# Biological aggregations from spatial memory and nonlocal advection

Di Liu
School of Mathematics
Harbin Institute of Technology
Harbin, China
&Yurij Salmaniw∗,
Department of Mathematical and Statistical Sciences
University of Alberta
Edmonton, Canada
<EMAIL_ADDRESS>
Jonathan R. Potts
School of Mathematics and Statistics
University of Sheffield
Sheffield, United Kingdom
&Junping Shi
Department of Mathematics
College of William $\&$ Mary
Williamsburg, VA.
&Hao Wang
Department of Mathematical and Statistical Sciences
University of Alberta
Edmonton, Alberta
Co-first authors.Corresponding author.

###### Abstract

We investigate a nonlocal single-species reaction-diffusion-advection model
that integrates the spatial memory of previously visited locations and
nonlocal detection in space, resulting in a coupled PDE-ODE system reflective
of several existing models found in spatial ecology. We prove the existence
and uniqueness of a Hölder continuous weak solution in one spatial dimension
under some general conditions, allowing for discontinuous kernels such as the
top-hat detection kernel. A robust spectral and bifurcation analysis is also
performed, providing the rigorous analytical study not yet found in the
existing literature. In particular, the essential spectrum is shown to be
entirely negative, and we classify the nature of the bifurcation near the
critical values obtained via a linear stability analysis. A pseudo-spectral
method is used to solve and plot the steady states near and far away from
these critical values, complementing the analytical insights.

_Keywords_ nonlocal advection $\cdot$ bifurcation analysis $\cdot$ spatial
memory $\cdot$ pattern formation

##  1 Introduction

###  1.1 Background and model formulation

Spatial memory is a key feature driving the movement of mobile organisms Fagan
et al. (2013). As organisms move, they gather information about where they
have been, building a map that informs future movement decisions. This process
generates a feedback mechanism, whereby previous visitations of favourable
locations can cause repeated visits, resulting in the organism confining
itself to certain specific areas of the landscape. In animal ecology, such
memory processes have been hypothesised to be foundational in the construction
of home ranges (Briscoe et al., 2002; Börger et al., 2008; Van Moorter et al.,
2009), small areas where an animal decides to perform its daily activities
instead of roaming more widely. Conversely, memory of unfavourable locations
can cause animals to relocate. For example, memory has been shown to play a
key role in migratory movements (Bracis and Mueller, 2017; Abrahms et al.,
2019) and avoiding conspecifics to form territories or home ranges (Potts and
Lewis, 2016a; Ellison et al., 2020).

Understanding how memory processes help to shape the space use of animals is
thus becoming a question of increasing interest in both empirical ecology
(Fagan et al., 2013; Merkle et al., 2014, 2019; Lewis et al., 2021) and
mathematical modelling (Potts and Lewis, 2016a; Shi et al., 2021; Wang and
Salmaniw, 2023a). From a modelling perspective, a key tool for modelling
movement in response to remembered space use is via an advection term in a
partial differential equation (PDE). This advection term is typically nonlocal
in space, for both biological and mathematical reasons. From a biological
perspective, nonlocality is important because organisms will generally sense
their surrounding environment – for example through sight, smell, or touch –
and make movement decisions accordingly (Benhamou, 2014; Martinez-Garcia et
al., 2020). Moreover, this nonlocality occurs not only in animals but also in
cells (Armstrong et al., 2009; Painter et al., 2023). Mathematically, nonlocal
advection is often crucial for well-posedness and avoiding blow-up of PDEs
(Bertozzi and Laurent, 2007; Giunta et al., 2022).

Alongside advection, mathematical models of organism movement typically have a
diffusive term, accounting for the aspects of movement that we are not
explicitly modelling (such as foraging), and may also have a reaction term
representing the births and deaths of organisms. This leads to the formalism
of reaction-diffusion-advection equations (RDAs). In a one dimensional spatial
domain $\Omega$, such an RDA might have the following general form

$\displaystyle u_{t}=du_{xx}+\alpha(ua_{x})_{x}+f(u),\quad x\in\Omega,\ t>0.$
(1.1)

Here, $d>0$ denotes the rate of diffusion (exploratory movement),
$\alpha\in\mathbb{R}$ denotes the rate of advection towards ($\alpha<0$) or
away from ($\alpha>0$) the environmental covariates described by the function
$a(x,t)$, and $f(u)$ describes population changes through birth/death
processes. When paired with an appropriate initial/boundary condition, we seek
to analyze dynamical behaviours of the solution $u(x,t)$ as it depends on
parameters appearing in the equation.

The aspect of memory then appears in the advection term $a(x,t)$. A recent
review paper by (Wang and Salmaniw, 2023b) covers in detail the development of
equations to model memory, as well as the related concept of learning, along
with a large collection of open problems and directions in this area. The
central idea is to model spatial memory as a map, $k(x,t)$, which evolves over
time as the organism learns about its environment (Fagan et al., 2013; Potts
and Lewis, 2016a, 2019). This map may represent something in the mind of a
specific animal, sometimes called a ‘cognitive map’ (Harten et al., 2020; Peer
et al., 2021), or it could represent memory of past animal locations embedded
in the environment, e.g. due to animals depositing scent marks or forging
trails.

Here, we seek to explore the influence of such a map on the space-use patterns
of a single population, $u$. To this end, we describe the evolution of
$k(x,t)$ through the ordinary differential equation for each $x\in\Omega$,
following Potts and Lewis (2016a)

$k_{t}=g(u)-(\mu+\beta u)k,\quad t>0.$ (1.2)

Here, the function $g(\cdot)$ describes the uptake rate of the map $k$ as it
depends on the population $u(x,t)$; $\mu\geq 0$ describes the rate at which
memories fade over time; and $\beta\geq 0$ describes a rate at which organisms
remove a location from their memory map on revisitation (e.g. if animals want
to avoid overuse of a location (Potts et al., 2022)).

Note that, for simplicity, we have assumed that all organisms in a population
share a common memory map. This makes it perhaps more amenable to modelling
the distribution of cues left on the environment, e.g. scent marks or visual
cues (Lewis and Murray, 1993; Moorcroft et al., 2006), rather than memory
contained in the minds of animals. Alternatively, if the population modelled
by $u(x,t)$ has some process of relatively-rapid information sharing, then we
can view $k(x,t)$ as a shared memory amongst the population (e.g. for social
insects this may be valid). As another example, if $u(x,t)$ is the probability
distribution of a single animal (in which case $f(u)=0$ per force) then
$k(x,t)$ can be used to model a map in the mind of an individual (Potts and
Lewis, 2016a).

Prototypical examples of the function $g(u)$ might be $g(u)=\rho u$, denoting
memory accruing in proportion to animal visitations, or $g(u)=\rho u^{2}$,
denoting uptake of memory when members of $u$ encounter one another (here,
$\rho$ is a constant). However, these can, in principle, both lead to
unbounded memory. Therefore we can either take another functional form, such
as $g(u)=\rho u^{2}/(1+cu)$ (cf. the Holling type II functional response
(Holling, 1965)), or modify Equation (1.2) to the following as in Potts and
Lewis (2016a):

$k_{t}=g(u)(\kappa-k)-(\mu+\beta u)k,\quad t>0,$ (1.3)

where $\kappa>0$ denotes a theoretical maximal memory capacity.

To combine this mechanism of spatial memory with nonlocal perception, we model
nonlocal effects through a spatial convolution:

$\overline{k}(x,t)=(G\ast
k)(x,t)=\frac{1}{\left|\Omega\right|}\int_{\Omega}G(x-y)k(y,t)dy.$ (1.4)

Here, the function $G(\cdot)$ is referred to as a perceptual kernel or
detection function, which describes how an animals’ ability to perceive
landscape information varies with distance Fagan et al. (2017); Wang and
Salmaniw (2023a). Common forms of the detection function $G(\cdot)$ include
the Gaussian detection function, the exponential detection function, or the
top-hat detection function, each taking the respective forms in
$\Omega=\mathbb{R}$:

$\displaystyle G(x)$ $\displaystyle:=\frac{1}{\sqrt{2\pi}R}e^{-x^{2}/2R^{2}},$
(1.5) $\displaystyle G(x)$ $\displaystyle:=\frac{1}{2R}e^{-\left|x\right|/R},$
(1.6) $\displaystyle G(x)$ $\displaystyle:=\begin{cases}\frac{1}{2R},\hskip
14.22636pt-R\leq x\leq R,\cr 0,\hskip 28.45274pt\text{otherwise}.\end{cases}$
(1.7)

Here, $R\geq 0$ is the perceptual radius, describing the maximum distance at
which landscape features can be distinguished Fagan et al. (2017). Roughly,
the Gaussian detection function provides the most information far away from
the location of observation, whereas the top-hat detection function gives a
strict limit on how far the organism can detect information. In general, it is
reasonable to assume the detection function satisfies

  1. i.)

$G(x)$ is symmetric about the origin;

  2. ii.)

$\int_{\mathbb{R}}G(x)dx=1$;

  3. iii.)

$\lim_{R\to 0^{+}}G(x)=\delta(x)$;

  4. iv.)

$G(x)$ is non-increasing from the origin.

Here, $\delta(x)$ denotes the Dirac-delta distribution. Each of the Gaussian,
exponential, and top-hat kernels satisfy these properties over $\mathbb{R}$;
appropriate modification is sometimes required in a bounded domain. Readers
are encouraged to review Fagan et al. (2017); Wang and Salmaniw (2023a) for
further discussion on detection kernels and some of the challenges in defining
nonlocal kernels near a boundary region.

Taking the advective potential $a(x,t)=\overline{k}(x,t)$, where $k$ solves
either (1.2) or (1.3), we combine the equation describing movement (1.1) with
a dynamic spatial map in $\Omega=(-L,L)$, $L>0$, to arrive at the following
two systems of equations subject to periodic boundary conditions:

$\begin{cases}u_{t}=du_{xx}+\alpha(u\overline{k}_{x})_{x}+f(u),&x\in(-L,L),\
t>0,\\\ k_{t}=g(u)-(\mu+\beta u)k,&x\in(-L,L),\ t>0,\end{cases}$ (1.8.a)

and

$\begin{cases}u_{t}=du_{xx}+\alpha(u\overline{k}_{x})_{x}+f(u),&x\in(-L,L),\
t>0,\\\ k_{t}=g(u)(\kappa-k)-(\mu+\beta u)k,&x\in(-L,L),\ t>0.\end{cases}$
(1.8.b)

In either case, we denote by $u(x,0)=u_{0}(x)$, $k(x,0)=k_{0}(x)$ the initial
data, chosen to be $2L$-periodic in $\Omega$. As discussed in Wang and
Salmaniw (2023a), boundary conditions in a nonlocal setting in a bounded
domain are highly non-trivial in general. More precisely, it is not clear how
to appropriately define the spatial convolution (1.4) near the boundary of the
domain while remaining analytically tractable. For this reason, we appeal to a
periodic boundary condition, which requires no further modification of (1.4)
near the boundary points $\\{-L,L\\}$. While problems (1.8.a) and (1.8.b)
appear similar in form, it is of interest to understand exactly when and how
these two formulations differ in their solution behaviours: should they be
identical, it seems reasonable to choose the more tractable model depending on
the goals; should they differ significantly, it is reasonable to determine
when and how they differ, which gives insights into the validity of either
case in a given context.

