Research article Topical Sections

The effect of nanocrystalline Ni-W coating on the tensile properties of copper

  • Received: 08 February 2016 Accepted: 09 March 2016 Published: 15 March 2016
  • Nanostructured Ni-W alloy coatings containing approximately 40 wt.% tungsten were electrodeposited onto copper substrates. The effect of the coatings thickness on the surface topography, microstructure and grain size was investigated with the aid of Atomic Force Microscopy (AFM), Scanning Electron Microscopy (SEM) and X-ray Diffraction (XRD) techniques respectively. In addition, this research work aims in understanding the influence and correlation between microstructure and thickness of these Ni-W coatings with the bulk mechanical properties of coated specimens. The experimental results indicated that the micro-hardness and Ultimate Tensile Strength (UTS) of the Ni-W coated copper were higher than that of bare copper, whereas both slightly increased with increasing coating thickness up to 21 μm. On the other hand, the ductility of Ni-W coated copper decreased significantly with increasing coating thickness. Thus it could be said that when applying Ni-W coatings there are certain limitations not only in terms of their composition, but their thickness, grain size and coating structure should be also taken into consideration, in order to obtain an understanding of their mechanical behavior.

    Citation: C. N. Panagopoulos, E. P. Georgiou, D.A. Lagaris, V. Antonakaki. The effect of nanocrystalline Ni-W coating on the tensile properties of copper[J]. AIMS Materials Science, 2016, 3(2): 324-338. doi: 10.3934/matersci.2016.2.324

    Related Papers:

    [1] Charles R Larson, Donald A Robin . Sensory Processing: Advances in Understanding Structure and Function of Pitch-Shifted Auditory Feedback in Voice Control. AIMS Neuroscience, 2016, 3(1): 22-39. doi: 10.3934/Neuroscience.2016.1.22
    [2] M. Rosario Rueda, Joan P. Pozuelos, Lina M. Cómbita, Lina M. Cómbita . Cognitive Neuroscience of Attention
    From brain mechanisms to individual differences in efficiency. AIMS Neuroscience, 2015, 2(4): 183-202. doi: 10.3934/Neuroscience.2015.4.183
    [3] Michaela D Curry, Abigail Zimmermann, Mohammadbagher Parsa, Mohammad-Reza A. Dehaqani, Kelsey L Clark, Behrad Noudoost . A Cage-Based Training System for Non-Human Primates. AIMS Neuroscience, 2017, 4(3): 102-119. doi: 10.3934/Neuroscience.2017.3.102
    [4] Christine A. Godwin, Ezequiel Morsella, Mark W. Geisler . The Origins of a Spontaneous Thought: EEG Correlates and Thinkers’ Source Attributions. AIMS Neuroscience, 2016, 3(2): 203-231. doi: 10.3934/Neuroscience.2016.2.203
    [5] Kenneth R. Paap, Hunter Myuz, Regina Anders-Jefferson, Lauren Mason, Brandon Zimiga . On the ambiguity regarding the relationship between sequential congruency effects, bilingual advantages in cognitive control, and the disengagement of attention. AIMS Neuroscience, 2019, 6(4): 282-298. doi: 10.3934/Neuroscience.2019.4.282
    [6] Evangelia G. Chrysikou, John S. Gero . Using neuroscience techniques to understand and improve design cognition. AIMS Neuroscience, 2020, 7(3): 319-326. doi: 10.3934/Neuroscience.2020018
    [7] Zoha Deldar, Carlos Gevers-Montoro, Ali Khatibi, Ladan Ghazi-Saidi . The interaction between language and working memory: a systematic review of fMRI studies in the past two decades. AIMS Neuroscience, 2021, 8(1): 1-32. doi: 10.3934/Neuroscience.2021001
    [8] Marian E. Berryhill . Longitudinal tDCS: Consistency across Working Memory Training Studies. AIMS Neuroscience, 2017, 4(2): 71-86. doi: 10.3934/Neuroscience.2017.2.71
    [9] Gianna Sepede, Francesco Gambi, Massimo Di Giannantonio . Insular Dysfunction in People at Risk for Psychotic Disorders. AIMS Neuroscience, 2015, 2(2): 66-70. doi: 10.3934/Neuroscience.2015.2.66
    [10] Mariana FP de Araújo, Wagner A de Castro, Hiroshi Nishimaru, Susumu Urakawa, Taketoshi Ono, Hisao Nishijo . Performance in a gaze-cueing task is associated with autistic traits. AIMS Neuroscience, 2021, 8(1): 148-160. doi: 10.3934/Neuroscience.2021007
  • Nanostructured Ni-W alloy coatings containing approximately 40 wt.% tungsten were electrodeposited onto copper substrates. The effect of the coatings thickness on the surface topography, microstructure and grain size was investigated with the aid of Atomic Force Microscopy (AFM), Scanning Electron Microscopy (SEM) and X-ray Diffraction (XRD) techniques respectively. In addition, this research work aims in understanding the influence and correlation between microstructure and thickness of these Ni-W coatings with the bulk mechanical properties of coated specimens. The experimental results indicated that the micro-hardness and Ultimate Tensile Strength (UTS) of the Ni-W coated copper were higher than that of bare copper, whereas both slightly increased with increasing coating thickness up to 21 μm. On the other hand, the ductility of Ni-W coated copper decreased significantly with increasing coating thickness. Thus it could be said that when applying Ni-W coatings there are certain limitations not only in terms of their composition, but their thickness, grain size and coating structure should be also taken into consideration, in order to obtain an understanding of their mechanical behavior.


