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Research article

Dynamics of a non-autonomous predator-prey system with Hassell-Varley-Holling Ⅱ function response and mutual interference

  • Received: 20 January 2021 Accepted: 19 March 2021 Published: 01 April 2021
  • MSC : 34K25, 34C27, 34D20, 92D25

  • In this paper, we establish a non-autonomous Hassell-Varley-Holling type predator-prey system with mutual interference. We construct some sufficient conditions for the permanence, extinction and globally asymptotic stability of system by use of the comparison theorem and an appropriate Liapunov function. Then the sufficient and necessary conditions for a periodic solution of the system are obtained via coincidence degree theorem. Finally, the correctness of the previous conclusions are demonstrated by some numerical cases.

    Citation: Luoyi Wu, Hang Zheng, Songchuan Zhang. Dynamics of a non-autonomous predator-prey system with Hassell-Varley-Holling Ⅱ function response and mutual interference[J]. AIMS Mathematics, 2021, 6(6): 6033-6049. doi: 10.3934/math.2021355

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  • In this paper, we establish a non-autonomous Hassell-Varley-Holling type predator-prey system with mutual interference. We construct some sufficient conditions for the permanence, extinction and globally asymptotic stability of system by use of the comparison theorem and an appropriate Liapunov function. Then the sufficient and necessary conditions for a periodic solution of the system are obtained via coincidence degree theorem. Finally, the correctness of the previous conclusions are demonstrated by some numerical cases.



    Since the functional response was proposed by Holling [1], many scholars have considered the dynamic behavior of systems with different functional response. The hybrid models combining Holling type and other functional response such as B-D and L-G type have received great attention (see e.g. [2,3,4,5,6,7,8,9,10,11]). Further, when several factors such as the growing process and gestation of population are taken into account, lots of models with time-delays and stage-structured have been investigated (see e.g. [12,13,14,15,16,17,18,19]). Some stochastic predator-prey systems have been studied due to the impact of environmental noise (see e.g. [20,21]).

    Hassell [22] considered a predator-prey system between parasite and host. It was found either one or both of them would leave from the meeting place when two predators meet. This phenomenon is known as the mutual interference of predatory behavior in single-species population. Subsequently to this discovery, many authors began to pay close attention to the dynamic behavior of systems with mutual interference(see e.g. [23,24,25]). Fu and Chen [25] studied the autonomous model with mutual interference:

    {dxdt=x(abxαcx2α)hxβym1+rxβ,dydt=y(d+fxβ1+rxβ), (1.1)

    where abxαcx2α is the nonlinear average growth of the prey due to environmental changes in the habitat, hxβ1+rxβ is the nonlinear saturated function response and m is interference parameter. It studied the persistence, the stability of the coexisting equilibrium point and the existence of limit cycles in [25].

    Obviously, hxβ1+rxβ is monotone increasing function for any x>0, that implies when the prey density increases, the predation rate increases. In other words, the predation rate is not affected by the number of predators. However, the predation rate can depend on its density in the real world. Hassell and Varley [26] found that the abundance of predators counteracts the predator rate by experiments, and obtained the functional response αxyσ which was named Hassell-Varley type functional response. The adaptive range of σ is (0,1] and the value of σ reflects the size of predator groups. Arditi [27] and Sutherland [28] combined Hassell-Varley type with Holling type functional response, and produced Hassell-Varley-Holling functional response αxyσ+hx (Ⅱ type) and αx2yσ+hx2 (Ⅲ type) respectively. Subsequently, some Hassell-Varley-Holling type predator-prey systems had been discussed (see e.g. [29,30,31]).

    In the paper, in order to better reflect the influence of predator groups on predation behavior, we choose Hassell-Varley-Holling Ⅱ functional response hxyσ+rx. According to the modeling mechanism of literature [25], we establish the following model:

    {dxdt=x(abxαcx2α)hxymrx+yσ,dydt=y(d+fxrx+yσ). (1.2)

    However, the biological and environmental parameters are changing over time. When these factors are considered, the corresponding model should be non-autonomous. Many authors focused on the permanence, stability and positive periodic solution about the non-autonomous models (see e.g. [2,5,7,12,13,14,15,18,23,24,29,32,33,34]). To our knowledge, there is no literature considering the non-autonomous model with Hassell-Varley-Holling Ⅱ and mutual interference.