There exist a number of works that consider a multi-species model of the form
taken in either (1.8.a) or (1.8.b), see, e.g., Potts and Lewis (2016a, 2019).
In these works, a linear stability analysis is performed to determine
conditions sufficient for pattern formation to occur. These models are
comparable in that they include a cognitive map through an additional, dynamic
equation, and they also incorporate nonlocal perception. Other models with
nonlocal advective operators have been studied by Ducrot et al. (2018); Giunta
et al. (2021); Jüngel et al. (2022), where some global existence results are
obtained. In Ducrot et al. (2018), fractional Sobolev spaces are utilized in a
one-dimensional torus to establish a global existence result which includes
the possibility of a top-hat kernel; however, the model does not incorporate a
dynamic cognitive map. In Giunta et al. (2021), a global existence result is
establish using a contraction mapping argument, but the regularity
requirements of the nonlocal kernel do not include the top-hat detection
function. In Jüngel et al. (2022), a global existence result is obtained for a
special case of the $n$-species cross-diffusion system considered in Giunta et
al. (2021) which includes the top-hat detection function, but the kernels are
assumed to be in detailed balance, which greatly reduces the applicability for
biological application. Other memory-based movement models has been
investigated in Shi et al. (2020); Song et al. (2022), where the cognitive map
is now given by a nonlocal integral operator in time. In such cases, the
problem is a delay partial differential equation. The stability of coupled
PDE-ODE models have also been studied in works such as Marciniak-Czochra et
al. (2017); Li et al. (2017).

With our models at hand, the major goals of this paper are as follows. First,
we seek to prove the well-posedness of models (1.8.a) and (1.8.b). In
particular, in Section 2 we prove the existence of a unique, global weak
solution when the detection function $G(\cdot)$ satisfies an $L^{p}$-embedding
type condition (see Hypotheses (H3)), which includes the discontinuous top-hat
detection function. This provides an answer to Open Problems 10 and 12 found
in Wang and Salmaniw (2023a), at least for the single species case. We then
shift our attention to the solution behaviour and the potential for pattern
formation at a steady state. In Sections 3-4, we perform a robust stability
and bifurcation analysis to understand the long term behaviour of the solution
as it depends on parameters $d$, $\alpha$, the uptake rate $g(\cdot)$ and the
kernel $G(\cdot)$. While an intuitive understanding of the relevant factors
influencing pattern formation can be gleaned from a less formal linear
stability analysis (see Section 3.2), further care is needed for nonlocal
advective operators as the essential spectrum may be non-empty. This is
different from standard reaction-diffusion systems where the essential
spectrum is empty. In our case, we find that the essential spectrum is
entirely negative, and so has no impact on changes of stability. Section 5 is
then dedicated to a detailed case study with the top-hat detection function.
To explore (perhaps subtle) differences in our formulations, we focus on three
particular forms of uptake $g(\cdot)$ to better understand differences in
fundamental assumptions for the function $g$. Numerical simulations using a
pseudo-spectral method Giunta et al. (2021); Wang and Salmaniw (2023a) are
presented to highlight these differences.

###  1.2 Preliminaries & Hypotheses

Denote $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. Recall that the following
eigenvalue problem

$\begin{cases}-\phi^{\prime\prime}(x)=l\phi(x),\quad\quad\quad\quad\quad
x\in(-L,L),\ \\\ \phi(-L)=\phi(L),\
\phi^{\prime}(-L)=\phi^{\prime}(L),&\end{cases}$ (1.8)

with eigenvalues and eigenfunctions

$\displaystyle l_{\pm n}=\frac{n^{2}\pi^{2}}{L^{2}},\ \ \phi_{\pm
n}(x)=e^{\frac{\pm in\pi}{L}x}=\cos\left(\frac{n\pi}{L}x\right)\pm
i\sin\left(\frac{n\pi}{L}x\right),\ n\in\mathbb{N}_{0}.$ (1.9)

We define the linear spaces

$\displaystyle L^{2}_{per}(-L,L)=\left\\{h\in
L^{2}(-L,L):h=\sum_{n=-\infty}^{\infty}c_{n}\phi_{n}\text{ with
}\sum_{n=-\infty}^{\infty}\lvert c_{n}\rvert^{2}<\infty\right\\},$

and

$H^{2}_{per}(-L,L)=\\{h\in L^{2}_{per}(-L,L):h^{\prime\prime}\in
L^{2}_{per}(-L,L)\\},$

where

$\displaystyle c_{n}=\langle
h,\phi_{n}\rangle=\frac{1}{2L}\int_{-L}^{L}h(x)\phi_{-n}(x)dx.$

We then denote by $X$ and $Y$ the spaces $H^{2}_{per}(-L,L)\times
L^{2}_{per}(-L,L)$ and $L^{2}_{per}(-L,L)\times L^{2}_{per}(-L,L)$,
respectively. We always assume the following for the spatial kernel $G$:

  1. (H0)

$\begin{cases}0\leq G(x)\in L^{1}_{per}(-L,L),\ G(-x)=G(x)\text{ for all
}x\in(-L,L),\cr(2L)^{-1}\int_{-L}^{L}G(y)dy=1.\end{cases}$

The Gaussian, exponential and top-hat detection functions all satisfy (H0) in
$(-L,L)$ with the following respective scaling prefactors:
$\operatorname{erf}(L/\sqrt{2}R)$, $1-e^{-L/R}$, and $2L$. For the stability
and bifurcation analysis performed in Sections 3-4, we assume that the growth
rates $f$ and $g$ satisfy

  1. (H1)

$f(u)\in C^{3}([0,\infty))$, $f(0)=f(1)=0,$ $f^{\prime}(0)>0$,
$f^{\prime}(1)<0$, $f(u)>0$ for $u\in(0,1)$ and $f(u)<0$ for $u>1.$

  2. (H2)

$g(u)\in C^{3}([0,\infty))$, $g(u)>0$ on $(0,\infty)$, $g(0)=0,$ and
$g(1)=\rho>0.$

To establish the well-posedness of the problem, we also assume in addition to
(H0) that the kernel $G(\cdot)$ satisfies the following $L^{p}$-type estimate
for any $R>0$ fixed:

  1. (H3)

$\left\|\overline{z}_{x}\right\|_{L^{p}(\Omega)}\leq
C\left\|z\right\|_{L^{p}(\Omega)}\quad\text{ for all }z\in L^{p}(\Omega),\
1\leq p\leq\infty$.

A prototypical example of $f(u)$ is the logistic function $f(u)=u(1-u)$,
foundational in models of population growth. Biologically-motivated examples
of $g(u)$ include $g(u)=\rho u$, $g(u)=\rho u^{2}$, and $g(u)=\rho
u^{2}/(1+cu)$, which were discussed in the paragraph prior to Equation (1.3).
Note that the hypotheses required in the bifurcation analysis are generally
stronger than those required for well-posedness; for this reason, we state the
sufficient hypotheses for the existence of a solution directly in the
statement of Theorem 1.1.

Throughout this paper, we denote the null space of a linear operator $L$ by
$\mathcal{N}(L)$, the domain of $L$ by $\mathcal{D}(L)$, the range of $L$ by
$\mathcal{R}(L)$, the resolvent set of $L$ by $\mathcal{\rho}(L)$, and the
spectrum of $L$ by $\mathcal{\sigma}(L)$. We always denote by
$Q_{T}:=\Omega\times(0,T)=(-L,L)\times(0,T)$.

###  1.3 Statement of Main Results

Our first result establishes the existence of a unique, nontrivial solution
$(u,k)$. Due to the weak regularity assumption (H0) on the kernel $G(\cdot)$,
we do not expect solutions to be classical necessarily. Denote by $h(u,k)$ the
right hand side of the equation for the map $k$ in either (1.8.a) or (1.8.b).
We call $(u,k)$ a weak solution to either (1.8.a) or (1.8.b) if, given any
test function $\phi_{i}\in L^{2}(0,T;H^{1}(\Omega))$, $i=1,2$, there holds

$\displaystyle\iint_{Q_{T}}u_{t}\phi_{1}{\rm d}x{\rm
d}t+\iint_{Q_{T}}(du_{x}+\alpha u\overline{k}_{x})(\phi_{1})_{x}{\rm d}x{\rm
d}t=\iint_{Q_{T}}f(u)\phi_{1}{\rm d}x{\rm d}t,$ (1.10)
$\displaystyle\iint_{Q_{T}}k_{t}\phi_{2}{\rm d}x{\rm
d}t=\iint_{Q_{T}}h(u,k)\phi_{2}{\rm d}x{\rm d}t,$ (1.11)

and the initial data is satisfied in the sense of $H^{1}(\Omega)$ (in fact,
the initial data will be satisfied in the sense of $C(\overline{\Omega})$ by
the Sobolev embedding). We call a weak solution a global weak solution if
(1.10)-(1.11) holds for any $T>0$.

We have the following well-posedness result for problems (1.8.a) and (1.8.b).

###### Theorem 1.1

Fix $T>0$, $\alpha\in\mathbb{R}\setminus\\{0\\}$, $d,R>0$,
$\mu,\beta,\kappa\geq 0$, and assume that the kernel $G(\cdot)$ satisfies (H0)
and (H3). Suppose that for some $\sigma\in(0,1)$, $f,g\in
C^{2+\sigma}(\mathbb{R}^{+})$ with $f(0)=g(0)=0$. Assume that $f$ satisfies
the bound

$f(z)\leq f^{\prime}(0)z\quad\text{ for all }\quad z\geq 0,$

while $g$ satisfies the bounds

$\displaystyle g(z)$ $\displaystyle\leq N(1+z^{q})\quad\text{ for all }\quad
z\geq 0,$ $\displaystyle\left|g^{\prime}(z)\right|$
$\displaystyle\leq\tilde{N}(1+z^{\tilde{q}})\quad\text{ for all }\quad z\geq
0,$

for some constants $N,\tilde{N}>0$, $q\geq 1$ and $\tilde{q}\geq 0$. Finally,
assume that the initial data $u_{0},k_{0}$ satisfy

$0<u_{0}(x),k_{0}(x)\in W^{1,2}(\Omega)\text{ are periodic in }\Omega.$

Then, there exists a unique, global weak solution $(u,k)$ solving problem
(1.8.a) in the sense of (1.10)-(1.11) satisfying $u\geq 0,k\geq 0$ so long as
there exists $M>0$ so that

$g(z)\leq M(\mu+\beta z)\quad\text{ for all }z\geq 0.$

For problem (1.8.b), there exists a unique, global weak solution in the sense
of (1.10)-(1.11) satisfying $u\geq 0,k\geq 0$ with no further restriction on
$g(\cdot)$ other than ((H2)). Moreover, in either case there holds

$\displaystyle u$ $\displaystyle\in L^{\infty}(0,T;L^{p}(\Omega))\cap
C^{\sigma,\sigma/2}(\overline{Q}_{T}),\quad u_{x},\ u_{t}\in
L^{2}(0,T;L^{2}(\Omega)),$ $\displaystyle k$ $\displaystyle\in
L^{\infty}(0,T;L^{p}(\Omega))\cap C^{\sigma,\sigma/2}(\overline{Q}_{T}),\quad
k_{x},\ k_{t}\in L^{\infty}(0,T;L^{2}(\Omega)),$

for any $1<p\leq\infty$, for some $\sigma\in(0,1/2)$, and the initial data is
satisfied in the sense of $C(\overline{\Omega})$.