    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 ($ \text{H}_2\text{S} $) 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) $. Organic matters including plant biomass in intertidal salt marshes can produce hydrogen sulfide (H$ _{2} $S). Assume the plant population produces hydrogen sulfide proportionally with the effective production rate $ \epsilon $. 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 $ D_P $. $ D_S $ 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 $ \frac{k_s}{P(x, t)+k_s} $. The parameter $ d $ is the maximum escape rate of sulfide through the mud-air interface. The parameter $ \xi $ 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 $ \frac{K-P}{K} $ 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, $ \tau $, 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 $ r\frac{K-P}{K} $ [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)\left(1-\frac{P(x, t)}{K}\right) $ for the plant population will be replaced by $ rP(x, t)(1-\frac{P(x, t-\tau)}{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 = \cfrac{P}{k_s},\; v = \cfrac{cS}{r},\; \hat{t} = rt,\; \hat{\tau} = r\tau,\; k = \cfrac{K}{k_s}, $
    $ a = \cfrac{c\epsilon \xi k_s}{r^2},\; b = \cfrac{d\xi}{r},\; d_1 = \cfrac{D_P}{r},\; d_2 = \cfrac{D_S}{r}. $

    Dropping the hats of $ \hat{t} $ and $ \hat{\tau} $, 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 $ \nu $ is the outward unit normal vector on $ \partial\Omega $.

    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: $ E_0 = (0, 0) $ and $ E^* = (u^*, v^*) $, where

    $ u^* = \frac{-(ak+b)+\sqrt{(ak+b)^2+4 abk^2}}{2ak},\; \; v^* = 1-\frac{u^*}{k}. $

    We claim that $ 0 < u^* < k $. In fact,

    $ k-u^* = \frac{2ak^2+ak+b-\sqrt{(ak+b)^2+4abk^2}}{2ak} = \frac{2ak^2(1+k)}{2ak^2+ak+b+\sqrt{(ak+b)^2+4abk^2}} > 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 $ \Omega\subset\mathbb{R}^2 $ be a bounded domain with smooth boundary $ \partial\Omega $, and denote

    $ u_1(x,t) = u(x,t),\; \; u_2(x,t) = v(x,t),\; \; U(x,t) = (u_1(t),u_2(t))^T. $

    In the phase space $ \texttt{C} = C([-\tau, \; 0], \; X) $, we can rewrite the system (1.3) as

    $ ˙U(t)=DΔU(t)+F(Ut),
    $
    (2.1)

    where $ D = \text{diag}(d_1, d_2) $, $ U_t(\cdot) = U(t+\cdot) $, and $ F:\texttt{C}\to X $ is defined by

    $ F(U_t) = \left(u1(t)(1u1(tτ)k)u1(t)u2(t)au1(t)bu2(t)1+u1(t)
    \right). $

    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: \texttt{C}\to X $ is given by

    $ L(\phi_t) = L_1 \phi(0)+L_2 \phi(-\tau) $

    and

    $ L_1 = \left(0ua+bv(1+u)2b1+u
    \right),\; \; L_2 = \left(uk000
    \right), $
    $ \phi(t) = (\phi_1(t),\phi_2(t))^T,\; \; \phi_t(\cdot) = (\phi_{1t}(\cdot),\phi_{2t}(\cdot))^T. $

    Let

    $ 0 = \mu_0 < \mu_1 < \mu_2 < \cdots $

    be all the eigenvalues of the operator $ -\Delta $ on $ \Omega $ 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 $ I_2 $ is the $ 2\times2 $ identity matrix and $ Q_n = \mu_n\text{diag}(d_1, d_2) $. That is, each eigenvalue $ \lambda $ 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 $ \tau = 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 $ \tau = 0 $.

    Theorem 2.1. When $ \tau = 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 $ T_n+B > 0, \; D_n+M_n > 0 $ for all $ n\in\mathbb{N}_0 $, 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

    $ T_n+B > T_{n-1}+B > \cdots > T_0+B = \frac{b}{1+u^*}+\frac{u^*}{k} > 0, $
    $ D_n+M_n > D_{n-1}+M_{n-1} > \cdots > D_0+M_0 = \frac{b u^*}{k(1+u^*)}+au^*+\frac{b u^*v^*}{(1+u^*)^2} > 0, $

    for any $ n\in\mathbb{N} $. So, we complete the proof.

    In the following, we will study the effect of delay on the stability of $ E^* $. Recall that $ D_n+M_n > 0 $ for any $ n\in\mathbb{N}_0 $, we can see that $ 0 $ is not the root of (2.4). Now, we will check whether there exist the critical values of $ \tau $ such that (2.4) has a pair of simple purely imaginary eigenvalues for some $ n\in\mathbb{N}_0 $. Let $ \pm \text{i}\omega(\omega > 0) $ be the solutions of the $ (n+1) $-th Eq (2.4), then we get

    $ -\omega^2+T_n\text{i}\omega+D_n+(B\text{i}\omega +M_n)\text{e}^{-\text{i}\omega\tau} = 0. $

    Separating the real and imaginary parts, and we obtain that $ \omega $ and $ \tau $ 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 = \omega^2 $, and then (2.7) becomes

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

    (2.9) has two roots:

    $ Z_n^\pm = \cfrac{(2D_n+B^2-T_n^2)\pm\sqrt{(T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2}}{2}. $

    From the proof of Theorem 2.1, we get $ D_n+M_n > 0 $ for any $ n\in\mathbb{N} $. So, the sign of $ D_n^2-M_n^2 $ 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 $ D_n-M_n $, $ T_n^2-B^2-2D_n $, and $ (T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2 $. It is not difficult to get the following Lemma.

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

    1) If $ T_n^2-B^2-2D_n > 0 $ and $ D_n-M_n > 0 $ or $ (T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2 < 0 $ for any $ n\in\mathbb{N}_0 $, then the Eq (2.9) has no positive root.

