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Research article Special Issues

Effect of antioxidant supplements on sperm parameters in infertile male smokers: a single-blinded clinical trial

  • Received: 19 September 2019 Accepted: 19 January 2020 Published: 11 February 2020
  • Citation: Safa Sadaghiani, Soghra Fallahi, Hossein Heshmati, Saeed Hosseini Teshnizi, Hooshang Abedini Chaijan, Fahimeh Farzad Amir Ebrahimi, Farhad Khorrami, Mina Poorrezaeian, Faezeh Alizadeh. Effect of antioxidant supplements on sperm parameters in infertile male smokers: a single-blinded clinical trial[J]. AIMS Public Health, 2020, 7(1): 92-99. doi: 10.3934/publichealth.2020009

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  • From the initial work by Lotka [1] and Volterra [2], predator-prey model has become and will continue to be one of the main themes in mathematical biology. In the interaction between predator and prey, the phenomenon of prey refuge always exists. It is assumed that prey species can live in two different regions. One is the prey refuge and the other is the predatory region. From biological view, prey refuge can exists and there are no predators in the prey refuge, so it can help increase the population density of the prey. Further, refuge is an effective strategy for reducing predation as a prey population evolved. For this reason, Gause and his partners [3,4] proposed the predator-prey model with a refuge. Moreover, Magalh˜aes et al. [5] studied the dynamics of thrips prey and their mite predators in a refuge, and they predicted the small effect of the refuge on the density of prey under the equilibrium state. Ghosh et al. [6] investigated the impact of additional food for predator on the dynamics of the predator-prey model with a prey refuge. When in a high-prey refuge ecological system, it was observed that the predator extinction possibility may be removed by supplying additional food [7,8] to predator population. Ufuktepe [9] investigated the stability of a prey refuge predator-prey model with Allee effects. Fractional-order factor was introduced into prey-predator model with prey refuge in Xie [10]. On the other hand, Holling [11] argued that the functional response is an important factor to affect the predator-prey model. The predator may reduce its feeding rate when it is fully saturated, and the feeding rate no longer varies with the increase in prey density. Thus he proposed three types of Holling functional responses. Among them, most of the researchers showed their interest in Holling type Ⅱ functional response [12,13,14,15,16,17,18,19,20,21,22,23,24,25]. For the above reasons, Jana [26] considered the following predator-prey system incorporating a prey refuge:

    {dxdt=r1x(1xk1)σ1x+σ2y,dydt=r2y(1yk2)+σ1xσ2yαyza+y,dzdt=βy(tτ)z(tτ)a+y(tτ)dzγz2, (1.1)

    where x and y denote the density of the prey in the refuge and in the predatory region at any time t, r1 and r2 denote intrinsic growth rate respectively for the prey population x and y at any time t. Further, the environment carrying capacity are denoted by k1 and k2 respectively for the prey x and y. Then, at any time t, σ1 denotes the per unit migration of the prey in the refuge to the predatory region and σ2 denotes from the predatory region to the refuge. Next, z denotes density of the predator in the predatory region at any time t. In addition, the predator consumes the prey at Holling type Ⅱ functional response αya+y, where α is the maximal predator per capita consumption rate and a is the half capturing saturation constant. Furthermore, d is the natural death rate of predator at any time t, γ is the density dependent mortality rate of predator. And β is the rate of the predator consumes prey (assume that 0<βα). Because the reproduction of predators after predating the prey is not instantaneous, we assumed that the time interval between the prey are killed and the corresponding increase in the number of predators are thought to be time delayed τ of the system (1.1).

    In the real world application, some authors argued that predators living in the predatory region are classified by two fixed ages [27,28,29,30,31], one is immature predator and the other is mature predator, the immature predator have no ability to attack prey.

    Motivated by the above mentioned works, in the present paper, we investigate the periodic solution of the following delayed model:

    {dxdt=r1(t)x(t)(1x(t)k1(t))σ1(t)x(t)+σ2(t)y(t),dydt=r2(t)y(t)(1y(t)k2(t))+σ1(t)x(t)σ2(t)y(t)α(t)y(t)z2(t)a+y(t),dz1dt=β(t)y(t)z2(t)a+y(t)β(tτ)y(tτ)z2(tτ)a+y(tτ)d1(t)z1(t),dz2dt=β(tτ)y(tτ)z2(tτ)a+y(tτ)d2(t)z2(t), (1.2)

    where z1(t) and z2(t) denote the density of immature predator and mature predator respectively at any time t. Next, r1(t),r2(t),k1(t),k2(t),σ1(t),σ2(t),α(t),β(t),d1(t)andd2(t) are continuously positive periodic functions with period ω. Moreover, d1(t)andd2(t) are the death rate of predator at any time t. The nomenclature y(tτ)z2(tτ)a+y(tτ) stands for the number of immature predator that were born at time (tτ) which still survive at time t and become mature predator. The initial conditions for the system (1.2) are

    (x(t),y(t),z1(t),z2(t))C+=C([τ,0],R4+),x(0)>0,y(0)>0,z1(0)>0,z2(0)>0.

