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Spatially localized sparse approximations of deep features for breast mass characterization


  • Received: 31 March 2023 Revised: 15 June 2023 Accepted: 05 July 2023 Published: 01 August 2023
  • We propose a deep feature-based sparse approximation classification technique for classification of breast masses into benign and malignant categories in film screen mammographs. This is a significant application as breast cancer is a leading cause of death in the modern world and improvements in diagnosis may help to decrease rates of mortality for large populations. While deep learning techniques have produced remarkable results in the field of computer-aided diagnosis of breast cancer, there are several aspects of this field that remain under-studied. In this work, we investigate the applicability of deep-feature-generated dictionaries to sparse approximation-based classification. To this end we construct dictionaries from deep features and compute sparse approximations of Regions Of Interest (ROIs) of breast masses for classification. Furthermore, we propose block and patch decomposition methods to construct overcomplete dictionaries suitable for sparse coding. The effectiveness of our deep feature spatially localized ensemble sparse analysis (DF-SLESA) technique is evaluated on a merged dataset of mass ROIs from the CBIS-DDSM and MIAS datasets. Experimental results indicate that dictionaries of deep features yield more discriminative sparse approximations of mass characteristics than dictionaries of imaging patterns and dictionaries learned by unsupervised machine learning techniques such as K-SVD. Of note is that the proposed block and patch decomposition strategies may help to simplify the sparse coding problem and to find tractable solutions. The proposed technique achieves competitive performances with state-of-the-art techniques for benign/malignant breast mass classification, using 10-fold cross-validation in merged datasets of film screen mammograms.

    Citation: Chelsea Harris, Uchenna Okorie, Sokratis Makrogiannis. Spatially localized sparse approximations of deep features for breast mass characterization[J]. Mathematical Biosciences and Engineering, 2023, 20(9): 15859-15882. doi: 10.3934/mbe.2023706

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  • We propose a deep feature-based sparse approximation classification technique for classification of breast masses into benign and malignant categories in film screen mammographs. This is a significant application as breast cancer is a leading cause of death in the modern world and improvements in diagnosis may help to decrease rates of mortality for large populations. While deep learning techniques have produced remarkable results in the field of computer-aided diagnosis of breast cancer, there are several aspects of this field that remain under-studied. In this work, we investigate the applicability of deep-feature-generated dictionaries to sparse approximation-based classification. To this end we construct dictionaries from deep features and compute sparse approximations of Regions Of Interest (ROIs) of breast masses for classification. Furthermore, we propose block and patch decomposition methods to construct overcomplete dictionaries suitable for sparse coding. The effectiveness of our deep feature spatially localized ensemble sparse analysis (DF-SLESA) technique is evaluated on a merged dataset of mass ROIs from the CBIS-DDSM and MIAS datasets. Experimental results indicate that dictionaries of deep features yield more discriminative sparse approximations of mass characteristics than dictionaries of imaging patterns and dictionaries learned by unsupervised machine learning techniques such as K-SVD. Of note is that the proposed block and patch decomposition strategies may help to simplify the sparse coding problem and to find tractable solutions. The proposed technique achieves competitive performances with state-of-the-art techniques for benign/malignant breast mass classification, using 10-fold cross-validation in merged datasets of film screen mammograms.



    Predator-prey interactions have been an interesting and challenging issue that is frequently discussed in marine ecosystems, especially in fish populations. Predator-prey interactions are the most important component of ecology, determining various factors such as community composition, species behavior and dynamics. Mathematical modeling helps to provide insights into the dynamics of the system, which was investigated in the early pioneering work of Lotka[1] and Volterra[2]. In dynamical systems, continuity, equilibrium stability, bifurcation, and control problems are also often studied[3,4,5,6,7]. Traditionally, predators could be distinguished as specialists or generalists based on whether they ate only one or more types of prey. In general, a predator-prey model can be described by the following ordinary differential equation for both specialists and generalist predators[8]:

    ˙P=F(P)PG(P,N)N,˙N=γυ(G(P,N))N+H(N)N,

    where P and N denote the prey and predator's population densities at moment t, respectively. F() and H() represent the growth of the species in the absence of the other one. G() is known as the functional response, which characterizes the average individual of prey consumed by a predator; γ represents the conversion rate from prey consumption to predator, υ() is a monotonically increasing function. For specialist predators, there is H(N)=d<0; and for generalist predators, it is required that γυ(G(0,N))+H(N)>0. In literature, different type of functional responses were adopted to model different species, which can be prey dependent[9,10] or prey and predator dependent[11,12,13,14,15].

    The classical view of population dynamics claims that the higher the population density, the lower the overall growth rate due to the competition for resources. The lower the density, the higher the overall growth rate. However, when the population density is low, Allee[16] introduced the opposite view that the lower the population density, the lower the overall growth rate, that is, the Allee effect. The Allee effect is a common phenomenon in marine populations[17,18,19]. When the population density is low, it may affect population development due to pairing restrictions, dispersal, habitat changes, cooperative foraging, cooperative defense, and predator saturation. Therefore, the study of the Allee effect on ecosystems has attracted the attention of many scholars. In general, the Allee effect can be represented by a multiplier of the form P-L[20,21,22], where L is the threshold for the Allee effect. When L<0, it is a weak Allee effect, and the Allee effect is always positive no matter how much the prey growth rate decreases. When L>0, it is a strong Allee effect. The strong Allee effect indicates that in order for the population to grow, the population size or density must be higher than L; otherwise, the population will die out. Scholars have analyzed the dynamics of the system with the Allee effect of the form P-L, and discussed the existence of the equilibrium of the system and various bifurcation phenomena such as saddle-node bifurcation and B-T bifurcation[23,24,25].

    Fish is a kind of important ecological resources. In view of the fish resources development problems, scholars studied population behavior by adding harvest items on the basis of continuous systems. However, fishing activities are not continuous, so continuous dynamic systems cannot accurately describe the actual fishing process. In the process of fish harvesting, the periodic harvesting of fish is a kind of human-controlled behavior that can be described by an impulse differential equation, and it has been found that the impulse differential equation is more accurate in describing and portraying the dynamical behavior of the population[26]. The theory of semi-continuous dynamic systems has been widely used in modeling research on pest management modeling[27,28,29,30,31]. Based on the analysis of the literature, due to the fact that impulsive differential systems (semi-continuous dynamical systems) have the characteristics of both continuous and discrete dynamical systems, there are some studies applying the theory to the development of fish populations in deterministic environments[32,33,34,35,36] and uncertain environment[37,38,39]. In addition, most of the studies considered fish harvesting or fish protection only (unilateral control); in this study, a bilateral state control[40,41,42] is considered, and a predator-prey model for conservation and harvesting of two fish species is developed by constructing a semi-continuous dynamical system. When designing the state feedback control strategy, the number of objective fish was used as the state variable for feedback control. On the one hand, when the number of prey fish is small, the Allee effect will lead to their extinction, which will lead to the lack of enough food for prey fish and destroy the balance of the ecosystem. Therefore, it is necessary to release a certain amount of prey fish when the number of prey fish decreases to a certain threshold. When the number of prey fish is high, it is necessary to catch prey fish from the economic point of view. Since the fishing behavior will also lead to the harvesting of some predator fish, in order to maintain the ecological balance and avoid the extinction of predator fish caused by the fishing behavior, it is necessary to release a certain amount of predator fish larvae at the same time. Based on the above two aspects, we propose a bilateral control strategy to maintain the population size of the two species in a suitable range.

    This paper considers a predator-prey model in which the predator is a generalist and the growth of the prey is affected by the Allee effect. The structure is as follows: In Section Ⅱ, we describe the fishery model, non-negativity, persistent survivability and discuss the existence and stability of equilibria of the model. In Section Ⅲ, the bifurcation dynamics of the model are discussed using bifurcation theory. In Section Ⅳ, based on this model, we analyze the model of marine fish harvesting and conservation with bilateral controls. The existence and stability of the order-1 and order-2 periodic solutions of the system are analyzed by using the geometry theory of differential equations. In the fifth section, we performed numerical simulations using MATLAB to verify the correctness of the results.

    In this paper, we present a predator-prey model in which predators are generalists, prey growth rates are logical and subject to strong Allee effects, and the functional response is a Holling-Ⅰ type, while the conversion from prey consumption to predator species is saturated,

    {dPdT=rP(1PK)(PL)APN,dNdT=e(AP1+BP)N+(s1+fNd)N, (2.1)

    where p(T), N(T) denote the prey and predator's densities at the moment of T; K denotes the prey's environmental holding capacity; L is the threshold of the prey's Allee effect; r and s represent the intrinsic growth rates of prey and predator, respectively; A is the capture rate of prey by the predator; e is the efficiency with which prey is converted to predator; B is the half-saturation constant; f is the intensity of predator density dependence, and d is the predator mortality rate. Since the predators are generalist, it requires that eABd, s>d, and all parameters of model (2.1) are positive.

    To facilitate the analysis, let x=PK, y=N, t=rKT, α=ArK, β=LK, γ=eAr, δ=BK, s1=srK, d1=drK. Then system (2.1) is simplified to system (2.2):

    {dxdt=x(1x)(xβ)αxy,dydt=γxy1+δx+(s11+fyd1)y, (2.2)

    and from the biological point of consideration, the model (2.2) is limited in the region

    Ω={(x,y)R2+|0x1,y0}.

    On the other hand, as a renewable resource, fish species are closely related to human life. In order to maintain the balance of the ecosystem during the fishing process, we consider a fish stock control strategy with a combination of fishing and investment. First, to avoid the distinction of prey fish caused by the Allee effect, a quantity (η) of juvenile prey fish is released when the prey density declines to the level x=h1. But at a higher level x=h2, a proportion a of prey fish together with a proportion b of predator fish will also be caught for economic purposes, and simultaneously, a quantity τ of juvenile predator fish is released into the system to maintain the level of predator fish. Based on the control measures, the model can be described as follows:

    {dxdt=x(1x)(xβ)αxydydt=γxy1+δx+(s11+fyd1)y}h1<x<h2,Δx=ηΔy=0}x=h1,Δx=axΔy=by+τ}x=h2, (2.3)

    where η, a, b, τ are all positive, and a,b(0,1).

    For a given planar model

    {dxdt=f1(x,y),dydt=f2(x,y)ω(x,y)0,Δx=I1(x,y),Δy=I2(x,y)ω(x,y)=0, (2.4)

    Definition 1 (Order-k periodic solution[32,33,36]) The solution ˜z(t)=(˜x(t),˜y(t)) is called periodic if there exists m(1) satisfying ˜zm=˜z0. Furthermore, ˜z is an order-k T-periodic solution with kmin{j|1jm,˜zj=˜z0}.

    Lemma 1 (Analogue of Poincaré Criterion[32,33,36]). The order-k T-periodic solution z(t)=(ξ(t),η(t))T is orbitally asymptotically stable if |μq|<1, where

    μk=kj=1Δjexp(T0[f1x+f2y](ξ(t),η(t))dt),

    with

    Δj=f+1(I2yωxI2xωy+ωx)+f+2(I1xωyI1yωx+ωy)f1ωx+f2ωy,

    f+1=f1(ξ(θ+j),η(θ+j)), f+2=f2(ξ(θ+j),η(θ+j)) and f1, f2, I1x, I1y, I2x, I2y, ωx, ωy are calculated at (ξ(θj),η(θj)).

    In this section, the bounded-ness of the solution for Model (2.2) is discussed. Moreover, the existence, type, and local stability of the equilibrium as well as the bifurcation properties are verified.

    Define

    g1(x,y)=(1x)(xβ)αy,f1(x,y)=xg1(x,y);
    g2(x,y)=γx1+δx+(s11+fyd1),f2(x,y)=yg2(x,y).

    Theorem 1. The solution of Model (2.2) with non-negative initial values will remain non-negative for all time and is bounded on R2+.

    Proof. By Eq (2.2), it can be obtained that

    x(t)=x(0)exp[t0g1(x(s),y(s))ds],y(t)=y(0)exp[t0g2(x(s),y(s))ds],

    for all t0, as long as x(0) and y(0) are non-negative, then x(t) and y(t) are also non-negative.

    Next, we define a function u(t)=γαx(t)+y(t). Then

    ˙u=γα˙x+˙y=γα[x(1x)(xβ)αxy]+γxy1+δx+(s11+fyd1)yγα[x(1x)(xβ)αxy]+γxy+(s11+fyd1)yγ(1x)(xβ)αx+s1fd1y[γα((1β)24+d1)+s1f]d1u,

    which implies that

    u(t)1d1[γα((1β)24+d1)+s1f]+(u01d1[γα((1β)24+d1)+s1f])ed1t,

    so long as u0=γαx0+y0 is bounded, u(t) is bounded in Ω. To sum up, any solution of Model (2.2) starting with a non-negative bounded initial condition is non-negative and bounded in Ω.

    Obviously, four equilibria always exist

    O(0,0),E1(0,1f(s1d1d1)),E2(β,0),E3(1,0).

    Define

    y1(x)=1α(1x)(xβ), (3.1)
    y2(x)=1f[s1δ(1δ+x)d1(γd1δ)x1] (3.2)

    and denote ¯γ1d1δ. Due to the assumptions eABd and s>d, then γ<¯γ1, i.e., y2(x)<s1δ¯γ1γ. The positional relationship between y1(x) and y2(x) for different cases is shown in Figure 1.

    Figure 1.  Illustration of the positional relationship between y1(x) and y2(x) for different values of f.

    Let y1(x)=y2(x). Then it has

    a1x3+a2x2+a3x+a4=:g(x)=0, (3.3)

    where

    a1=f(δd1γ)>0,a2=f[(γδd1)(β+1)+d1],a3=α[δ(s1d1)+γ]+f[β(δd1γ)d1(β+1)],a4=α(s1d1)+βfd1>0.

    Define

    ¯αfβ,¯δ1d1(1+β)s1β,¯δ2fd1(1+β)fβd1+α(s1d1),¯δ3(1+β)fd1αs1,
    ¯γ2αδ(s1d1)+fd1(βδ(1+β))fβα,xda223a1a3a23a1,ρg(xd).

    Theorem 2. For any of the following cases: C1) α<¯α, δ<¯δ2, γ<¯γ1; C2) α<¯α, ¯δ2<δ<¯δ3, ¯γ2<γ<¯γ1; C3) α>¯α, δ<¯δ2, 0<γ<¯γ2, if ρ<0 holds, there exists two interior equilibria in Model (2.2); if ρ=0, there exists a unique interior equilibrium in Model (2.2); if ρ>0, there doesn't exists interior equilibrium in Model (2.2).

    Proof. Clearly, the existence of an interior equilibrium is equivalent to that of a positive root of Eq (3.3) in the interval (0, 1). Since

    g(0)=a4>0,g(1)=a1+a2+a3+a4=α[(δ+1)(s1d1)+γ]>0,

    and

    g(x)=3a1x2+2a2x+a3, (3.4)

    then for any of the cases C1)C3), there is a3<0. Moreover, g(0)<0, g(xd)=0, g(xd)>0 and

    g(1)=(1β)f(δd1γ)+(1β)d1f+α[δ(s1d1)+γ]>0,

    so that for x(0,xd), g(x)<0, for x(xd,0), g(x)>0. If ρ<0, Eq (3.3) has two distinct positive roots xi(0,1), i=1,2. Denote yi=y1(xi), i=1,2. Then two interior equilibria exist in Model (2.2), denoted by E1(x1,y1) and E2(x2,y2). If ρ=0, Eq (3.3) has a unique positive root x=xd(0,1), then a unique positive equilibrium exists in Model (2.2), denoted by E(x,y1(x); when ρ>0, Eq (3.3) does not have positive root, and thus there does not exist interior equilibrium in Model (2.2).

    For any equilibrium ¯E(¯x,¯y), there is

    J(¯E)=[3¯x2+2(β+1)¯xβα¯yα¯xγ¯y(1+δ¯x)2γ¯x1+δ¯x+s1(1+f¯y)2d1],

    its characteristic equation is

    λ2TR(J(¯E))λ+DET(J(¯E))=0,

    where

    TR(J(¯E))=3¯x2+2(β+1)¯xβα¯y+γ¯x1+δ¯x+s1(1+f¯y)2d1,
    DET(J(¯E))=[3¯x+2(β+1)¯xβα¯y][γ¯x1+δ¯x+s1(1+f¯y)2d1]+α¯x(γ¯y(1+δ¯x)2).

    1) Boundary equilibria

    At O(0,0), there is

    J(O)=[β00s1d1],

    since λ1=β<0, λ2=s1d1>0, then O is an unstable higher-order singularity.

    At E1(0,1f(s1d1d1)), there is

    J(E1)=[βαf(s1d1d1)0γf(s1d1d1)0],

    since λ1=0, λ2=αf(s1d1d1)<0, so that E1 is locally stable.

    At E2(β,0), there is

    J(E2)=[β(1β)αβ0γβ1+δβ+s1d1],

    since λ1=β(1β)>0, λ2=γβ1+δβ+s1d1>0, then E2 is unstable.

    At E3(1,0), there is

    J(E3)=[β1α0γ1+δ+s1d1],

    since λ1=β1<0, λ2=γ1+δ+s1d1>0, then E3 is unstable.

    2) Interior equilibrium

    At E(x,y), there are g1(x,y)=0 and g2(x,y)=0, then

    J(E)=[(1+β)x2(x)2αxγy(1+δx)2s1fy(1+fy)2],

    thus,

    DET(J(E))=[xyg1yg2y(dy1dxdy2dx)](x,y),

    where xyg1yg2y|x,y>0, then the sign of DET(J(E)) is identical to that of dy1dx|x=xdy2dx|x=x. Next, it will discuss the sign of dy1dx|x=xdy2dx|x=x for different cases in Theorem 2.

    ⅰ) When ρ<0 holds, two interior equilibria E1(x1,y1) and E2(x2,y2) with 0<x1<x2<1 exists in Model (2.2), as illustrated in Figure 2(a). It can be easily checked that

    SIGN(J(E1))=[++],SIGN(J(E2))=[+].
    Figure 2.  Symbolic representation of Jacobian matrix elements for the case C1).

    Besides, at E1, there is dy1dx|x=x1>dy2dx|x=x1, thus

    DET(J(E1))=[xyf1yf2y(dy1dxdy2dx)](x1,y1)<0,

    i.e., E1 is unstable. Similarly, at E2, there is dy1dx|x=x2<dy2dx|x=x2, thus

    (λ1+λ2)|(x2,y2)=TR(J(E2))<0,
    (λ1λ2)|(x2,y2)=DET(J(E2))=[xyf1yf2y(dy1dxdy2dx)](x2,y2)>0,

    i.e., E2 is a locally asymptotically stable node.

    ⅱ) When ρ=0 holds, system (2.2) has a unique interior equilibrium E(x,y), as shown in Figure 2(b). Let X=xx,Y=yy, then E is converted to the origin O(0,0), and the model is written as

    {dXdt=a11X+a12Y+A1X2+A2XY,dYdt=a21X+a22Y+B1X2+B2XY+B3Y2+P3(X,Y), (3.5)

    where,

    a11=(1+β)x2(x)2,a12=αx,a21=γy(1+δx)2,a22=s1fy(1+fy)2,A1=β+13x,A2=α2,B1=γδy(1+δx)3,B2=γ(1+δx)2,B3=s1f(1+fy)3

    and P3(X,Y) is a function of (X,Y) with degree of three or higher. The Jacobian matrix at E is

    J(E)=[a11a12a21a22],

    thus

    DET(J(E))=a11a22a12a21=0,TR(J(E))=a11+a22.

    a) If TR(J(E))=a11+a22=0, then λ1=λ2=0. Make the transformation

    (XY)=(a110a211)(x1y1),

    then Model (3.5) is converted into the following standard form:

    {dx1dt=ˉa12y1+ˉA1x21+ˉA2x1y1,dy1dt=ˉB1x21+ˉB2x1y1+ˉB3y21+ˉP3(x1,y1),

    where

    ˉa12=a12a11,ˉA1=a11A1+a21A2,ˉA2=A2,
    ˉB1=a211B1+a11a21(B2A1)+a221(B3A2),ˉB2=a11B2+a21(2B3A2),ˉB3=B3,

    and ˉP3(X,Y) is a function of three or more degrees about (X,Y).

    Let τ=ˉa12t. For convenience, t is still used to represent τ, then it has

    {dx1dt=y1+˜A1x21+˜A2x1y1,dy1dt=˜B1x21+˜B2x1y1+˜B3y21+˜P3(x1,y1), (3.6)

    where

    ˜Ai=ˉAiˉa12,i=1,2,˜Bj=ˉBjˉa12,j=1,2,3.

    and ˜P3(X,Y) is a function of (X,Y) with three or higher degree. Model (3.6) can be transformed into the following form[43]:

    {dx1dt=y1,dy1dt=˜B1x21+(˜B2+2˜A1)x1y1+˜P3(x1,y1),

    and if ˜B10, then E is a cusp point.

    Meanwhile, if ˜B2+2˜A10, E is a cusp of codimension two. If ˜B2+2˜A1=0, E is a cusp with at least codimension three.

    b) If TR(J(E))=a11+a220, then λ1=0, λ20. Make the transformation

    (XY)=(a22a11a21a21)(x1y1),

    system (3.5) is converted into the following standard form

    {dx1dt=A1x21+A2x1y1+Ay21,dy1dt=a22y1+B1x21+B2x1y1+B3y21+P3(x1,y1),

    where

    A1=a21a222A1a221a22A2a11a222B1+a11a21a22B2a11a221B3a21(a11+a22),A2=2a211a22+a221a22A2a11a221A22a211a22B1a11a21a22B2+a11a221B2+2a11a221B3a21(a11+a22),A3=a211a21A1+a11a221A2a311B1a211a21B2a11a221B3a21(a11+a22),B1=a21a222A1a221a22A2+a322B1a21a222B2+a221a22B3a21(a11+a22),B2=2a11a21a22A2+a221a22A2a11a221A2+2a11a222B1+a21a222B2a11a21a22B22a221a22B3a21(a11+a22),B3=a211a21A1+a11a221A2+a11a222B1+a11a21a22B2+a221a22B3a21(a11+a22).

    Next, introduce a new variable τ=a22t (for convenience, is represented by t), then

    {dx1dt=ˆA1x21+ˆA2x1y1+ˆA3y21,dy1dt=y1+ˆB1x21+ˆB2x1y1+ˆB3y21+ˆP3(x1,y1),

    where, ˆAi=Aia22,ˆBi=Bia22,i=1,2,3.

    If ˆA10 then E is unstable. Meanwhile, E is a saddle node of attraction.

    To sum up, the following result hold.

    Theorem 3. For Model (2.2), 1) O(0,0) is unstable, E1(0,1f(s1d1d1)) is locally stable, E2(β,0), E3(1,0) is unstable; 2) For the interior equilibrium, when ρ>0, E1(x1,y1) is a saddle (unstable), and E2(x2,y2) is a stable node; when ρ=0, if TR(J(E))=0 and ˜B10, E(x,y) is a cusp of codimension two in case of ˜B2+2˜A10, and a cusp with at least codimension three in case of ˜B2+2˜A1=0; If TR(J(E))0 and ˆA10, E(x,y) is an attractive saddle node.

    Let α=α0 satisfy

    DET(J(E))=(λ1λ2)|(x,y)=0,TR(J(E))=(λ1+λ2)|(x,y)0.

    Denote

    ξ1=2(x)2(1+β)x,ξ2=γy(1+δx)2,ω1=αx,ω2=s1fy(1+fy)2

    and define

    Φ2Δ=2ω21ξ2x2γδy(1+δx)3ω21ξ12s1f2(1+fy)3ξ31.

    Theorem 4. Let the parameters of Model (2.2) satisfy TR(J(E))0 and ˆA10. If Φ20, system (2.2) undergoes a saddle node bifurcation near E(x,y) when α=α0.

    Proof. At E(x,y), there is

    J(E)=(ξ1ω1ξ2ω2).

    Let V=(V1,V2)T (W=(W1,W2)T) be the eigenvector corresponding to the zero eigenvalue of J(E) ((J(E))T). Then V=(ω1,ξ1)T, W=(ξ2,ξ1)T. Let g=(g1,g2)T. Then

    gα(E,α0)=gα(E,α0)=(y0).

    and

    D2g(E,α0)(V,V)=(2g1x2V21+22g1xyV1V2+2g1y2V222g2x2V21+22g2xyV1V2+2g2y2V22)=(2ω212γδ(1+δx)3ω212s1f2(1+fy)3ξ21).

    Since

    Φ1=WTgα(E,α0)=xξ10

    and

    Φ2=WT[D2g(E,α0)(V,V)]=2ω21ξ22γδ(1+δx)3ω21ξ12s1f2(1+fy)3ξ310,

    then according to Sotomayor's theorem[44], Model (2.2) undergoes a saddle-node bifurcation near E(x,y) when α=α0.

    Remark 1. For Model (2.2), it can be concluded that for a lager α, the interior equilibrium does not exist. When α decreases to α=α0, a unique equilibrium exists in the system. As α decreases below α0, the system undergoes a saddle-node bifurcation at the interior equilibrium E, giving rise to two interior equilibria E1 and E2.

    According to Theorem 3, when ρ=0, TR(J(E))=0, ˜B10 and ˜B2+2˜A10, Model (2.2) to undergo a Bogdanov-Takens bifurcation of codimension two near E when (α,β)=(α0,β0). Next, it will show the universal unfolding of the Bodmanov-Takens bifurcation of codimension two under parameter perturbation when α and β are taken as bifurcation parameters.

    Define

    b01=x(1x)(xϵ2)ϵ1xy,b11=[1+(β+ϵ2)]x2(x)2,b12=(α+ϵ1)x,b10=0,b21=γy(1+δx)2,b21=s1fy(1+fy)2,E1=(β+ϵ2)+13x,E2=(α+ϵ1)2,F1=γδy(1+δx)3,F2=γ(1+δx)2,F3=s1f(1+fy)3,H1=b212b10+b01b12b22b201F3b12,H2=b201E22+b212b10E22+b312b21b11b212b22+b01b212F2b212,H3=b11b12b01E2+b12b222b10F3b12,H4=b01b12E1E2b01b11E22b212b22E1b212F1+b11b212F2b211b12F3b212,H5=2b212E21b11b12E22b01E22+b212F22b11b12F3b212,H6=E2+F3b12,J1=H1,J2=H22H1H6,J3=H3J4=H1H262H2H6+H4,J5=H5H3H6,
    K1=J1J4,K2=J2J4,K3=J3J4,K4=J5J4,L1=K1+K224,L2=K3+K2K42,L3=K4,O1=L1L43,O2=L2L3,

    Let (ϵ1,ϵ2) be a parameter vector in a small neighborhood of (0,0). Then

    Theorem 5. Let the parameters of Model (2.2) satisfy ρ=0, TR(J(E))=0, ˜B10, ˜B2+2˜A10 and |(O1,O2)(ϵ1,ϵ2)|0. When (α,β) varies in the neighborhood of (α0,β0), Model (2.2) changes in the small neighborhood of E(x,y), and a codimensional 2 Bogdanov-Takens branching occurs.

    Proof. Consider the perturbation system

    {dxdt=x(1x)[x(γ+ϵ2)](α+ϵ1)xyF(x,y),dydt=γxy1+δx+(s11+fyd1)yG(x,y),

    For (α,β)=(α0,β0), there is DET(J(E))=0,TR(J(E))=0. With the transformation x1=xx, y1=yy, we can obtain

    {dx1dt=b01+b11x1+b12y1+E1x21+E2x1y1,dy1dt=b10+b21x1+b22y1+F1x21+F2x1y1+F3y21+N3(x1,y1), (4.1)

    where N3(x1,y1,ϵ1,ϵ2)C.

    Make the following transformation:

    {x2=x1,y2=b01+b11x1+b12y1+E1x21+E2x1y1,

    then Model (4.1) is converted to

    {dx2dt=y2,dy2dt=H1+H2x2+H3y2+H5x22+H5x2y2+H6y22+N3(x2,y2,ϵ1,ϵ2),

    where N3(x2,y2,ϵ1,ϵ2)C with coefficients depending smoothly on ϵ1,ϵ2.

    Next, introduce the variable τ, denoted as dt=(1H6x2)dτ (still denote τ as t), then

    {dx2dt=(1H6x2)y2,dy2dt=(1H6x2)(H1+H2x2+H3y2+H4x22+H5x2y2+H6y22+N3(x2,y2,λ1,λ2)),

    Let x3=x2,y3=(1H6x2)y2, then the above system of equations is transformed into

    {dx3dt=y3,dy3dt=J1+J2x3+J3y3+J4x23+J5x3y3+˜N3(x3,y3,ϵ1,ϵ2),

    where ˜N3(x3,y3,ϵ1,ϵ2)C with coefficients depending smoothly on ϵ1,ϵ2.

    (ⅰ) When J4<0, the following transformations are applied to the variables:

    x4=x3,y4=y3J4,τ=J4t,

    still denote τ as t, there is

    {dx4dt=y4,dy4dt=K1+K2x4+K3y4x24+K4x4y4+M3(x4,y4,ϵ1,ϵ2),

    where M3(x4,y4,ϵ1,ϵ2)C with at least third order.

    Let x5=x4K22, y5=y4, and obtain

    {dx5dt=y5,dy5dt=L1+L2y5x25+L3x5y5+M3(x5,y5,ϵ1,ϵ2),

    where M3(x5,y5,ϵ1,ϵ2)C with at least third order.

    If set J50, then L30. Define new variables: x6=L23x5, y6=L33y5, τ=tL3, and denote x6 by x, y6 by y, and τ by t, which yields that

    {dxdt=y,dydt=O1+O2y+x2+xy+˜M3(x,y,λ1,λ2), (4.2)

    where ˜N3(x2,y2,ϵ1,ϵ2)C with at least third order, and O1, O2 can be represented by ϵ1 and ϵ2.

    (ⅱ) When J4>0, the following transformations are applied

    x4=x3,y4=y3J4,τ=J4t,

    still denote τ by t, it has

    {dx4dt=y4,dy4dt=K1+K2x4+K3y4+x24+K4x4y4+N3(x4,y4,ϵ1,ϵ2),

    where N3(x4,y4,ϵ1,ϵ2)C with at least third order, and

    K1=J1J4,K2=J2J4,K3=J3J4,K4=J5J4.

    Let x5=x4+K2/2,y5=y4. Then it has

    {dx5dt=y5,dy5dt=L1+L2y5+x25+L3x5y5+N3(x5,y5,ϵ1,ϵ2),

    where N3(x5,y5,ϵ1,ϵ2)C with at least third order, and

    L1=K1K224,L2=K3K2K42,L3=K4.

    If set J50, then L30. Define new variables: x6=L32x5, y6=L33y5, τ=t/L3, and still denote x6 by x, y6 by y, τ by t, which yields that

    {dxdt=y,dydt=O1+O2y+x2+xy+˜N3(x2,y2,ϵ1,ϵ2), (4.3)

    where O1=L1L34,O2=L2L3, and ϵ1, ϵ2 can be represented by O1, O2.

    For convenience of discussion, O1, O2 is still denoted by O1,O2. When |(O1,O2)(ϵ1,ϵ2)|0, Models (4.2) and (4.3) are the cardinal folds of the Bogdanov-Takens bifurcation[44], when (α,β) varies in the vicinity of (α0,β0), Model (2.2) undergoes a codimension 2 Bogdanov-Takens bifurcation in a small neighborhood of E(x,y).

    It only focuses on the case of ρ<0 in Theorem 3. For Model (2.3), there are

    M1={(x,y)R2+:x=h1,y0},M2={(x,y)R2+:x=h2,y0},
    N1={(x,y)R2+:x=h1+η,y0},N2={(x,y)R2+:x=(1a)h2,yτ}.

    Denote l1=1α(1x)(xβ) as the prey isocline, l2=(s1d1)(1+δx)γxf[d1+(d1δγ)x] as the predator's isocline, {l_3} , {l_4} , {l_5} , {l_6} as the saddle point separatrix of E^*_1 in different directions. The intersection point between l_1 and \mathcal{N}_2 is denoted by A_3 . The intersection point between l_4 and \mathcal{N}_2 is denoted by M . The trajectory from A_3 intersects \mathcal{M}_2 at the point {B_3} . Define \overline{\tau}_2\triangleq y_{M}-(1-b)y_{B_3} . Denote

    {\Omega _1} = \left\{ {\left( {x, y} \right)\left| {0 \le x \le {x^*_{1}}, y \ge 0} \right.} \right\}, {\Omega _2} = \left\{ {\left( {x, y} \right)\left| {{x^*_{1}} \le x \le 1, y \ge 0} \right.} \right\}.

    Definition 2 (Successor function). For a point S\in \mathcal{N}_1 , if the trajectory from S intersects \mathcal{M}_1 , then denote the intersection point by S^{-}\in \mathcal{M}_1 . Under the impulse effect, the point S^{-} is mapped to S^{+}\in \mathcal{N}_1 . In such a case, we can define f^{\text{I}}_{\text{SOR}_1} : \mathcal{N}_1\rightarrow R , S\rightarrow f^{\text{I}}_{\text{SOR}_1}(S)\triangleq y_{S^{+}}-y_{S} . If the trajectory from S intersects \mathcal{M}_2 , then denote the intersection point by S^{-} . Under the impulse effect, the point S^{-} is mapped to S^{+}\in \mathcal{N}_2 . If the trajectory from S^{+} intersects \mathcal{M}_1 , denote the intersection point by S^{+-}\in \mathcal{M}_1 . Under the impulse effect, the point S^{+-} is mapped to S^{++}\in \mathcal{N}_1 . In such cases, we can define f^{\text{II}}_{\text{SOR}_1} : \mathcal{N}_1\rightarrow R , S\rightarrow f^{\text{II}}_{\text{SOR}_1}(S)\triangleq y_{S^{++}}-y_{S} . Similarly, For a point S'\in \mathcal{N}_2 , define f^{\text{I}}_{\text{SOR}_2} : \mathcal{N}_2\rightarrow R , S'\rightarrow f^{\text{I}}_{\text{SOR}_2}(S')\triangleq y_{S'^{+}}-y_{S'} .

    Theorem 6. For system (2.3) with model parameters satisfying any one of C_1) C_3) and \rho > 0 , if D-1) {h_1} + \eta < {x^*_{1}} < \left({1 - a} \right){h_2} < {h_2} < {x^*_{2}} , y_2(h_1)\geq y_1(h_1+\eta) and \tau < \overline{\tau}_2 , an order-1 periodic solution exists in each \Omega_i , i = 1, 2 ; if D-2) {h_1} < \left({1 - a} \right){h_2} < {h_1} + \eta < {x^*_{1}} < {h_2} < {x^*_{2}} and y_2(h_1)\geq y_1(h_1+\eta) , an order-1 periodic solution exists in \Omega_1 ; if D-3) {h_1} < {x^*_{1}} < {h_1} + \eta < \left({1 - a} \right){h_2} < {h_2} < {x^*_{2}} and \tau < \overline{\tau}_2 , an order-1 periodic solution exists in \Omega_2 .

    Proof. For case D-1) {h_1} + \eta < {x^*_{1}} < \left({1 - a} \right){h_2} < {h_2} < {x^*_{2}} and y_2(h_1)\geq y_1(h_1+\eta) , denote the intersection point between l_2 and \mathcal{N}_1 by A_0 . Select a point A_1\in \mathcal{N}_1 above A_0 ; the trajectory from A_1 intersects \mathcal{M}_1 at B_1 . Under the impulse effect, the point B_1 is mapped to A^{+}_1\in \mathcal{N}_1 . Since g_1(x_{A_1}, y_{A_1}) < 0 , g_2(x_{A_1}, y_{A_1}) < 0 , then we have f^{\text{I}}_{\text{SOR}_1}(A_1) = y_{A^{+}_1}-y_{A_1} = y_{B_1}-y_{A_1} < 0 . Besides, denote the intersection between l_1 and \mathcal{N}_1 by A_2 . Since g_1(x_{A_2}, y_{A_2}) = 0 , g_2(x_{A_2}, y_{A_2}) > 0 and g_1(x_S, y_S) < 0 for S\in U(A_1, \epsilon) with x_S < h_1+\eta and y_S < y_{A_1} , then we have f^{\text{I}}_{\text{SOR}_1}(A_2) = y_{A^{+}_2}-y_{A_2} = y_{B_2}-y_{A_2} > 0 due to y_2(h_1)\geq y_1(h_1+\eta) . The continuity of f^{\text{I}}_{\text{SOR}_1} implies that a point S\in \overline{A_1A_2} exists so that f^{\text{I}}_{\text{SOR}_1}(S) = 0 , i.e., an order-1 periodic solution exists in \Omega_1 (Figure 3(a)). Similarly, it can be proved that an order-1 periodic solution exists in \Omega_1 for case D-2).

    Figure 3.  Schematic diagram of the trajectory trend of the system's (2.3) for the case of D-1).

    Under the impulse effect, {B_3} is mapped to {A_3}^{+}\in{\mathcal{N}_2} , where {y_{{A_3}^ + }} = \left({1 - b} \right){y_{{B_3}}} + \tau . Define {\overline{\tau}_1} \triangleq {y_{{A_3}}} - \left({1 - b} \right){y_{{B_3}}} .

    ⅰ) If \tau = \overline{\tau}_1 , then f^{\text{I}}_{\text{SOR}_2}({A_3}) = {y_{{A_3}^ + }} - {y_{{A_3}}} = 0 , i.e., an order-1 periodic solution exists in \Omega_1 (Figure 3(a)).

    ⅱ) If \tau < \overline{\tau}_1 , then f^{\text{I}}_{\text{SOR}_2}({A_3}) = {y_{{A_3}^ + }} - {y_{{A_3}}} < 0 . On the other hand, for {A_4}((1-a)h_2, \tau) , there is f^{\text{I}}_{\text{SOR}_2}({A_4}) = {y_{{A_4}^ + }} - {y_{{A_4}}} > 0 . The continuity of f^{\text{I}}_{\text{SOR}_2} implies that a point S'\in \overline{A_3A_4} exists so that f^{\text{I}}_{\text{SOR}_2}(S') = 0 , i.e., an order-1 periodic solution exists in \Omega_2 (Figure 3(b)).

    ⅲ) If \overline{\tau}_1 < \tau < \overline{\tau}_2 , then f^{\text{I}}_{\text{SOR}_2}({A_3}) = {y_{{A_3}^ + }} - {y_{{A_3}}} > 0 . According to the trend of the trajectory and the fact that any two trajectories cannot be intersected, select D\in \mathcal{N}_2 above and sufficiently close to the {A_3} , then f^{\text{I}}_{\text{SOR}_2}(D) = {y_{{D^ + }}} - {y_D} > 0 . On the other hand, for C = {A^{+}_3} , there is f^{\text{I}}_{\text{SOR}_2}(C) = {y_{{C^ + }}} - {y_C} < {y_{{A^{+}_{3}}}} - {y_C} = 0 . The continuity of f^{\text{I}}_{\text{SOR}_2} implies that a point S'\in \overline{CD} exists so that f^{\text{I}}_{\text{SOR}_2}(S') = 0 , i.e., an order-1 periodic solution exists in \Omega_2 (Figure 3(c)).

    Similarly, system (2.3) has an order-1 periodic solution in \Omega_2 for case D - 3 ).

    To sum up, an order-1 periodic solution exists in Model (2.3) for case D-1).

    Let \mathbf{z}_i(t) = (\phi_i(t), \varphi_i(t)) (k-1)T_i\leq t\leq kT_i , k\in \mathbb{N} be the order-1 periodic solution in \Omega_i , i = 1, 2 . For \mathbf{z}_1(t) = (\phi_1(t), \varphi_1(t)) (k-1)T_1\leq t\leq kT_1 , denote

    \phi_1\left( {T_1^ + } \right) = \phi_1\left( {{T_1}} \right) + \eta = {h_1} + \eta, \varphi_1\left( {T_1^ + } \right) = \varphi_1\left( {{T_1}} \right) = {\delta_1}.

    For \mathbf{z}_2(t) = (\phi_2(t), \varphi_2(t)) (k-1)T_2\leq t\leq kT_2 , denote

    \phi_2\left( {T_2^ + } \right) = \left( {1 - a} \right)\phi_2 \left( {{T_2}} \right) = \left( {1 - a} \right){h_2}, \varphi_2 \left( {T_2^ + } \right) = \left( {1 - b} \right)\varphi_2 \left( {{T_2}} \right) + \tau = {\delta_2}.

    Theorem 7. For the model parameters with any one of C_1) C_3) , \rho > 0 and D-1), \mathbf{z}_i(t) = (\phi_i(t), \varphi_i(t)) (k-1)T_i\leq t\leq kT_i is orbitally asymptotically stable if \mu_i < 1 , where

    \begin{array}{l} \mu_1 = \left| {\frac{{\left[ {1 - \left( {{h_1} + \eta } \right)} \right]\left[ {\left( {{h_1} + \eta } \right) - \beta } \right] - \alpha {\delta _1}}}{{\left( {1 - {h_1}} \right)\left( {{h_1} - \beta } \right) - \alpha {\delta _1}}}} \right| \exp \left\{ {\int_{{0^ + }}^{{T_1}} {\left[ {\phi_1(t)\left( {\beta + 1 - 2\phi_1(t)} \right) - \frac{{{s_1}f\varphi_1(t)}}{{{{\left( {1 + f\varphi_1(t)} \right)}^2}}}} \right]} dt} \right\}, \\ \mu_2 = \left| {\frac{{\left\{ {\left[ {1 - \left( {1 - a} \right){h_2}} \right]\left[ {\left( {1 - a} \right){h_2} - \beta } \right] - \alpha \delta_2} \right\}{(\delta _2-\tau)}}}{{\delta _2\left[ {\left( {1 - b} \right)\left( {1 - {h_2}} \right)\left( {{h_2} - \beta } \right) - \alpha \left( {{\delta _2} - \tau } \right)} \right]}}} \right|\Lambda\left( t \right) \end{array}

    with \Lambda\left(t \right) = \exp \left\{ {\int_{{0^ + }}^{{T_2}} {\left[{\phi_2(t)\left({\beta + 1 - 2\phi_2(t)} \right) -\frac{{{s_1}f\varphi_2(t)}}{{{{\left({1 + f\varphi_2(t)} \right)}^2}}}} \right]} dt} \right\} .

    Proof. For system (2.3), there are

    {f_1}(x, y) = x(1 - x)(x - \beta ) - \alpha xy, {f_2}(x, y) = \frac{{\gamma xy}}{{1 + \delta x}} + \left( {\frac{{{s_1}}}{{1 + fy}} - {d_1}} \right)y,

    and {\omega_1}\left({x, y} \right) = x - {h_1}, {I_{11}}\left({x, y} \right) = \eta, {I_{21}}\left({x, y} \right) = 0 , So it can be concluded that

    \frac{{\partial {f_1}\left( {x, y} \right)}}{{\partial x}} = \frac{{\dot x}}{x} + x\left( {\beta + 1 - 2x} \right), \frac{{\partial {f_2}\left( {x, y} \right)}}{{\partial x}} = \frac{{\dot y}}{y} - \frac{{{s_1}fy}}{{{{\left( {1 + fy} \right)}^2}}},
    \frac{{\partial {\omega_1}\left( {x, y} \right)}}{{\partial x}} = 1, \frac{{\partial {\omega_1}\left( {x, y} \right)}}{{\partial y}} = \frac{{\partial {I_{11}}\left( {x, y} \right)}}{{\partial x}} = \frac{{\partial {I_{11}}\left( {x, y} \right)}}{{\partial y}} = \frac{{\partial {I_{21}}\left( {x, y} \right)}}{{\partial x}} = \frac{{\partial {I_{21}}\left( {x, y} \right)}}{{\partial y}} = 0.

    Denote {f_{{1^ + }}}\triangleq {f_1}\left({\phi_1\left({T_1^ + } \right), \varphi_1\left({T_1^ + } \right)} \right), {f_{{2^ + }}}\triangleq {f_2}\left({\phi_1\left({T_1^ + } \right), \varphi_1 \left({T_1^ + } \right)} \right) . Then by Lemma 1, there are

    \begin{align*} {\kappa_1} & = \frac{{\left( { \frac{{\partial {I_{21}}}}{{\partial x}} \cdot \frac{{\partial {\omega_1}}}{{\partial x}} - \frac{{\partial {\beta_1}}}{{\partial x}} \cdot \frac{{\partial {\omega_1}}}{{\partial y}} + \frac{{\partial {\omega_1}}}{{\partial x}}} \right){f_{{1^ + }}} + \left( { \frac{{\partial {I_{11}}}}{{\partial x}} \cdot \frac{{\partial {\omega_1}}}{{\partial y}} - \frac{{\partial {I_{11}}}}{{\partial y}} \cdot \frac{{\partial {\omega_1}}}{{\partial x}} + \frac{{\partial {\omega_1}}}{{\partial y}}} \right){f_{{2^ + }}}}}{{ \frac{{\partial {\omega_1}}}{{\partial x}}{f_1} + \frac{{\partial {\omega_1}}}{{\partial y}}{f_2}}}\\ & = \frac{{\left( {{h_1} + \eta } \right)\left[ {1 - \left( {{h_1} + \eta } \right)} \right]\left[ {\left( {{h_1} + \eta } \right) - \beta } \right] - \alpha \left( {{h_1} + \eta } \right){\delta _1}}}{{{h_1}\left( {1 - {h_1}} \right)\left( {{h_1} - \beta } \right) - \alpha {h_1}{\delta _1}}} \end{align*}

    and

    \begin{align*} \mu_1& = |{\kappa_1}|\exp \left[ {\int_{{0^ + }}^{{T_1}} {\left( {\frac{{\partial {f_1}}}{{\partial x}} + \frac{{\partial {f_2}}}{{\partial y}}} \right)dt} } \right]\\ & = \left|\frac{{\left[ {1 - \left( {{h_1} + \eta } \right)} \right]\left[ {\left( {{h_1} + \eta } \right) - \beta } \right] - \alpha {\delta _1}}}{{\left( {1 - {h_1}} \right)\left( {{h_1} - \beta } \right) - \alpha {\delta _1}}}\right|\exp \left\{ {\int_{{0^ + }}^{{T_1}} {\left[ {\phi_1(t)\left( {\beta + 1 - 2\phi_1(t)} \right) - \frac{{{s_1}f\varphi_1(t)}}{{{{\left( {1 + f\varphi_1(t)} \right)}^2}}}} \right]dt} } \right\}. \end{align*}

    If \mu_1 < 1 , the order-1 periodic solution \mathbf{z}_1(t) = (\phi_1(t), \varphi_1(t)) (k-1)T_1\leq t\leq kT_1 is orbitally asymptotically stable.

    Similarly, for the order-1 periodic solution \mathbf{z}_2(t) = (\phi_2(t), \varphi_2(t)) (k-1)T_2\leq t\leq kT_2 , there is

    \mu_2 = \left| {\frac{{\left\{ {\left[ {1 - \left( {1 - a} \right){h_2}} \right]\left[ {\left( {1 - a} \right){h_2} - \beta } \right] - \alpha \delta_2} \right\}{(\delta _2-\tau)}}}{{\delta _2\left[ {\left( {1 - b} \right)\left( {1 - {h_2}} \right)\left( {{h_2} - \beta } \right) - \alpha \left( {{\delta _2} - \tau } \right)} \right]}}} \right|\Lambda\left( t \right),

    where \Lambda\left(t \right) \triangleq \exp \left\{ {\int_{{0^ + }}^{{T_2}} {\left[{\phi_2(t)\left({\beta + 1 - 2\phi_2(t)} \right) - \frac{{{s_1}f\varphi_2(t)}}{{{{\left({1 + f\varphi_2(t)} \right)}^2}}}} \right]} dt} \right\} . By Lemma 1, if \mu_2 < 1 , the order-1 periodic solution \mathbf{z}_2(t) = (\phi_2(t), \varphi_2(t)) (k-1)T_2\leq t\leq kT_2 is orbitally asymptotically stable. The proof is completed.

    Here only the case for {h_1} < \left({1 - a} \right){h_2} < {x^*_{1}} < {h_1} + \eta < {h_2} < {x^*_{2}} is considered. Let {G_1}\left({{h_1} + \eta, {y_{{G_1}}}} \right) be the intersection point between {l_1} and {\mathcal{N}_1} , {G_2}\left({{h_1}, {y_{{G_2}}}} \right) be the intersection point between {l_2} and {\mathcal{M}_1} . The trajectory that starts with {G_1} intersects {\mathcal{M}_2} at {B_1}\left({{h_2}, {y_{{B_1}}}} \right) . Define \overline{\tau}_3\triangleq y_{G_1}-(1-b)y_{B_1} .

    Theorem 8. For the model parameters with any one of C_1) - C_3) , \rho > 0 and D-4) {h_1} < \left({1 - a} \right){h_2} < {x^*_{1}} < {h_1} + \eta < {h_2} < {x^*_{2}} , if by_{G_2}\leq \tau\leq \min\{y_{G_2}, \overline{\tau}_3\} holds, an order-2 periodic solution exists in system (2.3).

    Proof. For {G_1}\in \mathcal{N}_1 , the trajectory starting from {G_1} intersects with {\mathcal{M}_2} at {B_1} . Then {B_1} is mapped to {B''_1}\in{\mathcal{N}_2} , and next intersects with \mathcal{M}_1 at \hat{B}_1 , and then \hat{B}_1 is mapped to G^{+}_1 under impulse effect. Since \tau < \overline{\tau}_3 , then {y_{B''_1}} = \left({1 - b} \right){y_{{B_1}}} + \tau < {y_{{G_1}}} . Since g_1(B''_1) < 0 and g_2(B''_1) < 0 , it has {y_{{{\hat B}_1}}} < {y_{{{B''}_1}}} , then {y_{G_1^ + }} = y_{\hat{B}_1} < y_{B''_1} = \left({1 - b} \right){y_{{B_1}}} + \tau < {y_{{G_1}}} , i.e., f^{\text{II}}_{\text{SOR}1}({G_1}) = {y_{G_1^ + }} - {y_{{G_1}}} < 0 .

    On the other hand, since \tau\leq y_{G_2} , then G_2\in \mathcal{N}_1 . The trajectory starting from A_1\in \mathcal{N}_1 with y_{A_1} = y_{G_2} intersects with \mathcal{M}_2 at A^{-}_1 , and then A^{-}_1 is mapped to A^{+}_1 . Then it intersects with \mathcal{M}_1 at A^{+-}_1 , and next A^{+-}_1 is mapped to A^{++}_1 . Since y_{A^{+}_1} = (1-b)y_{A^{-}_1}+\tau > (1-b)y_{G_2}+\tau > y_{G_2} , so f^{\text{II}}_{\text{SOR}1}({A_1}) = y_{A^{++}_1} - y_{A_1} = y_{A^{+-}_1}-y_{G_2} > 0 .

    The continuity of f^{\text{II}}_{\text{SOR}_1} implies that a point S\in \overline{A_1G_1}\subset \mathcal{N}_1 exists so that f^{\text{II}}_{\text{SOR}_1}(S) = 0 , i.e., an order-2 periodic solution exists (Figure 4(a)).

    Figure 4.  Schematic diagram of the trajectory trend of the system's (2.3) for case D-4).

    Similarly, a point S'\in \overline{A_2B2}\subset \mathcal{N}_2 exists so that f^{\text{II}}_{\text{SOR}_2}(S') = 0 , i.e., an order-2 periodic solution exists (Figure 4(b)).

    Let \mathbf{z}_{3}(t) = (\phi_3(t), \varphi_3(t)) (k-1)(T_1+T_2)\leq t\leq k(T_1+T_2) , k\in \mathbb{N} be the order-2 periodic solution. Denote

    \phi_3(0) = {h_1} + \eta, \phi_3(T_1) = {h_2}, \phi_3(T_1^+) = (1 - a)h_2, \phi_3(T_1 + T_2) = {h_1}, \phi_3((T_1 + T_2)^+) = {h_1} + \eta,
    \varphi_3(0) = {\delta_3}, \varphi_3(T_1) = \delta_4, \varphi_3(T_1^+) = (1-b)\delta_4 + \tau, \varphi_3(T_1 +T_2) = \delta_3, \varphi_3((T_1+T_2)^+) = \delta_3.

    Define

    \Theta_0\triangleq \frac{\delta_4}{(1-b)\delta_4+\tau}\frac{\gamma_1\gamma_3}{\gamma_2\gamma_4},

    where

    \begin{array}{l} \gamma_1\triangleq (1-h_1-\eta)(h_1+\eta-\beta)-\alpha\delta_3, \gamma_2\triangleq (1-h_2)(h_2-\beta)-\alpha\delta_4, \\ \gamma_3\triangleq (1-(1-a)h_2)((1-a)h_2-\beta)-\alpha((1-b)\delta_4+\tau), \gamma_4\triangleq (1-h_1)(h_1-\beta)-\alpha\delta_3. \end{array}

    Theorem 9. For the model parameters with any one of C_1) C_3) , \rho > 0 and D-4) {h_1} < \left({1 - a} \right){h_2} < {x^*_{1}} < {h_1} + \eta < {h_2} < {x^*_{2}} , and by_{G_2}\leq \tau\leq \min\{y_{G_2}, \overline{\tau}_3\} , if

    \mu_3 = \Theta_0 \exp \left\{ {\int_{{0^ + }}^{{T_1} + {T_2}} {\left[ {\phi_3(t)\left( {\beta + 1 - 2\phi_3(t)} \right) -\frac{{{s_1}f\varphi_3(t)}}{{{{\left( {1 + f\varphi_3(t)} \right)}^2}}}} \right]} dt} \right\} < 1,

    then \mathbf{z}_{3}(t) = (\phi_3(t), \varphi_3(t)) (k-1)(T_1+T_2)\leq t\leq k(T_1+T_2) is orbitally asymptotically stable.

    Proof. For convenience, denote the intersection point between \mathbf{z}_{3}(t) = (\phi_3(t), \varphi_3(t)) and \mathcal{N}_1 ( \mathcal{M}_2 , \mathcal{N}_2 , \mathcal{M}_1 ) by L_1(h_1+\eta, \delta_3) ( L_2(h_2, \delta_4) , L_3((1-a)h_2, (1-b)\delta_4+\tau) , L_4(h_1, \delta_3) ). Then, according to analogue of Poincaré Criterion, there are

    {\kappa_1} = \frac{f_1(L_3)}{f_1(L_2)} = \frac{(1-a)h_3[(1-(1-a)h_2)((1-a)h_2-\beta)-\alpha((1-b)\delta_4+\tau)]}{h_2[(1-h_2)(h_2-\beta)-\alpha\delta_4]},
    {\kappa_2} = \frac{f_1(L_1)}{f_1(L_4)} = \frac{(h_1+\eta)[(1-h_1-\eta)(h_1+\eta-\beta)-\alpha\delta_3]}{h_1[(1-h_1)(h_1-\beta)-\alpha\delta_3}

    and

    \begin{array}{ll} \int_{{0^ + }}^{{T_1} + {T_2}} {\left( {\frac{{\partial {f_1}}}{{\partial x}} + \frac{{\partial {f_2}}}{{\partial y}}} \right)dt} & = \ln \left( {\frac{{{h_2}}}{{{h_1} + \eta }}} \right) + \ln \left( {\frac{{{\delta_4}}}{{{\delta_{3}}}}} \right) + \ln \left( {\frac{{{h_1}}}{{\left( {1 - a} \right){h_2}}}} \right) + \ln\left(\frac{\delta_3}{(1-b)\delta_4+\tau}\right)\\ & + \int_{{0^ + }}^{{T_1} + {T_2}} {\left[ {\phi_3(t)\left( {\beta + 1 - 2\phi_3(t)} \right) - \frac{{{s_1}f\varphi_3(t)}}{{{{\left( {1 + f\varphi_3(t)} \right)}^2}}}} \right]dt}. \end{array}

    Then

    \begin{array}{ll} \mu_3& = |\kappa_1\kappa_2|\exp\left(\int_{{0^ + }}^{{T_1} + {T_2}} {\left( {\frac{{\partial {f_1}}}{{\partial x}} + \frac{{\partial {f_2}}}{{\partial y}}} \right)dt}\right)\\ & = \frac{\delta_4}{(1-b)\delta_4+\tau}\frac{\gamma_1\gamma_3}{\gamma_2\gamma_4}\exp\left(\int_{{0^ + }}^{{T_1} + {T_2}} {\left[ {\phi_3(t)\left( {\beta + 1 - 2\phi_3(t)} \right) - \frac{{{s_1}f\varphi_3(t)}}{{{{\left( {1 + f\varphi_3(t)} \right)}^2}}}} \right]dt}\right). \end{array}

    If \mu < 1 , then \mathbf{z}_{3}(t) = (\phi_3(t), \varphi_3(t)) (k-1)(T_1+T_2)\leq t\leq k(T_1+T_2) is orbitally asymptotically stable.

    For system (2.2) with model parameters \alpha = 0.09 , \beta = 0.15 , \gamma = 0.6 , \delta = 0.9 , {s_1} = 1.12 , {d_1} = 1.1 , f = 0.21 , there is \rho < 0 , then two interior equilibria exist in the system, and the phase diagram is presented in Figure 5(a); For the model parameters \alpha = 0.219 , \beta = 0.051 , \gamma = 0.6 , \delta = 0.9 , {s_1} = 1.12 , {d_1} = 1.1 , f = 0.21 , there is \rho = 0 , then a unique equilibrium exists in system (2.2), which is a sharp point. The phase diagram of the system (2.2) for such a case is presented in Figure 5(b). Moreover, system (2.2) undergoes a Bogdanov-Takens bifurcation of codimension two in a very small neighborhood of the unique interior equilibrium.

    Figure 5.  Illustration of the trajectory trend in system (2.2) for different parameters.

    While for system (2.2) with model parameters \alpha = 0.09 , \beta = 0.29 , \gamma = 0.6 , \delta = 0.9 , {s_1} = 1.12 , {d_1} = 1.1 , f = 0.21 , there is also \rho = 0 , in such case a unique equilibrium exists in system (2.2), which is a saddle node, and the phase diagram is presented in Figure 5(c). System (2.2) undergoes a saddle-node bifurcation of codimension one in a very small neighborhood of the interior equilibrium.

    For system (2.2) with model parameters \alpha = 0.09 , \beta = 0.35, \gamma = 0.6 , \delta = 0.9 , {s_1} = 1.12 , {d_1} = 1.1 , f = 0.21 , there is \rho > 0 . In such a case, system (2.2) does not have interior equilibrium, and the phase diagram of the system (2.2) is presented in Figure 5(d).

    For system (2.2) with parameters \beta = 0.051 , \gamma = 0.6 , \delta = 0.9 , {s_1} = 1.12 , {d_1} = 1.1 , f = 0.21 , the bifurcation diagrams of the residual dimension 1 for the system (2.2) are shown in Figure 6(a), (b) when \alpha is selected as the bifurcation parameter. The result shows that for larger \alpha , the interior equilibrium does not exist. When \alpha decreases to \alpha = \alpha_0 , a unique equilibrium exists in the system. As \alpha decreases below \alpha_0 , the system undergoes a saddle-node bifurcation at the interior equilibrium {E^*} , giving rise to two interior equilibria {E^*_{{1}}} and {E^*_{{2}}} .

    Figure 6.  Schematic diagram of system (2.3) bifurcation when \alpha is selected as a key parameter.

    For system (2.3) with model parameters \alpha = 0.1 , \beta = 0.15 , \gamma = 0.6 , \delta = 0.9 , {s_1} = 1.12 , {d_1} = 1.1 , f = 0.21 , the control parameters: {h_1} = 0.07 , \eta = 0.1 , {h_2} = 0.6 , a = 0.35 , b = 0.38 , \tau = 0.51 , an order-1 periodic solution can be formed in both \Omega_1 and \Omega_2 (Theorem 6), as presented in Figure 7(a); For model parameters \alpha = 0.1 , \beta = 0.15 , \gamma = 0.6 , \delta = 0.9 , {s_1} = 1.12 , {d_1} = 1.1 , f = 0.21 and control parameters {h_1} = 0.07 , \eta = 0.128 , {h_2} = 0.44 , a = 0.7 , b = 0.08 , \tau = 0.51 , an order-1 periodic solution can be formed in \Omega_1 (Theorem 6), as presented in Figure 7(b); while for control parameters {h_1} = 0.14 , \eta = 0.16 , {h_2} = 0.6 , a = 0.25 , b = 0.38 , \tau = 0.51 , an order-1 periodic solution can be formed (Theorem 6), as presented in Figure 7(c).

    Figure 7.  Illustration of the order 1 periodic solution. Schematic diagram of system (2.3) under different parameters.

    For system (2.3) with given model parameters \alpha = 0.09 , \beta = 0.215 , \gamma = 0.6 , \delta = 0.9 , {s_1} = 1.12 , {d_1} = 1.1 , f = 0.21 and control parameters {h_1} = 0.07 , \eta = 0.1 , {h_2} = 0.4 , a = 0.65 , b = 2 , \tau = 3 , an order-2 periodic solution can be formed in system (2.3) (Theorem 3.2), as presented in Figure 8(a); while for the control parameters {h_1} = 0.07 , \eta = 0.4 , {h_2} = 0.68 , a = 0.65 , b = 0.85 , \tau = 0.5 , a different order-2 periodic solution can be formed in system (2.3) (Theorem 8), as presented in Figure 8(b).

    Figure 8.  Illustration of the order 2 periodic solution. Schematic diagram of system (2.3) under different parameters.

    Considering that the Allee effect is an important mechanism in ecosystems and a realistic description of the interaction between species, we presented a model of prey-predator system with prey's Allee effect and generalist predator in the context of fish resources (Models (2.1) or (2.2)). We investigated the dynamic properties of Model (2.2) such as the type and stability of the boundary equilibria as well as the existence and stability of the interior equilibrium in detail (Theorems 1–3, Figure 5).

    To show the influence of the parameters on the dynamics of Model (2.2), we analyzed the bifurcations in the predation system by selecting the capture rate of prey by predator and Allee threshold as key parameters. We showed that Model (2.2) will undergo a saddle-node bifurcation as changing of the capture rate \alpha (Figure 6), and undergo a Bogdanov-Takens bifurcation of codimension at least 2 and 3 as changing of (\alpha, \beta) .

    To achieve sustainable and efficient exploitation of fish stocks, we adopted a bilateral intervention strategy, i.e., to avoid the distinction of prey fish caused by the Allee effect, releasing juvenile prey fish is adopted at a lower-level x = h_1 ; while for economic purposes, capturing both prey and predator fish is adopted at a higher-level x = h_2 . We obtained the conditions for the existence and stability of the order-1 periodic solution (Theorems 6, 7, Figure 7) and order-2 periodic solution (Theorems 8, 9, Figure 8) of the control system (2.3). The results showed that the extinction can be prevented by control even when the prey density is low, while in the case of the prey density increasing to a certain extent, fishing activities can be taken in a periodic way (periodic solution) to obtain the fish resources. Therefore, as long as the fish stocks are properly managed, the number of fish stocks can be controlled within an appropriate range, and the sustainable development and utilization of biological resources can be realized.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The work was supported by the National Natural Science Foundation of China (No. 11401068).

    The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.



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