    2) If there exists some $ n\in\mathbb{N}_0 $ such that $ D_n-M_n < 0 $, then the Eq (2.9) has a positive root $ Z_n^+ $.

    3) If there exists some $ n\in\mathbb{N}_0 $ such that $ T_n^2-B^2-2D_n < 0 $, $ D_n-M_n > 0 $, and $ (T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2\geq0 $, then the Eq (2.9) has two positive roots $ Z_n^\pm $.

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

    $ T_0^2-B^2-2D_0 = \frac{1}{(1+u^*)^2}b^2-\frac{2 u^*v^*}{(1+u^*)^2}b-\frac{{u^*}^2}{k^2}-2au^*. $

    It is easy to see that $ T_0^2-B^2-2D_0 > 0 $ when

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

    holds. Similarly, we can get from (2.10) that $ D_0-M_0 > 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 > \underline{b} $. If $ b > \underline{b} $, we can obtain that $ D_0-M_0 < 0 $. It is easy to see that $ D_n-M_n \rightarrow \infty $ as $ n\rightarrow \infty $. So there must exist a $ N\in\mathbb{N} $ such that $ D_n-M_n < 0 $ for $ n < N $ and $ D_n-M_n\geq0 $ for $ n\geq N $. From Lemma 2.2, we know that the Eq (2.9) has positive roots $ Z_n^+ $ for $ n < N $.

    Case Ⅱ: $ \frac{u^*(1+u^*)d_2}{kd_1} < b < \min\{\underline{b}, \overline{b}\} $. If $ b < \underline{b} $, then $ D_0^2-M_0^2 > 0 $. Furthermore, we can obtain from (2.10) that $ D_n^2-M_n^2 > 0 $ for any $ n\in\mathbb{N}_0 $ when $ b > \frac{u^*(1+u^*)d_2}{kd_1} $. Notice that $ T_0^2-B^2-2D_0^2 < 0 $ when $ b < \overline{b} $, and then from (2.8), we know that there exists some $ N_1\in\mathbb{N} $ such that $ T_n^2-B^2-2D_n^2 < 0 $ for $ n < N_1 $, and $ T_n^2-B^2-2D_n^2\geq0 $ for $ n\geq N_1 $. 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 $ \lim_{n\rightarrow \infty} (T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2 = \infty $, and we can find a $ N_2\in\mathbb{N} $ such that $ (T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2\geq 0 $ for $ n\geq N_2 $. Therefore, when $ N_2 < N_1 $ and $ \frac{u^*(1+u^*)d_2}{kd_1} < b < \min\{\underline{b}, \overline{b}\} $ are satisfied, the Eq (2.9) has two positive roots $ Z_n^\pm $ for $ N_2\leq n < N_1 $. Otherwise, the Eq (2.9) does not have a positive root for any $ n\in\mathbb{N}_0 $.

    Case Ⅲ: $ \max\{\overline{b}, \frac{u^*d_2(1+u^*)}{kd_1}\} < b < \underline{b} $. We have from Case Ⅰ and Case Ⅱ that $ D_n^2-M_n^2 > 0 $ and $ T_n^2-B^2-2D_n^2 > 0 $ for any $ n\in\mathbb{N}_0 $ when $ \max\{\overline{b}, \frac{u^*d_2(1+u^*)}{kd_1}\} < b < \underline{b} $. Therefore, the Eq (2.9) has no positive root as $ \max\{\overline{b}, \frac{u^*d_2(1+u^*)}{kd_1}\} < b < \underline{b} $.

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

    Corollary 2.3. Denote $ \mathcal{D}_1 = \{n\in\mathbb{N}_0|T_n^2-B^2-2D_n^2 < 0\} $, $ \mathcal{D}_2 = \{n\in\mathbb{N}_0|(T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2\geq 0\} $, $ \omega_n^\pm = \sqrt{Z_n^\pm} $.

    1) If $ b > \underline{b} $, then there exists some $ N\in\mathbb{N}_0 $ such that the Eq (2.7) has a positive root $ \omega_n^+ $ for $ n < N $.

    2) If $ \frac{u^*(1+u^*)d_2}{kd_1} < b < \min\{\underline{b}, \overline{b}\} $ and $ \mathcal{D}_1\bigcap\mathcal{D}_2\neq\emptyset $, then the Eq (2.7) has two positive roots $ \omega_n^{\pm} $ for $ n\in\mathcal{D}_1\bigcap\mathcal{D}_2 $.

    3)If $ \max\{\overline{b}, \frac{u^*(1+u^*)d_2}{kd_1}\} < b < \underline{b} $, or $ \frac{u^*(1+u^*)d_2}{kd_1} < b < \min\{\underline{b}, \overline{b}\} $ and $ \mathcal{D}_1\bigcap\mathcal{D}_2 = \emptyset $, then the Eq (2.7) has no positive roots for any $ n\in\mathbb{N}_0 $.

    For simplicity, we define the following set

    $ \Gamma = \{n\in\mathbb{N}_0|T_n^2-B^2-2D_n < 0, D_n-M_n > 0\; \text{and}\; (T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2\geq0\} $

    It is easy to see that Eq (2.7) has a pair of positive roots $ \omega_n^\pm $ for $ n\in\Gamma\subset\mathbb{N}_0 $. Then the Eq (2.4) has a pair of purely imaginary roots $ \pm \text{i}\omega_n^\pm $ when $ \tau $ takes the critical values $ \tau_{n, j}^\pm $, 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 $ j\in \mathbb{N}_0 $. Let $ \lambda(\tau) = \alpha(\tau)+\text{i}\omega(\tau) $ be the root of (2.4) satisfying $ \text{Re}\lambda(\tau_{n, j}^\pm) = 0 $ and $ \text{Im}\lambda(\tau_{n, j}^\pm) = \omega_n^\pm $.

