Research article Special Issues

Revitalization of cultural heritage in the digital era: A case study in Taiwan

  • Cultural heritage organizations worldwide face daunting challenges, grappling with workforce shortages and financial constraints that often result in service closures, postponements, or cancellations. In response to these difficulties, we delved into the evolving interaction between cultural heritage sites and society, especially considering the profound socio-economic repercussions of the global pandemic at these sites. We scrutinized the dynamic heritage of community engagement, with a particular focus on pioneering methods to augment the participation and awareness of younger individuals. We focused on innovative methods to enhance the involvement and awareness of young individuals. Employing a comprehensive content analysis through a multiple case study approach, cultural heritage initiatives in Taiwan were investigated, emphasizing the pivotal role of technology and digital platforms in engaging young audiences. Using a comprehensive content analysis approach within a multiple case study framework, we examined various cultural heritage initiatives in Taiwan. Notably, we accentuated the crucial role played by technology and digital platforms in captivating younger audiences. Through theoretical sampling and triangulation methods, including semi-structured interviews, secondary sources, and participant observation, we sought to obtain a holistic understanding of the subject matter. The research findings underscore the pivotal importance of digital experiences as supplements to physical presence, providing a gateway to cultural heritage exploration. Moreover, we advocate for heritage sites to harness digital platforms effectively, encouraging collaborations with external partners to enrich visitor experiences. We also offer valuable recommendations aimed at enhancing customer engagement and communication with the younger demographic, thus making significant contributions to the cultural heritage sector in Asia. Furthermore, this research signifies a notable transition from traditional physical service design to online platforms, leveraging technology to inspire and engage diverse generations. By embracing digital tools, cultural heritage sites not only attract new visitors but also retain the interest of existing ones amidst an ever-evolving societal landscape. Ultimately, this study sheds light on the imperative nature of cultural heritage preservation and adaptation in the face of contemporary challenges, emphasizing the need for innovation and resilience in this vital sector.

    Citation: Wai-Kit Ng, Chun-Liang Chen, Yu-Hui Huang. Revitalization of cultural heritage in the digital era: A case study in Taiwan[J]. Urban Resilience and Sustainability, 2024, 2(3): 215-235. doi: 10.3934/urs.2024011

    Related Papers:

    [1] Huifeng Zhang, Xirong Xu, Ziming Wang, Qiang Zhang, Yuansheng Yang . (2n3)-fault-tolerant Hamiltonian connectivity of augmented cubes AQn. AIMS Mathematics, 2021, 6(4): 3486-3511. doi: 10.3934/math.2021208
    [2] Yanling Wang, Shiying Wang . Edge-fault-tolerant strong Menger edge connectivity of bubble-sort graphs. AIMS Mathematics, 2021, 6(12): 13210-13221. doi: 10.3934/math.2021763
    [3] Wenke Zhou, Guo Chen, Hongzhi Deng, Jianhua Tu . Enumeration of dissociation sets in grid graphs. AIMS Mathematics, 2024, 9(6): 14899-14912. doi: 10.3934/math.2024721
    [4] Yixin Zhang, Yanbo Zhang, Hexuan Zhi . A proof of a conjecture on matching-path connected size Ramsey number. AIMS Mathematics, 2023, 8(4): 8027-8033. doi: 10.3934/math.2023406
    [5] Dingjun Lou, Zongrong Qin . The structure of minimally 2-subconnected graphs. AIMS Mathematics, 2022, 7(6): 9871-9883. doi: 10.3934/math.2022550
    [6] Xiaohong Chen, Baoyindureng Wu . Gallai's path decomposition conjecture for block graphs. AIMS Mathematics, 2025, 10(1): 1438-1447. doi: 10.3934/math.2025066
    [7] Jinyu Zou, Haizhen Ren . Matching preclusion and conditional matching preclusion for hierarchical cubic networks. AIMS Mathematics, 2022, 7(7): 13225-13236. doi: 10.3934/math.2022729
    [8] Yuan Zhang, Haiying Wang . Some new results on sum index and difference index. AIMS Mathematics, 2023, 8(11): 26444-26458. doi: 10.3934/math.20231350
    [9] G. Nandini, M. Venkatachalam, Raúl M. Falcón . On the r-dynamic coloring of subdivision-edge coronas of a path. AIMS Mathematics, 2020, 5(5): 4546-4562. doi: 10.3934/math.2020292
    [10] Yanyi Li, Lily Chen . Injective edge coloring of generalized Petersen graphs. AIMS Mathematics, 2021, 6(8): 7929-7943. doi: 10.3934/math.2021460
  • Cultural heritage organizations worldwide face daunting challenges, grappling with workforce shortages and financial constraints that often result in service closures, postponements, or cancellations. In response to these difficulties, we delved into the evolving interaction between cultural heritage sites and society, especially considering the profound socio-economic repercussions of the global pandemic at these sites. We scrutinized the dynamic heritage of community engagement, with a particular focus on pioneering methods to augment the participation and awareness of younger individuals. We focused on innovative methods to enhance the involvement and awareness of young individuals. Employing a comprehensive content analysis through a multiple case study approach, cultural heritage initiatives in Taiwan were investigated, emphasizing the pivotal role of technology and digital platforms in engaging young audiences. Using a comprehensive content analysis approach within a multiple case study framework, we examined various cultural heritage initiatives in Taiwan. Notably, we accentuated the crucial role played by technology and digital platforms in captivating younger audiences. Through theoretical sampling and triangulation methods, including semi-structured interviews, secondary sources, and participant observation, we sought to obtain a holistic understanding of the subject matter. The research findings underscore the pivotal importance of digital experiences as supplements to physical presence, providing a gateway to cultural heritage exploration. Moreover, we advocate for heritage sites to harness digital platforms effectively, encouraging collaborations with external partners to enrich visitor experiences. We also offer valuable recommendations aimed at enhancing customer engagement and communication with the younger demographic, thus making significant contributions to the cultural heritage sector in Asia. Furthermore, this research signifies a notable transition from traditional physical service design to online platforms, leveraging technology to inspire and engage diverse generations. By embracing digital tools, cultural heritage sites not only attract new visitors but also retain the interest of existing ones amidst an ever-evolving societal landscape. Ultimately, this study sheds light on the imperative nature of cultural heritage preservation and adaptation in the face of contemporary challenges, emphasizing the need for innovation and resilience in this vital sector.



    Stochastic age-dependent population system has an increasingly important status in biomathematics, being the main research direction in biology and ecology in recent years. Especially, one of the most striking and meaningful problems in the study of stochastic age-dependent population system is its numerical scheme because of its nonlinear structure and non-existent explicit solutions. In [1], Li et al. first introduced Poisson jumps into stochastic age-dependent population system and proposed the following model:

    {dtPt=[Ptaμ(t,a)Pt+f(t,Pt)]dt+g1(t,Pt)dBt+h(t,Pt)dNt     (t,a)Q,P(0,a)=P0(a),   a[0,A],P(t,0)=A0β(t,a)P(t,a)da,    t[0,T], (1.1)

    where T>0, A is the maximal age of the population species and A>0, Q=(0,T)×(0,A). dtPt is the differential of Pt relative to t, i.e., dtPt=Pttdt. Pt=P(t,a) denotes the population density of age a at time t, μ(t,a) denotes the mortality rate of age a at time t, β(t,a) denotes the fertility rate of females of a at time t, f(t,Pt) denotes effects of external environment for population system, g1(t,Pt) is a diffusion coefficient, h(t,Pt) is a jump coefficient (it represents the size of the population systems increases or decreases drastically because brusque variations from earthquakes, floods, immigrants and so on), Bt is a Brownian motion, Nt is a Poisson process with intensity λ>0. Then, they investigated the convergence of Euler method for model (1.1). Since then, an increasing number of authors have analyzed stochastic age-dependent population models with Poisson jumps, and many significant results have been obtained (see e.g., [2,3,4,5,6,7,8,9,10,11]). For example, Wang and Wang [3] established the semi-implicit Euler method for stochastic age-dependent population models with Poisson jumps and discussed the convergence order of numerical solutions. Tan et al. [5] presented a split-step θ (SSθ) method of stochastic age-dependent population models with Poisson jumps, and the exponential stability of the model was established. Pei et al. [8] constructed two types of numerical methods for stochastic age-dependent population models with Poisson jumps, which are compensated and non-compensated. Then the asymptotic mean-square boundedness is discussed for numerical scheme.

    In the above-mentioned model (1.1), we can easily see that the effects of randomly environmental variations of parameter μ are described as a linear function of Gaussian white noise [12,13], that is μdtμdt+σdBt (where σ2 represents the intensity of Bt). Obviously, it is unreasonable to use linear function of Gaussian white noise to simulate parameters perturbation in a randomly varying environment. In [14], Duffie proposed to use a mean-reverting Ornstein-Uhlenbeck process to describe parameters which fluctuate around an average value. Up until now, the mean-reverting OU process have been extensively discussed in [15,16,17,18]. Zhao et al. [16] analyzed stationary distribution for stochastic competitive model incorporating the OU process. Wang et al. [17] introduced the OU process into the Susceptible-Infected-Susceptible (SIS) epidemic model and investigated its threshold. However, to the authors' knowledge, there is no literature to consider the OU process into the stochastic age-dependent population system with Poisson jumps. On the other hand, we find that the influence of time delay are not considered in the above papers [1,2,3,4,5,6,7,8,9,11]. There are very few papers in the literature that take time delay into account in the stochastic age-dependent population system so far (see e.g., [19,20]). Therefore, based on the above analysis, studying stochastic age-dependent delay population jumps equations, coupled with mean-reverting OU process have more practical significance.