    In the paper, let us discuss the model:

    {dxdt=x[a(t)b(t)xαc(t)x2α]h(t)xymr(t)x+yσ,(1.3a)dydt=y[d(t)+f(t)xr(t)x+yσ],(1.3b)

    where 0α1, 0<σ1, 0<m1. m is the mutual interference factor of predator. a(t) represents the intrinsic growth rate, b(t) measures the intra species competition rate, c(t) denotes the removal coefficient of the prey. f(t) and d(t) are the increasing coefficient and the death rate of predator respectively. h(t) and r(t) denote the ability and the unit time number to search for prey. If σ=m=1, the system (1.3) is well known as ratio-dependent predator-prey system. Especially, if α=σ=m=1,c(t)=0, the system (1.3) has been studied by Fan [32].

    One of our purpose is to obtain some conditions for the stability and periodic solutions of (1.3). The index m and σ of the term h(t)xymr(t)x+yσ prevent us from directly using the methods in the literature (see e.g. [5,18,23,24,29,32,33]). Here, we employ different methods in Section 2 to prove the stability and find the priori bound.

    The rest of this paper is organized as follow. Using the principle of comparison and constructing a suitable Liapunov function, we obtain the sufficient conditions for the permanence, non-permanence and globally asymptotic stability of system (1.3) in Section 2. In Section 3, the coincidence degree theorem is employed to find the conditions for the existence of positive periodic solutions. A sufficient and necessary condition is obtained when m>σ and some sufficient conditions are obtained when m=σ. Finally, we give some examples to demonstrate the validity of results.

    We suppose that all parameters are continuous and bounded functions in this section. Set

    R2+={(x,y)x0,y0},gl=inftRg(t),gu=suptRg(t).

    Clearly, (1.3) can be calculated by

    { x(t)=x(t0) exp{tt0(a(ξ)b(ξ)xα(ξ)c(ξ)x2α(ξ)h(ξ)x(ξ)ym(ξ)r(ξ)x(ξ)+yσ(ξ))dξ}, y(t)=y(t0) exp{tt0(d(ξ)+f(ξ)x(ξ)r(ξ)x(ξ)+yσ(ξ))dξ}. (2.1)

    Lemma 2.1 R2+ is positively invariant for system (1.3).

    From the view of the biological significance, we consider the initial condition satisfies x(t0)>0,y(t0)>0 in the following discussion.

    Theorem 2.2 If flrudu>0 and alhu(Mε2)mσ>0, then Γε is positively invariant for system (1.3), where

    Γε={(x,y)R2mε1xMε1,mε2yMε2},Mε1:=αaubl+ε,Mε2:=σ(fudlrl)Mε1dl+ε,mε1:=αalhu(Mε2)mσbu+cuMε1,mε2:=σ(flrudu)mε1du, (2.2)

    and ε0 is small enough to satisfy alhu(Mε2)mσ>0.

    Proof. According to (1.3a), we have

    dxdtx(t)(aublxα(t))x(t)(aubl+εxα).

    Using the comparison theorem, if 0<x(t0)Mε1, then x(t)Mε1 for any tt0.

    Similarly, From (1.3b), we can write

    dydt   y(t)(dl+fuMε1rlMε1+yσ(t)),                =dly(t)rlMε1+yσ(t)((fudlrl)Mε1dlyσ(t)).

    Thus, we obtain y(t)Mε2 for any tt0 when 0<y(t0)Mε2.

    Meanwhile, (1.3a) yields

    dxdt  x(t)(albuxα(t)cuMε1xα(t)hu(Mε2)mσ),=(bu+cuMε1)x(t)(alhu(Mε2)mσbu+cuMε1xα(t)).

    Hence, if x(t0)mε1, then x(t)mε1 for any tt0.

    Similarly, (1.3b) reduces to

    dydt  y(t)(du+flmε1rumε1+yσ(t)),                   =y(t)rumε1+yσ(t)((flrudu)mε1duyσ(t)).