###### Remark 1.2

In the theorem above, we generally require some global polynomial growth
control over the memory uptake function $g(\cdot)$. From this result, we see
that problem (1.8.a) is significantly more restrictive than problem (1.8.b) in
terms of further growth conditions on $g(\cdot)$. Indeed, the first case
requires that the memory uptake behaves roughly linearly, particularly for
large arguments, while the second case requires no further growth condition.
From our previous discussion of biologically-motivated forms of $g(\cdot)$, we
see that the forms $g(u)=\rho u$ and $g(u)=\rho u^{2}/(1+cu)$ satisfy the
necessary conditions for either system. On the other hand, quadratic growth
$g(u)=\rho u^{2}$ as described in Potts and Lewis (2016a) satisfies the
conditions for system (1.8.b) but not (1.8.a). This highlights an essential
key difference between these two problems in terms of their well-posedness.

For the detection function $G$ satisfying (H0), the Fourier coefficient
$C_{n}(G)$ is defined for any $n\in\mathbb{N}$ as follows:

$C_{n}(G)=\frac{1}{2L}\int_{-L}^{L}e^{-\frac{in\pi}{L}y}G(y)dy=\frac{1}{2L}\int_{-L}^{L}\cos\left(\frac{n\pi}{L}y\right)G(y)dy,$
(1.12)

For $n\in\mathbb{N}$, if $C_{n}(G)\neq 0$, define

$\alpha_{n}=\frac{-(\mu+\beta)^{2}}{[g^{\prime}(1)(\mu+\beta)-\beta\rho]C_{n}(G)}\left(d-\frac{f^{\prime}(1)}{l_{n}}\right).$
(1.13)

Note that $C_{n}(G)$ could be positive or negative. Define

$\displaystyle\Sigma^{+}:$ $\displaystyle=\\{n\in\mathbb{N}:\alpha_{n}>0\\},\
\Sigma^{-}:=\\{n\in\mathbb{N}:\alpha_{n}<0\\},$ (1.14)
$\displaystyle\alpha_{r}:$ $\displaystyle=\min_{n\in\Sigma^{+}}\alpha_{n},\ \
\ \alpha_{l}:=\max_{n\in\Sigma^{-}}\alpha_{n}.$

Note that $\alpha_{r}>0>\alpha_{l}$ as
$\sum_{-\infty}^{\infty}|C_{n}(G)|^{2}<\infty$. Then we have the following
theorem regarding the stability of the unique constant positive steady state
$U_{*}=(1,\rho/(\mu+\beta))$ with respect to (1.8.a).

###### Theorem 1.3

Assume that assumptions (H0)-(H2) are satisfied, and let
$\Sigma^{+},\Sigma^{-},\alpha_{l},\alpha_{r}$ be defined as in (1.14). Then

  1. (i)

The constant steady state solution $U_{*}$ is locally asymptotically stable
with respect to (1.8.a) if $\alpha_{l}<\alpha<\alpha_{r}$.

  2. (ii)

The constant steady state solution $U_{*}$ is unstable with respect to (1.8.a)
if $\alpha<\alpha_{l}$ or $\alpha>\alpha_{r}$.

Similarly we define

$\widehat{\alpha}_{n}=\frac{-(\rho+\mu+\beta)^{2}}{\kappa[g^{\prime}(1)(\mu+\beta)-\beta\rho]C_{n}(G)}\left(d-\frac{f^{\prime}(1)}{l_{n}}\right),$
(1.15)

and

$\displaystyle\widehat{\Sigma}^{+}:$
$\displaystyle=\\{n\in\mathbb{N}:\widehat{\alpha}_{n}>0\\},\
\widehat{\Sigma}^{-}:=\\{n\in\mathbb{N}:\widehat{\alpha}_{n}<0\\},$ (1.16)
$\displaystyle\widehat{\alpha}_{r}:$
$\displaystyle=\min_{n\in\widehat{\Sigma}^{+}}\widehat{\alpha}_{n},\ \ \
\widehat{\alpha_{l}}:=\max_{n\in\widehat{\Sigma}^{-}}\widehat{\alpha}_{n}.$

Then we have the stability results for the unique constant positive steady
state $\widehat{U_{*}}=(1,\rho\kappa/(\rho+\mu+\beta))$ with respect to
(1.8.b).

###### Theorem 1.4

Assume that assumptions (H0)-(H2) are satisfied, and let
$\widehat{\Sigma}^{+},\widehat{\Sigma}^{-},\widehat{\alpha}_{l},\widehat{\alpha}_{r}$
be defined as in (1.16). Then

  1. (i)

The constant steady state solution $\widehat{U}_{*}$ is locally asymptotically
stable with respect to (1.8.b) if
$\widehat{\alpha}_{l}<\alpha<\widehat{\alpha}_{r}$.

  2. (ii)

The constant steady state solution $\widehat{U}_{*}$ is unstable with respect
to (1.8.b) if $\alpha<\widehat{\alpha}_{l}$ or $\alpha>\widehat{\alpha}_{r}$.

The quantities $\alpha_{n}$ ($\widehat{\alpha}_{n}$) defined in (1.13)
((1.15)) are the critical parameter values such that the stability of the
spatially-constant steady state changes, and they are also bifurcation points
for (1.8.a) ((1.8.b)) where spatially non-homogeneous steady state solutions
bifurcate from the constant ones as found in the following Theorems.

###### Theorem 1.5

Assume that assumptions (H0)-(H2) are satisfied, and $n\in{\mathbb{N}}$ such
that $C_{n}(G)\neq 0$. Then near $(\alpha,U)=(\alpha_{n},U_{*})$, problem
(1.8.a) has a line of trivial solutions
$\Gamma_{0}:=\\{(\alpha,U_{*}):\alpha\in{\mathbb{R}}\\}$ and a family of non-
constant steady state solutions bifurcating from $\Gamma_{0}$ at
$\alpha=\alpha_{n}$ in a form of

$\Gamma_{n}:=\\{(\alpha_{n}(s),u_{n}(s,\cdot),k_{n}(s,\cdot)):-\delta<s<\delta\\}$
(1.17)

with

$\begin{cases}\alpha_{n}(s)=\alpha_{n}+\alpha_{n}^{\prime}(0)s+o(s),\\\
\displaystyle
u_{n}(s,x)=1+s\cos\left(\dfrac{n\pi}{L}x\right)+s^{2}z_{1n}(s,x),\\\
\displaystyle
k_{n}(s,x)=\dfrac{\rho}{\mu+\beta}+s\dfrac{g^{\prime}(1)(\mu+\beta)-\beta\rho}{(\mu+\beta)^{2}}\cos\left(\dfrac{n\pi}{L}x\right)+s^{2}z_{2n}(s,x),\end{cases}$
(1.18)

where $z_{n}(s)=(z_{1n}(s,\cdot),z_{2n}(s,\cdot))$ satisfies
$\displaystyle\lim\limits_{s\rightarrow 0}\lVert z_{n}(s)\rVert=0$. Moreover
the set of steady state solutions of (1.8.a) near $(\alpha_{n},U_{*})$
consists precisely of the curves $\Gamma_{0}$ and $\Gamma_{n}$.

###### Theorem 1.6

Assume that assumptions (H0)-(H2) are satisfied, and $n\in{\mathbb{N}}$ such
that $C_{n}(G)\neq 0$. Then near
$(\widehat{\alpha},\widehat{U})=(\widehat{\alpha}_{n},\widehat{U}_{*})$,
equation (1.8.b) has a line of trivial solutions
$\widehat{\Gamma}_{0}:=\\{(\alpha,\widehat{U}_{*}):\alpha\in{\mathbb{R}}\\}$
and a family of non-constant steady state solutions bifurcating from
$\widehat{\Gamma}_{0}$ at $\alpha=\widehat{\alpha}_{n}$ in a form of

$\widehat{\Gamma}_{n}:=\\{(\widehat{\alpha}_{n}(s),\widehat{u}_{n}(s,\cdot),\widehat{k}_{n}(s,\cdot)):-\delta<s<\delta\\}$
(1.19)

with

$\begin{cases}\widehat{\alpha}_{n}(s)=\widehat{\alpha}_{n}+\widehat{\alpha}_{n}^{\prime}(0)s+o(s),\\\
\displaystyle\widehat{u}_{n}(s,x)=1+s\cos\left(\dfrac{n\pi}{L}x\right)+s^{2}\widehat{z}_{1n}(s,x),\\\
\displaystyle\widehat{k}_{n}(s,x)=\dfrac{\rho}{\rho+\mu+\beta}+\kappa
s\dfrac{g^{\prime}(1)(\mu+\beta)-\beta\rho}{(\mu+\beta+\rho))^{2}}\cos\left(\dfrac{n\pi}{L}x\right)+s^{2}\widehat{z}_{2n}(s,x),\end{cases}$
(1.20)

where
$\widehat{z}_{n}(s)=(\widehat{z}_{1n}(s,\cdot),\widehat{z}_{2n}(s,\cdot))$
satisfies $\displaystyle\lim\limits_{s\rightarrow
0}\lVert\widehat{z}_{n}(s)\rVert=0$. Moreover the set of steady state
solutions of (1.8.b) near $(\widehat{\alpha}_{n},\widehat{U}_{*})$ consists
precisely of the curves $\widehat{\Gamma}_{0}$ and $\widehat{\Gamma}_{n}$.

We also classify the nature of the bifurcation at these critical values, see
Theorems 4.4 and 4.5. Together, these results show that the central quantity
governing spontaneous pattern formation is the advective strength towards or
away from memorised areas, encapsulated in $\alpha$. Since
$\alpha_{l}<0<\alpha_{r}$ holds necessarily from the assumptions made, the key
driver of pattern formation is that $\alpha$ is of sufficient magnitude. If
$\alpha$ is negative, then we have attraction towards remembered areas,
similar to many nonlocal models of biological aggregation (e.g. Carrillo et
al. (2019)). Some examples are found in Figures 2, 5 and 8. On the flip-side,
positive $\alpha$ indicates repulsion from remembered areas and leads to
patterns such as in Figures 3, 6 and 9. Interestingly, there is a lack of
symmetry in the sense that $-\alpha_{l}\neq\alpha_{r}$ in general. In fact,
$\left|\alpha_{l}\right|$ and $\alpha_{r}$ do not even remain ordered! This
can be seen in Figures 1, 4 and 7.

Moreover, Theorems 1.5 and 1.6 show that, close to the bifurcation point, the
steady state consists of a single cosine wave, and Theorems 4.4 and 4.5 give
conditions on their stability. An example of the stable case is shown in
Figures 5-6, where a cosine wave emerges just beyond the bifurcation point,
but diverges from this description as the bifurcation parameter is increased
further. For the unstable case we do not see the small-amplitude cosine wave
at all as the bifurcation threshold is crossed. Rather, the solution jumps to
a higher amplitude pattern as found in Figure 3.

##  2 Well-posedness

In this section we prove the existence of a unique global solution to a
problem more general than system (1.8.a) or (1.8.b) for detection functions
$G(\cdot)$ satisfying (H0) and (H3), which includes the top-hat kernel. The
restrictions on the growth terms $f(\cdot)$ and $g(\cdot)$ are compatible with
the choices made in Section 5. The challenge is in the treatment of the
(potentially) discontinuous kernel appearing inside the nonlocal advection
term. To overcome these difficulties, we abuse a useful ‘embedding’ property
of the top-hat kernel in one spatial dimension. This allows one to obtain a
priori estimates on the solution $k(x,t)$ from which we obtain appropriate
uniform bounds on $u(x,t)$ and higher derivatives. We begin with some
preliminary estimates.

