Research article

Multi-scale dynamics of predator-prey systems with Holling-IV functional response

  • Received: 07 November 2023 Revised: 21 December 2023 Accepted: 29 December 2023 Published: 09 January 2024
  • MSC : 34K18, 34K25, 34K27, 34D23, 34H05, 92B05, 92D25

  • In this paper, we propose a Holling-IV predator-prey system considering the perturbation of a slow-varying environmental capacity parameter. This study aims to address how the slowly varying environmental capacity parameter affects the behavior of the system. Based on bifurcation theory and the slow-fast analysis method, the critical condition for the Hopf bifurcation of the autonomous system is given. The oscillatory behavior of the system under different perturbation amplitudes is investigated, corresponding mechanism explanations are given, and it is found that the motion pattern of the non-autonomous system is closely related to the Hopf bifurcation and attractor types of the autonomous system. Meanwhile, there is a bifurcation hysteresis behavior of the system in bursting oscillations, and the bifurcation hysteresis mechanism of the system is analyzed by applying asymptotic theory, and its hysteresis time length is calculated. The final study found that the larger the perturbation amplitude, the longer the hysteresis time. These results can provide theoretical analyses for the prediction, regulation, and control of predator-prey populations.

    Citation: Kexin Zhang, Caihui Yu, Hongbin Wang, Xianghong Li. Multi-scale dynamics of predator-prey systems with Holling-IV functional response[J]. AIMS Mathematics, 2024, 9(2): 3559-3575. doi: 10.3934/math.2024174

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  • In this paper, we propose a Holling-IV predator-prey system considering the perturbation of a slow-varying environmental capacity parameter. This study aims to address how the slowly varying environmental capacity parameter affects the behavior of the system. Based on bifurcation theory and the slow-fast analysis method, the critical condition for the Hopf bifurcation of the autonomous system is given. The oscillatory behavior of the system under different perturbation amplitudes is investigated, corresponding mechanism explanations are given, and it is found that the motion pattern of the non-autonomous system is closely related to the Hopf bifurcation and attractor types of the autonomous system. Meanwhile, there is a bifurcation hysteresis behavior of the system in bursting oscillations, and the bifurcation hysteresis mechanism of the system is analyzed by applying asymptotic theory, and its hysteresis time length is calculated. The final study found that the larger the perturbation amplitude, the longer the hysteresis time. These results can provide theoretical analyses for the prediction, regulation, and control of predator-prey populations.



    In recent years, fractional differential equations have gained prominence due to their proven usefulness in several unrelated scientific and engineering fields. For example, the nonlinear oscillations of an earthquake can be characterized by a fractional derivative, and the fractional derivative of the traffic fluid dynamics model can solve the insufficiency resulting from the assumption of continuous traffic flows [1,3]. Numerous chemical processes, mathematical biology, engineering, and scientific problems [4,5,6,7] are also modeled with fractional differential equations. Nonlinear partial differential equations (NPDEs) characterize various physical, biological, and chemical phenomena. Current research is focused on developing precise traveling wave solutions for such equations. Exact and explicit solutions help scientists understand the complicated physical phenomena and dynamic processes portrayed by NPDEs [8,9,10]. In the past four decades, numerous essential methodologies for attaining accurate solutions to NPDEs have been proposed [11,12].

    The Helmholtz equation (HE) derives from the elliptic and wave equations. In a multi-dimensional nonhomogeneous isotropic standard with velocity c, the wave result is υ(ξ,ψ), which corresponds to a source of harmonic (ξ,ψ) vibrating at a given frequency and satisfying the Helmholtz equation in the area R. The classical order HE is

    D2ξυ(ξ,ψ)+D2ψυ(ξ,ψ)+ευ(ξ,ψ)=υ(ξ,ψ). (1.1)

    Here, υ is a suitable boundary differentiable term of R, is a known function, and the wave number with wavelength 2/ξ=0 renders Eq (1.1) homogeneous. If (1.1) is expressed as

    D2ξυ(ξ,ψ)+D2ψυ(ξ,ψ)ευ(ξ,ψ)=υ(ξ,ψ).

    Then it explains mass transfer with density biochemical processes of the 1st order. Equation (1.1) is investigated using the decomposition method [13], the finite element approach [14], the differential transform method [15], the Trefftz method [16], and the spectral collocation method [17], among others [3,4,5].

    The Helmholtz equation is a partial differential equation that describes wave phenomena in various fields of physics, such as electromagnetism, acoustics, and fluid mechanics. Traditionally, the Helmholtz equation has been formulated using integer-order derivatives. However, in recent years, there has been a growing interest in the use of fractional-order derivatives to describe complex phenomena more accurately. In particular, fractional-order space Helmholtz equations are derived directly from mathematical formulas that involve fractional derivatives, rather than being generalized from integer-order space derivative Helmholtz equations. These equations can provide a more accurate description of wave propagation in complex media, such as porous materials, biological tissues, and fractal structures. Fractional-order space Helmholtz equations have attracted significant attention due to their potential applications in a wide range of fields, including medical imaging, geophysics, and telecommunications. They offer a promising avenue for understanding the behavior of waves in complex media and developing new technologies for wave-based sensing and imaging [6,7,8].

    It is advantageous to utilize fractional differential equations in physical problems due to their nonlocal features. Non-locality characterizes fractional-order derivatives, whereas locality characterizes integer-order derivatives [24,25,26,27]. It demonstrates that the future state of the physical system depends on all of its previous states in addition to its current state. Consequently, fractional models are more accurate. In fractional differential equations, the response expression has a parameter that specifies the fractional derivative of the variable order, which may vary to achieve many responses [9,10,11].

    Standard HEs can be generalized to fractional-order Helmholtz equations by extending the Caputo fractional-order space derivative to the integer-order space derivative. The fractional Helmholtz equation in space is

    Dϱξυ(ξ,ψ)+D2ψυ(ξ,ψ)+ευ(ξ,ψ)=ψ(ξ,ψ),

    with υ(0,ψ)=g(ψ) as the initial condition (IC). Gupta et al. [31] solved the multi-dimensional fractional Helmholtz equation using the homotopy perturbation approach. In contrast, Abuasad et al. [14] recently solved a fractional model of the Helmholtz problem using the reduced differential transform method.

    This section describes the properties of the fractional derivatives and a few essential details concerning the Yang transform.

    Definition 2.1. The fractional derivative in terms of Caputo is as follows

     Dϱψυ(ξ,ψ)=1Γ(kϱ)ψ0(ψϱ)kϱ1υ(k)(ξ,ϱ)dϱ,k1<ϱk,kN. (2.1)

    Definition 2.2. The YT is represented as follows

    Y{υ(ψ)}=M(u)=0eψuυ(ψ)dψ,  ψ>0,  u(ψ1,ψ2), (2.2)

    having inverse YT as follows

    Y1{M(u)}=υ(ψ). (2.3)

    Definition 2.3. The nth derivative YT is stated as follows

    Y{υn(ψ)}=M(u)unn1k=0υk(0)unk1,    n=1,2,3, (2.4)

    Definition 2.4. The YT of derivative having fractional-order is stated as follows

    Y{υϱ(ψ)}=M(u)uϱn1k=0υk(0)uϱ(k+1),  0<ϱn. (2.5)

    Consider the general fractional partial differential equations,

    Dϱψυ(ξ,ψ)+Mυ(ξ,ψ)+Nυ(ξ,ψ)=h(ξ,ψ),ψ>0,0<ϱ1,υ(ξ,0)=g(ξ),ν. (3.1)

    Using Yang transform of Eq (3.1), we get

     Y[Dϱyυ(ξ,ψ)+Mυ(ξ,ψ)+Nυ(ξ,ψ)]=Y[h(ξ,ψ)],ψ>0,0<ϱ1,υ(ξ,ψ)=sg(ξ)+sϱY[h(ξ,ψ)]sϱY[Mυ(ξ,ψ)+Nυ(ξ,ψ)]. (3.2)

    Now, applying inverse Yang transform, we have

    υ(ξ,ψ)=F(ξ,ψ)Y1[sϱY{Mυ(ξ,ψ)+Nυ(ξ,ψ)}], (3.3)

    where

     F(ξ,ψ)=Y1[sg(ξ)+sϱY[h(ξ,ψ)]]=g(ν)+Y1[sϱY[h(ξ,ψ)]]. (3.4)

    The parameter p is perturbation technique and p[0,1] defined as

    υ(ξ,ψ)=k=0pkυk(ξ,ψ), (3.5)

    The nonlinear function is expressed as

    Nυ(ξ,ψ)=k=0pkHk(υk), (3.6)

    where Hn are He's polynomials in term of υ0,υ1,υ2,,υn, and can be calculated as

    Hn(υ0,υ1,,υn)=1ϱ(n+1)Dıp[N(ı=0pıυı)]p=0, (3.7)

    where Dıp=ıpı.

    Putting Eqs (3.6) and (3.7) in Eq (3.3), we achieved as

     ı=0pıυı(ξ,ψ)=F(ξ,ψ)p×[Y1{sϱY{Mı=0pıυı(ξ,ψ)+ı=0pıHı(υı)}}]. (3.8)

    Comparison both sides of coefficient p, we get

    p0:υ0(ξ,ψ)=F(ξ,ψ),p1:υ1(ξ,ψ)=Y1[sϱY(Mυ0(ξ,ψ)+H0(υ))],p2:υ2(ξ,ψ)=Y1[sϱY(Mυ1(ξ,ψ)+H1(υ))],pı:υı(ξ,ψ)=Y1[sϱY(Mυı1(ξ,ψ)+Hı1(υ))],ı>0,ıN. (3.9)

    Finally, present the obtained solution and check it with any available analytical or numerical solutions for the given PDE. The υı(ξ,ψ) components can be calculated easily which quickily converges to series form. We can get p1,

     υ(ξ,ψ)=limMMı=1υı(ξ,ψ). (3.10)

    Problem 4.1. Consider the space fractional Helmholtz equation

    ϱυ(ξ,ψ)ξϱ+2υ(ξ,ψ)ψ2υ(ξ,ψ)=0,1<ϱ2, (4.1)

    with the ICs

     υ(0,ψ)=ψ  andυξ(0,ψ)=0. (4.2)

    Using the Yang transform of Eq (4.1), we obtained as

     1sϱY[υ(ξ,ψ)]=υ(0,ψ)s1ϱY{2υ(ξ,ψ)ψ2υ(ξ,ψ)}, (4.3)
     Y[υ(ξ,ψ)]=sυ(0,ψ)sϱY{2υ(ξ,ψ)ψ2υ(ξ,ψ)}, (4.4)

    Taking inverse Yang Transformation, we have

    Y[υ(ξ,ψ)]=ψY1[sϱY{2υ(ξ,ψ)ψ2υ(ξ,ψ)}], (4.5)

    Implemented HPM in Eq (4.5), we can achieve as

     ı=0pıυı(ξ,ψ)=ψp[Y1{sϱY{(ı=0pıυı(ξ,ψ))ψψı=0pıυı(ξ,ψ)}}]. (4.6)

    On both sides comparing coefficients of p, we get

     p0:υ0(ξ,ψ)=ψ,p1:υ1(ξ,ψ)=Y1[sϱY{2υ0(ξ,ψ)ψ2υ0(ξ,ψ)}]=ξϱΓ(ϱ+1)ψ,p2:υ2(ξ,ψ)=Y1[sϱY{2υ1(ξ,ψ)ψ2υ1(ξ,ψ)}]=ξ2ϱΓ(2ϱ+1)ψ,p3:υ3(ξ,ψ)=Y1[sϱY{2υ2(ξ,ψ)ψ2υ2(ξ,ψ)}]=ξ3ϱΓ(3ϱ+1)ψ,p4:υ4(ξ,ψ)=Y1[sϱY{2υ3(ξ,ψ)ψ2υ3(ξ,ψ)}]=ξ4ϱΓ(4ϱ+1)ψ, (4.7)

    The series type result of the first problem example is

     υ(ξ,ψ)=υ0(ξ,ψ)+υ1(ξ,ψ)+υ2(ξ,ψ)+υ3(ξ,ψ)+υ4(ξ,ψ)+υ(ξ,ψ)=ψ[1+ξϱΓ(ϱ+1)+ξ2ϱΓ(2ϱ+1)+x3ϱΓ(3ϱ+1)+ξ4ϱΓ(4ϱ+1)+]. (4.8)

    The exact solution is

    υ(ξ,ψ)=ψcoshξ.

    Similarly y-space can be calculated as:

    ϱυ(ξ,ψ)ψϱ+2υ(ξ,ψ)ξ2υ(ξ,ψ)=0, (4.9)

    with the IC

    υ(ξ,0)=ξ. (4.10)

    Thus, the solution of the above Eq (4.9) is obtain as

    υ(ξ,ψ)=ξ(1+ψϱΓ(ϱ+1)+ψ2ϱΓ(2ϱ+1)+ψ3ϱΓ(3ϱ+1)+ψ4ϱΓ(4ϱ+1)+),

    in the case when ϱ=2, then the solution through HPTM is

    υ(ξ,ψ)=ξcoshψ. (4.11)

    Figure 1 illustrates the exact and HPTM solutions in two-dimensional plots for various values of ϱ ranging from 2 to 1.5, with ξ values ranging from 0 to 1 and ψ set to 1. The solutions are presented in Figure 1a and b for the exact and HPTM methods, respectively. Figure 2 displays the 3-dimensional plots of the exact and HPTM solutions for ϱ=2 and analyzes the point of intersection between the two solutions. Figure 2c and d depict the HPTM solutions at ϱ=1.8 and 1.6 respectively for Problem 4.1. The fractional results were also evaluated for their convergence towards an integer-order result for each problem. Similarly, the figures for the ψ-space can also be generated using the same approach.

    Figure 1.  The first graph show that the exact and approximate solution and second HPTM solution at the different fractional-order graph of Problem 4.1.
    Figure 2.  Exact and proposed method solution at various fractional orders of Problem 4.1.
    Table 1.  Exact and proposed method solution of Problem 4.1 at various fractional orders.
    (ξ,ψ) υ(ξ,ψ) at ϱ=1.5 υ(ξ,ψ) at ϱ=1.75 (HPTM) at ϱ= 2 Exact result
    (0.2, 0.1) 0.2102371 0.2100072 0.2100000 0.2100000
    (0.4, 0.1) 0.4104630 0.4100141 0.4100000 0.4100000
    (0.6, 0.1) 0.6106889 0.6100209 0.6100000 0.6100000
    (0.2, 0.2) 0.2103355 0.2100121 0.2100000 0.2100000
    (0.4, 0.2) 0.4106550 0.4100237 0.4100000 0.4100000
    (0.6, 0.2) 0.6109746 0.6100353 0.6100000 0.6100000
    (0.2, 0.3) 0.2104110 0.2100164 0.2100000 0.2100000
    (0.4, 0.3) 0.4108025 0.4100321 0.4100000 0.4100000
    (0.6, 0.3) 0.6111940 0.6100478 0.6100000 0.6100000
    (0.2, 0.4) 0.2104747 0.2100204 0.2100000 0.2100000
    (0.4, 0.4) 0.4109269 0.4100399 0.4100000 0.4100000
    (0.6, 0.4) 0.6113790 0.6100593 0.6100000 0.6100000
    (0.2, 0.5) 0.2105309 0.2100241 0.2100000 0.2100000
    (0.4, 0.5) 0.4110365 0.4100471 0.4100000 0.4100000
    (0.6, 0.5) 0.6115421 0.6100701 0.6100000 0.6100000

     | Show Table
    DownLoad: CSV

    Problem 4.2. Consider the space-fractional HE

    ϱυ(ξ,ψ)ξϱ+2υ(ξ,ψ)ψ2+5υ(ξ,ψ)=0,1<ϱ2, (4.12)

    with the ICs

     υ(0,ψ)=ψ  andυξ(0,ψ)=0. (4.13)

    Using the Yang transform of Eq (4.12), we obtain as

     1sϱY[υ(ξ,ψ)]=υ(0,ψ)s1ϱY{2υ(ξ,ψ)ψ2+5υ(ξ,ψ)}, (4.14)
     Y[υ(ξ,ψ)]=sυ(0,ψ)sϱY{2υ(ξ,ψ)ψ2+5υ(ξ,ψ)}. (4.15)

    Applying the inverse Yang Transform, we get

    Y[υ(ξ,ψ)]=ψY1[sϱY{2υ(ξ,ψ)ψ2+5υ(ξ,ψ)}], (4.16)

    Using the HPM in Eq (4.16), we obtained as

     ı=0pıυı(ξ,ψ)=ψp[Y1{sϱY{(ı=0pıυı(ξ,ψ))ψψ+5ı=0pıυı(ξ,ψ)}}]. (4.17)

    On both sides comparing coefficients of p, we get

     p0:υ0(ξ,ψ)=ψ,p1:υ1(ξ,ψ)=Y1[sϱY{2υ0(ξ,ψ)ψ2+5υ0(ξ,ψ)}]=5ψξϱΓ(ϱ+1),p2:υ2(ξ,ψ)=Y1[sϱY{2υ1(ξ,ψ)ψ2+5υ1(ξ,ψ)}]=25ψξ2ϱΓ(2ϱ+1),p3:υ3(ξ,ψ)=Y1[sϱY{2υ2(ξ,ψ)ψ2+5υ2(ξ,ψ)}]=125ξ3ϱΓ(3ϱ+1),p4:υ4(ξ,ψ)=Y1[sϱY{2υ3(ξ,ψ)ψ2+5υ3(ξ,ψ)}]=625ψξ4ϱΓ(4ϱ+1), (4.18)

    The series type result of second problem as

     υ(ξ,ψ)=υ0(ξ,ψ)+υ1(ξ,ψ)+υ2(ξ,ψ)+υ3(ξ,ψ)+υ4(ξ,ψ)+υ(ξ,ψ)=ψ[15ξϱΓ(ϱ+1)+25ξ2ϱΓ(2ϱ+1)125x3ϱΓ(3ϱ+1)+625ξ4ϱΓ(4ϱ+1)+]. (4.19)

    The exact solution is

    υ(ξ,ψ)=ψcos5ξ.

    Now similarly, the result of y-space can be calculated with the help of homotopy perturbation

    ϱυ(ξ,ψ)ψϱ+2υ(ξ,ψ)ξ2+5υ(ξ,ψ)=0, (4.20)

    with the IC

    υ(ξ,0)=ξ. (4.21)

    The solution of the Eq (4.20) is expressed as

    υ(ξ,ψ)=ξ(15ψϱΓ(ϱ+1)+25ψ2ϱΓ(2ϱ+1)125ψ3ϱΓ(3ϱ+1)+625ψ4ϱΓ(4ϱ+1)+).

    The exact solution is

    υ(ξ,ψ)=ξcos5ψ. (4.22)

    Figure 3 illustrates the solutions of exact and HPTM in two-dimensional plots, as shown in Figure 3a and b for different values of ϱ, ranging from 2 to 1.5, respectively. The interval considered for ξ is [0, 1], while ψ is constant at 1. The results obtained from the fractional-order model converge to the integer-order solution of the problem. In Figure 4, the 3-dimensional plots of exact and HPTM solutions are presented in Figures (a) and (b), respectively, for ϱ=2. The closed contact of the two solutions is analyzed. Additionally, Figure 4c and d depict the HPTM solutions at ϱ=1.8 and 1.6, respectively, for Problem 4.2. Similarly, graphs for ψ-space can also be generated.

    Figure 3.  Exact and proposed method solution at various fractional orders of Problem 4.2.
    Figure 4.  Exact and proposed method solution at various fractional orders of Problem 4.2.

    Problem 4.3. Consider the space-fractional HE

    ϱυ(ξ,ψ)ξϱ+2υ(ξ,ψ)ψ22υ(ξ,ψ)=(12ξ23ξ4)sinψ,1<ϱ2,0ψ2π, (4.23)

    with the ICs

     υ(0,ψ)=0  andυξ(0,ψ)=0. (4.24)

    Applying the Yang transform of Eq (4.23), we achieve

     1sϱY[υ(ξ,ψ)]=υ(0,ψ)s1ϱY{2υ(ξ,ψ)ψ22υ(ξ,ψ)}, (4.25)
     Y[υ(ξ,ψ)]=sυ(0,ψ)sϱY{2υ(ξ,ψ)ψ22υ(ξ,ψ)}. (4.26)

    Implementing inverse Yang transform, we get

    Y[υ(ξ,ψ)]=(ξ4ξ610)sinψY1[sϱY{2υ(ξ,ψ)ψ22υ(ξ,ψ)}]. (4.27)

    Applying Homotopy perturbation method in Eq (4.27), we achieved as

     ı=0pıυı(ξ,ψ)=ψp[Y1{sϱY{(ı=0pıυı(ξ,ψ))ψψ2ı=0pıυı(ξ,ψ)}}]. (4.28)

    Both sides on comparison coefficients of p, we obtain

     p0:υ0(ξ,ψ)=(ξ4ξ610)sinψ,p1:υ1(ξ,ψ)=Y1[sϱY{2υ0(ξ,ψ)ψ22υ0(ξ,ψ)}]=3(ξϱ+4Γ(ϱ+5)72ξϱ+6Γ(ϱ+7))sinψ,p2:υ2(ξ,ψ)=Y1[sϱY{2υ1(ξ,ψ)ψ22υ1(ξ,ψ)}]=3(ξ2ϱ+4Γ(2ϱ+5)216ξ2ϱ+6Γ(2ϱ+7))sinψ,p3:υ3(ξ,ψ)=Y1[sϱY{2υ2(ξ,ψ)ψ22υ2(ξ,ψ)}]=3(ξ3ϱ+4Γ(3ϱ+5)648ξ3ϱ+6Γ(3ϱ+7))sinψ,p4:υ4(ξ,ψ)=Y1[sϱY{2υ3(ξ,ψ)ψ22υ3(ξ,ψ)}]=3(ξ4ϱ+4Γ(4ϱ+5)1944ξ2ϱ+6Γ(2ϱ+7))sinψ, (4.29)

    The series type result of the third problem is

     υ(ξ,ψ)=υ0(ξ,ψ)+υ1(ξ,ψ)+υ2(ξ,ψ)+υ3(ξ,ψ)+υ4(ξ,ψ)+υ(ξ,ψ)=(ξ4ξ610)sinψ+3(ξϱ+4Γ(ϱ+5)72ξϱ+6Γ(ϱ+7))sinψ+3(ξ2ϱ+4Γ(2ϱ+5)216ξ2ϱ+6Γ(2ϱ+7))sinψ+3(ξ3γ+4Γ(3ϱ+5)648ξ3ϱ+6Γ(3ϱ+7))sinψ+3(ξ4ϱ+4Γ(4ϱ+5)1944ξ2ϱ+6Γ(2ϱ+7))sinψ+. (4.30)

    The exact solution is

    υ(ξ,ψ)=ξ4sinψ.

    Figure 5a and b display the exact and HPTM solutions, respectively, in a 3-dimensional plot at ϱ=2. The closed contact between the exact and HPTM solutions is examined. Figure 6 depicts the exact and HPTM solutions in two-dimensional plot for various values of ϱ=2,1.9,1.8,1.7,1.6,1.5 for ξ[0,1] and ψ=1. The fractional results are observed to approach an integer-order solution of the problem. Similarly, the graphs for ψ-space fractional-order derivative can also be plotted.

    Figure 5.  Exact and proposed method solution at various fractional orders of Problem 4.3.
    Figure 6.  Exact and proposed method solution at various fractional orders of Problem 4.3.

    In this study, fractional-order Helmholtz equations were solved using the Homotopy Perturbation Yang transform method. Due to the great agreement between the generated approximative solution and the precise solution, the homotopy perturbation Yang transform method was demonstrated to be a successful method for solving partial differential equations with Caputo operators. The computation size of the approach was compared to those required by other numerical methods to demonstrate how tiny it is. Additionally, the procedure's quick convergence demonstrates its dependability and marks a notable advancement in the way linear and non-linear fractional-order partial differential equations are solved.

    This research has been funded by Deputy for Research & Innovation, Ministry of Education through Initiative of Institutional Funding at University of Ha'il–Saudi Arabia through project number IFP-22 064.

    The authors declare that they have no competing interests.



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