    Obviously, the age-dependent stochastic delay population equations with mean-reverting OU process and Poisson jumps have no explicit solution. Thus, numerical approximation schemes should be developed as a essential and powerful tool to explore its properties. The numerical schemes of stochastic age-dependent population system have been extensively researched by many scholars, for example, [1,3,4,5,8,9,19,21,22,23]. However, due to the fact that the coefficients f, g1 and h of model (1.1) are particularly complex functions, so using the existing numerical methods to approximate the age-dependent stochastic delay population equations with mean-reverting OU process and Poisson jumps will result in slow convergence and very high computational cost. Recently, Jankovic and Ilic [24] introduced a Taylor approximation method for stochastic differential equations and proved that its convergence rate and computational cost is better than other numerical methods such as the Euler method, the semi-implicit Euler method and SSθ method mentioned in [1,2,3,4,5,6,7,8,9,11]. Motivated by Jankovic et al., we construct a Taylor approximation scheme for the age-dependent stochastic delay population equations with mean-reverting OU process and Poisson jumps in this paper. Furthermore, the convergence between the exact solutions and numerical solutions is investigated.

    The highlights of the present paper are summarized as follows:

    ● Age-dependent stochastic delay population equations with mean-reverting OU process and Poisson jumps are given.

    ● To improve the convergence speed and reduce cost, the Taylor approximation scheme for the age-dependent stochastic delay population equations with mean-reverting OU process and Poisson jumps is developed.

    ● The convergence and convergence order of Taylor approximation scheme are discussed.

    The arrangement of this paper is as follows. In section 2, we establish the age-dependent stochastic delay population equations with mean-reverting OU process and Poisson jumps, as well as introduce some notations and preliminaries. Then, the pth moments boundedness of exact solutions for age-dependent stochastic delay population equations with mean-reverting OU process and Poisson jumps are presented. In section 3, we propose a Taylor approximation scheme for a age-dependent stochastic delay population equations with mean-reverting OU process and Poisson jumps, and the convergence theory for the numerical method is proved. In section 4, we present some numerical simulations to demonstrate our theoretical results. Section 5 presents the conclusions of our research.

    In the above model (1.1), the authors took advantage of a traditional parameter perturbation method to reflect the effect of environmental noise, (μ(t,a)Pt+f(t,Pt))dt when it is stochastically perturbed with (μ(t,a)Pt+f(t,Pt))dt+g1(t,Pt)dBt. It is worth mentioning that Cai et al. [18] pointed out that due to environmental continuous fluctuations, the mortality rate μ(t,a) can not be described by a linear function of Gaussian white noise. In order to model the randomly varying environmental fluctuations in μ(t,a), we introduce the following mean-reverting Ornstein-Uhlenbeck process for μ(t,a) inspired by [16]:

    {μ(t,a)=μ1(t)μ2(a)dμ1(t)=η(μeμ1(t))dt+ξdBt (2.1)

    where we assume that μ1(t) represents the mortality rate at time t and μ2(a) represents the mortality rate at age a. All parameters η, μe and ξ are positive constants. η is the reversion rate, μe is the mean reversion level or long-run equilibrium of growth rate μ1(t), ξ is the intensity of volatility.

    For (2.1), applying the stochastic integral format, we obtain the explicit form of the solution as:

    μ1(t)=μe+(μ01μe)eηt+ξt0eη(ts)dBt (2.2)

    where (μ01:=μ1(0)). It is not difficult to see that the expected value of μ1(t) is

    E[μ1(t)]=μe+(μ01μe)eηt (2.3)

    and variance value of μ1(t) is

    Var[μ1(t)]=ξ22η(1e2ηt). (2.4)

    Combining (2.2), (2.3) and (2.4), we easily have that the term ξt0eη(ts)dBt satisfies the normal distribution E(0,ξ22η(1e2ηt)). Then we obtain ξt0eη(ts)dB(t) being equal to ξ2η1e2ηtdBtdt a.e..

    Therefore, we can rewrite (2.2) in the following form [17,18]

    μ1(t)=μe+(μ01μe)eηt+σ(t)dBtdt, (2.5)

    where σ(t)=ξ2η1e2ηt. A conceptual problem immediately occurs in that dBtdt is not defined except in a generalized sense.

    Replacing μ(t,a) in model (1.1) with (2.2) and (2.5) and rearranging leads to the following stochastic age-dependent population equation:

    {dtPt=[Pta(μe+(μ01μe)eηt)μ2(a)Pt+f(t,Pt)]dt+g(t,Pt)dBt+h(t,Pt)dNt      (t,a)(0,T)×(0,A),P(0,a)=P0(a), a[0,A],P(t,0)=A0β(t,a)P(t,a)da,  t[0,T], (2.6)

    where g(t,Pt)dBt contains g1(t,Pt)dBt and σ(t)μ2(a)PtdBt.

    On the other hand, due to the time delay is unavoidable in a real world. Motivated by [19] and [25], we derive the following system:

    {dtPt=[Pta(μe+(μ01μe)eηt)μ2(a)Pt+f(t,Pt,Ptτ)]dt+g(t,Pt,Ptτ)dBt+h(t,Pt,Ptτ)dNt,         (t,a)(0,T)×(0,A),P(t,a)=ϕ(t,a), (t,a)[τ,0]×[0,A],P(t,0)=A0β(t,a)P(t,a)da,   t[0,T], (2.7)

    where Pt=P(t,a) denotes the population density of age a at time t, Ptτ=P(tτ,a) denotes the population density of age a at time tτ, τ is time delay and τ>0. f(t,Pt,Ptτ) denotes the effects of external environment for the population system, g(t,Pt,Ptτ) is a diffusion coefficient, h(t,Pt,Ptτ) is a jump coefficient, ϕt=ϕ(t,a) denotes the histories of the population density of age a at time t, β(t, a) denotes the fertility rate of females of age a at time t. A is the maximal age of the population species, so P(t,a)=0,aA. Nt is a Poisson process with intensity λ>0. The explanation of the other symbols was given under Eq (2.1). In the following section, we concentrate on studying model (2.7).

    Let V=D1([0,A]){φ|φL2([0,A]),where φa represent the generalized partial derivatives}, V be a Sobolev space. D=L2[0,A] such that VDDV. V is the dual space of V. We denote by , and the norms in V,D and V respectively; by , the duality product between V and V, and by (,) the scalar product in D.

    For simplicity, we introduce some notations. Throughout this paper, unless otherwise, let (Ω,F,P) be a complete probability space with filtration {Ft}t0 satisfying the usual conditions (i.e. it is increasing and right continuous while F0 contains all P-null sets), and let E signify the expectation corresponding to P. For an operator HL(M,D) on the space of all bounded linear operators from M into D, we denote by H2 the Hilbert-Schmidt norm, i.e., B2=tr(HWH)T.

    Let τ>0 and C=C([τ,0];D) be the space of all continuous function from [0, T] into H with sup-norm ψC=supτs0|ψ(s)|, LPV=LP([0,T];V) and LPD=LP([0,T];D). Moreover, let F0-measurable, CbF0([τ,0];D) denote the family of all almost surely bounded, F0-measurable C=C([τ,0];D) -value random variables. For a pair of real numbers a and b, we use ab=max(a,b). If G is a set, its indicator function by 1G, namely 1G(x) = 1 if xG and 0 otherwise.

    The integer version of Eq (2.7) is given by

    Pt=P0t0Psadst0[μe+(μ01μe)eηs)]μ2(a)Psds+t0f(s,Ps,Psτ)ds+t0g(s,Ps,Psτ)dBs+t0h(s,Ps,Psτ)dNs. (2.8)

    For the existence and uniqueness of the solution, we assume that the following assumptions are satisfied:

    (A1) μ(t,a) and β(t,a) are nonnegative measurable, such that

    {0μ_μ2(a)<ˉμ<,0β(t,a)ˉβ<.

    (A2) f(t,0,0)=g(t,0,0)=h(t,0,0)=0.

    (A3) The Lipschitz and linear growth conditions: there exists a positive constant K such that

    |f(t,x1,y1)f(t,x2,y2)|g(t,x1,y1)g(t,x2,y2)2|h(t,x1,y1)h(t,x2,y2)|

    K(x1x2C+y1y2C),

    |f(t,x,y)|2g(t,x,y)22|h(t,x,y)|22K2(x2C+y2C)

    for x,y,x1,x2,y1,y2C.

    (A4) There exists constants ˜K,ˉK>0 and γ(0,1] such that for τs0, 0aA and r2

    E|ϕtϕs|r˜K(ts)γ.

    Consequently,

    E|ϕt|r<.

    (A5) f, g and h have Taylor approximations in the second argument, up to α1th, α2th and α3th derivatives, denoted as f(α1)x(t,x,y), g(α2)x(t,x,y) and h(α3)x(t,x,y), respectively.

