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Enhancing success in fundamental engineering courses: A case study on using team based learning to address high failure rates


  • This study examines the impact of team-based learning (TBL) on students' performance in mechanics of materials, a fundamental yet challenging course in engineering curricula. Traditional lecture-based instruction has often failed to fully engage students and allow them to enhance their critical thinking skills that can be applied in engineering. This study compares outcomes between traditional lecture-based classrooms and those incorporating TBL at California State University, Fullerton (CSUF). Data from 72 students in traditional settings and 80 students in TBL-integrated classrooms were analyzed over multiple semesters. The results reveal improvement in examination scores and a reduction in failure rates for TBL participants compared with traditional instruction. The survey responses indicate that students in TBL sessions reported increased confidence, improved critical thinking, and enhanced teamwork skills. While the reduction in failure rates did not achieve statistical significance, the positive trends suggested that TBL effectively addresses challenges in high-stakes courses. The study advocates for expanding TBL to other fundamental engineering courses and institutions to further explore its effectiveness and potential to improve student retention, particularly among underrepresented minorities in science, technology, engineering, and mathematics (STEM).

    Citation: Huda Munjy, Stephanie Botros, Rhonda Abouazra. Enhancing success in fundamental engineering courses: A case study on using team based learning to address high failure rates[J]. STEM Education, 2025, 5(1): 152-170. doi: 10.3934/steme.2025008

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  • This study examines the impact of team-based learning (TBL) on students' performance in mechanics of materials, a fundamental yet challenging course in engineering curricula. Traditional lecture-based instruction has often failed to fully engage students and allow them to enhance their critical thinking skills that can be applied in engineering. This study compares outcomes between traditional lecture-based classrooms and those incorporating TBL at California State University, Fullerton (CSUF). Data from 72 students in traditional settings and 80 students in TBL-integrated classrooms were analyzed over multiple semesters. The results reveal improvement in examination scores and a reduction in failure rates for TBL participants compared with traditional instruction. The survey responses indicate that students in TBL sessions reported increased confidence, improved critical thinking, and enhanced teamwork skills. While the reduction in failure rates did not achieve statistical significance, the positive trends suggested that TBL effectively addresses challenges in high-stakes courses. The study advocates for expanding TBL to other fundamental engineering courses and institutions to further explore its effectiveness and potential to improve student retention, particularly among underrepresented minorities in science, technology, engineering, and mathematics (STEM).



    Predator-prey is one of three major types of interactions between species besides symbiosis and competition. Understanding the interactions between predators and their prey has been one of the leading research interests in population dynamics [1]. A predator-prey system is dominated by two important factors: the population growth function and the functional response. Tanner [2] presented a predator-prey system in which the environmental carrying capacity of the predator is proportional to the prey population size, and the reduction rate of the prey is proportional to the predator size. It takes the following form:

    {dxdt=x[r(1xK)kyx+D],dydt=sy(1hyx), (1)

    where x(t) is the prey population and y(t) is the predator population at time t; r and s are the intrinsic growth rate of the prey and predator respectively; h,k,K and D are all positive parameters. The term kxx+D is called Holling-Ⅱ functional response, generally reflecting the reduction rate of the prey caused by per capita of the predator.

    By variable changes:

    u(τ)=x(t)K, v(τ)=hy(t)K, τ=rt, a=khr, b=sr, d=DK,

    system (1) can be rewritten in a nondimensional form:

    {dudτ=u(1u)auvu+d=f(u,v),dvdτ=bv(1vu)=g(u,v). (2)

    System (2) has been extensively studied [3,4,5,6,7]. Aziz-Alaoui and Okiye [8] considered the following system with alternative food sources for predators, which is called the modified Holling-Tanner model:

    {dxdt=x[r(1xK)kyx+D],dydt=sy(1yhx+K2), (3)

    where K2>0, hx+K2 in system (3) is the new carrying capacity for predators. K2 can be seen as an extra constant carrying capacity from all other food sources for predators. Several researchers [9,10,11,12,13,14,15] studied the existence of periodic solutions and bifurcation phenomena of system (3).

    Allee effect refers to the phenomenon that low population density inhibits growth. Bioresearch indicates that clustering benefits the growth and survival of species. However, extreme sparsity and overcrowding will prevent population growth and negatively affect reproduction [16,17,18,19]. Every species has its optimal density. Species with small population densities are generally vulnerable. Once the population density falls below a critical level, interactions within the species will diminish. Ye et al. [20] considered a predator-prey model with a strong Allee effect and a nonconstant mortality rate. They found that a strong Allee effect may guarantee the coexistence of the species. Hu and Cao [21] considered a predator-prey model with Michaelis-Menten type predator harvesting. Their model exhibits a Bogdanov-Takens bifurcation of codimension 2. Xiang et al. [22] studied the Holling-Tanner model with constant prey harvesting. They found a degenerate Bogdanov-Takens bifurcation of codimension 4 and at least three limit cycles. In [23], Xiang et al. considered the Holling-Tanner model with predator and prey refuge and proved that this model undergoes a Bogdanov-Takens bifurcation of codimension 3. Arancibia et al. [24] adjusted the Holling-Tanner model by adding a strong Allee effect to prey. They found a Bogdanov-Takens bifurcation of codimension 2 and a heteroclinic bifurcation. It seems that a strong Allee effect gives rise to the heteroclinic bifurcation. Jia et al. [25] studied a modified Leslie-Gower model with a weak Allee effect on prey. This model undergoes a degenerate Bogdanov-Takens bifurcation of codimension 3 and has at least two limit cycles. Zhang and Qiao [26] analyzed the SIR model with vaccination and proved that this model undergoes a Bogdanov-Takens bifurcation of codimension 3 in some specific cases.

    In this paper, we will analyze the following predator-prey model with the parameter M indicating strong Allee effect:

    {dxdt=rx(1xK)(xM)αxyx+m,dydt=sy(1βyx+m), (4)

    where α>0 and β>0 are real numbers. We shall assume MK from now on. If the original population of the prey is less than M, the prey will have a negative growth rate and become extinct ultimately. In system (4), m>0 measures the extent of protection to which the environment provides to prey (or, to predators). It means that the growth rate of prey and predators are not negative infinity when x reduces to zero.

    As a first step in analyzing system (4), we nondimensionalize system (4) by writing

    xKx, αyrK2y, rKtt,

    and it becomes

    {dxdt=x(1x)(xa)xyx+b,dydt=cy(1dyx+b), (5)

    where a=MK<1, b=mK, c=srK and  d=βrKα.

    This paper is organized as follows. Section 2 discusses the stability of the equilibria. Section 3 deals with possible bifurcations that system (5) undergoes, such as Hopf bifurcation and Bogdanov-Takens bifurcation. Section 4 summarizes our conclusions.

    The equilibria of system (5) satisfy the following equations

    {x(1x)(xa)xyx+b=0,cy(1dyx+b)=0.

    We get four boundary equilibria on the axes: E0=(0,0),E1=(1,0),E2=(a,0) and E3=(0,bd). These equilibria are all hyperbolic because corresponding linearized matrices at the equilibria Ei (i=0,1,2,3) are

    JE0=(a00c),  JE1=(a111+b0c),
    JE2=(a(1a)aa+b0c), JE3=(a1d0cdc).

    Because we have assumed 0<a<1, we get the stability of the boundary equilibria.

    Theorem 2.1. System (5) has four equilibria Ei (i=0,1,2,3) on the boundary. E0(0,0) and E1(1,0) are both hyperbolic saddles, E2(a,0) is a hyperbolic unstable node and E3(0,bd) is a hyperbolic stable node (see Figure 1(c)).

    Figure 1.  The number of equilibria: (a) 6 equilibria. (b) 5 equilibria. (c) 4 equilibria.

    Theorem 2.2. If Δ=(a1)24d>0, system (5) has two positive equilibria E4(x4,x4+bd) and E5(x5,x5+bd), where x4=1+aΔ2 and x5=1+a+Δ2. E4 is always a hyperbolic saddle. E5 may be a source, a sink or a center depending on the parametric values (see Figure 1(a)).

    Proof. The positive equilibria of system (5) satisfy the equations

    {x2(1+a)x+a+1d=0,y=x+bd.

    If Δ=(a1)24d>0, the equation x2(1+a)x+a+1d=0 has two roots xi (i=4,5).

    The linearized matrix of system (5) at Ei=(xi,xi+bd) (i=4,5) is

    JEi=(2x2i+(1+a)xi+xid(xi+b)xixi+bcdc),  i=4,5.

    The determinant of E4 is detJE4=cx4Δ<0, from which we get that E4 is a hyperbolic saddle.

    Because detJE5=cx5Δ>0 and trJE5=x5Δc+x5d(x5+b), the stability of E5 depends on the parameters a,b,c and d. That is, E5 is a sink if trJE5<0, a source if trJE5>0.

    Theorem 2.3. If Δ=(a1)24d=0, i.e., d=4(a1)2, system (5) has a unique positive equilibrium E6(x6,y6), where x6=1+a2 and y6=1+a+2b2d (see Figure 1(b)).

    1) If  c>1+ad(1+a+2b) (c<1+ad(1+a+2b)), E6 is an attracting (a repelling) saddle-node;

    2) If  c=1+ad(1+a+2b), E6 is a nilpotent cusp of codimension 2 (i.e., the Bogdanov-Takens singularity).

    The phase portraits are given in Figure 3.

    Figure 2.  A cusp at the origin.
    Figure 3.  (a) A cusp. (b) A saddlenode(λ2<0). (c) A saddle-node(λ2>0).

    Proof. The linearized matrix at the equilibrium E6 is

    JE6=(1+ad(1+a+2b)1+a1+a+2bcdc).

    E6 is not hyperbolic because detJE6=0. By a shift transformation xx6x and yy6y, system (5) becomes

    {dxdt=(1+a)(xdy)d(1+a+2b)+[4bd(1+a+2b)21+a2]x24b(1+a+2b)2xy+O(|x,y|3),dydt=cdxcy2cd(1+a+2b)x2+4c1+a+2bxy2cd1+a+2by2+O(|x,y|3). (6)

    The eigenvalues of JE6 are λ1=0 and λ2=1+ad(1+a+2b)c.

    Case 1. λ20, i.e., c1+ad(1+a+2b).

    By variable changes

    (xy)=(d1+ad(1+a+2b)1cd)(uv),

    system (6) can be rewritten as

    {dudt=cd2(1+a)2λ2u2+O(|u,v|3),dvdt=λ2v+O(|u,v|2). (7)

    By Theorem 7.1 in chapter 2 of [27], we obtain that E6 is a saddle-node. There is a parabolic sector neighborhood in which all trajectories approach to E6 when λ2<0, and leave it when λ2>0.

    Case 2. λ2=0, i.e., c=1+ad(1+a+2b).

    According to [28], the cusp is a kind of nonhyperbolic critical point. The cusp can be illustrated by the following example:

    {˙ξ=η,˙η=ξ2.

    The phase portrait for this system is shown in Figure 2. The neighborhood of the origin consists of two hyperbolic sectors and two separatrices.

    Now consider the case when the linearized matrix (denoted as A) has two zero eigenvalues, i.e., det A=0,tr A=0, but A0. In this case it is shown in [28], that the system can be put in the normal form:

    {˙ξ=η,˙η=akξk[1+h(ξ)]+bnξnη[1+g(ξ)]+η2R(ξ,η), (8)

    where h(ξ),g(ξ) and R(ξ,η) are analytic in a neighborhood of the origin, h(0)=g(0)=0,k2,ak0 and n1. The following lemma is given in [28].

    Lemma 2.1. Let k=2m with m1 in system (8). Then the type of the origin is given by Table 1.

    Table 1.  The relationship of bn,n,m and the type of the origin.
    The relationship of bn,n and m Type of the origin
    bn=0 Cusp
    bn0 nm
    n<m Saddle-node

     | Show Table
    DownLoad: CSV

    Next, we transform system (6) into a normal form by coordinate transformations. The corresponding linearized matrix of system (6) at E6 is

    JE6=(ccdcdc).

    By variable changes

    (xy)=(ddc10)(uv),

    system (6) can be rewritten as

    {dudt=v2d21+av2+O(|u,v|3)P(u,v),dvdt=(1+a)22(1+a+2b)u2d[(1+a)3+(3+a)(1+3a)b+4(1+a)b2](1+a+2b)2uvd2(ab+2a+b)1+av2+O(|u,v|3)Q(u,v). (9)

    Lemma 2.2. ([28]) System (10)

    {˙x=y+Ax2+Bxy+Cy2+O(|x,y|3),˙y=Dx2+Exy+Fy2+O(|x,y|3), (10)

    is equivalent to system (11) near the origin, where system (11) is

    {˙x=y,˙y=Dx2+(E+2A)xy+O(|x,y|3). (11)

    Therefore, by Lemma 2.2, we can transform system (9) into the following form:

    {dudt=v,dvdt=(1+a)22(1+a+2b)u2d((1+a)3+(3+a)(1+3a)b+4(1+a)b2)(1+a+2b)2uv+O(|u,v|3). (12)

    Using the notations of Lemma 2.1, we obtain k=2m, m=n=1, a2m=(1+a)22(1+a+2b) and bn=d[(1+a)3+(3+a)(1+3a)b+4(1+a)b2](1+a+2b)2. Consequently, by Lemma 2.1, we find that E6(x6,y6) is a degenerate critical point (cusp). More exactly, by the results in [28], E6 is a cusp of codimension 2.

    For example, we take a=0.1,b=0.1 and d=4.9383, which satisfy (a1)24d=0. E6 is a cusp for c=1+ad(1+a+2b)=0.1713. E6 is a saddle-node with parabolic sector approaching it for c=0.3. E6 is a saddle-node with parabolic sector repelling it for c=0.05 (see Figure 3).

    As stated in Theorem 2.2, the positive equilibrium E5 may be a center, source, or sink because detJE5>0. Considering c as the bifurcation parameter, Hopf bifurcation occurs when c=cH, where cH satisfies trJE5|c=cH=0. The local stability of E5 changes when c passes through c=cH. We summarize our results in the following theorem.

    Theorem 3.1. Hopf bifurcation occurs at E5 in system (5) when c=cH>0.

    Proof. Take c as the bifurcation parameter. By trJE5=x5Δ+x5d(x5+b)c, we have

    cH=x5Δ+x5d(x5+b),
    ctrJE5|c=cH=10,
    detJE5=cx5Δ>0.

    Hopf bifurcation may occur when c crosses cH.

    Next, we discuss the stability of E5 as c=cH. Moving E5 to (0,0) by X=x+x5 and Y=y+y5, the Taylor expansion of system (5) at E5 takes the form

    {dXdt=A10X+A01Y+A20X2+A11XY+A30X3+A21X2Y+O(|X,Y|4),dYdt=B10X+B01Y+B20X2+B11XY+B02Y2+B30X3+B21X2Y+B12XY2+O(|X,Y|4), (13)

    where

    A10=cH, A01=x5x5+b, A20=1+a3x5+by5(x5+b)3,A11=b(x5+b)2, A30=1by5(x5+b)4, A21=b(x5+b)3,B10=cHd, B01=cH, B20=cHy25d(b+x5)3, B11=2cHy5d(b+x5)2,B02=cHdb+x5, B30=cHy25d(b+x5)4, B21=2cHy5d(b+x5)3, B12=cHd(b+x5)2.

    With ω=A210A01B10>0 and

    (XY)=(ωA100B10)(UV),i.e.,  (UV)=(1ωdω01B10)(XY),

    we transform system (13) into the following system

    {dUdt=ωV+F(U,V),dVdt=ωU+G(U,V), (14)

    where F(U,V) and G(U,V) are the sum of those terms with orders not less than 2.

    The stability of O(0,0) relies on the number

    K=116(FUUU+FUVV+GUUV+GVVV)+116ω(FUVFUU+FUVFVV+FVVGVV)116ω(GUVGVV+GUVGUU+FUUGUU).

    The following simplified expression of K can be obtained by Maple:

    8KcH=cH[2A220x5Δ+2A20x5+b(2bd(x5+b)3+3)]+[2A220+2x5ΔA20x5+b3x5Δ(1+bd(x5+b)3)]+[cHb2(x5+b)4d2x5ΔA20bd(x5+b)23cHA20bx5Δd(x5+b)2]=I1+I2+I3,
    Δ=(a1)24d<(1a)2,   x5=1+a+Δ2>Δ,   cH=x5d(x5+b)x5Δ>0,A20=1+a3x5bd(x5+b)2<(x5+Δ)bd(x5+b)2<0,
    I1cH=2A220x5Δ+2A20x5+b(2bd(x5+b)3+3)=A20x5Δ(A20+2x5Δx5+b)+1x5Δ(A2203x5Δ2bx5Δd(x5+b)3)>A20x5Δ((x5+Δ)+2Δ)+1x5Δ((x5+Δ)23x5Δ)+2b(x5+Δx5Δx5+b)d(x5+b)2x5Δ>0,I2=2A220+2x5ΔA20x5+b3x5Δ(1+bd(x5+b)3)=A20(A20+2x5Δx5+b)+(A2203x5Δ(1+bd(x5+b)3))>A20((x5+Δ)+2Δ)+((x5+Δ)23x5Δ)+bd(x5+b)2(2(x5+Δ)3x5Δx5+b)>0,I3=cHb2(x5+b)4d2x5ΔA20bd(x5+b)23cHA20bx5Δd(x5+b)2.

    It is obvious that I3>0 because all the elements are positive. Therefore, E5 undergoes a subcritical bifurcation when c=cH. When c>cH and |ccH|ε, there is an unstable limit cycle (see Figure 4(b)).

    Figure 4.  Hopf Bifurcation of system (5): (a) E5 is an unstable focus. (b) E5 is a stable focus and there is an unstable closed orbit. (c) E5 is a linear center.
    Figure 5.  Bifurcation diagram of the parameter c and d with a=0.1 and b=0.1.

    Theorem 3.2. When c and d are selected as two bifurcation parameters, detJE5|(c,d)=(cBT,dBT)=0 and trJE5|(c,d)=(cBT,dBT)=0, system (5) undergoes a Bogdanov-Takens bifurcation of codimension 2 in a small neighborhood of E5 as (c,d) varies near (cBT,dBT)=((a1)241+a1+a+2b,4(a1)2).

    Proof. Perturb parameters c and d by c=cBT+ε1 and d=dBT+ε2, where (ε1,ε2) is sufficiently small, and system (5) takes the following form:

    {dxdt=x(1x)(xa)xyx+b,dydt=(cBT+ε1)y(1(dBT+ε2)yx+b). (15)

    The Taylor expansion of system (15) at E5(x5,y5) is

    {dxdt=p10x+p01y+p20x2+p11xy+O(|x,y,ε1,ε2|3),dydt=q00+q10x+q01y+q11xy+q02y2+O(|x,y,ε1,ε2|3), (16)

    where

    p10=2x25+(1+a)x5+x5d(x5+b), p01=x5x5+b, p20=1+a3x5+by5(x5+b)3, p11=b(x5+b)2, q00=ε2y5(cBT+ε1)dBT, q10=(cBT+ε1)(dBT+ε2)d2BT, q01=(cBT+ε1)(12(dBT+ε2)dBT), q11=2(cBT+ε1)(dBT+ε2)dBT(x5+b), q02=(cBT+ε1)(dBT+ε2)x5+b.

    Take a C change of coordinates around (0,0)

    u1=x, v1=dxdt,

    then system (16) is changed into the following form

    {˙u1=v1,˙v1=n00+n10u1+n01v1+n20u21+n11u1v1+n02v21+O(|u1,v1,ε1,ε2|3), (17)

    where

    n00=p01q00, n10=p01q10p10q01+p11q00, n01=p10+q01,n20=q20p01q11p10+p11q10q01p20+p210q02p01,n11=2p20+q11p11p10+2q02p10p01, n02=p11+q02p01.

    After rescaling the time by (1n02u1)tt, system (17) is rewritten as

    {˙u1=v1(1n02u1),˙v1=(1n02u1)[n00+n10u1+n01v1+n20u21+n11u1v1+n02v21+O(|u1,v1,ε1,ε2|3)]. (18)

    Letting u2=u1 and v2=v1(1n02u1), we get system (19) as follows

    {˙u2=v2,˙v2=θ00+θ10u2+θ01v2+θ20u22+θ11u2v2+O(|u2,v2,ε1,ε2|3), (19)

    where

    θ00=n00,θ10=n102n00n02,θ01=n01,[2mm]θ20=n202n10n02+n00n202,θ11=n11n01n02.

    Case 1: For small εi (i=1,2), if θ20>0, by the following change of variables

    u3=u2,v3=v2θ20,tθ20t,

    system (19) becomes

    {˙u3=v3,˙v3=s00+s10u3+s01v3+u23+s11u3v3+O(|u3,v3,ε1,ε2|3), (20)

    where

    s00=θ00θ20,  s10=θ10θ20,  s01=θ01θ20,  s11=θ11θ20.

    To eliminate the u3 term, letting u4=u3+s102 and v4=v3, we get system (21) as follows

    {˙u4=v4,˙v4=r00+r01v4+u24+r11u4v4+O(|u4,v4,ε1,ε2|3), (21)

    where

    r00=s00s2104,  r01=s01s10s112,  r11=s11. 

    Clearly, r11=s11=θ11θ200 if θ110.

    Setting u5=r211u4, v5=r311v4 and τ=1r11t, we obtain the universal unfolding of system (16)

    {˙u5=v5,˙v5=μ1+μ2v5+u25+u5v5+O(|u5,v5,ε1,ε2|3), (22)

    where

    μ1=r00r411,  μ2=r01r11. (23)

    Case 2: For small εi(i=1,2), if θ20<0, by the following change of variables

    u3=u2,  v3=v2θ20,  tθ20t,

    system (19) becomes

    {˙u3=v3,˙v3=s00+s10u3+s01v3u23+s11u3v3+O(|u3,v3,ε1,ε2|3), (24)

    where

    s00=θ00θ20,  s10=θ10θ20,  s01=θ01θ20,  s11=θ11θ20.

    To eliminate the u3 term, letting u4=u3s102 and v4=v3, we get system (25) as follows

    {˙u4=v4,˙v4=r00+r01v4u24+r11u4v4+O(|u4,v4,ε1,ε2|3), (25)

    where

    r00=s00+s2104,  r01=s01+s10s112,  r11=s11.

    Clearly, r11=s11=θ11θ200 if θ110.

    Setting u5=r211u4,  v5=r311v4 and τ=1r11t, we obtain the universal unfolding of system (16)

    {˙u5=v5,˙v5=μ1+μ2v5+u25+u5v5+O(|u5,v5,ε1,ε2|3), (26)

    where

    μ1=r00r411,  μ2=r01r11. (27)

    Retain μ1 and μ2 to denote μ1 and μ2. If the matrix |(μ1,μ2)(ε1,ε2)|ε1=ε2=0 is nonsingular, the parameter transformations (23) and (27) are homeomorphisms in a small neighborhood of (0,0), and μ1,μ2 are independent parameters. Direct computation shows that θ20=(1+a)2(a1)2(a2+2ab+2b2+2b+1)4(1+a+2b)3<0 when εi=0 (i=1,2) and

    |(μ1,μ2)(ε1,ε2)|ε1=ε2=0=2(a3+(3b+3)a2+(4b2+10b+3)a+4b2+3b+1)5(1+a+2b)(a2+2ab+2b2+2b+1)4(1+a)6(a1)20.

    By Perko [28], we know that systems (22) and (26) undergo the Bogdanov-Takens bifurcation when ε=(ε1,ε2) is in a small neighborhood of the origin. The local representations of the unfolding bifurcation curves are as follows ("+" for θ20>0, "-" for θ20<0):

    1) The saddle-node bifurcation curve SN ={(ε1,ε2):μ1(ε1,ε2)=0,μ2(ε1,ε2)0};

    2) The Hopf bifurcation curve H ={(ε1,ε2):μ2(ε1,ε2)=±μ1(ε1,ε2),μ1(ε1,ε2)<0};

    3) The homoclinic bifurcation curve HOM ={(ε1,ε2):μ2(ε1,ε2)=±57μ1(ε1,ε2),μ1(ε1,ε2)<0}.

    For example, in system (15), we can set a=0.1, b=0.1 and dBT=4(a1)24.93827. Further computation yields x5=1+a2=0.55, y5=x5+bd=0.131625 and cBT=2x25+(1+a)x5+x5d(x5+b)0.171346. Since

    |(μ1,μ2)(ε1,ε2)|ε1=ε2=0=|01.747427.546660.323622|13.18720,

    the parametric transformation (27) is nonsingular. Moreover, θ20=0.1394090.81361ε10.058426ε20.340982ε1ε2<0 and θ11=1.05207+1.81818ε1+0.126173ε2+8.97868ε21+1.67098ε1ε2+0.021619ε220 for small εi (i=1,2). The local representations of the bifurcation curves of system (15) up to second-order approximations are as follows. The details of the computation are given in Appendix.

    1) The saddle-node bifurcation curve SN, is expressed as {(ε1,ε2):ε2=0,ε1<0};

    2) The Hopf bifurcation curve H, is expressed as

    {(ε1,ε2):1.74742ε2+56.9521ε21+37.3603ε1ε2+3.09997ε22=0,ε1<0};

    3) The homoclinic bifurcation curve HOM, is expressed as

    {(ε1,ε2):1.74742ε2+111.626ε21+42.0495ε1ε2+3.20051ε22=0,ε1<0};

    4) The heteroclinic bifurcation curve HET, can be detected with MATCONT, the numerical bifurcation package.

    (a) When (ε1,ε2)=(0,0), E5 is a cusp of codimension 2 (see Figure 6(a)).

    Figure 6.  Phase portraits of system (5): (a) A cusp point. (b) There is no positive equilibrium. (c) A saddle and an unstable focus. (d) An unstable limit cycle and a stable focus. (e) An unstable homoclinic cycle. (f) A saddle and a stable focus. (g) E1 connects with E4. (h) E1 connects with E5.

    (b) When (ε1,ε2) is below SN, there are no positive equilibria (see Figure 6(b)).

    (c) When (ε1,ε2) crosses the SN curve and locates in the area between H and SN, there are two positive equilibria E4 and E5. E4 is a saddle and E5 is unstable (see Figure 6(c)).

    (d) When (ε1,ε2) crosses H, an unstable limit cycle will appear (see Figure 6(d)).

    (e) When (ε1,ε2) is on the curve HOM, there is an unstable homoclinic orbit (see Figure 6(e)).

    (f) When (ε1,ε2) is between the curve HOM and HET, E4 connects with E2 (see Figure 6(f)).

    (g) When (ε1,ε2) falls on the HET curve, E4 connects with E1 (see Figure 6(g)).

    (h) When (ε1,ε2) crosses the HET curve, E1 connects with E5 (see Figure 6(h)).

    This paper considers the modified Holling-Tanner model with a strong Allee effect. The aim is to explore the dynamical behaviors occurring in the predator-prey model with a strong Allee effect and alternative food sources for predators. Section 2 considers the equilibria and their stability. There exist four equilibria on the boundary. E0(0,0) and E1(1,0) are saddles, E2(a,0) is an unstable node and E3(0,bd) is a stable node. As for the positive equilibria, when d<4(a1)2, there is no positive equilibrium. When d=4(a1)2, saddle-node bifurcation occurs and there is a unique positive equilibrium E6, which is a cusp when c=(a1)241+a1+a+2b and a saddle-node when c(a1)241+a1+a+2b. When d>4(a1)2, the saddle-node E6 separates into two positive equilibria, a saddle E4 and a hyperbolic equilibrium E5. We examine the local stability of E5 and find that E5 is stable if c>cH and unstable if c<cH. In Section 3.1, we prove that system (5) undergoes a Hopf bifurcation near E5 when c=cH by showing that the constant K>0. In Section 3.2, we prove that system (5) exhibits a Bogdanov-Takens bifurcation of codimension 2 by calculating the universal unfolding near the cusp E6. Besides, we give the bifurcation diagram with a little perturbation (ε1,ε2) added to (cBT,dBT).

    Our main result is that, after adding a strong Allee effect and alternative food sources, system (5) allows the independent survival of predators. In addition, a strong Allee effect makes the system more stable. Since, in the original Holling-Tanner system (2), there are at least two limit cycles [29], while after adding a strong Allee effect, there seems to be a unique stable limit cycle [24].

    From the ecological viewpoint, alternative food sources for predators help them survive without prey. Besides, a strong Allee effect makes prey extinct at low density. Codimension of the Bogdanov-Takens bifurcation is at most two and the Hopf bifurcation is nondegenerate. We compare Holling-Tanner models with and without a strong Allee effect. It is found that a strong Allee effect increases and changes the dynamics. The first is that a strong Allee effect introduces saddle-node bifurcation. Second, heteroclinic bifurcation is brought about by the Allee effect, which means that the prey and predators may take different paths to reach distinct ultimate states. These indicate that the ecosystem may be sensitive to disturbances. It is essential to be aware of such bifurcations and protect the environment to weaken the Allee effect. In the future, we plan to study the system with different environmental protection for prey and predators. We will introduce m1 and m2 in place of b to system (5) and consider the following equation:

    {˙x=rx(1xK)(xM)αxyx+m1,˙y=sy(1βyx+m2). (28)

    If the functional response depends on the time as well as the prey and predator population, the model could exhibit more interesting dynamical behavior even chaos. We will do some research in our future works.

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

    This work was supported by National Natural Science Foundation of China (Grant No.62227810).

    The authors declare there is no conflict of interest.

    For a=0.1, b=0.1, system (16) takes the form

    {dxdt=p10x+p01y+p20x2+p11xy+O(|x,y,ε1,ε2|3),dydt=q00+q10x+q01y+q11xy+q02y2+O(|x,y,ε1,ε2|3), (A.1)

    where

    p10=0.171346,   p01=0.846154,   p20=0.502071,   p11=0.236686,q00=0.004567ε2,   q10=0.034698+0.2025ε1+0.007026ε2,q01=0.171346ε1+0.069395ε2,   q11=0.527219+3.076923ε1+0.106762ε2,q02=1.3017757.597341ε10.263609ε2.

    Take a C change of coordinates around (0,0)

    u1=x, v1=dxdt,

    then system (A.1) is changed into the following form

    {˙u1=v1,˙v1=n00+n10u1+n01v1+n20u21+n11u1v1+n02v21+O(|u1,v1,ε1,ε2|3), (A.2)

    where

    n00=0.003864ε2,   n10=0.007026ε2,   n01=ε10.069395ε2,n20=0.1394090.813609ε10.045651ε2,n11=1.052071,   n02=1.818182+8.978676ε1+0.311538ε2.

    Letting u2=u1,v2=v1(1n02u1), we get system (A.3) as follows

    {˙u2=v2,˙v2=θ00+θ10u2+θ01v2+θ20u22+θ11u2v2+O(|u2,v2,ε1,ε2|3), (A.3)

    where

    θ00=0.003864ε2,θ10=0.007026ε2,θ01=ε10.069395ε2,[2mm]θ20=0.1394090.813609ε10.058426ε2,θ11=1.052071+1.818182ε1+0.126173ε2.

    For small εi (i=1,2), θ20<0, by the following change of variables

    u3=u2,  v3=v2θ20,  tθ20t,

    system (A.3) becomes

    {˙u3=v3,˙v3=s00+s10u3+s01v3u23+s11u3v3+O(|u3,v3,ε1,ε2|3), (A.4)

    where

    s00=0.027720ε2,   s10=0.050400ε2,   s01=2.678273ε10.185859ε2,s11=2.817733+13.091929ε2+0.928379ε2.

    To eliminate the u3 term, letting u4=u3s102,  v4=v3, we get system (A.5) as follows

    {˙u4=v4,˙v4=r00+r01v4u24+r11u4v4+O(|u4,v4,ε1,ε2|3), (A.5)

    where

    r00=0.027720ε2,   r01=2.678273ε10.114852ε2,r11=2.817733+13.091929ε1+0.928379ε2.

    Setting u5=r211u4,  v5=r311v4,  τ=1r11t, we obtain the universal unfolding of system (A.6)

    {˙u5=v5,˙v5=μ1+μ2v5+u25+u5v5+O(|u5,v5,ε1,ε2|3), (A.6)

    where

    μ1=1.747416ε2,  μ2=7.546659ε10.323622ε2. (A.7)

    1) The saddle-node bifurcation curve SN = {(ε1,ε2):μ1(ε1,ε2)=0,μ2(ε1,ε2)0};

    2) The Hopf bifurcation curve H = {(ε1,ε2):μ2(ε1,ε2)=±μ1(ε1,ε2),μ1(ε1,ε2)<0};

    3) The homoclinic bifurcation curve HOM = {(ε1,ε2):μ2(ε1,ε2)=±57μ1(ε1,ε2),μ1(ε1,ε2)<0}.

    With the specific parameters, we have

    1) The saddle-node bifurcation curve, denoted SN, is expressed as

    {(ε1,ε2):ε2=0,ε1<0};

    2) The Hopf bifurcation curve, denoted H, is expressed as

    {(ε1,ε2):1.74742ε2+56.9521ε21+37.3603ε1ε2+3.09997ε22=0,ε1<0};

    3) The homoclinic bifurcation curve, denoted HOM, is expressed as

    {(ε1,ε2):1.74742ε2+111.626ε21+42.0495ε1ε2+3.20051ε22=0,ε1<0}.


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  • This article has been cited by:

    1. Aparna Das, Satyaram Mandal, Sankar Kumar Roy, Bifurcation analysis within a modified May-Holling-Tanner Prey-Predator system via Allee effect and harvesting on predator, 2024, 0, 2155-3289, 0, 10.3934/naco.2024043
  • Author's biography Dr. Huda Munjy is a professor of civil engineering at California State University, Fullerton. She specializes in structural engineering. Her research interests include earthquake engineering, and engineering and STEM education. She is TBL-certified through the TBLc. She is also a member of the American Society of Civil Engineers; Stephanie Botros is an alumna of California State University, Fullerton, where she received her Bachelor's degree in Civil and Environmental Engineering. She is currently studying to achieve her Master's degree in Structural Engineering at the same institute. She is a member of the American Society of Civil Engineers and serves in the student chapter. She assisted in this research project during her postbaccalaureate studies; Rhona Abouazra is an alumna of California State University, Fullerton. She worked on this research project during her undergraduate studies and obtained her Bachelor's degree in Civil and Environmental Engineering
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