∎

11institutetext: Yehu Lv 22institutetext: School of Mathematical Sciences,
Beijing Normal University, Beijing 100875, China
22email<EMAIL_ADDRESS>
# Inhomogeneous spatial patterns in diffusive predator-prey system with
spatial memory and predator-taxis

Yehu Lv

###### Abstract

In this paper, we consider a diffusive predator-prey system with spatial
memory and predator-taxis. Since in this system, the memory delay appears in
the diffusion term, and the diffusion term is nonlinear, the classical normal
form of Hopf bifurcation in the reaction-diffusion system with delay can’t be
applied to this system. Thus, in this paper, we first derive an algorithm for
calculating the normal form of Hopf bifurcation in this system. Then in order
to illustrate the effectiveness of our newly developed algorithm, we consider
the diffusive Holling-Tanner model with spatial memory and predator-taxis. The
stability and Hopf bifurcation analysis of this model are investigated, and
the direction and stability of Hopf bifurcation periodic solution are also
researched by using our newly developed algorithm for calculating the normal
form of Hopf bifurcation. At last, we carry out some numerical simulations,
two stable spatially inhomogeneous periodic solutions corresponding to the
mode-1 and mode-2 Hopf bifurcations are found, which verifies our theoretical
analysis results.

###### Keywords:

Predator-prey system Memory delay Predator-taxis Hopf bifurcation Normal form
Periodic solutions

###### MSC:

35B10 37G05 37L10 92D25

##  1 Introduction

The reaction-diffusion equations based on the Fick’s law have been widely used
in physics, chemistry and biology lv1 ; lv2 ; lv3 . More precisely, based on
the Fick’s law, that is the movement flux is in the direction of negative
gradient of the density distribution function, the diffusive Brusselator model
with gene expression time delay lv4 , the diffusive predator-prey model in
heterogeneous environment lv5 ; lv6 , the diffusive predator-prey model with a
protection zone lv7 , the diffusive predator-prey model with prey social
behavior lv8 , the diffusive predator-prey model with protection zone and
predator harvesting lv9  have been studied by many scholars. Furthermore, in
order to include the episodic-like spatial memory of animals, Shi et al. lv10
directed movement toward the negative gradient of the density distribution
function at past time, and they proposed the following diffusive model with
spatial memory

$\left\\{\begin{aligned} &\frac{\partial u(x,t)}{\partial t}=d_{1}\Delta
u(x,t)+d_{2}\left(u(x,t)u_{x}(x,t-\tau)\right)_{x}+f\left(u(x,t)\right),&x\in\Omega,~{}t>0,\\\
&\frac{\partial u}{\partial\mathbf{n}}(x,t)=0,&x\in\partial\Omega,~{}t>0,\\\
&u(x,t)=u_{0}(x,t),&x\in\Omega,~{}-\tau\leq t\leq 0,\end{aligned}\right.$
(1.1)

where $u(x,t)$ is the population density at spatial location $x$ and time $t$,
$d_{1}$ and $d_{2}$ are the Fickian diffusion coefficient and the memory-based
diffusion coefficient, respectively, $\Omega\subset\mathbb{R}$ is a smooth and
bounded domain, $u_{0}(x,t)$ is the initial function, $\Delta
u(x,t)=\partial^{2}u(x,t)/\partial x^{2}$, $u_{x}(x,t)=\partial
u(x,t)/\partial x$, $u_{x}(x,t-\tau)=\partial u(x,t-\tau)/\partial x$,
$u_{xx}(x,t-\tau)=\partial^{2}u(x,t-\tau)/\partial x^{2}$, and $\mathbf{n}$ is
the outward unit normal vector at the smooth boundary $\partial\Omega$. Here,
the time delay $\tau>0$ represents the averaged memory period, which is
usually called as the memory delay, and $f\left(u(x,t)\right)$ describes the
chemical reaction or biological birth/death. Moreover, in order to further
investigate the influence of memory delay on the stability of the positive
constant steady state, on the basis of model (1.1), Shi et al. lv11  studied
the spatial memory diffusion model with memory and maturation delays.
Furthermore, Song et al. lv12  proposed the following memory-based diffusion
system subjects to homogeneous Neumann boundary condition

$\left\\{\begin{aligned} &\frac{\partial u(x,t)}{\partial t}=d_{11}\Delta
u(x,t)+f\left(u(x,t),v(x,t)\right),&x\in\Omega,~{}t>0,\\\ &\frac{\partial
v(x,t)}{\partial t}=d_{22}\Delta
v(x,t)-d_{21}\left(v(x,t)u_{x}(x,t-\tau)\right)_{x}+g\left(u(x,t),v(x,t)\right),&x\in\Omega,~{}t>0,\\\
&u_{x}(0,t)=u_{x}(\ell\pi,t)=v_{x}(0,t)=v_{x}(\ell\pi,t)=0,&t\geq 0,\\\
&u(x,t)=u_{0}(x,t),&x\in\Omega,~{}-\tau\leq t\leq 0,\\\
&v(x,t)=v_{0}(x),&x\in\Omega,\end{aligned}\right.$ (1.2)

where $\Omega=(0,\ell\pi)$ with $\ell\in\mathbb{R}^{+}$, $u(x,t)$ and $v(x,t)$
represent the densities of prey and predator at the location $x$ and time $t$,
respectively, $d_{11}\geq 0$ and $d_{22}\geq 0$ are the random diffusion
coefficients, $d_{21}\geq 0$ is the memory-based diffusion coefficient,
$v_{0}(x)$ is also the initial function, and $f\left(u(x,t),v(x,t)\right)$ and
$g\left(u(x,t),v(x,t)\right)$ are the reaction terms. Furthermore, they
derived an algorithm for calculating the normal form of Hopf bifurcation in
the system (1.2). By numerical simulations, the stable spatially inhomogeneous
periodic solutions and the transition from the unstable mode-2 spatially
inhomogeneous periodic solution to the stable mode-1 spatially inhomogeneous
periodic solution are found.

For the general predator-prey models in ecology, in addition to random
diffusion of the predator and prey populations, the spatial movement of
predator and prey populations also occurs, which is usually shown by the
predator pursuing prey and prey escaping from predator lv13 . The pursuit and
evasion between the predator and prey populations also have a strong impact on
the movement pattern of the predator and prey populations lv14 ; lv15 ; lv16 .
Furthermore, by noticing that such movement is not random but directed, i.e.,
predator moves toward the gradient direction of prey distribution, which is
called prey-taxis, or prey moves opposite to the gradient of predator
distribution, which is called predator-taxis. Recently, the predator-prey
model with prey-taxis lv17 ; lv18 ; lv19 ; lv20 ; lv21 , the predator-prey
model with indirect prey-taxis lv22 ; lv23 ; lv24 , the predator-prey model
with predator-taxis lv25  and the predator-prey model with indirect predator-
taxis lv26  have been researched. Especially, Wang et al. lv13  considered the
following diffusive predator-prey model with both predator-taxis and prey-
taxis. The corresponding mathematical model is

$\left\\{\begin{aligned} &\frac{\partial u(x,t)}{\partial t}=d\Delta
u(x,t)+\xi\left(u(x,t)v_{x}(x,t)\right)_{x}+f(u(x,t),v(x,t)),&x\in\Omega,~{}t>0,\\\
&\frac{\partial v(x,t)}{\partial t}=\Delta
v(x,t)-\eta\left(v(x,t)u_{x}(x,t)\right)_{x}+g(u(x,t),v(x,t)),&x\in\Omega,~{}t>0,\\\
&\frac{\partial u(x,t)}{\partial\mathbf{n}}=\frac{\partial
v(x,t)}{\partial\mathbf{n}}=0,&x\in\partial\Omega,~{}t>0,\\\
&u(x,0)=u_{0}(x)\geq 0,~{}v(x,0)=v_{0}(x)\geq
0,&x\in\Omega,\end{aligned}\right.$ (1.3)

where $u_{0}(x)$ is the initial function, $d$ is the rescaled diffusion
coefficient for the prey population, and the diffusion coefficient of the
predator population is rescaled as 1. Furthermore, the term
$\xi\left(u(x,t)v_{x}(x,t)\right)_{x}$ represents the prey moves away from
predator, and $\xi\geq 0$ is the intrinsic predator-taxis rate. The term
$-\eta\left(v(x,t)u_{x}(x,t)\right)_{x}$ represents the predator moves towards
prey, and $\eta\geq 0$ is the intrinsic prey-taxis rate.

Therefore, based on the models (1.2) and (1.3), and by considering that the
spatial memory and predator-taxis, we proposed the following diffusive
predator-prey model with spatial memory and predator-taxis subjects to
homogeneous Neumann boundary condition

$\left\\{\begin{aligned} &\frac{\partial u(x,t)}{\partial t}=d_{11}\Delta
u(x,t)+\xi\left(u(x,t)v_{x}(x,t)\right)_{x}+f(u(x,t),v(x,t)),&x\in(0,\ell\pi),~{}t>0,\\\
&\frac{\partial v(x,t)}{\partial t}=d_{22}\Delta
v(x,t)-d_{21}\left(v(x,t)u_{x}(x,t-\tau)\right)_{x}+g(u(x,t),v(x,t)),&x\in(0,\ell\pi),~{}t>0,\\\
&u_{x}(0,t)=u_{x}(\ell\pi,t)=v_{x}(0,t)=v_{x}(\ell\pi,t)=0,&t\geq 0,\\\
&u(x,t)=u_{0}(x,t),&x\in(0,\ell\pi),~{}-\tau\leq t\leq 0,\\\
&v(x,t)=v_{0}(x),&x\in(0,\ell\pi).\end{aligned}\right.$ (1.4)

This paper is organized as follows. In Section 2, we derive an algorithm for
calculating the normal form of Hopf bifurcation in the model (1.4). In Section
3, we obtain the normal form of Hopf bifurcation truncated to the third-order
term by using our newly developed algorithm developed in Sec.2, and the
mathematical expressions of the corresponding coefficients are given. In
Section 4, we consider the diffusive Holling-Tanner model with spatial memory
and predator-taxis. The stability and Hopf bifurcation analysis in this model
are studied, and some numerical simulations are also given. In Section 5, we
give a brief conclusion and discussion.

##  2 Algorithm for calculating the normal form of Hopf bifurcation in the
system (1.4)

###  2.1 Characteristic equation at the positive constant steady state

Define the real-valued Sobolev space

$X:=\left\\{(u,v)^{T}\in\left(W^{2,2}(0,\ell\pi)\right)^{2}:\frac{\partial
u}{\partial x}=\frac{\partial v}{\partial x}=0\text{ at }x=0,\ell\pi\right\\}$

with the inner product defined by

$\left[U_{1},U_{2}\right]=\int_{0}^{\ell\pi}U_{1}^{T}U_{2}~{}dx\text{ for
}U_{1}=\left(u_{1},v_{1}\right)^{T}\in X\text{ and
}U_{2}=\left(u_{2},v_{2}\right)^{T}\in X,$

where the symbol $T$ represents the transpose of vector, and let
$\mathcal{C}:=C([-1,0];X)$ be the Banach space of continuous mappings from
$[-1,0]$ to $X$ with the sup norm. It is well known that the eigenvalue
problem

$\left\\{\begin{aligned}
&\widetilde{\varphi}^{\prime\prime}(x)=\widetilde{\lambda}\widetilde{\varphi}(x),~{}x\in(0,\ell\pi),\\\
&\widetilde{\varphi}^{\prime}(0)=\widetilde{\varphi}^{\prime}(\ell\pi)=0\end{aligned}\right.$

has eigenvalues $\widetilde{\lambda}_{n}=-n^{2}/\ell^{2}$ with corresponding
normalized eigenfunctions

$\beta_{n}^{(j)}=\gamma_{n}(x)e_{j},~{}\gamma_{n}(x)=\frac{\cos(nx/\ell)}{\left\|\cos(nx/\ell)\right\|_{L^{2}}}=\left\\{\begin{aligned}
&\frac{1}{\sqrt{\ell\pi}},&n=0,\\\
&\frac{\sqrt{2}}{\sqrt{\ell\pi}}\cos\left(\frac{nx}{\ell}\right),&n\geq
1,\end{aligned}\right.$ (2.1)

where $e_{j},~{}j=1,2$ is the unit coordinate vector of $\mathbb{R}^{2}$, and
$n\in\mathbb{N}_{0}=\mathbb{N}\cup\left\\{0\right\\}$ is often called wave
number, $\mathbb{N}_{0}$ is the set of all non-negative integers,
$\mathbb{N}=\left\\{1,2,...\right\\}$ represents the set of all positive
integers.

Without loss of generality, we assume that $E_{*}\left(u_{*},v_{*}\right)$ is
the positive constant steady state of system (1.4). The linearized equation of
(1.4) at $E_{*}\left(u_{*},v_{*}\right)$ is

$\left(\begin{aligned} &\frac{\partial u(x,t)}{\partial t}\\\ &\frac{\partial
v(x,t)}{\partial t}\end{aligned}\right)=D_{1}\left(\begin{aligned} &\Delta
u(x,t)\\\ &\Delta v(x,t)\end{aligned}\right)+D_{2}\left(\begin{aligned}
&\Delta u(x,t-\tau)\\\ &\Delta
v(x,t-\tau)\end{aligned}\right)+A\left(\begin{aligned} &u(x,t)\\\
&v(x,t)\end{aligned}\right),$ (2.2)

where

$D_{1}=\left(\begin{array}[]{cc}d_{11}&\xi u_{*}\\\
0&d_{22}\end{array}\right),~{}D_{2}=\left(\begin{array}[]{cc}0&0\\\
-d_{21}v_{*}&0\end{array}\right),~{}A=\left(\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right)$ (2.3)

and

$a_{11}=\frac{\partial f\left(u_{*},v_{*}\right)}{\partial
u},~{}a_{12}=\frac{\partial f\left(u_{*},v_{*}\right)}{\partial
v},~{}a_{21}=\frac{\partial g\left(u_{*},v_{*}\right)}{\partial
u},~{}a_{22}=\frac{\partial g\left(u_{*},v_{*}\right)}{\partial v}.$ (2.4)

Therefore, the characteristic equation of (2.2) is

$\prod_{n\in\mathbb{N}_{0}}\Gamma_{n}(\lambda)=0,$

where $\Gamma_{n}(\lambda)=\det\left(\mathcal{M}_{n}(\lambda)\right)$ with

$\mathcal{M}_{n}(\lambda)=\lambda
I_{2}+\frac{n^{2}}{\ell^{2}}D_{1}+\frac{n^{2}}{\ell^{2}}e^{-\lambda\tau}D_{2}-A.$
(2.5)

Here, $\det(.)$ represents the determinant of a matrix, $I_{2}$ is the
identity matrix of $2\times 2$, and $D_{1},D_{2},A$ are defined by (2.3). Then
we can obtain

$\Gamma_{n}(\lambda)=\operatorname{det}\left(\mathcal{M}_{n}(\lambda)\right)=\lambda^{2}-T_{n}\lambda+\widetilde{J}_{n}(\tau)=0,$
(2.6)

where

$\displaystyle T_{n}$
$\displaystyle=\operatorname{Tr}(A)-\operatorname{Tr}\left(D_{1}\right)\frac{n^{2}}{\ell^{2}},$
(2.7) $\displaystyle\widetilde{J}_{n}(\tau)$
$\displaystyle=(d_{11}d_{22}+d_{21}\xi
u_{*}v_{*}e^{-\lambda\tau})\frac{n^{4}}{\ell^{4}}-\left(d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}+d_{21}v_{*}a_{12}e^{-\lambda\tau}\right)\frac{n^{2}}{\ell^{2}}+\operatorname{Det}(A)$

with $\operatorname{Tr}(A)=a_{11}+a_{22}$,
$\operatorname{Tr}\left(D_{1}\right)=d_{11}+d_{22}$ and
$\operatorname{Det}(A)=a_{11}a_{22}-a_{12}a_{21}$.

###  2.2 Basic assumption and equation transformation

###### Assumption 2.1

Assume that at $\tau=\tau_{c}$, (2.6) has a pair of purely imaginary roots
$\pm i\omega_{n_{c}}$ with $\omega_{n_{c}}>0$ for $n=n_{c}\in\mathbb{N}$ and
all other eigenvalues have negative real parts. Let
$\lambda(\tau)=\alpha_{1}(\tau)\pm i\alpha_{2}(\tau)$ be a pair of roots of
(2.6) near $\tau=\tau_{c}$ satisfying $\alpha_{1}(\tau_{c})=0$ and
$\alpha_{2}(\tau_{c})=\omega_{n_{c}}$. In addition, the corresponding
transversality condition holds.