    (A6) f(α1+1)Pt(t,Pt,Ptτ), g(α2+1)Pt(t,Pt,Ptτ) and h(α3+1)Pt(t,Pt,Ptτ) are uniformly bounded, i.e. there exist positive constants K1, K2 and K3 satisfying

    {sup[0,T]×[0,A]|f(α1+1)Pt(t,Pt,Ptτ)|K1,sup[0,T]×[0,A]|g(α2+1)Pt(t,Pt,Ptτ)|K2,sup[0,T]×[0,A]|h(α3+1)Pt(t,Pt,Ptτ)|K3.

    Throughout the following analysis, for the purpose of simplicity, we will use C,C1,C2, to stand for generic constants that depend upon K and T, but not upon Δ. The precise value of these constants may be determined via the proof. Our first theorem shows the existence and uniqueness of the strong solution for the model (2.7).

    Theorem 2.1. Under the assumptions (A1)(A4), for t[0,T], Eq (2.7) has a unique strong solution.

    Proof. The proof of this theorem is standard (see Zhang et al. [26]) and hence is omitted.

    Moreover, the pth moment boundedness of the true solution Pt of the model (2.7) is proved in the following theorem.

    Theorem 2.2. Under the assumptions (A1)(A4), for each q2, there exists a constant C such that

    E[supτtT|Pt|q]C. (2.9)

    Proof. Form (2.8), applying Itô's formula [25] to |Pt|q yields

    |Pt|q= |P0|q+t0q|Ps|q2Psa(μe+(μ01μe)eηs)μ2(a)Ps,Psds+t0q|Ps|q2(f(s,Ps,Psτ),Ps)ds+t0q(q1)2|Ps|q2g(s,Ps,Psτ)22ds+t0q|Ps|q2(Ps,g(s,Ps,Psτ))dBs+t0q|Ps|q2(Ps,h(s,Ps,Psτ))dNs+t0q(q1)2|Ps|q2h(s,Ps,Psτ)22dNs= |P0|q+t0q|Ps|q2Psa(μe+(μ01μe)eηs)μ2(a)Ps,Psds+t0q|Ps|q2(f(s,Ps,Psτ),Ps)ds+t0q(q1)2|Ps|q2g(s,Ps,Psτ)22ds+t0q|Ps|q2(Ps,g(s,Ps,Psτ))dBs+t0q|Ps|q2(Ps,h(s,Ps,Psτ))dˉNs+λt0q|Ps|q2(Ps,h(s,Ps,Psτ))ds+t0q(q1)2|Ps|q2|h(s,Ps,Psτ)|2dˉNs+λt0q(q1)2|Ps|q2|h(s,Ps,Psτ)|2ds, (2.10)

    where ˉNs=Nsλs is a compensated Poisson process.

    Since

    Psa,Ps=A0Psda(Ps)= 12(A0β(t,a)Psda)2 12A0β2(t,a)daA0P2sda 12ˉβ2A2|Ps|2, (2.11)

    by the assumptions (A1)-(A3), we get that

    |Pt|q |P0|q+q(ˉβ2A22+μ01ˉμ)t0|Ps|qds+q(q1)K2t0|Ps|q2(Ps2C+Psτ2C)ds+Kqt0|Ps|q1(PsC+PsτC)ds+qt0|Ps|q2(Ps,g(s,Ps,Psτ))dBs+qt0|Ps|q2(Ps,h(s,Ps,Psτ))dˉNs+q(q1)2t0|Ps|q2|h(s,Ps,Psτ)|2dˉNs+λqKt0|Ps|q1(PsC+PsτC)ds+λq(q1)K2t0|Ps|q2(Ps2C+Psτ2C)ds |P0|q+q(ˉβ2A22+μ01ˉμ)t0|Ps|qds+2q(q1)K2t0supτus|Pu|qds+2Kqt0supτus|Pu|qds+qt0|Ps|q2(Ps,g(s,Ps,Psτ))dBs+qt0|Ps|q2(Ps,h(s,Ps,Psτ))dˉNs+2Kqλt0supτus|Pu|qds+q(q1)2t0|Ps|q2|h(s,Ps,Psτ)|2dˉNs+2λq(q1)K2t0supτus|Pu|qds |P0|q+q[(ˉβ2A22+μ01ˉμ)+2K+2(q1)K2+2Kλ+2λK2(q1)]t0supτus|Pu|qds+qt0|Ps|q2(Ps,g(s,Ps,Psτ))dBs+qt0|Ps|q2(Ps,h(s,Ps,Psτ))dˉNs+q(q1)2t0|Ps|q2|h(s,Ps,Psτ)|2dˉNs. (2.12)

    Note that for any t[0,T],

    E[supτut|Pu|q]=E[supτu0|Pu|q]E[sup0ut|Pu|q]. (2.13)

    Hence, we have

    E[supτut|Pu|q] E[supτu0|ϕu|q]+C1t0E[supτus|Pu|q]ds+qE[sup0stt0|Ps|q2(Ps,g(s,Ps,Psτ))dBs]+qE[sup0stt0|Ps|q2(Ps,h(s,Ps,Psτ))dˉNs]+q(q1)2E[sup0stt0|Ps|q2|h(s,Ps,Psτ)|2dˉNs], (2.14)

    where C1=q[(ˉβ2A22+μ01ˉμ)+2K+2(q1)K2+2Kλ+2λK2(q1)].

    Using the Burkholder-Davis-Gundy's inequality, we derive that

    E[sup0stt0|Ps|q2(Ps,g(s,Ps,Psτ))dBs]=E[sup0stt0|Ps|q2(Pq22s,g(s,Ps,Psτ))dBs]E[supτut|Pu|q2(t0(Pq22s,g(s,Ps,Psτ))dBs)]3E[supτut|Pu|q2(t0|Ps|q2g(s,Ps,Psτ)22ds)12]16qE[supτut|Pu|q]+C2E(t0|Ps|q2g(s,Ps,Psτ)22ds)16qE[supτut|Pu|q]+4C2K2(t0Esupτus|Pu|qds). (2.15)

    Similarly, we can obtain that

    E[sup0stt0|Ps|q2(Ps,h(s,Ps,Psτ))dˉNs]16qE[supτut|Pu|q]+4C3K2(t0Esupτus|Pu|qds) (2.16)

    and

    E[sup0stt0|Ps|q2|h(s,Ps,Psτ)|22dˉNs]13q(q1)E[supτut|Pu|q]+16C4K4(t0Esupτus|Pu|qds). (2.17)

    Substituting (2.15), (2.16) and (2.17) into (2.14) yields

    E[supτut|Pu|q] E[supτu0|ϕu|q]+12E[supτut|Pu|q]+(C1+4qC2K2+4qC3K2+8q(q1)C4K4)(t0Esupτus|Pu|qds). (2.18)

    Thus, the well-known Gronwall inequality obviously implies the desired equality (2.9).

    In this section, we will establish the Taylor approximation scheme for the stochastic age-dependent population Eq (2.7) and investigate the strong convergence between the true solutions and the numerical solutions derived from the Taylor approximation scheme.

    Let τj denote the jth jump of Ns occurrence time. For example, assume that jumps arrive at distinct, ordered times τ1<τ2<, let t1,t2,,tm be the deterministic grid points of [0,T]. We establish approximate solutions to (2.7) at a discrete set of times {τj}(j=1,2,). This set is the superposition of the random jump times of the Poisson process on [0,T] and a deterministic grid t1,t2,,tm and satisfy max{|τi+1τi|}<Δt (for the sake of simplicity, we denote Δt as Δ). It is quite clear that the random Poisson jump times can be computed without any knowledge of the realized path of (2.7).

    Next, we propose a Taylor approximation of the solutions of Eq (2.7). Without loss of any generality, given a step size Δ(0,1), we let tk=kΔ for k=0,1,2,3,,[τΔ], here [τΔ] is the integer part of τΔ. The continuous time Taylor approximate solution Qt=Q(t,a) to the stochastic age-dependent population Eq (2.7) can be defined by setting Q0=P0(a)=ϕ(0,a) and Q(t,0)=A0β(t,a)Qtda and forming

    {Qt=Q0t0Qsadst0(μe+(μ01μe)eηs)μ2(a)Qsds+t0α1j=0f(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)jds+t0α2j=0g(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)jdBs+t0α3j=0h(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)jdNs,     0tT,Qt=ϕ(t,a), τt0, (3.1)

    where Z1t=Z1(t,a)=[τΔ]k=0Qtk1[tk,tk+1)(t) and Z2t=Z2(t,a)=[τΔ]k=0Qtkτ1[tk,tk+1)(t) are step processes. That is, Z1t=Qtk and Z2t=Qtkτ for t[tk,tk+1) when k = 0, 1, 2, 3, , [τΔ].

    In this subsection, let us investigate the convergence of the Taylor approximate solutions of the stochastic age-dependent population Eq (2.7).

    In the following three lemmas we will show that Qt and Qtτ are close to Z1t and Z2t based on Lr, respectively.

    Lemma 3.1. For any q2, there exists a positive constant K4 such that

    Esupt[τ,T]|Qt|(α+1)2qK4, (3.2)

    where α=max{α1,α2,α3}.

