,32], we have introduced the competitive terms to describe two distinct blowfly populations that compete with each other. By constructing invariant sets and applying the fixed point theorem, we derived some sufficient conditions to ensure that the addressed system has a unique exponential stable positive almost periodic solution. Inspired by the latest Nicholson's blowfly models [3,4,5,6,7,8,9,10], our future works will be devoted to competitive and cooperative Nicholson's blowfly systems involving distinct delays, distributed delays and mixed delays.

    The author declares that he has not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by Natural Scientific Research Fund of Zhejiang Provincial of China (grant nos. LY18A010019, LY16A010018).

    The author declares no conflicts of interest.



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