    Thus, we obtain y(t)mε2 for any tt0 when y(t0)mε2. The proof is completed.

    Theorem 2.3 If flrudu>0 and alhu(M02)mσ>0 hold, system (1.3) is permanent.

    Proof. From (1.3a), we have

    dxdtx(t)(aublxα).

    By using the comparison theorem, it follows that

    limt+sup x(t)aubl:=M01.

    Meanwhile, for any ε>0, there exists T1>0 such that x(t)<M01+ε for all t>T0. Then, from (1.3b), we obtain

    dydty(t)(dl+fu(M01+ε)rl(M01+ε)+yσ(t)),                =dly(t)rl(M01+ε)+yσ(t)((fudlrl)(M01+ε)dlyσ(t)),

    for t>T1. Using the comparison theorem again, we show that

    limt+sup y(t)σ(fudlrl)(M01+ε)dl.

    Since the arbitrariness of ε, we have

    limt+sup y(t)σ(fudlrl)M01dl:=M02.

    Using a similar argument, it is easy to obtain that

    limt+inf x(t)m01,limt+inf y(t)m02.

    By the definition of persistence in [32], the conclusion is correct. The proof is completed.

    From the proof of Theorem 2.3, we easily know two facts that system (1.3) is ultimately bounded and the ultimate bound is Γε, which is asserted in the following theorem.

    Theorem 2.4 If flrudu>0 and alhu(M02)mσ>0, then system (1.3) is ultimately bounded, Γε in (2.1) is an ultimately bounded region.

    Remark 2.5 If m=σ=1, the above conclusions are refer to [32].

    Theorem 2.6 If furldl<0, then system (1.3) is not permanent.

    Proof. According to (1.3a), it is not difficult to have

    dydty(t)(dl+furl).

    Obviously, we have limt+y(t)=0.

    Theorem 2.7 If m=σ and hlru+1>au+σdu, then system (1.3) is not permanent.

    Proof. If m=σ and hlru+1>au+σdu, then we can obtain limt+x(t)=0 under certain initial conditions by the following argument.

    For hlru+1>au+σdu, there exists α>1, we have elruα+1=au+σdu. We can get limt+x(t)=0 when the initial value satisfies x(t0)yσ(t0)<α. Otherwise, there exists a first time t1, for t[t0,t1), we have x(t1)yσ(t1)=α and x(t)yσ(t)<α.

    For any t[t0,t1], we have

    dxdtx(t)(auhlrux(t)yσ(t)+1)x(t)(auhlruα+1)=σdu

    which yields

    x(t)x(t0)eσdu(tt0).

    However, for tt0, it leads to

    dydtduy(t),

    then

    yσ(t)yσ(t0)eσdu(tt0).

    Thus, for t[t0,t1], it produces

    x(t)yσ(t)x(t0)yσ(t0)<α.

    Obviously, it contradicts the existence of t1. Hence for tt0, it can be

    x(t)x(t0)eσdu(tt0),

    namely,

    limt+x(t)=0.

    The proof is completed.

    In fact, the growth of predator is entirely dependent on the amount of available prey. That is to say, when the prey goes extinct, so does the predator. Thus, both the prey and predator go extinct eventually when m=σ and hlru+1>au+σdu.

    Theorem 2.8 If (ˆx(t),ˆy(t))Lε is a solution and parameters satisfy the following conditions:

    (i) flrudu>0,alhu(M02)mσ>0,(ii) E1inftR{α(mε1)α1[b(t)+c(t)((mε1)α+ˆxα(t))]h(t)r(t)ˆym(t)+f(t)ˆyσ(t)(r(t)Mε1+(mε2)σ)(r(t)ˆx(t)+ˆyσ(t))}>0,    (iii) E2inftR{σ(mε2)σ1[f(t)ˆx(t)h(t)ˆym(t)]m(Mε2)m1h(t)(r(t)Mε1+(Mε2)σ(t))(r(t)Mε1+(mε2)σ)(r(t)ˆx(t)+ˆyσ(t))}>0.