    Proof. This proof is completed in Appendix A.

    Remark 3.1. If α1=α2=α3=0, then (A3) shows that the Taylor approximation solutions Qt admit finite moments (see, [27,28]).

    Lemma 3.2. Under the assumptions (A1),(A3),(A5), (A6), Lemma 3.1 and E|Qsa|r<K5 hold, K5 is a positive constant. For 2r(α+1)q, we have

    E|QtZ1t|rCΔr2. (3.3)

    Proof. For any t0, there exists an integer k0 such that t[tk,tk+1), we have

    QtZ1t=QtQtk=ttkQsadsttk(μe+(μ01μe)eηs)μ2(a)Qsds+ttkX1(s,Qs,Z1s,Z2s)ds+ttkX2(s,Qs,Z1s,Z2s)dBs+ttkX3(s,Qs,Z1s,Z2s)dNs,

    where

    X1(t,Qt,Z1t,Z2t)=α1j=0f(j)Z1t(t,Z1t,Z2t)j!(QtZ1t)j,X2(t,Qt,Z1t,Z2t)=α2j=0g(j)Z1t(t,Z1t,Z2t)j!(QtZ1t)j,X3(t,Qt,Z1t,Z2t)=α3j=0h(j)Z1t(t,Z1t,Z2t)j!(QtZ1t)j.

    By the elementary inequality, we further have

    E|QtZt|r 5r1[E|ttkQsads|r+E|ttk(μe+(μ01μe)eηs)μ2(a)Qsds|r+E|ttkX1(s,Qs,Z1s,Z2s)|r+E|ttkX2(s,Qs,Z1s,Z2s)dBs|r+E|ttkX3(s,Qs,Z1s,Z2s)dNs|r].

    Applying the Hölder inequality and moment inequality, we obtain

    E|QtZt|r 5r1[Δr1ttkE|Qsa|rds+(μ01ˉμ)rΔr1ttkE|Qs|rds+Δr1ttkE|X1(s,Qs,Z1s,Z2s)|rds+C1Δr21ttkEX2(s,Qs,Z1s,Z2s)r2dBs+E|ttkX3(s,Qs,Z1s,Z2s)dNs|r]. (3.4)

    For the jump integer, by virtue of the elementary inequality and Doob's inequality, we derive

    E|ttkX3(s,Qs,Z1s,Z2s)dNs|r= E|ttkX3(s,Qs,Z1s,Z2s)dˉNs+λttkX3(s,Qs,Z1s,Z2s)ds|r 2r1E|ttkX3(s,Qs,Z1s,Z2s)dˉNs|r+2r1E|λttkX3(s,Qs,Z1s,Z2s)ds|r C22r1Δr21ttkE|X3(s,Qs,Z1s,Z2s)|rds+2r1λrΔr1ttkE|X3(s,Qs,Z1s,Z2s)|rds. (3.5)

    Moreover, by the well-known mean value theorem, we observe that there exists a θ(0,1) such that

    ttkE|X1(s,Qs,Z1s,Z2s)|rds= ttkE|f(s,Qs,Z2s)[f(s,Qs,Z2s)X1(s,Qs,Z1s,Z2s)]|rds= ttkE|f(s,Qs,Z2s)f(α1+1)P(s,Z1s+θ(QsZ1s),Z2s)(α1+1)!(QsZ1s)α1+1|rds.

    Then, by the assumptions (A1),(A3),(A5), (A6) and Lemma 3.1, we obtain

    ttkE|X1(s,Qs,Z1s,Z2s)|rds 2r1ttk[E(|f(s,Qs,Z2s)|2)r2+Kr1[(a1+1)!]rE|QsZ1s|(α1+1)r]ds 2r1ttk[2r1KrE(|Qs|r+|Z2s|r)+Kr12(α1+1)r1[(a1+1)!]r(E|Qs|(α1+1)r+|Z1s|(α1+1)r)]ds 2r1ttk[2rKrK4+Kr12(α1+1)r[(a1+1)!]rK4]ds C2Δ. (3.6)

    In the same way as (3.6) was derived, we can show that

    ttkEX2(s,Qs,Z1s,,Z2s)r2dsC3Δ (3.7)

    and

    ttkE|X3(s,Qs,Z1s,Z2s)|rdsC4Δ. (3.8)

    Substituting (3.5), (3.6), (3.7) and (3.8) into (3.4) yields

    E|QtZt|r 5r1[K5Δr+K4(μ01ˉμ)rΔr+C2Δr+C3Δr2+C2C42r1Δr2+C42r1λrΔr] CΔr2,

    which is the required inequality (3.3).

    Lemma 3.3. Under the assumptions (A1)-(A6), Lemma 3.1 and E|Qsa|r<K5 holds. For 2r(α+1)q, there exists a γ(0,1] such that

    E|QtτZ2t|rCΔγ,t0. (3.9)

    Proof. For any t0, there exists an integer k0 such that t[tk,tk+1). We divide the whole proof into the following three cases.

    ● If τtkτtτ0. Then, by the assumption (A4), we have

    E|QtτZ2t|r=E|QtτQtkτ|r=E|ϕtτϕtkτ|r˜KΔγ. (3.10)

    ● If 0tkτtτ. Then, by Lemma 3.2, we have

    E|QtτZ2t|rCΔr2. (3.11)

    ● If τtkτ0tτ. Then, we have

    E|QtτZ2t|r2r1E|Qtτϕ0|r+2r1E|Qtkτϕ0|r. (3.12)

    Then together with (3.10) and (3.11), we have the following results immediately,

    E|QtτZ2t|rC(Δr2+Δγ). (3.13)

    Summarizing the above three cases, we therefore derive that

    E|QtτZ2t|rCΔγ

    for 2r(α+1)q and γ(0,1], which is the desired inequality (3.9).

    We can now begin to prove the following theorem which reveals the convergence of the Taylor approximate solutions to the true solutions.

    Theorem 3.1. , Let the assumptions (A1)(A6) and Lemma 3.1 hold. Then for any q2 and γ(0,1],

    Esup0tT|PtQt|qCΔγ. (3.14)

    Consequently

    limΔ0E[sup0tT|PtQt|q]=0. (3.15)

    Proof. By the (2.2) and (3.1), it is not difficult to show that

    PtQt=t0(PsQs)adst0(μe+(μ01μe)eηs)μ2(a)(PsQs)ds+ttk(f(s,Ps,Psτ)X1(s,Qs,Z1s,Z2s))ds+ttk(g(s,Ps,Psτ)X2(s,Qs,Zs,Z2s))dBs+ttk(h(s,Ps,Psτ)X3(s,Qs,Zs,Z2s))dNs.

    We write

    e(t)=PtQt,I1(s)=f(s,Ps,Psτ)X1(s,Qs,Z1s,Z2s),I2(s)=g(s,Ps,Psτ)X2(s,Qs,Z1s,Z2s),I3(s)=h(s,Ps,Psτ)X2(s,Qs,Z1s,Z2s)

    for simplicity. For all t[0,T], using Itô's formula to |e(t)|q and copying the analysis of (2.11) to (2.13), we have

    |e(t)|q (qˉβ2A2+2μ01ˉμq)2t0|e(s)|qds+t0q|e(s)|q1|I1(s)|ds+q(q1)2t0|e(s)|q2I2(s)22ds+t0q|e(s)|q1|I2(s)|dBs+t0q|e(s)|q1|I3(s)|dˉNs+q(q1)2t0|e(s)|q2|I3(s)|2dˉNs+λqt0|e(s)|q1|I3(s)|ds+λq(q1)2t0|e(s)|q2|I3(s)|2ds.

    The Young inequality yields

    |a|c|b|d|a|c+d+dc+d[cε(c+d)]cd|b|c+d (3.16)

    for a,bR and c,d,ε>0. We hence have

    Esup0tT|e(t)|q K6Esup0tTt0|e(s)|qds+Esup0tTt0q(|e(s)|q+K7|I1(s)|q)ds+q(q1)2Esup0tTt0(|e(s)|q+K8I2(s)q2)ds+λq(q1)2Esup0tTt0(|e(s)|q+K8|I3(s)|q)ds.+λqEsup0tTt0(|e(s)|q+K7|I3(s)|q)ds+q(q1)2Esup0tTt0|e(s)|q2|I3(s)|2dˉNs+Esup0tTt0q|e(s)|q2(e(s),I2(s))dBs+Esup0tTt0q|e(s)|q2(e(s),I3(s))dˉNs, (3.17)

    where K6=(qˉβ2A2+2μ01ˉμq)2, K7=1q[q1εq]q1 and K8=2q[q2εq]q22.

    Applying the Burkholder-Davis-Gundy's inequality [29] and Young inequality, we obtain that

    Esup0tTt0|e(s)|q2(e(s),I2(s))dBs 16qE[sup0tT|e(t)|q]+C1E(t0|e(s)|q2I2(s)22ds) 16qE[sup0tT|e(t)|q]+C1Esup0tTt0(|e(s)|q+K8I2(s)q2)ds. (3.18)

    Continuing this approach, we have

    Esup0tTt0|e(s)|q2(e(s),I3(s))dˉNs16qE[sup0tT|e(t)|q]+C2Esup0tTt0(|e(s)|q+K8|I3(s)|q)ds (3.19)

    and

    Esup0tTt0|e(s)|q2|I3(s)|2dˉNs13q(q1)E[sup0tT|e(t)|q]+C3Esup0tTt0(|e(s)|q+K9|I3(s)|q)ds, (3.20)

    where K9=4q[q4εq]q44.