###  2.1 Preliminary Estimates

In general, we assume $G(\cdot)$ satisfies (H3); we first show this holds for
the top-hat kernel.

###### Lemma 2.1

Let $1\leq p\leq\infty$ and fix $T>0$. Suppose $z(\cdot,t)\in L^{p}(\Omega)$
is periodic in $\Omega$ for almost every $t\in[0,T]$. Denote by
$\overline{z}_{x}(\cdot,t)$ the spatial convolution (1.4) of $z_{x}$ with the
top-hat detection function (1.7). Then, for almost every $t\in[0,T]$ there
holds

$\displaystyle\left\|\overline{z}_{x}(\cdot,t)\right\|_{L^{p}(\Omega)}$
$\displaystyle\leq\frac{1}{R}\left\|z(\cdot,t)\right\|_{L^{p}(\Omega)}.$ (2.1)

In particular, we have that

$\displaystyle\operatorname*{ess\,sup}_{t\in(0,T)}\left\|\overline{z}_{x}(\cdot,t)\right\|_{L^{p}(\Omega)}$
$\displaystyle\leq
R^{-1}\operatorname*{ess\,sup}_{t\in(0,T)}\left\|z(\cdot,t)\right\|_{L^{p}(\Omega)},$
(2.2)

for any $T>0$ fixed.

###### Remark 2.2

If $z(\cdot,t)$ is continuous, we may replace “almost every" with “every" and
$\operatorname*{ess\,sup}$ with $\sup$. This will be the case in the
forthcoming results.

###### Proof 1

First, we drop the dependence on $t$ for notational brevity. The result
essentially follows from an elementary inequality and the fact that

$\overline{z}_{x}=\frac{z(x+R)-z(x-R)}{2R}$

when $G(\cdot)$ is the top hat detection function. Consequently,

$\displaystyle\left\|\overline{z}_{x}\right\|_{L^{p}(\Omega)}^{p}$
$\displaystyle=\frac{1}{(2R)^{p}}\int_{\Omega}\left|z(x+R)-z(x-R)\right|^{p}{\rm
d}x$
$\displaystyle\leq\frac{2^{p-1}}{(2R)^{p}}\int_{\Omega}\left(\left|z(x+R)\right|^{p}+\left|z(x-R)\right|^{p}\right){\rm
d}x\leq\frac{1}{R^{p}}\left\|z\right\|_{L^{p}(\Omega)}^{p},$ (2.3)

where we have used the periodicity of $z$ in $\Omega$. This proves (2.1).
Taking the $p^{\text{th}}$ roots of both sides followed by the supremum over
$t\in(0,T)$ yields (2.2).

###### Remark 2.3

The same $L^{p}$-type estimate holds for the Gaussian and exponential
detection functions (1.5)-(1.6) as well. These cases are easier since the
kernels themselves are appropriately differentiable and bounded. We omit the
details.

Next, we obtain $L^{p}(\Omega)$ bounds on a function $k(x,t)$ when $k$ solves
a linear, first order differential equation for each $x\in\Omega$.

###### Lemma 2.4

Let $0\lneqq w(x,t)\in C^{1,1}(\overline{Q}_{T})\cap L^{1,1}(Q_{T})$ be
periodic in $\Omega$ for all $t\in(0,T)$ and assume $1<p\leq\infty$. For each
$x\in\Omega$, let $k(x,\cdot)$ solve the ordinary differential equation

$\frac{dk}{dt}=g_{1}(w)-g_{2}(w)k$ (2.4)

where $g_{1},g_{2}\in C^{1}(\mathbb{R}^{+})$ are nonnegative and
$k(x,0)=k_{0}(x)\in W^{1,2}(\Omega)$. Then, if there exists $M>0$ such that

$g_{1}(z)\leq Mg_{2}(z)\quad\text{ for all }z\geq 0,$ (2.5)

there holds

$\sup_{t\in[0,T]}\left\|k(\cdot,t)\right\|_{L^{\infty}(\Omega)}\leq
M+\left\|k_{0}\right\|_{L^{\infty}(\Omega)}$ (2.6)

###### Proof 2

First, note that by solving the differential equation directly, $k(x,\cdot)\in
C^{1}([0,T])$. By the smoothness of $g_{i}(\cdot)$, $i=1,2$ and the
boundedness of $w(x,t)$ in $Q_{T}$, $k(\cdot,t)\in L^{p}(\Omega)$ for any
$p>1$, for all $t\in(0,T)$. Taking the time derivative of
$\tfrac{1}{p}\left\|k(\cdot,t)\right\|_{L^{p}(\Omega)}^{p}$ gives

$\displaystyle\frac{1}{p}\frac{d}{dt}\int_{\Omega}k^{p}{\rm d}x$
$\displaystyle=\int_{\Omega}k^{p-1}g_{1}(w){\rm
d}x-\int_{\Omega}k^{p}g_{2}(w){\rm d}x.$ (2.7)

We now apply Young’s inequality. To this end, we carefully rewrite as

$\displaystyle k^{p-1}g_{1}(w)$
$\displaystyle=k^{p-1}g_{2}(w)^{(p-1)/p}\cdot\frac{g_{1}(w)}{g_{2}(w)^{(p-1)/p}}$
$\displaystyle\leq\frac{1}{p_{1}}\left(k^{p-1}g_{2}(w)^{(p-1)/p}\right)^{p_{1}}+\frac{1}{q_{1}}\left(\frac{g(w)}{g_{2}(w)^{(p-1)/p}}\right)^{q_{1}},$
(2.8)

where $p_{1},q_{1}>1$ satisfy $p_{1}^{-1}+q_{1}^{-1}=1$. Choosing
$p_{1}=p/(p-1)$ and $q_{1}=p$, equation (2.7) then becomes

$\displaystyle\frac{d}{dt}\int_{\Omega}k^{p}{\rm d}x$
$\displaystyle\leq\int_{\Omega}\left(\frac{g_{1}(w)}{g_{2}(w)}\right)^{p}g_{2}(w){\rm
d}x$ (2.9)

Using bound (2.5) we then have

$\displaystyle\frac{d}{dt}\int_{\Omega}k^{p}{\rm d}x$ $\displaystyle\leq
M^{p}\left\|g_{2}(w(\cdot,t))\right\|_{L^{1}(\Omega)}.$ (2.10)

Integrating both sides from $0$ to $t$ yields

$\displaystyle\left\|k(\cdot,t)\right\|_{L^{p}(\Omega)}^{p}$
$\displaystyle\leq
M^{p}\left\|g_{2}(w)\right\|_{L^{1,1}(Q_{T})}+\left\|k_{0}\right\|_{L^{p}(\Omega)}^{p}.$
(2.11)

Taking $p^{th}$ roots of both sides and sending $p\to\infty$ leaves

$\displaystyle\left\|k(\cdot,t)\right\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq M+\left\|k_{0}\right\|_{L^{\infty}(\Omega)}.$ (2.12)

Taking the supremum over $t\in(0,T)$ yields (2.6), completing the proof.

We also highlight the following properties of $k(x,t)$ inherited by the
function $w(x,t)$, a simple consequence of solving the ordinary differential
equation.

###### Proposition 2.5

Suppose $k(x,\cdot)$ solves (2.4) with $w\in C^{1,1}(\overline{Q}_{T})$
periodic in $\Omega$. Then, $k(x,t)\in C^{1,2}(\overline{Q}_{T})$. Moreover,
$k(\cdot,t)$ is periodic in $\Omega$ for all $t>0$.

Finally, we obtain $L^{p}$ estiates on the time/space derivatives $k_{t}$ and
$k_{x}$.

###### Theorem 2.6

Assume the same conditions as in Lemma 2.4 hold. Assume also that there exists
$N>0$ and $q\geq 1$ fixed so that

$g_{2}(z)\leq N(1+z^{q})\quad\text{for all }z\geq 0.$ (2.13)

Assume in addition that $w_{x}\in L^{2,2}(Q_{T})$ and
$\sup_{t\in(0,T)}\left\|w(\cdot,t)\right\|_{L^{p}(\Omega)}<\infty$ for all
$p\geq 1$. Then there holds

$\displaystyle\sup_{t\in(0,T)}\left\|k_{t}(\cdot,t)\right\|_{L^{p}(\Omega)}\leq
4N\left(M+\sup_{t\in(0,T)}\left\|k(\cdot,t)\right\|_{L^{\infty}(\Omega)}\right)\left(\left|\Omega\right|^{1/p}+\sup_{t\in(0,T)}\left\|w(\cdot,t)\right\|_{L^{pq}(\Omega)}^{q}\right)$
(2.14)

for any $p\in(1,\infty)$. Moreover, if for some $\tilde{N}>0$, $\tilde{q}\geq
0$ fixed we have that

$\displaystyle\left|g_{1}^{\prime}(z)\right|,\left|g_{2}^{\prime}(z)\right|\leq\tilde{N}(1+z^{\tilde{q}})\quad\text{
for all }z\geq 0,$ (2.15)

then there exists a constant $C>0$ depending only on $T$, $\tilde{N}$, and
$\tilde{q}$ so that

$\displaystyle\sup_{t\in(0,T)}\left\|k_{x}(\cdot,t)\right\|_{L^{p}(\Omega)}$
$\displaystyle\leq
C\sup_{t\in(0,T)}\left\|(1+w^{\tilde{q}})(\cdot,t)\right\|_{L^{2p/(2-p)}(\Omega)}\left\|w_{x}\right\|_{L^{2,2}(Q_{T})}+\left\|(k_{0})_{x}\right\|_{L^{p}(\Omega)},$
(2.16)

for any $p\in(1,2)$ whenever $\tilde{q}>0$, and any $p\in(1,2]$ whenever
$\tilde{q}=0$.

###### Proof 3

Integrating $\left|k_{t}\right|^{p}$ over $\Omega$, applying an elementary
inequality, and using (2.5) yields

$\displaystyle\int_{\Omega}\left|k_{t}\right|^{p}{\rm d}x$
$\displaystyle=\int_{\Omega}\left|g_{1}(w)-g_{2}(w)k\right|^{p}{\rm d}x$
$\displaystyle\leq
2^{p-1}\int_{\Omega}\left(\left|g_{1}(w)\right|^{p}+\left|g_{2}(w)\right|^{p}\left|k\right|^{p}\right){\rm
d}x.$ $\displaystyle\leq
2^{p-1}\int_{\Omega}\left(M^{p}+\left|k\right|^{p}\right)\left|g_{2}(w)\right|^{p}{\rm
d}x$ (2.17)

Estimating further, we use the bound on $k$ along with bound (2.13) and the
same elementary inequality to see that

$\displaystyle\int_{\Omega}\left|k_{t}\right|^{p}{\rm d}x$ $\displaystyle\leq
2^{p-1}\left(M^{p}+\left\|k(\cdot,t)\right\|_{L^{\infty}(\Omega)}^{p}\right)\int_{\Omega}\left|g_{2}(w)\right|^{p}{\rm
d}x$ $\displaystyle\leq
4^{p-1}N^{p}\left(M^{p}+\left\|k(\cdot,t)\right\|_{L^{\infty}(\Omega)}^{p}\right)\int_{\Omega}\left(1+w^{pq}\right){\rm
d}x$ $\displaystyle\leq
4^{p}N^{p}\left(M^{p}+\left\|k(\cdot,t)\right\|_{L^{\infty}(\Omega)}^{p}\right)\left(\left|\Omega\right|+\left\|w(\cdot,t)\right\|^{pq}_{L^{pq}(\Omega)}\right).$

Thus, we take the $p^{th}$ roots of both sides followed by the supremum over
$t\in(0,T)$ to obtain (2.14). This completes the first part of the proof.

Next, we obtain $L^{p}$ bounds on $k_{x}$ for any $p\in(1,2)$. Solving the
ordinary differential equation, we may compute $k_{x}(x,t)$ as follows:

$\displaystyle k_{x}(x,t)$ $\displaystyle=\frac{\partial}{\partial
x}\left(\int_{0}^{t}e^{-\int_{s}^{t}g_{2}(w){\rm d}\xi}g_{1}(w){\rm
d}s+k_{0}(x)\right)$ $\displaystyle=\int_{0}^{t}e^{-\int_{s}^{t}g_{2}(w){\rm
d}\xi}\left(g_{1}^{\prime}(w)w_{x}-\int_{s}^{t}g_{2}^{\prime}(w)w_{x}d\xi\right){\rm
d}s+(k_{0})_{x}.$ (2.18)

Therefore, estimating crudely and using bound (2.15) there holds

$\displaystyle\left|k_{x}\right|$
$\displaystyle\leq\int_{0}^{T}\left(\left|g_{1}^{\prime}(w)\right|\left|w_{x}\right|+\int_{0}^{T}\left|g^{\prime}_{2}(w)\right|\left|w_{x}\right|{\rm
d}\xi\right){\rm d}s+\left|(k_{0})_{x}\right|$
$\displaystyle\leq\int_{0}^{T}\left(\left|g_{1}^{\prime}(w)\right|+T\left|g^{\prime}_{2}(w)\right|\right)\left|w_{x}\right|{\rm
d}s+\left|(k_{0})_{x}\right|$
$\displaystyle\leq\tilde{N}(1+T)\int_{0}^{T}\left(1+w^{\tilde{q}}\right)\left|w_{x}\right|{\rm
d}s+\left|(k_{0})_{x}\right|.$ (2.19)

Raising both sides to the power $p$, integrating over $\Omega$ and estimating
via an elementary application of Hölder’s inequality in the temporal domain
yields

$\displaystyle\int_{\Omega}\left|k_{x}\right|^{p}{\rm d}x$
$\displaystyle\leq\tilde{N}^{p}(1+T)^{p}\int_{\Omega}\left(\int_{0}^{T}(1+w^{\tilde{q}})\left|w_{x}\right|{\rm
d}s\right)^{p}{\rm d}x+\left\|(k_{0})_{x}\right\|^{p}_{L^{p}(\Omega)}$
$\displaystyle\leq\tilde{N}^{p}(1+T)^{p}T^{p-1}\iint_{Q_{T}}(1+w^{\tilde{q}})^{p}\left|w_{x}\right|^{p}{\rm
d}x{\rm d}s+\left\|(k_{0})_{x}\right\|^{p}_{L^{p}(\Omega)}.$ (2.20)

We now apply Hölder’s inequality in the spatial domain as follows:

$\int_{\Omega}(1+w^{\tilde{q}})^{p}\left|w_{x}\right|^{p}{\rm
d}x\leq\left(\int_{\Omega}\left|w_{x}\right|^{pp_{1}}{\rm
d}x\right)^{1/p_{1}}\left(\int_{\Omega}(1+w^{\tilde{q}})^{pq_{1}}{\rm
d}x\right)^{1/q_{1}},$

where we again choose $p_{1}=2/p>1$ so that $q_{1}=2/(2-p)>1$. Simplifying and
taking the supremum over $t\in(0,T)$ for the lower order term, we find

$\int_{\Omega}(1+w^{\tilde{q}})^{p}\left|w_{x}\right|^{p}{\rm
d}x\leq\left\|w_{x}(\cdot,t)\right\|_{L^{2}(\Omega)}^{p}\sup_{t\in(0,T)}\left\|(1+w^{\tilde{q}})(\cdot,t)\right\|_{L^{2p/(2-p)}(\Omega)}^{p}$

(3) then becomes

$\displaystyle\int_{\Omega}\left|k_{x}\right|^{p}{\rm d}x$
$\displaystyle\leq\tilde{N}^{p}(1+T)^{p}T^{p-1}\sup_{t\in(0,T)}\left\|(1+w^{\tilde{q}})(\cdot,t)\right\|_{L^{2p/(2-p)}(\Omega)}^{p}\int_{0}^{T}\left\|w_{x}(\cdot,s)\right\|_{L^{2}(\Omega)}^{p}{\rm
d}s+\left\|(k_{0})_{x}\right\|_{L^{p}(\Omega)}.$ (2.21)

Finally, we apply Hölder’s inequality in the temporal domain once more as
follows:

$\int_{0}^{T}\left\|w_{x}(\cdot,s)\right\|_{L^{2}(\Omega)}^{p}{\rm d}s\leq
T^{1-p/2}\left\|w_{x}\right\|_{L^{2,2}(Q_{T})}^{p},$

whence (2.21) becomes

$\displaystyle\int_{\Omega}\left|k_{x}\right|^{p}{\rm d}x$
$\displaystyle\leq\tilde{N}^{p}(1+T)^{p}T^{p/2}\sup_{t\in(0,T)}\left\|(1+w^{\tilde{q}})(\cdot,t)\right\|_{L^{2p/(2-p)}(\Omega)}^{p}\left\|w_{x}\right\|_{L^{2,2}(Q_{T})}^{p}+\left\|(k_{0})_{x}\right\|_{L^{p}(\Omega)}^{p}.$
(2.22)

Taking the $p^{th}$ roots of both sides followed by the supremum over
$t\in(0,T)$ yields (2.16), valid for any $p\in(1,2)$. Finally, if
$\tilde{q}=0$, the dependence on $w^{\tilde{q}}$ vanishes and the bound holds
for $p=2$, completing the proof.

###  2.2 Existence & Uniqueness

We are now prepared to use these preliminary estimates to prove the existence
of a weak solution. Much of the heavy lifting is now complete. What remains is
to construct an appropriate sequence of approximate solutions and use our
previously obtained estimates to extract a convergent subsequence.

######  Proof 4 (Proof of Theorem 1.1)

We prove the existence of a global weak solution to the following general
system subject to periodic boundary conditions:

$\displaystyle\begin{cases}u_{t}=du_{xx}+\alpha(u\overline{k}_{x})_{x}+f(u),\quad&\text{
in }Q_{T},\cr k_{t}=g_{1}(u)-g_{2}(u)k\quad&\text{ in }Q_{T},\end{cases}$
(2.23)

where systems (1.8.a) and (1.8.b) are obtained by choosing $g_{1}(u):=g(u)$,
$g_{2}(u):=\mu+\beta u$ or $g_{1}(u):=\kappa g(u)$, $g_{2}(u):=\mu+\beta
u+g(u)$, respectively. First we construct a sequence of approximate solutions
via the following iteration scheme:

$\displaystyle\begin{cases}(u_{n})_{t}=d(u_{n})_{xx}+\alpha(u_{n}(\overline{k_{n}})_{x})_{x}+f(u_{n}),\quad&\text{
in }Q_{T},\cr(k_{n})_{t}=g_{1}(u_{n-1})-g_{2}(u_{n-1})k_{n}\quad&\text{ in
}Q_{T},\end{cases}$ (2.24)

for $n\geq 2$, where we choose $(u_{n}(x,0),k_{n}(x,0))=(u_{0}(x),k_{0}(x))$
for each $n$. Note carefully that $u_{n}=u_{n}(x,t)$ is a function defined in
$\overline{Q}_{T}$ for all $n\geq 1$, whereas $u_{0}=u_{0}(x)$ denotes the
fixed initial data of the original problem. The same holds for
$\\{k_{n}\\}_{n\geq 2}$, each of which are defined over $\overline{Q}_{T}$;
notice also that we do not refer to $k_{1}(x,t)$ as we require only
$u_{1}(x,t)$ to initiate.

Through this construction, we generate a sequence of solutions
$\\{(u_{n},k_{n})\\}_{n\geq 2}$. More precisely, we choose the initial iterate
$0<u_{1}(x,t)\in C^{2+\sigma,1+\sigma/2}(\overline{Q}_{T})$ for some
$\sigma\in(0,1)$. By solving differential equation for $k_{2}(x,t)$, the
dependence of $k_{2}(x,t)$ on the sufficiently regular functions $u_{1}(x,t)$
and $g_{i}(\cdot)$, $i=1,2$, ensures that $k_{2}\in
C^{2+\sigma,1+\sigma/2}(\overline{Q}_{T})$ for some (possibly smaller)
$\sigma\in(0,1)$ as well. Furthermore, the positivity of $u_{1}$ and
$u_{1}(x,0)=u_{0}(x)$ ensures that $k_{2}\gneqq 0$ in $Q_{T}$. Then, the
existence of a nonnegative, nontrivial solution $u_{2}(x,t)\in
C^{2+\sigma,1+\sigma/2}(\overline{Q}_{T})$ follows from the classical theory
of parabolic equations since it is a second order, semi-linear parabolic
equation with Hölder continuous coefficients Ladyzhenskaiâ et al. (1968).
Therefore, there exists a nonnegative, nontrivial classical solution pair
$(u_{2},k_{2})$ each belonging to $C^{2+\sigma,1+\sigma/2}(\overline{Q}_{T})$
for some $\sigma\in(0,1)$. One may then proceed inductively, proving the
existence of a nonnegative, nontrivial classical solution pair $(u_{n},k_{n})$
for any $n\geq 3$ using the regularity of the previous iterate $u_{n-1}(x,t)$.
We now seek uniform bounds in a weaker setting.

To this end, fix $n\geq 2$. It is easy to obtain $L^{1}$-bounds on $u_{n}$ as
follows:

$\displaystyle\frac{{\rm d}}{{\rm d}t}\int_{\Omega}u_{n}{\rm
d}x=\int_{\Omega}f(u_{n}){\rm d}x\leq f^{\prime}(0)\int_{\Omega}u_{n}{\rm
d}x,$ (2.25)

where we have integrated by parts, applied the boundary conditions, and used
the assumed bound $f(z)\leq f^{\prime}(0)z$ for all $z\geq 0$. Grönwall’s
inequality implies that

$\left\|u_{n}(\cdot,t)\right\|_{L^{1}(\Omega)}\leq
e^{f^{\prime}(0)T}\left\|u_{0}\right\|_{L^{1}(\Omega)}.$

Thus, integrating over $(0,T)$ yields

$\displaystyle\left\|u_{n}\right\|_{L^{1,1}(Q_{T})}\leq
Te^{f^{\prime}(0)T}\left\|u_{0}\right\|_{L^{1}(\Omega)},$ (2.26)

and so $\\{u_{n}\\}_{n\geq 1}$ is uniformly bounded in $L^{1,1}(Q_{T})$ for
any $T>0$ fixed.

Return now to the equation for $k_{n}$. The smoothness of the iterates $u_{n}$
allows us to apply Lemma 2.4, giving us

$\displaystyle\sup_{t\in(0,T)}\left\|k_{n}(\cdot,t)\right\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq M+\left\|k_{0}\right\|_{L^{\infty}(\Omega)}.$ (2.27)

Note that $k_{0}\in W^{1,2}(\Omega)\Rightarrow k_{0}\in L^{\infty}(\Omega)$ by
the Sobolev embedding. Then, Lemma 2.1 paired with estimate (2.27) implies
that

$\displaystyle\sup_{t\in(0,T)}\left\|(\overline{k_{n}})_{x}(\cdot,t)\right\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq
R^{-1}\sup_{t\in(0,T)}\left\|k_{n}(\cdot,t)\right\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq
R^{-1}\left(M+\left\|k_{0}\right\|_{L^{\infty}(\Omega)}\right)=:C_{1}$ (2.28)

Now we seek $L^{p}$-bounds on the iterates $u_{n}$. Fix $p\geq 2$. Taking the
time derivative of
$\tfrac{1}{p}\left\|u_{n}(\cdot,t)\right\|_{L^{p}(\Omega)}^{p}$, integrating
by parts yields and using the bound for $f(\cdot)$ yields

$\displaystyle\frac{1}{p}\frac{d}{dt}\int_{\Omega}u_{n}^{p}{\rm d}x$
$\displaystyle=\int_{\Omega}u_{n}^{p-1}(d(u_{n})_{x}+\alpha
u_{n}(\overline{k_{n}})_{x})_{x}{\rm d}x+\int_{\Omega}u_{n}^{p-1}f(u_{n}){\rm
d}x$ $\displaystyle\leq f^{\prime}(0)\int_{\Omega}u_{n}^{p}{\rm
d}x-d(p-1)\int_{\Omega}u_{n}^{p-2}\left|(u_{n})_{x}\right|^{2}{\rm d}x$
$\displaystyle\
+\left|\alpha\right|(p-1)\int_{\Omega}u_{n}^{p-1}\left|(u_{n})_{x}\right|\left|(\overline{k_{n}})_{x}\right|{\rm
d}x.$ (2.29)

We now use (4) and Cauchy’s inequality with epsilon to control the third term
by the second term on the right hand side of (4). To this end, we estimate

$\displaystyle\left|\alpha\right|u_{n}^{p-1}\left|(u_{n})_{x}\right|\left|(\overline{k}_{n})_{x}\right|$
$\displaystyle\leq\left|\alpha\right|C_{1}u_{n}^{(p-2)/2}\left|(u_{n})_{x}\right|\left|u_{n}\right|^{p/2}$
$\displaystyle\leq\left|\alpha\right|C_{1}\left(\frac{\varepsilon}{2}u_{n}^{p}+\frac{1}{2\varepsilon}u_{n}^{p-2}\left|(u_{n})_{x}\right|^{2}\right),$
(2.30)

where we choose $\varepsilon=d^{-1}\left|\alpha\right|C_{1}$. Paired with (4),
this leaves

$\displaystyle\frac{1}{p}\frac{d}{dt}\int_{\Omega}u_{n}^{p}{\rm d}x$
$\displaystyle\leq-\frac{d}{2}\int_{\Omega}u_{n}^{p-2}\left|(u_{n})_{x}\right|^{2}{\rm
d}x+(f^{\prime}(0)+C_{1}^{2}\alpha^{2}d^{-1}(p-1))\int_{\Omega}u_{n}^{p}{\rm
d}x$ (2.31)

Therefore, dropping the negative term and applying Grönwall’s inequality
yields

$\left\|u_{n}(\cdot,t)\right\|^{p}_{L^{p}(\Omega)}\leq
e^{p(f^{\prime}(0)+C_{1}^{2}\alpha^{2}d^{-1}(p-1))T}\left\|u_{0}\right\|^{p}_{L^{p}(\Omega)}.$

Taking $p^{\text{th}}$ roots followed by the supremum over $t\in(0,T)$ yields
the estimate

$\displaystyle\sup_{t\in(0,T)}\left\|u_{n}(\cdot,t)\right\|_{L^{p}(\Omega)}$
$\displaystyle\leq
e^{(f^{\prime}(0)+C_{1}^{2}\alpha^{2}d^{-1}(p-1))T}\left\|u_{0}\right\|_{L^{p}(\Omega)}$
$\displaystyle=:C_{2},$ (2.32)

noting that the exponent depends critically on $p$. Next, we return to (2.31)
for the case $p=2$. Upon rearrangement, we apply estimate (4) to obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Omega}u_{n}^{2}{\rm
d}x+\frac{d}{2}\int_{\Omega}\left|(u_{n})_{x}\right|^{2}{\rm d}x$
$\displaystyle\leq\left(f^{\prime}(0)+C_{1}^{2}\alpha^{2}d^{-1}\right)\int_{\Omega}u_{n}^{2}{\rm
d}x$ $\displaystyle\leq
C_{2}^{2}\left(f^{\prime}(0)+C_{1}^{2}\alpha^{2}d^{-1}\right)$
$\displaystyle=:C_{3}.$ (2.33)

Integrating both sides from $0$ to $T$ yields

$\displaystyle\frac{1}{2}\left(\left\|u_{n}(\cdot,T)\right\|_{L^{2}(\Omega)}^{2}-\left\|u_{0}\right\|_{L^{2}(\Omega)}^{2}+d\left\|(u_{n})_{x}\right\|_{L^{2,2}(Q_{T})}^{2}\right)\leq
C_{3}T.$ (2.34)

Ignoring the positive term on the left hand side, we extract the desired
estimate for $(u_{n})_{x}$:

$\displaystyle\left\|(u_{n})_{x}\right\|_{L^{2,2}(Q_{T})}^{2}$
$\displaystyle\leq
d^{-1}\left(2C_{3}T+\left\|u_{0}\right\|_{L^{2}(\Omega)}\right)=:C_{4}^{2}.$
(2.35)

We now immediately have the boundedness of $(k_{n})_{x}$ and $(k_{n})_{t}$ for
any $n\geq 2$ in some $L^{p}$ spaces. Indeed, by Theorem 2.6, estimates
(2.27), (4) and (2.35) imply the existence of a constant $C_{5}$, independent
of $n$, such that

$\displaystyle\sup_{t\in(0,T)}\left\|(k_{n})_{t}(\cdot,t)\right\|_{L^{p}(\Omega)},\
\sup_{t\in(0,T)}\left\|(k_{n})_{x}(\cdot,t)\right\|_{L^{p}(\Omega)}\leq
C_{5},$ (2.36)

for any $p\in(1,2)$.

We now appeal to standard $L^{p}$-estimates for parabolic equations and the
Sobolev embedding to improve our estimates on $u_{n}$. If we expand the
equation for $u_{n}$ it reads

$(u_{n})_{t}-d(u_{n})_{xx}=\alpha\left(\overline{(k_{n})}_{x}(u_{n})_{x}+u_{n}\overline{(k_{n})}_{xx}\right)+f(u_{n}).$

Obviously, $f(z)\leq f^{\prime}(0)z$ and $u_{n}\in L^{p,p}(Q_{T})$ for any
$p\geq 1$ gives us that $f(u_{n})\in L^{p,p}(Q_{T})$ as well. Then,
$L^{p}$-estimates for strong solutions (see, e.g., Wang (2003)) ensures that
there holds

$\displaystyle\left\|u_{n}\right\|_{W^{2,1}_{r}(Q_{T})}$ $\displaystyle\leq
C\left(\left\|\overline{(k_{n})}_{x}(u_{n})_{x}\right\|_{L^{r}(Q_{T})}+\left\|u_{n}\overline{(k_{n})}_{xx}\right\|_{L^{r}(Q_{T})}+\left\|u_{n}\right\|_{L^{r}(Q_{T})}\right),$
(2.37)

for some $C>0$, for any $r>1$. Choosing $r\in(1,p)$, Hölder’s inequality gives

$\displaystyle\left\|u_{n}\overline{(k_{n})}_{xx}\right\|_{L^{r}(Q_{T})}\leq\left\|u_{n}\right\|_{L^{pr/(p-r)}(Q_{T})}\left\|\overline{(k_{n})}_{xx}\right\|_{L^{p}(Q_{T})}.$
(2.38)

By Lemma (2.1), the bound (2.36) and (4), we may further estimate as

$\displaystyle\left\|u_{n}\right\|_{L^{pr/(p-r)}(Q_{T})}\left\|\overline{(k_{n})}_{xx}\right\|_{L^{p}(Q_{T})}$
$\displaystyle\leq
R^{-1}C_{2}\sup_{t\in(0,T)}\left\|(k_{n})_{x}(\cdot,t)\right\|_{L^{p}(\Omega)}$
$\displaystyle\leq R^{-1}C_{2}C_{5}=:C_{6},$ (2.39)

for any $r\in(1,p)$, where $C_{6}$ does not depend on $n$. Similarly, there
holds

$\left\|\overline{(k_{n})}_{x}(u_{n})_{x}\right\|_{L^{r}(Q_{T})}\leq C_{7},$

where $C_{7}$ does not depend on $n$. Hence,

$\left\|u_{n}\right\|_{W^{2,1}_{r}(Q_{T})}\leq C(C_{2}+C_{5}+C_{6}),$

and so $\\{u_{n}\\}_{n\geq 2}$ is bounded in $W^{2,1}_{r}(Q_{T})$ for any
$r\in(1,p)$. Since $p$ can be chosen as close to $2$ as we like, we choose
$r\in(\tfrac{3}{2},2)$ and apply the Sobolev embedding to conclude that in
fact

$\displaystyle\left\|u_{n}\right\|_{C^{\sigma,\sigma/2}(\overline{Q}_{T})}\leq\tilde{C}\left\|u_{n}\right\|_{W^{2,1}_{r}(Q_{T})}\leq\tilde{C}C(C_{2}+C_{5}+C_{6}),$
(2.40)

for any $\sigma\in(0,\tfrac{1}{2})$, for some $\tilde{C}>0$. In particular,
$u_{n}$ is uniformly bounded in $\overline{Q}_{T}$, independent of $n$.

Now we are ready to obtain bounds on the time derivative $(u_{n})_{t}$. While
the previous step gives $L^{p}$-bounds on the time derivative for $p\in(1,2)$
only, with a bit of extra work we can show that it also holds for $p=2$. These
estimates follow from standard arguments used in the development of the
$L^{2}$-theory of parabolic equations (see, e.g., (Wu et al., 2006, Ch. 3.3)),
using all previous bounds. We show the key details only.

First, note that bound (2.16) in Theorem 2.6 paired with the uniform
boundedness of the iterates $\\{u_{n}\\}_{n\geq 2}$ over $\overline{Q}_{T}$
obtained in (2.40) implies that in fact $\\{(k_{n})_{x}\\}_{n\geq 2}$ is
uniformly bounded in $L^{2,2}(Q_{T})$. Multiplying the equation for $u_{n}$ by
$(u_{n})_{t}$ and integrating over $\Omega$ gives

$\displaystyle\int_{\Omega}\left|(u_{n})_{t}\right|^{2}{\rm d}x$
$\displaystyle=\int_{\Omega}(u_{n})_{t}\left((d(u_{n})_{x}+\alpha
u_{n}\overline{(k_{n})}_{x})_{x}+f(u_{n})\right){\rm d}x$ (2.41)

By the regularity of the iterates $(u_{n},k_{n})$ for fixed $n$, we may
exchange the order of differentiation and integrate by parts to obtain

$\displaystyle\int_{\Omega}\left|(u_{n})_{t}\right|^{2}{\rm d}x$
$\displaystyle=-\frac{d}{2}\int_{\Omega}\left(\left|(u_{n})_{x}\right|^{2}\right)_{t}{\rm
d}x+\int_{\Omega}(u_{n})_{t}f(u_{n}){\rm d}x$
$\displaystyle+\alpha\int_{\Omega}(u_{n})_{t}\left((u_{n})_{x}\overline{(k_{n})}_{x}+u_{n}\overline{(k_{n})}_{xx}\right){\rm
d}x.$ (2.42)

Integrating from $0$ to $T$ and dropping the negative term, we are left with

$\displaystyle\iint_{Q_{T}}\left|(u_{n})_{t}\right|^{2}{\rm d}x{\rm d}t$
$\displaystyle\leq\frac{d}{2}\left\|(u_{0})_{x}\right\|_{L^{2}(\Omega)}^{2}+\iint_{Q_{T}}(u_{n})_{t}f(u_{n}){\rm
d}x$
$\displaystyle+\alpha\iint_{Q_{T}}(u_{n})_{t}\left((u_{n})_{x}\overline{(k_{n})_{x}}+u_{n}\overline{(k_{n})_{xx}}\right){\rm
d}x.$ (2.43)

We then estimate crudely as follows: since $u_{n}$ and
$\overline{(k_{n})}_{x}$ are uniformly bounded in $\overline{Q}_{T}$ by (2.40)
and (4), and since $(u_{n})_{x}$ and $\overline{(k_{n})}_{xx}$ are uniformly
bounded in $L^{2,2}(Q_{T})$ by (2.35) and preceding arguments, $f(u_{n})$,
$(u_{n})_{x}\overline{(k_{n})}_{x}$ and $u_{n}\overline{(k_{n})}_{xx}$ are all
uniformly bounded in $L^{2,2}(Q_{T})$. Hence, a simple application of Cauchy’s
inequality with epsilon yields the existence of a constant
$\tilde{C}^{\prime}>0$, independent of $n$, such that

$\displaystyle\frac{1}{2}\int_{Q_{T}}\left|(u_{n})_{t}\right|^{2}{\rm d}x{\rm
d}t$
$\displaystyle\leq\frac{d}{2}\left\|(u_{0})_{x}\right\|_{L^{2}(\Omega)}^{2}+\tilde{C}^{\prime}.$
(2.44)

We now summarize the uniform estimates we have obtained and complete the
limiting process.

$\displaystyle u_{n}$ $\displaystyle\in L^{\infty}(0,T;L^{p}(\Omega))\cap
C^{\sigma,\sigma/2}(\overline{Q}_{T}),$ $\displaystyle(u_{n})_{x}$
$\displaystyle\in L^{2}(0,T;L^{2}(\Omega)),\quad(u_{n})_{t}\in
L^{2}(0,T;L^{2}(\Omega)),$ $\displaystyle k_{n}$ $\displaystyle\in
L^{\infty}(0,T;L^{\infty}(\Omega))\cap C^{\sigma,\sigma/2}(\overline{Q}_{T}),$
$\displaystyle(k_{n})_{x}$ $\displaystyle\in
L^{2}(0,T;L^{2}(\Omega)),\quad(k_{n})_{t}\in L^{2}(0,T;L^{2}(\Omega)).$ (2.45)

Hence, there exists a limit function $(u_{\infty},k_{\infty})$ so that for any
$1\leq p\leq\infty$ and any $0<\sigma^{\prime}<\sigma<1/2$, there holds

$\displaystyle u_{n}\to u_{\infty},\ k_{n}\to k_{\infty}\quad\text{ strongly
in }$ $\displaystyle L^{p,p}(Q_{T})\cap
C^{\sigma^{\prime},\sigma^{\prime}/2}(\overline{Q}_{T}),$
$\displaystyle(u_{n})_{x}\to(u_{\infty})_{x}\quad\text{ strongly in }$
$\displaystyle L^{2,2}(Q_{T}),$
$\displaystyle(u_{n})_{t}\to(u_{\infty})_{t}\quad\text{ weakly in }$
$\displaystyle L^{2,2}(Q_{T}),$ $\displaystyle(k_{n})_{t}\to(k_{\infty})_{t},\
(k_{n})_{x}\to(k_{\infty})_{x}\quad\text{ weakly in }$ $\displaystyle
L^{2,2}(Q_{T}).$ (2.46)

It is not difficult to verify that $(u_{\infty},k_{\infty})$ is indeed a weak
solution to the original problem (2.23) in the sense of (1.10)-(1.11) and
satisfies the initial data in the classical sense. Since $u_{n},k_{n}$ are
nonnegative for all $n\geq 2$, we find that $0\leq u_{\infty},k_{\infty}$ in
$\overline{Q}_{T}$. Furthermore, the solution $u_{\infty}$ is nontrivial since
$f^{\prime}(0)>0$, whence $k_{\infty}$ is also nontrivial. We now write
$(u,k)$ for the solution obtained.

Uniqueness given initial data $(u_{0},k_{0})$ follows from standard arguments
and using the fact that $u$ and $\overline{k}_{x}$ are uniformly bounded over
$Q_{T}$. Indeed, if there were two solution pairs $(u,k)$ and
$(\tilde{u},\tilde{k})$ satisfying the same initial data, an application of
Cauchy’s inequality with epsilon paired with the uniform boundedness of the
solutions over $Q_{T}$, the smoothness of the functions $g_{i}(\cdot)$,
$i=1,2$, $f(\cdot)$ (Lipschitz continuity is sufficient), and the linearity of
the spatial convolution operation yields

$\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\int_{\Omega}\left((u-\tilde{u})^{2}+(k-\tilde{k})^{2}\right){\rm d}x\leq
C\int_{\Omega}\left((u-\tilde{u})^{2}+(k-\tilde{k})^{2}\right){\rm d}x,$

and so Grönwall’s inequality implies that
$\left\|(u-\tilde{u})(\cdot,t)\right\|_{L^{2}(\Omega)}=\left\|(k-\tilde{k})(\cdot,t)\right\|_{L^{2}(\Omega)}=0$
for any $t\in(0,T)$, and uniqueness is proved.

Hence, for problem (1.8.a), there exists a unique, global weak solution in the
sense of (1.10)-(1.11) so long as $g(u)$ satisfies the bound

$g(z)\leq M(\mu+\beta z)\quad\forall z\geq 0,$

for $\mu,\ \beta\geq 0$ fixed, for some $M>0$. For problem (1.8.b),
$g(z)\leq\mu+\beta z+g(z)$ holds trivially, and no further condition on $g$ is
required, concluding the proof.

##  3 Stability of spatially-constant steady states

###  3.1 Spatially-constant steady states

Under assumptions (H0)-(H2), system (1.8.a) has two constant steady-states
$(0,0)$ and $(1,\dfrac{\rho}{\mu+\beta}).$ For simplicity, denote

$\displaystyle h(u,k):=g(u)-(\mu+\beta u)k,\ U_{0}:=(0,0),\
U_{*}:{=}(1,\frac{\rho}{\mu+\beta}).$

The ODE kinetic system corresponding to (1.8.a) is given by

$\begin{cases}u^{\prime}=f(u),&\ t>0,\\\ k^{\prime}=g(u)-(\mu+\beta u)k,&\
t>0.\end{cases}$ (3.1)

Then the Jacobian matrices $J_{0}$ at $U_{0}$ and $J_{*}$ at $U_{*}$ of (3.1)
are given by

$\displaystyle J_{0}=\left(\begin{array}[]{cc}f_{u0}&0\\\
0&-\mu\end{array}\right),\ J_{*}=\left(\begin{array}[]{cc}f_{u*}&0\\\
h_{u*}&h_{k*}\end{array}\right),$

where

$\begin{split}f_{u0}&:{=}f^{\prime}(0)>0,\;\;h_{u*}:{=}h_{u}(U_{*})=\frac{g^{\prime}(1)(\mu+\beta)-\beta\rho}{\mu+\beta},\\\
f_{u*}&:{=}f^{\prime}(1)<0,\;\;h_{k*}:{=}h_{k}(U_{*})=-(\mu+\beta)<0.\end{split}$
(3.2)

Let $Tr(J_{*})$ denotes the trace of $J_{*}$, and let $Det(J_{*})$ be the
determinant of $J_{*}$. Then we have

$Tr(J_{*})=f_{u*}+h_{k*}<0,\ \ Det(J_{*})=f_{u*}h_{k*}>0.$ (3.3)

Hence $U_{*}$ is a locally asymptotically stable steady state with respect to
(3.1), and $U_{0}$ is linearly unstable with (3.1) and also (1.8.a).

###  3.2 Linear stability analysis

In Section 3.3 we will use spectral analysis to determine rigorously the
regions of stability for the constant steady state. However, the formalism
required for spectral analysis can obscure the central message. Therefore it
is valuable first to perform linear stability analysis using an ansatz. This
gives a quick route to an answer that relies on said ansatz, which we then
make rigorous via a more detailed spectral analysis.

The ansatz we use is to assume that non-constant perturbations of the constant
steady state have the following form at arbitrarily small times

$\displaystyle\tilde{u}=u_{0}{\rm e}^{{\rm i}q_{n}x+\lambda
t},\quad\tilde{k}=k_{0}{\rm e}^{{\rm i}q_{n}x+\lambda t},\quad
u\approx\tilde{u}+1,\quad k\approx\tilde{k}+\frac{\rho}{\mu+\beta},$ (3.4)

where $u_{0},k_{0},\lambda\in{\mathbb{R}}$ are constants and
$q_{n}=\sqrt{l_{n}}=n\pi/L$ for $n\in{\mathbb{N}}$. These particular
wavenumbers, $q_{n}$, are chosen as they satisfy the periodic boundary
conditions. Then, neglecting nonlinear terms and applying Fourier theory, the
PDEs in system (1.8.a) become

$\displaystyle\lambda\left(\begin{array}[]{c}\tilde{u}\\\
\tilde{k}\end{array}\right)$
$\displaystyle=M_{n}\left(\begin{array}[]{c}\tilde{u}\\\
\tilde{k}\end{array}\right),$ (3.9)

where

$\displaystyle M_{n}$
$\displaystyle=\left(\begin{array}[]{cc}-l_{n}d+f^{\prime}(1)&-l_{n}\alpha
C_{n}(G)\\\
g^{\prime}(1)-\frac{\beta\rho}{\mu+\beta}&-\mu-\beta\end{array}\right),$
(3.12)

and $C_{n}(G)$ (the Fourier coefficient of $G$) is defined in (1.12).

Stability requires that the trace of $M_{n}$ is negative and the determinant
positive. For the determinant to be positive, we require

$\displaystyle\alpha
C_{n}(G)[g^{\prime}(1)(\mu+\beta)-\beta\rho]+(\mu+\beta)^{2}\left(d-\frac{f^{\prime}(1)}{l_{n}}\right)>0.$
(3.13)

(3.13) holds when $\alpha C_{n}(G)$ is small or zero as $f^{\prime}(1)<0$ by
(H0), and (3.13) is true for all $n$ if $\alpha$ (positive or negative) is
sufficiently close to $0$. For the trace to be negative, we require

$\displaystyle f^{\prime}(1)<l_{n}d+\mu+\beta,$ (3.14)

which is always true as the right-hand side is positive and $f^{\prime}(1)<0$
by (H0). As long as Equations (3.13)-(3.14) are satisfied, system (1.8.a) will
be stable to perturbations of the exponential functional form given in
Equation (3.4) at wavenumber $q_{n}$. A similar process gives the analogous
result for system (1.8.b), which we leave as an exercise for the reader. In
the next section, we generalise this result to arbitrary perturbations, for
both systems (1.8.a) and (1.8.b) (see Theorems 1.3 and 1.4).

