Research article Special Issues

The mechanism of resistance-reducing/anti-adhesion and its application on biomimetic disc furrow opener

  • Received: 26 May 2020 Accepted: 01 July 2020 Published: 07 July 2020
  • The black soil of Northeast China is sticky and agglomerates easily, which often adheres to the surface of a traditional furrow opener during the furrowing process. In this paper, biomimetic design principles in resistance-reducing, anti-adhesion and resistance-reducing mechanism of biomimetic disc furrow opener were studied. Nine kinds of singular convex hull, nine kinds of singular wedge and nine kinds of mixed convex hull and wedge structural biomimetic disc furrow opener were designed, and the furrowing process with the soil simulated by finite element method (FEM).Three types of biomimetic disc furrow opener with less resistance were manufactured by laser processing for comparative test in soil bin based on the simulation results. The test results showed that the resistance of the biomimetic disc furrow opener was less than that of the ordinary disc. The soil-disc stress, influence of biomimetic structures, moisture content and furrow speeds on resistance were discussed. The resistance-reducing rate of D-BC-3 reached the maximum value 15.36% at the furrow speed of 0.6 m/s and the soil moisture content of 20%. It is believed that the biomimetic design principles can provide the significant inspirations for the future design of disc furrow opener with drag reduction.

    Citation: Jiyu Sun, Yueming Wang, Shujun Zhang, Yunhai Ma, Jin Tong, Zhijun Zhang. The mechanism of resistance-reducing/anti-adhesion and its application on biomimetic disc furrow opener[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 4657-4677. doi: 10.3934/mbe.2020256

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  • The black soil of Northeast China is sticky and agglomerates easily, which often adheres to the surface of a traditional furrow opener during the furrowing process. In this paper, biomimetic design principles in resistance-reducing, anti-adhesion and resistance-reducing mechanism of biomimetic disc furrow opener were studied. Nine kinds of singular convex hull, nine kinds of singular wedge and nine kinds of mixed convex hull and wedge structural biomimetic disc furrow opener were designed, and the furrowing process with the soil simulated by finite element method (FEM).Three types of biomimetic disc furrow opener with less resistance were manufactured by laser processing for comparative test in soil bin based on the simulation results. The test results showed that the resistance of the biomimetic disc furrow opener was less than that of the ordinary disc. The soil-disc stress, influence of biomimetic structures, moisture content and furrow speeds on resistance were discussed. The resistance-reducing rate of D-BC-3 reached the maximum value 15.36% at the furrow speed of 0.6 m/s and the soil moisture content of 20%. It is believed that the biomimetic design principles can provide the significant inspirations for the future design of disc furrow opener with drag reduction.


    Spatial patterns are widespread in ecological and chemical systems, such as salt marshes [1,2,3], predator-prey systems [4,5,6], the Brusselator model [7,8,9], Sel'kov model [10,11], Lengyel-Epstein system [12,13] and Degn-Harrison system [14,15]. It was proposed that spatial patterns may unveil underlying mechanisms that drive ecological resilience, and may serve as signals for environmental changes in many ecological systems (see [16,17,18]). It is assumed that self-organization theory is of great significance in helping us to understand spatial patterns, which was first proposed by Alan Turing in his seminal work [19]. Mathematically, asymptotic dynamics, which corresponds to persistent patterns, has been intensively studied, and the underlying mechanism was proposed as scale-dependent alternation between facilitation and inhibitory interactions, also called scale-dependent feedback [20]. There are many interesting persistent patterns which have been discovered, such as stable fairy circles and ring type patterns ([21,22,23,24,25,26]). There also is an increasing recognition that dynamics on ecological time scales, called transient patterns, may be of some significance. The impermanence of transient patterns means that an ecological system in a transient state may change abruptly, even without any underlying change in environmental conditions, while the possibility of transient patterns implies that an ecological system may remain far from its asymptotic patterns for some period of time [27]. In a recent study conducted by Zhao et al. [28], transient fairy circles of S. alterniflora were observed in salt marsh pioneer zones in the Yangtze estuary north branch in eastern China coasts. The authors proposed that hydrogen sulfide (H2S) may serve as a feedback regulator to drive such transient fairy circles, and proposed the following mathematical model to demonstrate the sulfide feedback mechanism:

    {P(x,t)t=DPΔP(x,t)+rP(x,t)(1P(x,t)K)cPS,S(x,t)t=DSΔS(x,t)+ξ[ϵP(x,t)dksP(x,t)+ksS]. (1.1)

    The plant biomass concentration at the location x and time t is denoted by P(x,t), where x is a point on the plane. Assume the plant population growth follows the Verhulst-Pearl logistic pattern with the intrinsic growth rate r and the carrying capacity K. The sulfide concentration is denoted by S(x,t). Organic matters including plant biomass in intertidal salt marshes can produce hydrogen sulfide (H2S). Assume the plant population produces hydrogen sulfide proportionally with the effective production rate ϵ. Hydrogen sulfide is toxic to plants and can lead plants to die off. It was assumed that the loss of plants is to increase with an increase of the sulfide concentration and plant biomass, represented by cPS. As field studies about salt marsh plant species S. mariqueter and S. alterniflora show that plant lateral expansion through vegetative growth can be described by random walk, the diffusion process without drifts was employed to model plant dispersal with diffusion coefficient DP. DS is the planar dispersion rate of the sulfide concentration. In addition, plants produce dissolved organic carbon which promote bioactivities of sulfate-reducing bacteria which, in turn, promote sulfide enrichment. This effect is represented by the term ksP(x,t)+ks. The parameter d is the maximum escape rate of sulfide through the mud-air interface. The parameter ξ is dimensionless, which controls the time scale between the plant biomass and sulfide concentration. For a more detailed description of the model (1.1), we refer the reader to [28,29,30]. The conclusion of the study [28] is that transient fairy circle patterns in intertidal salt marshes can both infer the underlying ecological mechanisms and provide a measure of ecosystem resilience.

    In this study, we would like to investigate how the self-correcting mechanism of the plant population influence transient patterns of fairy circles in salt marshes. For a single population growth, Hutchinson considered that for KPK formally describing a self-regulatory mechanism, the only formal conditions that must be imposed on the biologically possible mechanism is that they operate so rapidly that the lag, τ, is negligible between t when any given value of P is reached, and the establishment of the appropriately corrected value of the effective reproductive rate rKPK [31]. Now, we would like to consider the "negligible time lag" for the plant population. It should be noticed that the time lag may not be negligible since different populations may have different growth properties. Particularly, for plant populations within constricted environments, the self-correcting adjustment with time lag may be necessary for understanding the underlying ecological mechanisms. Thus, the time delay will be incorporated in the self-regulatory mechanism. That is, the logistic growth term rP(x,t)(1P(x,t)K) for the plant population will be replaced by rP(x,t)(1P(x,tτ)K). Thus, we will have a delayed plant-sulfide feedback model as follows:

    {P(x,t)t=DPΔP(x,t)+rP(x,t)(1P(x,tτ)K)cP(x,t)S(x,t),S(x,t)t=DSΔS(x,t)+ξ[ϵP(x,t)dksP(x,t)+ksS(x,t)]. (1.2)

    For simplicity, we introduce the following non-dimensional variables

    u=Pks,v=cSr,ˆt=rt,ˆτ=rτ,k=Kks,
    a=cϵξksr2,b=dξr,d1=DPr,d2=DSr.

    Dropping the hats of ˆt and ˆτ, we get the system with Neumann boundary conditions and initial conditions as follows

    {u(x,t)t=d1Δu+u(x,t)(1u(x,tτ)k)uv,xΩ,t>0,v(x,t)t=d2Δv+aubvu+1,xΩ,t>0,u(x,t)ν=v(x,t)ν=0,xΩ,t>0,u(x,t)=u0(x,t)0,v(x,t)=v0(x,t)0,x¯Ω,τt0, (1.3)

    where ν is the outward unit normal vector on Ω.

    The aim of this article is to conduct a detailed analysis about the effect of the time delay on the dynamics of the system (1.3). Our analysis shows that there is a critical value of the time delay. When the time delay is greater than the critical value, the system will have temporal periodic solutions. Since there are no analytical methods to study transient patterns yet, we will use numerical simulations to study transient patterns. We show that there are transient fairy circles for any time delay. However, there are different types of fairy circles and rings occurring in this system.

    The rest of this paper is organized as follows. In Section 2, we mainly study the stability of the positive steady state of the system (1.3) by discussing the distribution of the eigenvalues, and give the sufficient conditions for the occurrence of Hopf bifurcations induced by the time delay. In Section 3, by using the center manifold theory and normal form theory for partial differential equations, we analyze the properties of Hopf bifurcations, and obtain the formulas determining the direction of Hopf bifurcation and stability of bifurcating periodic solutions. In Section 4, we conduct numerical studies to demonstrate transient patterns, and give some simulations to illustrate our theoretical results. The paper is closed with a brief discussion.

    Clearly, the system (1.3) has two nonnegative constant steady states: E0=(0,0) and E=(u,v), where

    u=(ak+b)+(ak+b)2+4abk22ak,v=1uk.

    We claim that 0<u<k. In fact,

    ku=2ak2+ak+b(ak+b)2+4abk22ak=2ak2(1+k)2ak2+ak+b+(ak+b)2+4abk2>0.

    Notice that a,b,k>0, and we can obtain that E=(u,v) is the unique positive steady state of the system (1.3).

    Let ΩR2 be a bounded domain with smooth boundary Ω, and denote

    u1(x,t)=u(x,t),u2(x,t)=v(x,t),U(x,t)=(u1(t),u2(t))T.

    In the phase space C=C([τ,0],X), we can rewrite the system (1.3) as

    ˙U(t)=DΔU(t)+F(Ut), (2.1)

    where D=diag(d1,d2), Ut()=U(t+), and F:CX is defined by

    F(Ut)=(u1(t)(1u1(tτ)k)u1(t)u2(t)au1(t)bu2(t)1+u1(t)).

    By calculation, we have the linearization of system (2.1) at E, which can be written as

    dU(t)dt=DU(t)+L(Ut), (2.2)

    where L:CX is given by

    L(ϕt)=L1ϕ(0)+L2ϕ(τ)

    and

    L1=(0ua+bv(1+u)2b1+u),L2=(uk000),
    ϕ(t)=(ϕ1(t),ϕ2(t))T,ϕt()=(ϕ1t(),ϕ2t())T.

    Let

    0=μ0<μ1<μ2<

    be all the eigenvalues of the operator Δ on Ω with the Neumann boundary conditions. Using the techniques in [32], we obtain that the characteristic equation of the system (2.2) can be expressed as

    det(λI2+QnL1L2eλτ)=0,nN0=N{0}={0,1,2,3,}, (2.3)

    where I2 is the 2×2 identity matrix and Qn=μndiag(d1,d2). That is, each eigenvalue λ should satisfy the equation as follows

    λ2+Tnλ+Dn+(B+Mn)eλτ=0,nN0, (2.4)

    where

    Tn=(d1+d2)μn+b1+u,Dn=d1d2μ2n+bd11+uμn+au+buv(1+u)2,B=uk>0,Mn=uk(μnd2+b1+u).

    When τ=0, the characteristic equation (2.4) becomes

    λ2+(Tn+B)λ+Dn+Mn=0,nN0. (2.5)

    We give the following conclusion about the stability of the positive steady state E as τ=0.

    Theorem 2.1. When τ=0, all of the roots of the Eq (2.4) have negative real parts. That is, the positive steady state E of the system (1.3) without delay is locally asymptotically stable.

    Proof. If Tn+B>0,Dn+Mn>0 for all nN0, then all roots of the Eq (2.5) have negative real parts. By calculation, we have that 0<u<k and v>0. Recall that k>0,a>0, and b>0, we can get

    Tn+B>Tn1+B>>T0+B=b1+u+uk>0,
    Dn+Mn>Dn1+Mn1>>D0+M0=buk(1+u)+au+buv(1+u)2>0,

    for any nN. So, we complete the proof.

    In the following, we will study the effect of delay on the stability of E. Recall that Dn+Mn>0 for any nN0, we can see that 0 is not the root of (2.4). Now, we will check whether there exist the critical values of τ such that (2.4) has a pair of simple purely imaginary eigenvalues for some nN0. Let ±iω(ω>0) be the solutions of the (n+1)-th Eq (2.4), then we get

    ω2+Tniω+Dn+(Biω+Mn)eiωτ=0.

    Separating the real and imaginary parts, and we obtain that ω and τ should satisfy

    {ω2Dn=Mncosωτ+Bωsinωτ,Tnω=MnsinωτBωcosωτ. (2.6)

    It yields

    ω4+(T2nB22Dn)ω2+D2nM2n=0, (2.7)

    where

    T2nB22Dn=(d21+d22)μ2n+2bd21+uμn+b2(1+u)2u2k22au2buv(1+u)2,D2nM2n=(Dn+Mn)[d1d2μ2n+(bd11+uud2k)μn+au+buv(1+u)2buk(1+u)],Dn+Mn=d1d2μ2n+(bd11+u+ud2k)μn+au+buv(1+u)2+buk(1+u). (2.8)

    Denote Z=ω2, and then (2.7) becomes

    Z2+(T2nB22Dn)Z+D2nM2n=0. (2.9)

    (2.9) has two roots:

    Z±n=(2Dn+B2T2n)±(T2nB2)(T2nB24Dn)+4M2n2.

    From the proof of Theorem 2.1, we get Dn+Mn>0 for any nN. So, the sign of D2nM2n is the same to that of

    DnMn=d1d2μ2n+(bd11+uud2k)μn+au+buv(1+u)2buk(1+u). (2.10)

    To study the existence of positive roots of (2.9), we only need to discuss the sign of DnMn, T2nB22Dn, and (T2nB2)(T2nB24Dn)+4M2n. It is not difficult to get the following Lemma.

    Lemma 2.2. For the Eq (2.9), the following conclusions hold.

    1) If T2nB22Dn>0 and DnMn>0 or (T2nB2)(T2nB24Dn)+4M2n<0 for any nN0, then the Eq (2.9) has no positive root.

    2) If there exists some nN0 such that DnMn<0, then the Eq (2.9) has a positive root Z+n.

    3) If there exists some nN0 such that T2nB22Dn<0, DnMn>0, and (T2nB2)(T2nB24Dn)+4M2n0, then the Eq (2.9) has two positive roots Z±n.

    Next, we discuss some sufficient conditions for Lemma 2.2 to hold. Using the first equation of (2.8), we have

    T20B22D0=1(1+u)2b22uv(1+u)2bu2k22au.

    It is easy to see that T20B22D0>0 when

    b>(1+u)(uk)2+2au(1+2u)1+udef=¯b (2.11)

    holds. Similarly, we can get from (2.10) that D0M0>0 provided that

    b<ak(1+2k+2k2+k+1)3def=b_. (2.12)

    To illustrate the existence of positive roots of the Eq (2.9), we mainly study three cases as follows.

    Case Ⅰ: b>b_. If b>b_, we can obtain that D0M0<0. It is easy to see that DnMn as n. So there must exist a NN such that DnMn<0 for n<N and DnMn0 for nN. From Lemma 2.2, we know that the Eq (2.9) has positive roots Z+n for n<N.

    Case Ⅱ: u(1+u)d2kd1<b<min{b_,¯b}. If b<b_, then D20M20>0. Furthermore, we can obtain from (2.10) that D2nM2n>0 for any nN0 when b>u(1+u)d2kd1. Notice that T20B22D20<0 when b<¯b, and then from (2.8), we know that there exists some N1N such that T2nB22D2n<0 for n<N1, and T2nB22D2n0 for nN1. Through calculation, we can obtain

    (T2nB2)(T2nB24Dn)+4M2n=p0μ4n+p1μ3n+p2μ2n+p3μn+p4, (2.13)

    where

    p0=(d21d22)2,p1=4d2(d22d21)b1+u,p2=2(3d22d21)(b1+u)2+(d22d21)(uk)24(au+buv(1+u)2)(d1+d2)2,p3=4b1+u[(b1+u)2(uk)22(au+buv(1+u)2)(d1+d2)],p4=[(b1+u)2+(uk)2]2+4(au+buv(1+u)2)[(uk)2(b1+u)2].

    Notice that limn(T2nB2)(T2nB24Dn)+4M2n=, and we can find a N2N such that (T2nB2)(T2nB24Dn)+4M2n0 for nN2. Therefore, when N2<N1 and u(1+u)d2kd1<b<min{b_,¯b} are satisfied, the Eq (2.9) has two positive roots Z±n for N2n<N1. Otherwise, the Eq (2.9) does not have a positive root for any nN0.

    Case Ⅲ: max{¯b,ud2(1+u)kd1}<b<b_. We have from Case Ⅰ and Case Ⅱ that D2nM2n>0 and T2nB22D2n>0 for any nN0 when max{¯b,ud2(1+u)kd1}<b<b_. Therefore, the Eq (2.9) has no positive root as max{¯b,ud2(1+u)kd1}<b<b_.

    Combining Lemma 2.2 with the above analysis, we have the following result.

    Corollary 2.3. Denote D1={nN0|T2nB22D2n<0}, D2={nN0|(T2nB2)(T2nB24Dn)+4M2n0}, ω±n=Z±n.

    1) If b>b_, then there exists some NN0 such that the Eq (2.7) has a positive root ω+n for n<N.

    2) If u(1+u)d2kd1<b<min{b_,¯b} and D1D2, then the Eq (2.7) has two positive roots ω±n for nD1D2.

    3)If max{¯b,u(1+u)d2kd1}<b<b_, or u(1+u)d2kd1<b<min{b_,¯b} and D1D2=, then the Eq (2.7) has no positive roots for any nN0.

    For simplicity, we define the following set

    Γ={nN0|T2nB22Dn<0,DnMn>0and(T2nB2)(T2nB24Dn)+4M2n0}

    It is easy to see that Eq (2.7) has a pair of positive roots ω±n for nΓN0. Then the Eq (2.4) has a pair of purely imaginary roots ±iω±n when τ takes the critical values τ±n,j, which can be determined from (2.6), given by

    τ±n,j={1ω±n[arccos((MnTnB)ω±n2DnMnM2n+B2ω±n2)+2jπ],sinω±nτ>0,1ω±n[arccos((MnTnB)ω±n2DnMnM2n+B2ω±n2)+2(j+1)π],sinω±nτ<0, (2.14)

    for jN0. Let λ(τ)=α(τ)+iω(τ) be the root of (2.4) satisfying Reλ(τ±n,j)=0 and Imλ(τ±n,j)=ω±n.

    Lemma 2.4. Assume that the condition 2 or 3 of Lemma 2.2 holds, then

    1) Reλ(τ±n,j)=0, when (T2nB2)(T2nB24Dn)+4M2n=0.

    2) Reλ(τ+n,j)>0, Reλ(τn,j)<0, when (T2nB2)(T2nB24Dn)+4M2n>0.

    Proof. Differentiating the two sides of the Eq (2.4) with respect to τ, we have

    (2λ+Tn+Beλτ)dλdτ(Bλ+Mn)(λ+τdλdτ)eλτ=0.

    Thus,

    (dλdτ)1=(2λ+Tn)eλτ+Bλ(Bλ+Mn)τλ.

    Following the techniques in Cooke and Grossman [33], and using the Eqs (2.4) and (2.6), we obtain that

    Re(dλdτ)1|τ=τ±n,j=Re[Bλ(Bλ+Mn)2λ+Tnλ(λ2+Tnλ+Dn)τλ]τ=τ±n,j=±[(T2nB2)(T2nB24Dn)+4M2n]12B2ω±n2+M2n.

    Note that

    sign{Re(dλdτ)|τ=τ±n,j}=sign{Re(dλdτ)1|τ=τ±n,j},

    and we can complete the proof.

    Combining Theorem 2.1 and Lemma 2.4, we have that τ+n,0<τn,0 holds true for nΓ. Then, we define the smallest critical value such that the stability of E will change, which can be given by

    τdef=τ+n0,0=minΓ{τ+n,0,τn,0} (2.15)

    Combined the above analysis with Corollary 2.4 in Ruan & Wei [34], we know that all of the roots of (2.4) have negative real parts when τ[0,τ), and the (n+1)-th equation of (2.4) has a pair of simply purely imaginary roots when τ=τ±n,j. Moreover, we see that (2.4) has at least one pair of conjugate complex roots with positive real parts when τ>τ. Based on the above discussion, we can obtain the following conclusion about the stability of E.

    Theorem 2.5. For τ defined in (2.15), the following statements about system (1.3) hold true.

    1) If T2nB22Dn>0 and DnMn>0 or (T2nB2)(T2nB24Dn)+4M2n<0 hold for any nN0, then the positive steady state E is locally asymptotically stable for any τ0.

    2) If DnMn<0 holds for some nN0, then

    (a) the positive steady state E is locally asymptotically stable for τ[0,τ), and unstable for τ>τ.

    (b) the system (1.3) undergoes a Hopf bifurcation at E when τ=τ+n,j for jN0.

    3) If T2nB22Dn<0, DnMn>0, and (T2nB2)(T2nB24Dn)+4M2n>0 hold for some nN0, then the positive steady state E is locally asymptotically stable for τ[0,τ). Moreover, the system (1.3) undergoes a Hopf bifurcation at E when τ=τ±n,j for jN0, where τ±n,j is defined in (2.14).

    Remark 2.6. From Lemma 2.4 and Theorem 2.5, we have that

    Reλ(τ+n,j)>0,Reλ(τn,j)<0,

    provided that T2nB22Dn<0, DnMn>0, and (T2nB2)(T2nB24Dn)+4M2n>0 are satisfied. In this case, the stability switch may exist.

    In this section, we study the stability and direction of the Hopf bifurcations by applying the center manifold theorem and the normal formal theory of partial functional differential equations [32,35]. First, the system (1.3) can be represented as an abstract ODE system. Second, on the center manifold of the ODE system corresponding to E, the normal form or Taylor expansion of the ODE system will be computed. Then, using the techniques in [36], the coefficients of the first 4 terms of the normal form will reveal all the properties of the periodical solutions. These analytical results will also be used in numerical studies.

    In this section, we choose Ω=[0,lπ]×[0,lπ]. To write the system (1.3) as an ODE system, we define a function space

    X={(u1,u2):uiW2,2(Ω),uiν=0,xΩ,i=1,2},

    where u1(,t)=u(,τt)u,u2(,t)=v(,τt)v, and U(t)=(u1(,t),u2(,t))T. Then the system (1.3) can be written as

    dU(t)dt=τDΔU(t)+L(τ)(Ut)+f(Ut,τ), (3.1)

    in the function space C=C([1,0],X), where D=diag(d1,d2), L(τ)():CX and f:C×RCX are given, respectively, by

    L(τ)(φ)=τL1φ(0)+τL2φ(1),f(φ,τ)=τ(f1(φ,τ),f2(φ,τ))T,

    with

    f1(φ,τ)=a1φ1(0)φ2(0)+a2φ1(0)φ1(1),

    f2(φ,τ)=a3φ21(0)+a4φ1(0)φ2(0)+a5φ31(0)+a6φ21(0)φ2(0)+O(4),

    for φ=(φ1,φ2)TC,

    where

    a1=1,a2=1k,a3=au(1+u)2,a4=b(1+u)2,a5=au(1+u)3,a6=b(1+u)3.

    Let τ=τ+μ, and then (3.1) can be rewritten as

    dU(t)dt=τDΔU(t)+L(τ)(Ut)+F(Ut,μ), (3.2)

    where

    F(φ,μ)=μDΔφ(0)+L(μ)(φ)+f(φ,τ+μ),

    for φC.

    From the analysis in Section 2, we know that system (3.2) undergoes Hopf bifurcation at the equilibrium (0,0) when μ=0 (i.e., τ=τ). We assume that when μn0=j20+k20l2, (2.4) has roots ±iω as τ=τ. Moreover, we also have that ±iωτ are simply purely imaginary eigenvalues of the linearized system of (3.2) at the origin:

    dU(t)dt=(τ+μ)DΔU(t)+L(τ+μ)(Ut), (3.3)

    as μ=0 and all other eigenvalues of (3.3) at μ=0 have negative real parts.

    The eigenvalues of τDΔ on X are τd1j2+k2l2 and τd2j2+k2l2,j,kN0, with corresponding eigenfunctions β1j,k(x)=(γj,k(x),0)T and β2j,k(x)=(0,γj,k(x))T, where x=(x1,x2), γj,k(x)=cosjx1lcoskx2llπ0cos2jx1ldx1lπ0cos2kx2ldx2.

    We define a space as Mj,k=span{φ,βij,kβij,k:φC,i=1,2},j,kN0, and the inner product , is defined by

    u,v=ΩuTvdx,foru,vX.

    Then, on Mj,k, the Eq (3.3) is equivalent to the ODE on R2:

    dU(t)dt=(τ+μ)j2+k2l2DU(t)+L(τ+μ)(Ut). (3.4)

    Now, we compute the normal form in the center manifold. There are several steps. We first compute eigenvectors of the infinitesimal generator of the semigroup defined by the linearized system at τ=τ. From the Riesz representation theorem, there exists a bounded variation function ηj,k(μ,θ) for θ[1,0], such that

    (τ+μ)j2+k2l2Dφ(0)+L(τ+μ)(φ)=01dηj,k(μ,θ)φ(θ) (3.5)

    for φC([1,0],R2). In fact, we can choose

    ηj,k(μ,θ)={(τ+μ)(L1j2+k2l2D),θ=0,0,θ(1,0),(τ+μ)L2,θ=1.

    Let A denote the infinitesimal generator of the semigroup defined by (3.4) with μ=0,j=j0,k=k0 and A denote the formal adjoint of A under the bilinear form

    (ψ,ϕ)j,k=ψ(0)ϕ(0)01θ0ψ(ξθ)dηj,k(0,θ)ϕ(ξ)dξ (3.6)

    for ϕC([1,0],R2) and ψC([0,1],R2T). Then, we know that ±iωτ are simply purely imaginary eigenvalues of A, and they are also eigenvalues of A. By direct computations, we get q(θ)=q(0)eiωτθ=(1,q1)Teiωτθ(θ[1,0]) is eigenvector of A corresponding to iωτ, where

    q1=(a+bv(1+u)2)(iω+d2(j20+k20)l2+b1+u)1.

    Similarly, we have q(s)=eiωτs(1,q2)(s[0,1]) is eigenvector of A corresponding to iωτ, where

    q2=u(iω+d2(j20+k20)l2+b1+u)1.

    Let Φ=(Φ1,Φ2)=(Req,Imq) and Ψ=(Ψ1,Ψ2)T=(Req,Imq)T. Denote

    (Ψ,Φ)j0,k0=((Ψ1,Φ1)j0,k0(Ψ1,Φ2)j0,k0(Ψ2,Φ1)j0,k0(Ψ2,Φ2)j0,k0),

    where

    (Ψ1,Φ1)j0,k0=1+Req1Req2τu2k(cosωτ+sinωτωτ),(Ψ1,Φ2)j0,k0=Imq1Req2+τu2ksinωτ=(Ψ2,Φ1)j0,k0,(Ψ2,Φ2)j0,k0=Imq1Imq2+τu2k(cosωτ+sinωτωτ).

    Let Ψ=(Ψ1,Ψ2)T=(Ψ,Φ)1j0,k0Ψ, (Ψ,Φ)j0,k0=I2, and I2 is a 2×2 identity matrix.

    We now write the reduced equation on the center manifold. The center subspace of linear equation (3.3) with μ=0 is given by PCNC, where

    PCNφ=φ(Ψ,φ,βj0,k0)j0,k0βj0,k0,φC,

    with βj0,k0=(β1j0,k0,β2j0,k0) and cβj0,k0=c1β1j0,k0+c2β2j0,k0 for c=(c1,c2)TC. Let PSC denote the stable subspace of linear equation (3.3) with μ=0, and then C=PCNCPSC.

    Using the decomposition C=PCNCPSC and following [32], the flow of (3.2) with μ=0 in the center manifold is given by the following formulae:

    (y1(t),y2(t))T=(Ψ,Ut,βj0,k0)j0,k0,
    Ut=Φ(y1(t),y2(t))Tβj0,k0+h(y1,y2,0), (3.7)
    (˙y1(t)˙y2(t))=(0ωτωτ0)(y1(t)y2(t))+Ψ(0)F(Ut,0),βj0,k0, (3.8)

    with h(0,0,0)=0 and Dh(0,0,0)=0.

    Let us write the reduced equation in complex form. Set z=y1iy2 and Ψ(0)=(Ψ1(0),Ψ2(0))T, and then q=Φ1+iΦ2 and Φ(y1(t),y2(t))Tβj0,k0=(qz+¯q¯z)βj0,k0/2. Thus, (3.7) can be written as

    Ut=12(qz+¯q¯z)βj0,k0+w(z,¯z), (3.9)

    where

    w(z,¯z)=h(z+¯z2,i(z¯z)2,0).

    From (3.8) and (3.9), we obtain that z satisfies

    ˙z=iωτz+g(z,¯z), (3.10)

    where

    g(z,¯z)=(Ψ1(0)iΨ2(0))F(Ut,0),βj0,k0=(Ψ1(0)iΨ2(0))f(Ut,τ),βj0,k0.

    Now, let us compute g(z,¯z). Set

    g(z,¯z)=g20z22+g11z¯z+g02¯z22+g21z2¯z2+,w(z,¯z)=w20z22+w11z¯z+w02¯z22+. (3.11)

    Let (ψ1,ψ2)=Ψ1(0)iΨ2(0). From (3.7), (3.9) and (3.10), we can get the following quantities:

    g20=τ2Ωγ3j0,k0dx[(a1q1+a2eiωτ)ψ1+(a3+a4q1)ψ2],g02=τ2Ωγ3j0,k0dx[(a1¯q1+a2eiωτ)ψ1+(a3+a4¯q1)ψ2],g11=τ4Ωγ3j0,k0dx{[a1(q1+¯q1)+a2(eiωτ+eiωτ)]ψ1+[2a3+a4(q1+¯q1)]ψ2},

    and

    g21=τ4Ωγ4j0,k0dx[3a5+a6(¯q1+2q1)]ψ2+τ2[a1(2w(1)11(0)q1+w(1)20(0)¯q1+2w(2)11(0)+w(2)20(0))+a2(2w(1)11(0)eiωτ+w(1)20(0)eiωτ+2w(1)11(1)+w(1)20(1))]γj0,k0,γj0,k0ψ1+τ2[a3(4w(1)11(0)+2w(1)20(0))+a4(2w(1)11(0)q1+w(1)20(0)¯q1+2w(2)11(0)+w(2)20(0))]γj0,k0,γj0,k0ψ2.

    To obtain g21, we need to compute w11 and w20. The calculation of w11 and w20 is somewhat tedious. Let AU denote the generator of the semigroup generated by the linear system (3.3) with μ=0. From (3.9) and (3.10), we have

    ˙w=˙Ut12(q˙z+¯q˙¯z)βj0,k0={AUw12(qg+¯q¯g)βj0,k0,θ[1,0),AUw12(qg+¯q¯g)βj0,k0+f(12(q˙z+¯qz)βj0,k0+w,τ),θ=0,=AUw+H(z,¯z,θ), (3.12)

    where

    H(z,ˉz,θ)=H20(θ)z22+H11(θ)zˉz+H02(θ)ˉz22+.

    Let

    f(12(q˙z+¯qz)βj0,k0+w,τ)=fz2z22+fz¯zz¯z+f¯z2¯z22+.

    Furthermore, by comparing the coefficients, we obtain that

    H20(θ)={12(q(θ)g20+¯q(θ)¯g02)βj0,k0,θ[1,0),12(q(θ)g20+¯q(θ)¯g02)βj0,k0+fz2,θ=0,H11(θ)={12(q(θ)g11+¯q(θ)¯g11)βj0,k0,θ[1,0),12(q(θ)g11+¯q(θ)¯g11)βj0,k0+fz¯z,θ=0. (3.13)

    By using the chain rule,

    ˙w=w(z,¯z)z˙z+w(z,¯z)¯z˙¯z,

    and we obtain, from (3.11) and (3.12), that

    {H20=(2iωτAU)w20,H11=AUw11. (3.14)

    As 2iωτ and 0 are not characteristic values of (3.3), (3.14) has unique solutions w20 and w11 in PSC, given by

    {w20=(2iωτAU)1H20,w11=A1UH11. (3.15)

    Using the definition of AU, we get, from the first equation (3.13) and (3.14), that for θ[1,0],

    ˙w20=2iωτw20(θ)+12(q(θ)g20+¯q(θ)¯g02)βj0,k0.

    Therefore

    w20(θ)=12[ig20ωτq(θ)+i¯g023ωτ¯q(θ)]βj0,k0+Ee2iωτθ,

    where E is a 2-dimensional vector in X. According to the definition of βij,k(i=1,2) and q(θ)(θ[1,0]), we have

    τDΔq(0)βj0,k0+L(τ)(q(θ)βj0,k0)=iωq(0)βj0,k0,
    τDΔ¯q(0)βj0,k0+L(τ)(¯q(θ)βj0,k0)=iω¯q(0)βj0,k0.

    From (3.14), we get that

    2iωτEτDΔEL(τ)(Ee2iωτθ)=fz2. (3.16)

    Representing E and fz2 by series:

    E=j,k=0Ej,kβj,k=j,k=0Ej,kγj,k(Ej,kR2),
    fz2=j,k=0fz2,βj,kβj,k=j,k=0fz2,βj,kγj,k.

    We get from (3.16) that

    2iωτEj,k+τj2+k2l2DEj,kL(τ)(Ej,ke2iωτ)=fz2,βj,k,j,kN0.

    So, Ej,k could be calculated by

    Ej,k=˜E1j,kfz2,βj,k,

    where

    ˜Ej,k=τ(2iω+d1(j2+k2)l2+uke2iωτuabv(1+u)22iω+d2(j2+k2)l2+b1+u),
    fz2,βj,k={1lπ˜fz2,j=k=0,12π˜fz2,j=2j00,k=k0=0orj=j0=0,k=2k00,142π˜fz2,j=2j00,k=2k00,0,other,

    with

    ˜fz2=τ2(a1+a2eiωτa3+a4q1).

    Similarly, we get

    w11(θ)=12[ig11ωτq(θ)+i¯g11ωτ¯q(θ)]βj0,k0+F,
    F=j,k=0Fj,kγj,k(Fj,kR2),Fj,k=˜F1j,k<fz¯z,βj,k>,

    where

    ˜Fj,k=τ(d1(j2+k2)l2+uku(a+bv(1+u)2)d2(j2+k2)l2+b1+u),
    fz¯z,βj,k={1lπ˜fz¯z,j=k=0,12π˜fz¯z,j=2j00,k=k0=0orj=j0=0,k=2k00,142π˜fz¯z,j=2j00,k=2k00,0,other,

    with

    ˜fz¯z=τ4(a1(q1+¯q1)+a2(eiωτ+eiωτ)2a3+a4(q1+¯q1)).

    Then, the coefficient g21 is completely determined.

    Let λ(τ)=α(τ)+iω(τ) denote the eigenvalues of (3.3). Thus we can compute the following quantities:

    c1(0)=i2ωτ(g20g112|g11|213|g02|2)+12g21,μ2=Re(c1(0))α(τ),β2=2Re(c1(0)),T2=1ωτ(Im(c1(0))+μ2ω(τ)). (3.17)

    According to the Hopf bifurcation theory (see [36]), we write our derivation as a theorem which is well-known.

    Theorem 3.1. The quantity μ2 determines the direction of the Hopf bifurcation (forward if μ2>0, backward if μ2<0). The quantity β2 determines the stability of the bifurcating periodic solutions (stable if β2<0, unstable if β2>0). The quantity T2 determines the period of the bifurcating periodic solutions (the period increases if T2>0, decreases if T2<0).

    In this section, we perform two sets of numerical studies. According to theoretical derivations in the previous sections, we choose parameter values to produce numerical simulations. We observe several different shapes of transient fairy circles, and stable temporal periodic solutions.

    Example 4.1. We choose x=(x1,x2)Ω=[0,2π]×[0,2π],d1=1,d2=3,a=0.1,b=0.4,k=1. This set of parameters satisfies Case Ⅰ of Section 2. Through calculation, we get E(0.7016,0.2984), and only for

    μ0=0,μ1=0.25,μ2=0.5,

    (2.4) has pure roots ±iω+n,n=0,1,2, where

    ω+00.8132,ω+10.6718,ω+20.4447,

    and the corresponding bifurcation values τ+k,j are

    τ+0,j1.9885+7.7260j,τ+1,j3.0398+9.3522j,τ+2,j5.5770+14.1290j,

    respectively. When τ<τ=1.9885, the positive equilibrium E(0.7016,0.2984) of (1.3) is asymptotically stable (see Figure 1). Before the system reaches this stable equilibrium solution, there are transient fairy circles. The plant population and sulfide concentration have similar-shaped fairy circles.

    Figure 1.  Numerical simulations of the system (1.3) for Example 4.1 with τ=1. There are transient fairy circles before the system reaches its asymptotically stable positive equilibrium E(0.7016,0.2984).

    By the formulas derived in the previous section, we get c1(0)0.5506+1.3320i. Because Rec1(0)<0, we know that when τ>τ=1.9885, there exist orbitally stable periodic solutions (see Figure 2). We observe that there are transient fairy circles before the system tends to stable periodic solutions. However, the shapes of the fairy circles for the plant population and sulfide concentration are different. In the numerical simulations for Figures 1 and 2, the initial conditions are

    u(x,t)={0.70.7sin(5x1)sin(5x2),x1[2π5,3π5],x2[2π5,3π5],0.7+0.7sin(5x1)sin(5x2),x1[2π5,3π5],x2[7π5,8π5],0.7+0.7sin(5x1)sin(5x2),x1[7π5,8π5],x2[2π5,3π5],0.70.7sin(5x1)sin(5x2),x1[7π5,8π5],x2[7π5,8π5],0,other,
    v(x,t)={0.3+0.3sin(5x1)sin(5x2),x1[2π5,3π5],x2[2π5,3π5],0.30.3sin(5x1)sin(5x2),x1[2π5,3π5],x2[7π5,8π5],0.30.3sin(5x1)sin(5x2),x1[7π5,8π5],x2[2π5,3π5],0.3+0.3sin(5x1)sin(5x2),x1[7π5,8π5],x2[7π5,8π5],0,other,

    for t[τ,0].

    Figure 2.  Numerical simulations of the system (1.3) for Example 4.1 with τ=2.5. The fairy circles are transient and the bifurcating periodic solutions are stable.

    Example 4.2. We choose x=(x1,x2)Ω=[0,2π]×[0,2π],d1=0.5,d2=0.1,a=0.2,b=0.2,k=1. This set of parameters satisfies Case Ⅱ of Section 2. Through calculation, we get E(0.4142,0.5858), and only for

    μ0=0,μ1=0.25,μ2=0.5,

    (2.4) has pure roots ±iω±n,n=0,1,2, where

    ω+00.5851,ω+10.5517,ω+20.4431,
    ω00.1533,ω10.1953,ω20.3004,

    and the corresponding bifurcation values τ+k,j are

    τ+0,j2.8577+10.7394j,τ+1,j3.6560+11.3898j,τ+2,j5.6984+14.1787j,
    τ0,j24.2297+40.9961j,τ1,j17.6208+32.1697j,τ2,j10.0789+20.9129j,

    respectively. When τ<τ=2.8577, the positive equilibrium E(0.7016,0.2984) of (1.3) is asymptotically stable (see Figure 3). We observe that there are transient fairy circles before the system reaches the stable equilibrium point. The shapes of the fairy circles for the plant population and sulfide concentration are different.

    Figure 3.  Numerical simulations of the system (1.3) for Example 4.2 with τ=2. The fairy circles are transient, and the positive equilibrium E(0.4142,0.5858) is asymptotically stable.

    By the formulas derived in the previous section, we get c1(0)1.3262+1.2114i. Because Rec1(0)<0, we know that when τ>τ=2.8577, there exist orbitally stable periodic solutions (see Figure 4). We observe that there are transient fairy circles. The shapes of the fairy circles for the plant population and that for the sulfide concentration are the same as that in the equilibrium point case, respectively, although the shapes of the fairy circles for the plant population and sulfide concentration are different. In the numerical simulations for Figures 3 and 4, the initial conditions are

    u(x,t)={0.40.4sin(5x1)sin(5x2),x1[2π5,3π5],x2[2π5,3π5],0.4+0.4sin(5x1)sin(5x2),x1[2π5,3π5],x2[7π5,8π5],0.4+0.4sin(5x1)sin(5x2),x1[7π5,8π5],x2[2π5,3π5],0.40.4sin(5x1)sin(5x2),x1[7π5,8π5],x2[7π5,8π5],0,other,
    v(x,t)={0.6+0.6sin(5x1)sin(5x2),x1[2π5,3π5],x2[2π5,3π5],0.60.6sin(5x1)sin(5x2),x1[2π5,3π5],x2[7π5,8π5],0.60.6sin(5x1)sin(5x2),x1[7π5,8π5],x2[2π5,3π5],0.6+0.6sin(5x1)sin(5x2),x1[7π5,8π5],x2[7π5,8π5],0,other,

    for t[τ,0].

    Figure 4.  Numerical simulations of the system (1.3) for Example 4.2 with τ=3. The fairy circles are transient, and the bifurcating periodic solutions are stable.

    Transient spatial patterns in ecosystems have gained more attention in recent research. Several mathematical models were proposed to understand fairy circles and rings observed in salt marshes. One conclusion drawn from this research was that transient fairy circle patterns can infer the underlying ecological mechanisms and provide a measure of resilience of salt marsh ecosystems. The underlying mechanism proposed was plant-sulfide feedbacks instead of scale-dependent feedbacks. It is assumed that any population of a species in a location has a self-regulatory mechanism, and the time of mechanism action has some time lag. In this research, we consider how the time delay in the self-regulatory mechanism influences fairy circle formations in plant-sulfide feedbacks. Based on a mathematical model [28], we proposed a delay plant-sulfide feedback system. We performed a detailed investigation of the plant-sulfide feedback system subject to Neumann boundary conditions, and identified the parameter ranges of stability of the positive equilibrium and the existence of Hopf bifurcation. There is a critical value of the time delay. When the time delay is smaller than the critical value, the system will approach the spatial homogeneous equilibrium state asymptotically. At the critical value, Hopf bifurcations occur. When the time delay is greater than the critical value, the system will have temporal periodic solutions. Since we do not have any method to analyze transient patterns yet, we numerically demonstrated that there always are transient fairy circles for any time delay. We also found that there are different shapes of fairy circles or rings. This confirms that transient fairy circle patterns in intertidal salt marshes can infer the underlying ecological mechanisms and provide a measure for ecosystem resilience.

    In [28], simulations were conducted with several points as initial values, and several circles progressed to one circle and to a uniform distribution. It is easy to see that the number and shape of fairy circles may depend on initial values. In our numerical study, we observed several different patterns of transient fairy circles. The number and shape of these fairy circles seem also to depend on initial distributions. This may be reasonable in reality since a natural species population usually starts with various situations and then progresses to its asymptotical patterns.

    It is conventional that a natural population will obey the self-regulatory mechanism which is different from the scale-dependent feedback mechanism. The scale-dependent feedback mechanism assumes that scale-dependent feedbacks between localized facilitation and large-scale inhibition induce spatial self-organization. The self-regulatory mechanism assumes that the population growth rate is limited by its total population. The former may be considered as a special case of the latter. However, when mathematical models are constructed for those mechanisms, different terms should be taken in the equations. This is the reason we incorporate the self-regulatory mechanism with time delay to the plant-sulfide feedback mechanism.

    In [28], two additional models, the nutrient depletion model and scale-dependent model, were also proposed to explain transient fairy circle patterns. We may consider to study time delayed versions of those models in order to compare how different mechanisms with natural time delay influence transient spatial patterns in the future.

    In general, it is difficult to characterize transient patterns in dynamical systems. One direction to attempt may be time-transformation. Given that we want to know the dynamics of a system during a finite period of time, we make a time-transformation that changes this finite period of time to infinity, and then study the transformed system. However, it is required that the time period is given. It seems that we need to develop new analytical tools for transient patterns. This is an interesting mathematical question in dynamical systems for the future.

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

    This research is supported by the National Natural Science Foundation of China (Nos.11901172, 12271144) and Fundamental Research Fund for Heilongjiang Provincial Colleges and Universities (Nos. 2021-KYYWF-0017, 2022-KYYWF-1043).

    The authors declare that there is no conflict of interest.



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