    Next, under the assumptions (A3), (A6), Lemma 3.2 and Lemma 3.3, we then compute

    Esup0tTt0|I1(s)|qds= Esup0tTt0|f(s,Ps,Psτ)X1(s,Qs,Z1s,Z2s)|qds= Esup0tTt0|f(s,Ps,Psτ)f(s,Qs,Z2s)+f(s,Qs,Z2s)X1(s,Qs,Z1s,Z2s)|qds 2q1E[T0(|f(s,Ps,Psτ)f(s,Qs,Z2s)|q+|f(s,Qs,Z2s)X1(s,Qs,Z1s,Z2s)|q)ds] 2q1[22q2KqT0E(|PsQs|q+|PsτQsτ|q+|QsτZ2s|q)ds+T0E|f(α1+1)P(s,Z1s+θ(QsZ1s),Z2s)(a1+1)!(QsZ1s)α1+1|qds] 23q2Kq[T0Esup0us|e(u)|qds]+23q3KqTCΔγ+2q1Kq1TC[(a1+1)!]qΔ(α1+1)q2. (3.21)

    Moreover, we can similarly compute

    Esup0tTt0|I2(s)|qds= Esup0tTt0(g(s,Ps,Psτ)X2(s,Qs,Z1s,Z2s))q2ds 23q2Kq[T0Esup0us|e(u)|qds]+23q3KqTCΔγ+2q1Kq2TC[(a2+1)!]qΔ(α2+1)q2 (3.22)

    and

    Esup0tTt0|I3(s)|qds= Esup0tTt0|h(s,Ps,Psτ)X3(s,Qs,Z1s,Z2s)|qds 23q2Kq[T0Esup0us|e(u)|qds]+23q3KqTCΔγ+2q1Kq3TC[(a3+1)!]qΔ(α3+1)q2. (3.23)

    Substituting (3.18) to (3.23) into (3.17) we obtain that

    Esup0tT|e(t)|qCΔ(α+1)q2+CΔγ+Ct0Esup0us|e(u)|qds

    and the required result (3.14) and (3.15) follows from the Gronwall inequality.

    In this section, we present some numerical experiments to demonstrate the theoretical result. Let us consider the following age-dependent stochastic delay population equations with OU process and Poisson jumps:

    {dtPt=[Pta(0.65+(0.50.65)e0.75t)11aPt+f(t,Pt,Pt1)]dt+g(t,Pt,Pt1)dBt+h(t,Pt,Pt1)dNt,     (t,a)(0,1)×(0,1),P(t,a)=exp(11a), (t,a)[1,0]×[0,1],P(t,0)=A01(1a)2P(t,a)da,    t[0,1], (4.1)

    where A=1, T=1, τ=1, μe = 0.65, μ01 = 0.5, η = 0.75, μ2(a)=11a, Bt is a scalar Brownian motion, Nt is a Poisson process with intensity λ=1, ϕ(t,a)=exp(11a) and β(t,a)=1(1a)2.

    Now, we employ MATLAB for numerical simulations. First, we compare the convergence speed of the Taylor approximation scheme and backward Euler methods (BEM) mentioned in [30]. Let T=1, Δt=5×104 and Δa=0.05. For f, g and h of model (4.1), we choose three different groups of functions as examples. Obviously, it is easy to verify that assumptions (A1)(A6) are satisfied. By averaging over all of the 500 samples, on the computer running at Intel Core i5-4570 CPU 3.20 GHz, the runtimes of the Taylor approximation scheme (where f, g and h are approximated up to the 5th order) and the backward Euler methods for model (4.1) are given in Table 1.

    Table 1.  Runtimes for the Taylor approximation scheme and backward Euler method.
    f(t,Pt,Ptτ) g(t,Pt,Ptτ) h(t,Pt,Ptτ) Taylor approximation BEM
    sin2Pt+14sin4Pt1 P2t+1+Pt1 sinPtPt1 17.561282 29.227642
    expPt+P2t11 (lnPt)1P2t1 cosPt+sin2Pt1 19.325948 31.497162
    2PtlnPt+P2t1 sin3PtcosPt+Pt1 2PtlnPt+1 25.367845 34.894251

     | Show Table
    DownLoad: CSV

    Form the fist group f, g and h functions in Table 1, we observe that the runtime of the Taylor approximation scheme (3.1) is about 17.561282 seconds while the runtime of backward Euler method is about 29.227642 seconds on the same computer, and conclude that the convergence speed of the Taylor approximation scheme is 1.664 times faster than that of the backward Euler methods. As the theoretical results, Table 1 reveals that the rate of convergence for the Taylor approximation scheme is faster than the backward Euler methods.

    Next, we explore the convergence of the Taylor approximation scheme (3.1). By Theorem 3.1, we obtain that the numerical solution of the Taylor approximation scheme will converge to the exact solution with γq, where γ(0,1] and q2. Since the age-dependent stochastic delay population equations with OU process and Poisson jumps (4.1) cannot be solved analytically, we use more precise numerical solutions to obtain the exact solution. We take T=1, Δt=0.005, Δa=0.05, f(t,Pt,Ptτ)=sin2Pt+14sin4Pt1, g(t,Pt,Ptτ)=P2t+1+Pt1 and h(t,Pt,Ptτ)=sinPtPt1. Based on [5], the "explicit solutions" P(t,a) to model (4.1) can be given by the numerical solution of the SSθ method with θ=0.2.

    In Figure 1, we show the paths of the "explicit solutions" P(t,a), the numerical solution Q(t,a) of the Taylor approximation scheme (where f, g and h approximated up to the 10th order) and error simulations between them. In Figure 2, the relative difference between "explicit solutions" P(t, a) and numerical solutions Q(t,a) is presented. Moreover, we can see that the maximum value of the error square is less than 0.04 from Figure 1 and the maximum value of the the relative difference is less than 0.2 from Figure 2. Clearly the numerical solution Q(t,a) converge to exact solution in the mean sense.

    Figure 1.  The upper left corner shows the path of the "explicit solutions" P(t, a). The upper right corner displays the path of the Taylor approximation of solution of Q(t, a). The lower left and lower right diagrams represent mean and mean-square error simulations between "explicit solutions" P(t, a) and numerical solutions Q(t,a) based on the Taylor approximation, respectively.
    Figure 2.  The relative difference between "explicit solutions" P(t, a) and numerical solutions Q(t,a).

    To further demonstrate the convergence of the Taylor approximation scheme, we show the errors between the "explicit solutions" P(t,a) and numerical solutions Q(t,a) at different values of time step Δt, expansion order x and age step Δa in Table 2. Table 2(a) shows that for fixed expansion order x=10 and time step Δt=0.005, the corresponding value of (P(t,a)Q(t,a))2 when Δa take 0.04, 0.05, 0.2, 0.25 and 0.5, separately. In Table 2(b), for Taylor approximations of the coefficients f, g and h, we choose the expansion order to take 5, 8, 10, 15 and 20, separately. Then the values of (P(t,a)Q(t,a))2 are given. In Table 2(c), for fixed expansion order x=10 and age step Δa=0.05, we give the corresponding value of (P(t,a)Q(t,a))2 when Δt take 0.005, 0.001, 0.0005, 0.0001 and 0.00005, separately. To illustrate our results more succinctly and forcefully, we use log-log plot Figure 3(a)–(c) to simulate the data of Table 2(a)–(c), respectively. In Figure 3(b), when expansion order x=10 and time step Δt=0.005, as the value of age step Δa increases, the value of (P(t,a)Q(t,a))2 increases. In Figure 3(b), as one would expect, as the expansion order increases, the value of (P(t,a)Q(t,a))2 is getting smaller with time step Δt=0.005, age step Δa=0.05. From Figure 3(c), it clearly reveals the fact that for fixed expansion order x=10 and age step Δa=0.05, the value of (P(t,a)Q(t,a))2 will tend to decrease when the increments of time Δt smaller. Thus, based on the above numerical analysis, we conclude that the Taylor approximation scheme is a simple and efficient numerical method for the age-dependent stochastic delay population equations with OU process and Poisson jumps.

    Table 2.  Error simulation between P(t,a) and Q(t,a) at different values of Δa, x, and Δt.
    (a) x=10,Δt=0.005
    Δa0.04 0.05 0.2 0.25 0.5
    (P(t,a)Q(t,a))2 0.03 0.04 0.07 0.08 0.2
    (b)Δt=0.005,Δa=0.05
    x 5 8 10 20 50
    (P(t,a)Q(t,a))2 0.08 0.07 0.04 0.01 0.005
    (c)x=10,Δa=0.05
    Δt 0.005 0.001 0.0005 0.0001 0.00005
    (P(t,a)Q(t,a))2 0.04 0.04 0.03 0.02 0.01

     | Show Table
    DownLoad: CSV
    Figure 3.  Error simulation between P(t,a) and Q(t,a) at different values of order Δa, x, and Δt.

