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Research article

Analysis of a stochastic fear effect predator-prey system with Crowley-Martin functional response and the Ornstein-Uhlenbeck process

  • This paper studied a stochastic fear effect predator-prey model with Crowley-Martin functional response and the Ornstein-Uhlenbeck process. First, the biological implication of introducing the Ornstein-Uhlenbeck process was illustrated. Subsequently, the existence and uniqueness of the global solution were then established. Moreover, the ultimate boundedness of the model was analyzed. Then, by constructing the Lyapunov function and applying Itˆo's formula, the existence of the stationary distribution of the model was demonstrated. In addition, sufficient conditions for species extinction were provided. Finally, numerical simulations were performed to demonstrate the analytical results.

    Citation: Jingwen Cui, Hao Liu, Xiaohui Ai. Analysis of a stochastic fear effect predator-prey system with Crowley-Martin functional response and the Ornstein-Uhlenbeck process[J]. AIMS Mathematics, 2024, 9(12): 34981-35003. doi: 10.3934/math.20241665

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  • This paper studied a stochastic fear effect predator-prey model with Crowley-Martin functional response and the Ornstein-Uhlenbeck process. First, the biological implication of introducing the Ornstein-Uhlenbeck process was illustrated. Subsequently, the existence and uniqueness of the global solution were then established. Moreover, the ultimate boundedness of the model was analyzed. Then, by constructing the Lyapunov function and applying Itˆo's formula, the existence of the stationary distribution of the model was demonstrated. In addition, sufficient conditions for species extinction were provided. Finally, numerical simulations were performed to demonstrate the analytical results.



    1. Introduction

    G. Caginalp proposed in [3] and [4] two phase-field system, namely,

    utΔu+f(u)=T, (1.1)
    TtΔT=ut, (1.2)

    called nonconserved system, and

    ut+Δ2uΔf(u)=ΔT, (1.3)
    TtΔT=ut, (1.4)

    called concerved system (in the sense that, when endowed with Neumann boundary conditions, the spacial average of u is conserved). In this context, u is the order parameter, T is the relative temperature (defined as T=˜TTE, where ˜T is the absolute temperature and TE is the equilibrium melting temperature) and f is the derivative of a double-well potential F (a typical choice is F(s)=14(s21)2, hence the usual cubic nonlinear term f(s)=s3s). Furthermore, we have set all physical parameters equal to one. These systems have been introduced to model phase transition phenomena, such as melting-solidication phenomena, and have been much studied from a mathematical point of view. We refer the reader to, e.g., [3,4,5,8,9,10,12,13,14,15,16,18,19,21,22,23,25].

    Both systems are based on the (total Ginzburg-Landau) free energy

    ΨGL=Ω(12|u|2+F(u)uT12T2)dx, (1.5)

    where Ω is the domain occupied by the system (we assume here that it is a bounded and regular domain of R3, with boundary Γ), and the enthalpy

    H=u+T. (1.6)

    As far as the evolution equations for the order parameter are concerned, one postulates the relaxation dynamics (with relaxation parameter set equal to one)

    uu=DΨGLDu, (1.7)

    for the nonconserved model, and

    uu=ΔDΨGLDu, (1.8)

    for the conserved one, where DDu denotes a variational derivative with respect to u, which yields (1.1) and (1.3), respectively. Then, we have the energy equation

    Ht=divq, (1.9)

    where q is the heat flux. Assuming finally the usual Fourier law for heat conduction,

    q=T, (1.10)

    we obtain (1.2).

    In (1.5), the term |u|2 models short-ranged interactions. It is however interesting to note that such a term is obtained by truncation of higher-order ones; it can also be seen as a first-order approximation of a nonlocal term accounting for long-ranged interactions [11].

    G. Caginalp and Esenturk recently proposed in [6] (see also [20]) higher-order phase-field models in order to account for anisotropic interfaces (see also [7] for other approaches which, however, do not provide an explicit way to compute the anisotropy). More precisely, these autors proposed the following modified (total) free energy

    ΨHOGL=Ω(12ki=1|β|=iaβ|Dβu|2+F(u)uT12T2)dx,kN, (1.11)

    where, for β=(k1,k2,k3)(N{0})3,

    |β|=k1+k2+k3

    and, for β(0,0,0),

    Dβ=|β|xk11xk22xk33

    (we agree that D(0,0,0)v=v).

    A. Miranville studied in [17] the corresponding nonconserved higher-order phase-field system.

    As far as the conserved case is concerned, the above generalized free energy yields, procceding as above, the following evolution equation for the order parameter u:

    utΔki=1(1)i|β|=iaβD2βuΔf(u)=Δ(αtΔαt), (1.12)

    In particular, for k = 1 (anisotropic conserved Caginalp phase-field), we have an equation of the form

    ut+Δ3i=1ai2ux2iΔf(u)=Δ(αtΔαt)

    and, for k = 2 (fourth-order anisotropic conserved Caginalp phase-field system), we have an equation of the form

    utΔ3i,j=1aij4ux2ix2j+Δ3i=1bi2ux2iΔf(u)=Δ(αtΔαt).

    L. Cherfils A. Miranville and S. Peng have studied in [8] the corresponding higher-order isotropic equation (without the coupling with the temperature), namely, the equation

    utΔP(Δ)uΔf(u)=0,

    where

    P(s)=ki=1aisi,ak>0,k1,

    endowed with the Dirichlet/Navier boundary conditions

    u=Δu=...=Δku=0onΓ.

    Our aim in this paper is to study the model consisting of the higher-order anisotropic equation (1.12) and the temperature equation

    2αt2Δ2αt2ΔαtΔα=ut. (1.13)

    In particular, we obtain the existence and uniqueness of solutions.


    2. Setting of the problem

    We consider the following initial and boundary value problem, for kN, k2 (the case k = 1 can be treated as in the original conserved system; see, e.g., [23]):

    utΔki=1(1)i|β|=iaβD2βuΔf(u)=Δ(αtΔαt), (2.1)
    2αt2Δ2αt2ΔαtΔα=ut, (2.2)
    Dβu=α=0onΓ,|β|k, (2.3)
    u|t=0=u0,α|t=0=α0,αt|t=0=α1. (2.4)

    We assume that

    aβ>0,|β|=k, (2.5)

    and we introduce the elliptic operator Ak defined by

    Akv,wHk(Ω),Hk0(Ω)=|β|=kaβ((Dβv,Dβw)), (2.6)

    where Hk(Ω) is the topological dual of Hk0(Ω). Furthermore, ((., .)) denotes the usual L2-scalar product, with associated norm .. More generally, we denote by .X the norm on the Banach space X; we also set .1=(Δ)12., where (Δ)1 denotes the inverse minus Laplace operator associated with Dirichlet boudary conditions. We can note that

    (v,w)Hk0(Ω)2|β|=kaβ((Dβv,Dβw))

    is bilinear, symmetric, continuous and coercive, so that

    Ak:Hk0(Ω)Hk(Ω)

    is indeed well defined. It then follows from elliptic regularity results for linear elliptic operators of order 2k (see [1] and [2]) that Ak is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

    D(Ak)=H2k(Ω)Hk0(Ω),

    where, for vD(Ak),

    Akv=(1)k|β|=kaβD2βv.

    We further note that D(A12k)=Hk0(Ω) and, for (v,w)D(A12k)2,

    ((A12kv,A12kw))=|β|=kaβ((Dβv,Dβw)).

    We finally note that (see, e.g., [24]) Ak. (resp., A12k.) is equivalent to the usual H2k-norm (resp., Hk-norm) on D(Ak) (resp., D(A12k)).

    Similarly, we can define the linear operator ¯Ak=ΔAk

    ˉAk:Hk+10(Ω)Hk1(Ω)

    which is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

    D(ˉAk)=H2k+2(Ω)Hk+10(Ω),

    where, for vD(ˉAk),

    ˉAkv=(1)k+1Δ|β|=kaβD2βv.

    Furthermore, D(ˉA12k)=Hk+10(Ω) and, for (v,w)D(ˉA12k),

    ((ˉA12kv,ˉA12kw))=|β|=kaβ((Dβv,Dβw)).

    Besides ˉAk. (resp., ˉA12k.) is equivalent to the usual H2k+2-norm (resp., Hk+1-norm) on D(ˉAk) (resp., D(ˉA12k)).

    We finally consider the operator ˜Ak=(Δ)1Ak, where

    ˜Ak:Hk10(Ω)Hk+1(Ω);

    note that, as Δ and Ak commute, then the same holds for (Δ)1 and Ak, so that ˜Ak=Ak(Δ)1.