Let $\tau=\tau_{c}+\mu,~{}|\mu|\ll 1$ such that $\mu=0$ corresponds to the
Hopf bifurcation value for system (1.4). Moreover, we shift
$E_{*}(u_{*},v_{*})$ to the origin by setting

$U(x,t)=\left(U_{1}(x,t),U_{2}(x,t)\right)^{T}=\left(u(x,t),v(x,t)\right)^{T}-\left(u_{*},v_{*}\right)^{T},$

and normalize the delay by rescaling the time variable $t\rightarrow t/\tau$.
Furthermore, we rewrite $U(t)$ for $U(x,t)$, and $U_{t}\in\mathcal{C}$ for
$U_{t}(\theta)=U(x,t+\theta),~{}-1\leq\theta\leq 0$. Then the system (1.4)
becomes the compact form

$\frac{dU(t)}{dt}=d(\mu)\Delta(U_{t})+L(\mu)(U_{t})+F(U_{t},\mu),$ (2.8)

where for
$\varphi=\left(\varphi^{(1)},\varphi^{(2)}\right)^{T}\in\mathcal{C}$,
$d(\mu)\Delta$ is given by

$d(\mu)\Delta(\varphi)=d_{0}\Delta(\varphi)+F^{d}(\varphi,\mu)$

with

$\displaystyle d_{0}\Delta(\varphi)$
$\displaystyle=\tau_{c}D_{1}\Delta\varphi(0)+\tau_{c}D_{2}\Delta\varphi(-1),$
(2.9) $\displaystyle F^{d}(\varphi,\mu)$
$\displaystyle=\left(\begin{array}[]{c}\xi(\tau_{c}+\mu)(\varphi_{x}^{(1)}(0)\varphi_{x}^{(2)}(0)+\varphi^{(1)}(0)\varphi_{xx}^{(2)}(0))\\\
-d_{21}(\tau_{c}+\mu)(\varphi_{x}^{(1)}(-1)\varphi_{x}^{(2)}(0)+\varphi_{xx}^{(1)}(-1)\varphi^{(2)}(0))\end{array}\right)$
$\displaystyle+\mu\left(\begin{array}[]{c}d_{11}\varphi_{xx}^{(1)}(0)+\xi
u_{*}\varphi_{xx}^{(2)}(0)\\\
-d_{21}v_{*}\varphi_{xx}^{(1)}(-1)+d_{22}\varphi_{xx}^{(2)}(0)\end{array}\right).$

Furthermore, $L(\mu):\mathcal{C}\rightarrow X$ is given by

$L(\mu)(\varphi)=\left(\tau_{c}+\mu\right)A\varphi(0),$ (2.10)

and $F:\mathcal{C}\times\mathbb{R}^{2}\rightarrow X$ is given by

$F(\varphi,\mu)=\left(\tau_{c}+\mu\right)\left(\begin{aligned}
f\left(\varphi^{(1)}(0)+u_{*},\varphi^{(2)}(0)+v_{*}\right)\\\
g\left(\varphi^{(1)}(0)+u_{*},\varphi^{(2)}(0)+v_{*}\right)\end{aligned}\right)-L(\mu)(\varphi).$
(2.11)

In what follows, we assume that $F(\varphi,\mu)$ is $C^{k}(k\geq 3)$ function,
which is smooth with respect to $\varphi$ and $\mu$. Notice that $\mu$ is the
perturbation parameter and is treated as a variable in the calculation of
normal form. Moreover, by (2.10), if we denote
$L_{0}(\varphi)=\tau_{c}A\varphi(0)$, then (2.8) can be rewritten as

$\frac{dU(t)}{dt}=d_{0}\Delta(U_{t})+L_{0}(U_{t})+\widetilde{F}\left(U_{t},\mu\right),$
(2.12)

where the linear and nonlinear terms are separated, and

$\widetilde{F}(\varphi,\mu)=\mu
A\varphi(0)+F(\varphi,\mu)+F^{d}(\varphi,\mu).$ (2.13)

Thus, the linearized equation of (2.12) can be written as

$\frac{dU(t)}{dt}=d_{0}\Delta(U_{t})+L_{0}(U_{t}).$ (2.14)

Moreover, the characteristic equation for the linearized equation (2.14) is

$\prod_{n\in\mathbb{N}_{0}}\widetilde{\Gamma}_{n}(\lambda)=0,$ (2.15)

where
$\widetilde{\Gamma}_{n}(\lambda)=\operatorname{det}\left(\widetilde{\mathcal{M}}_{n}(\lambda)\right)$
with

$\widetilde{\mathcal{M}}_{n}(\lambda)=\lambda
I_{2}+\tau_{c}\frac{n^{2}}{\ell^{2}}D_{1}+\tau_{c}\frac{n^{2}}{\ell^{2}}e^{-\lambda}D_{2}-\tau_{c}A.$
(2.16)

By comparing (2.16) with (2.5), we know that (2.15) has a pair of purely
imaginary roots $\pm i\omega_{c}$ for $n=n_{c}\in\mathbb{N}$, and all other
eigenvalues have negative real parts, where
$\omega_{c}=\tau_{c}\omega_{n_{c}}$. In order to write (2.12) as an abstract
ordinary differential equation in a Banach space, follows by lv27 , we can
take the enlarged space

$\mathcal{BC}:=\left\\{\widetilde{\psi}:[-1,0]\rightarrow
X:\widetilde{\psi}\text{ is continuous on
}[-1,0),\exists\lim_{\theta\rightarrow 0^{-}}\widetilde{\psi}(\theta)\in
X\right\\},$

then the equation (2.12) is equivalent to an abstract ordinary differential
equation on $\mathcal{BC}$

$\frac{dU_{t}}{dt}=\widetilde{A}U_{t}+X_{0}\widetilde{F}\left(U_{t},\mu\right).$

Here, $\widetilde{A}$ is a operator from
$\mathcal{C}_{0}^{1}=\\{\varphi\in\mathcal{C}:\dot{\varphi}\in\mathcal{C},\varphi(0)\in\operatorname{dom}(\Delta)\\}$
to $\mathcal{BC}$, which is defined by

$\widetilde{A}\varphi=\dot{\varphi}+X_{0}\left(\tau_{c}D_{1}\Delta\varphi(0)+\tau_{c}D_{2}\Delta\varphi(-1)+L_{0}(\varphi)-\dot{\varphi}(0)\right),$

and $X_{0}=X_{0}(\theta)$ is given by

$X_{0}(\theta)=\left\\{\begin{aligned} &0,&-1\leq\theta<0,\\\
&1,&\theta=0.\end{aligned}\right.$

In the following, the method given in lv27  is used to complete the
decomposition of $\mathcal{BC}$. Let
$C:=C\left([-1,0],\mathbb{R}^{2}\right),~{}C^{*}:=C\left([0,1],\mathbb{R}^{2*}\right)$,
where $\mathbb{R}^{2*}$ is the two-dimensional space of row vectors, and
define the adjoint bilinear form on $C^{*}\times C$ as follows

$\langle\Psi(s),\Phi(\theta)\rangle=\Psi(0)\Phi(0)-\int_{-1}^{0}\int_{0}^{\theta}\Psi(\xi-\theta)dM_{n}(\theta)\Phi(\xi)d\xi$

for $\Psi\in C^{*},~{}\Phi\in C$ and $\xi\in[-1,0]$, where $M_{n}(\theta)$ is
a bounded variation function from $[-1,0]$ to $\mathbb{R}^{2\times 2}$, i.e.,
$M_{n}(\theta)\in BV\left([-1,0];\mathbb{R}^{2\times 2}\right)$, such that for
$\Phi(\theta)\in\mathcal{C}$, one has

$-\tau_{c}\frac{n^{2}}{\ell^{2}}D_{1}\Phi(0)-\tau_{c}\frac{n^{2}}{\ell^{2}}D_{2}\Phi(-1)+L_{0}(\Phi(\theta))=\int_{-1}^{0}dM_{n}(\theta)\Phi(\theta).$

By choosing

$\Phi(\theta)=(\phi(\theta),\overline{\phi}(\theta)),~{}\Psi(s)=\operatorname{col}\left(\psi^{T}(s),\overline{\psi}^{T}(s)\right),$

where the $\operatorname{col}(.)$ represents the column vector,
$\phi(\theta)=\operatorname{col}\left(\phi_{1}(\theta),\phi_{2}(\theta)\right)=\phi
e^{i\omega_{c}\theta}\in\mathbb{C}^{2}$ with
$\phi=\operatorname{col}\left(\phi_{1},\phi_{2}\right)$ is the eigenvector of
(2.14) associated with the eigenvalue $i\omega_{c}$, and
$\psi(s)=\operatorname{col}\left(\psi_{1}(s),\psi_{2}(s)\right)=\psi
e^{-i\omega_{c}s}\in\mathbb{C}^{2}$ with
$\psi=\operatorname{col}\left(\psi_{1},\psi_{2}\right)$ is the corresponding
adjoint eigenvector such that

$\left\langle\Psi(s),\Phi(\theta)\right\rangle=I_{2},$

where

$\phi=\left(\begin{array}[]{c}1\\\
\frac{a_{11}-i\omega_{n_{c}}-d_{11}(n_{c}^{2}/\ell^{2})}{\xi
u_{*}(n_{c}^{2}/\ell^{2})-a_{12}}\end{array}\right),~{}\psi=\eta\left(\begin{array}[]{c}1\\\
\frac{a_{12}-\xi
u_{*}(n_{c}^{2}/\ell^{2})}{i\omega_{n_{c}}+d_{22}(n_{c}^{2}/\ell^{2})-a_{22}}\end{array}\right)$

and

$\eta=\frac{i\omega_{n_{c}}+\left(n_{c}/\ell\right)^{2}d_{22}-a_{22}}{2i\omega_{n_{c}}+\left(n_{c}/\ell\right)^{2}d_{11}-a_{11}+\left(n_{c}/\ell\right)^{2}d_{22}-a_{22}+\tau_{c}a_{12}d_{21}v_{*}\left(n_{c}/\ell\right)^{2}e^{-i\omega_{c}}}.$

According to lv27 , the phase space $\mathcal{C}$ can be decomposed as

$\mathcal{C}=\mathcal{P}\oplus\mathcal{Q},~{}\mathcal{P}=\operatorname{Im}\pi,~{}\mathcal{Q}=\operatorname{Ker}\pi,$

where for $\widetilde{\phi}\in\mathcal{C}$, the projection
$\pi:\mathcal{C}\rightarrow\mathcal{P}$ is defined by

$\pi(\widetilde{\phi})=\left(\Phi\left\langle\Psi,\left(\begin{aligned}
&\left[\widetilde{\phi}(\cdot),\beta_{n_{c}}^{(1)}\right]\\\
&\left[\widetilde{\phi}(\cdot),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right)\right\rangle\right)^{T}\beta_{n_{c}}.$
(2.17)

Therefore, by following the method in lv27 , $\mathcal{BC}$ can be divided
into a direct sum of center subspace and its complementary space, that is

$\mathcal{BC}=\mathcal{P}\oplus\operatorname{ker}\pi,$ (2.18)

where $\operatorname{dim}\mathcal{P}=2$. It is easy to see that the projection
$\pi$ which is defined by (2.17), is extended to a continuous projection
(which is still denoted by $\pi$), that is,
$\pi:\mathcal{BC}\mapsto\mathcal{P}$. In particular, for
$\alpha\in\mathcal{C}$, we have

$\pi\left(X_{0}(\theta)\alpha\right)=\left(\Phi(\theta)\Psi(0)\left(\begin{aligned}
&\left[\alpha,\beta_{n_{c}}^{(1)}\right]\\\
&\left[\alpha,\beta_{n_{c}}^{(2)}\right]\end{aligned}\right)\right)^{T}\beta_{n_{c}}.$
(2.19)

By combining with (2.18) and (2.19), $U_{t}(\theta)$ can be decomposed as

$\displaystyle U_{t}(\theta)$
$\displaystyle=\left(\Phi(\theta)\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)\right)^{T}\left(\begin{array}[]{c}\beta_{n_{c}}^{(1)}\\\
\beta_{n_{c}}^{(2)}\end{array}\right)+w=\left(z_{1}\phi
e^{i\omega_{c}\theta}+z_{2}\overline{\phi}e^{-i\omega_{c}\theta}\right)\gamma_{n_{c}}(x)+w$
(2.20)
$\displaystyle=\left(\phi(\theta)~{}~{}\overline{\phi}(\theta)\right)\left(\begin{array}[]{c}z_{1}\gamma_{n_{c}}(x)\\\
z_{2}\gamma_{n_{c}}(x)\end{array}\right)+\left(\begin{array}[]{c}w_{1}\\\
w_{2}\end{array}\right),$

where $w=\operatorname{col}\left(w_{1},w_{2}\right)$ and

$\displaystyle\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)=\left\langle\Psi(0),\left(\begin{aligned}
&\left[U_{t}(\theta),\beta_{n_{c}}^{(1)}\right]\\\
&\left[U_{t}(\theta),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right)\right\rangle.$

If we assume that

$\Phi(\theta)=\left(\phi(\theta),\overline{\phi}(\theta)\right),~{}z_{x}=\left(z_{1}\gamma_{n_{c}}(x),z_{2}\gamma_{n_{c}}(x)\right)^{T},$

then (2.20) can be rewritten as

$U_{t}(\theta)=\Phi(\theta)z_{x}+w\text{ with
}w\in\mathcal{C}_{0}^{1}\cap\operatorname{Ker}\pi:=\mathcal{Q}^{1}.$ (2.21)

Then by combining with (2.21), the system (2.12) is decomposed as a system of
abstract ordinary differential equations (ODEs) on
$\mathbb{R}^{2}\times\text{Ker}~{}\pi$, with finite and infinite dimensional
variables are separated in the linear term. That is

$\left\\{\begin{aligned} &\dot{z}=Bz+\Psi(0)\left(\begin{aligned}
&\left[\widetilde{F}\left(\Phi(\theta)z_{x}+w,\mu\right),\beta_{n_{c}}^{(1)}\right]\\\
&\left[\widetilde{F}\left(\Phi(\theta)z_{x}+w,\mu\right),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right),\\\
&\dot{w}=A_{\mathcal{Q}^{1}}w+(I-\pi)X_{0}(\theta)\widetilde{F}\left(\Phi(\theta)z_{x}+w,\mu\right),\end{aligned}\right.$
(2.22)

where $I$ is the identity matrix, $z=\left(z_{1},z_{2}\right)^{T}$,
$B=\text{diag}\left\\{i\omega_{c},-i\omega_{c}\right\\}$ is the diagonal
matrix, and
$A_{\mathcal{Q}^{1}}:\mathcal{Q}^{1}\rightarrow\operatorname{Ker}\pi$ is
defined by

$A_{\mathcal{Q}^{1}}w=\dot{w}+X_{0}(\theta)\left(\tau_{c}D_{1}\Delta
w(0)+\tau_{c}D_{2}\Delta w(-1)+L_{0}(w)-\dot{w}(0)\right).$

Consider the formal Taylor expansion

$\widetilde{F}(\varphi,\mu)=\sum_{j\geq
2}\frac{1}{j!}\widetilde{F}_{j}(\varphi,\mu),~{}F(\varphi,\mu)=\sum_{j\geq
2}\frac{1}{j!}F_{j}(\varphi,\mu),~{}F^{d}(\varphi,\mu)=\sum_{j\geq
2}\frac{1}{j!}F_{j}^{d}(\varphi,\mu).$

From (2.13), we have

$\widetilde{F}_{2}(\varphi,\mu)=2\mu
A\varphi(0)+F_{2}(\varphi,\mu)+F_{2}^{d}(\varphi,\mu)$ (2.23)

and

$\widetilde{F}_{j}(\varphi,\mu)=F_{j}(\varphi,\mu)+F_{j}^{d}(\varphi,\mu),~{}j=3,4,\cdots.$
(2.24)

By combining with (2.19), the system (2.22) can be rewritten as

$\left\\{\begin{aligned} &\dot{z}=Bz+\sum_{j\geq
2}\frac{1}{j!}f_{j}^{1}(z,w,\mu),\\\ &\dot{w}=A_{\mathcal{Q}^{1}}w+\sum_{j\geq
2}\frac{1}{j!}f_{j}^{2}(z,w,\mu),\end{aligned}\right.$

where

$\displaystyle f_{j}^{1}(z,w,\mu)=\Psi(0)\left(\begin{aligned}
&\left[\widetilde{F}_{j}\left(\Phi(\theta)z_{x}+w,\mu\right),\beta_{n_{c}}^{(1)}\right]\\\
&\left[\widetilde{F}_{j}\left(\Phi(\theta)z_{x}+w,\mu\right),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right),$
(2.25) $\displaystyle
f_{j}^{2}(z,w,\mu)=(I-\pi)X_{0}(\theta)\widetilde{F}_{j}\left(\Phi(\theta)z_{x}+w,\mu\right).$

In terms of the normal form theory of partial functional differential
equations lv27 , after a recursive transformation of variables of the form

$(z,w)=(\widetilde{z},\widetilde{w})+\frac{1}{j!}\left(U_{j}^{1}(\widetilde{z},\mu),U_{j}^{2}(\widetilde{z},\mu)\right),~{}j\geq
2,$ (2.26)

where $z,\widetilde{z}\in\mathbb{R}^{2}$, $w,\widetilde{w}\in\mathcal{Q}^{1}$
and $U_{j}^{1}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}$,
$U_{j}^{2}:\mathbb{R}^{3}\rightarrow\mathcal{Q}^{1}$ are homogeneous
polynomials of degree $j$ in $\widetilde{z}$ and $\mu$, a locally center
manifold for (2.12) satisfies $w=0$ and the flow on it is given by the two-
dimensional ODEs

$\dot{z}=Bz+\sum_{j\geq 2}\frac{1}{j!}g_{j}^{1}(z,0,\mu),$

which is the normal form as in the usual sense for ODEs.