    The aim of this paper is to obtain some sufficient conditions for the existence of positive periodic solution of system (1.2). However, we encounter with some difficulties when we use Mawhin's coincidence degree theory to obtain the periodic solutions. Firstly, the forth equation of system (1.2) has a term y(tτ)z2(tτ), rather than y(tτ)z2(t). If we follow the skill in [32], it will lead us to u3(ξ3)ˉα1(a+l+)[(ˉr2ˉσ2)+l1ˉσ1eu1(η1)], which contains u1(η1). Consequently, we can not get the bound of u3 or u1. We will overcome this difficulty in the following paper.

    The rest of our paper is organized as follows: section 2 is to prove the existence of the positive periodic solution of system (1.2). In section 3, an example is to demonstrate the obtained result.

    It is not difficult to see that we can separate the third equation of (1.2) from the whole system and obtain the following subsystem of (1.2):

    {dxdt=r1(t)x(t)(1x(t)k1(t))σ1(t)x(t)+σ2(t)y(t),dydt=r2(t)y(t)(1y(t)k2(t))+σ1(t)x(t)σ2(t)y(t)α(t)y(t)z(t)a+y(t),dz2dt=β(tτ)y(tτ)z2(tτ)a+y(tτ)d2(t)z2(t). (2.1)

    The initial conditions for the system (2.1) are

    (x(t),y(t),z2(t))C+=C([τ,0],R3+),x(0)>0,y(0)>0,z2(0)>0.

    In order to obtain the positive periodic solution of (2.1), we need some known and preliminary results.

    Lemma 2.1. Let ΩU be an open bounded set. Let L be a Fredholm operator of index zero and let N be L-compact on ˉΩ. Suppose that the following conditions are satisfied:

    (a) for each λ(0,1),uΩDomL,LuλNu;

    (b) for each uΩkerL,QNu0;

    (c) deg[JQN,ΩkerL,0]0.

    Then Lu=Nu has at least one solution in ˉΩDomL.

    For the notations, concepts and further details of Lemma2.1, one can refer to [33,34,35,36].

    Lemma 2.2. If f(t) is a continuously periodic function with period ω, then

    t+ωωf(s)ds=t0f(s)ds,forany t.

    Proof. Let j=sω, then

    t+ωωf(s)ds=t0f(j+ω)dj=t0f(j)dj=t0f(s)ds.

    Lemma 2.3. [37] If u(t) is a continuously differentiable periodic function with period ω, then there is a ˜t[0,ω] such that

    |u(t)||u(˜t)|+ω0|˙u(s)|dsor|u(t)||u(˜t)|ω0|˙u(s)|ds.

    For convenience, we adopt the following notations in our paper:

    ˉf=1ωω0f(t)dt,fL=mint[0,ω]f(t),fM=maxt[0,ω]f(t),

    l=adL2βMe2ˉσ2ωdL2,l+=adM2βLe2ˉσ2ωdM2,u0=adL2ˉβa,

    b1=12(¯k1r1){(ˉr1ˉσ1)+[(ˉr1ˉσ1)2+4ˉσ2l+(¯r1k1)]12},b2=(¯k1r1)(ˉr1ˉσ1),

    b3=ˉα1a[(ˉr2ˉσ2)(¯r2k2)l+],b4=ˉα1(a+l+)[(ˉr2ˉσ2)+ˉσ2leB1],

    b5=12(¯k1r1){(ˉr1ˉσ1)+[(ˉr1ˉσ1)2+4ˉσ2u0(¯r1k1)]12},

    b6=ˉα1(a+u0)[(ˉr2ˉσ2)(¯r1k1)u0+ˉσ2D6u0],

    B1=max{|B11|,|B12|},B2=max{|B21|,|B22|},B3=max{|B31|,|B32|}.