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

    1) $ \mathit{\text{Re}}\lambda'(\tau_{n, j}^\pm) = 0 $, when $ (T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2 = 0 $.

    2) $ \mathit{\text{Re}}\lambda'(\tau_{n, j}^+) > 0 $, $ \mathit{\text{Re}}\lambda'(\tau_{n, j}^-) < 0 $, when $ (T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2 > 0 $.

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

    $ (2\lambda+T_n +B e^{-\lambda \tau})\frac{\text{d} \lambda}{\text{d} \tau}-(B \lambda+M_n)(\lambda+\tau \frac{\text{d} \lambda}{\text{d}\tau})e^{-\lambda\tau} = 0. $

    Thus,

    $ \left(\frac{\text{d} \lambda}{\text{d}\tau}\right)^{-1} = \frac{(2\lambda+T_n)e^{\lambda\tau}+B}{\lambda(B \lambda+M_n)}-\frac{\tau}{\lambda}. $

    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

    $ \text{sign}\left \{\text{Re}\left.\left(\frac{\text{d}\lambda}{\text{d}\tau}\right)\right|_{\tau = \tau_{n,j}^\pm}\right\} = \text{sign}\left \{\text{Re}\left.\left(\frac{\text{d}\lambda}{\text{d}\tau}\right)^{-1}\right|_{\tau = \tau_{n,j}^\pm}\right\}, $

    and we can complete the proof.

    Combining Theorem 2.1 and Lemma 2.4, we have that $ \tau_{n, 0}^+ < \tau_{n, 0}^- $ holds true for $ n\in\Gamma $. 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 $ \tau\in [0, \tau^*) $, and the $ (n+1) $-th equation of (2.4) has a pair of simply purely imaginary roots when $ \tau = \tau_{n, j}^\pm $. Moreover, we see that (2.4) has at least one pair of conjugate complex roots with positive real parts when $ \tau > \tau^* $. Based on the above discussion, we can obtain the following conclusion about the stability of $ E^* $.

    Theorem 2.5. For $ \tau^* $ defined in (2.15), the following statements about system (1.3) hold true.

    1) If $ T_n^2-B^2-2D_n > 0 $ and $ D_n-M_n > 0 $ or $ (T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2 < 0 $ hold for any $ n\in\mathbb{N}_0 $, then the positive steady state $ E^* $ is locally asymptotically stable for any $ \tau\geq0 $.

    2) If $ D_n-M_n < 0 $ holds for some $ n\in\mathbb{N}_0 $, then

    (a) the positive steady state $ E^* $ is locally asymptotically stable for $ \tau \in [0, \tau^*) $, and unstable for $ \tau > \tau^* $.

    (b) the system (1.3) undergoes a Hopf bifurcation at $ E^* $ when $ \tau = \tau_{n, j}^+ $ for $ j\in\mathbb{N}_0 $.

    3) If $ T_n^2-B^2-2D_n < 0 $, $ D_n-M_n > 0 $, and $ (T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2 > 0 $ hold for some $ n\in\mathbb{N}_0 $, then the positive steady state $ E^* $ is locally asymptotically stable for $ \tau \in [0, \tau^*) $. Moreover, the system (1.3) undergoes a Hopf bifurcation at $ E^* $ when $ \tau = \tau_{n, j}^\pm $ for $ j\in\mathbb{N}_0 $, where $ \tau_{n, j}^\pm $ is defined in (2.14).

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

    $ \text{Re}\lambda'(\tau_{n,j}^+) > 0, \; \; \text{Re}\lambda'(\tau_{n,j}^-) < 0, $

    provided that $ T_n^2-B^2-2D_n < 0 $, $ D_n-M_n > 0 $, and $ (T_n^2-B^2)(T_n^2-B^2-4D_n)+4M_n^2 > 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 $ \Omega = [0, l\pi]\times[0, l\pi] $. To write the system (1.3) as an ODE system, we define a function space

    $ X = \left\{(u_1,u_2):u_i\in W^{2,2}(\Omega),\; \frac{\partial u_i}{\partial \nu} = 0,\; x\in\partial\Omega,\; i = 1,2 \right\}, $

    where $ u_1(\cdot, t) = u(\cdot, \tau t)-u^*, \; u_2(\cdot, t) = v(\cdot, \tau t)-v^* $, and $ U(t) = (u_1(\cdot, t), u_2(\cdot, 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 $ \texttt{C} = C([-1, 0], X) $, where $ D = \text{diag}(d_1, d_2) $, $ L(\tau)(\cdot):\texttt{C}\to X $ and $ f:\texttt{C}\times\mathbb{R}\texttt{C}\to X $ are given, respectively, by

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

    with

    $ f_1(\varphi, \tau) = a_1\varphi_1(0)\varphi_2(0)+a_2\varphi_1(0)\varphi_1(-1), $

    $f_2(\varphi, \tau) = a_3\varphi_1^2(0)+a_4\varphi_1(0)\varphi_2(0)+a_5\varphi_1^3(0)+a_6\varphi_1^2(0)\varphi_2(0)+\mathcal{O}(4), $

    for $ \varphi = (\varphi_1, \varphi_2)^T\in\texttt{C} $,

    where

    $ a_1 = -1,\; a_2 = -\frac{1}{k},\; a_3 = -\frac{au^*}{(1+u^*)^2},\; a_4 = \frac{b}{(1+u^*)^2},\; a_5 = \frac{au^*}{(1+u^*)^3},\; a_6 = -\frac{b}{(1+u^*)^3}. $

    Let $ \tau = \tau^*+\mu $, and then (3.1) can be rewritten as

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

    where

    $ F(\varphi,\mu) = \mu D\Delta\varphi(0)+L(\mu)(\varphi)+f(\varphi,\tau^*+\mu), $

    for $ \varphi\in\texttt{C} $.