    Then system (1.3) is globally asymptotically stable.

    Proof. Let (x(t),y(t)) be any solution, there exists T1>t0, we have (x(t),y(t))Lε for any t>T1.

    Let us define the Liapunov function

    V(t)=|lnx(t)lnˆx(t)|+|lny(t)lnˆy(t)|.

    The D+V(t) along the solution for t>T1 is calculated as follows:

    D+V(t)=sgn{x(t)ˆx(t)}[b(t)(xα(t)ˆxα(t))c(t)(x2α(t)ˆx2α(t))(h(t)ym(t)r(t)x(t)+yσ(t)h(t)ˆym(t)r(t)ˆx(t)+ˆyσ(t))]   +sgn{y(t)ˆy(t)}(f(t)x(t)r(t)x(t)+yσ(t)f(t)ˆx(t)r(t)ˆx(t)+ˆyσ(t))],=[b(t)c(t)(xα(t)+ˆxα(t))]|xα(t)ˆxα(t)|sgn{x(t)ˆx(t)}   h(t)[(r(t)ˆx(t)+yσ(t))(ym(t)ˆym(t))r(t)ˆym(t)(x(t)ˆx(t))ˆym(t)(yσ(t)ˆyσ(t))(r(t)x(t)+yσ(t))(r(t)ˆx(t)+ˆyσ(t))]   +sgn{y(t)ˆy(t)}f(t)yσ(t)(x(t)ˆx(t))ˆx(t)(yσ(t)ˆyσ(t))(r(t)x(t)+yσ(t))(r(t)ˆx(t)+ˆyσ(t)),[b(t)c(t)(xα(t)+ˆxα(t))]|xα(t)ˆxα(t)|+(h(t)r(t)ˆym(t)+f(t)ˆyσ(t))(r(t)x(t)+yσ(t))(r(t)ˆx(t)+ˆyσ(t))|x(t)ˆx(t)|   +h(t)r(t)ˆx(t)+ˆyσ(t)|ym(t)ˆym(t)|+h(t)ˆym(t)f(t)ˆx(t)(r(t)x(t)+yσ(t))(r(t)ˆx(t)+ˆyσ(t))|yσ(t)ˆyσ(t)|,={αξα1(t)[b(t)+c(t)(xα(t)+ˆx(t)α(t))]h(t)r(t)ˆym(t)+f(t)ˆyσ(t)(r(t)x(t)+yσ(t))(r(t)ˆx(t)+ˆyσ(t))}|x(t)ˆx(t)|   σησ12(t)[f(t)ˆx(t)h(t)ˆym(t)]mηm11(t)h(t)(r(t)x(t)+yσ(t))(r(t)x(t)+yσ(t))(r(t)ˆx(t)+ˆyσ(t))|y(t)ˆy(t)|,{α(mε1)α1[b(t)+c(t)((mε1)α+ˆxα(t))]h(t)r(t)ˆym(t)+f(t)ˆyσ(t)(r(t)(mε1)+(mε2)σ)(r(t)ˆx(t)+ˆyσ(t))}|x(t)ˆx(t)|   σ(mε2)σ1[f(t)ˆx(t)h(t)ˆym(t)]m(Mε2)m1h(t)(r(t)Mε1+(Mε2)σ)(r(t)mε1+(mε2)σ)(r(t)ˆx(t)+ˆyσ(t))|y(t)ˆy(t)|,

    where ξ(t) lies between x(t) and ˆx(t), η1(t) and η2(t) lie between y(t) and ˆy(t) respectively.

    Let G(t)|x(t)ˆx(t))|+|y(t)ˆy(t)| and λ=min{E1,E2}, then for t>T1, it follows that

    D+V(t)λG(t). (2.3)

    We integrate both sides with (2.3) form T1 to t, then

    V(t)V(T1)λtT1G(u)du,

    namely,

    tT1G(u)du1λV(T1).

    Obviously, we have

    G(t)L1([T1,+]).

    For t>T1, we know that x(t), y(t), ˆx(t) and ˆy(t) are all bounded, it implies that their derivatives are bounded. Hence, G(t) is uniformly continuous. We have

    limt+G(t)=0.