    And we assume that:

    (H1):dL2βMe2ˉσ2ω;

    (H2):βL(dL2)1dM2βM;

    (H3):ˉr2<ˉαa1eB3+(¯r2k2)l++ˉσ2.

    Now, we are in a position to state our main results.

    Theorem 2.1. If system (1.2) satisfies: (H1), (H2) and (H3), then it has at least one positive periodic solution.

    Proof. We prove this theorem for two steps.

    Step 1: We prove subsystem (2.1) has at least one periodic solution.

    Letting

    u1(t)=lnx(t),u2(t)=lny(t),u3(t)=lnz2(t),

    then we have

    {˙u1(t)=r1(t)(1eu1(t)k1(t))+σ2(t)eu2(t)u1(t)σ1(t),˙u2(t)=r2(t)(1eu2(t)k2(t))+σ1(t)eu1(t)u2(t)α(t)eu3(t)a+eu2(t)σ2(t),˙u3(t)=β(tτ)eu2(tτ)+u3(tτ)u3(t)a+eu2(tτ)d2(t). (2.2)

    Take

    U=V={u=(u1,u2,u3)C(R,R3)|u(t+ω)=u(t)}.

    It is easy to see that U,V are both Banach Spaces with the norm ||||,

    ||u||=maxt[0,ω]|u1|+maxt[0,ω]|u2|+maxt[0,ω]|u3|,u=(u1,u2,u3)UorV.

    For any u=(u1,u2,u3)U, by the periodicity of the coefficients of system (2.2). We can check that:

    r1(t)(1eu1(t)k1(t))+σ2(t)eu2(t)u1(t)σ1(t):=Θ1(u,t),

    r2(t)(1eu2(t)k2(t))+σ1(t)eu1(t)u2(t)α(t)eu3(t)a+eu2(t)σ2(t):=Θ2(u,t)

    and

    β(tτ)eu2(tτ)+u3(tτ)u3(t)a+eu2(tτ)d2(t):=Θ3(u,t)

    are all ω-periodic functions.

    In fact,

    Θ1(u(t+ω),t+ω)=r1(t+ω)(1eu1(t+ω)k1(t+ω))+σ2(t+ω)eu2(t+ω)u1(t+ω)σ1(t+ω)=r1(t)(1eu1(t)k1(t))+σ2(t)eu2(t)u1(t)σ1(t)=Θ1(u,t).

    In a similar way, one can obtain

    Θ2(u(t+ω),t+ω)=Θ2(u,t),Θ3(u(t+ω),t+ω)=Θ3(u,t).

    Set

    L:DomLUL(u1(t),u2(t),u3(t))=(du1(t)dt,du2(t)dt,du3(t)dt),

    where DomL={(u1,u2,u3)C(R,R3)} and N:UU is defined by

    N(u1u2u3)=(Θ1(u,t)Θ2(u,t)Θ3(u,t)),

    Define

    P(u1u2u3)=Q(u1u2u3)=(1ωω0u1(t)dt1ωω0u2(t)dt1ωω0u3(t)dt), (u1u2u3)U=V.

    It is not difficult to know that

    kerL={uU|u=C0,C0R3} and ImL={vV|ω0v(t)dt=0}.

    Consequently, dimkerL=codimImL=3<+, and P and Q are continuous projectors such that

    ImP=kerL,kerQ=ImL=Im(IQ).

    It follows that L is a Fredholm mapping of index zero. Moreover, the generalized inverse(of L) Kp:ImLDomLkerP exists and is given by

    Kp(v)=t0v(s)ds1ωω0t0v(s)dsdt,

    then,

    QNu=(1ωω0Θ1(u,t)dt1ωω0Θ2(u,t)dt1ωω0Θ3(u,t)dt)

    and

    Kp(IQ)Nu=t0Nu(s)ds1ωω0t0Nu(s)dsdt(tω12)ω0Nu(s)ds.

    Obviously, QN and Kp(IQ)N are continuous. By using the Arzela-Ascoli theorem, it's not difficult to see that N is L-compact on ˉΩ with any open bounded set ΩU.