    We have the (see [17])

    Lemme 2.1. The operator ˜Ak is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

    D(˜Ak)=H2k2(Ω)Hk10(Ω),

    where, for vD(˜Ak)

    ˜Akv=(1)k|β|=kaβD2β(Δ)1v.

    Furthermore, D(˜A12k)=Hk10(Ω) and, for (v,w)D(˜A12k),

    ((˜A12kv,˜A12kw))=|β|=kaβ((Dβ(Δ)12v,Dβ(Δ)12w)).

    Besides ˜Ak. (resp., ˜A12k.) is equivalent to the usual H2k2-norm (resp., Hk1-norm) on D(˜Ak) (resp., D(˜A12k)).

    Proof. We first note that ˜Ak clearly is linear and unbounded. Then, since (Δ)1 and Ak commute, it easily follows that ˜Ak is selfadjoint.

    Next, the domain of ˜Ak is defined by

    D(˜Ak)={vHk10(Ω),˜AkvL2(Ω)}.

    Noting that ˜Akv=f,fL2(Ω),vD(˜Ak), is equivalent to Akv=Δf, where ΔfH2(Ω), it follows from the elliptic regularity results of [1] and [2] that vH2k2(Ω), so that D(˜Ak)=H2k2(Ω)Hk10(Ω).

    Noting then that ˜A1k maps L2(Ω) onto H2k2(Ω) and recalling that k2, we deduce that ˜Ak has compact inverse.

    We now note that, considering the spectral properties of Δ and Ak (see, e.g., [24]) and recalling that these two operators commute, Δ and Ak have a spectral basis formed of common eigenvectors. This yields that, s1,s2R, (Δ)s1 and As2k commute.

    Having this, we see that ˜A12k=(Δ)12A12k, so that D(˜A12k)=Hk10(Ω), and for (v,w)D(˜A12k)2,

    ((˜A12kv,˜A12kw))=|β|=kaβ((Dβ(Δ)12v,Dβ(Δ)12w)).

    Finally, as far as the equivalences of norms are concerned, we can note that, for instance, the norm ˜A12k. is equivalent to the norm (Δ)12.Hk(Ω) and, thus, to the norm (Δ)k12..

    Having this, we rewrite (2.1) as

    utΔAkuΔBkuΔf(u)=Δ(αtΔαt), (2.7)

    where

    Bkv=k1i=1(1)i|β|=iaβD2βv.

    As far as the nonlinear term f is concerned, we assume that

    fC2(R),f(0)=0, (2.8)
    fc0,c00, (2.9)
    f(s)sc1F(s)c2c3,c1>0,c2,c30,sR, (2.10)
    F(s)c4s4c5,c4>0,c50,sR, (2.11)

    where F(s)=s0f(τ)dτ. In particular, the usual cubic nonlinear term f(s)=s3s satisfies these assumptions.

    Throughout the paper, the same letters c, c' and c" denote (generally positive) constants which may vary from line to line. Similary, the same letter Q denotes (positive) monotone increasing (with respect to each argument) and continuous functions which may vary from line to line.


    3. A priori estimates

    We multiply (2.7) by (Δ)1ut and (2.2) by αtΔαt, sum the two resulting equalities and integrate over Ω and by parts. This gives

    ddt(A12ku2+B12k[u]+2ΩF(u)dx+α2+Δα2+αtΔαt2)+2ut21+2αt2+2Δαt2=0 (3.1)

    (note indeed that αt2+2αt2+Δαt2=αtΔαt2), where

    B12k[u]=k1i=1|β|=iaβDβu2 (3.2)

    (note that B12k[u] is not necessarily nonnegative). We can note that, owing to the interpolation inequality

    B12k[u]=k1i=1|β|=iaβDβu2 (3.3)
    (Δ)i2vc(i)(Δ)m2vimv1im,

    there holds

    vHm(Ω),i{1,...,m1},mN,m2, (3.4)

    This yields, employing (2.11),

    |B12k[u]|12A12ku2+cu2.

    whence

    A12ku2+B12k[u]+2ΩF(u)dx12A12ku2+ΩF(u)dx+cu4L4(Ω)cu2c", (3.5)

    nothing that, owing to Young's inequality,

    A12ku2+B12k[u]+2ΩF(u)dxc(u2Hk(Ω)+ΩF(u)dx)c,c>0, (3.6)

    We then multiply (2.7) by (Δ)1u and have, owing to (2.10) and the interpolation inequality (3.3),

    u2ϵu4L4(Ω)+c(ϵ),ϵ>0.

    hence, proceeding as above and employing, in particular, (2.11)

    ddtu21+c(u2Hk(Ω)+ΩF(u)dx)c(u2+αt2+Δαt2)+c", (3.7)

    Summing (3.1) and δ1 times (3.7), where δ1>0 is small enough, we obtain a differential inegality of the form

    ddtu21+c(u2Hk(Ω)+ΩF(u)dx)c(αt2+Δαt2)+c,c>0. (3.8)

    where

    ddtE1+c(u2Hk(Ω)+ΩF(u)dx+ut21+αt2H2(Ω))c,c>0,

    satisfies, owing to (3.5)

    E1=A12ku2+B12k[u]+2ΩF(u)dx+α2+Δα2+αtΔαt2+δ1u21 (3.9)

    Multiplying (2.2) by Δα, we then obtain

    E1c(u2Hk(Ω)+ΩF(u)dx+α2H2(Ω)+αt2H2(Ω))c,c>0.

    which yields, employing the interpolation inequality

    ddt(Δα22((αt,Δα))+2((Δαt,Δα)))+Δα2ut2+αt2+Δαt2, (3.10)

    the differential inequality, with 0<ϵ<<1 is small enough

    v2cv1vH1(Ω),vH10(Ω), (3.11)

    We now differentiate (2.7) with respect to time to find, owing to (2.2),

    ddt(Δα22((αt,Δα))+2((Δαt,Δα)))+cα2H2(Ω)c(ut21+ϵut2H1(Ω)+αt2H2(Ω)),c>0. (3.12)

    together with the boundary condition

    tutΔAkutΔBkutΔ(f(u)ut)=Δ(Δαt+Δαut), (3.13)

    We multiply (3.11) by (Δ)1ut and obtain, owing to (2.9) and the interpolation inequality (3.3),

    Dβut=0onΓ,|β|k.

    hence, owing to (3.10), the differential inequality

    ddtut21+cut2Hk(Ω)c(ut2+Δα2+Δαt2),c>0, (3.14)

    Summing finally (3.8), δ2 times (3.11) and δ3 times (3.14), where δ2,δ3>0 are small enough, we find a differential inequality of the form

    ddtut21+cut2Hk(Ω)c(ut21+α2H2(Ω)+αt2H2(Ω)),c>0. (3.15)

    where

    dE2dt+c(E2+ut2Hk(Ω))c,c>0,

    Owing to the continuous embedding H2k+1(Ω)C(ˉΩ), we deduce that

    E2=E1+δ2(Δα22((αt,Δα))+2((Δαt,Δα)))+δ3ut21.

    and since

    |ΩF(u0)dx|Q(u0H2k+1(Ω))

    we see that (Δ)12ut(0)L2(Ω) and

    (Δ)12ut(0)=(Δ)12Aku0(Δ)12Bku0(Δ)12f(u0)+(Δ)12(α1Δα1), (3.16)

    Furthermore E2 satisfies

    ut(0)1Q(u0H2k+1(Ω),α1H3(Ω)). (3.17)

    It thus follows from (3.15), (3.16), (3.17) and Growall's lemma that

    E2c(u2Hk(Ω)+ut21+ΩF(u)dx+α2H2(Ω)+αt2H2(Ω))c,c>0. (3.18)

    and

    u(t)2Hk(Ω)+ut(t)21+α(t)2H2(Ω)+αt(t)2H2(Ω)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c>0,t0, (3.19)

    r>0 given.

    Multiplying next (2.7) by ˜Aku, we find, owing to the interpolation inequality (3.3),

    t+rtut2Hk(Ω)dsectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c(r),c>0,t0,

    hence, since f and F are continuous and owing to (3.18),

    ddt˜A12ku2+cu2H2k(Ω)c(u2+f(u)2+αt2+Δαt2),c>0, (3.20)

    Summing (3.15) and (3.22), we have a differential inequality of the form

    ddt˜A12ku2+cu2H2k(Ω)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c",c,c>0,t0. (3.21)

    where

    dE3dt+c(E3+u2H2k(Ω)+ut2Hk(Ω))ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c",c,c>0,t0,

    satisfies

    E3=E2+˜A12ku2 (3.22)

    In particular, it follows from (3.21)-(3.22) that

    E3c(u2Hk(Ω)+ut21+ΩF(u)dx+α2H2(Ω)+αt2H2(Ω))c,c>0. (3.23)

    r>0 given.