    The proof is completed.

    In this section, we suppose that all parameters are periodic functions with period ω and denote that ˉp=1ωω0p(t)dt.

    Lemma 3.1 (see [35]) Let L be a Fredholm operator of index zero and N be L compact on Ω. If

    (i) For each λ(0,1), any xΩ is such that LxλNx.

    (ii) QNx0 for each xΩKerL and the Brouwer degree:

    deg{JQN,ΩKerL,0}0.

    Then Lx=Nx has at least one solution on DomLΩ.

    Theorem 3.2 If m>σ, then the sufficient and necessary condition of system (1.3) which has at least one positive solution with period ω is ¯(fr)ˉd>0.

    Proof. We prove the necessity first. Integrating (1.3b) over one period ω, we obtain

    ˉd=1ωt+ωtf(s)˜x(s)r(s)˜x(s)+˜yσ(s)ds<1ωt+ωtf(s)r(s)ds=¯(fr).

    Here, we assume that (˜x(t),˜y(t))T is a positive solution with period ω.

    Next, we proceed to prove the sufficiency via Lemma 3.1. The following notations can refer to (see [32,35]). Let

    ˆx(t)=lnx(t),ˆy(t)=lny(t).

    System (1.3) is rewritten as

    {ˆx(t)=a(t)b(t)exp{αˆx(t)}c(t)exp{2αˆx(t)}h(t)exp{mˆy(t)}r(t)exp{ˆx(t)}+exp{σˆy(t)},ˆy(t)=d(t)+f(t)exp{ˆx(t)}r(t)exp{ˆx(t)}+exp{σˆy(t)}. (3.1)

    Let

    X=Z={v(t)=(ˆx(t),ˆy(t))TC(R,R2)v(t)=v(t+ω)},
    v(t)∥=maxt[0,ω]|ˆx(t)|+maxt[0,ω]|ˆy(t)|.

    It can be seen that X and Z are Banach spaces.

    (Nv)(t)=B(t)=[N1(t)N2(t)]=[a(t)b(t)exp{αˆx(t)}c(t)exp{2αˆx(t)}h(t)exp{mˆy(t)}r(t)exp{ˆx(t)}+exp{σˆy(t)}d(t)+f(t)exp{ˆx(t)}r(t)exp{ˆx(t)}+exp{σˆy(t)}                       ],
    Lv=v(t),  Pv=Qv=1ωω0v(t)dt,  vX.

    Clearly, KerL={vXvR2}, ImL={vZω0v(t)dt=0} is closed in Z. Meanwhile, dimKerL=CodimImL=2. So L is a Freedom mapping of index zero.

    On the other hand, P, Q are continuous projectors and satisfy P2=P, Q2=Q, ImP=KerL, ImL=KerQ=Im(IQ). Hence, there is a mapping Kp:ImLDomLKerP and given by

    Kpu=t0v(ξ)dξ1ωω0t0v(ξ)dξdt.

    Thus

    QNv=1ωω0B(t)dt,
    Kp(IQ)Nv=t0B(ξ)dξ1ωω0t0B(ξ)dξdt(tω12)ω0B(t)dt.

    Obviously, QN and Kp(IQ)N are continuous mapping. Based on Arzela-Ascoli theorem, we have Kp(IQ)N(ˉΩ) is compact and QN(ˉΩ) is bounded. Then N is L compact on ˉΩ.

    Next, we look for a set Ω which satisfies the coincidence degree theorem.

    According to the above definition, the equation Lx=λNx can be written as

    {ˆx(t)=λ(a(t)b(t)exp{αˆx(t)}c(t)exp{2αˆx(t)}h(t)exp{mˆy(t)}r(t)exp{ˆx(t)}+exp{σˆy(t)}),ˆy(t)=λ(d(t)+f(t)exp{ˆx(t)}r(t)exp{ˆx(t)}+exp{σˆy(t)}). (3.2)

    For a certain λ, let (ˆx(t),ˆy(t))TX be a solution of (3.2). By integrating over [0,ω], we have

    {ˉaω=ω0(b(t)exp{αˆx(t)}+c(t)exp{2αˆx(t)}+h(t)exp{mˆy(t)}r(t)exp{ˆx(t)}+exp{σˆy(t)})dt,(3.3a)ˉdω=ω0(f(t)exp{ˆx(t)}r(t)exp{ˆx(t)}+exp{σˆy(t)})dt.(3.3b)

    By (3.2) and (3.3), we have

    ω0|ˆx(t)|dt2ˉaω,
    ω0|ˆy(t)|dt2ˉdω.

    Let

    ˆx(ξ1)=mint[0,ω]ˆx(t),ˆx(η1)=maxt[0,ω]ˆx(t),ˆy(ξ2)=mint[0,ω]ˆy(t),ˆy(η2)=mint[0,ω]ˆy(t). (3.4)

    From (3.3a), we have

    ˉa1ωω0(b(t)exp{αˆx(ξ1)}+c(t)exp{2αˆx(ξ1)})=ˉbexp{αˆx(ξ1)}+ˉcexp{2αˆx(ξ1)}

    which yields

    ˆx(ξ1)1αlnˉb2+4ˉaˉcˉb2ˉc,

    then

    ˆx(t)ˆx(ξ1)+ω0|ˆx(t)|1αlnˉb2+4ˉaˉcˉb2ˉc+2ˉaω:=H1. (3.5)

    We transform (3.3a) again and obtain

    ˉa1ωω0(b(t)exp{αˆx(η1)}+c(t)exp{2αˆx(η1)}+h(t)exp{(mσ)ˆy(η2)})dt=ˉbexp{αˆx(η1)}+ˉcexp{2αˆx(η1)}+ˉhexp{(mσ)ˆy(η2)}. (3.6)

    If ˆx(η1)ˆy(η2), the inequality (3.6) reduces to

    ˉbexp{αˆx(η1)}+ˉcexp{2αˆx(η1)}+ˉhexp{(mσ)ˆx(η1)}ˉa.

    Using the function

    g(u)=ˉbuα+ˉcu2α+ˉhumσˉa,

    then g(0)=ˉa, limu+g(u)=0, and g(u) is strictly monotone increasing function over the interval (0,+). Therefore, there exists δ1>0 such that ˆx(η1)lnδ1.

    If x(η1)<y(η2), by the inequality (3.6) again, we have

    ˉbexp{αˆx(η2)}+ˉcexp{2αˆx(η2)}+ˉhexp{(mσ)ˆx(η2)}ˉa.

    Similarly, it exists δ2>0 such that ˆy(η2)lnδ2. Then

    ˆy(t)ˆy(η2)ω0|ˆy(t)|dt=lnδ22ˉdω.

    On the other hand, (3.3b) reduces to

    ˉd1ωω0f(t)exp{ˆx(η1)}exp{σˆy(ξ2)}dte2σˉdωˉfexp{ˆx(η1)}δ2σ,

    thus

    ˆx(η1)lnˉdδ2σe2σˉdωˉf.

    Taking δ=max{δ1,ˉdσδ2e2ˉdωˉf}, then ˆx(η1)lnδ. Therefore

    ˆx(t)ˆx(η1)ω0|ˆx(t)|dtlnδ2ˉaω:=H2. (3.7)

    (3.3b) can also produce

    ˉd1ωω0f(t)exp{σˆx(η1)}exp{σˆy(ξ2)}dtˉfeσH1exp{σˆy(ξ2)},

    then

    ˆy(ξ2)1σlnˉfeσH1ˉd.

    Therefore

    ˆy(t)ˆy(ξ2)+ω0|ˆy(t)|dt1σlnˉfeσH1ˉd+2ˉdω:=H3. (3.8)

    Similarly, we also have

    ˉd1ωω0(f(t)eH2r(t)eH2+exp{σˆy(η2)})dt¯(fr)eH2eH2+1rlexp{σˆy(η2)},

    then

    ˆy(η2)1σln(¯(fr)ˉd)rleH2ˉd.

    Hence

    ˆy(t)ˆy(η2)ω0|ˆy(t)|dt1σln(¯(fr)ˉd)rleH2ˉd2ˉdω:=H4. (3.9)

    Obviously, H1,H2,H3 and H4 are independent of λ. Let

    H=max{|H1|,|H2|}+max{|H3|,|H4|},
    Ω={(ˆx(t),ˆy(t))TX|(ˆx(t),ˆy(t))<H}.

    QNv(0,0)T for any vΩKerL. Otherwise, there exists a constant vector v=(v1,v2)ΩR2 such that QNv=(0,0)T, that is

    {ˉaˉbeαv1ˉce2αv11ωω0h(t)emv2r(t)ev1+eσv2dt=0,ˉd+1ωω0f(t)ev1r(t)ev1+eσv2dt=0. (3.10)

    This contradicts the previous result which H1v1H2 and H3v2H4. We define the mapping as follows

    φ(v1,v2,θ)=θ[ˉaˉbeαv1ˉce2αv11ωω0h(t)emv2r(t)ev1+eσv2dtˉd+1ωω0f(t)ev1r(t)ev1+eσv2dt]+(1θ)[ˉaˉbeαv1ˉce2αv1ˉd+1ωω0f(t)ev1r(t)ev1+eσv2dt].

    for any θ[0,1]. Obviously, if v=(v1,v2)KerLΩ, then φ(v1,v2,θ)0. We claim that φ is a homotopic mapping. Taking J=I, then

    deg{JQN,ΩKerL,(0,0)T}=deg{φ(v1,v2,0),ΩKerL,(0,0)T}.

    However, φ(v1,v2,0)=0 implies

    {ˉaˉbeαv1ˉce2αv1=0,ˉd+1ωω0f(t)ev1r(t)ev1+eσv2dt=0. (3.11)

    It is easy to know that (3.11) has a single solution (v1,v2)T.

    deg{JQN,ΩKerL,(0,0)T}=sign[ˉbeαv1+ˉce2αv1ωω0σf(t)eαv1e(σ1)v2(r(t)eαv1+eσv2)2dt]=1.

    We get a set Ω which satisfies the conditions in coincidence degree theorem. Therefore, (3.1) has at least a solution (ˆx(t),ˆy(t))T with period ω, corresponding to (1.3) has a solution (exp{ˆx(t)},exp{ˆy(t)})T. The proof is completed.

    Theorem 3.3 If the conditions (i)m=σ, (ii) ¯(fr)ˉd>0, (iii) ˉa>ˉh hold, then system (1.3) has at least one positive periodic solution.

    Proof. If m=σ, then (3.6) reduces to

    ˉaˉbexp{αˆx(η1)}+ˉcexp{2αˆx(η1)}+ˉh.

    Taking

    g(u)=ˉbuα+ˉcu2α+ˉhˉa.

    Then g(0)=ˉhˉa<0, limu+g(u)=+. Based on the analysis in Theorem 3.2, there exists δ>0 such that ˆx(η1)lnδ. Therefore,

    ˆx(t)ˆx(η1)ω0|ˆx(t)|dtlnδ2ˉaω.

    The rest of the proof is completely the same as Theorem 3.2. The proof is completed.

    Combine Theorem 2.8 and Theorem 3.2 (or Theorem 3.3), the following theorem is obvious.

    Theorem 3.4 If the conditions in Theorem 2.8 and Theorem 3.2 (or Theorem 3.3) hold simultaneously, then the periodic solution of (1.3) is unique and globally asymptotically stable.

    Remark 3.5 Theorem 3.3 admits that if m>σ, then the existence of positive periodic solution only depends on the relationship of the average intrinsic growth rate of prey, the average unit time number to search for prey and the average increasing coefficient of predator.

    Remark 3.6 There is an interesting phenomenon: We derive the priori bounds for the solution of Lx=λNx in the same way, however, we obtain sufficient and necessary conditions when m>σ and only get sufficient conditions when m=σ. Especially, if m=σ = 1, the result corresponds to results in [32].

    The following theorem shows the properties of a boundary solution.