    Next, our aim is to search for an appropriate open bounded subset Ω for the application of the continuation theorem. Corresponding to the operator equation Lu=λNu,λ(0,1), we have

    ˙u1(t)=λ[r1(t)(1eu1(t)k1(t))+σ2(t)eu2(t)u1(t)σ1(t)], (2.3)
    ˙u2(t)=λ[r2(t)(1eu2(t)k2(t))+σ1(t)eu1(t)u2(t)α(t)eu3(t)a+eu2(t)σ2(t)], (2.4)
    ˙u3(t)=λ[β(tτ)eu2(tτ)+u3(tτ)u3(t)a+eu2(tτ)d2(t)]. (2.5)

    Suppose u=(u1(t),u2(t),u3(t))TU is a solution of (2.3), (2.4) and (2.5), for a certain λ(0,1). Integrating (2.3), (2.4) and (2.5) over the interval [0,ω], we obtain

    ˉσ1ω=ω0[σ2(t)eu2(t)u1(t)+r1(t)r1(t)k1(t)eu1(t)]dt, (2.6)
    ˉσ2ω=ω0[σ1(t)eu1(t)u2(t)α(t)eu3(t)a+eu2(t)+r2(t)r2(t)k2(t)eu1(t)]dt, (2.7)
    ˉd2ω=ω0β(tτ)eu2(tτ)+u3(tτ)u3(t)a+eu2(tτ)dt. (2.8)

    From the Eqs (2.3) and (2.6), we have

    ω0|˙u1(t)|dt=λω0|r1(t)(1eu1(t)k1(t))+σ2(t)eu2(t)u1(t)σ1(t)|dt<ω0|r1(t)r1(t)k1(t)eu1(t)+σ2(t)eu2(t)u1(t)|dt+ω0|σ1(t)|dt<ˉσ1ω+ˉσ1ω=2ˉσ1ω,

    that is

    ω0|˙u1(t)|dt<2ˉσ1ω. (2.9)

    Similarly, it follows from (2.4) and (2.7), (2.5) and (2.8) that

    ω0|˙u2(t)|dt<2ˉσ2ω, (2.10)
    ω0|˙u3(t)|dt<2ˉd2ω, (2.11)

    Since (u1(t),u2(t),u3(t))U, there exists ξi,ηi[0,ω], such that

    ui(ξi)=mint[0,ω]ui(t),ui(ηi)=maxt[0,ω]ui(t),i=1,2,3.

    Multiplying (2.5) by eu3(t) and integrating over [0,ω], we have

    ω0d2(t)eu3(t)dt=ω0β(tτ)eu2(tτ)+u3(tτ)a+eu2(tτ)dt.

    Now we make the change of a variable j=tτ and Lemma2, we obtain

    ω0β(tτ)eu2(tτ)+u3(tτ)a+eu2(tτ)dt=ωττβ(j)eu2(j)+u3(j)a+eu2(j)dj=ω0β(t)eu2(t)+u3(t)a+eu2(t)dt,

    that is

    ω0d2(t)eu3(t)dt=ω0β(t)eu2(t)+u3(t)a+eu2(t)dt. (2.12)

    From the Eq (2.12), we obtain

    dL2ω0eu3(t)dtω0d2(t)eu3(t)dt=ω0β(t)eu2(t)+u3(t)a+eu2(t)dtβMeu2(η2)a+eu2(ξ2)ω0eu3(t)dt,

    which is

    dL2βMeu2(η2)a+eu2(ξ2),

    thus

    u2(η2)lndL2(a+eu2(ξ2))βM. (2.13)

    It follows from (2.10), (2.13) and Lemma3 that we have

    u2(t)u2(η2)ω0|˙u2(t)|dt>lndL2(a+eu2(ξ2))βM2ˉσ2ω:=B21. (2.14)

    In particular, we obtain

    u2(ξ2)>lndL2(a+eu2(ξ2))βM2ˉσ2ω,

    or

    (βMe2ˉσ2ωdL2)eu2(ξ2)adL2>0.

    And in view of (H1), we have

    u2(ξ2)>lnadL2βMe2ˉσ2ωdL2=lnl.

    In a similar way, from the Eq (2.12) we obtain

    dM2βLeu2(ξ2)a+eu2(η2),

    therefore

    u2(ξ2)lndM2(a+eu2(η2))βL. (2.15)

    It follows from (2.10), (2.15) and Lemma3 that we have

    u2(t)u2(ξ2)+ω0|˙u2(t)|dt<lndM2(a+eu2(η2))βL+2ˉσ2ω:=B22. (2.16)

    In particular, we have

    u2(η2)<lndM2(a+eu2(η2))βL+2ˉσ2ω,

    or

    (βLe2ˉσ2ωdM2)eu2(η2)adM2<0.

    In view of (H1), we have

    u2(η2)<lnadM2βLe2ˉσ2ωdM2=lnl+.