By following lv27  and lv28 , we have

$g_{2}^{1}(z,0,\mu)=\operatorname{Proj}_{\operatorname{Ker}\left(M_{2}^{1}\right)}f_{2}^{1}(z,0,\mu)$
(2.27)

and

$g_{3}^{1}(z,0,\mu)=\operatorname{Proj}_{\operatorname{Ker}\left(M_{3}^{1}\right)}\widetilde{f}_{3}^{1}(z,0,\mu)=\operatorname{Proj}_{S}\widetilde{f}_{3}^{1}(z,0,0)+O\left(\mu^{2}|z|\right),$
(2.28)

where $\operatorname{Proj}_{p}(q)$ represents the projection of $q$ on $p$,
and $\widetilde{f}_{3}^{1}(z,0,\mu)$ is vector and its element is the cubic
polynomial of $(z,\mu)$ after the variable transformation of (2.26), and it is
determined by (2.38),

$\displaystyle\operatorname{Ker}\left(M_{2}^{1}\right)=\operatorname{Span}\left\\{\left(\begin{array}[]{c}\mu
z_{1}\\\ 0\end{array}\right),\left(\begin{array}[]{c}0\\\ \mu
z_{2}\end{array}\right)\right\\},$ (2.29)
$\displaystyle\operatorname{Ker}\left(M_{3}^{1}\right)=\operatorname{Span}\left\\{\left(\begin{array}[]{c}z_{1}^{2}z_{2}\\\
0\end{array}\right),\left(\begin{array}[]{c}\mu^{2}z_{1}\\\
0\end{array}\right),\left(\begin{array}[]{c}0\\\
z_{1}z_{2}^{2}\end{array}\right),\left(\begin{array}[]{c}0\\\
\mu^{2}z_{2}\end{array}\right)\right\\},$

and

$S=\operatorname{Span}\left\\{\left(\begin{array}[]{c}z_{1}^{2}z_{2}\\\
0\end{array}\right),\left(\begin{array}[]{c}0\\\
z_{1}z_{2}^{2}\end{array}\right)\right\\}.$ (2.30)

In the following, for notational convenience, we let

$\mathcal{H}\left(\alpha
z_{1}^{q_{1}}z_{2}^{q_{2}}\mu\right)=\left(\begin{array}[]{c}\alpha
z_{1}^{q_{1}}z_{2}^{q_{2}}\mu\\\
\overline{\alpha}z_{1}^{q_{2}}z_{2}^{q_{1}}\mu\end{array}\right),~{}\alpha\in\mathbb{C}.$

We then calculate $g_{j}^{1}(z,0,\mu),~{}j=2,3$ step by step.

###  2.3 Algorithm for calculating the normal form of Hopf bifurcation

####  2.3.1 Calculation of $g_{2}^{1}(z,0,\mu)$

From the second mathematical expression in (2.9), we have

$F_{2}^{d}(\varphi,\mu)=F_{20}^{d}(\varphi)+\mu F_{21}^{d}(\varphi)$ (2.31)

and

$F_{3}^{d}(\varphi,\mu)=\mu
F_{31}^{d}(\varphi),~{}F_{j}^{d}(\varphi,\mu)=(0,0)^{T},~{}j=4,5,\cdots,$
(2.32)

where

$\left\\{\begin{aligned}
&F_{20}^{d}(\varphi)=2\left(\begin{array}[]{c}\xi\tau_{c}\left(\varphi_{x}^{(1)}(0)\varphi_{x}^{(2)}(0)+\varphi^{(1)}(0)\varphi_{xx}^{(2)}(0)\right)\\\
-d_{21}\tau_{c}\left(\varphi_{x}^{(1)}(-1)\varphi_{x}^{(2)}(0)+\varphi_{xx}^{(1)}(-1)\varphi^{(2)}(0)\right)\end{array}\right),\\\
&F_{21}^{d}(\varphi)=2D_{1}\Delta\varphi(0)+2D_{2}\Delta\varphi(-1),\\\
&F_{31}^{d}(\varphi)=6\left(\begin{array}[]{c}\xi\mu\left(\varphi_{x}^{(1)}(0)\varphi_{x}^{(2)}(0)+\varphi^{(1)}(0)\varphi_{xx}^{(2)}(0)\right)\\\
-d_{21}\mu\left(\varphi_{x}^{(1)}(-1)\varphi_{x}^{(2)}(0)+\varphi_{xx}^{(1)}(-1)\varphi^{(2)}(0)\right)\end{array}\right).\end{aligned}\right.$
(2.33)

Furthermore, it is easy to verify that

$\displaystyle\left(\begin{aligned} &\left[2\mu
A(\Phi(0)z_{x}),\beta_{n_{c}}^{(1)}\right]\\\ &\left[2\mu
A(\Phi(0)z_{x}),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right)=2\mu
A\left(\Phi(0)\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)\right),$ (2.34) $\displaystyle\left(\begin{aligned}
&\left[\mu
F_{21}^{d}\left(\Phi(\theta)z_{x}\right),\beta_{n_{c}}^{(1)}\right]\\\
&\left[\mu
F_{21}^{d}\left(\Phi(\theta)z_{x}\right),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right)=-\frac{2n_{c}^{2}}{\ell^{2}}\mu\left(D_{1}\left(\Phi(0)\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)\right)+D_{2}\left(\Phi(-1)\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)\right)\right).$

From (2.11), we have
$F_{2}\left(\Phi(\theta)z_{x},\mu\right)=F_{2}\left(\Phi(\theta)z_{x},0\right)$
for all $\mu\in\mathbb{R}$. It follows from the first mathematical expression
in (2.25) that

$f_{2}^{1}(z,0,\mu)=\Psi(0)\left(\begin{aligned}
&\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},\mu\right),\beta_{n_{c}}^{(1)}\right]\\\
&\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},\mu\right),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right).$

This, together with (2.23), (2.27), (2.29), (2.31), (2.32), (2.33) and (2.34),
yields to

$g_{2}^{1}(z,0,\mu)=\operatorname{Proj}_{\operatorname{Ker}\left(M_{2}^{1}\right)}f_{2}^{1}(z,0,\mu)=\mathcal{H}\left(B_{1}\mu
z_{1}\right),$ (2.35)

where

$B_{1}=2\psi^{T}(0)\left(A\phi(0)-\frac{n_{c}^{2}}{\ell^{2}}\left(D_{1}\phi(0)+D_{2}\phi(-1)\right)\right).$
(2.36)

####  2.3.2 Calculation of $g_{3}^{1}(z,0,\mu)$

In this subsection, we calculate the third term $g_{3}^{1}(z,0,\mu)$ in terms
of (2.28). Denote

$\displaystyle f_{2}^{(1,1)}(z,w,0)=\Psi(0)\left(\begin{aligned}
&\left[F_{2}\left(\Phi(\theta)z_{x}+w,0\right),\beta_{n_{c}}^{(1)}\right]\\\
&\left[F_{2}\left(\Phi(\theta)z_{x}+w,0\right),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right),$
(2.37) $\displaystyle f_{2}^{(1,2)}(z,w,0)=\Psi(0)\left(\begin{aligned}
&\left[F_{2}^{d}\left(\Phi(\theta)z_{x}+w,0\right),\beta_{n_{c}}^{(1)}\right]\\\
&\left[F_{2}^{d}\left(\Phi(\theta)z_{x}+w,0\right),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right).$

It follows from (2.35) that $g_{2}^{1}(z,0,0)=(0,0)^{T}$. Then
$\widetilde{f}_{3}^{1}(z,0,0)$ is determined by

$\displaystyle\widetilde{f}_{3}^{1}(z,0,0)$
$\displaystyle=f_{3}^{1}(z,0,0)+\frac{3}{2}\left((D_{z}f_{2}^{1}(z,0,0))U_{2}^{1}(z,0)+(D_{w}f_{2}^{(1,1)}(z,0,0))U_{2}^{2}(z,0)\right.$
(2.38)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+(D_{w,w_{x},w_{xx}}f_{2}^{(1,2)}(z,0,0))U_{2}^{(2,d)}(z,0)(\theta)\right),$

where $f_{2}^{1}(z,0,0)=f_{2}^{(1,1)}(z,0,0)+f_{2}^{(1,2)}(z,0,0)$,

$\displaystyle
D_{w,w_{x},w_{xx}}f_{2}^{(1,2)}(z,0,0)=\left(D_{w}f_{2}^{(1,2)}(z,0,0),D_{w_{x}}f_{2}^{(1,2)}(z,0,0),D_{w_{xx}}f_{2}^{(1,2)}(z,0,0)\right),$
(2.39) $\displaystyle
U_{2}^{1}(z,0)=\left(M_{2}^{1}\right)^{-1}\operatorname{Proj}_{\operatorname{Im}\left(M_{2}^{1}\right)}f_{2}^{1}(z,0,0),~{}U_{2}^{2}(z,0)(\theta)=\left(M_{2}^{2}\right)^{-1}f_{2}^{2}(z,0,0),$

and

$U_{2}^{(2,d)}(z,0)(\theta)=\operatorname{col}\left(U_{2}^{2}(z,0)(\theta),U_{2x}^{2}(z,0)(\theta),U_{2xx}^{2}(z,0)(\theta)\right).$
(2.40)

We calculate $\operatorname{Proj}_{S}\widetilde{f}_{3}^{1}(z,0,0)$ in the
following four steps.

Step 1: Calculation of $\operatorname{Proj}_{S}f_{3}^{1}(z,0,0)$

Writing $F_{3}\left(\Phi(\theta)z_{x},0\right)$ as follows

$\displaystyle
F_{3}\left(\Phi(\theta)z_{x},0\right)=\sum_{q_{1}+q_{2}=3}A_{q_{1}q_{2}}z_{1}^{q_{1}}z_{2}^{q_{2}}\gamma_{n_{c}}^{3}(x),$
(2.41)

where $A_{q_{1}q_{2}}=\overline{A_{q_{2}q_{1}}}$ with
$q_{1},q_{2}\in\mathbb{N}_{0}$. From (2.24) and (2.32), we have
$\widetilde{F}_{3}\left(\Phi(\theta)z_{x},0\right)=F_{3}\left(\Phi(\theta)z_{x},0\right)$,
and thus

$\operatorname{Proj}_{S}f_{3}^{1}(z,0,0)=\mathcal{H}\left(B_{21}z_{1}^{2}z_{2}\right),$

where

$B_{21}=\frac{3}{2\ell\pi}\psi^{T}A_{21}.$ (2.42)

Step 2: Calculation of
$\operatorname{Proj}_{S}\left((D_{z}f_{2}^{1}(z,0,0))U_{2}^{1}(z,0)\right)$

Form (2.23) and (2.31), we have

$\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right)=F_{2}\left(\Phi(\theta)z_{x},0\right)+F_{20}^{d}\left(\Phi(\theta)z_{x}\right).$
(2.43)

By (2.11), we write

$\displaystyle
F_{2}\left(\Phi(\theta)z_{x}+w,\mu\right)=F_{2}\left(\Phi(\theta)z_{x}+w,0\right)$
(2.44)
$\displaystyle=\sum_{q_{1}+q_{2}=2}A_{q_{1}q_{2}}z_{1}^{q_{1}}z_{2}^{q_{2}}\gamma_{n_{c}}^{2}(x)+\mathcal{S}_{2}\left(\Phi(\theta)z_{x},w\right)+O\left(|w|^{2}\right),$

where $\mathcal{S}_{2}\left(\Phi(\theta)z_{x},w\right)$ is the product term of
$\Phi(\theta)z_{x}$ and $w$.

By (2.31) and (2.33), we write

$F_{2}^{d}\left(\Phi(\theta)z_{x},0\right)=F_{20}^{d}\left(\Phi(\theta)z_{x}\right)=\frac{n_{c}^{2}}{\ell^{2}}\sum_{q_{1}+q_{2}=2}A_{q_{1}q_{2}}^{d}z_{1}^{q_{1}}z_{2}^{q_{2}}\left(\xi_{n_{c}}^{2}(x)-\gamma_{n_{c}}^{2}(x)\right),$
(2.45)

where
$\xi_{n_{c}}(x)=(\sqrt{2}/\sqrt{\ell\pi})\sin\left((n_{c}/\ell)x\right)$, and

$\displaystyle\left\\{\begin{aligned}
&A_{20}^{d}=\left(\begin{array}[]{c}2\xi\tau_{c}\phi_{1}(0)\phi_{2}(0)\\\
-2d_{21}\tau_{c}\phi_{1}(-1)\phi_{2}(0)\end{array}\right)=\overline{A_{02}^{d}},\\\
&A_{11}^{d}=\left(\begin{array}[]{c}4\xi\tau_{c}\operatorname{Re}\left\\{\phi_{1}(0)\overline{\phi_{2}}(0)\right\\}\\\
-4d_{21}\tau_{c}\operatorname{Re}\left\\{\phi_{1}(-1)\overline{\phi_{2}}(0)\right\\}\end{array}\right).\end{aligned}\right.$
(2.46)

From (2.1), it is easy to verify that

$\int_{0}^{\ell\pi}\gamma_{n_{c}}^{3}(x)dx=\int_{0}^{\ell\pi}\xi_{n_{2}}^{2}(x)\gamma_{n_{c}}(x)dx=0.$

Then, from (2.43), (2.44) and (2.45), we have

$f_{2}^{1}(z,0,0)=\Psi(0)\left(\begin{aligned}
&\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right),\beta_{n_{c}}^{(1)}\right]\\\
&\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right)=\left(\begin{array}[]{c}0\\\
0\end{array}\right).$ (2.47)

Thus, by combining with (2.30) and (2.47), we have

$\operatorname{Proj}_{S}\left((D_{z}f_{2}^{1}(z,0,0))U_{2}^{1}(z,0)\right)=\left(\begin{array}[]{c}0\\\
0\end{array}\right).$

Step 3: Calculation of
$\operatorname{Proj}_{S}\left((D_{w}f_{2}^{(1,1)}(z,0,0))U_{2}^{2}(z,0)(\theta)\right)$

Let

$U_{2}^{2}(z,0)(\theta)\triangleq
h(\theta,z)=\sum_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\gamma_{n}(x),$ (2.48)

where
$h_{n}(\theta,z)=\sum_{q_{1}+q_{2}=2}h_{n,q_{1}q_{2}}(\theta)z_{1}^{q_{1}}z_{2}^{q_{2}}$.
Then, we have

$\displaystyle\left(\begin{aligned}
\left[\mathcal{S}_{2}\left(\Phi(\theta)z_{x},\sum_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\gamma_{n}(x)\right),\beta_{n_{c}}^{(1)}\right]\\\
\left[\mathcal{S}_{2}\left(\Phi(\theta)z_{x},\sum_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\gamma_{n}(x)\right),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right)$
$\displaystyle=\sum_{n\in\mathbb{N}_{0}}b_{n}\left(\mathcal{S}_{2}\left(\phi(\theta)z_{1},h_{n}(\theta,z)\right)+\mathcal{S}_{2}\left(\overline{\phi}(\theta)z_{2},h_{n}(\theta,z)\right)\right),$

where

$b_{n}=\int_{0}^{\ell\pi}\gamma_{n_{c}}^{2}(x)\gamma_{n}(x)dx=\left\\{\begin{aligned}
&\frac{1}{\sqrt{\ell\pi}},&n=0,\\\ &\frac{1}{\sqrt{2\ell\pi}},&n=2n_{c},\\\
&0,&\text{ otherwise }.\end{aligned}\right.$ (2.49)