    From the analysis in Section 2, we know that system (3.2) undergoes Hopf bifurcation at the equilibrium $ (0, 0) $ when $ \mu = 0 $ (i.e., $ \tau = \tau^* $). We assume that when $ \mu_{n_0} = \frac{j_0^2+k_0^2}{l^2} $, (2.4) has roots $ \pm\text{i} \omega^* $ as $ \tau = \tau^* $. Moreover, we also have that $ \pm \text{i} \omega^*\tau^* $ 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 $ \mu = 0 $ and all other eigenvalues of (3.3) at $ \mu = 0 $ have negative real parts.

    The eigenvalues of $ \tau D\Delta $ on $ X $ are $ -\tau d_1\frac{j^2+k^2}{l^2} $ and $ -\tau d_2\frac{j^2+k^2}{l^2}, \; j, k\in\mathbb{N}_0 $, with corresponding eigenfunctions $ \beta_{j, k}^1(x) = (\gamma_{j, k}(x), 0)^T $ and $ \beta_{j, k}^2(x) = (0, \gamma_{j, k}(x))^T $, where $ x = (x_1, x_2) $, $ \gamma_{j, k}(x) = \frac{\cos\frac{jx_1}{l}\cos\frac{kx_2}{l}}{\sqrt{\int_0^{l\pi}\cos^2\frac{jx_1}{l}\text{d}x_1\int_0^{l\pi}\cos^2\frac{kx_2}{l}\text{d}x_2}}. $

    We define a space as $ M_{j, k} = \text{span}\{\langle\varphi, \beta_{j, k}^i\rangle\beta_{j, k}^i:\varphi\in\texttt{C}, i = 1, 2\}, \; j, k\in\mathbb{N}_0 $, and the inner product $ \langle\cdot, \cdot\rangle $ is defined by

    $ \langle u,v\rangle = \displaystyle {\int}_\Omega u^Tv\text{d}x,\; \text{for} \; u,v\in X. $

    Then, on $ M_{j, k} $, the Eq (3.3) is equivalent to the ODE on $ \mathbb{R}^2 $:

    $ 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 $ \tau = \tau^* $. From the Riesz representation theorem, there exists a bounded variation function $ \eta_{j, k}(\mu, \theta) $ for $ \theta\in[-1, 0] $, such that

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

    for $ \varphi\in C([-1, 0], \mathbb{R}^2) $. 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 $ \mu = 0, \; j = j_0, \; k = k_0 $ 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 $ \phi\in C([-1, 0], \mathbb{R}^2) $ and $ \psi\in C([0, 1], \mathbb{R}^{2^T}) $. Then, we know that $ \pm \text{i}\omega^*\tau^* $ are simply purely imaginary eigenvalues of $ A $, and they are also eigenvalues of $ A^* $. By direct computations, we get $ q(\theta) = q(0)\text{e}^{\text{i}\omega^*\tau^*\theta} = (1, q_1)^T\text{e}^{\text{i}\omega^*\tau^*\theta}(\theta\in[-1, 0]) $ is eigenvector of $ A $ corresponding to $ \text{i}\omega^*\tau^* $, where

    $ q_1 = \left(a+\frac{b v^*}{(1+u^*)^2}\right)\left(\text{i}\omega^*+\frac{d_2(j_0^2+k_0^2)}{l^2}+\frac{b}{1+u^*}\right)^{-1}. $

    Similarly, we have $ q^*(s) = \text{e}^{-\text{i}\omega^*\tau^* s}(1, q_2)(s\in[0, 1]) $ is eigenvector of $ A^* $ corresponding to $ \text{i}\omega^*\tau^* $, where

    $ q_2 = -u^*\left(\text{i}\omega^*+\frac{d_2(j_0^2+k_0^2)}{l^2}+\frac{b}{1+u^*}\right)^{-1}. $

    Let $ \Phi = (\Phi_1, \Phi_2) = (\text{Re}q, \text{Im}q) $ and $ \Psi^* = (\Psi_1^*, \Psi_2^*)^T = (\text{Re}q^*, \text{Im}q^*)^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 $ \Psi = (\Psi_1, \Psi_2)^T = (\Psi^*, \Phi)_{j_0, k_0}^{-1}\Psi^* $, $ (\Psi, \Phi)_{j_0, k_0} = I_2 $, and $ I_2 $ is a $ 2\times2 $ identity matrix.

    We now write the reduced equation on the center manifold. The center subspace of linear equation (3.3) with $ \mu = 0 $ is given by $ P_{CN}\texttt{C} $, where

    $ P_{CN}\varphi = \varphi(\Psi,\langle\varphi,\beta_{j_0,k_0}\rangle)_{j_0,k_0}\cdot\beta_{j_0,k_0},\; \varphi\in\texttt{C}, $

    with $ \beta_{j_0, k_0} = (\beta_{j_0, k_0}^1, \beta_{j_0, k_0}^2) $ and $ c\cdot\beta_{j_0, k_0} = c_1\beta_{j_0, k_0}^1+c_2\beta_{j_0, k_0}^2 $ for $ c = (c_1, c_2)^T\in\texttt{C} $. Let $ P_S\texttt{C} $ denote the stable subspace of linear equation (3.3) with $ \mu = 0 $, and then $ \texttt{C} = P_{CN}\texttt{C}\bigoplus P_S\texttt{C} $.

    Using the decomposition $ \texttt{C} = P_{CN}\texttt{C}\bigoplus P_S\texttt{C} $ and following [32], the flow of (3.2) with $ \mu = 0 $ in the center manifold is given by the following formulae:

    $ (y_1(t),y_2(t))^T = (\Psi,\langle U_t,\beta_{j_0,k_0}\rangle)_{j_0,k_0}, $
    $ 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 = y_1-\text{i}y_2 $ and $ \Psi(0) = (\Psi_1(0), \Psi_2(0))^T $, and then $ q = \Phi_1+\text{i}\Phi_2 $ and $ \Phi(y_1(t), y_2(t))^T\cdot \beta_{j_0, k_0} = (qz+\overline q\overline z)\cdot \beta_{j_0, k_0}/2 $. Thus, (3.7) can be written as

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

    where

    $ w(z,\overline z) = h\left(\frac{z+\overline z}{2},\frac{\text{i}(z-\overline z)}{2},0\right). $

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

    $ ˙z=iωτz+g(z,¯z),
    $
    (3.10)

    where

    $ g(z,\overline z) = (\Psi_1(0)-\text{i}\Psi_2(0))\langle F(U_t,0),\beta_{j_0,k_0}\rangle = (\Psi_1(0)-\text{i}\Psi_2(0))\langle f(U_t,\tau^*),\beta_{j_0,k_0}\rangle. $

    Now, let us compute $ g(z, \overline z) $. Set

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

    Let $ (\psi_1, \psi_2) = \Psi_1(0)-\text{i}\Psi_2(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 $ g_{21} $, we need to compute $ w_{11} $ and $ w_{20} $. The calculation of $ w_{11} $ and $ w_{20} $ is somewhat tedious. Let $ A_U $ denote the generator of the semigroup generated by the linear system (3.3) with $ \mu = 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(\frac{1}{2}(q\dot z+\overline qz)\cdot\beta_{j_0,k_0}+w,\tau^*) = f_{z^2}\frac{z^2}{2}+f_{z\overline z}z\overline z+f_{\overline z^2}\frac{\overline z^2}{2}+\cdots. $

    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,

    $ \dot w = \frac{\partial w(z,\overline z)}{\partial z}\dot z+\frac{\partial w(z,\overline z)}{\partial \overline z}\dot{\overline z}, $

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

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

    As $ 2\text{i}\omega^*\tau^* $ and 0 are not characteristic values of (3.3), (3.14) has unique solutions $ w_{20} $ and $ w_{11} $ in $ P_S\texttt{C} $, given by

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

    Using the definition of $ A_U $, we get, from the first equation (3.13) and (3.14), that for $ \theta\in[-1, 0] $,

    $ \dot w_{20} = 2\text{i}\omega^*\tau^*w_{20}(\theta)+\frac{1}{2}(q(\theta)g_{20}+\overline q(\theta)\overline g_{02})\cdot\beta_{j_0,k_0}. $

    Therefore

    $ w_{20}(\theta) = \frac{1}{2}\left[\frac{\text{i}g_{20}}{\omega^*\tau^*}q(\theta)+\frac{\text{i}\overline g_{02}}{3\omega^*\tau^*}\overline q(\theta)\right]\cdot\beta_{j_0,k_0}+E\text{e}^{2\text{i}\omega^*\tau^*\theta}, $

    where $ E $ is a 2-dimensional vector in $ X $. According to the definition of $ \beta_{j, k}^i(i = 1, 2) $ and $ q(\theta)(\theta\in[-1, 0]) $, we have

    $ \tau^*D \Delta q(0)\cdot\beta_{j_0,k_0}+L(\tau^*)(q(\theta)\cdot\beta_{j_0,k_0}) = \text{i}\omega^*q(0)\cdot\beta_{j_0,k_0}, $
    $ \tau^*D \Delta \overline q(0)\cdot\beta_{j_0,k_0}+L(\tau^*)(\overline q(\theta)\cdot\beta_{j_0,k_0}) = -\text{i}\omega^*\overline q(0)\cdot\beta_{j_0,k_0}. $

    From (3.14), we get that

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

    Representing $ E $ and $ f_{z^2} $ by series:

    $ E = \sum\limits_{j,k = 0}^\infty E_{j,k}\cdot\beta_{j,k} = \sum\limits_{j,k = 0}^\infty E_{j,k} \gamma_{j,k}\; (E_{j,k}\in\mathbb{R}^2), $
    $ f_{z^2} = \sum\limits_{j,k = 0}^\infty \langle f_{z^2},\beta_{j,k}\rangle\cdot\beta_{j,k} = \sum\limits_{j,k = 0}^\infty \langle f_{z^2},\beta_{j,k}\rangle\gamma_{j,k}. $

    We get from (3.16) that

    $ 2\text{i}\omega^*\tau^*E_{j,k}+\tau^*\frac{j^2+k^2}{l^2}D E_{j,k}-L(\tau^*)(E_{j,k}\text{e}^{2\text{i}\omega^*\tau^*\cdot}) = \langle f_{z^2},\beta_{j,k}\rangle,\; j,k\in\mathbb{N}_0. $

    So, $ E_{j, k} $ could be calculated by

    $ E_{j,k} = \tilde E^{-1}_{j,k}\langle f_{z^2},\beta_{j,k}\rangle, $

    where

    $ \tilde E_{j,k} = \tau^*\left(2iω+d1(j2+k2)l2+uke2iωτuabv(1+u)22iω+d2(j2+k2)l2+b1+u
    \right), $
    $ \langle f_{z^2}, \beta_{j,k} \rangle = {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

    $ \tilde f_{z^2} = \frac{\tau^*}{2} \left(a1+a2eiωτa3+a4q1
    \right). $

    Similarly, we get

    $ w_{11}(\theta) = \frac{1}{2}\left[\frac{-\text{i}g_{11}}{\omega^*\tau^*}q(\theta)+\frac{\text{i}\overline g_{11}}{\omega^*\tau^*}\overline q(\theta)\right]\cdot\beta_{j_0,k_0}+F, $
    $ F = \sum\limits_{j,k = 0}^\infty F_{j,k}\gamma_{j,k}\; (F_{j,k}\in\mathbb{R}^2),\; \; F_{j,k} = \tilde F^{-1}_{j,k} < f_{z\overline z},\beta_{j,k} > , $

    where

    $ \tilde F_{j,k} = \tau^*\left(d1(j2+k2)l2+uku(a+bv(1+u)2)d2(j2+k2)l2+b1+u
    \right), $
    $ \langle f_{z\overline z}, \beta_{j,k} \rangle = {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

    $ \tilde f_{z\overline z} = \frac{\tau^*}{4}\left(a1(q1+¯q1)+a2(eiωτ+eiωτ)2a3+a4(q1+¯q1)
    \right). $

    Then, the coefficient $ g_{21} $ is completely determined.

    Let $ \lambda(\tau) = \alpha(\tau)+\text{i}\omega(\tau) $ 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 $ \mu_2 $ determines the direction of the Hopf bifurcation (forward if $ \mu_2 > 0 $, backward if $ \mu_2 < 0 $). The quantity $ \beta_2 $ determines the stability of the bifurcating periodic solutions (stable if $ \beta_2 < 0 $, unstable if $ \beta_2 > 0 $). The quantity $ T_2 $ determines the period of the bifurcating periodic solutions (the period increases if $ T_2 > 0 $, decreases if $ T_2 < 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 = (x_1, \; x_2)\in\Omega = [0, 2\pi]\times[0, 2\pi], \; d_1 = 1, \; d_2 = 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

    $ \mu_0 = 0,\; \mu_1 = 0.25,\; \mu_2 = 0.5, $

    (2.4) has pure roots $ \pm\text{i}\omega_n^+, \; n = 0, 1, 2 $, where

    $ \omega_0^+\approx0.8132,\; \omega_1^+\approx0.6718,\; \omega_2^+\approx0.4447, $

    and the corresponding bifurcation values $ \tau_{k, j}^+ $ are

    $ \tau_{0,j}^+\approx1.9885+ 7.7260j,\; \tau_{1,j}^+\approx3.0398+9.3522j,\; \tau_{2,j}^+\approx5.5770+14.1290j, $

    respectively. When $ \tau < \tau^* = 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 $ \tau = 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 $ c_1(0)\approx -0.5506 + 1.3320\text{i} $. Because $ \text{Re}c_1(0) < 0 $, we know that when $ \tau > \tau^* = 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\in[-\tau, 0] $.

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

    Example 4.2. We choose $ x = (x_1, \; x_2)\in\Omega = [0, 2\pi]\times[0, 2\pi], \; d_1 = 0.5, \; d_2 = 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

    $ \mu_0 = 0,\; \mu_1 = 0.25,\; \mu_2 = 0.5, $

    (2.4) has pure roots $ \pm\text{i}\omega_n^{\pm}, \; n = 0, 1, 2 $, where

    $ \omega_0^+\approx0.5851,\; \omega_1^+\approx0.5517,\; \omega_2^+\approx0.4431, $
    $ \omega_0^-\approx0.1533,\; \omega_1^-\approx0.1953,\; \omega_2^-\approx0.3004, $

    and the corresponding bifurcation values $ \tau_{k, j}^+ $ are

    $ \tau_{0,j}^+\approx2.8577+10.7394j,\; \tau_{1,j}^+\approx3.6560+11.3898j,\; \tau_{2,j}^+\approx5.6984+14.1787j, $
    $ \tau_{0,j}^-\approx24.2297+40.9961j,\; \tau_{1,j}^-\approx17.6208+32.1697j,\; \tau_{2,j}^-\approx10.0789+20.9129j, $

    respectively. When $ \tau < \tau^* = 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 $ \tau = 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 $ c_1(0)\approx -1.3262 + 1.2114\text{i} $. Because $ \text{Re}c_1(0) < 0 $, we know that when $ \tau > \tau^* = 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\in[-\tau, 0] $.

    Figure 4.  Numerical simulations of the system (1.3) for Example 4.2 with $ \tau = 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.