    This paper discuss a Taylor approximation scheme for a class of stochastic age-dependent population equations. In order to obtain a more realistic and improved model compared to those in the literature [1,2,3,4,5,6,7,8,9,11], we introduce the mean-reverting Ornstein-Uhlenbeck (OU) process, time delay and Poisson jumps into equation and form a new system (2.7). We investigate the pth moments boundedness of exact solutions of age-dependent stochastic delay population equations with mean-reverting OU process and Poisson jumps (2.7). When the drift and diffusion coefficients satisfies Taylor approximations, we construct a Taylor approximation scheme for Eq (2.7) and reveal that the Taylor approximation solutions converge to the exact solutions for the equations. Furthermore, we estimate the order of the convergence. We also utilize a numerical example to confirm our theoretical results. In our future work, we will consider the effect of variable delay for stochastic age-dependent population equations and investigate the convergence of numerical methods for stochastic age-dependent population equations with OU process and variable delay.

    The authors are very grateful to the anonymous reviewers for their insightful comments and helpful suggestions. This research was funded by the "Major Innovation Projects for Building First-class Universities in China's Western Region" (ZKZD2017009).

    All authors declare no conflicts of interest in this paper.

    Proof of Lemma 3.1. For the purpose of simplification, let l=(α+1)2q. Applying Itô's formula to |Qt|l, we have

    |Qt|l= |Q0|l+t0l|Qs|l2Qsa(μe+(μ01μe)eηs)μ2(a)Qs,Qsds+t0l|Qs|l2(α1j=0f(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j,Qs)ds+t0l(l1)2|Qs|l2α2j=0g(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j22ds+t0l|Qs|l2(Qs,α2j=0g(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j)dBs+t0l|Qs|l2(Qs,α3j=0h(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j)dNs+t0l(l1)2|Qs|l2α3j=0h(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j22dNs= |Q0|l+t0l|Qs|l2Qsa(μe+(μ01μe)eηs)μ2(a)Qs,Qsds+t0l|Qs|l2(α1j=0f(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j,Qs)ds+t0l(l1)2|Qs|l2α2j=0g(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j22ds+t0l|Qs|l2(Qs,α2j=0g(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j)dBs+t0l|Qs|l2(Qs,α3j=0h(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j)dˉNs+λt0l|Qs|l2(Qs,α3j=0h(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j)ds+t0l(l1)2|Qs|l2|α3j=0h(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j|2dˉNs+λt0l(l1)2|Qs|l2|α3j=0h(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j|2ds, (5.1)

    where ˉNs=Nsλs is a compensated Poisson process. Since

    Qsa,Qs=A0Qsda(Qs)= 12(A0β(t,a)Qsda)2 12A0β2(t,a)daA0Q2sda 12ˉβ2A2|Qs|2, (5.2)

    by the assumptions (A1)-(A3), we get that

    |Qt|l |Q0|l+l(ˉβ2A22+μ01ˉμ)t0|Qs|lds+t0l|Qs|l2(α1j=0f(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j,Qs)ds+t0l(l1)2|Qs|l2α2j=0g(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j22ds+t0l|Qs|l2(Qs,α2j=0g(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j)dBs+t0l|Qs|l2(Qs,α3j=0h(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j)dˉNs+λt0l|Qs|l2(Qs,α3j=0h(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j)ds+t0l(l1)2|Qs|l2|α3j=0h(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j|2dˉNs+λt0l(l1)2|Qs|l2|α3j=0h(j)Z1s(s,Z1s,Z2s)j!(QsZ1s)j|2ds. (5.3)

    Using the well-known mean value theorem, we derive that there exists a θ(0,1) such that

    |Qt|l |Q0|l+l(ˉβ2A22+μ01ˉμ)t0|Qs|lds+t0l|Qs|l2(f(s,Qs,Z2s)f(α1+1)Z1s(s,Z1s+θ(QsZ1s),Z2s)(α1+1)!(QsZ1s)α1+1,Qs)ds+t0l(l1)2|Qs|l2g(s,Qs,Z2s)g(α2+1)Z1s(s,Z1s+θ(QsZ1s),Z2s)(α2+1)!(QsZ1s)α2+122ds
    +t0l|Qs|l2(Qs,g(s,Qs,Z2s)g(α2+1)Z1s(s,Z1s+θ(QsZ1s),Z2s)(α2+1)!(QsZ1s)α2+1)dBs+t0l|Qs|l2(Qs,h(s,Qs,Z2s)h(α3+1)Z1s(s,Z1s+θ(QsZ1s),Z2s)(α3+1)!(QsZ1s)α3+1)dˉNs+λt0l|Qs|l2(Qs,h(s,Qs,Z2s)h(α3+1)Z1s(s,Z1s+θ(QsZ1s),Z2s)(α3+1)!(QsZ1s)α3+1)ds+t0l(l1)2|Qs|l2|h(s,Qs,Z2s)h(α3+1)Z1s(s,Z1s+θ(QsZ1s),Z2s)(α3+1)!(QsZ1s)α3+1|2dˉNs+λt0l(l1)2|Qs|l2|h(s,Qs,Z2s)h(α3+1)Z1s(s,Z1s+θ(QsZ1s),Z2s)(α3+1)!(QsZ1s)α3+1|2ds. (5.4)

    Denotes

    H1(s,Qs,Z1s,Z2s)=f(α1+1)Z1s(s,Z1s+θ(QsZ1s),Z2s),H2(s,Qs,Z1s,Z2s)=g(α2+1)Z1s(s,Z1s+θ(QsZ1s),Z2s),H3(s,Qs,Z1s,Z2s)=h(α3+1)Z1s(s,Z1s+θ(QsZ1s),Z2s).

    Therefore, we obtain that

    t0l|Qs|l2(f(s,Qs,Z2s)H1(s,Qs,Z1s,Z2s)(α1+1)!(QsZ1s)α1+1,Qs)ds+t0l(l1)2|Qs|l2g(s,Qs,Z2s)H2(s,Qs,Z1s,Z2s)(α2+1)!(QsZ1s)α2+122ds+λt0l|Qs|l2(Qs,h(s,Qs,Z2s)H3(s,Qs,Z1s,Z2s)(α3+1)!(QsZ1s)α3+1)ds+λt0l(l1)2|Qs|l2|h(s,Qs,Z2s)H3(s,Qs,Z1s,Z2s)(α3+1)!(QsZ1s)α3+1|2ds t0l|Qs|l2(K(Qs+Z2s)+K1(α1+1)!2Qsα1+1,Qs)ds+t0l(l1)|Qs|l2(2K2Qs2+(K2(α2+1)!2α2+1)2Qs2α2+2)ds+λt0l(l1)|Qs|l2(2K2Qs2+(K3(α3+1)!2α3+1)2Qs2α3+2)ds+λt0l|Qs|l2(K(Qs+Z2s)+K3(α3+1)!2Qsα3+1,Qs)ds (1+λ)t0l|Qs|l2(2KQs2+K1(α1+1)!2α+1Qsα+2)ds+(1+λ)t0(2l(l1)K2|Qs|l+l(l1)(C(ˉα+1)!2α+1)2Qsl+2α)ds (1+λ)t0((2lK+2l(l1)K2)|Qs|l+(lK1(α1+1)!2α+1+l(l1)(C(ˉα+1)!2α+1)2)Qsl+2α)ds, (5.5)

    where ˉα=min{α1,α2,α3}.