    We now multiply (2.7) by u and obtain, employing (2.9) and the interpolation inequality (3.3)

    t+rtu2H2k(Ω)dsectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c(r),c>0,t0,

    whence, proceeding as above,

    ddtu2+cu2Hk+1(Ω)c(u2H1(Ω)+αt2+Δαt2),c>0, (3.24)

    We also multiply (2.7) by ut and find

    ddtu2+cu2Hk+1(Ω)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c,c>0.

    where

    ddt(ˉA12ku2+ˉB12k[u])+cut2cΔf(u)22((Δut,αtΔαt)),

    Since f is of class C2, it follows from the continuous embedding H2(Ω)C(ˉΩ) that

    ˉB12k[u]=k1i=1|β|=iaβDβu2.

    hence, owing to (3.18),

    Δf(u)2Q(uH2(Ω)), (3.25)

    Multiply next (2.2) by Δ(αtΔαt), we have

    ddt(ˉA12ku2+ˉB12k[u])+cut2ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))2((Δut,αtΔαt))+c,c,c>0. (3.26)

    (note indeed that αt2+2Δαt2+Δαt2=αtΔαt2).

    Summing (3.25) and (3.26), we obtain

    ddt(Δα2+Δα2+αtΔαt2)+c(Δαt2+Δαt2)2((Δut,αtΔαt)),c>0 (3.27)

    Summing finally (3.21), (3.24) and (3.27), we find a differential inegality of the form

    ddt(ˉA12ku2+ˉB12k[u]+Δα2+Δα2+αtΔαt2)+c(ut2+Δαt2+Δαt2)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c,c>0. (3.28)

    where

    dE4dt+c(E3+u2Hk+1(Ω)+u2H2k(Ω)+ut2+ut2Hk(Ω)+αt2H3(Ω))ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c,c>0,t0

    satisfies, owing to (2.11) and the interpolation inegality (3.3)

    E4=E3+u2+ˉA12ku2+ˉB12k[u]+Δα2+Δα2+αtΔαt2 (3.29)

    In particular, it follows from (3.28)-(3.29) that

    E4c(u2Hk+1(Ω)+ut21+ΩF(u)dx+α2H3(Ω)+αt2H3(Ω))c,c>0. (3.30)

    and

    u(t)Hk+1(Ω)+α(t)H3(Ω)+αt(t)H3(Ω)ectQ(u0H2k+1(Ω),α0H3(Ω),α1H3(Ω))+c,c>0,t0, (3.31)

    r given.

    We finally rewrite (2.7) as an elliptic equation, for t > 0 fixed,

    t+rt(ut2+αt2H3(Ω))dsectQ(u0H2k+1(Ω),α0H3(Ω),α1H3(Ω))+c(r),c>0,t0, (3.32)

    Multiplying (3.32) by Aku, we obtain, owing to the interpolation inequality (3.3),

    Aku=(Δ)1utBkuf(u)+αtΔαt,Dβu=0onΓ,|β|k1.

    hence, since f is continuous and owing to (3.18)

    Aku2c(u2+f(u)2+ut21+αt2+Δαt2), (3.33)

    4. Existence and uniqueness of solutions

    We first have the following theorem.

    Theorem 4.1. (i) We assume that (u0,α0,α1)Hk0(Ω)×(H2(Ω)H10(Ω))×(H2(Ω)H10(Ω)), with ΩF(u0)dx<+. Then, (2.1)(2.4) possesses at last one solution (u,α,αt) such that, T>0, u(0)=u0, α(0)=α0, αt(0)=α1,

    u(t)2H2k(Ω)cectQ(u0H2k+1(Ω),α0H3(Ω),α1H3(Ω))+c,c>0t0.
    uL(R+;Hk0(Ω))L2(0,T;H2k(Ω)Hk0(Ω)),
    utL(R+;H1(Ω))L2(0,T;Hk0(Ω)),

    and

    α,αtL(R+;H2(Ω)H10(Ω))
    ddt((Δ)1u,v))+ki=1|β|=iai((Dβu,Dβv))+((f(u),v))=ddt(((u,v))+((u,v))),vCc(Ω),

    in the sense of distributions.

    (ii) If we futher assume that (u0,α0,α1)(Hk+1(Ω)Hk0(Ω))×(H3(Ω)H10(Ω))×(H3(Ω)H10(Ω)), then, T>0,

    ddt(((αt,w))+((αt,w))+((α,w)))+((α,w))=ddt((u,w)),wCc(Ω),
    uL(R+;Hk+1(Ω)Hk0(Ω))L2(R+;Hk+1(Ω)Hk0(Ω))
    utL2(R+;L2(Ω)),

    and

    αL(R+;H3(Ω)H10(Ω))

    The proofs of existence and regularity in (i) and (ii) follow from the a priori estimates derived in the previous section and, e.g., a standard Galerkin scheme.

    We then have the following theorem.

    Theorem 4.2. The system (1.1)-(1.4) possesses a unique solution with the above regularity.

    proof. Let (u(1),α(1),α(1)t) and (u(2),α(2),α(2)t) be two solutions to (2.1)-(2.3) with initial data (u(1)0,α(1)0,α(1)1) and (u(2)0,α(2)0,α(2)1), respectively. We set

    αtL(R+;H3(Ω)H10(Ω))L2(0,T;H3(Ω)H10(Ω))

    and

    (u,α,αt)=(u(1),α(1),α(1)t)(u(2),α(2),α(2)t)

    Then, (u,α) satisfies

    (u0,α0,α1)=(u(1)0,α(1)0,α(1)1)(u(2)0,α(2)0,α(2)1). (4.1)
    utΔAkuΔBkuΔ(f(u(1))f(u(2)))=Δ(αtΔαt), (4.2)
    2αt2Δ2αt2ΔαtΔα=ut, (4.3)
    Dβu=α=0 on Γ,|β|k, (4.4)

    Multiplying (4.1) by (Δ)1u and integrating over Ω, we obtain

    u|t=0=u0,α|t=0=α0,αt|t=0=α1.

    We note that

    ddtu21+cu2Hk(Ω)c(u2+αtΔαt2)2((f(u(1))f(u(2),u)).

    with l defined as

    f(u(1))f(u(2))=l(t)u,

    Owing to (2.9), we have

    l(t)=10f(su(1)(t)+(1s)u(2)(t))ds.

    and we obtain owing to the intepolation inequalities (3.3) and (3.10),

    2((f(u(1))f(u(2),u))2c0u2                                      cu2 (4.5)

    Next, multiplying (4.2) by (Δ)1(u+αtΔαt), we find

    ddtu21+cu2Hk(Ω)c(u21+αtΔαt2),c>0. (4.6)

    Summing then δ4 times (4.5) and (4.6), where δ4>0 is small enough, we have, employing once more the interpolation inequality (3.10), a differential inequality of the form

    ddt(α2+α2+u+αtΔαt21)+c(αt2+αt2H1(Ω))c(u2+α2). (4.7)

    where

    dE5dtcE5,

    satisfies

    E5=δ4u21+α2+α2+u+αtΔαt21 (4.8)

    It follows from (4.7)-(4.8) and Gronwall's lemma that

    E5c(u21+α2H1(Ω)+αtΔαt2),c>0. (4.9)

    hence the uniquess, as well as the continuous dependence with respect to the initial data in H1×H1×H1-norm.


    Conflict of Interest

    All authors declare no conflicts of interest in this paper.




    [1] A. J. Lotka, Eelements of physical biology, Science Progress in the Twentieth Century, 21 (1926), 341–343.
    [2] V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118 (1926), 558–560. https://doi.org/10.1038/118558a0 doi: 10.1038/118558a0
    [3] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331–340. https://doi.org/10.2307/3866 doi: 10.2307/3866
    [4] D. L. DeAngelis, R. A. Goldstein, R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881–892. https://doi.org/10.2307/1936298
    [5] P. H. Leslie, J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219–234. https://doi.org/10.2307/2333294
    [6] R. Subarna, K. T. Pankaj, Bistability in a predator-prey model characterized by the Crowley-Martin functional response: Effects of fear, hunting cooperation, additional foods and nonlinear harvesting, Math. Comput. Simulat., 228 (2025), 274–297. https://doi.org/10.1016/j.matcom.2024.09.001
    [7] S. L. Pimm, J. H. Lawton, On feeding on more than one trophic level, Nature, 275 (1978), 542–544. https://doi.org/10.1038/275542a0
    [8] R. M. May, Stability and complexity in model ecosystems, Princeton: Princeton University Press, 2019.
    [9] S. G. Mortoja, P. Panja, S. K. Mondal, Dynamics of a predator-prey model with nonlinear incidence rate, Crowley-Martin type functional response and disease in prey population, Ecological Genetics and Genomics, 10 (2019), 100035. https://doi.org/10.1016/j.egg.2018.100035
    [10] P. H. Crowley, E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 8 (1989), 211–221. https://doi.org/10.2307/1467324 doi: 10.2307/1467324
    [11] N. Sk, B. Mondal, A. A. Thirthar, M. A. Alqudah, T. Abdeljawad, Bistability and tristability in a deterministic prey-predator model: Transitions and emergent patterns in its stochastic counterpart, Chaos Soliton. Fract., 176 (2023), 114073. https://doi.org/10.1016/j.chaos.2023.114073 doi: 10.1016/j.chaos.2023.114073
    [12] X. Y. Meng, H. F. Huo, H. Xiang, Q. Y. Yin, Stability in a predator-prey model with Crowley-Martin function and stage structure for prey, Appl. Math. Comput., 232 (2014), 810–819. https://doi.org/10.1016/j.amc.2014.01.139 doi: 10.1016/j.amc.2014.01.139
    [13] S. Creel, D. Christianson, Relationships between direct predation and risk effects, Trends Ecol. Evol., 23 (2008), 194–201. https://doi.org/10.1016/j.tree.2007.12.004 doi: 10.1016/j.tree.2007.12.004
    [14] W. Cresswell, Predation in bird populations, J. Ornithol., 152 (2011), 251–263. https://doi.org/10.1007/s10336-010-0638-1 doi: 10.1007/s10336-010-0638-1
    [15] X. Y. Wang, X. F. Zhou, Modeling the fear effect in predator-prey interactions with adaptive avoidance of predators, Bull. Math. Biol., 79 (2017), 1325–1359. https://doi.org/10.1007/s11538-017-0287-0 doi: 10.1007/s11538-017-0287-0
    [16] A. Das, G. P. Samanta, Modeling the fear effect on a stochastic prey-predator system with additional food for the predator, J. Phys. A: Math. Theor., 51 (2018), 465601. https://doi.org/10.1088/1751-8121/aae4c6 doi: 10.1088/1751-8121/aae4c6
    [17] V. Kumar, N. Kumari, Stability and bifurcation analysis of Hassell-Varley prey-predator system with fear effect, Int. J. Appl. Comput. Math., 6 (2020), 150. https://doi.org/10.1007/s40819-020-00899-y doi: 10.1007/s40819-020-00899-y
    [18] H. M. Li, Y. Tian, Dynamic behavior analysis of a feedback control predator-prey model with exponential fear effect and Hassell-Varley functional response, J. Franklin I., 360 (2023), 3479–3498. https://doi.org/10.1016/j.jfranklin.2022.11.030 doi: 10.1016/j.jfranklin.2022.11.030
    [19] K. Sarkar, S. Khajanchi, Impact of fear effect on the growth of prey in a predator-prey interaction model, Ecol. Complex., 42 (2020), 100826. https://doi.org/10.1016/j.ecocom.2020.100826 doi: 10.1016/j.ecocom.2020.100826
    [20] Y. K. Zhang, X. Z. Meng, Dynamics of a stochastic predation model with fear effect and Crowley-Martin functional response, Journal of Shandong University (Natural Science), 58 (2023), 54–66. https://doi.org/10.6040/j.issn.1671-9352.0.2022.635 doi: 10.6040/j.issn.1671-9352.0.2022.635
    [21] X. R. Mao, G. Marion, E. Renshaw, Environmental brownian noise suppresses explosions in population dynamics, Stoch. Proc. Appl., 97 (2002), 95–110. https://doi.org/10.1016/S0304-4149(01)00126-0 doi: 10.1016/S0304-4149(01)00126-0
    [22] D. Gravel, F. Massol, M. A. Leibold, Stability and complexity in model meta-ecosystems, Nat. Commun., 7 (2016), 12457. https://doi.org/10.1038/ncomms12457 doi: 10.1038/ncomms12457
    [23] Q. Wang, L. Zu, D. Q. Jiang, D. O'Regan, Study on dynamic behavior of a stochastic predator-prey system with Beddington-DeAngelis functional response and regime switching, Mathematics, 11 (2023), 2735. https://doi.org/10.3390/math11122735 doi: 10.3390/math11122735
    [24] B. Mondal, A. Sarkar, S. S. Santra, D. Majumder, T. Muhammad, Sensitivity of parameters and the impact of white noise on a generalist predator-prey model with hunting cooperation, Eur. Phys. J. Plus, 138 (2023), 1070. https://doi.org/10.1140/epjp/s13360-023-04710-x doi: 10.1140/epjp/s13360-023-04710-x
    [25] P. Ghosh, P. Das, D. Mukherjee, Persistence and stability of a seasonally perturbed three species of salmonoid aquaculture, Differ. Equ. Dyn. Syst., 27 (2019), 449–465. https://doi.org/10.1007/s12591-016-0283-0 doi: 10.1007/s12591-016-0283-0
    [26] A. Das, G. P. Samanta, Modelling the effect of resource subsidy on a two-species predator-prey system under the influence of environmental noises, Int. J. Dynam. Control, 9 (2021), 1800–1817. https://doi.org/10.1007/s40435-020-00750-8 doi: 10.1007/s40435-020-00750-8
    [27] E. Allen, Environmental variability and mean-reverting processes, Discrete Cont. Dyn.-B, 21 (2016), 2073–2089. https://doi.org/10.3934/dcdsb.2016037 doi: 10.3934/dcdsb.2016037
    [28] X. R. Mao, Stochastic differential equations and applications, 2 Eds., Amsterdam: Elsevier, 2007.
    [29] Q. Liu, D. Q. Jiang, Analysis of a stochastic within-host model of Dengue infection with immune response and Ornstein-Uhlenbeck process, J. Nonlinear Sci., 34 (2024), 28. https://doi.org/10.1007/s00332-023-10004-4 doi: 10.1007/s00332-023-10004-4
    [30] Q. Liu, A stochastic predator-prey model with two competitive preys and Ornstein-Uhlenbeck process, J. Biol. Dynam., 17 (2023), 2193211. https://doi.org/10.1080/17513758.2023.2193211 doi: 10.1080/17513758.2023.2193211
    [31] B. Q. Zhou, D. Q. Jiang, T. Hayat, Analysis of a stochastic population model with mean-reverting Ornstein-Uhlenbeck process and Allee effects, Commun. Nonlinear Sci., 111 (2022), 106450. https://doi.org/10.1016/j.cnsns.2022.106450 doi: 10.1016/j.cnsns.2022.106450
    [32] Q. Liu, D. Q. Jiang, Analysis of a stochastic logistic model with diffusion and Ornstein-Uhlenbeck process, J. Math. Phys., 63 (2022), 053505. https://doi.org/10.1063/5.0082036 doi: 10.1063/5.0082036
    [33] R. Khasminskii, Stochastic stability of differential equations, 2 Eds., Heidelberg: Springer-Verlag Berlin, 2012. https://doi.org/10.1007/978-3-642-23280-0
    [34] D. Y. Xu, Y. M. Huang, Z. G. Yang, Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Cont. Dyn.-A, 24 (2009), 1005–1023. https://doi.org/10.3934/dcds.2009.24.1005 doi: 10.3934/dcds.2009.24.1005
    [35] Q. Luo, X. R. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334 (2007), 69–84. https://doi.org/10.1016/j.jmaa.2006.12.032 doi: 10.1016/j.jmaa.2006.12.032
    [36] X. Z. Chen, B. D. Tian, X. Xu, H. L. Zhang, D. Li, A stochastic predator-prey system with modified LG-Holling type II functional response, Math. Comput. Simulat., 203 (2023), 449–485. https://doi.org/10.1016/j.matcom.2022.06.016 doi: 10.1016/j.matcom.2022.06.016
    [37] S. E. Jørgensen, Handbook of environmental data and ecological parameters: environmental sciences and applications, Amsterdam: Elsevier, 2013.
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