Hence, we have

$\displaystyle(D_{w}f_{2}^{(1,1)}(z,0,0))U_{2}^{2}(z,0)(\theta)$
$\displaystyle=\Psi(0)\left(\sum_{n=0,2n_{c}}b_{n}\left(\mathcal{S}_{2}\left(\phi(\theta)z_{1},h_{n}(\theta,z)\right)+\mathcal{S}_{2}\left(\overline{\phi}(\theta)z_{2},h_{n}(\theta,z)\right)\right)\right),$

and

$\operatorname{Proj}_{S}\left((D_{w}f_{2}^{(1,1)}(z,0,0))U_{2}^{2}(z,0)(\theta)\right)=\mathcal{H}\left(B_{22}z_{1}^{2}z_{2}\right),$

where

$\displaystyle B_{22}$
$\displaystyle=\frac{1}{\sqrt{\ell\pi}}\psi^{T}\left(\mathcal{S}_{2}\left(\phi(\theta),h_{0,11}(\theta)\right)+\mathcal{S}_{2}\left(\overline{\phi}(\theta),h_{0,20}(\theta)\right)\right)$
(2.50)
$\displaystyle+\frac{1}{\sqrt{2\ell\pi}}\psi^{T}\left(\mathcal{S}_{2}\left(\phi(\theta),h_{2n_{c},11}(\theta)\right)+\mathcal{S}_{2}\left(\overline{\phi}(\theta),h_{2n_{c},20}(\theta)\right)\right).$

Step 4: Calculation of
$\operatorname{Proj}_{S}\left((D_{w,w_{x},w_{xx}}f_{2}^{(1,2)}(z,0,0))U_{2}^{(2,d)}(z,0)(\theta)\right)$

Denote
$\varphi(\theta)=\left(\varphi^{(1)}(\theta),\varphi^{(2)}(\theta)\right)^{T}=\Phi(\theta)z_{x}$,

$\displaystyle
F_{2}^{d}\left(\varphi(\theta),w,w_{x},w_{xx}\right)=F_{2}^{d}(\varphi(\theta)+w,0)=F_{20}^{d}(\varphi(\theta)+w)$
$\displaystyle=2\left(\begin{array}[]{c}\xi\tau_{c}\left((\varphi_{x}^{(1)}(0)+w_{x}^{(1)}(0))(\varphi_{x}^{(2)}(0)+w_{x}^{(2)}(0))+(\varphi^{(1)}(0)+w^{(1)}(0))(\varphi_{xx}^{(2)}(0)+w_{xx}^{(2)}(0))\right)\\\
-d_{21}\tau_{c}\left((\varphi_{x}^{(1)}(-1)+w_{x}^{(1)}(-1))(\varphi_{x}^{(2)}(0)+w_{x}^{(2)}(0))+(\varphi_{xx}^{(1)}(-1)+w_{xx}^{(1)}(-1))(\varphi^{(2)}(0)+w^{(2)}(0))\right)\end{array}\right)$

and

$\displaystyle\widetilde{\mathcal{S}}_{2}^{(d,1)}(\varphi(\theta),w)=2\left(\begin{array}[]{c}\xi\tau_{c}\varphi_{xx}^{(2)}(0)w^{(1)}(0)\\\
-d_{21}\tau_{c}\varphi_{xx}^{(1)}(-1)w^{(2)}(0)\end{array}\right),$
$\displaystyle\widetilde{\mathcal{S}}_{2}^{(d,2)}(\varphi(\theta),w_{x})=2\left(\begin{array}[]{c}\xi\tau_{c}(\varphi_{x}^{(2)}(0)w_{x}^{(1)}(0)+\varphi_{x}^{(1)}(0)w_{x}^{2}(0))\\\
-d_{21}\tau_{c}(\varphi_{x}^{(2)}(0)w_{x}^{(1)}(-1)+\varphi_{x}^{(1)}(-1)w_{x}^{(2)}(0))\end{array}\right),$
$\displaystyle\widetilde{\mathcal{S}}_{2}^{(d,3)}(\varphi(\theta),w_{xx})=2\left(\begin{array}[]{c}\xi\tau_{c}\varphi^{(1)}(0)w_{xx}^{(2)}(0)\\\
-d_{21}\tau_{c}\varphi^{(2)}(0)w_{xx}^{(1)}(-1)\end{array}\right).$

By combining with (2.1) and (2.48), we have

$\left\\{\begin{aligned}
&U_{2x}^{2}(z,0)(\theta)=h_{x}(\theta,z)=-\frac{n}{\ell}\sum_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\xi_{n}(x),\\\
&U_{2xx}^{2}(z,0)(\theta)=h_{xx}(\theta,z)=-\frac{n^{2}}{\ell^{2}}\sum_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\gamma_{n}(x).\end{aligned}\right.$

Then we have

$\displaystyle\left(D_{w,w_{x},w_{xx}}F_{2}^{d}\left(\varphi(\theta),w,w_{x},w_{xx}\right)\right)U_{2}^{(2,d)}(z,0)(\theta)$
$\displaystyle=\widetilde{\mathcal{S}}_{2}^{(d,1)}(\varphi(\theta),h(\theta,z))+\widetilde{\mathcal{S}}_{2}^{(d,2)}(\varphi(\theta),h_{x}(\theta,z))+\widetilde{\mathcal{S}}_{2}^{(d,3)}(\varphi(\theta),h_{xx}(\theta,z))$

and

$\displaystyle\left(\begin{aligned}
&\left[\widetilde{S}_{2}^{(d,1)}(\varphi(\theta),h(\theta,z)),\beta_{n_{c}}^{(1)}\right]\\\
&\left[\widetilde{S}_{2}^{(d,1)}(\varphi(\theta),h(\theta,z)),\beta_{n_{c}}^{(2)}\right]\\\
\end{aligned}\right)$
$\displaystyle=-\left(n_{c}/\ell\right)^{2}\sum_{n\in\mathbb{N}_{0}}b_{n}\left(\mathcal{S}_{2}^{(d,1)}\left(\phi(\theta)z_{1},h_{n}(\theta,z)\right)+\mathcal{S}_{2}^{(d,1)}\left(\overline{\phi}(\theta)z_{2},h_{n}(\theta,z)\right)\right),$
$\displaystyle\left(\begin{aligned}
&\left[\widetilde{S}_{2}^{(d,2)}\left(\varphi(\theta),h_{x}(\theta,z)\right),\beta_{n_{c}}^{(1)}\right]\\\
&\left[\widetilde{S}_{2}^{(d,2)}\left(\varphi(\theta),h_{x}(\theta,z)\right),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right)$
$\displaystyle=\left(n_{c}/\ell\right)\sum_{n\in\mathbb{N}_{0}}(n/\ell)c_{n}\left(\mathcal{S}_{2}^{(d,2)}\left(\phi(\theta)z_{1},h_{n}(\theta,z)\right)+\mathcal{S}_{2}^{(d,2)}\left(\overline{\phi}(\theta)z_{2},h_{n}(\theta,z)\right)\right),$
$\displaystyle\left(\begin{aligned}
&\left[\widetilde{S}_{2}^{(d,3)}\left(\varphi(\theta),h_{xx}(\theta,z)\right),\beta_{n_{c}}^{(1)}\right]\\\
&\left[\widetilde{S}_{2}^{(d,3)}\left(\varphi(\theta),h_{xx}(\theta,z)\right),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right)$
$\displaystyle=-\sum_{n\in\mathbb{N}_{0}}(n/\ell)^{2}b_{n}\left(\mathcal{S}_{2}^{(d,3)}\left(\phi(\theta)z_{1},h_{n}(\theta,z)\right)+\mathcal{S}_{2}^{(d,3)}\left(\overline{\phi}(\theta)z_{2},h_{n}(\theta,z)\right)\right),$

where $b_{n}$ is defined by (2.49) and

$c_{n}=\int_{0}^{\ell\pi}\xi_{n_{c}}(x)\xi_{n}(x)\gamma_{n_{c}}(x)dx=\left\\{\begin{aligned}
&\frac{1}{\sqrt{2\ell\pi}},&n=2n_{c},\\\ &0,&\text{ otherwise
},\end{aligned}\right.$

and for
$\phi(\theta)=\left(\phi_{1}(\theta),\phi_{2}(\theta)\right)^{T},~{}y(\theta)=\left(y_{1}(\theta),y_{2}(\theta)\right)^{T}\in
C\left([-1,0],\mathbb{R}^{2}\right)$, we have

$\left\\{\begin{aligned}
&\mathcal{S}_{2}^{(d,1)}(\phi(\theta),y(\theta))=2\left(\begin{array}[]{c}\xi\tau_{c}\phi_{2}(0)y_{1}(0)\\\
-d_{21}\tau_{c}\phi_{1}(-1)y_{2}(0)\end{array}\right),\\\
&\mathcal{S}_{2}^{(d,2)}(\phi(\theta),y(\theta))=2\left(\begin{array}[]{c}\xi\tau_{c}(\phi_{2}(0)y_{1}(0)+\phi_{1}(0)y_{2}(0))\\\
-d_{21}\tau_{c}(\phi_{2}(0)y_{1}(-1)+\phi_{1}(-1)y_{2}(0))\end{array}\right),\\\
&\mathcal{S}_{2}^{(d,3)}(\phi(\theta),y(\theta))=2\left(\begin{array}[]{c}\xi\tau_{c}\phi_{1}(0)y_{2}(0)\\\
-d_{21}\tau_{c}\phi_{2}(0)y_{1}(-1)\end{array}\right).\end{aligned}\right.$

Furthermore, from (2.37), (2.39) and (2.40), we have

$\displaystyle(D_{w,w_{x},w_{xx}}f_{2}^{(1,2)}(z,0,0))U_{2}^{(2,d)}(z,0)(\theta)$
$\displaystyle=\Psi(0)\left(\begin{aligned}
&\left[D_{w,w_{x},w_{xx}}F_{2}^{d}\left(\varphi(\theta),w,w_{x},w_{xx}\right)U_{2}^{(2,d)}(z,0)(\theta),\beta_{n_{c}}^{(1)}\right]\\\
&\left[D_{w,w_{x},w_{xx}}F_{2}^{d}\left(\varphi(\theta),w,w_{x},w_{xx}\right)U_{2}^{(2,d)}(z,0)(\theta),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right),$

and then we obtain

$\operatorname{Proj}_{S}\left((D_{w,w_{x},w_{xx}}f_{2}^{(1,2)}(z,0,0))U_{2}^{(2,d)}(z,0)(\theta)\right)=\mathcal{H}\left(B_{23}z_{1}^{2}z_{2}\right),$

where

$\displaystyle B_{23}=$
$\displaystyle-\frac{1}{\sqrt{\ell\pi}}(n_{c}/\ell)^{2}\psi^{T}\left(\mathcal{S}_{2}^{(d,1)}\left(\phi(\theta),h_{0,11}(\theta)\right)+\mathcal{S}_{2}^{(d,1)}\left(\overline{\phi}(\theta),h_{0,20}(\theta)\right)\right)$
(2.51)
$\displaystyle+\frac{1}{\sqrt{2\ell\pi}}\psi^{T}\sum_{j=1,2,3}b_{2n_{c}}^{(j)}\left(\mathcal{S}_{2}^{(d,j)}\left(\phi(\theta),h_{2n_{c},11}(\theta)\right)+\mathcal{S}_{2}^{(d,j)}\left(\overline{\phi}(\theta),h_{2n_{c},20}(\theta)\right)\right)$

with

$b_{2n_{c}}^{(1)}=-\frac{n_{c}^{2}}{\ell^{2}},~{}b_{2n_{c}}^{(2)}=\frac{2n_{c}^{2}}{\ell^{2}},~{}b_{2n_{c}}^{(3)}=-\frac{(2n_{c})^{2}}{\ell^{2}}.$

##  3 Normal form of the Hopf bifurcation and the corresponding coefficients

According to the algorithm developed in Section 2, we can obtain the normal
form of the Hopf bifurcation truncated to the third-order term

$\dot{z}=Bz+\frac{1}{2}\left(\begin{array}[]{c}B_{1}z_{1}\mu\\\
\overline{B}_{1}z_{2}\mu\end{array}\right)+\frac{1}{3!}\left(\begin{array}[]{c}B_{2}z_{1}^{2}z_{2}\\\
\overline{B}_{2}z_{1}z_{2}^{2}\end{array}\right)+O\left(|z|\mu^{2}+\left|z\right|^{4}\right),$
(3.1)

where

$\displaystyle B_{1}$
$\displaystyle=2\psi^{T}(0)\left(A\phi(0)-\frac{n_{c}^{2}}{\ell^{2}}\left(D_{1}\phi(0)+D_{2}\phi(-1)\right)\right),$
$\displaystyle B_{2}$
$\displaystyle=B_{21}+\frac{3}{2}\left(B_{22}+B_{23}\right).$

Here, $B_{1}$ is determined by (2.36), $B_{21}$, $B_{22}$ and $B_{23}$ are
determined by (2.42), (2.50), (2.51), respectively, and they can be calculated
by using the MATLAB software. The normal form (3.1) can be written in real
coordinates through the change of variables
$z_{1}=v_{1}-iv_{2},~{}z_{2}=v_{1}+iv_{2}$, and then changing to polar
coordinates by $v_{1}=\rho\cos\Theta,~{}v_{2}=\rho\sin\Theta$, where $\Theta$
is the azimuthal angle. Therefore, by the above transformation and removing
the azimuthal term $\Theta$, (3.1) can be rewritten as

$\dot{\rho}=K_{1}\mu\rho+K_{2}\rho^{3}+O\left(\mu^{2}\rho+\lvert(\rho,\mu)\rvert^{4}\right),$

where

$K_{1}=\frac{1}{2}\operatorname{Re}\left(B_{1}\right),~{}K_{2}=\frac{1}{3!}\operatorname{Re}\left(B_{2}\right).$

According to lv29 , the sign of $K_{1}K_{2}$ determines the direction of the
Hopf bifurcation, and the sign of $K_{2}$ determines the stability of the Hopf
bifurcation periodic solution. More precisely, we have the following results

(i) when $K_{1}K_{2}<0$, the Hopf bifurcation is supercritical, and the Hopf
bifurcation periodic solution is stable for $K_{2}<0$ and unstable for
$K_{2}>0$;

(ii) when $K_{1}K_{2}>0$, the Hopf bifurcation is subcritical, and the Hopf
bifurcation periodic solution is stable for $K_{2}<0$ and unstable for
$K_{2}>0$.

From (2.42), (2.50) and (2.51), it is obvious that in order to obtain the
value of $K_{2}$, we still need to calculate
$h_{0,20}(\theta),h_{0,11}(\theta),h_{2n_{c},20}(\theta),h_{2n_{c},11}(\theta)$
and $A_{ij}$.

###  3.1 Calculations of
$h_{0,20}(\theta),h_{0,11}(\theta),h_{2n_{c},20}(\theta)$ and
$h_{2n_{c},11}(\theta)$

From lv27 , we have

$M_{2}^{2}\left(h_{n}(\theta,z)\gamma_{n}(x)\right)=D_{z}\left(h_{n}(\theta,z)\gamma_{n}(x)\right)Bz-
A_{\mathcal{Q}^{1}}\left(h_{n}(\theta,z)\gamma_{n}(x)\right),$

which leads to

$\displaystyle\left(\begin{aligned}
\left[M_{2}^{2}\left(h_{n}(\theta,z)\gamma_{n}(x)\right),\beta_{n}^{(1)}\right]\\\
\left[M_{2}^{2}\left(h_{n}(\theta,z)\gamma_{n}(x)\right),\beta_{n}^{(2)}\right]\end{aligned}\right)$
(3.2)
$\displaystyle=2i\omega_{c}\left(h_{n,20}(\theta)z_{1}^{2}-h_{n,02}(\theta)z_{2}^{2}\right)-\left(\dot{h}_{n}(\theta,z)+X_{0}(\theta)\left(\mathscr{L}_{0}\left(h_{n}(\theta,z)\right)-\dot{h}_{n}(0,z)\right)\right),$

where

$\mathscr{L}_{0}\left(h_{n}(\theta,z)\right)=-\tau_{c}(n/\ell)^{2}\left(D_{1}h_{n}(0,z)+D_{2}h_{n}(-1,z)\right)+\tau_{c}Ah_{n}(0,z).$

By (2.19) and the second mathematical expression in (2.25), we have

$\displaystyle f_{2}^{2}(z,0,0)$
$\displaystyle=X_{0}(\theta)\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right)-\pi\left(X_{0}(\theta)\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right)\right)$
(3.3)
$\displaystyle=X_{0}(\theta)\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right)-\Phi(\theta)\Psi(0)\left(\begin{aligned}
\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right),\beta_{n_{c}}^{(1)}\right]\\\
\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right),\beta_{n_{c}}^{(2)}\right]\end{aligned}\right)\gamma_{n_{c}}(x).$

Furthermore, by (2.43), (2.44) and (2.45), we have

$\left(\begin{aligned} \left[f_{2}^{2}(z,0,0),\beta_{n}^{(1)}\right]\\\
\left[f_{2}^{2}(z,0,0),\beta_{n}^{(2)}\right]\end{aligned}\right)=\left\\{\begin{aligned}
&\frac{1}{\sqrt{\ell\pi}}X_{0}(\theta)\left(A_{20}z_{1}^{2}+A_{02}z_{2}^{2}+A_{11}z_{1}z_{2}\right),&n=0,\\\
&\frac{1}{\sqrt{2\ell\pi}}X_{0}(\theta)\left(\widetilde{A}_{20}z_{1}^{2}+\widetilde{A}_{02}z_{2}^{2}+\widetilde{A}_{11}z_{1}z_{2}\right),&n=2n_{c},\end{aligned}\right.$
(3.4)

where $\widetilde{A}_{j_{1}j_{2}}$ is defined as follows

$\left\\{\begin{aligned}
&\widetilde{A}_{j_{1}j_{2}}=A_{j_{1}j_{2}}-2\left(n_{c}/\ell\right)^{2}A_{j_{1}j_{2}}^{d},\\\
&j_{1},j_{2}=0,1,2,~{}j_{1}+j_{2}=2,\end{aligned}\right.$ (3.5)

where $A_{j_{1}j_{2}}^{d}$ is determined by (2.46), and $A_{j_{1}j_{2}}$ will
be calculated in the following section. Therefore, from (2.39), (3.2), (3.3),
(3.4), and by matching the coefficients of $z_{1}^{2}$ and $z_{1}z_{2}$, we
have

$n=0,~{}\left\\{\begin{aligned} &z_{1}^{2}:\left\\{\begin{aligned}
&\dot{h}_{0,20}(\theta)-2i\omega_{c}h_{0,20}(\theta)=(0,0)^{T},\\\
&\dot{h}_{0,20}(0)-L_{0}\left(h_{0,20}(\theta)\right)=\frac{1}{\sqrt{\ell\pi}}A_{20},\end{aligned}\right.\\\
&z_{1}z_{2}:\left\\{\begin{aligned} &\dot{h}_{0,11}(\theta)=(0,0)^{T},\\\
&\dot{h}_{0,11}(0)-L_{0}\left(h_{0,11}(\theta)\right)=\frac{1}{\sqrt{\ell\pi}}A_{11}\end{aligned}\right.\end{aligned}\right.$
(3.6)

and

$n=2n_{c},~{}\left\\{\begin{aligned} &z_{1}^{2}:\left\\{\begin{aligned}
&\dot{h}_{2n_{c},20}(\theta)-2i\omega_{c}h_{2n_{c},20}(\theta)=(0,0)^{T},\\\
&\dot{h}_{2n_{c},20}(0)-\mathscr{L}_{0}\left(h_{2n_{c},20}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{20},\end{aligned}\right.\\\
&z_{1}z_{2}:\left\\{\begin{aligned} &\dot{h}_{2n_{c},11}(\theta)=(0,0)^{T},\\\
&\dot{h}_{2n_{c},11}(0)-\mathscr{L}_{0}\left(h_{2n_{c},11}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{11}.\end{aligned}\right.\end{aligned}\right.$
(3.7)

Next, by combining with (3.6) and (3.7), we will give the mathematical
expressions of $h_{0,20}(\theta)$, $h_{0,11}(\theta)$, $h_{2n_{c},20}(\theta)$
and $h_{2n_{c},11}(\theta)$.

(1) Calculations of $h_{0,20}(\theta)$ and $h_{0,11}(\theta)$

(i) Notice that

$\left\\{\begin{aligned}
\dot{h}_{0,20}(\theta)-2i\omega_{c}h_{0,20}(\theta)&=(0,0)^{T},\\\
\dot{h}_{0,20}(0)-L_{0}\left(h_{0,20}(\theta)\right)&=\frac{1}{\sqrt{\ell\pi}}A_{20},\end{aligned}\right.$
(3.8)

then from (3.8), we have $h_{0,20}(\theta)=e^{2iw_{c}\theta}h_{0,20}(0)$ and
$\dot{h}_{0,20}(0)-2i\omega_{c}h_{0,20}(0)=(0,0)^{T}$. Notice that
$L_{0}\left(h_{0,20}(\theta)\right)=\tau_{c}Ah_{0,20}(0)$, then we have

$(2iw_{c}I-\tau_{c}A)h_{0,20}(0)=\frac{1}{\sqrt{\ell\pi}}A_{20},$

and hence $h_{0,20}(\theta)=e^{2iw_{c}\theta}C_{1}$ with

$C_{1}=(2iw_{c}I-\tau_{c}A)^{-1}\frac{1}{\sqrt{\ell\pi}}A_{20}.$

(ii) Notice that

$\left\\{\begin{aligned} &\dot{h}_{0,11}(\theta)=(0,0)^{T},\\\
&\dot{h}_{0,11}(0)-L_{0}\left(h_{0,11}(\theta)\right)=\frac{1}{\sqrt{\ell\pi}}A_{11},\end{aligned}\right.$
(3.9)

then from (3.9), we have $h_{0,11}(\theta)=h_{0,11}(0)$ and
$\dot{h}_{0,11}(0)=(0,0)^{T}$. Notice that
$L_{0}\left(h_{0,11}(\theta)\right)=\tau_{c}Ah_{0,11}(0)$, then we have

$-\tau_{c}Ah_{0,11}(0)=\frac{1}{\sqrt{\ell\pi}}A_{11},$

and hence $h_{0,11}(\theta)=C_{2}$ with

$C_{2}=(-\tau_{c}A)^{-1}\frac{1}{\sqrt{\ell\pi}}A_{11}.$

(2) Calculations of $h_{2n_{c},20}(\theta)$ and $h_{2n_{c},11}(\theta)$

(i) Notice that

$\left\\{\begin{aligned}
&\dot{h}_{2n_{c},20}(\theta)-2i\omega_{c}h_{2n_{c},20}(\theta)=(0,0)^{T},\\\
&\dot{h}_{2n_{c},20}(0)-\mathscr{L}_{0}\left(h_{2n_{c},20}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{20},\end{aligned}\right.$
(3.10)

then from (3.10), we have
$h_{2n_{c},20}(\theta)=e^{2iw_{c}\theta}h_{2n_{c},20}(0)$, and hence
$h_{2n_{c},20}(-1)=e^{-2iw_{c}}h_{2n_{c},20}(0)$. Furthermore, from (3.10) and

$\mathscr{L}_{0}\left(h_{2n_{c},20}(\theta)\right)=-\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}\left(D_{1}h_{2n_{c},20}(0)+D_{2}h_{2n_{c},20}(-1)\right)+\tau_{c}Ah_{2n_{c},20}(0),$

we have

$2i\omega_{c}h_{2n_{c},20}(0)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{20}-\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}\left(D_{1}h_{2n_{c},20}(0)+D_{2}h_{2n_{c},20}(-1)\right)+\tau_{c}Ah_{2n_{c},20}(0).$
(3.11)

Therefore, by combining with $h_{2n_{c},20}(-1)=e^{-2iw_{c}}h_{2n_{c},20}(0)$
and (3.11), we can obtain

$(2iw_{c}I+\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}D_{1}+\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}D_{2}e^{-2iw_{c}}-\tau_{c}A)h_{2n_{c},20}(0)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{20},$

and hence

$h_{2n_{c},20}(\theta)=e^{2iw_{c}\theta}C_{3}$

with

$C_{3}=(2iw_{c}I+\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}D_{1}+\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}D_{2}e^{-2iw_{c}}-\tau_{c}A)^{-1}\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{20}.$

Here, $\widetilde{A}_{20}$ and $A_{20}^{d}$ are defined by (3.5) and (2.46),
respectively.

(ii) Notice that

$\left\\{\begin{aligned} &\dot{h}_{2n_{c},11}(\theta)=(0,0)^{T},\\\
&\dot{h}_{2n_{c},11}(0)-\mathscr{L}_{0}\left(h_{2n_{c},11}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{11},\end{aligned}\right.$
(3.12)

then from (3.12), we have $h_{2n_{c},11}(\theta)=h_{2n_{c},11}(0)$, and hence
$h_{2n_{c},11}(-1)=h_{2n_{c},11}(0)$. Furthermore, from (3.12) and

$\mathscr{L}_{0}\left(h_{2n_{c},11}(\theta)\right)=-\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}\left(D_{1}h_{2n_{c},11}(0)+D_{2}h_{2n_{c},11}(-1)\right)+\tau_{c}Ah_{2n_{c},11}(0),$

we have

$(0,0)^{T}=-\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}\left(D_{1}h_{2n_{c},11}(0)+D_{2}h_{2n_{c},11}(-1)\right)+\tau_{c}Ah_{2n_{c},11}(0)+\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{11}.$
(3.13)

Therefore, by combining with $h_{2n_{c},11}(-1)=h_{2n_{c},11}(0)$ and (3.13),
we can obtain

$\left(\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}D_{1}+\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}D_{2}-\tau_{c}A\right)h_{2n_{c},11}(0)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{11},$

and hence

$h_{2n_{c},11}(\theta)=C_{4}$

with

$C_{4}=\left(\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}D_{1}+\tau_{c}\frac{4n_{c}^{2}}{\ell^{2}}D_{2}-\tau_{c}A\right)^{-1}\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{11}.$

Here, $\widetilde{A}_{11}$ and $A_{11}^{d}$ are defined by (3.5) and (2.46),
respectively.

###  3.2 Calculations of $A_{i,j}$ and $\mathcal{S}_{2}(\Phi(\theta)z_{x},w)$

In this subsection, let
$F(\varphi,\mu)=\left(F^{(1)}(\varphi,\mu),F^{(2)}(\varphi,\mu)\right)^{T}$
and $\varphi=(\varphi_{1},\varphi_{2})^{T}\in\mathcal{C}$, and we write

$\frac{1}{j!}F_{j}(\varphi,\mu)=\sum_{j_{1}+j_{2}=j}\frac{1}{j_{1}!j_{2}!}f_{j_{1}j_{2}}\varphi_{1}^{j_{1}}(0)\varphi_{2}^{j_{2}}(0),$
(3.14)

where
$f_{j_{1}j_{2}}=\operatorname{col}\left(f_{j_{1}j_{2}}^{(1)},f_{j_{1}j_{2}}^{(2)}\right)$
with

$f_{j_{1}j_{2}}^{(k)}=\frac{\partial^{j_{1}+j_{2}}F^{(k)}(0,0)}{\partial\varphi_{1}^{j_{1}}(0)\partial\varphi_{2}^{j_{2}}(0)},~{}k=1,2.$

Then from (3.14), we have

$\displaystyle F_{2}(\varphi,\mu)$
$\displaystyle=F_{2}(\varphi,0)=2\sum_{j_{1}+j_{2}=2}\frac{1}{j_{1}!j_{2}!}f_{j_{1}j_{2}}\varphi_{1}^{j_{1}}(0)\varphi_{2}^{j_{2}}(0)$
(3.15)
$\displaystyle=f_{20}\varphi_{1}^{2}(0)+f_{02}\varphi_{2}^{2}(0)+2f_{11}\varphi_{1}(0)\varphi_{2}(0)$

and

$\displaystyle F_{3}(\varphi,0)$
$\displaystyle=6\sum_{j_{1}+j_{2}=3}\frac{1}{j_{1}!j_{2}!}f_{j_{1}j_{2}}\varphi_{1}^{j_{1}}(0)\varphi_{2}^{j_{2}}(0)$
(3.16)
$\displaystyle=f_{30}\varphi_{1}^{3}(0)+f_{03}\varphi_{2}^{3}(0)+3f_{21}\varphi_{1}^{2}(0)\varphi_{2}(0)+3f_{12}\varphi_{1}(0)\varphi_{2}^{2}(0).$

Notice that

$\displaystyle\varphi(\theta)=\Phi(\theta)z_{x}=\phi(\theta)z_{1}(t)\gamma_{n_{c}}(x)+\overline{\phi}(\theta)z_{2}(t)\gamma_{n_{c}}(x)$
(3.17)
$\displaystyle=\left(\begin{array}[]{c}\phi_{1}(\theta)z_{1}(t)\gamma_{n_{c}}(x)+\overline{\phi}_{1}(\theta)z_{2}(t)\gamma_{n_{c}}(x)\\\
\phi_{2}(\theta)z_{1}(t)\gamma_{n_{c}}(x)+\overline{\phi}_{2}(\theta)z_{2}(t)\gamma_{n_{c}}(x)\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{c}\varphi_{1}(\theta)\\\
\varphi_{2}(\theta)\end{array}\right),$

and similar to (2.41), we have

$F_{2}\left(\Phi(\theta)z_{x},0\right)=\sum_{q_{1}+q_{2}=2}A_{q_{1}q_{2}}\gamma_{n_{c}}^{q_{1}+q_{2}}(x)z_{1}^{q_{1}}z_{2}^{q_{2}},$
(3.18)

then by combining with (3.15), (3.17) and (3.18), we have

$\displaystyle A_{20}$
$\displaystyle=f_{20}\phi_{1}^{2}(0)+f_{02}\phi_{2}^{2}(0)+2f_{11}\phi_{1}(0)\phi_{2}(0),$
$\displaystyle A_{02}$
$\displaystyle=f_{20}\overline{\phi_{1}}^{2}(0)+f_{02}\overline{\phi_{2}}^{2}(0)+2f_{11}\overline{\phi_{1}}(0)\overline{\phi_{2}}(0),$
$\displaystyle A_{11}$
$\displaystyle=2f_{20}\phi_{1}(0)\overline{\phi_{1}}(0)+2f_{02}\phi_{2}(0)\overline{\phi_{2}}(0)+2f_{11}(\phi_{1}(0)\overline{\phi_{2}}(0)+\overline{\phi_{1}}(0)\phi_{2}(0)).$

Furthermore, from (2.41), (3.16) and (3.17), we have

$\displaystyle A_{30}$
$\displaystyle=f_{30}\phi_{1}^{3}(0)+f_{03}\phi_{2}^{3}(0)+3f_{21}\phi_{1}^{2}(0)\phi_{2}(0)+3f_{12}\phi_{1}(0)\phi_{2}^{2}(0),$
$\displaystyle A_{03}$
$\displaystyle=f_{30}\overline{\phi_{1}}^{3}(0)+f_{03}\overline{\phi_{2}}^{3}(0)+3f_{21}\overline{\phi_{1}}^{2}(0)\overline{\phi_{2}}(0)+3f_{12}\overline{\phi_{1}}(0)\overline{\phi_{2}}^{2}(0),$
$\displaystyle A_{21}$
$\displaystyle=3f_{30}\phi_{1}^{2}(0)\overline{\phi_{1}}(0)+3f_{03}\phi_{2}^{2}(0)\overline{\phi_{2}}(0)+3f_{21}(\phi_{1}^{2}(0)\overline{\phi_{2}}(0)+2\phi_{1}(0)\overline{\phi_{1}}(0)\phi_{2}(0))$
$\displaystyle+3f_{12}(2\phi_{1}(0)\phi_{2}(0)\overline{\phi_{2}}(0)+\overline{\phi_{1}}(0)\phi_{2}^{2}(0)),$
$\displaystyle A_{12}$
$\displaystyle=3f_{30}\phi_{1}(0)\overline{\phi_{1}}^{2}(0)+3f_{03}\phi_{2}(0)\overline{\phi_{2}}^{2}(0)+3f_{21}(2\phi_{1}(0)\overline{\phi_{1}}(0)\overline{\phi_{2}}(0)+\overline{\phi_{1}}^{2}(0)\phi_{2}(0))$
$\displaystyle+3f_{12}(\phi_{1}(0)\overline{\phi_{2}}^{2}(0)+2\overline{\phi_{1}}(0)\phi_{2}(0)\overline{\phi_{2}}(0)).$

Moreover, from (3.14), we have

$\displaystyle F_{2}(\varphi(\theta)+w,\mu)$
$\displaystyle=F_{2}(\varphi(\theta)+w,0)=2\sum_{j_{1}+j_{2}=2}\frac{1}{j_{1}!j_{2}!}f_{j_{1}j_{2}}(\varphi_{1}(0)+w_{1}(0))^{j_{1}}(\varphi_{2}(0)+w_{2}(0))^{j_{2}}$
(3.19)
$\displaystyle=f_{20}(\varphi_{1}(0)+w_{1}(0))^{2}+f_{02}(\varphi_{2}(0)+w_{2}(0))^{2}+2f_{11}(\varphi_{1}(0)+w_{1}(0))(\varphi_{2}(0)+w_{2}(0)).$

Notice that

$\displaystyle\varphi(\theta)+w$
$\displaystyle=\Phi(\theta)z_{x}+w=\phi(\theta)z_{1}(t)\gamma_{n_{c}}(x)+\overline{\phi}(\theta)z_{2}(t)\gamma_{n_{c}}(x)+w$
(3.20)
$\displaystyle=\left(\begin{array}[]{c}\phi_{1}(\theta)z_{1}(t)\gamma_{n_{c}}(x)+\overline{\phi}_{1}(\theta)z_{2}(t)\gamma_{n_{c}}(x)+w_{1}\\\
\phi_{2}(\theta)z_{1}(t)\gamma_{n_{c}}(x)+\overline{\phi}_{2}(\theta)z_{2}(t)\gamma_{n_{c}}(x)+w_{2}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{c}\varphi_{1}(\theta)+w_{1}\\\
\varphi_{2}(\theta)+w_{2}\end{array}\right)$

and

$\displaystyle F_{2}\left(\Phi(\theta)z_{x}+w,\mu\right)$
$\displaystyle=F_{2}\left(\Phi(\theta)z_{x}+w,0\right)$ (3.21)
$\displaystyle=\sum_{q_{1}+q_{2}=2}A_{q_{1}q_{2}}\gamma_{n_{c}}^{q_{1}+q_{2}}(x)z_{1}^{q_{1}}z_{2}^{q_{2}}+\mathcal{S}_{2}\left(\Phi(\theta)z_{x},w\right)+O\left(|w|^{2}\right),$

then by combining with (3.19), (3.20) and (3.21), we have

$\displaystyle\mathcal{S}_{2}\left(\Phi(\theta)z_{x},w\right)$
$\displaystyle=2f_{20}(\phi_{1}(0)z_{1}(t)\gamma_{n_{c}}(x)+\overline{\phi_{1}}(0)z_{2}(t)\gamma_{n_{c}}(x))w_{1}(0)$
$\displaystyle+2f_{02}(\phi_{2}(0)z_{1}(t)\gamma_{n_{c}}(x)+\overline{\phi_{2}}(0)z_{2}(t)\gamma_{n_{c}}(x))w_{2}(0)$
$\displaystyle+2f_{11}\left((\phi_{1}(0)z_{1}(t)\gamma_{n_{c}}(x)+\overline{\phi_{1}}(0)z_{2}(t)\gamma_{n_{c}}(x))w_{2}(0)+(\phi_{2}(0)z_{1}(t)\gamma_{n_{c}}(x)+\overline{\phi_{2}}(0)z_{2}(t)\gamma_{n_{c}}(x))w_{1}(0)\right).$

##  4 Application to Holling-Tanner model with spatial memory and predator-
taxis

In this section, we apply our newly developed algorithm in Section 2 to the
Holling-Tanner model with spatial memory and predator-taxis, i.e., in the
model (1.4), we let

$\displaystyle f\left(u(x,t),v(x,t)\right)=u(x,t)\left(1-\beta
u(x,t)\right)-\frac{mu(x,t)v(x,t)}{1+u(x,t)},$ (4.1) $\displaystyle
g\left(u(x,t),v(x,t)\right)=sv(x,t)\left(1-\frac{v(x,t)}{u(x,t)}\right),$

where $\beta>0$, $m>0$ and $s>0$.

Thus, (1.4) becomes the following model

$\left\\{\begin{aligned} &\frac{\partial u(x,t)}{\partial t}=d_{11}\Delta
u(x,t)+\xi\left(u(x,t)v_{x}(x,t)\right)_{x}+u(x,t)\left(1-\beta
u(x,t)\right)-\frac{mu(x,t)v(x,t)}{1+u(x,t)},&x\in(0,\ell\pi),~{}t>0,\\\
&\frac{\partial v(x,t)}{\partial t}=d_{22}\Delta
v(x,t)-d_{21}\left(v(x,t)u_{x}(x,t-\tau)\right)_{x}+sv(x,t)\left(1-\frac{v(x,t)}{u(x,t)}\right),&x\in(0,\ell\pi),~{}t>0,\\\
&u_{x}(0,t)=u_{x}(\ell\pi,t)=v_{x}(0,t)=v_{x}(\ell\pi,t)=0,&t\geq
0.\end{aligned}\right.$ (4.2)

The Holling-tanner model is one of the typical predator-prey models. For the
ordinary differential equation (4.2) with $d_{11}=\xi=d_{21}=d_{22}=0$, it has
been completely analyzed in lv30 . For the diffusive model (4.2) with
$\xi=d_{21}=0$, the global stability of the positive constant steady state was
proved in lv31 ; lv32 , and the Hopf bifurcation and Turing instability have
been studied in lv33 .

###  4.1 Stability and Hopf bifurcation analysis

The system (4.2) has the positive constant steady state
$E_{*}\left(u_{*},v_{*}\right)$, where

$u_{*}=v_{*}=\frac{1}{2\beta}\left(\sqrt{R^{2}+4\beta}-R\right)$ (4.3)

with $R=\beta+m-1$. By combining with $E_{*}\left(u_{*},v_{*}\right)$, (2.4)
and (4.1), we have

$\displaystyle a_{11}=1-2\beta
u_{*}-\frac{mu_{*}}{(1+u_{*})^{2}},~{}a_{12}=-\frac{mu_{*}}{1+u_{*}}<0,$ (4.4)
$\displaystyle a_{21}=s>0,~{}a_{22}=-s<0.$

Moreover, by combining with (2.3), (2.5), (2.6) and (4.4), the characteristic
equation of system (4.2) can be written as

$\Gamma_{n}(\lambda)=\det\left(\mathcal{M}_{n}(\lambda)\right)=\lambda^{2}-T_{n}\lambda+\widetilde{J}_{n}(\tau)=0,$
(4.5)

where

$\displaystyle T_{n}$
$\displaystyle=\operatorname{Tr}(A)-\operatorname{Tr}(D_{1})\frac{n^{2}}{\ell^{2}},$
(4.6) $\displaystyle\widetilde{J}_{n}(\tau)$
$\displaystyle=(d_{11}d_{22}+d_{21}\xi
u_{*}v_{*}e^{-\lambda\tau})\frac{n^{4}}{\ell^{4}}-\left(d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}+d_{21}v_{*}a_{12}e^{-\lambda\tau}\right)\frac{n^{2}}{\ell^{2}}+\operatorname{Det}(A).$

Notice that the mathematical expression in (4.6) the same as in (2.7).

Furthermore, when $\tau=0$, the characteristic equation (4.5) becomes

$\lambda^{2}-T_{n}\lambda+\widetilde{J}_{n}(0)=0,$ (4.7)

where

$\widetilde{J}_{n}(0)=(d_{11}d_{22}+d_{21}\xi
u_{*}v_{*})\frac{n^{4}}{\ell^{4}}-\left(d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}+d_{21}v_{*}a_{12}\right)\frac{n^{2}}{\ell^{2}}+\operatorname{Det}(A).$
(4.8)

A set of sufficient and necessary condition that all roots of (4.7) have
negative real parts is $T_{n}<0,~{}\widetilde{J}_{n}(0)>0$, which is always
holds provided that $a_{11}<0$, i.e.,

$(C_{0}):~{}1-2\beta u_{*}-\frac{mu_{*}}{(1+u_{*})^{2}}<0.$

This implies that when $\tau=0$ and the condition $(C_{0})$ holds, the
positive steady state $E_{*}(u_{*},v_{*})$ is asymptotically stable for
$d_{11}\geq 0$, $\xi\geq 0$, $d_{21}\geq 0$ and $d_{22}\geq 0$. Meanwhile, if
we let $d_{21}=0$, then by (4.8), we have

$\widetilde{J}_{n}:=d_{11}d_{22}\frac{n^{4}}{\ell^{4}}-\left(d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}\right)\frac{n^{2}}{\ell^{2}}+\operatorname{Det}(A).$

It is easy to verify that $T_{n}<0$ and $\widetilde{J}_{n}>0$ provided that
the condition $(C_{0})$ holds. This implies that when $\tau=0$, $d_{21}=0$ and
the condition $(C_{0})$ holds, the positive steady state $E_{*}(u_{*},v_{*})$
is asymptotically stable for $d_{11}\geq 0$, $\xi\geq 0$ and $d_{22}\geq 0$.
Furthermore, since $\Gamma_{n}(0)=\widetilde{J}_{n}(0)>0$ under the condition
$(C_{0})$, this implies that $\lambda=0$ is not a root of (4.5).

In the following, we let

$\displaystyle
J_{n}=d_{11}d_{22}\frac{n^{4}}{\ell^{4}}-\left(d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}\right)\frac{n^{2}}{\ell^{2}}+\operatorname{Det}(A).$

Furthermore, let $\lambda=i\omega_{n}(\omega_{n}>0)$ be a root of (4.5). By
substituting it along with expressions in (4.6) into (4.5), and separating the
real part from the imaginary part, we have

$\left\\{\begin{aligned} &\omega_{n}^{2}-J_{n}=\left(d_{21}\xi
u_{*}v_{*}\frac{n^{4}}{\ell^{4}}-d_{21}v_{*}a_{12}\frac{n^{2}}{\ell^{2}}\right)\cos(\omega_{n}\tau),\\\
&-T_{n}\omega_{n}=\left(d_{21}\xi
u_{*}v_{*}\frac{n^{4}}{\ell^{4}}-d_{21}v_{*}a_{12}\frac{n^{2}}{\ell^{2}}\right)\sin(\omega_{n}\tau),\end{aligned}\right.$
(4.9)

which yields

$\omega_{n}^{4}+P_{n}\omega_{n}^{2}+Q_{n}=0,$ (4.10)

where

$P_{n}=T_{n}^{2}-2J_{n}=\left(d_{11}^{2}+d_{22}^{2}\right)\frac{n^{4}}{\ell^{4}}-2\left(d_{11}a_{11}+d_{22}a_{22}+a_{21}\xi
u_{*}\right)\frac{n^{2}}{\ell^{2}}+a_{11}^{2}+a_{22}^{2}+2a_{12}a_{21},$

and

$Q_{n}=\left(J_{n}+\left(d_{21}\xi
u_{*}v_{*}\frac{n^{4}}{\ell^{4}}-d_{21}v_{*}a_{12}\frac{n^{2}}{\ell^{2}}\right)\right)\left(J_{n}-\left(d_{21}\xi
u_{*}v_{*}\frac{n^{4}}{\ell^{4}}-d_{21}v_{*}a_{12}\frac{n^{2}}{\ell^{2}}\right)\right).$
(4.11)

Notice that for (4.10), it is easy to see that if

$\text{ either }P_{n}>0\text{ and }Q_{n}>0\text{ or }P_{n}^{2}-4Q_{n}<0,$

then (4.10) has no positive root. Suppose that

$Q_{n}>0,~{}P_{n}<0\text{ and }P_{n}^{2}-4Q_{n}>0,$

then (4.10) has two positive roots. In addition, if

$\text{ either }Q_{n}<0\text{ or }P_{n}<0\text{ and }P_{n}^{2}-4Q_{n}=0,$

then (4.10) has only one positive root.

###### Case 1

It is easy to see that if the conditions $(C_{0})$ and

$(C_{1}):~{}P_{n}>0\text{ and }Q_{n}>0\text{ or }P_{n}^{2}-4Q_{n}<0,$

hold, then (4.10) has no positive roots. Hence, by combining with the
assumption 2.1, we know that all roots of (4.5) have negative real parts when
$\tau\in[0,+\infty)$ under the conditions $(C_{0})$ and $(C_{1})$.

In the following, we mainly consider the case of $Q_{n}<0$, that is (4.10) has
only one positive root $\omega_{n}$. In the following, we will discuss the
case which is used to guarantee $Q_{n}<0$ under the condition $(C_{0})$.

When $\tau>0$, according to (4.5) and (4.11), we can define
$Q_{n}=\Gamma_{n}(0)\widetilde{Q}_{n}$ with

$\Gamma_{n}(0)=\widetilde{J}_{n}(0)=(d_{11}d_{22}+d_{21}\xi
u_{*}v_{*})\frac{n^{4}}{\ell^{4}}-\left(d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}+d_{21}v_{*}a_{12}\right)\frac{n^{2}}{\ell^{2}}+\operatorname{Det}(A)$

and

$\widetilde{Q}_{n}=(d_{11}d_{22}-d_{21}\xi
u_{*}v_{*})\frac{n^{4}}{\ell^{4}}-(d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}-d_{21}v_{*}a_{12})\frac{n^{2}}{\ell^{2}}+\operatorname{Det}(A),$ (4.12)

and then by a simple analysis, we have $\Gamma_{n}(0)=\widetilde{J}_{n}(0)>0$
for any $n\in\mathbb{N}_{0}$. Therefore, the sign of $Q_{n}$ coincides with
that of $\widetilde{Q}_{n}$, and in order to guaranteeing $Q_{n}<0$, we only
need to study the case of $\widetilde{Q}_{n}<0$.

###### Case 2

If $(d_{11}d_{22}-d_{21}\xi u_{*}v_{*})>0$ and the conditions $(C_{0})$,

$\displaystyle(C_{2}):~{}\operatorname{Det}(A)>0,~{}d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}-d_{21}v_{*}a_{12}>0$ $\displaystyle(d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}-d_{21}v_{*}a_{12})^{2}-4(d_{11}d_{22}-d_{21}\xi
u_{*}v_{*})\operatorname{Det}(A)>0$

hold, then (4.12) has two positive roots. Without loss of generality, we
assume that the two positive roots of (4.12) are $\widetilde{x}_{1}$ and
$\widetilde{x}_{2}$, i.e.,

$\widetilde{x}_{1,2}=\frac{\widetilde{A}_{1}\mp\sqrt{\widetilde{A}_{3}}}{2\widetilde{A}_{2}},$
(4.13)

where

$\displaystyle\widetilde{A}_{1}$
$\displaystyle=d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}-d_{21}v_{*}a_{12},~{}\widetilde{A}_{2}=d_{11}d_{22}-d_{21}\xi
u_{*}v_{*},$ (4.14) $\displaystyle\widetilde{A}_{3}$
$\displaystyle=\widetilde{A}_{1}^{2}-4\widetilde{A}_{2}\operatorname{Det}(A).$

Since $\widetilde{x}_{1}=n_{1}^{2}/\ell^{2}$ and
$\widetilde{x}_{2}=n_{2}^{2}/\ell^{2}$, then
$n_{1}=\ell\sqrt{\widetilde{x}_{1}}$ and $n_{2}=\ell\sqrt{\widetilde{x}_{2}}$.
By using a geometric argument, we can conclude that

$Q_{n}=\Gamma_{n}(0)\widetilde{Q}_{n}\left\\{\begin{aligned}
&<0,&n_{1}<n<n_{2},\\\ &\geq 0,&n\leq n_{1}\text{ or }n\geq
n_{2},\end{aligned}\right.$

where $n\in\mathbb{N}$. Therefore, (4.10) has one positive root $\omega_{n}$
for $n_{1}<n<n_{2}$ with $n\in\mathbb{N}$, where

$\omega_{n}=\sqrt{\frac{-P_{n}+\sqrt{P_{n}^{2}-4Q_{n}}}{2}}.$ (4.15)

Furthermore, by combining with the second mathematical expression in (4.9),
and notice that $a_{12}<0$, $T_{n}<0$ under the condition $(C_{0})$, then we
have $\sin(\omega_{n}\tau)>0$. Thus, from the first mathematical expression in
(4.9), we can set

$\tau_{n,j}=\frac{1}{\omega_{n}}\left\\{\arccos\left\\{\frac{\omega_{n}^{2}-J_{n}}{d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})-d_{21}v_{*}a_{12}(n^{2}/\ell^{2})}\right\\}+2j\pi\right\\},~{}n\in\mathbb{N},~{}j\in\mathbb{N}_{0}.$
(4.16)

Next, we continue to verify the transversality conditions for the Case 4.2.

###### Lemma 1

Suppose that $(d_{11}d_{22}-d_{21}\xi u_{*}v_{*})>0$, the conditions
$(C_{0})$, $(C_{2})$ hold, and $n_{1}<n<n_{2}$ with $n\in\mathbb{N}$, then we
have

$\left.\frac{d\operatorname{Re}(\lambda(\tau))}{d\tau}\right|_{\tau=\tau_{n,j}}>0,$

where $\operatorname{Re}(\lambda(\tau))$ represents the real part of
$\lambda(\tau)$.

###### Proof

By differentiating the two sides of

$\Gamma_{n}(\lambda)=\operatorname{det}\left(\mathcal{M}_{n}(\lambda)\right)=\lambda^{2}-T_{n}\lambda+\widetilde{J}_{n}(\tau)=0$

with respect to $\tau$, where $T_{n}$ and $\widetilde{J}_{n}(\tau)$ are
defined by (4.6), we have

$\left(\frac{d\lambda(\tau)}{d\tau}\right)^{-1}=\frac{\left(2\lambda-
T_{n}\right)e^{\lambda\tau}}{-\lambda
d_{21}v_{*}a_{12}(n^{2}/\ell^{2})+\lambda d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})}-\frac{\tau}{\lambda}.$ (4.17)

Therefore, by (4.17), we have

$\displaystyle\operatorname{Re}\left(\left.\frac{d\lambda(\tau)}{d\tau}\right|_{\tau=\tau_{n,j}}\right)^{-1}$
$\displaystyle=\operatorname{Re}\left(\frac{\left(2i\omega_{n}-T_{n}\right)e^{i\omega_{n}\tau_{n,j}}}{-i\omega_{n}d_{21}v_{*}a_{12}(n^{2}/\ell^{2})+i\omega_{n}d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})}\right)$ (4.18)
$\displaystyle=\operatorname{Re}\left(\frac{\left(2i\omega_{n}-T_{n}\right)\left(\cos(\omega_{n}\tau_{n,j})+i\sin(\omega_{n}\tau_{n,j})\right)}{-i\omega_{n}d_{21}v_{*}a_{12}(n^{2}/\ell^{2})+i\omega_{n}d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})}\right)$
$\displaystyle=\operatorname{Re}\left(\frac{\left(2i\omega_{n}-T_{n}\right)\left(\cos(\omega_{n}\tau_{n,j})+i\sin(\omega_{n}\tau_{n,j})\right)}{i\omega_{n}(d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})-d_{21}v_{*}a_{12}(n^{2}/\ell^{2}))}\right)$
$\displaystyle=\frac{2\cos(\omega_{n}\tau_{n,j})}{d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})-d_{21}v_{*}a_{12}(n^{2}/\ell^{2})}-\frac{T_{n}\sin(\omega_{n}\tau_{n,j})}{\omega_{n}(d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})-d_{21}v_{*}a_{12}(n^{2}/\ell^{2}))}.$

Furthermore, according to (4.9), we have

$\displaystyle\sin(\omega_{n}\tau_{n,j})=\frac{-T_{n}\omega_{n}}{d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})-d_{21}v_{*}a_{12}(n^{2}/\ell^{2})},$ (4.19)
$\displaystyle\cos(\omega_{n}\tau_{n,j})=\frac{\omega_{n}^{2}-J_{n}}{d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})-d_{21}v_{*}a_{12}(n^{2}/\ell^{2})}.$

Moreover, by combining with (4.18), (4.19) and

$\omega_{n}=\sqrt{\frac{-P_{n}+\sqrt{P_{n}^{2}-4Q_{n}}}{2}}>0,~{}Q_{n}<0,~{}a_{12}<0,$

we have

$\displaystyle\operatorname{Re}\left(\left.\frac{d\lambda(\tau)}{d\tau}\right|_{\tau=\tau_{n,j}}\right)^{-1}$
$\displaystyle=\frac{2\cos(\omega_{n}\tau_{n,j})}{d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})-d_{21}v_{*}a_{12}(n^{2}/\ell^{2})}-\frac{T_{n}\sin(\omega_{n}\tau_{n,j})}{\omega_{n}(d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})-d_{21}v_{*}a_{12}(n^{2}/\ell^{2}))}$
$\displaystyle=\frac{2\omega_{n}^{3}+\omega_{n}(T_{n}^{2}-2J_{n})}{\omega_{n}\left(d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})-d_{21}v_{*}a_{12}(n^{2}/\ell^{2})\right)^{2}}$
$\displaystyle=\frac{\sqrt{P_{n}^{2}-4Q_{n}}}{\left(d_{21}\xi
u_{*}v_{*}(n^{4}/\ell^{4})-d_{21}v_{*}a_{12}(n^{2}/\ell^{2})\right)^{2}}>0.$

This, together with the fact that

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

completes the proof, where $\operatorname{sign}(.)$ represents the sign
function.

Moreover, according to the above analysis, we have the following results.

###### Lemma 2

If the condition $(C_{0})$ is satisfied, then we have the following
conclusions

(i) if the condition $(C_{1})$ holds, then the positive constant steady state
$E_{*}\left(u_{*},v_{*}\right)$ of system (4.2) is asymptotically stable for
all $\tau\geq 0$;

(ii) if $(d_{11}d_{22}-d_{21}\xi u_{*}v_{*})>0$ and the condition $(C_{2})$
holds, by denoting
$\tau_{*}=\min\left\\{\tau_{n,0}:n_{1}<n<n_{2},~{}n\in\mathbb{N}\right\\}$,
then the positive constant steady state $E_{*}\left(u_{*},v_{*}\right)$ of
system (4.2) is asymptotically stable for $0\leq\tau<\tau_{*}$ and unstable
for $\tau>\tau_{*}$. Furthermore, system (4.2) undergoes mode-$n$ Hopf
bifurcations at $\tau=\tau_{n,0}$ for $n\in\mathbb{N}$.

###  4.2 Numerical simulations

In this section, we verify the analytical results given in the previous
sections by some numerical simulations and investigate the direction and
stability of Hopf bifurcation. We use the following initial conditions for the
system (4.2)

$u(x,t)=u_{0}(x),~{}v(x,t)=v_{0}(x),~{}t\in\left[-\tau,0\right].$

####  4.2.1 Mode-1 Hopf bifurcation

(a) (b)

(c) (d)

Figure 1: For the parameters
$\ell=2,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8$,
when $\tau=3<\tau_{1,0}=6.1498$, the positive constant steady state
$E_{*}\left(u_{*},v_{*}\right)=(1.4142,1.4142)$ is locally asymptotically
stable. The initial values are $u_{0}(x)=1.4142-0.1\cos(x/2)$ and
$v_{0}(x)=1.4142+0.1\cos(x/2)$.

(a) (b)

(c) (d)

Figure 2: For the parameters
$\ell=2,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8$,
when $\tau=8>\tau_{1,0}=6.1498$, there exists a stable spatially inhomogeneous
periodic solution. (a) and (b) are the transient behaviours for $u(x,t)$ and
$v(x,t)$, respectively, (c) and (d) are long-term behaviours for $u(x,t)$ and
$v(x,t)$, respectively. The initial values are $u_{0}(x)=1.4142-0.1\cos(x/2)$
and $v_{0}(x)=1.4142+0.1\cos(x/2)$.

(a) (b)

Figure 3: For the parameters
$\ell=2,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8$,
when $\tau=8>\tau_{1,0}=6.1498$, there exists a stable spatially inhomogeneous
periodic solution. The initial values are $u_{0}(x)=1.4142-0.1\cos(x/2)$ and
$v_{0}(x)=1.4142+0.1\cos(x/2)$.

If we set the parameters as follows

$\ell=2,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8,$

then we can easily obtain that

$\displaystyle a_{11}=1-2\beta
u_{*}-\frac{mu_{*}}{(1+u_{*})^{2}}=-0.5355<0,~{}d_{11}d_{22}-d_{21}\xi
u_{*}v_{*}=3.84>0,$
$\displaystyle\operatorname{Det}(A)=a_{11}a_{22}-a_{12}a_{21}=0.6627>0,~{}d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}-d_{21}v_{*}a_{12}=4.1814>0,$
$\displaystyle(d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}-d_{21}v_{*}a_{12})^{2}-4(d_{11}d_{22}-d_{21}\xi
u_{*}v_{*})\operatorname{Det}(A)=7.3041>0.$

Therefore, the conditions $(C_{0})$ and $(C_{2})$ are satisfied under the
above parameters settings. In the following, we mainly verify the conclusion
in Lemma 4.4 (ii). According to (4.3) and (4.4), we have
$E_{*}\left(u_{*},v_{*}\right)=(1.4142,1.4142)$,

$a_{11}=-0.5355,~{}a_{12}=-0.2929,~{}a_{21}=0.8,~{}a_{22}=-0.8.$

By combining with (4.13), (4.14), (4.15) and (4.16), we have $n_{1}=0.8776$,
$n_{2}=1.8935$, and consider that $n\in\mathbb{N}$, we have
$\omega_{c}=\omega_{1}=0.418$ and $\tau_{c}=\tau_{1,0}=6.1498$. Moreover, by
Lemma 4.4 (ii), we have the following proposition.

###### Proposition 1

For system (4.2) with the parameters
$\ell=2,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8$,
the positive constant steady state $E_{*}\left(u_{*},v_{*}\right)$ of system
(4.2) is asymptotically stable for $0\leq\tau<\tau_{1,0}=6.1498$ and unstable
for $\tau>\tau_{1,0}=6.1498$. Furthermore, system (4.2) undergoes mode-1 Hopf
bifurcation at $\tau=\tau_{1,0}=6.1498$.

For the parameters
$\ell=2,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8$,
according to Proposition 4.5, we know that system (4.2) undergoes a Hopf
bifurcation at $\tau_{1,0}=6.1498$. Furthermore, the direction and stability
of Hopf bifurcation can be determined by calculating $K_{1}K_{2}$ and $K_{2}$
using the procedures developed in Section 2. After a direct calculation using
MATLAB software, we obtain

$K_{1}=0.016>0,~{}K_{2}=-0.9283<0,~{}K_{1}K_{2}=-0.0148<0,$

which implies that the Hopf bifurcation at $\tau_{1,0}=6.1498$ is
supercritical and stable.

When $\tau=3<\tau_{1,0}=6.1498$, Fig.1 (a)-(d) illustrate the evolution of the
solution of system (4.2) starting from the initial values
$u_{0}(x)=1.4142-0.1\cos(x/2)$ and $v_{0}(x)=1.4142+0.1\cos(x/2)$, finally
converging to the positive constant steady state
$E_{*}\left(u_{*},v_{*}\right)$. Furthermore, when $\tau=8>\tau_{1,0}=6.1498$,
Fig.2 (a)-(d) illustrate the existence of the spatially homogeneous periodic
solution with the initial values $u_{0}(x)=1.4142-0.1\cos(x/2)$ and
$v_{0}(x)=1.4142+0.1\cos(x/2)$. When $\tau=8>\tau_{1,0}=6.1498$, the space-
time diagrams for the prey population $u(x,t)$ and the predator population
$v(x,t)$ are given in Fig.3 (a) and (b), respectively.

####  4.2.2 Mode-2 Hopf bifurcation

(a) (b)

(c) (d)

Figure 4: For the parameters
$\ell=3,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8$,
when $\tau=2<\tau_{2,0}=3.5361$, the positive constant steady state
$E_{*}\left(u_{*},v_{*}\right)=(1.4142,1.4142)$ is locally asymptotically
stable. The initial values are $u_{0}(x)=1.4142-0.1\cos(2x/3)$ and
$v_{0}(x)=1.4142+0.1\cos(2x/3)$.

(a) (b)

(c) (d)

Figure 5: For the parameters
$\ell=3,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8$,
when $\tau=6>\tau_{2,0}=3.5361$, there exists a stable spatially inhomogeneous
periodic solution. (a) and (b) are the transient behaviours for $u(x,t)$ and
$v(x,t)$, respectively, (c) and (d) are long-term behaviours for $u(x,t)$ and
$v(x,t)$, respectively. The initial values are $u_{0}(x)=1.4142-0.1\cos(2x/3)$
and $v_{0}(x)=1.4142+0.1\cos(2x/3)$.

(a) (b)

Figure 6: For the parameters
$\ell=3,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8$,
when $\tau=6>\tau_{2,0}=3.5361$, there exists a stable spatially inhomogeneous
periodic solution. The initial values are $u_{0}(x)=1.4142-0.1\cos(2x/3)$ and
$v_{0}(x)=1.4142+0.1\cos(2x/3)$.

If we set the parameters as follows

$\ell=3,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8,$

then we can also easily obtain that

$\displaystyle a_{11}=1-2\beta
u_{*}-\frac{mu_{*}}{(1+u_{*})^{2}}=-0.5355<0,~{}d_{11}d_{22}-d_{21}\xi
u_{*}v_{*}=3.84>0,$
$\displaystyle\operatorname{Det}(A)=a_{11}a_{22}-a_{12}a_{21}=0.6627>0,~{}d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}-d_{21}v_{*}a_{12}=4.1814>0,$
$\displaystyle(d_{11}a_{22}+d_{22}a_{11}-a_{21}\xi
u_{*}-d_{21}v_{*}a_{12})^{2}-4(d_{11}d_{22}-d_{21}\xi
u_{*}v_{*})\operatorname{Det}(A)=7.3041>0.$

Therefore, the conditions $(C_{0})$ and $(C_{2})$ are satisfied under the
above parameters settings. In the following, we mainly verify the conclusion
in Lemma 4.4 (ii). According to (4.3) and (4.4), we have
$E_{*}\left(u_{*},v_{*}\right)=(1.4142,1.4142)$,

$a_{11}=-0.5355,~{}a_{12}=-0.2929,~{}a_{21}=0.8,~{}a_{22}=-0.8.$

By combining with (4.13), (4.14), (4.15) and (4.16), we have $n_{1}=1.3164$,
$n_{2}=2.8403$, and consider that $n\in\mathbb{N}$, we have
$\omega_{c}=\omega_{2}=0.6870$ and $\tau_{c}=\tau_{2,0}=3.5361$. Moreover, by
Lemma 4.4 (ii), we have the following proposition.

###### Proposition 2

For system (4.2) with the parameters
$\ell=3,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8$,
the positive constant steady state $E_{*}\left(u_{*},v_{*}\right)$ of system
(4.2) is asymptotically stable for $0\leq\tau<\tau_{2,0}=3.5361$ and unstable
for $\tau>\tau_{2,0}=3.5361$. Furthermore, system (4.2) undergoes mode-2 Hopf
bifurcation at $\tau=\tau_{2,0}=3.5361$.

For the parameters
$\ell=3,~{}d_{11}=2,~{}d_{22}=3,~{}d_{21}=18,~{}\xi=0.06,~{}\beta=0.5,~{}m=0.5,~{}s=0.8$,
according to Proposition 4.6, we know that system (4.2) undergoes a Hopf
bifurcation at $\tau_{2,0}=3.5361$. Furthermore, the direction and stability
of Hopf bifurcation can be determined by calculating $K_{1}K_{2}$ and $K_{2}$
using the procedures developed in Section 2. After a direct calculation using
MATLAB software, we obtain

$K_{1}=0.0410>0,~{}K_{2}=-1.3669<0,~{}K_{1}K_{2}=-0.0561<0,$

which implies that the Hopf bifurcation at $\tau_{2,0}=3.5361$ is
supercritical and stable.

When $\tau=2<\tau_{2,0}=3.5361$, Fig.4 (a)-(d) illustrate the evolution of the
solution of system (4.2) starting from the initial values
$u_{0}(x)=1.4142-0.1\cos(2x/3)$ and $v_{0}(x)=1.4142+0.1\cos(2x/3)$, finally
converging to the positive constant steady state
$E_{*}\left(u_{*},v_{*}\right)$. Furthermore, when $\tau=6>\tau_{2,0}=3.5361$,
Fig.5 (a)-(d) illustrate the existence of the spatially homogeneous periodic
solution with the initial values $u_{0}(x)=1.4142-0.1\cos(2x/3)$ and
$v_{0}(x)=1.4142+0.1\cos(2x/3)$. When $\tau=6>\tau_{2,0}=3.5361$, the space-
time diagrams for the prey population $u(x,t)$ and the predator population
$v(x,t)$ are given in Fig.6 (a) and (b), respectively.

##  5 Conclusion and discussion

In this paper, the diffusive predator-prey system with spatial memory and
predator-taxis is proposed, and we derive an algorithm for calculating the
normal form of Hopf bifurcation in this system. As a real application, we
consider the Holling-Tanner model with spatial memory and predator-taxis. By
carrying out the stability and Hopf bifurcation analysis for the Holling-
Tanner model with spatial memory and predator-taxis, the inhomogeneous spatial
patterns, i.e., two stable spatially inhomogeneous periodic solutions are
found. Furthermore, the supercritical and stable mode-1 and mode-2 Hopf
bifurcation periodic solutions are found by using our newly developed
algorithm. Numerical simulations verify the results of our theoretical
analysis and give us a more intuitive display.

