
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(a−bxα−cx2α)−hxβym1+rxβ,dydt=y(−d+fxβ1+rxβ), | (1.1) |
where a−bxα−cx2α 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(a−bxα−cx2α)−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)xα−c(t)x2α]−h(t)xymr(t)x+yσ,(1.3a)dydt=y[−d(t)+f(t)xr(t)x+yσ],(1.3b) |
where 0≤α≤1, 0<σ≤1, 0<m≤1. 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)∣x≥0,y≥0},gl=inft∈Rg(t),gu=supt∈Rg(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 fl−rudu>0 and al−hu(Mε2)m−σ>0, then Γε is positively invariant for system (1.3), where
Γε={(x,y)∈R2∣mε1≤x≤Mε1,mε2≤y≤Mε2},Mε1:=α√aubl+ε,Mε2:=σ√(fu−dlrl)Mε1dl+ε,mε1:=α√al−hu(Mε2)m−σbu+cuMε1,mε2:=σ√(fl−rudu)mε1du, | (2.2) |
and ε≥0 is small enough to satisfy al−hu(Mε2)m−σ>0.
Proof. According to (1.3a), we have
dxdt≤x(t)(au−blxα(t))≤x(t)(aubl+ε−xα). |
Using the comparison theorem, if 0<x(t0)≤Mε1, then x(t)≤Mε1 for any t≥t0.
Similarly, From (1.3b), we can write
dydt ≤y(t)(−dl+fuMε1rlMε1+yσ(t)), =dly(t)rlMε1+yσ(t)((fu−dlrl)Mε1dl−yσ(t)). |
Thus, we obtain y(t)≤Mε2 for any t≥t0 when 0<y(t0)≤Mε2.
Meanwhile, (1.3a) yields
dxdt ≥x(t)(al−buxα(t)−cuMε1xα(t)−hu(Mε2)m−σ),=(bu+cuMε1)x(t)(al−hu(Mε2)m−σbu+cuMε1−xα(t)). |
Hence, if x(t0)≤mε1, then x(t)≥mε1 for any t≥t0.
Similarly, (1.3b) reduces to
dydt ≥y(t)(−du+flmε1rumε1+yσ(t)), =y(t)rumε1+yσ(t)((fl−rudu)mε1−duyσ(t)). |
Thus, we obtain y(t)≤mε2 for any t≥t0 when y(t0)≥mε2. The proof is completed.
Theorem 2.3 If fl−rudu>0 and al−hu(M02)m−σ>0 hold, system (1.3) is permanent.
Proof. From (1.3a), we have
dxdt≤x(t)(aubl−xα). |
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
dydt≤y(t)(−dl+fu(M01+ε)rl(M01+ε)+yσ(t)), =dly(t)rl(M01+ε)+yσ(t)((fu−dlrl)(M01+ε)dl−yσ(t)), |
for t>T1. Using the comparison theorem again, we show that
limt→+∞sup y(t)≤σ√(fu−dlrl)(M01+ε)dl. |
Since the arbitrariness of ε, we have
limt→+∞sup y(t)≤σ√(fu−dlrl)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 fl−rudu>0 and al−hu(M02)m−σ>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 fu−rldl<0, then system (1.3) is not permanent.
Proof. According to (1.3a), it is not difficult to have
dydt≤y(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
dxdt≤x(t)(au−hlrux(t)yσ(t)+1)≤x(t)(au−hlruα+1)=−σdu |
which yields
x(t)≤x(t0)e−σdu(t−t0). |
However, for t≥t0, it leads to
dydt≥−duy(t), |
then
yσ(t)≥yσ(t0)e−σdu(t−t0). |
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 t≥t0, it can be
x(t)≤x(t0)e−σdu(t−t0), |
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) fl−rudu>0,al−hu(M02)m−σ>0,(ii) E1≡inft∈R{α(mε1)α−1[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) E2≡inft∈R{σ(mε2)σ−1[f(t)ˆx(t)−h(t)ˆym(t)]−m(Mε2)m−1h(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)|,=−{αξα−1(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)| −σησ−12(t)[f(t)ˆx(t)−h(t)ˆym(t)]−mηm−11(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)α−1[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)σ−1[f(t)ˆx(t)−h(t)ˆym(t)]−m(Mε2)m−1h(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)≤−λ∫tT1G(u)du, |
namely,
∫tT1G(u)du≤1λV(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=1ω∫ω0p(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 x∈∂Ω is such that Lx≠λNx.
(ii) QNx≠0 for each x∈∂Ω∩KerL and the Brouwer degree:
deg{JQN,Ω∩KerL,0}≠0. |
Then Lx=Nx has at least one solution on DomL∩−Ω.
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=1ω∫t+ωtf(s)˜x(s)r(s)˜x(s)+˜yσ(s)ds<1ω∫t+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))T∈C(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=1ω∫ω0v(t)dt, v∈X. |
Clearly, KerL={v∈X∣v∈R2}, ImL={v∈Z∣∫ω0v(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(I−Q). Hence, there is a mapping Kp:ImL→DomL∩KerP and given by
Kpu=∫t0v(ξ)dξ−1ω∫ω0∫t0v(ξ)dξdt. |
Thus
QNv=1ω∫ω0B(t)dt, |
Kp(I−Q)Nv=∫t0B(ξ)dξ−1ω∫ω0∫t0B(ξ)dξdt−(tω−12)∫ω0B(t)dt. |
Obviously, QN and Kp(I−Q)N are continuous mapping. Based on Arzela-Ascoli theorem, we have Kp(I−Q)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))T∈X 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)|dt≤2ˉaω, |
∫ω0|ˆy′(t)|dt≤2ˉdω. |
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
ˉa≥1ω∫ω0(b(t)exp{αˆx(ξ1)}+c(t)exp{2αˆx(ξ1)})=ˉbexp{αˆx(ξ1)}+ˉcexp{2αˆx(ξ1)} |
which yields
ˆx(ξ1)≤1αln√ˉb2+4ˉaˉc−ˉb2ˉc, |
then
ˆx(t)≤ˆx(ξ1)+∫ω0|ˆx′(t)|≤1αln√ˉb2+4ˉaˉc−ˉb2ˉc+2ˉaω:=H1. | (3.5) |
We transform (3.3a) again and obtain
ˉa≤1ω∫ω0(b(t)exp{αˆx(η1)}+c(t)exp{2αˆx(η1)}+h(t)exp{(m−σ)ˆy(η2)})dt=ˉbexp{αˆx(η1)}+ˉcexp{2αˆx(η1)}+ˉhexp{(m−σ)ˆy(η2)}. | (3.6) |
If ˆx(η1)≥ˆy(η2), the inequality (3.6) reduces to
ˉbexp{αˆx(η1)}+ˉcexp{2αˆx(η1)}+ˉhexp{(m−σ)ˆx(η1)}≥ˉa. |
Using the function
g(u)=ˉbuα+ˉcu2α+ˉhum−σ−ˉa, |
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{(m−σ)ˆx(η2)}≥ˉa. |
Similarly, it exists δ2>0 such that ˆy(η2)≥lnδ2. Then
ˆy(t)≥ˆy(η2)−∫ω0|ˆy′(t)|dt=lnδ2−2ˉdω. |
On the other hand, (3.3b) reduces to
ˉd≤1ω∫ω0f(t)exp{ˆx(η1)}exp{σˆy(ξ2)}dt≤e2σˉdωˉfexp{ˆx(η1)}δ2σ, |
thus
ˆx(η1)≥lnˉdδ2σe2σˉdωˉf. |
Taking δ=max{δ1,ˉdσδ2e2ˉdωˉf}, then ˆx(η1)≥lnδ. Therefore
ˆx(t)≥ˆx(η1)−∫ω0|ˆx′(t)|dt≥lnδ−2ˉaω:=H2. | (3.7) |
(3.3b) can also produce
ˉd≤1ω∫ω0f(t)exp{σˆx(η1)}exp{σˆy(ξ2)}dt≤ˉfeσH1exp{σˆy(ξ2)}, |
then
ˆy(ξ2)≤1σlnˉfeσH1ˉd. |
Therefore
ˆy(t)≤ˆy(ξ2)+∫ω0|ˆy′(t)|dt≤1σlnˉfeσH1ˉd+2ˉdω:=H3. | (3.8) |
Similarly, we also have
ˉd≥1ω∫ω0(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)|dt≤1σln(¯(fr)−ˉd)rleH2ˉd−2ˉdω:=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))T∈X|‖(ˆx(t),ˆy(t))‖<H}. |
QNv≠(0,0)T for any v∈∂Ω∩KerL. Otherwise, there exists a constant vector v=(v1,v2)∈∂Ω∩R2 such that QNv=(0,0)T, that is
{ˉa−ˉbeαv1−ˉce2αv1−1ω∫ω0h(t)emv2r(t)ev1+eσv2dt=0,−ˉd+1ω∫ω0f(t)ev1r(t)ev1+eσv2dt=0. | (3.10) |
This contradicts the previous result which H1≤v1≤H2 and H3≤v2≤H4. We define the mapping as follows
φ(v1,v2,θ)=θ[ˉa−ˉbeαv1−ˉce2αv1−1ω∫ω0h(t)emv2r(t)ev1+eσv2dt−ˉd+1ω∫ω0f(t)ev1r(t)ev1+eσv2dt]+(1−θ)[ˉa−ˉbeαv1−ˉce2αv1−ˉd+1ω∫ω0f(t)ev1r(t)ev1+eσv2dt]. |
for any θ∈[0,1]. Obviously, if v=(v1,v2)∈KerL∩∂Ω, then φ(v1,v2,θ)≠0. We claim that φ is a homotopic mapping. Taking J=I, then
deg{JQN,Ω∩KerL,(0,0)T}=deg{φ(v1,v2,0),Ω∩KerL,(0,0)T}. |
However, φ(v1,v2,0)=0 implies
{ˉa−ˉbeαv1−ˉce2αv1=0,−ˉd+1ω∫ω0f(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,Ω∩KerL,(0,0)T}=sign[ˉbeαv1∗+ˉce2αv1∗ω∫ω0σf(t)eαv1∗e(σ−1)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
ˉa≤ˉbexp{αˆx(η1)}+ˉcexp{2αˆx(η1)}+ˉh. |
Taking
g(u)=ˉbuα+ˉcu2α+ˉh−ˉa. |
Then g(0)=ˉh−ˉa<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)|dt≥lnδ−2ˉaω. |
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)xα−c(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)−x∗α(t))−c(t)(x2α(t)−x∗2α(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)+x∗α(t))]|xα(t)−x∗α(t)|−(d(t)−f(t)r(t)−h(t))ym−σ(t),≤−bl|xα(t)−x∗α(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.
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.
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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