    In view of (H2), we have

    adM2βLe2ˉσ2ωdM2adL2βMe2ˉσ2ωdL2.

    It follows from (2.14) and (2.16) that

    maxt[0,ω]|u2(t)|=maxt[0,ω]{|B21|,|B22|}:=B2.

    It follows from (2.6) that

    ˉσ1ωˉσ2ωelnl+eu1(ξ1)+ˉr1ω¯(r1k1)ωeu1(ξ1),

    thus

    u1(ξ1)ln{12(¯k1r1)[(ˉr1ˉσ1)+(ˉr1ˉσ1)2+4ˉσ2l+(¯r1k1)]}=lnb1.

    This combined with (2.9), give

    u1(t)u1(ξ1)+ω0|˙u1(t)|dt<lnb1+2ˉσ1ω:=B11. (2.17)

    Similarly, we have

    ˉσ1ωˉr1ω¯(r1k1)ωeu1(η1),

    therefore

    u1(η1)ln[(¯k1r1)(ˉr1ˉσ1)]=lnb2.

    This together with (2.9), gives

    u1(t)u1(η1)ω0|˙u1(t)|dt>lnb22ˉσ1ω:=B12. (2.18)

    It follows from (2.17) and (2.18) that

    maxt[0,ω]|u1(t)|=maxt[0,ω]{|B11|,|B12|}:=B1.

    It follows from that (2.7) that

    ˉσ2ωa1ˉαωeu3(η3)+ˉr2ω¯(r2k2)ωl+,

    thus

    u3(η3)ln{aˉα1[(ˉr2ˉσ2)(¯r2k2)l+]}=lnb3.

    This combined with (2.11), give

    u3(t)u3(η3)ω0|˙u3(t)|dt>lnb32ˉd2ω:=B31, (2.19)

    Similarly,

    ˉσ2ωˉαωeu3(ξ3)a+l++ˉσ1ωleu1(η1)+ˉr2ω,

    therefore

    eu3(ξ3)(a+l+)[(ˉr2ˉσ2)+ˉσ1leu1(η1)],

    or

    u3(ξ3)<ln{ˉα1(a+l+)[(ˉr2ˉσ2)+ˉσ2leB1]}=lnb4.

    Combining with (2.11), give

    u3(t)<u3(ξ3)+ω0|˙u3(t)|dt<lnb4+2ˉd2ω:=B32. (2.20)

    It follows from (2.19) and (2.20) that

    maxt[0,ω]|u3(t)|=maxt[0,ω]{|B31|,|B32|}:=B3.

    Next, let's consider QNu with u=(u1,u2,u3)R3. Note that

    QN(u1,u2,u3)=[(ˉr1ˉσ1)+ˉσ2eu2(t)u1(t)¯(r1k1)eu1(t),
    (ˉr2ˉσ2)+ˉσ1eu1(t)u2(t)¯(r2k2)eu2(t)ˉαeu3(t)a+eu2(t),ˉd2+ˉβeu2(t)a+eu2(t)].

    In view of (H1),(H2),(H3), QN(u1,u2,u3)=0 has a solution ˜u=(lnb5,lnu0,lnb6). Take B=max{B1+C,B2+C,B3+C}, where C>0 is taken sufficiently large such that (lnb5,lnu0,lnb6)<C. Define Ω={u(t)=(u1(t),u2(t),u3(t))TU:u<B}. Then Ω is a bounded open subset of U, therefore Ω satisfies the requirement (a) in Lemma1. Moreover, it's not difficult to verify QNu0 for uΩkerL=ΩR3. A direct computation gives deg{JQN,ΩkerL,0}0. Therefore, system (2.1) has at least one ω periodic solution ˜u.

    Step 2: We prove that the third equation of system (1.2) has a unique ω-periodic solution associated with the obtained ˜u. Letting

    h(t)=β(t)y(t)z2(t)a+y(t)β(tτ)y(tτ)z2(tτ)a+y(tτ),

    then the third equation of (1.2) is

    dz1dt=d1(t)z1(t)+h(t). (2.21)

    Obviously,

    d1(t+ω)=d1(t)

    and

    h(t+ω)=β(t+ω)y(t+ω)z2(t+ω)a+y(t+ω)β(t+ωτ)y(t+ωτ)z2(t+ωτ)a+y(t+ωτ)=β(t)y(t)z2(t)a+y(t)β(tτ)y(tτ)z2(tτ)a+y(tτ)=h(t).

    Since d1(t) is nonnegative, ˉd1>0, it follows that

    dz1dt=d1(t)z1(t), (2.22)

    admits exponential dichotomy. Therefore, we have

    z1(t)=tetsd1(σ)dσh(s)ds.

    Consequently, (lnx(t),lny(t),z1(t),lnz2(t)) is a ω-periodic solution of system (1.2). This completes the proof.

    As an example, corresponding to the model (1.2), we have the following stage-structured predator-prey model with Holling type Ⅱ functional response incorporating prey refuge with actual biological parameters:

    {dxdt=(1.2+sin20πt)x(t)(1x(t)10+sin20πt)0.2x(t)+0.15y(t),dydt=(1.5+sin20πt)y(t)(1y(t)15+sin20πt)+0.2x(t)0.15y(t)(2+sin20πt)y(t)z2(t)2+y(t),dz1dt=(1.5+sin20πt)y(t)z2(t)2+y(t)(1.5+sin20π(t1))y(t1)z2(t1)2+y(t1)0.2z1(t),dz2dt=(1.5+sin20π(t1))y(t1)z2(t1)2+y(t1)0.1z2(t), (3.1)

    where r1(t)=1.2+sin20πt and r2(t)=1.5+sin20πt denote intrinsic growth rates, they are directly proportional to the the density of the prey x and y; k1(t)=10+sin20πt and k2(t)=15+sin20πt denote the environment carrying capacity for the prey x and y; σ1(t)=0.2 and σ2(t)=0.15 denote the per unit migration of the prey in the refuge to the predatory region and the opposite of it; α(t)=2+sin20πt denotes the maximal predator per capita consumption rate and β(t)=1.5+sin20πt denotes the rate of the predator consumes prey; (2+sin20πt)y(t)2+y(t) denotes Holling type Ⅱ functional response, which reflects the capture ability of the predator; d1(t)=0.2 and d2(t)=0.1 are the death rate of predator at any time t; (1.5+sin20πt)y(t1)z2(t1)/(2+y(t1)) stands for the number of immature predator born at time (t1) which still survive at time t and become mature predator.

    Since τ=1, a=2, d2=0.1, σ2=0.15, r2(t)=1.5+sin20πt, k(t)=15+sin20πt, α(t)=2+sin20πt, β(t)=1.5+sin20πt, we have ˉd2=dM2=dL2=0.1,ˉσ2=0.15,ˉr2=1.5,¯(r2k2)=0.1,ˉα=2,βL=0.5,βM=2.5,ω=0.1. Simple computation shows

    βMe2ˉσ2ω=2.5×e2×0.15×0.1=2.5×e0.03>2.5>0.1=dL2,
    (dL2)1dM2βM=10×0.1×2.5=2.5>0.5=βL

    and

    ˉαa1eB3+(¯r2k2)l++ˉσ2
    =12×2×emax{|B31|,|B32|}+110×2×0.10.5×e0.030.1+0.15
    =eB32+250e0.0310+0.15
    >(1.35+(2.5e0.030.01)emax{|B11|,|B12|})e0.02+250e0.0310+0.15
    >(1.35+(2.5e0.030.01)e0)e0.02+250e0.0310+0.15>1.5=ˉr2.

    The above inequalities show that system (3.1) satisfies the hypothesis (H1),(H2),(H3) in Theorem1. Therefore, system (3.1) has at least one positive periodic solution.

    In mathematical biology, dynamic relationship between predator and prey is always and will continue to be one of the main themes, many researchers have contributed to the study and improvement for the predator-prey model [17,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65]. The model in the present paper mimics the dynamic nature of the refuge. The populations varies due to the rates of emergence from and re-entry into refuge, which incorporates simultaneous effects of the refuge and migration of the population from the refuge area to the predatory area. For instance, it may happen in the birds migration. In fact, the dynamic nature of the refuge is an effective strategy for reducing predation as a prey population evolved. For this reason, a delayed stage-structured predator-prey model with a prey refuge is considered in this paper.

    This work was supported by the National Natural Science Foundation of China under Grant (No. 11931016), Natural Science Foundation of Zhejiang Province under Grant (No. LY20A010016). The authors thank the editor and anonymous referees for their valuable suggestions and comments, which improved the presentation of this paper.

    The authors declare that there is no conflict of interests regarding the publication of this article.


    Acknowledgments



    We are sincerely thankful to our counsellors in Clinical Research Development Center of ShahidMohammadi Hospital.

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