    [1] Edelstein AS, Cammaratra RC (1996) Nanomaterials: synthesis, properties and applications, Bristol: Institute of physics.
    [2] Metikoš-Huković M, Grubač Z, Radić N, et al. (2006) Sputter deposited nanocrystalline Ni and Ni-W films as catalysts for hydrogen evolution. J Mol Catal A Chem 249: 172–180. doi: 10.1016/j.molcata.2006.01.020
    [3] Brenner A (1963) Electrodeposition of Alloys: Principles and Practice, Vol. 2, Publ. Academic Press Inc., London.
    [4] Panagopoulos CN, Plainakis GD, Tsoutsouva MG (2015) Corrosion of nanocrystalline Ni-W coated copper. J Surf Eng Mater Adv Technol 5: 65–72.
    [5] Sriraman KR, Raman SGS, Seshadri SK (2006) Synthesis and evaluation of hardness and sliding wear resistance of electrodeposited nanocrystalline Ni–W alloys. Mater Sci Eng A 418: 303–311. doi: 10.1016/j.msea.2005.11.046
    [6] Quiroga Arganaraz MP, Ribotta SB, Folquer ME, et al. (2011) Ni–W coatings electrodeposited on carbon steel: Chemical composition, mechanical properties and corrosion resistance. Electrochim Acta 56: 5898–5903. doi: 10.1016/j.electacta.2011.04.119
    [7] Iwasaki H, Higashi K, Nieh TG (2004) Tensile deformation and microstructure of a nanocrystalline Ni–W alloy produced by electrodeposition. Scripta Mater 50: 395–399.
    [8] Matsui I, Takigawa Y, Uesugi T, et al. (2013) Effect of orientation on tensile ductility of electrodeposited bulk nanocrystalline Ni–W alloys. Mater Sci Eng A 578: 318–322. doi: 10.1016/j.msea.2013.04.114
    [9] Panagopoulos CN, Plainakis GD, Lagaris DA (2011) Nanocrystalline Ni–W coatings on copper. Mater Sci Eng B 176: 477–479. doi: 10.1016/j.mseb.2010.03.058
    [10] Wu Y, Chang D, Kim D, et al. (2003) Influence of boric acid on the electrodepositing process and structures of Ni–W alloy coating. Surf Coat Technol 173: 259–264.
    [11] Hou KH, Chang Y-F, Chang S-M, et al. (2010) The heat treatment effect on the structure and mechanical properties of electrodeposited nano grain size Ni–W alloy coatings. Thin Solid Films 518: 7535–7540.
    [12] Mizushima I, Tang PT, Hansen HN, et al. (2006) Residual stress in Ni–W electrodeposits. Electrochim Acta 51: 6128–6134. doi: 10.1016/j.electacta.2005.11.053
    [13] Panagopoulos CN, Georgiou EP (2007) The effect of hydrogen charging on the mechanical behaviour of 5083 wrought aluminum alloy. Corros Sci 49: 4443–4451. doi: 10.1016/j.corsci.2007.03.047
    [14] Metals Handbook. 9th Ed., Alloy Phase Diagrams, ASM, USA, 1992.
    [15] Cullity BD (1959) Elements of X-ray Diffraction, 1st Edition, Addison-Wesley Publishing Company, Massachusetts.
    [16] Auerswald J, Fecht H-J (2010) Nanocrystalline Ni-W for wear-resistant coatings and electroforming. J Electrochem Soc 157: 199–205.
    [17] Mizushima I, Tang PT, Hansen HN, et al. (2006) Residual stress in Ni–W electrodeposits. Electrochim Acta 51: 6128–6134. doi: 10.1016/j.electacta.2005.11.053
    [18] Dieter GE (1988) Mechanical Metallurgy, SI Metric Edition, Publ. McGraw-Hill, London.
    [19] Kumar KS, Van Swygenhoven H, Suresh S (2003) Mechanical behavior of nanocrystalline metals and alloys. Acta Mater 51: 5743–5774. doi: 10.1016/j.actamat.2003.08.032
    [20] Hahn H, Padmanabhan KA (1995) Mechanical response of nanostructured materials. Nanostruct Mater 6: 191–200. doi: 10.1016/0965-9773(95)00042-9
    [21] Wang N, Wang Z, Aust KT, et al. (1995) Effect of grain size on mechanical properties of nanocrystalline materials. Acta Metall Mater 43: 519–528.
    [22] Flinn RA, Trojan PK (1994) Engineering materials and their applications, 4th ed., Wiley-VCH, Boston.
    [23] Meyers MA, Mishra A, Benson DJ (2006) Mechanical properties on nanocrystalline materials. Progr Mater Sci 51: 427–556. doi: 10.1016/j.pmatsci.2005.08.003
    [24] Panagopoulos CN, Pelegri AA (1993) Tensile properties of zinc coated aluminium. Surf Coat Technol 57: 203–206. doi: 10.1016/0257-8972(93)90041-L
    [25] Armstrong DEJ, Haseeb ASMA, Roberts SG, et al. (2012) Nanoindentation and micro-mechanical fracture toughness of electrodeposited nanocrystalline Ni–W alloy films. Thin Solid Films 520: 4369–4372.
    [26] Panagopoulos CN, Papachristos VD, Sigalas G (1999) Tensile behaviour of as deposited and heat-treated electroless Ni–P deposits. J Mater Sci 34: 2587–2600.
  • This article has been cited by:

    1. Wei Dou, Allison K. Allen, Hyein Cho, Sabrina Bhangal, Alexander J. Cook, Ezequiel Morsella, Mark W. Geisler, EEG Correlates of Involuntary Cognitions in the Reflexive Imagery Task, 2020, 11, 1664-1078, 10.3389/fpsyg.2020.00482
    2. Katelyn Gardner, Erica B. Walker, Yanming Li, Adam Gazzaley, Ezequiel Morsella, Involuntary attentional shifts as a function of set and processing fluency, 2020, 203, 00016918, 103009, 10.1016/j.actpsy.2020.103009
    3. Elizabeth Heinrichs-Graham, Elizabeth A. Walker, Jacob A. Eastman, Michaela R. Frenzel, Ryan W. McCreery, Amount of Hearing Aid Use Impacts Neural Oscillatory Dynamics Underlying Verbal Working Memory Processing for Children With Hearing Loss, 2022, 43, 1538-4667, 408, 10.1097/AUD.0000000000001103
    4. Christina Y. Wong, Alejandro Heredia Cedillo, Ezequiel Morsella, The priming of stimulus-elicited involuntary mental imagery, 2024, 246, 00016918, 104250, 10.1016/j.actpsy.2024.104250
  • Reader Comments
  • © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(6567) PDF downloads(1142) Cited by(4)

Figures and Tables

Figures(11)  /  Tables(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog