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A discrete extension of the Xgamma random variable: mathematical framework, estimation methods, simulation ranking, and applications to radiation biology and industrial engineering data

  • Received: 29 December 2024 Revised: 03 March 2025 Accepted: 06 March 2025 Published: 19 March 2025
  • MSC : 62E99, 62E15

  • Count data modeling and its practical applications have garnered significant attention in recent research, owing to its relevance in a wide range of fields. This study specifically explores a novel discrete distribution characterized by two parameters, which is derived using the survival discretization method. The statistical properties of this distribution are thoroughly explained in closed forms, with several key mathematical attributes also derived. These characteristics underscore the distribution's effectiveness in modeling data that exhibit (right-skewed) asymmetry and have extended heavy tails, making it particularly suitable for such real-world applications. Furthermore, the failure rate function corresponding to this distribution is particularly appropriate for scenarios characterized by an increasing or bathtub-shaped failure rate over time. The model is also highly versatile, offering valuable insights into probabilistic modeling for datasets that display over dispersion, under dispersion, or equi dispersion. The study introduces several estimation techniques, including the maximum product of spacings, Anderson–Darling, right–tail Anderson–Darling, maximum likelihood estimation, least squares, weighted least squares, Cramer–Von–Mises, and percentile methods. Each of these methods is explained in detail, providing a comprehensive understanding of their application. A ranking simulation study is conducted to evaluate the performance of these estimators across varying sample sizes, using ranking techniques to identify the most effective estimator in different scenarios. The analysis of real-world datasets from biotechnology and industrial engineering further demonstrates the practical utility and relevance of the proposed model. The results highlight the model's ability to offer accurate and insightful analyses, reinforcing its significance in count data modeling and its wide-ranging applications.

    Citation: Mohamed S. Algolam, Mohamed S. Eliwa, Mohamed El-Dawoody, Mahmoud El-Morshedy. A discrete extension of the Xgamma random variable: mathematical framework, estimation methods, simulation ranking, and applications to radiation biology and industrial engineering data[J]. AIMS Mathematics, 2025, 10(3): 6069-6101. doi: 10.3934/math.2025277

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  • Count data modeling and its practical applications have garnered significant attention in recent research, owing to its relevance in a wide range of fields. This study specifically explores a novel discrete distribution characterized by two parameters, which is derived using the survival discretization method. The statistical properties of this distribution are thoroughly explained in closed forms, with several key mathematical attributes also derived. These characteristics underscore the distribution's effectiveness in modeling data that exhibit (right-skewed) asymmetry and have extended heavy tails, making it particularly suitable for such real-world applications. Furthermore, the failure rate function corresponding to this distribution is particularly appropriate for scenarios characterized by an increasing or bathtub-shaped failure rate over time. The model is also highly versatile, offering valuable insights into probabilistic modeling for datasets that display over dispersion, under dispersion, or equi dispersion. The study introduces several estimation techniques, including the maximum product of spacings, Anderson–Darling, right–tail Anderson–Darling, maximum likelihood estimation, least squares, weighted least squares, Cramer–Von–Mises, and percentile methods. Each of these methods is explained in detail, providing a comprehensive understanding of their application. A ranking simulation study is conducted to evaluate the performance of these estimators across varying sample sizes, using ranking techniques to identify the most effective estimator in different scenarios. The analysis of real-world datasets from biotechnology and industrial engineering further demonstrates the practical utility and relevance of the proposed model. The results highlight the model's ability to offer accurate and insightful analyses, reinforcing its significance in count data modeling and its wide-ranging applications.



    Overfishing occurs when fish are harvested at a rate faster than the resource can replenish itself. This practice has consequences that extend beyond the depletion of single-species fish populations, potentially leading to global food insecurity, ecological extinction, and other devastating effects[1,2,3,4]. Marine ecosystems and their associated livelihoods worldwide are threatened by overfishing. Nicholas et al. [5] have shown that overfishing is pushing over one-third of all sharks and rays towards global extinction. They suggest that marine life could face a future similar to that of land biodiversity, where human pressures have resulted in the loss of numerous species, possibly triggering a sixth mass extinction. According to IUCN Red List criteria [6], they estimate that one-quarter of 1,041 chondrichthyan fishes are threatened due to overfishing (both targeted and incidental). Therefore, it is urgent to establish the optimal harvesting strategy (OHS) for overfishing to prevent extinctions and promote population recovery. Consequently, investigating OHS has significant implications for marine ecosystems.

    Mathematical models are widely utilized to analyze the OHS of populations [7,8,9,10,11,12,13]. Clark [7] proposes several deterministic population systems with harvesting and establishes the OHS. Martin and Ruan [8] investigate the predator-prey models with delay and prey harvesting. However, nature is full of random phenomena, and various populations in the ecosystem experience different forms of random perturbations. May [9] indicates that environmental noise affects competition coefficient, environmental capacity, the growth rate and other parameters of population systems. Lande et al.[10] demonstrate that the extinction of numerous species caused by over-harvesting may be largely due to the ignorance of randomness. Therefore, it is crucial to investigate the OHS of population system in random environment. Li and Wang [11] establish a general stochastic logistic population model with white noises and explore the optimal harvesting policy. Wang et al.[12] propose a stochastic predator-prey model with time delays and shows how randomness influences the extinction and OHS. Therefore, exploring the OHS of stochastic marine population systems is meaningful.

    Lévy noise, another type of environmental perturbations, is considered below. During the process of species evolution, population growth is often affected by random disturbances in the environment, such as toxic pollutants, epidemics, earthquakes, floods and other factors [14,15,16,17]. For example, the Sandoz chemical accident in 1986 resulted in the death of over 600,000 fish in the Rhine [14], while the 2010 Gulf of Mexico oil spill caused a devastating impact on marine life in the Gulf of Mexico [15]. Bao et al. [16] propose that non-Gaussian Lévy noise is suitable for concluding these phenomena and revealed that Lévy noise can inhibit the explosive increase of population. Zhang et al. [17] also proposed to use Lévy noise to describe the random perturbations.

    A lot of work has been done to study the stochastic food chain population dynamics model so far. Our previous work has investigated the dynamics of a stochastic three species food chain model in [18], the dynamics of a four-species stochastic food chain model with delays in [19] and the dynamic behaviors of a stochastic n-species food chain model with Lévy noises and white noises in [20] as follows:

    {dx1(t)=x1(t)[r1a11x1(t)a12x2(t)]dt+σ1x1(t)dB1(t)+Zx1(t)γ1(u)˜N(dt,du),dxi(t)=xi(t)[riaiixi(t)+aii1xi1(t)aii+1xi+1(t)]dt+σixi(t)dBi(t)+Zxi(t)γi(u)˜N(dt,du),2in1,dxn(t)=xn(t)[rnannxn(t)+ann1xn1(t)]dt+σnxn(t)dBn(t)+Zxn(t)γn(u)˜N(dt,du). (1.1)

    By introducing a new research method, that is algebraic approach, the sufficient and necessary criteria for the stochastic extinction, persistence in mean and global stability are established in [20]. These conclusions directly generalize some research results on three and four species stochastic food chain dynamics models in [18,19]. However, due to the high dimensionality of model (1.1) and technical difficulties, the stability in distribution and optimal harvesting problems have not been discussed until now. In this paper, we consider the effects of harvesting and Lévy noises, then address these issues by proposing a stochastic n-species food chain model as follows:

    {dx1(t)=x1(t)[r1h1a11x1(t)a12x2(t)]dt+σ1x1(t)dB1(t)+Zx1(t)γ1(u)˜N(dt,du),dxi(t)=xi(t)[rihiaiixi(t)+aii1xi1(t)aii+1xi+1(t)]dt+σixi(t)dBi(t)+Zxi(t)γi(u)˜N(dt,du),2in1,dxn(t)=xn(t)[rnhiannxn(t)+ann1xn1(t)]dt+σnxn(t)dBn(t)+Zxn(t)γn(u)˜N(dt,du). (1.2)

    On the basis of the research results in [20], our primary objective is to establish criteria for the stability in distribution of model (1.2). Additionally, we aim to establish criteria for the optimal harvesting strategy in the sense of sustainable yield. Lastly, we will verify the theoretical results through numerical examples and propose some open questions and propositions.

    This paper is organized as follows. In Section 2, we provide the basic assumption, and further introduce several important lemmas that will be used to prove the major results. In Section 3, we state and prove the stability in distribution of model (1.2). In Section 4, we outline the non-existence and existence criteria for OHS and MESY. Finally, in Section 5, we illustrate our major results through numerical examples.

    Denote by (Ω,{Ft}t0,P) the complete probability space, and its filtration {Ft}t0 satisfies the usual conditions. In this paper, we assume that model (1.2) is defined on the probability space (Ω,{Ft}t0,P). In addition, for an integrable function f(t) defined for t[0,+), we define the mean value of f(t) on interval [0,t] for any t>0 by f(t)=1tt0f(s)ds.

    In models (1.1) and (1.2), parameter r1>0 is the intrinsic growth rate of species x1, ri0 represents the death rate of species xi for i=2,3,,n, hi0 represents the harvesting rate of human for the species xi for i=1,2,,n, xi(t) is the left limit of xi(t) at time t, aii>0(i=1,2,,n) is the density dependent coefficient of species xi, aii+10(i=2,3,,n1) is the capture rate of which species xi+1 preys on species xi, aii10(i=2,3,,n) stands for efficiency of food conversion from species xi1 to species xi. Bi(t)(i=1,2,,n) are independent standard Brownian motion, and σ2i>0 is the intensity of Bi(t). Lévy noises ˜N(dt,du)=N(dt,du)λ(du)dt, where N(dt,du) is a poisson counting measure with characteristic measure λ on a Borel-measurable subset Z of (0,) satisfying λ(Z)<+. Functions γi(u):ZR(i=1,2,,n) represent the effects of random jumps for species xi that are assumed to be continuous on Z. In this paper we further assume that Lévy noises ˜N(dt,du) and Brownian motion Bi(t)(i=1,2,,n) are also independent.

    For any solution (x1(t),,xn(t)) of model (1.2), the initial value is given as follows.

    xi(0)=xi0,i=1,2,,n, (2.1)

    where xi0(i=1,2,,n) are positive constants.

    Denote bi=ri+z[ln(1+γi(u))γi(u)]λ(du)σ2i2 and ci=bihi for i=1,2,,n, and αk=(c1,c2,,ck)T for k=1,2,,n. Furthermore, we denote

    Ak=[a11a120000a21a22a23000000ak1k2ak1k1ak1k0000akk1akk].

    Particularly, we have

    A1=[a11],A2=[a11a12a21a22].

    Denote by Hk the determinant of Ak. Let A(i)k be the matrix gained from matrix Ak by changing its i-th column to αk, and D(i)k is the determinant of A(i)k. Furthermore, for any square matrix P we denote by P(i,j) the algebraic complement corresponding to the element of i-th row and j-th column of P.

    For the Lévy noises in models (1.1) and (1.2), we always assume that the following condition is satisfied.

    (H) For i=1,2,,n and uZ, 1+γi(u)>0 holds, and a constant z>0 exists such that

    Z[ln(1+γi(u))]2λ(du)<z.

    Using the same method as in [20], the following conclusions can be established for model (1.2).

    Lemma 1. Hk>0 for every k=1,2,,n.

    Lemma 2. For any initial values (x10,x20,,xn0)Rn+, model (1.2) possesses a globally unique positive solution x(t)=(x1(t),,xn(t))Rn+a.s. for all t0 satisfying the initial condition (2.1). Furthermore, for any constant q>0, there are constants Ci(q)>0(i=1,2,,n) such that

    lim suptE[xqi(t)]Ci(q),i=1,2,,n.

    Similar to [20], we define

    N1=a12a11A(2)k(3,2)Hk(3,k),Ni=aii1aiiNi1+aii+1aiiA(i+1)k(i+2,i+1)Hk(i+2,k),i=1,2,,k2,Nk1=ak1k2ak1k1Nk2.

    Lemma 3. For any positive solution (x1(t),,xn(t)) of model (1.2), assume that D(n)n>0 and (1+ann1annNn1)>0 if n3, then limtxi(t)=D(i)nHn a.s., i=1,2,,n.

    A simple calculation we can obtain that for any integers k,i,j{1,2,,n} with i,jk

    A(i)k(j,i)=Ak(j,i).

    Therefore, we further have

    N1=a12a11Ak(3,2)Hk(3,k),Ni=aii1aiiNi1+aii+1aiiAk(i+2,i+1)Hk(i+2,k),i=1,2,,k2,Nk1=ak1k2ak1k1Nk2. (2.2)

    Theorem 1. Let (x1(t;ψ),x2(t;ψ),,xn(t;ψ)) and (z1(t;ψ),z2(t;ψ),,zn(t;ψ)) be two solutions of model (1.2) with initial values ψ,ψRn+. Assume that there are positive constants wi(i=1,2,,n) such that

    w1a11w2a21>0,wiaiiwi1ai1iwi+1ai+1i>0(i=2,3,,n1),wnannwn1an1n>0.

    Then we have

    limtE(ni=1|xi(t,ψ)zi(t,ψ)|2)12=0.

    Proof. We only need to show

    limtE|xi(t;ψ)zi(t;ψ)|=0,i=1,2,,n. (3.1)

    Define the functions as follows

    Vi(xi)=|lnxi(t;ψ)lnzi(t;ψ)|,i=1,2,,n.

    Applying Itô formula, then

    LVi(xi)aii|xi(t;ψ)zi(t;ψ)|+aii1|xi1(t;ψ)zi1(t;ψ)|+aii+1|xi+1(t;ψ)zi+1(t;ψ)|,i=1,2,,n. (3.2)

    where a10=ann+1=0 are stupulated. Define the following function

    V(t)=ni=1wiVi(xi). (3.3)

    From (3.1) we obtain

    LV(t)=ni=1wiLVi(xi)ni=1(wiaiiwi1ai1iwi+1ai+1i)|xi(t;ψ)zi(t;ψ)|,

    where a01=an+1n=0 and ω0=ωn+1=0 are stipulated. Therefore,

    E[V(t)]E[V(0)]ni=1(wiaiiwi1ai1iwi+1ai+1i)t0E[|xi(s;ψ)zi(s;ψ)]|ds.

    which implies

    0E[|xi(s;ψ)zi(s;ψ)|]ds<+,i=1,2,,n. (3.4)

    Define the functions

    Fi(t)=E[|xi(t;ψ)zi(t;ψ)|],i=1,2,,n.

    Then, for any t1,t2[0,+) and each i=1,2,,n,

    |Fi(t2)Fi(t1)|=|E[|xi(t2;ψ)zi(t2;ψ)||xi(t1;ψ)zi(t1;ψ)|]|E[|xi(t2;ψ)xi(t1;ψ)|]+E[|zi(t2;ψ)zi(t1;ψ)|]. (3.5)

    Applying Itô formula to model (1.2), then

    xi(t2;ϕ)xi(t1;ϕ)=t2t1xi(s;ϕ)[rihi+aii1xi1(s;ϕ)aiixi(s)aii+1xi+1(s;ϕ)]ds+t2t1σixi(s;ϕ)dBi(s)+t2t1Zxi(s)γi(u)˜N(ds,du),i=1,2,...,n. (3.6)

    For any t2>t1 and q>1, from the first equation of (3.6) and Hölder's inequality, then

    (E[|x1(t2;ψ)x1(t1;ψ)|])qE[(t2t1x1(s;ψ)|r1h1a11x1(s;ψ)a12x2(s;ψ)|ds+|t2t1σ1x1(s;ψ)dB1(s)|)q]+|t2t1Zx1(s)γ1(u)˜N(ds,du)|3qE[(t2t1x1(s;ψ)|r1h1a11x1(s;ψ)a12x2(s;ψ)|ds)q]+3qE[|t2t1σ1x1(s;ψ)dB1(s)|q]+3qE[|t2t1Zxi(s)γi(u)˜N(ds,du)|q]. (3.7)

    Similarly,

    E[(t2t1x1(s;ψ)|r1h1a11x1(s;ψ)a12x2(s;ψ)|ds)q](t2t1)q1E[t2t1(|r1h1|x1(s;ψ)+a11x21(s;ψ)+a12x1(s;ψ)x2(s;ψ))qds]3q(t2t1)q1|r1h1|qt2t1E[xq1(s;ψ)]ds+3qaq11(t2t1)q1t2t1E[x2q1(s;ψ)]ds+3q(t2t1)q1E[t2t1(a12x1(s;ψ)x2(s;ψ))qds]. (3.8)

    and

    E[t2t1(a12x1(s;ψ)x2(s;ψ))qds]E[t2t1(12a12x21(s;ψ)+12a12x22(s;ψ))qds]aq12t2t1E[x2q1(s;ψ)]ds+aq12t2t1E[x2q2(s;ψ)]ds. (3.9)

    In view of Theorem 7.1 in [21] on the Burkholder-Davis-Gundy inequality, for 1<q2 and any t2>t1, then

    E[|t2t1σ1x1(s;ψ)dB1(s)|q]DqE[t2t1|σ1x1(s;ψ)|2ds]q/2Dqσq1(t2t1)q22t2t1E[xq1(s;ψ)]ds. (3.10)

    Using the Kunita first inequality in [21], there exists a constant Kq>0 and from assumption (H) we can obtain

    E[|t2t1Zx1(s;ψ)γ1(u)˜N(ds,du)|q]KqE[t2t1Z|x1(s)γ1(u)|2λ(du)ds]q/2+E[t2t1Z|x1(s)γ1(u)|qλ(du)ds]KqLq/2(t2t1)q22t2t1E[xq1(s;ψ)]ds+KqLq/2t2t1E[xq1(s;ψ)]ds. (3.11)

    From Lemma 2, there exists Ci(q)>0 for i=1,2,,n such that suptγE[xqi(t)]Ci(q). Therefore, from (3.7)–(3.10) there is a δ>0 satisfies that for 1<q2 and any t10,t20 with |t2t1|δ, such that

    (E[|x1(t2;ψ)x1(t1;ψ)|])q3q[Dqσq1(t2t1)q2C1(q)]+(t2t1)q23q[KqLq/2C1(q)+KqLq/2(t2t1)1q/2C1(q)]+3q[3q(t2t1)q|r1h1|qC1(q)+3qaq11(t2t1)qC1(2q)]+32q(t2t1)qaq12[C1(2q)+C2(2q)]M1|t2t1|q2,

    where

    M1=3qDqσq1C1(q)+3qKqLq/2[1+δ2q2]C1(q)+[81δ]q2[|r1h1|qC1(q)+aq11C1(2q)]+[81δ]q2aq12[C1(2q)+C2(2q)].

    Similarly, we also obtain

    (E[|z1(t2;ψ)z1(t1;ψ)|])qM1|t2t1|q2.

    for any t10,t20 with |t2t1|δ and 1<q2. Thus, from (3.5), we obtain

    |F1(t2)F1(t1)|E[|x1(t2;ψ)x1(t1;ψ)|]+E[|z1(t2;ψ)z1(t1;ψ)|]2(M1)1q|t2t1|. (3.12)

    Using similar argument, for Fi(t)(i=2,3,,n), we also can obtain that there is a δ>0 for any t10,t20 with |t2t1|δ and 1<q2

    |Fi(t2)Fi(t1)|2(Mi)1q|t2t1|,i=2,3,,n, (3.13)

    where

    Mi=3qDqσqiCi(q)+3qKqLq/2[1+δ2q2]Ci(q)+[144δ]q2[|ri+hi|qCi(q)+aqiiCi(2q)]+[144δ]q2aqii+1[Ci(2q)+Ci+1(2q)]+[144δ]q2aqii1[Ci1(2q)+Ci+1(2q)],i=2,3,,n.

    From (3.12) and (3.13), we obtain that Fi(t),i=1,2,,n for t(0,) are uniformly continuous. Therefore, from (3.4) and Barbalat lemma in [22] we finally obtain that (3.1) holds. This completes the proof.

    Let P(Rn+) be the space of all probability measures on C(Rn+). For Q1,Q2P(Rn+), we define

    dLB(Q1,Q2)=supfLB|Rn+f(y)Q1(dy)Rn+f(y)Q2(dy)|, (3.14)

    where

    LB={f:Rn+R:|f(y1)f(y2)|y1y2,|f()|1}. (3.15)

    Denote by p(t,ϕ,dx) the transition probability of process x(t)=(x1(t),x2(t),,xn(t)). We have the following results.

    Theorem 2. Assume that there exists constants wi>0(i=1,2,,n) such that

    w1a11w2a21>0,wiaiiwi1ai1iwi+1ai+1i>0(i=2,3,,n1),wnannwn1an1n>0.

    Then model (1.2) is asymptotically stable in distribution, that is, there is a probability measure v() satisfies that for any initial function ψRn+, the transition probability p(t,ψ,) of solution (x1(t,ψ),x2(t,ψ),,xn(t,ψ)) such that

    limtdLB(p(t,ψ,),v())=0.

    Proof. For any initial data ψRn+, the family probability p(t,ψ,) is tight which obtained from Theorem 1 and Chebyshev's inequality. For any fLB and t,s>0, we compute

    Ef(y(t+s;ψ))p(t+s,ψ,dy)Ef(y(t;ψ))p(t,ψ,dy)=E[E(f(y(t+s;ψ))Fs)]E[f(y(t;ψ))]=Rn+E[f(y(t;ϕ))]p(s,ψ,dϕ)E[f(y(t;ψ))]∣≤Rn+Ef(y(t;ϕ))Ef(y(t;ψ))p(s,ψ,dϕ).

    It is clear that there exists a constant T>0 from (3.1), such that for all tT,

    supfLBEf(y(t;ϕ))Ef(y(t;ψ))∣≤ε.

    Consequently

    Ef(y(t+s;ψ))Ef(y(t;ψ))∣≤ε.

    Since f is arbitrary, we gain

    supfLBEf(y(t+s;ψ))Ef(y(t;ψ))∣≤ε.

    That is,

    dLB(p(t+s,ψ,),p(t,ψ,))ε.

    Therefore {p(t,ψ,):t0} is Cauchy in the space P(Rn+). Hence, there is a unique v()P(Rn+) such that

    limtdLB(p(t,ψ0,),v())=0.

    where ψ0=(c10,c20,...,cn0)TRn+, and c10,,cn0 are positive constants. It then follows from (3.1) that

    limtdLB(p(t,ψ,),p(t,ψ0,)=0.

    Hence,

    limtdLB(p(t,ψ,),v()limtdLB(p(t,ψ,),p(t,ψ0,))+limtdLB(p(t,ψ0,),v())=0.

    This completes the proof.

    Since Hn>0 in Lemma 1, the inverse matrix A1n exists. Let B=(b1,b2,,bn)T and H=(h1,h2,,hn)T. Moreover, let E be the unit matrix, and H=(An(A1n)T+E)1B(h1,h2,,hn)T.

    Theorem 3. Assume that the following conditions hold

    (1) (1+ann1annNn1)>0 if n3;

    (2) there exists constants wi>0(i=1,2,,n) such that

    w1a11w2a21>0,wiaiiwi1ai1iwi+1ai+1i>0(i=2,3,,n1),wnannwn1an1n>0.

    Then the following conclusions hold.

    (A1) If D(n)n|hi=hi>0, A1n+(A1n)T is positive semi-definite and hi0 (i=1,2,,n), then model (1.2) has the OHS H=H, and

    MESYS(H)=(H)TA1n(BH). (4.1)

    (A2) If any of the following conditions stands, then the OHS for model (1.2) does not exist.

    (B1) D(n)n|hi=hi,i=1,2,,n0;

    (B2) There is a hi with hi<0;

    (B3) A1n+(A1n)T is not positive semi-definite.

    Proof. Let Y={H=(h1,h2,,hn)TRn:D(n)n>0,hi0,i=1,2,,n}, which means that for any HY, the conclusion of Lemma 3 stands.

    First, we prove the conclusion (A1). Since HY, the set Y is not empty. A unique probability measure v() is gained from Theorem 2. Then, v() is also strong mixing by Corollary 3.4.3 [23]. From Theorem 3.2.6[23], we gain that the measure v() is ergodic. For any initial value ψRn+, model (1.2) possess a positive gloabl solution x(t)=(x1(t),x2(t),,xn(t)). It yields by Theorem 3.3.1 [23] that

    limt1tt0HTx(s)ds=Rn+HTxv(dx), (4.2)

    where H=(h1,h2,,hn)TY. Let ϱ(z) be the stationary probability density of model (1.2), then we have

    S(H)=limtE[ni=1hixi(t)]=limtE[HTx(t)]=Rn+HTxϱ(x)dx. (4.3)

    In view of the invariant measure is sole, there also exists a one-to-one correspondence among ϱ(z) and its corresponding invariant measure. Hence, we have

    Rn+HTxϱ(x)dx=Rn+HTxv(dx). (4.4)

    Combining (4.2)–(4.4) with the conclusion of Lemma 3, then

    S(H)=limt+1tt0HTx(s)ds=h1limt+1tt0x1(s)ds+h2limt+1tt0x2(s)ds++hnlimt+1tt0xn(s)ds=h1D(1)nHn+h2D(2)nHn++hnD(n)nHn.

    Moreover, S(H)=HT(An)1(BH) is derived by careful calculation. Computing the gradient of S(H), then

    S(H)H=HTH(An)1(BH)+(BH)TH((An)1)TH.

    Since HTH=E, then

    S(H)H=(An)1(BH)((An)1)TH=(An)1B((An)1+((An)1)T)H.

    By solving S(H)H=0, the following critical value is gained

    H=((An)1+((An)1)T)1(An)1B.

    That is,

    H=[(An)1((An)((An)1)T+E)]1(An)1B=((An)((An)1)T+E)1B=H.

    Calculating the Hessian matrix of S(H), we obtain

    H(S(H)H)=((An)1+((An))T)HH=((An)1+((An)1)T). (4.5)

    Since (An)1+((An)1)T is positive semi-definit, from the existence principle of extremum value of multi-variable function we obtain that S(H) has a global maximum value at H=H. Obviously, H is also sole. Therefore, if HY, that is hi0(i=1,2,,n) and D(n)n|hi=hi,i=1,2,,n>0, then we finally obtain that H is an OHS, and MESY shown in (20).

    Next, we prove the conclusion (A2). We first assume that (B1) or (B2) stands. Suppose that ˜Γ=(γ1,γ2,,γn) is the OHS for model (1.2), thus ΓY, that is

    D(n)n|hi=γi,i=1,2,,n>0,γi0,i=1,2,,n. (4.6)

    On the other hand, if ΓY is the OHS, then Γ is a unique solution of equation S(H)H=0. Therefore, we have (h1,h2,,hn)=(γ1,γ2,,γn). Thus, condition (4.6) becomes into

    D(n)n|hi=hi,i=1,2,,n>0,hi0,i=1,2,,n.

    However, this is impossible.

    Finally, consider (B3). It can be assumed that (B1) and (B2) fail to stand in this case. Therefore, hi0(i=1,2,,n) and D(n)n|hi=hi,i=1,2,,n>0. Thus, Y is not empty, i.e., (4.2)(4.4) hold. Denote (An)1+((An)1)T=(hij)n×n. Then, by calculating we have

    h11=2(a22a33ann++a22a34ann1+a23a32ann)Hn

    Obviously, h11>0, which means that (An)1+((An)1)T is not negative semi-definite. It can be further derived from the conditions in (B3) that (An)1+((An)1)T is indefinite. Hence, there is not the OHS if B3 holds. This completes the proof.

    Remark 1. In Theorem 3 we see that the conditions (1) and (2) are required. From the following argumentations we will find that when n=3 and n=4 condition (2) can bring out condition (1). Therefore, we propose the following proposition. However, when n>4 whether condition (2) can bring out condition (1) still is an open problem. Thus, the following conjecture can be proposed.

    Proposition 1. When n=3 and n=4, if condition (2) holds then condition (1) also holds.

    Proof. Let k=3, then condition (2) becomes to

    w1a11w2a21>0,w2a22w1a12w3a32>0w3a33w2a23>0. (4.7)

    From (2.2), by calculating we have

    N1=a12a23a11a22+a12a21,N2=a12a21a23a22(a11a22+a12a21),

    and

    1+a32a33N2=a22a33(a11a22+a12a21)a12a21a23a32a22a33(a11a22+a12a21).

    From (4.7) we derive

    a221w2(w1a12)+w3a32,a33w2a23w3.

    Therefore, we further have

    a22a33(a11a22+a12a21)a12a21a23a32>[1w2(w1a12+w3a32)1w3w2a23(a11a22+a12a21)]a12a21a23a32=(w1a12w2w3+a32w2)w2a23(a11a22+a12a21)a12a21a23a32=(w1a12a23w3+a23a32)(a11a22+a12a21)a12a21a23a32=w1a12a23w3(a11a22+a12a21)+a11a22a23a32>0.

    This shows that 1+a32a33N2>0.

    Let k=4, then condition (2) becomes to

    w1a11w2a21>0,w2a22w1a12w3a32>0,w3a33w2a23w4a43>0,w4a44w3a34>0. (4.8)

    From (2.2), by calculating, we have

    N1=a12a23a44a43H2,N2=(a12a21a23a44H3+a23a34a43H22)a22a43H2H3,N3=a32(a12a21a23a44H3+a23a34a43H22)a33a22a43H2H3,

    and

    1+a43a44N3=1(a12a21a23a32a44H3+a23a32a34a43H22)a22a33a44H2H3=a22a33a44H2H3a12a21a23a32a44H3a23a32a34a43H22a22a33a44H2H3.

    From (4.8) we obtain

    a11>w2w1a21,aii>1wi(wi1ai1i+wi+1ai+1i),i=2,3,a44>w3w4a34.

    Therefore,

    (a22a33H2a12a21a23a32)a44H3>(1w2[w1a12+w3a32]1w3[w2a23+w4a43][a11a22+a12a21]a12a21a23a32)a44H3((1w2w1a121w3[w2a23+w4a43]+w4w2a32a43)(a11a22+a12a21))a44H3w4w2a32a43(a11a22+a12a21)w3w4a34H3.

    Since

    H3>(a11a22+a12a21)a33>(a11a22+a12a21)w2w3a23,

    we further obtain

    (a22a33H2a12a21a23a32)a44H3>w4w2a32a43(a11a22+a12a21)w3w4a34(a11a22+a12a21)w2w3a23=a23a32a34a43H22.

    This shows that 1+a43a44N3>0.

    Conjecture 1. When n>4, if condition (2) holds then condition (1) also holds.

    To support our analysis results, we will provide the numerical examples for model (1.2). The numerical simulations can be performed qualitatively, and parameters are selected in an epidemiologically feasible way.

    Example 1. In model (1.2) we take n=4, r1=1.5, r2=0.6, r3=0.08, r4=0.01, γ1=0.1, γ2=0.15, γ3=0.15, γ4=0.1, a11=1.6, a12=0.2, a22=2.2, a21=2.5, a23=0.1, a32=2.5, a33=2, a34=0.3, a43=1.3, a44=2, σ1=0.1, σ2=0.2, σ3=0.1 and σ4=0.1

    Choose w1=2.5, w2=1.3, w3=0.8 and w4=0.5, by calculating we have w1a11w2a21=0.7500>0, w2a22w1a12w3a32=0.3600>0, w3a33w2a23w4a43=0.8200>0 and m4a44m3a34=0.7600>0. Proposition 1 shows that 1+a43a44N3>0. Thus, all four species in model (1.2) are persistent in mean which is shown in Figure 1 below. Furthermore, h1=0.3250, h2=0.3914, h3=0.3264, h4=0.1281 and D(4)4|hi=hi,i=1,2,3,4=0.5201>0 are obtained by calculating, all leading principal minors of A4+(A14)T are positive, which shows that the A4+(A14)T is positive semi-definite. Thus, the condition of conclusion (A1) is satisfied, then the OHS: H=(0.32500.39140.32640.1281)T and MESY=0.4321 are gained, as Figure 1 shows below.

    Figure 1.  The sample path for the solution and MESY illustrate that species xi(t)(i=1,2,3,4) is persistent in the mean and MESY exists. The brown line represent the solution of the corresponding undisturbed system and the blue lines represent the solution of model (1.2).

    Example 2. In model (1.2) we take n=4, r1=1.5, r2=0.5, r3=0.03, r4=0.01; γ1=0.15, γ2=0.2, γ3=0.1 and γ4=0.05. The rest parameters in model (1.2) are shown in the following three cases.

    Case 2.1. a11=1.6, a12=0.2, a22=2, a21=2.5, a23=1, a32=2.5, a33=2.1, a34=0.2, a43=0.1, a44=2, σ1=0.1, σ2=0.2, σ3=0.2 and σ4=0.1.

    Case 2.2. a11=1.8, a12=0.2, a22=2, a21=1.5, a23=0.2, a32=1.5, a33=2.1, a34=0.8, a43=1.1, a44=2.5, σ1=0.2, σ2=0.2, σ3=0.3 and σ4=0.1.

    Case 2.3. a11=0.8, a12=0.2, a22=2, a21=1, a23=0.15, a32=1.5, a33=2, a34=0.1, a43=1, a44=0.08, σ1=0.2, σ2=0.3, σ3=0.3 and σ4=0.1.

    For Case 2.1, choose w1=2.5, w2=1.3, w3=0.8 and w4=1.4, then we have w1a11w2a21=0.75>0, w2a22w1a12w3a32=0.10>0, w3a33w2a23w4a43=0.24>0 ºÍ m4a44m3a34=2.64>0. Furthermore, h1=0.3621, h2=0.4773, h3=0.4665, h4=0.0051 are gained by calculating. Meanwhile, D(4)4|hi=hi,i=1,2,3,4=0.1475<0. Thus the condition (B1) of (A2) is satisfied. Therefore, the OHS do not exist. The numerical simulations are given in Figure 2.

    Figure 2.  The sample path for the solution shows that species xi(t)(i=1,2,3) is persistent in the mean and x4 goes to extinction.

    For Case 2.2, choose w1=2.2, w2=1.1, w3=1.1 and w4=1.4, then we have w1a11w2a21=2.31>0, w2a22w1a12w3a32=0.11>0, w3a33w2a23w4a43=0.55>0 ºÍ m4a44m3a34=2.62>0. Furthermore, h1=0.9445>0, h_2^* = 0.2174 > 0 , h_3^* = 0.0039 > 0 and h_4^* = -0.0124 < 0 are obtained by calculating. Thus the condition ( \mathcal{B}_2 ) of conclusion ( \mathcal{A}_2 ) is satisfied. Therefore, the OHS do not exist. The numerical simulations are given in Figure 3.

    Figure 3.  The sample path for the solution shows that species x_i(t)\; (i = 1, 2, 3) is persistent in the mean and x_4 goes to extinction.

    For Case 2.3, choose w_1 = 2.8 , w_2 = 2 , w_3 = 2 and w_4 = 2.8 , then we have w_1a_{11}-w_2a_{21} = 0.24 > 0 , w_2a_{22}-w_1a_{12}-w_3a_{32} = 0.44 > 0 , w_3a_{33}-w_2a_{23}-w_4a_{43} = 0.90 > 0 ºÍ m_4a_{44}-m_3a_{34} = 0.024 > 0 . Furthermore, by calculating we obtain that the fourth leading principle minor of A_4+(A_4^{-1})^T is equal to -6.2127 < 0 , which means that A_4+(A_4^{-1})^T is negative semi-definite. Thus the condition ( \mathcal{B}_3 ) is satisfied. Therefore, the optimal harvesting strategy do not exist. The numerical simulations are given in Figure 4.

    Figure 4.  The sample path for the solution shows that species x_i(t)\; (i = 1, 2, 3) is persistent in the mean and x_4 goes to extinction.

    In this paper, based on the effect of harvesting and Lévy noises, a stochastic marine n -species system is investigated. On the basis of the research results in [23], we first establish the criterion for the stability in distribution of model (1.2). Then, the criteria for existence and nonexistence of OHS and the corresponding MESY are derived. This shows that the environmental noises affect the OHS and MESY.

    Compared with the existing results, our first contribution is to investigate the stochastic stability and optimal harvesting problems for a stochastic marine n -species food chain model with Lévy noises and white noises. Second, the criteria on the stability in distribution and optimal harvesting strategy for stochastic n -species food chain model are established for the first time. Third, the noises intensity can easily affect the dynamics and MESY of marine populations, and further lead to the imbalances of marine ecology. Finally, we find that for better manage and protect marine resources, the establishment of an optimal harvesting strategy should fully consider the impact of noise intensity.

    There are still some remained problems that need to be followed up. For example, in Conjecture 1, we assume if w_1a_{11}-w_2a_{21} > 0\text{, } \; w_ia_{ii}-w_{i-1}a_{i-1i}-w_{i+1}a_{i+1i} > 0\; (i = 2\text{, }3\text{, }\cdots\text{, } n-1)\text{, }\; w_na_{nn}-w_{n-1}a_{n-1n} > 0 , then 1+\frac{a_{nn-1}}{a_{nn}}N_{n-1} > 0 , how to prove is an open problem. Second, marine population are easily affected by climate change, consider marine population model with regime switching is necessary. Finally, more complex and realistic stochastic system with nonlinear functional response, delays and Allee effect need to be investigated. These issues may be investigated in the future.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by the National Natural Science Foundation of China (Grant No. 12061079), Foundation of Fujian Educational Committee (Grant No. JAT220189), Startup Foundation of Jimei University(Grant No.ZQ2022025).

    The authors declare there is no conflicts of interest.



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