    By the Burkholder-Davis-Gundy's inequality, we then have

    E[sup0stt0l|Qs|l2(Qs,g(s,Qs,Z2s)H2(s,Qs,Z1s,Z2s)(α2+1)!(QsZ1s)α2+1)dBs]+E[sup0stt0l|Qs|l2(Qs,h(s,Qs,Z2s)H3(s,Qs,Z1s,Z2s)(α3+1)!(QsZ1s)α3+1)dˉNs]+E[sup0stt0l(l1)2|Qs|l2|h(s,Qs,Z2s)H3(s,Qs,Z1s,Z2s)(α3+1)!(QsZ1s)α3+1|2dˉNs] E[sup0stt0l|Qs|l2(Ql22s,g(s,Qs,Z2s)H2(s,Qs,Z1s,Z2s)(α2+1)!(QsZ1s)α2+1)dBs]+E[sup0stt0l|Qs|l2(Ql22s,h(s,Qs,Z2s)H3(s,Qs,Z1s,Z2s)(α3+1)!(QsZ1s)α3+1)dˉNs]+E[sup0stt0l(l1)2|Qs|l2|h(s,Qs,Z2s)H3(s,Qs,Z1s,Z2s)(α3+1)!(QsZ1s)α3+1|2dˉNs] lE[supτut|Qu|l2(t0(Ql22s,g(s,Qs,Z2s)H2(s,Qs,Z1s,Z2s)(α2+1)!(QsZ1s)α2+1)dBs)]+lE[supτut|Qu|l2(t0(Ql22s,h(s,Qs,Z2s)H3(s,Qs,Z1s,Z2s)(α3+1)!(QsZ1s)α3+1)dˉNs)]+E[sup0stt0l(l1)2|Qs|l2|h(s,Qs,Z2s)H3(s,Qs,Z1s,Z2s)(α3+1)!(QsZ1s)α3+1|2dˉNs] 3lE[supτut|Qu|l2(t0|Qs|l2g(s,Qs,Z2s)H2(s,Qs,Z1s,Z2s)(α2+1)!(QsZ1s)α2+122ds)12]+3lE[supτut|Qu|l2(t0|Qs|l2h(s,Qs,Z2s)H3(s,Qs,Z1s,Z2s)(α3+1)!(QsZ1s)α3+122ds)12]+E[sup0stt0l(l1)2|Qs|l2|h(s,Qs,Z2s)H3(s,Qs,Z1s,Z2s)(α3+1)!(QsZ1s)α3+1|2dˉNs] 3lE[supτut|Qu|l2(t0|Qs|l2(2K2Qs2+(K2(α2+1)!2α2+1)2Qs2α2+2)ds)12]+3lE[supτut|Qu|l2(t0|Qs|l2(2K2Qs2+(K3(α3+1)!2α3+1)2Qs2α3+2)ds)12]+E[sup0stt0l(l1)2|Qs|l2|h(s,Qs,Z2s)H3(s,Qs,Z1s,Z2s)(α3+1)!(QsZ1s)α3+1|2dˉNs] 16E[supτut|Qu|l]+CE(t0|Qs|l2(2K2Qs2+(K2(α2+1)!2α2+1)2Qs2α2+2)ds)+16E[supτut|Qu|l]+CE(t0|Qs|l2(2K2Qs2+(K3(α3+1)!2α3+1)2Qs2α3+2)ds)+16E[supτut|Qu|l]+CE(t0|Qs|l2(2K2Qs2+(K3(α3+1)!2α3+1)2Qs2α3+2)ds) 12E[supτut|Qu|l]+CE(t0(|Qs|l+Qs2α+l)ds). (5.6)

    Note that for any t[0,T],

    E[supτut|Qu|l]=E[supτu0|Qu|l]E[sup0ut|Qu|l].

    Combining (5.4), (5.5) and (5.6), we can show that

    E[supτut|Qu|l] E[supτu0|ϕu|l]+l(ˉβ2A22+μ01ˉμ+(1+λ)(2K+2(l1)K2))t0Esupτus|Qu|lds+(1+λ)(lK1(α1+1)!2α+1+l(l1)(C(ˉα+1)!2α+1)2)t0EsupτusQul+2αds+12E[supτut|Qu|l]+C(t0(Esupτus|Qu|l+EsupτusQu2α+l)ds) E[supτu0|ϕu|l]+Ct0Esupτus|Qu|lds+Ct0(EsupτusQu2α+l)ds E[supτu0|ϕu|l]+Ct0(EsupτusQu2l)ds.

    Finally, by Generalization of the Bellman lemma [31], we obtain the desired result (3.2).



    [1] Bajec JF (2019) The interpretation and utilization of cultural heritage and its values by young people in Slovenia. Is heritage really boring and uninteresting? Etnoloska Tribina 49: 173–193. https://doi.org/10.15378/1848-9540.2019.42.07
    [2] Halu ZY, Küçükkaya AG (2016) Public participation of young people for architectural heritage conservation. Procedia Soc Behav Sci 225: 166–179. https://doi.org/10.1016/j.sbspro.2016.06.017 doi: 10.1016/j.sbspro.2016.06.017
    [3] Carbone F, Oosterbeek L, Costa C, et al. (2020) Extending and adapting the concept of quality management for museums and cultural heritage attractions: A comparative study of southern European cultural heritage managers' perceptions. Tour Manag Perspect 35: 100698. https://doi.org/10.1016/j.tmp.2020.100698 doi: 10.1016/j.tmp.2020.100698
    [4] Wang H, Zhang B, Qiu H (2022) How a hierarchical governance structure influences cultural heritage destination sustainability: A context of red tourism in China. J Hosp Tour Manag 50: 421–432. https://doi.org/10.1016/j.jhtm.2022.02.002 doi: 10.1016/j.jhtm.2022.02.002
    [5] Lin JH, Fan DXF, Tsaur SH, et al. (2021) Tourists' cultural competence: A cosmopolitan perspective among Asian tourists. Tour Manag 83: 104207. https://doi.org/10.1016/j.tourman.2020.104207 doi: 10.1016/j.tourman.2020.104207
    [6] Kotler P, Kartajaya H, Setiawan I (2017) Marketing 4.0: Do Tradicional Ao Digital. Rio de Janeiro: Sextante.
    [7] Clark C (2006) Defence heritage moves on to civilian futures. WIT Trans Ecol Environ 94: 197–207. https://doi.org/10.2495/BF060191 doi: 10.2495/BF060191
    [8] Hausman AV, Siekpe JS (2009) The effect of web interface features on consumer online purchase intentions. J Bus Res 62: 5–13. https://doi.org/10.1016/j.jbusres.2008.01.018 doi: 10.1016/j.jbusres.2008.01.018
    [9] Hall CM, Williams AM (2019) Tourism and Innovation. London: Routledge. https://doi.org/10.4324/9781315162836
    [10] Taormina F, Baraldi SB (2022) Museums and digital technology: a literature review on organizational issues. Eur Plan Stud 30: 1676–1694. https://doi.org/10.1080/09654313.2021.2023110 doi: 10.1080/09654313.2021.2023110
    [11] Su X, Li X, Kang Y (2019) A bibliometric analysis of research on intangible cultural heritage using CiteSpace. Sage Open 9: 2158244019840119. https://doi.org/10.1177/2158244019840119 doi: 10.1177/2158244019840119
    [12] Calvo-Porral C, Lévy-Mangin JP (2021) Examining the influence of store environment in hedonic and utilitarian shopping. Adm Sci 11: 6. https://www.mdpi.com/2076-3387/11/1/6
    [13] Samaroudi M, Echavarria KR, Perry L (2020) Heritage in lockdown: digital provision of memory institutions in the UK and US of America during the COVID-19 pandemic. Mus Manag Curatorship 35: 337–361. https://doi.org/10.1080/09647775.2020.1810483 doi: 10.1080/09647775.2020.1810483
    [14] Zollo L, Rialti R, Marrucci A, et al. (2022) How do museums foster loyalty in tech-savvy visitors? The role of social media and digital experience. Curr Issues Tour 25: 2991–3008. https://doi.org/10.1080/13683500.2021.1896487 doi: 10.1080/13683500.2021.1896487
    [15] Trunfio M, Lucia MD, Campana S, et al. (2022) Innovating the cultural heritage museum service model through virtual reality and augmented reality: The effects on the overall visitor experience and satisfaction. J Herit Tour 17: 1–19. https://doi.org/10.1080/1743873X.2020.1850742 doi: 10.1080/1743873X.2020.1850742
    [16] Sudbury-Riley L, Hunter-Jones P, Al-Abdin A, et al. (2020) The trajectory touchpoint technique: A deep dive methodology for service innovation. J Serv Res 23: 229–251. https://doi.org/10.1177/1094670519894642 doi: 10.1177/1094670519894642
    [17] Vargo SL, Lusch RF (2004) Evolving to a new dominant logic for marketing. J Mark 68: 1–17. https://doi.org/10.1509/jmkg.68.1.1.24036 doi: 10.1509/jmkg.68.1.1.24036
    [18] Lemon KN, Verhoef PC (2016) Understanding customer experience throughout the customer journey. J Mark 80: 69–96. https://doi.org/10.1509/jm.15.0420 doi: 10.1509/jm.15.0420
    [19] Wetter-Edman K, Vink J, Blomkvist J (2018) Staging aesthetic disruption through design methods for service innovation. Des Stud 55: 5–26. https://doi.org/10.1016/j.destud.2017.11.007 doi: 10.1016/j.destud.2017.11.007
    [20] Chen CL (2022) Strategic sustainable service design for creative-cultural hotels: A multi-level and multi-domain view. Local Environ 27: 46–79. https://doi.org/10.1080/13549839.2021.2001796 doi: 10.1080/13549839.2021.2001796
    [21] Chen JS, Kerr D, Chou CY, et al. (2017) Business co-creation for service innovation in the hospitality and tourism industry. Int J Contemp Hosp Manag 29: 1522–1540. https://doi.org/10.1108/IJCHM-06-2015-0308 doi: 10.1108/IJCHM-06-2015-0308
    [22] Roigé X, Arrieta-Urtizberea I, Seguí J (2021) The sustainability of intangible heritage in the COVID-19 era—resilience, reinvention, and challenges in Spain. Sustainability 13: 5796. https://doi.org/10.3390/su13115796 doi: 10.3390/su13115796
    [23] Marshall MT, Dulake N, Ciolfi L, et al. (2016) Using tangible smart replicas as controls for an interactive museum exhibition, in Proceedings of the TEI'16: Tenth International Conference on Tangible, Embedded, and Embodied Interaction, 159–167. https://doi.org/10.1145/2839462.2839493
    [24] Huang H, Mbanyele W, Fan S, et al. (2022) Digital financial inclusion and energy-environment performance: What can learn from China. Struct Chang Econ Dyn 63: 342–366. https://doi.org/10.1016/j.strueco.2022.10.007 doi: 10.1016/j.strueco.2022.10.007
    [25] Scholz J, Smith AN (2016) Augmented reality: Designing immersive experiences that maximize consumer engagement. Bus Horiz 59: 149–161. https://doi.org/10.1016/j.bushor.2015.10.003 doi: 10.1016/j.bushor.2015.10.003
    [26] Navarrete T (2019) Digital heritage tourism: Innovations in museums. World Leis J 61: 200–214. https://doi.org/10.1080/16078055.2019.1639920 doi: 10.1080/16078055.2019.1639920
    [27] Ng WK, Hsu FT, Chao CF, et al. (2023) Sustainable competitive advantage of cultural heritage sites: Three destinations in East Asia. Sustainability 15: 8593. https://doi.org/10.3390/su15118593 doi: 10.3390/su15118593
    [28] Hoyer WD, Kroschke M, Schmitt B, et al. (2020) Transforming the customer experience through new technologies. J Interact Mark 51: 57–71. https://doi.org/10.1016/j.intmar.2020.04.001 doi: 10.1016/j.intmar.2020.04.001
    [29] Huang H, Mbanyele W, Wang F, et al. (2023) Nudging corporate environmental responsibility through green finance? Quasi-natural experimental evidence from China. J Bus Res 167: 114147. https://doi.org/10.1016/j.jbusres.2023.114147 doi: 10.1016/j.jbusres.2023.114147
    [30] Huang H, Mo R, Chen X (2021) New patterns in China's regional green development: An interval Malmquist–Luenberger productivity analysis. Struct Chang Econ Dyn 58: 161–173. https://doi.org/10.1016/j.strueco.2021.05.011 doi: 10.1016/j.strueco.2021.05.011
    [31] Boukis A, Kabadayi S (2020) A classification of resources for employee-based value creation and a future research agenda. Eur Manag J 38: 863–873. https://doi.org/10.1016/j.emj.2020.05.001 doi: 10.1016/j.emj.2020.05.001
    [32] Clinehens JL (2019) CX That Sings: An Introduction to Customer Journey Mapping. Jennifer Clinehens. Available from: https://books.google.com.tw/books?id = qgPaDwAAQBAJ.
    [33] Mustak M (2019) Customer participation in knowledge intensive business services: Perceived value outcomes from a dyadic perspective. Ind Mark Manag 78: 76–87. https://doi.org/10.1016/j.indmarman.2017.09.017 doi: 10.1016/j.indmarman.2017.09.017
    [34] Wassler P, Fan DXF (2021) A tale of four futures: Tourism academia and COVID-19. Tour Manag Perspect 38: 100818. https://doi.org/10.1016/j.tmp.2021.100818 doi: 10.1016/j.tmp.2021.100818
    [35] Smyth L (2016) The disorganized family: Institutions, practices and normativity. Br J Sociol 67: 678–696. https://doi.org/10.1111/1468-4446.12217 doi: 10.1111/1468-4446.12217
    [36] Geus SD, Richards G, Toepoel V (2016) Conceptualisation and operationalisation of event and festival experiences: Creation of an event experience scale. Scand J Hosp Tour 16: 274–296. https://doi.org/10.1080/15022250.2015.1101933 doi: 10.1080/15022250.2015.1101933
    [37] Strauss A, Corbin J (1990) Basics of Qualitative Research. Newbury Park, CA: Sage publications.
    [38] Ridder HG (2017) The theory contribution of case study research designs. Bus Res 10: 281–305. https://doi.org/10.1007/s40685-017-0045-z doi: 10.1007/s40685-017-0045-z
    [39] Yin RK (2003) Designing case studies, In: Qualitative Research Methods, Sage, 5: 359–386.
    [40] Baxter P, Jack S (2008) Qualitative case study methodology: Study design and implementation for novice researchers. Qual Rep 13: 544–559. Available from: http://www.nova.edu/ssss/QR/QR13-4/baxter.pdf.
    [41] Etikan I, Musa SA, Alkassim RS (2016) Comparison of convenience sampling and purposive sampling. Am J Theor Appl Stat 5: 1–4. https://doi.org/10.11648/j.ajtas.20160501.11 doi: 10.11648/j.ajtas.20160501.11
    [42] Merriam SB, Tisdell EJ (2015) Qualitative Research: A Guide to Design and Implementation. Hoboken: John Wiley & Sons.
    [43] Nightingale AJ (2020) Triangulation, In: International Encyclopedia of Human Geography, 2 Eds., Elsevier, 477–480. https://doi.org/https://doi.org/10.1016/B978-0-08-102295-5.10437-8
    [44] Kurtmollaiev S, Fjuk A, Pedersen PE, et al. (2018) Organizational transformation through service design: The institutional logics perspective. J Serv Res 21: 59–74. https://doi.org/10.1177/1094670517738371 doi: 10.1177/1094670517738371
    [45] Hu M, Zhang M, Wang Y (2017) Why do audiences choose to keep watching on live video streaming platforms? An explanation of dual identification framework. Comput Hum Behav 75: 594–606. https://doi.org/10.1016/j.chb.2017.06.006 doi: 10.1016/j.chb.2017.06.006
    [46] Anshu K, Gaur L, Singh G (2022) Impact of customer experience on attitude and repurchase intention in online grocery retailing: A moderation mechanism of value Co-creation. J Retail Consum Serv 64: 102798. https://doi.org/10.1016/j.jretconser.2021.102798 doi: 10.1016/j.jretconser.2021.102798
    [47] Blocker CP, Barrios A (2015) The transformative value of a service experience. J Serv Res 18: 265–283. https://doi.org/10.1177/1094670515583064 doi: 10.1177/1094670515583064
    [48] Mele E, Filieri R, De Carlo M (2023) Pictures of a crisis. Destination marketing organizations' Instagram communication before and during a global health crisis. J Bus Res 163: 113931. https://doi.org/10.1016/j.jbusres.2023.113931
    [49] Ferreira J, Sousa B (2020) Experiential marketing as leverage for growth of creative tourism: A co-creative process, In: Advances in Tourism, Technology and Smart Systems. Smart Innovation, Systems and Technologies, Singapore: Springer, 171. https://doi.org/10.1007/978-981-15-2024-2_49
    [50] Falk JH, Dierking LD (2016) The Museum Experience. New York: Routledge. https://doi.org/10.4324/9781315417899
    [51] Della Corte V, Aria M (2016) Coopetition and sustainable competitive advantage. The case of tourist destinations. Tour Manag 54: 524–540. https://doi.org/10.1016/j.tourman.2015.12.009
    [52] Chang AYP, Hung KP (2021) Development and validation of a tourist experience scale for cultural and creative industries parks. J Dest Mark Manage 20: 100560. https://doi.org/10.1016/j.jdmm.2021.100560 doi: 10.1016/j.jdmm.2021.100560
    [53] Tran TP, Mai ES, Taylor EC (2021) Enhancing brand equity of branded mobile apps via motivations: A service-dominant logic perspective. J Bus Res 125: 239–251. https://doi.org/10.1016/j.jbusres.2020.12.029 doi: 10.1016/j.jbusres.2020.12.029
    [54] Sayer F (2024) Heritage and Wellbeing: The Impact of Heritage Places on Visitors' Wellbeing. Oxford: Oxford University Press, 142–168. https://doi.org/10.1093/9780191914539.003.0006
    [55] Settimini E (2021) Cultural landscapes: Exploring local people's understanding of cultural practices as "heritage". J Cult Herit Manage Sustain Dev 11: 185–200. https://doi.org/10.1108/JCHMSD-03-2020-0042 doi: 10.1108/JCHMSD-03-2020-0042
    [56] Kostka G, Mol APJ (2013) Implementation and participation in China's local environmental politics: Challenges and innovations. J Environ Policy Plan 15: 3–16. https://doi.org/10.1080/1523908X.2013.763629 doi: 10.1080/1523908X.2013.763629
    [57] Echavarria KR, Samaroudi M, Dibble L, et al. (2022) Creative experiences for engaging communities with cultural heritage through place-based narratives. ACM J Comput Cult Heritage (JOCCH) 15: 1–19. https://doi.org/10.1145/3479007 doi: 10.1145/3479007
    [58] Raffaelli R, Franch M, Menapace L, et al. (2022) Are tourists willing to pay for decarbonizing tourism? Two applications of indirect questioning in discrete choice experiments. J Environ Plan Manag 65: 1240–1260. https://doi.org/10.1080/09640568.2021.1918651 doi: 10.1080/09640568.2021.1918651
  • This article has been cited by:

    1. Huili Wei, Wenhe Li, Dynamical behaviors of a Lotka-Volterra competition system with the Ornstein-Uhlenbeck process, 2023, 20, 1551-0018, 7882, 10.3934/mbe.2023341
    2. Meng Gao, Xiaohui Ai, A stochastic Gilpin-Ayala nonautonomous competition model driven by mean-reverting OU process with finite Markov chain and Lévy jumps, 2024, 32, 2688-1594, 1873, 10.3934/era.2024086
    3. Meng Gao, Xiaohui Ai, A stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps, 2024, 21, 1551-0018, 4117, 10.3934/mbe.2024182
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2181) PDF downloads(66) Cited by(0)

Figures and Tables

Figures(8)  /  Tables(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog