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Review

Understanding potato with the help of genomics

  • Potato (Solanum tuberosum L.) is an important staple crop with a highly heterozygous and complex genome. Despite its cultural and economic significance, potato improvement efforts have been held back by the relative lack of genetic resources available to producers and breeders. The publication of the potato reference genome and advances in high-throughput sequencing technologies have led to the development of a wide range of genomic and transcriptomic resources. An overview these new tools, from the updated versions of the potato reference genome and transcriptome, to more recent gene expression, regulatory motif, re-sequencing and SNP genotyping analyses, paints a picture of modern potato research and how it will change our understanding of potato as well as other tuber producing Solanaceae.

    Citation: José Héctor Gálvez, Helen H. Tai, Noelle A. Barkley, Kyle Gardner, David Ellis, Martina V. Strömvik. Understanding potato with the help of genomics[J]. AIMS Agriculture and Food, 2017, 2(1): 16-39. doi: 10.3934/agrfood.2017.1.16

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  • Potato (Solanum tuberosum L.) is an important staple crop with a highly heterozygous and complex genome. Despite its cultural and economic significance, potato improvement efforts have been held back by the relative lack of genetic resources available to producers and breeders. The publication of the potato reference genome and advances in high-throughput sequencing technologies have led to the development of a wide range of genomic and transcriptomic resources. An overview these new tools, from the updated versions of the potato reference genome and transcriptome, to more recent gene expression, regulatory motif, re-sequencing and SNP genotyping analyses, paints a picture of modern potato research and how it will change our understanding of potato as well as other tuber producing Solanaceae.


    Over the past several decades, the predator-prey interaction has become a hot point of studies in biomathematics [1,2,3,4,5,6,7,8,9,10]. Because differential equations can assume that generations overlap and that populations vary continously in time, the general model for predator-prey interaction may be written as

    {dxdt=f(x)xg(x,y)y,dydt=h(x,y)ymy, (1.1)

    where x and y are expressed as prey and predator population sizes (or densities), respectively, f(x) denotes the growth rate of prey with the absence of predator, g(x,y) represents the amount of prey consumed per predator per unit time, h(x,y) is on behalf of per capita predator production, and m is the intrinsic death rate of predator. See also [1].

    Due to the realistic meaning of f(x), one can assume that the prey grows logistically with growth rate r and carrying capacity k in the absence of predator (i.e., f(x)=r(1xk)). Hence the system (1.1) can be written as

    {dxdt=rx(1xk)g(x,y)y,dydt=eg(x,y)ymy, (1.2)

    where e is the conversion effciency.

    As for the functional response g(x,y), there are many different kinds of forms. Bian et al. proposed a system with the Beddington-DeAngelis funcional response [5]; De Assis et al. proposed a system with the square-root functional response [7] and so on. Notice the fact that in the natural ecosystem, many species may gather together and form herds to either search for food resources or to defend the predators, which means that all members of a group do not interact at one time. This behaviour is often called herd behaviour. In this paper, one talks about the following system [6,7]:

    {dxdt=rx(1xk)axyx+˜h,dydt=eaxyx+˜hmy. (1.3)

    Here, the funcional response axx+˜h can be expressed as the function of the ratio of prey to predator, where ˜h is a threshold for the transition between herd grouping and solitary behaviour and a is the maximum value of prey consumed by each predator per unit time. In this system, all parameters are positive. The biological meanings for the parameters r, k, e, and m are the same as in (1.2).

    For the sake of simplicity of mathematical analysis, let xkx,mtt,yeky,rmγ,aekmβ,˜hkh, then one can derive an equivalent form of the system (1.3) as follows:

    {dxdt=x(γ(1x)βyx+h),dydt=y(βxx+h1). (1.4)

    This continuous system has been discussed in [6,7], but its discrete version has not been investgated as of yet. To be honest, it is very difficult to solve a complicate continuous equation or system without using computer. Therefore, one natuarally wishes to consider the corresponding discrete version of a continuous model. One tries to use various methods to derive the discrete model of the system (1.4) to make it easily studied [8,9,10,11,12,13,14,15,16]. In this paper, we adopt a semi-discretazation method, which does not need to consider the step size, to derive its discrete model. For this, suppose that [t] denotes the greatest integer not exceeding t. Consider the average change rate of the system (1.4) at integer number points

    {1x(t)dxdt=γ(1x([t]))βy([t])x([t])+h,1y(t)dydt=βx([t])x([t])+h1. (1.5)

    It is easy to see that the system (1.5) has piecewise constant arguments, and that a solution (x(t),y(t)) of the system (1.5) for t[0,+) has the following characteristics:

    1) on the interval [0,+),x(t) and y(t) are continuous;

    2) when t[0,+) except possibly for the points {0,1,2,3,}, dx(t)dt and dy(t)dt exist.

    The following system can be obtained by integrating the system (1.5) over the interval [n, t] for any t[n,n+1) and n=0,1,2,

    {x(t)=xneγ(1xn)βynxn+h(tn),y(t)=yneβxnxn+h1(tn), (1.6)

    where xn=x(n) and yn=y(n). Letting t(n+1) in the system (1.6) produces

    {xn+1=xneγ(1xn)βynxn+h,yn+1=yneβxnxn+h1, (1.7)

    where the parameters h,β,γ>0, and their biological meanings are the same as in (1.4). The system (1.7) will be considered in the sequel.

    The rest of the paper is organized as follows. In Section 2, we investigate the existence and stability of the fixed points of the system (1.7) in detail. In Section 3, we derive the sufficient conditions for transcritical bifurcation, period-doubling bifurcation, and Neimark-Sacker bifurcation of the system (1.7) to occur. In Section 4, numerical simulations are performed to illustrate the above theoretical results. In the end, some brief conclusions are stated in Section 5.

    Considering the biological meaning of the system (1.7), we discuss the existence and stability of non-negative fixed points of the system (1.7) in this section. By solving the equations of fixed points of system (1.7)

    x=xeγ(1x)βyx+h,y=yeβxx+h1,

    it's easy to find that there are three nonnegative fixed points E0=(0,0), E1=(1,0), and E2=(x0,y0) for β>h+1, where

    x0=1+1+4hβ22β2,y0=γx0(1x0).

    The Jacobian matrix of the system (1.7) at a fixed point E(x,y) is

    J(E)=(eγ(1x)βyh+x(1γx+βxy2(h+x)3/2)βxeγ(1x)βyh+xh+xyeβxh+x1(βh+xβx2(h+x)3/2)eβxh+x1),

    whose charactertistic polynomial reads as

    F(λ)=λ2Tr(J(E))λ+Det(J(E)),

    where

    Tr(J(E))=eγ(1x)βyh+x(1γx+βxy2(h+x)3/2)+eβxh+x1,
    Det(J(E))=eγ(1x)βyh+x+βxh+x1(1γx+βxy2(h+x)3/2+β2xyh+xβ2x2y2(h+x)2).

    In order to analyze the properties of the fixed points of the system (1.7), we utilize the Appendix definition and Lemma [17,18,19].

    By using Definition 5.1 and Lemma 5.2 in the Appendix, the following conclusions can be obtained.

    Theorem 2.1. The fixed point E0=(0,0) of the system (1.7) is a saddle.

    The proof for this theorem is simple and omitted here.

    Theorem 2.2. The type of the fixed point E1=(1,0) of the system (1.7) complies with the following results:

    Proof. The Jacobian matrix J(E1) of the system (1.7) at the fixed point E1 reads

    J(E1)=(1γβh+10eβh+11).

    Obviously, λ1=1γ and λ2=eβh+11.

    When 0<γ<2, |λ1|<1. If 0<β<h+1, then |λ2|<1, so E1is a sink; if β=h+1, then |λ2|=1, therefore E1 is non-hyperbolic; if β>h+1, meaning |λ2|>1, then E1 is a saddle.

    When γ=2, which reads |λ1|=1, E1 is non-hyperbolic.

    When γ>2, |λ1|>1. If 0<β<h+1, then |λ2|<1, so E1 is a saddle; if β=h+1, then |λ2|=1, therefore E1 is non-hyperbolic; if β>h+1, implying |λ2|>1, then E1 is a source. The proof is complete.

    We can easily derive the following result.

    Lemma 2.3. Consider the function f(x)=4x24x+7+(2x7)4x2+20x+1 with x(1,). Then f(x) is strictly increasing for x(1,), Furthermore, f(x) has a unique positive root X0 in (2, 2.5).

    Proof. Evidently, f(x)=4(2x1)+16x2+32x684x2+20x+1 and f(x)=8+64x3+520x2+912x712(4x2+20x+1)3>0, so, for x>1, f(x)>f(1)=0. Hence, f(x) is strictly increasing for x(1,). Again, f(2)=15357<0 and f(2.5)=22276>0. Therefore, f(x) has a unique positive root X0 in (2, 2.5).

    Now consider the stability of the fixed point E2.

    Theorem 2.4. For β>h+1, E2=(x0,y0)=(1+1+4hβ22β2,γ1+1+4hβ22β2(11+1+4hβ22β2)) is a positive fixed point of the system (1.7).

    Let X0 be the unique positive root of the function f(x)=4x24x+7+(2x7)4x2+20x+1 in (2, 2.5). Put β0=4h+2 and h0=4β44β2+7+(2β27)4β4+20β2+172β2. Denote γ0=8β2(1+1+4hβ2)3(1+4hβ2)+(72β2)1+4hβ2+4(1β2), where h>h0. Then the following consequences hold about the fixed point E2 illustrated in the Table 2.

    Table 1.  Properties of the positive fixed point E1.
    Conditions Eigenvalues Properties
    0<γ<2 0<β<h+1 |λ1|<1,|λ2|<1 sink
    β=h+1 |λ1|<1,|λ2|=1 nonhyperbolic
    β>h+1 |λ1|<1,|λ2|>1 saddle
    γ=2 |λ1|=1 nonhyperbolic
    γ>2 0<β<h+1 |λ1|>1,|λ2|<1 saddle
    β=h+1 |λ1|>1,|λ2|=1 nonhyperbolic
    β>h+1 |λ1|>1,|λ2|>1 source

     | Show Table
    DownLoad: CSV
    Table 2.  Properties of the fixed point E2.
    Conditions Eigenvalues Properties
    0<hX024 h+1<βX0 0<γ<γ0 β<β0 |λ1|<1,|λ2|<1 sink
    β=β0 |λ1|=1,|λ2|=1 nonhyperbolic
    β>β0 |λ1|>1,|λ2|>1 source
    γ=γ0 λ1=1,λ21 nonhyperbolic
    γ>γ0 |λ1|>1,|λ2|<1 saddle
    β>X0 hh0 |λ1|>1,|λ2|>1 source
    h>h0 0<γ<γ0 |λ1|>1,|λ2|>1 source
    γ=γ0 λ1=1,λ21 nonhyperbolic
    γ>γ0 |λ1|>1,|λ2|<1 saddle
    h>X024 h+1<βX0 0<γ<γ0 |λ1|<1,|λ2|<1 sink
    γ=γ0 λ1=1,λ21 nonhyperbolic
    γ>γ0 |λ1|>1,|λ2|<1 saddle
    β>X0 hh0 β<β0 |λ1|<1,|λ2|<1 sink
    β=β0 |λ1|=1,|λ2|=1 nonhyperbolic
    β>β0 |λ1|>1,|λ2|<1 source
    h>h0 0<γ<γ0 β<β0 |λ1|<1,|λ2|<1 sink
    β=β0 |λ1|=1,|λ2|=1 nonhyperbolic
    β>β0 |λ1|>1,|λ2|<1 source
    γ=γ0 λ1=1,λ21 nonhyperbolic
    γ>γ0 |λ1|>1,|λ2|<1 saddle

     | Show Table
    DownLoad: CSV

    Proof. The Jacobian matrix J(E2) of the system (1.7) at the fixed point E2 is

    J(E2)=(1γx0+γ(1x0)2β2x01γ(1x0)(112β2x0)1),

    whose characteristic polynomial can be written as

    F(λ)=λ2pλ+q, (2.1)

    where

    p=2γx0+γ(1x0)2β2x0,q=1+γ(12x0).

    Note that x0=1+1+4hβ22β2,y0=γx0(1x0), and E2=(x0,y0) is a positive fixed point, so 0<x0<1. It's easy to calculate that

    F(1)=γ(1x0)1+4hβ21+1+4hβ2>0,F(1)=4+γ[13x0+1x02β2x0]=4γ(6β2x20(2β21)x01)2β2x0=4γ(3(1+4hβ2)2+(72β2)1+4hβ2+4(1β2))2β2(1+1+4hβ2).

    If 6β2x2(2β21)x1=0 and x>0, then x=2β21+4β4+20β2+112β2. Simultaneously, it is easy to prove x<12.

    Notice that 0<h<β21. Moreover, x0>(=,<)xh>(=,<)h0. Additionally, β21h0=68β468β27(2β27)4β4+20β2+172β2. Set x=β2>1. Denote

    f(x)=4x24x+7+(2x7)4x2+20x+1

    and

    g(x)=68x268x7(2x7)4x2+20x+1.

    Lemma 2.3 tells us that f(x) is strictly increasing for x>1 and has a unique positive root X0 in (2, 2.5). From this one can see

    h0<(=,>)0f(β2)<(=,>)0β<(=,>)X0.

    Obiviously, g(1)=18>0, and g(x)=68(2x1)4x2+20x+116x232x+684x2+20x+1>68(2x1)(2x+1)16x232x+684x2+20x+1=256x232x4x2+20x+1>0. So, g(x)>g(1)>0 for x>1. This implies that h0<β21 always holds.

    It is easy to see x0>(=,<)x6β2x20(2β21)x01>(=,<)0. From F(1)=4γ(6β2x20+2β2x0+x01)2β2x0=0, one has

    γ=8β2x06β2x20(2β21)x01=:γ0=8β2(1+1+4hβ2)3(1+4hβ2)+(72β2)1+4hβ2+4(1β2).

    Again, β>(=,<)β0=4h+2x0<(=,>)1+1+4hβ202β20=12.

    Now, one considers the following two cases:

    1) Case Ⅰ: 0<hX024. Then β0=4h+2X0.

    (a) Subcase 1: h+1<βX0. Then h00<h, implying x<x0 and γ0>0.

    ⅰ. If 0<γ<γ0, then F(1)>0.

    ● For β<β0, q<1, which reads |λ1|<1 and |λ2|<1 by Lemma 6.2(i.1). So, E2 is a sink;

    ● For β=β0, q=1. Lemma 6.2(i.5) shows that |λ1|=|λ2|=1, so E2 is non-hyperbolic;

    ● For β>β0, q>1. Lemma 6.2(i.4) shows that |λ1|>1 and |λ2|>1, so E2 is a source.

    ⅱ. If γ=γ0, then F(1)=0. In other words, 1 is a root of the characteristic polynomial, namely E2 is non-hyperbolic.

    ⅲ. If γ>γ0, then F(1)<0. Using Lemma 6.2(i.3), we conclude that |λ1|<1 and |λ2|>1, so E2 is a saddle.

    (b) Subcase 2: β>X0. Then h0>0.

    ⅰ. If 0<hh0, then 0<x0x, implying that 6β2x20(2β21)x010. So, F(1)4>0. From β>X0β0, we see q>1. Lemma 6.2(i.4) shows that |λ1|>1 and |λ2|>1, so E2 is a source.

    ⅱ. If h>h0, then x<x0<1, implying that γ0>0.

    A. If 0<γ<γ0, then F(1)>0. For β>X0β0, q>1. Lemma 6.2(i.4) shows that |λ1|>1 and |λ2|>1, so E2 is a source.

    B. If γ=γ0, then F(1)=0. In other words, 1 is one root of the characteristic polynomial, namely, E2 is non-hyperbolic.

    C. If γ>γ0, then F(1)<0. Lemma 6.2(i.3) shows that |λ1|<1 and |λ2|>1, so E2 is a saddle.

    2) Case Ⅱ: h>X024. Then β0=4h+2>X0.

    (a) Subcase 1: h+1<βX0. Then h00<h, so, x<x0 and hence r0>0.

    ⅰ. If 0<γ<γ0, then F(1)>0. For h+1<βX0<β0, q<1, which reads |λ1|<1 and |λ2|<1 by Lemma 6.2(i.1). Therefore, E2 is a sink.

    ⅱ. If γ=γ0, then F(1)=0. Hence, E2 is non-hyperbolic.

    ⅲ. If γ>γ0, then F(1)<0. Lemma 6.2(i.3) shows that E2 is a saddle.

    (b) Subcase 2: β>X0. Then h0>0.

    ⅰ. If X024<hh0, then x0x, so, F(1)4>0.

    ● For X0<β<β0, q<1, which reads |λ1|<1 and |λ2|<1 by Lemma 6.2(i.1), thus, E2 is a sink;

    ● For β=β0, q=1. Lemma 6.2(i.5) shows that |λ1|=|λ2|=1, so E2 is non-hyperbolic;

    ● For β>β0, q>1. It follows from Lemma 6.2(i.4) that |λ1|>1 and |λ2|>1, hence E2 is a source.

    ⅱ. If h>h0, then x<x0, so, γ0>0.

    A. If 0<γ<γ0, then F(1)>0.

    ● For X0<β<β0, q<1. Lemma 6.2(i.1) tells us E2 is a sink;

    ● For β=β0, q=1. Therefore, E2 is non-hyperbolic;

    ● For β>β0, q>1. Lemma 6.2(i.4) shows that E2 is a source.

    B. If γ=γ0, then F(1)=0, which shows E2 is non-hyperbolic.

    C. If γ>γ0, then F(1)<0. Using Lemma 6.2(i.3), we conclude that |λ1|<1 and |λ2|>1, so E2 is a saddle.

    Summarizing the above analysis, the proof is complete.

    In this section, we apply the center manifold theorem and bifurcation theory to investigate the local bifurcation problems of the system at the fixed points E1 and E2.

    It follows from Eq (1.4) that the fixed point E1 always exists, regardless of what values the parameters β and γ take. One can see from Theorem 2.2 that the fixed point E1 is a non-hyperbolic fixed point when β=h+1 or γ=2. As soon as the parameters β or γ goes through corresponding critical values, the dimensional numbers for the stable manifold and the unstable manifold of the fixed point E1 vary. Therefore, a bifurcation probably occurs. Now, the considered parameter case is divided into the following three subcases:

    Case Ⅰ: β=h+1, γ2;

    Case Ⅱ: βh+1, γ=2;

    Case Ⅲ: β=h+1, γ=2.

    First we consider Case Ⅰ: β=h+1, γ2, i.e., the parameters (h,β,γ)Ω1={(h,β,γ)R3+h>0,β>0,γ>0,γ2}. Then, the following result is obtained.

    Theorem 3.1. Suppose the paramenters (h,β,γ)Ω1. Let β1=h+1. If the parameter β varies in a small neighborhood of the critical value β1, then the system (1.7) experiences a transcritical bifurcation at the fixed point E1 when the parameter β goes through the critical value β1.

    Proof. First, assume that un=xn1,vn=yn0, which transforms the fixed point E1 to the origin, and the system (1.7) to

    {un+1=(un+1)eγunβvnun+h+11,vn+1=vneβ(un+1)un+h+11. (3.1)

    Second, giving a small perturbation β of the parameter β around β1, i.e., β=ββ1 with 0<|β|1, and letting βn+1=βn=β, the system (3.1) is perturbed into

    {un+1=(un+1)eγun(βn+β1)vnun+h+11,vn+1=vne(βn+β1)(un+1)un+h+11,βn+1=βn. (3.2)

    By the Taylor expansion, the system (3.2) at (un,vn,βn)=(0,0,0) can be written as

    (unvnβn)(1γ10010001)(unvnβn)+(g1(un,vn,βn)+o(ρ31)g2(un,vn,βn)+o(ρ31)0), (3.3)

    where ρ1=u2n+v2n+βn2,

    g1(un,vn,βn)=u2n(γ22γ)+v2n2+unvn(γ1+12(h+1))vnβnh+1v3n6+u3n(γ36+γ22)+u2nvn(12(h+1)38(h+1)2+γγ2(h+1)γ22)+unv2n(1212(h+1)γ2)+v2nβnh+1+unvnβn(γh+11h+1+12(h+1)32),g2(un,vn,βn)=unvn(112(h+1))+vnβnh+1+u2nvn2(11h+1)2+vnβn22(h+1)+unvnβn(2h+11(h+1)32).

    It is easy to derive the three eigenvalues of the matrix

    A=(1γ10010001),

    to be λ1=1γ and λ2=λ3=1 with corresponding eigenvectors

    ξ1=(100),ξ2=(1γ10),ξ3=(001).

    Notice 0<γ2 implies that |λ1|1.

    Set T=(ξ1,ξ2,ξ3), i.e.,

    T=(11γ0010001),

    then,

    T1=(11γ0010001).

    Taking the following transformation

    (un,vn,βn)T=T(Xn,Yn,δn)T,

    the system (3.3) is changed into the following form

    (XnYnδn)(1γ00010001)(XnYnδn)+(g3(Xn,Yn,δn)+o(ρ32)g4(Xn,Yn,δn)+o(ρ32)0), (3.4)

    where ρ2=X2n+Y2n+δ2n,

    g3(Xn,Yn,δn)=g1(Xn1γYn,Yn,δn)+1γg2(Xn1γYn,Yn,δn),g4(Xn,Yn,δn)=g2(Xn1γYn,Yn,δn).

    Assume on the center manifold

    Xn=h(Yn,δn)=a20Y2n+a11Ynδn+a02δ2n+o(ρ23),

    where ρ3=Y2n+δ2n, then, from

    Xn+1=(1γ)h(Yn,δn)+g1(h(Yn,δn)1γYn,Yn,δn)+1γg2(h(Yn,δn)1γYn,Yn,δn)+o(ρ23),h(Yn+1,δn+1)=a20Y2n+1+a11Yn+1δn+1+a02δ2n+1+o(ρ23)=a20(Yn+g2(Xn1γYn,Yn,δn)2+a11(Yn+g2(Xn1γYn,Yn,δn)δn+a02δ2n+o(ρ23),

    and Xn+1=h(Yn+1,δn+1), we obtain the center manifold equation

    (1γ)h(Yn,δn)+g1(h(Yn,δn)1γYn,Yn,δn)+1γg2(h(Yn,δn)1γYn,Yn,δn)+o(ρ23)=a20(Yn+g2(Xn1γYn,Yn,δn)2+a11(Yn+g2(Xn1γYn,Yn,δn)δn+a02δ2n+o(ρ23).

    By comparing the corresponding coefficients of terms with the same order in the above center manifold equation, it is easy to derive that

    a20=2hγ1γ3(2h+2),a11=1γγ2h+1,a02=0.

    Therefore, the system (3.4) restricted to the center manifold is given by

    Yn+1=f1(Yn,δn):=Yn+g2(h(Yn,δn)1γYn,Yn,δn)+o(ρ33)=Yn+2h+1γ(2h+2)Y2nYnδnh+1+o(ρ23)

    It is not difficult to calculate

    f1(Yn,δn)|(0,0)=0,f1Yn|(0,0)=1,f1δn|(0,0)=0,2f1Ynδn|(0,0)=1h+10,2f1Y2n|(0,0)=2h+1γ(h+1)0.

    According to (21.1.43)–(21.1.46) in [24, p507], for a transcritical bifurication to occur, all conditions hold, hence, the system (1.7) undergoes a transcritical bifurcation at the fixed point E1. The proof is over.

    Next we consider Case Ⅱ: βh+1, γ=2. By Theorem 2.2, one can see that λ1=1 and |λ2|1when βh+1, γ=2. Thereout, the following result can be derived.

    Theorem 3.2. Let γ1=2. Suppose the paramenters (h,β,γ)Ω2={(h,β,γ)R3+h>0,β>0,βh+1,γ>0}. If the parameter γ varies in a small neighborhood of the critical value γ1, then the system (1.7) undergoes a period-doubling bifurcation at the fixed point E1 when the parameter γ goes through the critical value γ1.

    Proof. Shifting E1=(1,0) to the origin O(0,0) and giving a small perturbation γ of the parameter γ at the critical value γ1 with 0<|γ|1, the system (3.1) is changed into the following form:

    {un+1=(un+1)e(γ+2)unβvnun+h+11,vn+1=vneβ(un+1)un+h+11. (3.5)

    Set γn+1=γn=γ, then (3.5) can be seen as

    {un+1=(un+1)e(γ+2)unβvnun+h+11,vn+1=vneβ(un+1)un+h+11,γn+1=γn. (3.6)

    By the Taylor expansion, the system (3.6) at (un,vn,βn)=(0,0,0) can be expended into

    (unvnβn)(1βh+100eβh+110001)(unvnβn)+(g5(un,vn,γn)+o(ρ34)g6(un,vn,γn)+o(ρ34)0), (3.7)

    where ρ4=u2n+v2n+γn2,

    g5(un,vn,γn)=v2nβ22(h+1)+unvn(βh+1+β2(h+1)32)vnγn+2u3n3v3nβ36(h+1)32unvn(β2(h+1)323β8(h+1)52)+u2nγnunv2n(β22(h+1)2+β22(h+1))+unvnγnβh+1,g6(un,vn,γn)=unvn[β(1h+112(h+1)32) e(βh+11)]u2nvne(βh+11)[β(12(h+1)3238(h+1)52)β2(1h+112(h+1)32)22].

    It is not difficult to derive the three eigenvalues of the matrix

    A=(1βh+100eβh+110001),

    to be λ1=1, λ2=e(βh+11) and λ3=1 with corresponding eigenvectors

    ξ1=(100),ξ2=(β(eβh+11+1)h+110),ξ3=(001).

    Notice βh+1 implies |λ2|1.

    Set T=(ξ1,ξ2,ξ3), i.e.,

    T=(1β(eβh+11+1)h+10010001),

    then,

    T1=(1β(eβh+11+1)h+10010001).

    Taking the following transformation

    (un,vn,γn)T=T(Xn,Yn,δn)T,

    the system (3.7) is changed into the following form

    (XnYnδn)(1000eβh+110001)(XnYnδn)+(g7(Xn,Yn,δn)+o(ρ35)g8(Xn,Yn,δn)+o(ρ35)0), (3.8)

    where ρ5=X2n+Y2n+δ2n,

    g7(Xn,Yn,δn)=g5(Xnβ(eβh+11+1)h+1Yn,Yn,δn)+β(eβh+11+1)h+1g6(Xnβ(eβh+11+1)h+1Yn,Yn,δn),g8(Xn,Yn,δn)=g6(Xnβ(eβh+11+1)h+1Yn,Yn,δn).

    Suppose on this center manifold

    Yn=h(Xn,δn)=b20X2n+b11Xnδn+b02δ2n+o(ρ26),

    where ρ6=X2n+δ2n, which must satisfy

    Yn+1=eβh+11h(Yn,δn)+g8(Xn,h(Yn,δn),δn)+o(ρ36).

    Similar to Case Ⅰ, one can establish the corresponding center manifold equation. Comparing the corresponding coefficients of terms with the same type in the equation produces

    b20=0,b11=0,b02=0.

    That is to say, Yn=h(Xn,δn)=o(ρ26). Therefore, the center manifold equation is given by

    Xn+1=f2(Xn,δn):=Xn+g7(Xn,h(Yn,δn)=Xn+g5(Xnβ(eβh+11+1)h+1h(Xn,δn),h(Xn,δn),δn)+β(eβh+11+1)h+1g6(Xnβ(eβh+11+1)h+1h(Xn,δn),h(Xn,δn),δn)+o(ρ36)=XnXnδn+X2nδn+23X3n+o(ρ36).

    Thereout, one has

    f22(Xn,δn)=f2(f2(Xn,δn),δn)=Xn+2Xnδn+Xnδ2n43X3n+o(ρ36).

    Therefore, the following results are derived:

    f2(0,0)=0,f2Xn|(0,0)=1,f22δn|(0,0)=0,2f22X2n|(0,0)=0,2f22Xnδn|(0,0)=20,3f22Xn3|(0,0)=80,

    which, according to (21.2.17)–(21.2.22) in [24, p516], satisfy all conditions for a period-doubling bifurcation to occur. Therefore, the system (1.7) undergoes a period-doubling bifurcation at E1. Again,

    3f22X3n|(0,0)/2f22Xnδn|(0,0)=4(>0).

    Therefore, the period-two orbit bifurcated from E1 lies on the right of γ1=2.

    Of course, one can also compute the following two quantities, which are the transversal condition and non-degenerate condition for judging the occurrence and stability of a period-doubling bifurcation, respectively (see [3,15,16,17,18]),

    α1=(2f2Xnδn+12f2δn2f2Xn2)|(0,0),α2=(163f2Xn3+(122f2Xn2)2)|(0,0).

    It is easy to say α1=1 and α2=23. Due to α2>0, the period-two orbit bifurcated from E1 is stable. The proof is complete.

    Finally, we consider Case Ⅲ: β=h+1, γ=2. At this time, the two eigenvalues of the linearized matrix evaluated at this fixed point E1 are λ1=1 and λ2=1. The bifurcation problem in this case is very complicated and will be considered future work.

    Consider the bifurcation of the system (1.7) at the fixed point E2. The parameters are divided into the following three cases:

    Case Ⅰ: β=4h+2, γ8(4h+2)4h+1;

    Case Ⅱ: β4h+2, γ=8β2(1+1+4hβ2)3(1+4hβ2)2+(72β2)1+4hβ2+4(1β2);

    Case Ⅲ: β=4h+2, γ=8(4h+2)4h+1.

    According to our calculations, there is no bifurcation under Case Ⅱ. Additionally, the bifurcation problem in case Ⅲ is very complicated and will be considered future work. Therefore, we only consider Case Ⅰ.

    Suppose the paramenters

    (h,β,γ)Ω3={(h,β,γ)R3+h>0,β>0,γ>0,γ8(4h+2)4h+1}.

    Then the following result may be obtained.

    Theorem 3.3. Suppose the paramenters (h,β,δ)Ω3 and meet γ<8(4h+2)4h+1. Let β2=4h+2. Then the system (1.7) undergoes a Neimark-Sacker bifurcation at the fixed point E2 when the parament β varies in a small neighborhood of the critical value β2. Moreover, if L<(>)0 in (3.13), then a (an) stable (unstable) invariant closed orbit is bifurcated out from the fixed point E2 of system (1.7) when β>(<)β2.

    Proof. First, give a small perturbation β of the parameter β around β2 in the system (3.1), i.e., β=ββ2 with 0<|β|1, and set x01=x01(β)=1+1+4h(β+β2)22(β+β2)2 and y01=γx01(1x01). Under the perturbation, the system (3.1) reads

    {un+1=(un+x01)eγ(1(unx01))(β+β2)[vn+γx01(1x01)]un+x01+hx01,vn+1=(vn+y01)e((β+β2)(un+x01)un+x01+h1)y01. (3.9)

    The characteristic equation of the linearized equation of the system (3.9) at the origin (0, 0) is

    F(λ)=λ2p(β)λ+q(β)=0, (3.10)

    where

    p(β)=2γx01+γ(1x01)2(β+β2)2x01,q(β)=1+γ(12x01).

    Notice β2=4h+2. For γ<8(4h+2)4h+1, 2<p(0)<2, q(0)=1, so p2(0)4q(0)<0, and hence the two roots of F(λ)=0 are

    λ1,2(β)=ω±μi,

    where ω=12p(β), μ=124q(β)p2(β).

    It is easy to obseve that |λ1,2(β)|=q(β) and (|λ1,2(β)|)|β=0=q(0)=1. Therefore, a Neimark-Sacker bifurcation probably occurs.

    The occurrence of the Neimark-Sacker bifurcation requires the following two conditions to be satisfied:

    1) (d|λ1,2(β)|dβ)|β=00;

    2) λi1,2(0)1,i=1,2,3,4.

    Notice

    (d|λ1,2(β)|dβ)|β=0=γ(2h+1)(4h+1)4h+20.

    Obviously λi1,2(0)1 for i=1,2,3,4, so the two conditions are satisfied.

    Second, in order to derive the normal form of the system (3.9), one expands (3.9) in power series up to the third-order term around the origin to get

    {un+1=a10un+a01vn+a20u2n+a11unvn+a02v2n+a30u3n+a21u2nvn+a12unv2n+a03v3n+o(ρ37),vn+1=b10un+b01vn+b20u2n+b11unvn+b02v2n+b30u3n+b21u2nvn+b12unv2n+b03v3n+o(ρ37), (3.11)

    where ρ7=u2n+v2n,

    a10=γ8h+4γ2+1,a01=1,a20=(γ2γ8h+4)2γ+γ4h+23γ32(h+12)2,a11=γ+γ4h+2+12h+12,a02=1,a30=(γγ4h+2)(γ2γ8h+4)3γ16(h+12)2+3γ(γ2γ8h+4)32(h+12)2+5γ64(h+12)3(γγ4h+2)((γγ4h+2)(γ6γ24h+12)3γ32(h+12)2)2,
    a21=2γ(γγ4h+2)(γ6γ24h+12)+1h+1238(h+12)2γ2h+1γ2γ8h+42h+1(γγ4h+2)(2γ3γ6h+3+12h+1)2+3γ16(h+12)2,a12=γ4h+21h+12γ+2,a03=23,b10=γ(114h+2)2,b01=1,b20=γ[2(114h+2)21h+1238(h+12)2]4,b02=0,b11=218h+4,b03=0,b21=2(114h+2)21(h+12)+38(h+12)2,b12=0,
    b30=γ4(212h+1)(2(114h+2)2312h+1+316(h+12)2,)γ4(2h+1212(h+12)3/2)(12(h+12)316(h+12)3/2)+γ4(34(h+12)2516(h+12)3).

    Take matrix

    T=(0a01μ1ω), then T1=(ω1μa011μ1a010).

    Make a change of variables

    (u,v)T=T(X,Y)T,

    then the system (3.11) is changed to the following form:

    (XY)(ωμμω)(XY)+(F(X,Y)+o(ρ48)G(X,Y)+o(ρ48)), (3.12)

    where ρ8=X2+Y2,

    F(X,Y)=c20u2+c11uv+c02v2+c30u3+c21u2v+c12uv2+c03v3,G(X,Y)=d20u2+d11uv+d02v2+d30u3+d21u2v+d12uv2+d03v3,u=a01Y,v=μX+(1ω)Y,c20=a20(ω1)μa01+b20μ,c11=a11(ω1)μa01+b11μ,c02=a02(ω1)μa01+b02μ,c30=a30(ω1)μa01+b30μ,c21=a21(ω1)μa01+b21μ,c12=a12(ω1)μa01+b12μ,c03=a03(ω1)μa01+b03μ,d20=a20a01,d11=a11a01,d02=a02a01,d30=a30a01,d21=a21a01,d12=a12a01,d03=a03a01

    Furthermore,

    FXX|(0,0)=2c02μ3,FXY|(0,0)=c11a01μ+2c02μ(1ω),FYY|(0,0)=2c02a201+2c11a01(1ω),FXXX|(0,0)=6c03μ3,FXXY|(0,0)=2c21a01μ2+6c03μ2(1ω),FXYY|(0,0)=2c21a201μ+4c12a01μ(1ω)+6c03μ(1ω)2,FYYY|(0,0)=4(1ω)3+6c30a301+4c21a201(1ω)+6c12a01(1ω)2,GXX|(0,0)=2d02μ3,GXY|(0,0)=d11a01μ+2d02μ(1ω),GYY|(0,0)=2d02a201+2d11a01(1ω),GXXX|(0,0)=6c03μ3,GXXY|(0,0)=2d21a01μ2+6d03μ2(1ω),GXYY|(0,0)=2d21a201μ+4d12a01μ(1ω)+6d03μ(1ω)2,GYYY|(0,0)=4(1ω)3+6d30a301+4d21a201(1ω)+6d12a01(1ω)2.

    To determine the stability and direction of the bifurcation curve (closed orbit) for the system (1.7), the discriminating quantity L should be calculated and not to be zero, where

    L=Re((12λ1)λ221λ1ζ20ζ11)12|ζ11|2|ζ02|2+Re(λ2ζ21), (3.13)
    ζ20=18[FXXFYY+2GXY+i(GXXGYY2FXY)]|(0,0),ζ11=14[FXX+FYY+i(GXX+GYY)]|(0,0),ζ02=18[FXXFYY2GXY+i(GXXGYY+2FXY)]|(0,0),ζ21=116[FXXX+FXYY+GXXY+GYYY+i(GXXX+GXYYFXXYFYYY)]|(0,0).

    Based on [24,25,26], we see that if L<(>)0, then an attracting (a repelling) invariant closed curve bifurcates from the fixed point for β>(<)β2.

    The proof of this theorem is complete.

    In this section, we utilize Matlab to perform numerical simulations to validate the above theoretical analysis through utilizing bifurcation diagrams, phase portraits, maximum Lyapunov expoents, and fractal dimensions of the system (1.7) at the fixed point E2.

    Consider the fixed point E2. Vary β in the range (1.4,1.85), and fix γ=2,h=0.2 with the initial value (x0,y0)=(0.4,0.5). Figure 1(a) shows that the existence of a Neimark-Sacker bifurcation at the fixed point E2=(0.5,0.5) when β=β2=2.81.6733. Figure 1(b) describes the spectrum of maximum Lyapunov exponents, which are positive for the parameter β(1.4,1.85), which leads to chaos in system (1.7). For this, the interested readers may refer to [28] to create an electronic emulator to get immediate results.

    Figure 1.  Bifurcation of the system (1.7) in (β,x)-plane and maximal Lyapunov exponents.

    The phase portraits associated with Figure 1(a) are drawn in Figure 2. When β increases, a circular curve enclosing the fixed point E2 appears.

    Figure 2.  Phase portraits for the system (1.7) with γ=2,h=0.2 and different β with the initial value (x0,y0)=(0.4,0.5) outside the closed orbit.

    By choosing a different initial value (x0,y0) = (0.52, 0.48) and three same values of β, the correspending phase portraits are plotted in Figure 3. Figure 2 implies that the closed curve is stable outside, while Figure 3 indicates that the closed curve is stable inside. That is to say, a stable invariant closed curve around the fixed point E2 occurs. This agrees with the conclusion in Theorem 3.3.

    Figure 3.  Phase portraits for the system (1.7) with γ=2,h=0.2 and different β with the initial value (x0,y0)=(0.52,0.48) inside the closed orbit.

    In this paper, we consider a predator–prey model with the prey individual behaviour and herd behaviour. By using the semi-discretization method, the continuous system (1.4) is transformed to the discrete system (1.7). Under the given parametric conditions, we demonstrate the existence and stability of three nonnegative fixed points E0=(0,0), E1=(1,0) and E2=(1+1+4hβ22β2,γ1+1+4hβ22β2(11+1+4hβ22β2)). By using the center manifold theory, we determine the existence conditions of transcritical bifurcation and period-doubling bifurcation in the fixed point E1 and the Neimark-Sacker bifurcation at the fixed point E2 of system (1.7). we also derive that E2 is asymptotically stable when β>β2=4h+2 and unstable when β<β2. Additionally, the system (1.7) undergoes a Neimark-Sacker bifurcation when the parameter β goes through the critical value β2. The occurrence for this phenomenon of Neimark-Sacker bifurcation indicates the coexistence of prey and predator when the parameter β=β2.

    Our findings indicate that the proposed discrete model shows a behaviour similar to the one found in the corresponding continuous model [27]. In particular, it gives rise to stable populations limit cycles. Ecologically, this means that the suggested response function may be adequate if we want to model the prey herd behaviour that takes place only for a sizable population, namely when the population level settles in a certain threshold (critical value).

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

    This work is partly supported by the National Natural Science Foundation of China (61473340), the Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Province (F703108L02), and the Natural Science Foundation of Zhejiang University of Science and Technology (F701108G14). The authors are specially thankful to the anonymous referee for his/her presenting us the electric version of the monograph [28], which is really helpful to us.

    The authors declare that they have no competing interests. All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

    We here give a definition and a key Lemma.

    Definition 5.1. Let E(x,y) be a fixed piont of the system (1.7) with multipliers λ1 and λ2.

    (i) If |λ1|<1 and |λ2|<1, E(x,y) is called sink, so a sink is locally asymptotically stable.

    (ii) If |λ1|>1 and |λ2|>1, E(x,y) is called source, so a source is locally asymptotically unstable.

    (iii) If |λ1|<1 and |λ2|>1 (or |λ1|>1 and |λ2|<1), E(x,y) is called saddle.

    (iv) If either |λ1|=1 or |λ2|=1, E(x,y) is called to be non-hyperbolic.

    Lemma 5.2. Let F(λ)=λ2+Bλ+C, where B and C are two real constants. Suppose λ1 and λ2 are two roots of F(λ)=0. Then the following statements hold.

    (i) If F(1)>0, then

    (i.1) |λ1|<1 and |λ2|<1 if and only if F(1)>0 and C<1;

    (i.2) λ1=1 and λ21 if and only if F(1)=0 and B2;

    (i.3) |λ1|<1 and |λ2|>1 if and only if F(1)<0;

    (i.4) |λ1|>1 and |λ2|>1 if and only if F(1)>0 and C>1;

    (i.5) λ1 and λ2 are a pair of conjugate complex roots and, |λ1|=|λ2|=1 if and only if 2<B<2 and C=1;

    (i.6) λ1=λ2=1 if and only if F(1)=0 and B=2.

    (ii) If F(1)=0, namely, 1 is one root of F(λ)=0, then another root

    λ satisfies |λ|=(<,>)1 if and only if |C|=(<,>)1.

    (iii) If F(1)<0, then F(λ)=0 has one root lying in (1,). Moreover,

    (iii.1) the other root λ satisfies λ<(=)1 if and only if F(1)<(=)0;

    (iii.2) the other root 1<λ<1 if and only if F(1)>0.

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