In the paper, a Leslie-Gower predator-prey system with harvesting and fear effect is considered. The existence and stability of all possible equilibrium points are analyzed. The bifurcation dynamic behavior at key equilibrium points is investigated to explore the intrinsic driving mechanisms of population interaction modes. It is shown that the system undergoes various bifurcations, including transcritical, saddle-node, Hopf and Bogdanov-Takens bifurcations. The numerical simulation results show that harvesting and fear effect can seriously affect the dynamic evolution trend and coexistence mode. Furthermore, it is particularly worth pointing out that harvesting not only drives changes in population coexistence mode, but also has a certain degree delay. Finally, it is anticipated that these research results will be beneficial for the vigorous development of predator-prey system.
Citation: Rongjie Yu, Hengguo Yu, Chuanjun Dai, Zengling Ma, Qi Wang, Min Zhao. Bifurcation analysis of Leslie-Gower predator-prey system with harvesting and fear effect[J]. Mathematical Biosciences and Engineering, 2023, 20(10): 18267-18300. doi: 10.3934/mbe.2023812
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In the paper, a Leslie-Gower predator-prey system with harvesting and fear effect is considered. The existence and stability of all possible equilibrium points are analyzed. The bifurcation dynamic behavior at key equilibrium points is investigated to explore the intrinsic driving mechanisms of population interaction modes. It is shown that the system undergoes various bifurcations, including transcritical, saddle-node, Hopf and Bogdanov-Takens bifurcations. The numerical simulation results show that harvesting and fear effect can seriously affect the dynamic evolution trend and coexistence mode. Furthermore, it is particularly worth pointing out that harvesting not only drives changes in population coexistence mode, but also has a certain degree delay. Finally, it is anticipated that these research results will be beneficial for the vigorous development of predator-prey system.
Recently, fractional calculus methods became of great interest, because it is a powerful tool for calculating the derivation of multiples systems. These methods study real world phenomena in many areas of natural sciences including biomedical, radiography, biology, chemistry, and physics [1,2,3,4,5,6,7]. Abundant publications focus on the Caputo fractional derivative (CFD) and the Caputo-Hadamard derivative. Additionally, other generalization of the previous derivatives, such as Ψ-Caputo, study the existence of solutions to some FDEs (see [8,9,10,11,12,13,14]).
In general, an m-point fractional boundary problem involves a fractional differential equation with fractional boundary conditions that are specified at m different points on the boundary of a domain. The fractional derivative is defined using the Riemann-Liouville fractional derivative or the Caputo fractional derivative. Solving these types of problems can be challenging due to the non-local nature of fractional derivatives. However, there are various numerical and analytical methods available for solving such problems, including the spectral method, the finite difference method, the finite element method, and the homotopy analysis method. The applications of m-point fractional boundary problems can be found in various fields, including physics, engineering, finance, and biology. These problems are useful in modeling and analyzing phenomena that exhibit non-local behavior or involve memory effects (see [15,16,17,18]).
Pantograph equations are a set of differential equations that describe the motion of a pantograph, which is a mechanism used for copying and scaling drawings or diagrams. The equations are based on the assumption that the pantograph arms are rigid and do not deform during operation, we can simply say that see [19]. One important application of the pantograph equations is in the field of drafting and technical drawing. Before the advent of computer-aided design (CAD) software, pantographs were commonly used to produce scaled copies of drawings and diagrams. By adjusting the lengths of the arms and the position of the stylus, a pantograph can produce copies that are larger or smaller than the original [20], electrodynamics [21] and electrical pantograph of locomotive [22].
Many authors studied a huge number of positive solutions for nonlinear fractional BVP using fixed point theorems (FPTs) such as SFPT, Leggett-Williams and Guo-Krasnosel'skii (see [23,24]). Some studies addressed the sign-changing of solution of BVPs [25,26,27,28,29].
In this work, we use Schauder's fixed point theorem (SFPT) to solve the semipostone multipoint Ψ-Caputo fractional pantograph problem
Dν;ψrϰ(ς)+F(ς,ϰ(ς),ϰ(r+λς))=0, ς in (r,ℑ) | (1.1) |
ϰ(r)=ϑ1, ϰ(ℑ)=m−2∑i=1ζiϰ(ηi)+ϑ2, ϑi∈R, i∈{1,2}, | (1.2) |
where λ∈(0,ℑ−rℑ),Dν;ψr is Ψ-Caputo fractional derivative (Ψ-CFD) of order ν, 1<ν≤2, ζi∈R+(1≤i≤m−2) such that 0<Σm−2i=1ζi<1, ηi∈(r,ℑ), and F:[r,ℑ]×R×R→R.
The most important aspect of this research is to prove the existence of a positive solution of the above m-point FBVP. Note that in [30], the author considered a two-point BVP using Liouville-Caputo derivative.
The article is organized as follows. In the next section, we provide some basic definitions and arguments pertinent to fractional calculus (FC). Section 3 is devoted to proving the the main result and an illustrative example is given in Section 4.
In the sequel, Ψ denotes an increasing map Ψ:[r1,r2]→R via Ψ′(ς)≠0, ∀ ς, and [α] indicates the integer part of the real number α.
Definition 2.1. [4,5] Suppose the continuous function ϰ:(0,∞)→R. We define (RLFD) the Riemann-Liouville fractional derivative of order α>0,n=[α]+1 by
RLDα0+ϰ(ς)=1Γ(n−α)(ddς)n∫ς0(ς−τ)n−α−1ϰ(τ)dτ, |
where n−1<α<n.
Definition 2.2. [4,5] The Ψ-Riemann-Liouville fractional integral (Ψ-RLFI) of order α>0 of a continuous function ϰ:[r,ℑ]→R is defined by
Iα;Ψrϰ(ς)=∫ςr(Ψ(ς)−Ψ(τ))α−1Γ(α)Ψ′(τ)ϰ(τ)dτ. |
Definition 2.3. [4,5] The CFD of order α>0 of a function ϰ:[0,+∞)→R is defined by
Dαϰ(ς)=1Γ(n−α)∫ς0(ς−τ)n−α−1ϰ(n)(τ)dτ, α∈(n−1,n),n∈N. |
Definition 2.4. [4,5] We define the Ψ-CFD of order α>0 of a continuous function ϰ:[r,ℑ]→R by
Dα;Ψrϰ(ς)=∫ςr(Ψ(ς)−Ψ(τ))n−α−1Γ(n−α)Ψ′(τ)∂nΨϰ(τ)dτ, ς>r, α∈(n−1,n), |
where ∂nΨ=(1Ψ′(ς)ddς)n,n∈N.
Lemma 2.1. [4,5] Suppose q,ℓ>0, and ϰinC([r,ℑ],R). Then ∀ς∈[r,ℑ] and by assuming Fr(ς)=Ψ(ς)−Ψ(r), we have
1) Iq;ΨrIℓ;Ψrϰ(ς)=Iq+ℓ;Ψrϰ(ς),
2) Dq;ΨrIq;Ψrϰ(ς)=ϰ(ς),
3) Iq;Ψr(Fr(ς))ℓ−1=Γ(ℓ)Γ(ℓ+q)(Fr(ς))ℓ+q−1,
4) Dq;Ψr(Fr(ς))ℓ−1=Γ(ℓ)Γ(ℓ−q)(Fr(ς))ℓ−q−1,
5) Dq;Ψr(Fr(ς))k=0, k=0,…,n−1, n∈N, qin(n−1,n].
Lemma 2.2. [4,5] Let n−1<α1≤n,α2>0, r>0, ϰ∈L(r,ℑ), Dα1;Ψrϰ∈L(r,ℑ). Then the differential equation
Dα1;Ψrϰ=0 |
has the unique solution
ϰ(ς)=W0+W1(Ψ(ς)−Ψ(r))+W2(Ψ(ς)−Ψ(r))2+⋯+Wn−1(Ψ(ς)−Ψ(r))n−1, |
and
Iα1;ΨrDα1;Ψrϰ(ς)=ϰ(ς)+W0+W1(Ψ(ς)−Ψ(r))+W2(Ψ(ς)−Ψ(r))2+⋯+Wn−1(Ψ(ς)−Ψ(r))n−1, |
with Wℓ∈R, ℓ∈{0,1,…,n−1}.
Furthermore,
Dα1;ΨrIα1;Ψrϰ(ς)=ϰ(ς), |
and
Iα1;ΨrIα2;Ψrϰ(ς)=Iα2;ΨrIα1;Ψrϰ(ς)=Iα1+α2;Ψrϰ(ς). |
Here we will deal with the FDE solution of (1.1) and (1.2), by considering the solution of
−Dν;ψrϰ(ς)=h(ς), | (2.1) |
bounded by the condition (1.2). We set
Δ:=Ψ(ℑ)−Ψ(r)−Σm−2i=1ζi(Ψ(ηi)−Ψ(r)). |
Lemma 2.3. Let ν∈(1,2] and ς∈[r,ℑ]. Then, the FBVP (2.1) and (1.2) have a solution ϰ of the form
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)h(τ)Ψ′(τ)dτ, |
where
ϖ(ς,τ)=1Γ(ν){[(Ψ(ℑ)−Ψ(r))ν−1−Σm−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1]Ψ(ς)−Ψ(r)Δ−(Ψ(ς)−Ψ(τ))ν−1,τ≤ς,ηi−1<τ≤ηi,[(Ψ(ℑ)−Ψ(τ))ν−1−Σm−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1]Ψ(ℑ)−Ψ(r)Δ,ς≤τ,ηi−1<τ≤ηi, | (2.2) |
i=1,2,...,m−2.
Proof. According to the Lemma 2.2 the solution of Dν;ψrϰ(ς)=−h(ς) is given by
ϰ(ς)=−1Γ(ν)∫ςr(Ψ(ς)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+c0+c1(Ψ(ς)−Ψ(r)), | (2.3) |
where c0,c1∈R. Since ϰ(r)=ϑ1 and ϰ(ℑ)=∑m−2i=1ζiϰ(ηi)+ϑ2, we get c0=ϑ1 and
c1=1Δ(−1Γ(ν)m−2∑i=1ζi∫ηjr(Ψ(ηi)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+1Γ(ν)∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+ϑ1[m−2∑i=1ζi−1]+ϑ2). |
By substituting c0,c1 into Eq (2.3) we find,
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2−1Γ(ν)(∫ςr(Ψ(ς)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+(Ψ(ς)−Ψ(r))Δm−2∑i=1ζi∫ηjr(Ψ(ηi)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ−Ψ(ς)−Ψ(r)Δ∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2+∫ℑrϖ(ς,τ)h(τ)Ψ′(τ)dτ, |
where ϖ(ς,τ) is given by (2.2). Hence the required result.
Lemma 2.4. If 0<∑m−2i=1ζi<1, then
i) Δ>0,
ii) (Ψ(ℑ)−Ψ(τ))ν−1−∑m−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1>0.
Proof. i) Since ηi<ℑ, we have
ζi(Ψ(ηi)−Ψ(r))<ζi(Ψ(ℑ)−Ψ(r)), |
−m−2∑i=1ζi(Ψ(ηi)−Ψ(r))>−m−2∑i=1ζi(Ψ(ℑ)−Ψ(r)), |
Ψ(ℑ)−Ψ(r)−m−2∑i=1ζi(Ψ(ηi)−Ψ(r))>Ψ(ℑ)−Ψ(r)−m−2∑i=1ζi(Ψ(ℑ)−Ψ(r))=(Ψ(ℑ)−Ψ(r))[1−m−2∑i=1ζi]. |
If 1−Σm−2i=1ζi>0, then (Ψ(ℑ)−Ψ(r))−Σm−2i=1ζi(Ψ(ηi)−Ψ(r))>0. So we have Δ>0.
ii) Since 0<ν−1≤1, we have (Ψ(ηi)−Ψ(τ))ν−1<(Ψ(ℑ)−Ψ(τ))ν−1. Then we obtain
m−2∑j=iζj(Ψ(ηj)−Ψ(τ))ν−1<m−2∑j=iζj(Ψ(ℑ)−Ψ(τ))ν−1≤(Ψ(ℑ)−Ψ(τ))ν−1m−2∑i=1ζi<(Ψ(ℑ)−Ψ(τ))ν−1, |
and so
(Ψ(ℑ)−Ψ(τ))ν−1−m−2∑j=iζj(Ψ(ηj)−Ψ(τ))ν−1>0. |
Remark 2.1. Note that ∫ℑrϖ(ς,τ)Ψ′(τ)dτ is bounded ∀ς∈[r,ℑ]. Indeed
∫ℑr|ϖ(ς,τ)|Ψ′(τ)dτ≤1Γ(ν)∫ςr(Ψ(ς)−Ψ(τ))ν−1Ψ′(τ)dτ+Ψ(ς)−Ψ(r)Γ(ν)Δm−2∑i=1ζi∫ηir(Ψ(ηj)−Ψ(τ))ν−1Ψ′(τ)dτ+Ψ(ς)−Ψ(r)ΔΓ(ν)∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1Ψ′(τ)dτ=(Ψ(ς)−Ψ(r))νΓ(ν+1)+Ψ(ς)−Ψ(r)ΔΓ(ν+1)m−2∑i=1ζi(Ψ(ηi)−Ψ(r))ν+Ψ(ς)−Ψ(r)ΔΓ(ν+1)(Ψ(ℑ)−Ψ(r))ν≤(Ψ(ℑ)−Ψ(r))νΓ(ν+1)+Ψ(ℑ)−Ψ(r)ΔΓ(ν+1)m−2∑i=1ζi(Ψ(ηi)−Ψ(r))ν+(Ψ(ℑ)−Ψ(r))ν+1ΔΓ(ν+1)=M. | (2.4) |
Remark 2.2. Suppose Υ(ς)∈L1[r,ℑ], and w(ς) verify
{Dν;ψrw(ς)+Υ(ς)=0,w(r)=0, w(ℑ)=Σm−2i=1ζiw(ηi), | (2.5) |
then w(ς)=∫ℑrϖ(ς,τ)Υ(τ)Ψ′(τ)dτ.
Next we recall the Schauder fixed point theorem.
Theorem 2.1. [23] [SFPT] Consider the Banach space Ω. Assume ℵ bounded, convex, closed subset in Ω. If ϝ:ℵ→ℵ is compact, then it has a fixed point in ℵ.
We start this section by listing two conditions which will be used in the sequel.
● (Σ1) There exists a nonnegative function Υ∈L1[r,ℑ] such that ∫ℑrΥ(ς)dς>0 and F(ς,ϰ,v)≥−Υ(ς) for all (ς,ϰ,v)∈[r,ℑ]×R×R.
● (Σ2) G(ς,ϰ,v)≠0, for (ς,ϰ,v)∈[r,ℑ]×R×R.
Let ℵ=C([r,ℑ],R) the Banach space of CFs (continuous functions) with the following norm
‖ϰ‖=sup{|ϰ(ς)|:ς∈[r,ℑ]}. |
First of all, it seems that the FDE below is valid
Dν;ψrϰ(ς)+G(ς,ϰ∗(ς),ϰ∗(r+λς))=0, ς∈[r,ℑ]. | (3.1) |
Here the existence of solution satisfying the condition (1.2), such that G:[r,ℑ]×R×R→R
G(ς,z1,z2)={F(ς,z1,z2)+Υ(ς), z1,z2≥0,F(ς,0,0)+Υ(ς), z1≤0 or z2≤0, | (3.2) |
and ϰ∗(ς)=max{(ϰ−w)(ς),0}, hence the problem (2.5) has w as unique solution. The mapping Q:ℵ→ℵ accompanied with the (3.1) and (1.2) defined as
(Qϰ)(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)G(ς,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ, | (3.3) |
where the relation (2.2) define ϖ(ς,τ). The existence of solution of the problems (3.1) and (1.2) give the existence of a fixed point for Q.
Theorem 3.1. Suppose the conditions (Σ1) and (Σ2) hold. If there exists ρ>0 such that
[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≤ρ, |
where L≥max{|G(ς,ϰ,v)|:ς∈[r,ℑ], |ϰ|,|v|≤ρ} and M is defined in (2.4), then, the problems (3.1) and (3.2) have a solution ϰ(ς).
Proof. Since P:={ϰ∈ℵ:‖ϰ‖≤ρ} is a convex, closed and bounded subset of B described in the Eq (3.3), the SFPT is applicable to P. Define Q:P→ℵ by (3.3). Clearly Q is continuous mapping. We claim that range of Q is subset of P. Suppose ϰ∈P and let ϰ∗(ς)≤ϰ(ς)≤ρ, ∀ς∈[r,ℑ]. So
|Qϰ(ς)|=|[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)G(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ|≤[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≤ρ, |
for all ς∈[r,ℑ]. This indicates that ‖Qϰ‖≤ρ, which proves our claim. Thus, by using the Arzela-Ascoli theorem, Q:ℵ→ℵ is compact. As a result of SFPT, Q has a fixed point ϰ in P. Hence, the problems (3.1) and (1.2) has ϰ as solution.
Lemma 3.1. ϰ∗(ς) is a solution of the FBVP (1.1), (1.2) and ϰ(ς)>w(ς) for every ς∈[r,ℑ] iff the positive solution of FBVP (3.1) and (1.2) is ϰ=ϰ∗+w.
Proof. Let ϰ(ς) be a solution of FBVP (3.1) and (1.2). Then
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)G(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)(F(τ,ϰ∗(τ),ϰ∗(r+λτ))+p(τ))Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ+1Γ(ν)∫ℑrϖ(ς,τ)p(τ)Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)G(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ+w(ς). |
So,
ϰ(ς)−w(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ. |
Then we get the existence of the solution with the condition
ϰ∗(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ. |
For the converse, if ϰ∗ is a solution of the FBVP (1.1) and (1.2), we get
Dν;ψr(ϰ∗(ς)+w(ς))=Dν;ψrϰ∗(ς)+Dν;ψrw(ς)=−F(ς,ϰ∗(ς),ϰ∗(r+λς))−p(ς)=−[F(ς,ϰ∗(ς),ϰ∗(r+λς))+p(ς)]=−G(ς,ϰ∗(ς),ϰ∗(r+λς)), |
which leads to
Dν;ψrϰ(ς)=−G(ς,ϰ∗(ς),ϰ∗(r+λς)). |
We easily see that
ϰ∗(r)=ϰ(r)−w(r)=ϰ(r)−0=ϑ1, |
i.e., ϰ(r)=ϑ1 and
ϰ∗(ℑ)=m−2∑i=1ζiϰ∗(ηi)+ϑ2, |
ϰ(ℑ)−w(ℑ)=m−2∑i=1ζiϰ(ηi)−m−2∑i=1ζjw(ηi)+ϑ2=m−2∑i=1ζi(ϰ(ηi)−w(ηi))+ϑ2. |
So,
ϰ(ℑ)=m−2∑i=1ζiϰ(ηi)+ϑ2. |
Thus ϰ(ς) is solution of the problem FBVP (3.1) and (3.2).
We propose the given FBVP as follows
D75ϰ(ς)+F(ς,ϰ(ς),ϰ(1+0.5ς))=0, ς∈(1,e), | (4.1) |
ϰ(1)=1, ϰ(e)=17ϰ(52)+15ϰ(74)+19ϰ(115)−1. | (4.2) |
Let Ψ(ς)=logς, where F(ς,ϰ(ς),ϰ(1+12ς))=ς1+ςarctan(ϰ(ς)+ϰ(1+12ς)).
Taking Υ(ς)=ς we get ∫e1ςdς=e2−12>0, then the hypotheses (Σ1) and (Σ2) hold. Evaluate Δ≅0.366, M≅3.25 we also get |G(ς,ϰ,v)|<π+e=L such that |ϰ|≤ρ, ρ=17, we could just confirm that
[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≅16.35≤17. | (4.3) |
By applying the Theorem 3.1 there exit a solution ϰ(ς) of the problem (4.1) and (4.2).
In this paper, we have provided the proof of BVP solutions to a nonlinear Ψ-Caputo fractional pantograph problem or for a semi-positone multi-point of (1.1) and(1.2). What's new here is that even using the generalized Ψ-Caputo fractional derivative, we were able to explicitly prove that there is one solution to this problem, and that in our findings, we utilize the SFPT. The results obtained in our work are significantly generalized and the exclusive result concern the semi-positone multi-point Ψ-Caputo fractional differential pantograph problem (1.1) and (1.2).
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Small Groups (RGP.1/350/43).
The authors declare no conflict of interest.
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