    Theorem 3.7 System (1.3) has at least a boundary period solution, namely (x(t),0). Moreover, if d(t)f(t)r(t)h(t)>0, then (x(t),0) is globally asymptotically stable.

    Proof. For the equation dxdt=x(a(t)b(t)xαc(t)x2α), it is easy to obtain the first part of conclusion by using the proof method of Theorem 3.2. Since d(t)f(t)r(t)h(t)>0 implies d(t)f(t)r(t)>0, it follows that limt+y(t)=0 from system (1.3). Therefore, we just prove limt+x(t)=x(t). Define the following Liapunov function

    V(t)=|lnx(t)lnx(t)|+y(t).

    The D+V(t) along the solution can be written as

    D+V(t)=sgn{x(t)x(t)}[b(t)(xα(t)xα(t))c(t)(x2α(t)x2α(t)) h(t)ym(t)r(t)x(t)+yσ(t)]+y(t)[d(t)+f(t)x(t)r(t)x(t)+yσ(t)],[b(t)+c(t)(xα(t)+xα(t))]|xα(t)xα(t)|(d(t)f(t)r(t)h(t))ymσ(t),bl|xα(t)xα(t)|.

    The remaining proof details are similar as Theorem 2.7. This completes the proof.

    Several cases demonstrate the correctness of the previous conclusions in this section. We let ρ=¯(fr)ˉd, x(0)=1.5, y(0)=1.4. Choosing the parameters in (1.3) as follows:

    a(t)=2.2+0.2sin2t,  b(t)=0.25+0.05cos2t,  c(t)=0.8+0.1sin2t,                d(t)=0.42+0.12sin2t,  r(t)=0.7+0.1sin2t,   α=0.5.               (4.1)

    Example 4.1 Let σ=0.5,m=0.75, h(t)=1.1+0.1sin2t, others parameters are the same in (4.1). The solutions and phase portraits of (1.3) are shown in Figure 1.

    Figure 1.  Solution curves and phase portrait of (1.3).

    Example 4.2 Let σ=m=0.5, f(t)=0.6+0.1sin2t,ρ=0.4375, others parameters are the same in (4.1). The solutions and phase portraits of (1.3) are shown in Figure 2.

    Figure 2.  Solution curves and phase portrait of (1.3).

    The simulation results in Figure 1 shows that the following conclusions: If ρ>0 and ρ gets more and more small, the predator curve oscillates at a lower density and the prey curve oscillates at a higher density. If ρ<0, then limt+y(t)=0, namely, the positive periodic solution disappears. It also shows that ρ=0 is the threshold, which confirms Theorem 3.2.

    From (c, d) in Figure 2, we know ˉa>ˉh is not necessary for positive periodic solution of system (1.3) but only sufficient when σ=m.

    In this paper, we discussed a non-autonomous Hassell-Varley-Holling type predator-prey system with mutual interference. Compared with that in [25], we believe our model reflects the influence of predator groups on predation behavior.

    Firstly, we focused on permanence, extinction and globally asymptotic stability of the model by using the principle of comparison and a suitable Liapunov function and differential mean value theorem. The investigation showed that the shorter the time to search for prey is, the more favorable permanence is under the conditions of the Theorem 2.3. However, the conditions for globally asymptotic stability in Theorem 2.8 is too complex to be applied directly.

    Secondly, we studied some conditions for the existence of a positive periodic solution by using the coincidence degree theorem and illustrate with some examples. When m>σ, we obtain a sufficient and necessary condition in Theorem 3.2, that is a perfect result. Figure 1 confirms this result. When m=σ, we only obtain some sufficient conditions in Theorem 3.3, but Figure 2 shows that the condition ˉa>ˉh is not necessary. In addition, we give some sufficient conditions for the globally asymptotic stability of a boundary periodic solution.

    This paper leaves a seemingly difficult problem that we can not solve: what are the conditions for the existence of a positive periodic solution when m<σ?

    The authors thank the editor and referees for their valuable suggestions and comments, which improved the presentation of this manuscript. The research is supported by the Startup Foundation for Introducing Talent of Wuyi University (No. YJ201802).

    For the publication of this article, no conflict of interest among the authors is disclosed.



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