It is worth mentioning that in this paper, for our proposed diffusive
predator-prey system with spatial memory and predator-taxis, the delay only
occurs in the diffusion term, not in the reaction form. However, for the
general predator-prey models, the gestation, hunting, migration and maturation
delays, etc., often occur in the reaction term. By noticing this point, on the
basis of system (1.4), the system

$\left\\{\begin{aligned} &\frac{\partial u(x,t)}{\partial t}=d_{11}\Delta
u(x,t)+\xi\left(u(x,t)v_{x}(x,t)\right)_{x}+f(u(x,t),v(x,t),u(x,t-\sigma),v(x,t-\sigma)),&x\in(0,\ell\pi),~{}t>0,\\\
&\frac{\partial v(x,t)}{\partial t}=d_{22}\Delta
v(x,t)-d_{21}\left(v(x,t)u_{x}(x,t-\tau)\right)_{x}+g(u(x,t),v(x,t),u(x,t-\sigma),v(x,t-\sigma)),&x\in(0,\ell\pi),~{}t>0,\end{aligned}\right.$

which needs further research, where $\sigma>0$ is the delay occurs in the
reaction term.

## Acknowledgments

The author is grateful to the anonymous referees for their useful suggestions
which improve the contents of this article.

## Disclosures and declarations

This research did not involve human participants and animals.

Funding: This research did not receive any specific grant from funding
agencies in the public, commercial, or not-for-profit sectors.

Conflicts of interest: The author declares that there is not conflict of
interest, whether financial or non-financial.

Availability of data and material: This research didn’t involve the private
data, and the involving data and material are all available.

Code availability: The numerical simulations in this paper are carried by
using the MATLAB software.

Authors’ contributions: This manuscript is investigated and written by Yehu
Lv.

## Appendix A

###### Remark 1

Assume that at $\tau=\tau_{c}$, system (4.2) has a pair of purely imaginary
roots $\pm i\omega_{n_{c}}$ with $\omega_{n_{c}}>0$ for $n=n_{c}\in\mathbb{N}$
and all other eigenvalues have negative real parts. Let
$\lambda(\tau)=\alpha_{1}(\tau)\pm i\alpha_{2}(\tau)$ be a pair of roots of
system (4.2) near $\tau=\tau_{c}$ satisfying $\alpha_{1}(\tau_{c})=0$ and
$\alpha_{2}(\tau_{c})=\omega_{n_{c}}$. In addition, the corresponding
transversality condition holds.

The normal form of Hopf bifurcation for the system (4.2) can be calculated by
using our newly developed algorithm in Section 2. Here, we give the detail
calculation procedures of $B_{1},B_{21},B_{22},B_{23}$ steps by steps.

  1. Step 1:

$B_{1}=2\psi^{T}(0)\left(A\phi(0)-\frac{n_{c}^{2}}{\ell^{2}}\left(D_{1}\phi(0)+D_{2}\phi(-1)\right)\right)$

with

$D_{1}=\left(\begin{array}[]{cc}d_{11}&\xi u_{*}\\\
0&d_{22}\end{array}\right),~{}D_{2}=\left(\begin{array}[]{cc}0&0\\\
-d_{21}v_{*}&0\end{array}\right),~{}A=\left(\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right).$

Here,

$\phi=\left(\begin{array}[]{c}1\\\
\frac{a_{11}-i\omega_{n_{c}}-d_{11}(n_{c}^{2}/\ell^{2})}{\xi
u_{*}(n_{c}^{2}/\ell^{2})-a_{12}}\end{array}\right),~{}\psi=\eta\left(\begin{array}[]{c}1\\\
\frac{a_{12}-\xi
u_{*}(n_{c}^{2}/\ell^{2})}{i\omega_{n_{c}}+d_{22}(n_{c}^{2}/\ell^{2})-a_{22}}\end{array}\right)$

with

$\eta=\frac{i\omega_{n_{c}}+\left(n_{c}/\ell\right)^{2}d_{22}-a_{22}}{2i\omega_{n_{c}}+\left(n_{c}/\ell\right)^{2}d_{11}-a_{11}+\left(n_{c}/\ell\right)^{2}d_{22}-a_{22}+\tau_{c}a_{12}d_{21}v_{*}\left(n_{c}/\ell\right)^{2}e^{-i\omega_{c}}}.$

  1. Step 2:

$B_{21}=\frac{3}{2\ell\pi}\psi^{T}A_{21}$

with

$\displaystyle A_{21}$
$\displaystyle=3f_{30}\phi_{1}^{2}(0)\overline{\phi_{1}}(0)+3f_{03}\phi_{2}^{2}(0)\overline{\phi_{2}}(0)+3f_{21}(\phi_{1}^{2}(0)\overline{\phi_{2}}(0)+2\phi_{1}(0)\overline{\phi_{1}}(0)\phi_{2}(0))$
$\displaystyle+3f_{12}(2\phi_{1}(0)\phi_{2}(0)\overline{\phi_{2}}(0)+\overline{\phi_{1}}(0)\phi_{2}^{2}(0)).$

Here,

$\displaystyle f^{(1)}_{30}$
$\displaystyle=-6\tau_{c}m(1+u_{*})^{-4}v_{*},~{}f^{(2)}_{30}=6\tau_{c}su_{*}^{-4}v_{*}^{2},$
$\displaystyle f^{(1)}_{21}$
$\displaystyle=2\tau_{c}m(1+u_{*})^{-3},~{}f^{(2)}_{21}=-4\tau_{c}su_{*}^{-3}v_{*},$
$\displaystyle f^{(1)}_{12}$
$\displaystyle=0,~{}f^{(2)}_{12}=2\tau_{c}su_{*}^{-2},$ $\displaystyle
f^{(1)}_{03}$ $\displaystyle=0,~{}f^{(2)}_{03}=0.$

  1. Step 3:

$\displaystyle B_{22}$
$\displaystyle=\frac{1}{\sqrt{\ell\pi}}\psi^{T}\left(\mathcal{S}_{2}\left(\phi(\theta),h_{0,11}(\theta)\right)+\mathcal{S}_{2}\left(\overline{\phi}(\theta),h_{0,20}(\theta)\right)\right)$
$\displaystyle+\frac{1}{\sqrt{2\ell\pi}}\psi^{T}\left(\mathcal{S}_{2}\left(\phi(\theta),h_{2n_{c},11}(\theta)\right)+\mathcal{S}_{2}\left(\overline{\phi}(\theta),h_{2n_{c},20}(\theta)\right)\right)$

with

$\displaystyle\mathcal{S}_{2}\left(\phi(\theta),h_{0,11}(\theta)\right)$
$\displaystyle=2f_{20}\phi_{1}(0)h^{(1)}_{0,11}(0)+2f_{02}\phi_{2}(0)h^{(2)}_{0,11}(0)$
$\displaystyle+2f_{11}\left(\phi_{1}(0)h^{(2)}_{0,11}(0)+\phi_{2}(0)h^{(1)}_{0,11}(0)\right),$
$\displaystyle\mathcal{S}_{2}\left(\overline{\phi}(\theta),h_{0,20}(\theta)\right)$
$\displaystyle=2f_{20}\overline{\phi}_{1}(0)h^{(1)}_{0,20}(0)+2f_{02}\overline{\phi}_{2}(0)h^{(2)}_{0,20}(0)$
$\displaystyle+2f_{11}\left(\overline{\phi}_{1}(0)h^{(2)}_{0,20}(0)+\overline{\phi}_{2}(0)h^{(1)}_{0,20}(0)\right),$
$\displaystyle\mathcal{S}_{2}\left(\phi(\theta),h_{2n_{c},11}(\theta)\right)$
$\displaystyle=2f_{20}\phi_{1}(0)h^{(1)}_{2n_{c},11}(0)+2f_{02}\phi_{2}(0)h^{(2)}_{2n_{c},11}(0)$
$\displaystyle+2f_{11}\left(\phi_{1}(0)h^{(2)}_{2n_{c},11}(0)+\phi_{2}(0)h^{(1)}_{2n_{c},11}(0)\right),$
$\displaystyle\mathcal{S}_{2}\left(\overline{\phi}(\theta),h_{2n_{c},20}(\theta)\right)$
$\displaystyle=2f_{20}\overline{\phi}_{1}(0)h^{(1)}_{2n_{c},20}(0)+2f_{02}\overline{\phi}_{2}(0)h^{(2)}_{2n_{c},20}(0)$
$\displaystyle+2f_{11}\left(\overline{\phi}_{1}(0)h^{(2)}_{2n_{c},20}(0)+\overline{\phi}_{2}(0)h^{(1)}_{2n_{c},20}(0)\right).$

Here,

$\displaystyle f^{(1)}_{20}$
$\displaystyle=-2\tau_{c}\beta+2\tau_{c}m(1+u_{*})^{-3}v_{*},~{}f^{(2)}_{20}=-2\tau_{c}su_{*}^{-3}v_{*}^{2},$
$\displaystyle f^{(1)}_{11}$
$\displaystyle=-\tau_{c}m(1+u_{*})^{-2},~{}f^{(2)}_{11}=2\tau_{c}su_{*}^{-2}v_{*},$
$\displaystyle f^{(1)}_{02}$
$\displaystyle=0,~{}f^{(2)}_{02}=-2\tau_{c}su_{*}^{-1}.$

Furthermore, we have

$\left\\{\begin{aligned}
h_{0,20}(\theta)&=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}\left(2i\omega_{c}\right)\right)^{-1}A_{20}e^{2i\omega_{c}\theta},\\\
h_{0,11}(\theta)&=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}(0)\right)^{-1}A_{11}\end{aligned}\right.$

and

$\left\\{\begin{aligned}
h_{2n_{c},20}(\theta)&=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{c}}\left(2i\omega_{c}\right)\right)^{-1}\widetilde{A}_{20}e^{2i\omega_{c}\theta},\\\
h_{2n_{c},11}(\theta)&=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{c}}(0)\right)^{-1}\widetilde{A}_{11}\end{aligned}\right.$

with

$\widetilde{\mathcal{M}}_{n}(\lambda)=\lambda
I_{2}+\tau_{c}(n/\ell)^{2}D_{1}+\tau_{c}(n/\ell)^{2}e^{-\lambda}D_{2}-\tau_{c}A.$

Here,

$\displaystyle A_{20}$
$\displaystyle=f_{20}\phi_{1}^{2}(0)+f_{02}\phi_{2}^{2}(0)+2f_{11}\phi_{1}(0)\phi_{2}(0),$
$\displaystyle A_{11}$
$\displaystyle=2f_{20}\phi_{1}(0)\overline{\phi_{1}}(0)+2f_{02}\phi_{2}(0)\overline{\phi_{2}}(0)+2f_{11}(\phi_{1}(0)\overline{\phi_{2}}(0)+\overline{\phi_{1}}(0)\phi_{2}(0))$

and

$\left\\{\begin{aligned}
\widetilde{A}_{20}&=A_{20}-2\left(n_{c}/\ell\right)^{2}A_{20}^{d},\\\
\widetilde{A}_{11}&=A_{11}-2\left(n_{c}/\ell\right)^{2}A_{11}^{d}\end{aligned}\right.$

with

$\left\\{\begin{aligned}
A_{20}^{d}&=\left(\begin{array}[]{c}2\xi\tau_{c}\phi_{1}(0)\phi_{2}(0)\\\
-2d_{21}\tau_{c}\phi_{1}(-1)\phi_{2}(0)\end{array}\right)=\overline{A_{02}^{d}},\\\
A_{11}^{d}&=\left(\begin{array}[]{c}4\xi\tau_{c}\operatorname{Re}\left\\{\phi_{1}(0)\overline{\phi_{2}}(0)\right\\}\\\
-4d_{21}\tau_{c}\operatorname{Re}\left\\{\phi_{1}(-1)\overline{\phi_{2}}(0)\right\\}\end{array}\right).\end{aligned}\right.$

  1. Step 4:

$\displaystyle B_{23}=$
$\displaystyle-\frac{1}{\sqrt{\ell\pi}}\left(n_{c}/\ell\right)^{2}\psi^{T}\left(\mathcal{S}_{2}^{(d,1)}\left(\phi(\theta),h_{0,11}(\theta)\right)+\mathcal{S}_{2}^{(d,1)}\left(\overline{\phi}(\theta),h_{0,20}(\theta)\right)\right)$
$\displaystyle+\frac{1}{\sqrt{2\ell\pi}}\psi^{T}\sum_{j=1,2,3}b_{2n_{c}}^{(j)}\left(\mathcal{S}_{2}^{(d,j)}\left(\phi(\theta),h_{2n_{c},11}(\theta)\right)+\mathcal{S}_{2}^{(d,j)}\left(\overline{\phi}(\theta),h_{2n_{c},20}(\theta)\right)\right)$

with

$b_{2n_{c}}^{(1)}=-\frac{n_{c}^{2}}{\ell^{2}},~{}b_{2n_{c}}^{(2)}=\frac{2n_{c}^{2}}{\ell^{2}},~{}b_{2n_{c}}^{(3)}=-\frac{(2n_{c})^{2}}{\ell^{2}}$

and

$\left\\{\begin{aligned}
&\mathcal{S}_{2}^{(d,1)}\left(\phi(\theta),h_{0,11}(\theta)\right)=2\left(\begin{array}[]{c}\xi\tau_{c}\phi_{2}(0)h_{0,11}^{(1)}(0)\\\
-d_{21}\tau_{c}\phi_{1}(-1)h_{0,11}^{(2)}(0)\end{array}\right),\\\
&\mathcal{S}_{2}^{(d,1)}\left(\overline{\phi}(\theta),h_{0,20}(\theta)\right)=2\left(\begin{array}[]{c}\xi\tau_{c}\overline{\phi}_{2}(0)h_{0,20}^{(1)}(0)\\\
-d_{21}\tau_{c}\overline{\phi}_{1}(-1)h_{0,20}^{(2)}(0)\end{array}\right),\\\
&\mathcal{S}_{2}^{(d,1)}\left(\phi(\theta),h_{2n_{c},11}(\theta)\right)=2\left(\begin{array}[]{c}\xi\tau_{c}\phi_{2}(0)h_{2n_{c},11}^{(1)}(0)\\\
-d_{21}\tau_{c}\phi_{1}(-1)h_{2n_{c},11}^{(2)}(0)\end{array}\right),\\\
&\mathcal{S}_{2}^{(d,2)}(\phi(\theta),h_{2n_{c},11}(\theta))=2\left(\begin{array}[]{c}\xi\tau_{c}(\phi_{2}(0)h_{2n_{c},11}^{(1)}(0)+\phi_{1}(0)h_{2n_{c},11}^{(2)}(0))\\\
-d_{21}\tau_{c}(\phi_{2}(0)h_{2n_{c},11}^{(1)}(-1)+\phi_{1}(-1)h_{2n_{c},11}^{(2)}(0))\end{array}\right),\\\
&\mathcal{S}_{2}^{(d,3)}(\phi(\theta),h_{2n_{c},11}(\theta))=2\left(\begin{array}[]{c}\xi\tau_{c}\phi_{1}(0)h_{2n_{c},11}^{(2)}(0)\\\
-d_{21}\tau_{c}\phi_{2}(0)h_{2n_{c},11}^{(1)}(-1)\end{array}\right),\\\
&\mathcal{S}_{2}^{(d,1)}\left(\overline{\phi}(\theta),h_{2n_{c},20}(\theta)\right)=2\left(\begin{array}[]{c}\xi\tau_{c}\overline{\phi}_{2}(0)h_{2n_{c},20}^{(1)}(0)\\\
-d_{21}\tau_{c}\overline{\phi}_{1}(-1)h_{2n_{c},20}^{(2)}(0)\end{array}\right),\\\
&\mathcal{S}_{2}^{(d,2)}(\overline{\phi}(\theta),h_{2n_{c},20}(\theta))=2\left(\begin{array}[]{c}\xi\tau_{c}(\overline{\phi}_{2}(0)h_{2n_{c},20}^{(1)}(0)+\overline{\phi}_{1}(0)h_{2n_{c},20}^{(2)}(0))\\\
-d_{21}\tau_{c}(\overline{\phi}_{2}(0)h_{2n_{c},20}^{(1)}(-1)+\overline{\phi}_{1}(-1)h_{2n_{c},20}^{(2)}(0))\end{array}\right),\\\
&\mathcal{S}_{2}^{(d,3)}(\overline{\phi}(\theta),h_{2n_{c},20}(\theta))=2\left(\begin{array}[]{c}\xi\tau_{c}\overline{\phi}_{1}(0)h_{2n_{c},20}^{(2)}(0)\\\
-d_{21}\tau_{c}\overline{\phi}_{2}(0)h_{2n_{c},20}^{(1)}(-1)\end{array}\right).\end{aligned}\right.$

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