Research article

Optimal harvesting for a periodic n-dimensional food chain model with size structure in a polluted environment


  • Received: 12 March 2022 Revised: 19 April 2022 Accepted: 02 May 2022 Published: 19 May 2022
  • This study examines an optimal harvesting problem for a periodic n-dimensional food chain model that is dependent on size structure in a polluted environment. This is closely related to the protection of biodiversity, as well as the development and utilization of renewable resources. The model contains state variables representing the density of the ith population, the concentration of toxicants in the ith population, and the concentration of toxicants in the environment. The well-posedness of the hybrid system is proved by using the fixed point theorem. The necessary optimality conditions are derived by using the tangent-normal cone technique in nonlinear functional analysis. The existence and uniqueness of the optimal control pair are verified via the Ekeland variational principle. The finite difference scheme and the chasing method are used to approximate the nonnegative T-periodic solution of the state system corresponding to a given initial datum. Some numerical tests are given to illustrate that the numerical solution has good periodicity. The objective functional here represents the total profit obtained from harvesting n species.

    Citation: Tainian Zhang, Zhixue Luo, Hao Zhang. Optimal harvesting for a periodic n-dimensional food chain model with size structure in a polluted environment[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 7481-7503. doi: 10.3934/mbe.2022352

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  • This study examines an optimal harvesting problem for a periodic n-dimensional food chain model that is dependent on size structure in a polluted environment. This is closely related to the protection of biodiversity, as well as the development and utilization of renewable resources. The model contains state variables representing the density of the ith population, the concentration of toxicants in the ith population, and the concentration of toxicants in the environment. The well-posedness of the hybrid system is proved by using the fixed point theorem. The necessary optimality conditions are derived by using the tangent-normal cone technique in nonlinear functional analysis. The existence and uniqueness of the optimal control pair are verified via the Ekeland variational principle. The finite difference scheme and the chasing method are used to approximate the nonnegative T-periodic solution of the state system corresponding to a given initial datum. Some numerical tests are given to illustrate that the numerical solution has good periodicity. The objective functional here represents the total profit obtained from harvesting n species.



    In today's world of industrial pollution, toxicants are pervading the air, ecological problems have become increasingly prominent, and environmental pollution has become a major problem. SARS, Ebola virus, AIV, H1N1 influenza, and COVID-19 are threatening the ecological balance as well as the survival of human beings and other creatures. It is necessary to study the effects of toxicants on the ecosystem. Hallam et al. proposed using a dynamic methodology to examine ecotoxicology in [1,2,3]. They established a model of the interaction between toxicants and population, and provided sufficient conditions for the persistence and extinction of a population stressed by a toxicant. Researchers have been studying ecotoxicology since the 1980s, and a large amount of literature has been devoted to problems in the area. Luo et al. [4,5,6] studied a new age-dependent model of toxicant population in an environment with a small capacity for toxicants. The threshold between persistence in the mean and extinction was obtained for each species in a polluted environment [7,8]. The results of thresholding in [9] were then extended to a stochastic Lotka–Volterra cooperative model for n-species.

    The effects of environmental pollution on biological population, the dynamical behavioral analysis of ecosystem models, and the control problem have attracted the attention of many scholars. A large number of ecological studies have shown that differences in individual size structures have a more important effect on population development than those in the age structure. This kind of model has achieved remarkable results through theory, numerical calculations, and experimental methods. Among them, the maximum principle and bang-bang structure of the optimal control were established in [10] for a size-structured forestry model with the benefit of carbon sequestration. In [11,12,13], several authors investigated size-structured population models with separable mortality rates. He and Liu [14] studied an optimal birth control problem for a size-structured population model that takes fertility as the control variable. Later, they studied a nonlinear egg-juvenile-adult model in which eggs and juveniles were structured by age while adults were structured by size [15]. In [16], Liu et al. proposed a unified size-structured PDE model for the growth of metastatic tumors and provided several numerical examples. In [17], Mansal et al. studied the dynamics of the predator and the prey in a fishery model by using the fractional-order derivative. In [18], Thiao et al. discussed the fractional optimal economic control of a continuous game theory model described by the fractional-order derivative. Optimal harvesting problems for a size-structured population model were analyzed in [19,20]. However, most of these studies have focused on a single species, and few have examined interactions among species, especially predator-prey models of two species. In addition, due to the influence of seasonal changes and other factors, the living environment of populations often undergoes periodic changes. Research on optimal harvesting problems dependent on the model of individual size in a periodic environment has been reported in [21,22]. In this study, we establish mathematical population models to consider the size factor to render them reasonable.

    Few studies to date have examined optimal control problems of size-dependent population models and periodic effects in a polluted environment. In order to bridge this gap, we discuss optimal harvesting for a periodic, n-dimensional food chain model dependent on size structure in a polluted environment, and detail a simulation and an example based on the population density.

    The remainder of this paper is organized as follows: Section 2 describes a population model with size structure in a polluted environment. Its well-posedness is proved in Section 3 and its optimality conditions are established in Section 4. In Section 5, we discuss the existence and uniqueness of the optimal control pair. Section 6 is devoted to approximating the T-periodic solution by a numerical algorithm. A discussion of the results and the conclusions of this study are provided in Section 7.

    In [1,2,3], Hallam et al. proposed the following dynamic population model with toxicant effects:

    {dxdt=x[r0r1C0fx],dC0dt=kCEgC0mC0,dCEdt=k1CEx+g1C0xhCE+u, (2.1)

    where x=x(t) is the population biomass at time t; C0=C0(t) is the concentration of toxicants in the organism at time t; CE=CE(t) is the concentration of toxicants in the environment of the population at time t. The exogenous rate of input of toxicants into the environment was represented by u. They investigated the persistence and extinction of a population in a polluted environment.

    Luo et al. [23] studied optimal birth control for the following age-dependent n-dimensional food chain model:

    {p1t+p1a=μ1(a,t)p1λ1(a,t)P2(t)p1,pit+pia=μi(a,t)pi+λ2i2(a,t)Pi1(t)piλ2i1(a,t)Pi+1(t)pi,pnt+pna=μn(a,t)pn+λ2n2(a,t)Pn1(t)pn,pi(0,t)=βi(t)a2a1mi(a,t)pi(a,t)da,pi(a,0)=pi0(a),Pi(t)=a+0pi(a,t)da,(a,t)Q, (2.2)

    where Q=(0,a+)×(0,+), [a1,a2] is the fertility interval. pi(a,t) represents the density of the ith population of age a at time t, and a+ is the life expectancy of individuals. The control variable βi(t) is the average fertility of the ith population. The vitality rate μi(a,t) denotes the average mortality of the ith population. λk(a,t) is the interaction coefficient, and mi(a,t) is the ratio of females in the ith population. Maximum principles for the control problems with free terminal states, infinite horizons and target sets have been derived.

    By combining (2.1) and (2.2), we propose the following periodic, n-dimensional food chain model with size structure in a polluted environment:

    {p1t+(V1(x,t)p1)x=f1(x,t)μ1(x,c10(t))p1λ1(x,t)P2(t)p1u1(x,t)p1,pit+(Vi(x,t)pi)x=fi(x,t)μi(x,ci0(t))pi+λ2i2(x,t)Pi1(t)piλ2i1(x,t)Pi+1(t)piui(x,t)pi,i=2,3,n1,pnt+(Vn(x,t)pn)x=fn(x,t)μn(x,cn0(t))pn+λ2n2(x,t)Pn1(t)pnun(x,t)pn,dci0dt=k1ce(t)g1ci0(t)mci0(t),dcedt=k2ce(t)ni=1Pi(t)+g2ni=1ci0(t)Pi(t)h1ce(t)+v(t),Vi(0,t)pi(0,t)=l0βi(x,ci0(t))pi(x,t)dx,pi(x,t)=pi(x,t+T),Pi(t)=l0pi(x,t)dx,(x,t)Q, (2.3)

    where Q=(0,l)×R+,lR+ is the maximal size of an individual in the population, TR+ is the period of evolution of the population. The meanings of the other parameters are as follows: pi(x,t): the density of the ith population of size x at time t; ci0(t): the concentration of toxicants in the ith population; ce(t): the concentration of toxicants in the environment; Vi(x,t): the growth rate of the ith population; μi(x,ci0(t)),βi(x,ci0(t)): the mortality and fertility rates of the ith population, respectively; λk(x,t): the interaction coefficient (k=1,2,,2n2); v(t): the input rate of exogenous toxicants; Pi(t): total number of individuals in the ith population; fi(x,t): the immigration rate of the ith population; ui(x,t): function of the harvesting efforts of the ith population. k1,g1,m,h1,k2, and g2 are nonnegative constants. For the sake of convenience, if there is no special description, i=1,2,,n. The model represents the population dynamics, and couples toxicants with the population. Moreover, it is more realistic for the fertility and mortality rates to depend on the size and the concentration of toxicants in an organism.

    This paper considers the following objective functional:

    max{J(u,v):u=(u1(x,t),u2(x,t),,un(x,t),v=v(t),(u,v)Ω)}, (2.4)

    where

    J(u,v)=ni=1T0l0wi(x,t)ui(x,t)pi(x,t)dxdt12ni=1T0l0ciu2i(x,t)dxdt12T0cn+1v2(t)dt,

    represents the revenue obtained from harvesting less the costs of harvesting and curbing environmental pollution. The weight function wi(x,t) is the selling price factor of an individual belonging to the ith population. The positive constants ci and cn+1 are the cost factors of the ith harvested population and the curbing of environmental pollution, respectively. Therefore, the objective function represents the total profit from the harvested populations during period T. To utilize the resources of the species over a long time, we must develop them rationally and manage them scientifically. We should not only consider the maximization of current economic benefits, but should also consider the ecological balance to ensure the maximization of long-term economic benefits. Our optimal control pair (u,v) in Ω satisfies J(u,v)=max(u,v)ΩJ(u,v). The admissible control set Ω is as follows:

    Ω={(u,v)[LT(Q)]n×LT(R+):0ui(x,t)Ni,a.e.(x,t)Q,0v0v(t)v1,a.e.tR+},

    where

    LT(Q)={ηL(Q):η(x,t)=η(x,t+T),a.e.(x,t)Q},
    LT(R+)={ηL(R+):η(t)=η(t+T),a.e.t∈∈R+}.

    This paper makes the following assumptions:

    (A1) Vi:[0,l)×R+R+ are bounded continuous functions, Vi(x,t)>0 and Vi(x,t)=Vi(x,t+T) for (x,t)Q, limxlVi(x,t)=0, and Vi(0,t)=1 for tR+. There are Lipschitz constants LVi such that

    |Vi(x1,t)Vi(x2,t)|LVi|x1x2|forx1,x2[0,l],tR+.

    (A2) 0βi(x,ci0(t))=βi(x,ci0(t+T))¯βi,¯βi are constants.

    (A3) {μi(x,ci0(t))=μ0(x)+¯μi(x,ci0(t))a.e.(x,t)Q,whereμ0L1loc([0,l)),μ0(x)0a.e.x[0,l),l0μ0(x)dx=+,¯μiL(Q),¯μi(x,ci0(t))0and¯μi(x,ci0(t))=¯μi(x,ci0(t+T))a.e.(x,t)Q.

    (A4) fiL(Q),0fi(x,t)=fi(x,t+T),0wi(x,t)wi(x,t+T)¯wi,0λi(x,t)¯λi,Pi(t) M0,¯wi, ¯λi and M0 are constants.

    (A5) βi(x,c1i0(t))βi(x,c2i0(t))∣≤Lβc1i0(t)c2i0(t),μi(x,c1i0(t))μi(x,c2i0(t)) Lμc1i0(t)c2i0(t).

    (A6) g1k1g1+m,v1h1. (see [24])

    This section discusses the existence and uniqueness of the solution to the state system (2.3). For convenience of research, the following definitions are introduced:

    Definition 3.1 For i=1,2,,n, the unique solution x=φi(t;t0,xi0) of the initial value problem x(t)=Vi(x,t),x(t0)=xi0 is said to be a characteristic curve of (2.3) through (t0,xi0). In particular, let zi(t):=φi(t;0,0) be the characteristic curve through (0, 0) in the xt plane.

    Definition 3.2 Suppose g(x) is an essentially bounded measurable function on E; let

    g=infE0E(supEE0|g(x)|),

    where the infimum is taken for all zero sets E0 in E that make f(x) a bounded function on EE0. This is also denoted by

    EsssupxE|g(x)|.

    For any point (x,t)[0,l)×[0,T] such that xzi(t), define the initial time τ:=τ(t,x); then, φi(t;τ,0)=xφi(τ;t,x)=0. The solution of (2.3) is

    pi(x,t)=pi(0,tz1i(x))Πi(x;x,t)+x0fi(r,φ1i(r;t,x))Vi(r,φ1i(r;t,x))Πi(x;x,t)Πi(r;x,t)dr, (3.1)

    where

    Π1(s;x,t)=exp{s0μ1(r,c10(φ11(r;t,x)))+λ1(r,φ11(r;t,x))P2(φ11(r;t,x))V1(r,φ11(r;t,x))=+u1(r,φ11(r;t,x))+V1x(r,φ11(r;t,x))V1(r,φ11(r;t,x))dr},
    Πi(s;x,t)=exp{s0μi(r,ci0(φ1i(r;t,x)))λ2i2(r,φ1i(r;t,x))Pi1(φ1i(r;t,x))Vi(r,φ1i(r;t,x))=+λ2i1(r,φ1i(r;t,x))Pi+1(φ1i(r;t,x))+ui(r,φ1i(r;t,x))+Vix(r,φ1i(r;t,x))Vi(r,φ1i(r;t,x))dr}i=2,3,,n1,
    Πn(s;x,t)=exp{s0μn(r,cn0(φ1n(r;t,x)))λ2n2(r,φ1n(r;t,x))Pn1(φ1n(r;t,x))Vn(r,φ1n(r;t,x))=+un(r,φ1n(r;t,x))+Vnx(r,φ1n(r;t,x))Vn(r,φ1n(r;t,x))dr}.
    ci0(t)=ci0(0)exp{(g1+m)t}+k1t0ce(σ)exp{(σt)(g1+m)}dσ. (3.2)
    ce(t)=ce(0)exp{t0[k2ni=1Pi(τ)+h1]dτ}+t0[g2ni=1ci0(σ)Pi(σ)+v(σ)]exp{σt[k2ni=1Pi(τ)+h1]dτ}dσ. (3.3)

    Theorem 3.1 Assume that (A1)(A6) hold; then, the hybrid system (2.3) has a nonnegative and unique solution (p1(x,t),p2(x,t),,pn(x,t),c10(t),c20(t),,cn0(t),ce(t)), such that

    (i)(pi(x,t),ci0(t),ce(t))L(Q)×L(0,T)×L(0,T).

    (ii)0ci0(t)1,0ce(t)1,t(0,T),0pi(x,t),l0pi(x,t)dxM,(x,t)Q.

    Proof. Without loss of generality, we assume that ui(x,t)0. p(x,t)=(p1(x,t),p2(x,t),,pn(x,t)), c0(t)=(c10(t),c20(t),,cn0(t)), X=[LT(R+,L1(0,l))]n×[L(R+)]n×L(R+). Then, the state space is defined as follows:

    Y={(p,c0,ce)Xpi(x,t)0a.e.inQ,l0pi(x,t)dxM,0ci0(t)1,0ce(t)1.},

    where

    M:=max{¯β1Tf1(,)L1(Q)exp{¯β1T}+f1(,)L1(Q),¯βiTexp{¯λ2i2M0T}fi(,)L1(Q)=exp{¯βiTexp{¯λ2i2M0T}}+exp{¯λ2i2M0T}fi(,)L1(Q),i=2,3,,n}.

    Define a mapping

    G:YX,G(p,c0,ce)=(G1(p,c0,ce),G2(p,c0,ce),,G2n+1(p,c0,ce)).

    where

    Gi(p,c0,ce)(x,t)=pi(0,tz1i(x))Πi(x;x,t)+x0fi(r,φ1i(r;t,x))Vi(r,φ1i(r;t,x))Πi(x;x,t)Πi(r;x,t)dr,i=1,2,,n.
    Gj(p,c0,ce)(t)=cj0(0)exp{(g1+m)t}+k1t0ce(σ)exp{(σt)(g1+m)}dσ,j=n+1,n+2,,2n.
    G2n+1(p,c0,ce)(t)=ce(0)exp{t0[k2ni=1Pi(τ)+h1]dτ}+t0[g2ni=1ci0(σ)Pi(σ)+v(σ)]exp{σt[k2ni=1Pi(τ)+h1]dτ}dσ.

    By assumption (A1), we have Vi(0,t)=1. Let bi(t)=pi(0,t), then, noting φ1i=tz1i(x), we have

    b1(t)=V1(0,t)p1(0,t)=l0β1(x,c10(t))p1(x,t)dx=l0β1(x,c10(t))b1(φ11(0;t,x))dx+l0β1(x,c10(t))x0f1(r,φ11(r;t,x))V1(r,φ11(r;t,x))drdx=l0β1(x,c10(t))b1(tz11(x))dx+l0β1(x,c10(t))tφ11(0;t,x)f1(φ1(σ;t,x),σ)dσdx¯β1l0b1(tz11(x))dx+¯β1l0t0f1(φ1(σ;t,x),σ)dσdx¯β1t0b1(σ)dσ+¯β1f1(,)L1(Q).

    It follows from Bellman's lemma that

    b1(t)¯β1f1(,)L1(Q)exp{¯β1T}.

    Then we can see that

    l0G1(p,c0,ce)(x,t)dx=l0p1(0,φ11(0;t,x))Π1(x;x,t)dx+l0x0f1(r,φ11(r;t,x))V1(r,φ11(r;t,x))Π1(x;x,t)Π1(r;x,t)drdxl0p1(0,φ11(0;t,x))dx+l0x0f1(r,φ11(r;t,x))V1(r,φ11(r;t,x))drdx=l0b1(tz11)dx+l0tφ11(0;t,x)f1(φ1(σ;t,x),σ)dσdxt0b1(σ)dσ+f1(,)L1(Q)ˉβ1Tf1(,)L1(Q)exp{ˉβ1T}+f1(,)L1(Q).

    Similarly, we have

    l0Gi(p,c0,ce)(x,t)dx¯βiTexp{¯λ2i2M0T}fi(,)L1(Q)exp{¯βiTexp{¯λ2i2M0T}}+exp{¯λ2i2M0T}fi(,)L1(Q),i=2,3,,n.

    It follows that G is a mapping from Y to Y. We now discuss the compressibility of the mapping G.

    G1(p,c0,ce)(x,t)=b1(φ11(0;t,x))E1(φ11(0;t,x);x,t,P2(t))+lφ11(0;t,x)f1(φ1(σ;t,x),σ)E1(σ;x,t,P2(t))dσ,

    where

    E1(r;x,t,P2(t))=exp{trμ1(φ1(σ;t,x),c10(σ))+λ1(φ1(σ;t,x),σ)P2(t)+V1x(φ1(σ;t,x),σ)dσ}.

    Then, we have

    l0G1(p1,c10,c1e)G1(p2,c20,c2e)dx=l0b11(φ11(0;t,x))E11(φ11(0;t,x);x,t,P12(t))b21(φ11(0;t,x))E21(φ11(0;t,x);x,t,P22(t))dx+l0tφ11(0;t,x)f1(φ1(σ;t,x),σ)E11(σ;x,t,P12(t))f1(φ1(σ;t,x),σ)E21(σ;x,t,P22(t))dσdxl0b11(φ11(0;t,x))b21(φ11(0;t,x))dx+l0b11(φ11(0;t,x))tφ11(0;t,x)μ1(φ1(σ;t,x),c110(σ))μ1(φ1(σ;t,x),c210(σ))dσdx+l0b11(φ11(0;t,x))tφ11(0;t,x)¯λ1P12(t)P22(t)dσdx+l0tφ11(0;t,x)f1(φ1(σ;t,x),σ)tσμ1(φ1(σ;t,x),c110(σ))μ1(φ1(σ;t,x),c210(σ))drdσdx+l0tφ11(0;t,x)f1(φ1(σ;t,x),σ)tσP12(t)P22(t)drdσdxt0b11(σ)b21(σ)dσ+MLμt0c110(σ)c210(σ)dσ+¯λ1MTl0p12(x,σ)p22(x,σ)dx+lLμf1(,)L1(Q)t0c110(σ)c210(σ)dσ+lTf1(,)L1(Q)l0p12(x,σ)p22(x,σ)dx¯β1t0l0p11(x,σ)p21(x,σ)dxdσ+[MLμ+lLμf1(,)L1(Q)+MLβ]t0c110(σ)c210(σ)dσ+[¯λ1MT+lTf1(,)L1(Q)]l0p12(x,σ)p22(x,σ)dx.

    Similarly, we have

    l0Gi(p1,c10,c1e)Gi(p2,c20,c2e)dxexp{¯λ2i2M0T}{¯βit0l0p1i(x,σ)p2i(x,σ)dxdσ=+[MLμ+lLμf(,)L1(Q)+MLβ]t0c1i0(σ)c2i0(σ)dσ=+[¯λ2i1MT+lTf(,)L1(Q)]l0p1i+1(x,σ)p2i+1(x,σ)dx},i=2,3,,n1.
    l0Gn(p1,c10,c1e)Gn(p2,c20,c2e)dxexp{¯λ2n2M0T}{¯βnt0l0p1n(x,σ)p2n(x,σ)dxdσ=+[MLμ+lLμf(,)L1(Q)+MLβ]t0c1n0(σ)c2n0(σ)dσ}.

    Thus, we have

    l0Gi(p1,c10,c1e)Gi(p2,c20,c2e)dx,M1(ni=1t0l0p1i(x,σ)p2i(x,σ)dxdσ+t0c1i0(σ)c2i0(σ)dσ), (3.4)

    where

    M1=max{¯β1,[1+¯λ1MT+lTf(,)L1(Q)]¯β2exp{¯λ2M0T},,=[1+¯λ2n3MT+lTf(,)L1(Q)]¯βnexp{¯λ2n2M0T},[MLμ+lLμf(,)L1(Q)+MLβ]}.

    By (3.2) and (3.3), we obtain

    Gj(p1,c10,c1e)Gj(p2,c20,c2e)(t)(j=n+1,n+2,,2n)M2t0c1e(σ)c2e(σ)dσ, (3.5)

    where M2=k1.

    G2n+1(p1,c10,c1e)G2n+1(p2,c20,c2e)(t)M3(ni=1t0l0p1i(x,σ)p2i(x,σ)dxdσ+t0c1i0(σ)c2i0(σ)dσ), (3.6)

    where M3=max{k2+g2+k2h1T+k2g2MT,g2M}.

    We now show that the mapping G has a unique fixed point. Due to the periodicity of elements in the set Y, we consider the case [0,T] only. We define an equivalent norm in L(0,T) as follows:

    (p,c0,ce)=Esssupt[0,T]eλt{ni=1l0pi(x,t)dx+ni=1ci0(t)+ce(t)},

    where λ>0 is large enough. Then, we have

    G(p1,c10,c1e)G(p2,c20,c2e)M4Esssupt[0,T]eλtt0{ni=1l0p1i(x,σ)p2i(x,σ)dx=+ni=1c1i0(σ)c2i0(σ)+c1e(σ)c2e(σ)}dσM4Esssupt[0,T]eλtt0eλσ{eλσ[ni=1l0p1i(x,σ)p2i(x,σ)dx=+ni=1c1i0(σ)c2i0(σ)+c1e(σ)c2e(σ)]}dσM4(p1p2,c10c20,c1ec2e)Esssupt[0,T]eλtt0eλσdσM4λ(p1p2,c10c20,c1ec2e),

    where M4=max{M1,M2,M3}. Thus, choosing λ>M4 yields G as a strict contraction on the space of (Y,). By the Banach fixed point theory, G has a unique fixed point, which is the solution of the system (2.3).

    Theorem 3.2 If T is small enough, then there are constants Kj(T) with limT0Kj(T)>0,j=1,2, such that

    ni=1p1ip2iL(0,T;L1(0,l))+ni=1c1i0c2i0L(0,T)+c1ec2eL(0,T)K1(T)T(ni=1u1iu2iL(0,T;L1(0,l))+v1v2L(0,T)). (3.7)
    ni=1p1ip2iL1(Q)+ni=1c1i0c2i0L1(0,T)+c1ec2eL1(0,T)K2(T)T(ni=1u1iu2iL1(Q)+v1v2L1(0,T)). (3.8)

    Proof. Let (pj,cj0,cje) be the solution of (2.3) corresponding to (uj,vj),j=1,2, it follows from (3.1) that

    l0p11(x,t)p21(x,t)dx=l0b11(φ11(0;t,x))Π11(x;x,t)b21(φ11(0;t,x))Π21(x;x,t)dx+l0x0|f1(r,φ11(r;t,x))V1(r,φ11(r;t,x))Π11(x;x,t)Π11(r;x,t)f1(r,φ11(r;t,x))V1(r,φ11(r;t,x))Π21(x;x,t)Π21(r;x,t)|drdxl0b11(φ11(0;t,x))b11(φ11(0;t,x))dx+l0b11(φ11(0;t,x))[tφ11(0;t,x)μ1(φ1(r;t,x),c110(r))μ1(φ1(r;t,x),c210(r))dr]dx+l0b11(φ11(0;t,x))[tφ11(0;t,x)λ1(φ1(r;t,x),r)(P12(t)P22(t))dr]dx+l0b11(φ11(0;t,x))[tφ11(0;t,x)u11(φ1(r;t,x),r)u21(φ1(r;t,x),r)dr]dx+l0tφ11(0;t,x)f1(φ1(σ;t,x),σ)[tσμ1(φ1(σ;t,x),c110(σ))μ1(φ1(σ;t,x),c210(σ))dr]dσdx+l0tφ11(0;t,x)f1(φ1(σ;t,x),σ)[tσλ1(φ1(σ;t,x),σ)(P12(t)P22(t))dr]dσdx+l0tφ11(0;t,x)f1(φ1(σ;t,x),σ)[tσu11(φ1(σ;t,x),σ)u21(φ1(σ;t,x),σ)dr]dσdxt0b11(σ)b21(σ)dx+¯β1MLμt0c110(σ)c210(σ)dσ+¯β1¯λ1MTl0p12(x,σ)p12(x,σ)dx+¯β1Mt0l0u11(φ1(σ;t,x),σ)u21(φ1(σ;t,x),σ)dxdσ+lLμf1(,)L1(Q)t0c110(σ)c210(σ)dσ+¯λ1lTf1(,)L1(Q)l0p12(x,σ)p22(x,σ)dx+f1(,)L1(Q)t0l0u11(φ1(σ;t,x),σ)u21(φ1(σ;t,x),σ)dxdσ¯β1t0l0p11(x,σ)p21(x,σ)dxdσ+¯λ1[MT+lTf1(,)L1(Q)]l0p12(x,σ)p22(x,σ)dx+[¯β1MLμ+lLμf1(,)L1(Q)+MLβ]t0c110(σ)c210(σ)dσ+[¯β1M+f1(,)L1(Q)]t0l0u11(φ1(σ;t,x),σ)u21(φ1(σ;t,x),σ)dxdσ. (3.9)

    Similarly, we can see that

    l0p1i(x,t)p2i(x,t)dx¯βiexp{¯λ2i2M0T}t0l0p1i(x,σ)p2i(x,σ)dxdσ+exp{¯λ2i2M0T}[¯βiMLμ+lLμfi(,)L1(Q)+MLβ]t0c1i0(σ)c2i0(σ)dσ+¯λiexp{¯λ2i2M0T}[MT+lTfi(,)L1(Q)]l0p1i+1(x,σ)p2i+1(x,σ)dx+exp{¯λ2i2M0T}[¯βiM+fi(,)L1(Q)]t0l0u1i(φi(σ;t,x),σ)u2i(φi(σ;t,x),σ)dxdσ,i=2,3,,n1. (3.10)
    l0p1n(x,t)p2n(x,t)dx¯βnexp{¯λ2n2M0T}t0l0p1n(x,σ)p2n(x,σ)dxdσ+exp{¯λ2n2M0T}[¯βnMLμ+lLμfn(,)L1(Q)+MLβ]t0c1n0(σ)c2n0(σ)dσ+exp{¯λ2n2M0T}[¯βnM+fn(,)L1(Q)]t0l0u1n(φn(σ;t,x),σ)u2n(φn(σ;t,x),σ)dxdσ. (3.11)

    From (3.2) and (3.3), we obtain

    c1i0(t)c2i0(t)∣≤k1t0c1e(s)c2e(s)ds. (3.12)
    c1e(t)c2e(t)∣≤(k2+g2+k2h1T+k2g2MT)ni=1l0t0p1i(x,s)p2i(x,s)dsdx=+g2Mni=1t0c1i0(s)c2i0(s)ds+t0v1(s)v2(s)ds. (3.13)

    Finally, from (3.9)–(3.13), we can deduce (3.7), where K1(T) is a constant depending on the bounds of parameters in (2.3). In addition, combined with (3.9)–(3.13), there exists K2(T), which enables us to obtain (3.8) of the standard norm in L1 space.

    In this section, we employ tangent-normal cone techniques in nonlinear functional analysis to deduce the necessary conditions for the optimal control pair.

    Theorem 4.1 If (u,v) is an optimal control pair and (p,c0,ce) is the corresponding optimal state, then

    ui(x,t)=Fi([wi(x,t)ξi(x,t)]pi(x,t)ci), (4.1)
    v(t)=Fn+1(ξ2n+1(t)cn+1), (4.2)

    in which Fj are given by

    Fj(η)={0,η<0,η,0ηNj,j=1,2,,n+1,Nj,η>Nj, (4.3)

    and (ξ1,ξ2,,ξ2n+1) is the solution of the following adjoint system corresponding to (u,v):

    {ξ1t+V1ξ1x=[μ1(x,c10(t))+λ1P2(t)+u1]ξ1+[k2ce(t)g2c10(t)]ξ2n+1ξ1(0,t)β1(x,c10(t))l0(λ2p2ξ2)(x,t)dx+w1u1,ξit+Viξix=[μi(x,ci0(t))λ2i2Pi1(t)+λ2i1Pi+1(t)+ui]ξi+[k2ce(t)g2ci0(t)]ξ2n+1ξi(0,t)βi(x,ci0(t))+l0(λ2i3pi1ξi1λ2ipi+1ξi+1)(x,t)dx+wiui,i=2,3,n1,ξnt+Vnξnx=[μn(x,cn0(t))λ2n2Pn1(t)+un]ξn+[k2ce(t)g2cn0(t)]ξ2n+1ξn(0,t)βn(x,cn0(t))+l0(λ2n3pn1ξn1)(x,t)dx+wnun,dξn+idt=l0μi(x,ci0(t))ci0piξidt+(g1+m)ξn+ig2Pi(t)ξ2n+1ξi(0,t)l0βi(x,ci0(t))ci0pidx,dξ2n+1dt=k1ni=1ξi+n+[k2ni=1Pi(t)+h1]ξ2n+1,ξi(l,t)=0,ξi(x,t)=ξi(x,t+T),ξj(T)=0,j=n+1,n+2,,2n+1. (4.4)

    Proof. The existence of a unique, bounded solution to the system (4.4) can be treated in the same manner as that for (2.3). For any given (ν1,ν2)TΩ(u,v) (the tangent cone of Ω at (u,v)), ν1=(ν11,ν21,,νn1), such that (u+εν1,v+εν2)Ω, where ε>0 is small enough. Then, from J(u+εν1,v+εν2)J(u,v), we derive

    ni=1T0l0wi(ui+ενi1)pεidxdt12ni=1T0l0ci(ui+ενi1)2dxdt12T0cn+1(v+εν2)2dtni=1T0l0wiuipidxdt12ni=1T0l0ciu2idxdt12T0cn+1v2dt,

    that is

    ni=1T0l0wiuipεipiεdxdt+ni=1T0l0wiνi1pεidxdtni=1T0l0ciuiνi1dxdt+T0cn+1vν2dt+ni=1T0l0ciεν2i1dxdt+T0cn+1εν22dt,

    as ε0+,

    ni=1T0l0wi(uizi+νi1pi)dxdtni=1T0l0ciuiνi1dxdtT0cn+1vν2dt0, (4.5)

    where

    1ε(pεipi)zi,1ε(cεi0ci0)zi+n,1ε(cεece)z2n+1,asε0.

    By Theorem 3.2 we see that z1,z2,,z2n+1 makes sense ([25], p.18). (pε,cε0,cεe) is the state corresponding to (u+εν1,v+εν2). It follows from (2.3) that (z1, z2,, z2n+1) satisfies

    {z1t+V1z1x=[μ1(x,c10(t))+λ1P2(t)+V1x+u1]z1λ1Z2(t)p1μ1(x,c10(t))c10p1zn+1v11p1,zit+Vizix=[μi(x,ci0(t))λ2i2Pi1(t)+λ2i1Pi+1(t)+Vix+ui]zi+λ2i2Zi1(t)piλ2i1(x,t)Zi+1(t)piμi(x,ci0(t))ci0pizn+ivi1pi,i=2,3,n1,znt+Vnznx=[μn(x,cn0(t))λ2n2Pn1(t)+Vnx+un]zn+λ2n2Zn1(t)pnμn(x,cn0(t))cn0pnz2nvnpn,dzn+idt=k1z2n+1g1zn+imzn+i,dz2n+1dt=k2ce(t)ni=1Zi(t)+g2ni=1(ci0Zi(t)+zn+iPi(t))[k2ni=1Pi(t)+h1]z2n+1+v2,Vi(0,t)zi(0,t)=l0βi(x,ci0(t))zidx+l0βi(a,ci0(t))ci0pizn+idx,zi(x,t)=zi(x,t+T),zn+i(0)=z2n+1(0)=0,Pi(t)=l0pi(x,t)dx,Zi(t)=l0zi(x,t)dx. (4.6)

    We multiply the first five equations in (4.6) by ξi(i=1,2,,2n+1) respectively, and integrate on Q and (0,T). By using (4.4), we have

    ni=1T0l0wiuizidxdt=ni=1T0l0νi1ξipidxdt+T0ν2ξ2n+1dt. (4.7)

    Substituting (4.7) and (4.5) gives

    ni=1T0l0[(wiξi)piciui]νi1dxdt+T0(cn+1v+ξ2n+1)ν2dt0,

    for any (ν1,ν2)TΩ(u,v). Consequently, the structure of normal cone tells us that [(wiξi)piciui,cn+1v+ξ2n+1]NΩ(u,v), (the normal cone of Ω at (u,v)), which gives the desired result.

    Theorem 4.2 If T is small enough, then there is a constant K3 such that

    ni=1ξ1iξ2iL(Q)+2nq=n+1ξ1qξ2qL(0,T)+ξ12n+1ξ22n+1L(0,T)K3(T)T(ni=1u1iu2iL(Q)+v1v2L(0,T)). (4.8)

    The proof process of Theorem 4.2 is similar to that of Theorem 3.2, and is omitted here.

    In order to show that there exists a unique optimal control pair by means of the Ekeland variational principle, we embed the functional ˜J(u,v) into [L1(Q)]n×L1(0,T). We define

    ˜J(u,v)={J(u,v),(u,v)Ω,,otherwise.

    Lemma 5.1([6], Lemma 4.1) ˜J(u,v) is upper semi-continuous in [L1(Q)]n×L1(0,T).

    Theorem 5.1 If T is sufficiently small, there exists one and only one optimal control pair (u,v), which is in the feedback, and is determined by (4.1)–(4.4) and (2.3), where C1 and C2 are the supremum of |pi| and |ξj|,i=1,2,,n,j=1,2,,2n+1, respectively.

    Proof. Define the mapping L:ΩΩ as follows:

    L(u,v)=F((w1ξ1)p1c1,(w2ξ2)p2c2,,(wnξn)pncn,ξ2n+1cn+1)=(F1((w1ξ1)p1c1),F2((w2ξ1)p2c2),,Fn((wnξn)pncn),Fn+1(ξ2n+1cn+1)),

    where (p,c0,ce) and (ξ1,ξ2,,ξ2n+1) are the state and adjoint state, respectively, corresponding to the control (u,v). We show that L admits a unique fixed point, which maximizes the functional ˜J.

    From Lemma 5.1 and the Ekeland variational principle ([26], p.180), for any given ε>0, there exists (uε,vε)Ω[L1(Q)]n×L1(0,T) such that

    ˜J(uε,vε)sup(u,v)Ω˜J(u,v)ε, (5.1)
    ˜J(uε,vε)sup(u,v)Ω{˜J(u,v)ε(ni=1uεiuiL1(Q)+vεvL1(0,T))}, (5.2)

    Thus, the perturbed functional

    ˜Jε(u,v)=˜J(u,v)ε(ni=1uεiuiL1(Q)+vεvL1(0,T)),

    attains its supremum at (uε,vε). Then, we argue as in Theorem 4.1:

    (uε,vε)=L(uε,vε)=(F1((w1ξε1)pε1+εθε1c1),F2((w2ξε2)pε2+εθε2c2),,=Fn((wnξεn)pεn+εθεncn),Fn+1(ξε2n+1+εθεn+1cn+1)), (5.3)

    where (pε,cε0,cεe) and (ξε1,ξε2,,ξε2n+1) are the state and adjoint state, respcetively, corresponding to the control (uε,vε), θε1,θε2,,θεnL(Q), and θεn+1L(0,T) with θεi∣≤1,i=1,2,,n+1.

    First, we show that L has only one fixed point. Let (pj,cj0,cje) and (ξj1,ξj2,,ξj2n+1) be the state and adjoint state corresponding to the control (uj,vj),j=1,2. By (3.7) and (4.8), we have

    L(u1,v1)L(u2,v2)=ni=1Fi((wiξ1i)p1ici)Fi((wiξ2i)p2ici)L(Q)+Fn+1(ξ12n+1cn+1)Fn+1(ξ12n+1cn+1)L(0,T)ni=1(wiξ1i)p1ici(wiξ1i)p1iciL(Q)+ξ12n+1cn+1ξ22n+1cn+1L(0,T)ni=1wi(p1ip2i)ci+ξ1i(p1ip2i)ci+p2i(ξ1iξ2i)ciL(Q)+ξ12n+1ξ22n+1cn+1L(0,T)T(ni=11ci(¯wiK1+C2K1+C1K3)+K3)(ni=1u1iu2iL(Q)+v1v2L(0,T)).

    Clearly, L is a contraction if T is sufficiently small. Hence, L has a unique fixed point (u,v).

    Next, we prove (uε,vε)(u,v) as ε0+. The relations (4.1), (4.2), and (5.3) lead to

    L(uε,vε)(uε,vε=(F1((w1ξε1)pε1c1),F2((w2ξε2)pε2c2),,Fn((wnξεn)pεncn),Fn+1(ξε2n+1cn+1))(F1((w1ξε1)pε1c1+εθε1c1),F2((w2ξε2)pε2c2+εθε2c2),,Fn((wnξεn)pεncn+εθεncn),Fn+1(ξε2n+1cn+1+εθεn+1cn+1))ni=1(Fi((wiξεi)pεici)Fi((wiξεi)pεici+εθεici))L(Q)+Fn+1(ξε2n+1cn+1)Fn+1(ξε2n+1cn+1+εθεn+1cn+1)L(0,T)=ni=1(wiξεi)pεici(wiξεi)pεiciεθεiciL(Q)+ξε2n+1cn+1ξε2n+1cn+1εθεn+1cn+1L(0,T)εni=1θεi(x,t)L(Q)ci+εθεn+1(t)L(0,T)cn+1εn+1i=11ci,

    it is easy to derive that

    (u,v)(vε,vε)L((u,v))L(uε,vε)+L(uε,vε)(uε,vε)T(ni=11ci(¯wiK1+C2K1+C1K3)+K3)(ni=1uiuεiL(Q)+vvεL(0,T))+εn+1i=11ci.

    So, if T is small enough, the following result holds:

    ni=1uiuεiL(Q)+vvεL(0,T)εn+1i=11ci1T(ni=11ci(¯wiK1+C2K1+C1K3)+K3).

    which gives the desired result.

    Finally, passing to the limit ε0+ in the inequality of (5.2) and using Lemma 5.1 yield ˜J(u,v)lim sup(u,v)Ω˜J(u,v), which finishes the proof.

    Our goal is to obtain a numerical approximation for the nonnegative T-periodic solution of the system (2.3). We discuss the problem of evolution of a single species in a polluted environment. Suppose the computational domain ˜Q=[0,l]×[0,˜T] is divided into an J×N mesh with the spatial step size h=lJ=0.01 in the x direction and time step size τ=˜TN=0.03. The grid points (xj,tn) are defined by

    xj=jh,j=0,1,2,,J;
    tn=nτ,n=0,1,2,,N,

    where J and N are two integers. The notation pnj denotes the solution p(jh,nτ) of the difference equation.

    Based on the above analysis, the finite difference scheme of system (2.3) can be written as follows:

    pnjpn1jτ+Vpnjpnj1h+Vxpnj+μpnjfnj=0, (6.1)

    where j=1,2,,J;n=1,2,,N. It follows from (6.1) that

    λVpnj1+[1+λV+τ(Vx+μ)]pnj=pn1j+τfnj, (6.2)

    where λ=τh.

    The boundary and initial conditions can be discretized as

    {p0j=p0j,pn0=Jj=1βjpjh. (6.3)

    From (6.2) and (6.3), we have

    APn=Pn1+τF, (6.4)

    where

    A=[1+λV+τ(Vx+μ)λVβhλVβhλVβhλVβhλV1+λV+τ(Vx+μ)λV1+λV+τ(Vx+μ)λV],
    Pn=(pn1,pn2,,pnJ)T,F=(fn1,fn2,,fnJ)T.

    Note that A is an upper triangular matrix, so the nonlinear algebraic equation (6.4) have solutions.

    We first prove the stability of the implicit difference scheme (6.1) by using the Fourier method. We suppose that f0. The domain of the function defined on the grid point is extended according to the usual method, that is, when x(xjh2,xj+h2), pn(x)=pnj; then, we have

    λVpn(xh)+[1+λV+τ(Vx+μ)]pn(x)=pn1(x),xR. (6.5)

    Taking the Fourier transform of both sides of (6.5), we obtain

    λVˆUn(k)eikh+[1+λV+τ(Vx+μ)]ˆUn(k)=ˆUn1(k). (6.6)

    By (6.6), we get

    ˆUn(k)=ˆUn1(k)λVeikh+1+λV+τ(Vx+μ).

    Thus, the growth factor is as follows:

    G(τ,k)=1λVeikh+1+λV+τ(Vx+μ)=11+λV(1coskh)+τ(Vx+μ)+λVisinkh.

    Then, we have

    |G(τ,k)|2=1[1+λV(1coskh)+τ(Vx+μ)]2+λ2V2sin2kh=1[1+2λVsin2kh2+τ(Vx+μ)]2+4λ2V2sin2kh2(1sin2kh2).

    If Vx+μ>0, there is |G(τ,k)|1, that is, the von Neumann condition is satisfied, the difference scheme (6.1) is stable under condition Vx+μ>0.

    Next, we analyze the convergence of the difference scheme (6.1). Note that the difference scheme is compatible, we thus use the Lax equivalence theorem. The difference scheme must be convergent and have first-order accuracy.

    We chose the following parameters:

    {β(x,c0(t))=100x2(1x)(1+sin(πx))|sin2πc0(t)T|,μ(x,c0(t)=e4x(1x)1.4(2+cos2πc0(t)T),V(t,x)=1x,f(x,t)=2+(1+x)sin(2πtT),p0(x)=ex,u(x,t)=0,x=1,T=13,˜T=3.

    In this paper, we used the backward difference scheme and chasing method, and (6.4) was solved through programming. The fertility rate, mortality rate, and immigration rate were T-periodic and were all greater than zero, which is consistent with the assumptions. We considered T=13. Their graphs are given in Figures 13, respectively. The fertility rate was the highest when the size was half and the mortality rate was the highest when the size was the maximum, which conformed to the empirical situation. Therefore, the selection of parameters β, μ, and f was reasonable.

    Figure 1.  Fertility rate of the population.
    Figure 2.  Mortality rate of the population.
    Figure 3.  Immigration rate of the population.

    The graph of the numerical solution p is given in Figure 4. Over time, solution p showed T-periodic changes. We take the numerical solution of system (2.3), corresponding to an arbitrary positive initial datum p0, on some interval [kT,(k+1)T], where k is large enough. We can then get the periodic solution of system (2.3) by extending the numerical solution p. Over time, the density of the population increased first and then decreased. With respect to size, the density of the population gradually decreased, and finally tended to be stable. The maximum value was reached at x=0. On such an interval with a sufficiently large k, the numerical solution was already stable. During computation we found that any positive initial datum p0 was appropriate for use.

    Figure 4.  Numerical solution of the system.

    In this paper, we considered the problem of optimal harvesting for a periodic n-dimensional food chain model dependent on the size structure, that combined toxicants with the population. In the foregoing, we have discussed the existence and uniqueness of a nonnegative solution of the state system. The necessary optimality conditions were provided, and the existence of the optimal policy was investigated. Some numerical results were finally given. The results implied that the solution of (2.3) always maintains the pattern of increasing periodically, and any positive initial datum p0 is appropriate. From Theorem 4.1, the optimal strategy had a bang-bang structure and provided threshold conditions for the optimal control problems (2.3) and (2.4). The bang-bang structure of solutions is much more common in optimal population management.

    We now comment on the differences between the methods and results of this paper as well as closely related work. The existence of the optimal strategy was proved by compactness and the extreme sequence in [13]. In this paper, the corresponding problem was treated by using Ekeland variational principle. The authors [10] established the maximum principle and bang-bang structure for optimal control but paid no attention to existence and uniqueness results. The existence of the optimal harvesting rate in the harvesting problem was only given in [11]. Furthermore, if Vi(x,t)=1 for Q=(0,l)×R+,i=1,2,n, the state system degenerates into an age-structured model, and our results cover the corresponding results [4,5,6].

    Note that the individual price factor wi(x,t) plays an important role in the structure of the optimal controller (4.1). However, as we do not have a clear biological meaning for the solutions ξi(i=1,2,,2n+1) of the adjoint system (4.4), it is difficult to give a precise explanation of the threshold conditions (4.1) and (4.2). In specific applications, the optimal population density and optimal policy are calculated by combining the state system and the adjoint system. In this way, we enable the objective functional (2.4) to maximize the total profit during period T. This is a challenging problem, and future work in the area should address it. In addition, inspired by [17,18], we intend to consider the problem of fractional optimal control problem of the population model in future research.

    The authors thank the referees for their valuable comments and suggestions on the original manuscript that helped improve its quality. The work was supported by the National Natural Science Foundation of China under grant 11561041.

    None of the authors has a conflict of interest in the publication of this paper.



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