###  3.3 Spectral analysis

We now provide a detailed spectral analysis to confirm that the insights in
Section 3.2 hold. Equation (3.6) is the eigenvalue problem to be examined here
but requires further justification, which is done in Lemma 3.3 and 3.4. This
ensures that the Fourier analysis utlized is robust. Of note is the symmetry
of the kernel $G(\cdot)$ about the origin, which guarantees that the
coefficients $C_{n}(G)$ are real-valued. If $G$ is not even, then $C_{n}(G)$
could be complex-valued, and may lead to a Hopf bifurcation. We do not explore
this any further in the present work. Finally Theorem 3.6 shows other than the
eigenvalues from (3.6), there is an essential spectral point. In this case,
that essential spectral point is entirely negative and so will not affect
stability. In general reaction-diffusion systems, the essential spectrum will
not occur, but for such coupled PDE-ODE system, it may, and so we rule out the
possibility.

The linearized equation of system (1.8.a) at a constant steady state
$U_{*}=(u_{*},k_{*})$ is given by

$\begin{cases}\widetilde{u}_{t}=d\widetilde{u}_{xx}+f_{u*}\widetilde{u}+\alpha
u_{*}(G\ast\widetilde{k})_{xx},&x\in(-L,L),\ t>0,\\\
\widetilde{k}_{t}=h_{u*}\widetilde{u}+h_{k*}\widetilde{k},&x\in(-L,L),\
t>0,\\\ \widetilde{u}(-L,t)=\widetilde{u}(L,t)=0,&t>0,\\\
\widetilde{u}_{x}(-L,t)=\widetilde{u}_{x}(L,t)=0,&t>0.\end{cases}$ (3.15)

Define the linearized operator $\mathcal{L_{*}}(\alpha):X\rightarrow Y$ in
(3.15) by

$\mathcal{L_{*}}(\alpha)\left[\begin{array}[]{c}\phi\\\
\psi\end{array}\right]=\left(\begin{array}[]{cc}d\phi_{xx}+f_{u*}\phi+\alpha
u_{*}(G\ast\psi)_{xx}\\\ h_{u*}\phi+h_{k*}\psi\end{array}\right).$ (3.16)

For further spectral analysis, we first recall the following definitions and
give some lemmas.

###### Definition 3.1

(Magal and Ruan, 2018, Definition 2.2.1) Let $A:\mathcal{D}(L)\subset
X\rightarrow X$ be a linear operator on a $\mathbb{K}$-Banach space $X$ with
$\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. The resolvent set $\rho(A)$ of $A$ is
the set of all points $\lambda\in\mathbb{K}$ such that $(\lambda I-A)^{-1}$ is
a bijection from $\mathcal{D}(A)$ into $X$ and the inverse $(\lambda
I-A)^{-1}$, called the resolvent of $A$, is a bounded linear operator from $X$
into itself.

###### Definition 3.2

(Magal and Ruan, 2018, Definition 4.2.1) Let $A:\mathcal{D}(L)\subset
X\rightarrow X$ be a linear operator on a complex Banach space $X$. The
spectrum of the operator $A$ is defined as the complement of the resolvent set
$\sigma(A)=\mathbb{C}\setminus\rho(A)$. Consider the following three
conditions:

  * (1)

$(\lambda I-A)^{-1}$ exists;

  * (2)

$(\lambda I-A)^{-1}$ is bounded;

  * (3)

the domain of $(\lambda I-A)^{-1}$ is dense in $X$.

The spectrum $\sigma(A)$ can be further decomposed into three disjoint
subsets.

  * (a)

The point spectrum is the set

$\sigma_{p}(A):=\\{\lambda\in\sigma(A):\mathcal{N}(\lambda
I-A)\neq\\{0\\}\\}.$

Elements of the point spectrum $\sigma_{p}(A)$ are called eigenvalues. If
$\lambda\in\sigma_{p}(A)$, elements $x\in\mathcal{N}(\lambda I-A)$ are called
eigenvectors or eigenfunctions. The dimension of $\mathcal{N}(\lambda I-A)$ is
the multiplicity of $\lambda$.

  * (b)

The continuous spectrum is the set

$\sigma_{c}(A):=\\{\lambda\in\sigma(A):(1)\ \text{and}\ (3)\ \text{hold but}\
(2)\ \text{does not}\\}.$

  * (c)

The residual spectrum is the set

$\sigma_{r}(A):=\\{\lambda\in\sigma(A):(\lambda I-A)^{-1}\ \text{exists but}\
\overline{\mathcal{R}(\lambda I-A)}\neq X\\}.$

Furthermore, we have the following spectrum decomposition:

$\sigma(A)=\sigma_{p}(A)\cup\sigma_{c}(A)\cup\sigma_{r}(A).$

###### Lemma 3.3

Assume $G$ satisfies (H0), and $\phi\in H^{2}_{per}(-L,L)$. Then

$(G\ast\phi)_{xx}=G\ast(\phi_{xx}).$ (3.17)

###### Proof 5

By using the symmetry of $G$, the periodicity of $G$ and $\phi$, we get

$\displaystyle(G\ast\phi)_{xx}=$
$\displaystyle\dfrac{1}{2L}\int_{-L}^{L}G_{xx}(x-y)\phi(y)dy=\dfrac{1}{2L}\int_{-L}^{L}G_{yy}(x-y)\phi(y)dy$
$\displaystyle=$
$\displaystyle\dfrac{1}{2L}(G_{y}(x+L)\phi(-L)-G_{y}(x-L)\phi(L)-\int_{-L}^{L}G_{y}(x-y)\phi^{\prime}(y)dy)$
$\displaystyle=$
$\displaystyle\dfrac{1}{2L}(G_{y}(x+L)\phi(-L)-G_{y}(x-L)\phi(L)-G(x-L)\phi^{\prime}(L)+G(x+L)\phi^{\prime}(-L)+\int_{-L}^{L}G(x-y)\phi^{\prime\prime}(y)dy)$
$\displaystyle=$
$\displaystyle\dfrac{1}{2L}(G^{\prime}(x+L)\phi(-L)-G^{\prime}(x-L)\phi(L)-G(x-L)\phi^{\prime}(L)+G(x+L)\phi^{\prime}(-L))+G\ast(\phi_{xx})$
$\displaystyle=$ $\displaystyle\ G\ast(\phi_{xx}).$

###### Lemma 3.4

Assume $G$ satisfies (H0). Then

$G\ast\phi_{n}=C_{n}(G)\phi_{n},\ \ G\ast\phi_{-n}=C_{n}(G)\phi_{-n}$ (3.18)

where $C_{n}(G)$ is defined in (1.12) and $\phi_{n},\phi_{-n}$ are defined in
(1.9).

###### Proof 6

For $x\in[-L,L]$,

$\displaystyle G\ast\phi_{n}(x)=$ $\displaystyle\phi_{n}\ast
G(x)=\frac{1}{2L}\int_{-L}^{L}\phi_{n}(x-y)G(y)dy=\frac{1}{2L}\int_{-L}^{L}e^{\frac{in\pi}{L}(x-y)}G(y)dy$
$\displaystyle=$
$\displaystyle\frac{1}{2L}\int_{-L}^{L}e^{-\frac{in\pi}{L}y}G(y)dye^{\frac{in\pi}{L}x}=\frac{1}{2L}\int_{-L}^{L}\cos\left(\frac{n\pi}{L}y\right)G(y)dy$
$\displaystyle=$ $\displaystyle\ C_{n}(G)\phi_{n}(x).$

Note that $C_{n}\in\mathbb{R}$ as $G$ is an even function from (H0). Thus the
eigenspace corresponding to the eigenvalue $\lambda=C_{n}(G)$ is

$\displaystyle
V_{n}:=\operatorname{span}\left\\{\cos\left(\dfrac{n\pi}{L}x\right),\
\sin\left(\dfrac{n\pi}{L}x\right)\right\\}.$

Following a similar approach to the proof of (Ducrot et al., 2018, Proposition
2.1), we obtain the following lemma.

###### Lemma 3.5

The spectrum of the linear operator $\mathcal{A}:D(\mathcal{A})\subset
L^{2}_{per}(-L,L)\rightarrow L^{2}_{per}(-L,L)$ defined as

$\begin{cases}D(\mathcal{A})=H^{2}_{per}(-L,L),\\\
\mathcal{A}\phi=a\phi^{\prime\prime}+b(G\ast\phi^{\prime\prime}),\end{cases}$

is

$\sigma(\mathcal{A})=\\{\mu_{n}=\mu_{-n}:=-l_{n}(a+bC_{n}(G)),\
n\in\mathbb{N}_{0}\\},$

and the corresponding eigenfunctions are $\phi_{n}(x)$ and $\phi_{-n}(x)$ for
$n\in\mathbb{N}_{0}$, where $a,b$ are constants, and $l_{n}$, $\phi_{n}$,
$\phi_{-n}$ are defined in (1.9).

Now we can determine the spectral set of the linearized operator
$\mathcal{L_{*}}(\alpha)$.

###### Theorem 3.6

Assume that assumptions (H0)-(H2) are satisfied. Let $l_{n}$ and $\phi_{n}$ be
the eigenvalues and eigenfunctions of problem (1.8). Then

$\sigma(\mathcal{L_{*}}(\alpha))=\sigma_{p}(\mathcal{L_{*}}(\alpha))=\\{\lambda_{n}^{\pm}\\}_{n\in\mathbb{Z}}\bigcup\\{h_{k*}\\},$
(3.19)

where

$\begin{split}&\lambda_{n}^{\pm}=\lambda_{-n}^{\pm}=\dfrac{B_{n}\pm\sqrt{B_{n}^{2}-4C_{n}}}{2},\\\
&B_{n}=Tr(J_{*})-dl_{n},\;\;C_{n}=Det(J_{*})+(\alpha
u_{*}h_{u*}C_{n}(G)-dh_{k*})l_{n},\end{split}$ (3.20)

for $n\in\mathbb{N}_{0}$, and

$\displaystyle\left(\phi_{n,\pm},\psi_{n,\pm}\right)=\left(\phi_{n}(x),-\dfrac{h_{u*}}{h_{k*}-\lambda_{n}^{\pm}}\phi_{n}(x)\right),$
(3.21)
$\displaystyle\left(\phi_{-n,\pm},\psi_{-n,\pm}\right)=\left(\phi_{-n}(x),-\dfrac{h_{u*}}{h_{k*}-\lambda_{n}^{\pm}}\phi_{-n}(x)\right)$

are the eigenfunctions corresponding to $\lambda_{n}^{\pm}$ and
$\lambda_{-n}^{\pm}$, where $\phi_{n}$, $\phi_{-n}$, $h_{u*}$, $h_{k*}$,
$Tr(J_{*})$, $Det(J_{*})$ and $C_{n}(G)$ are defined in (1.9), (3.2), (3.3)
and (1.12), respectively. Furthermore, $\lambda_{n}^{\pm}$ and
$\lambda_{-n}^{\pm}$ are eigenvalues of $\mathcal{L_{*}}(\alpha)$ of finite
multiplicity, and $h_{k*}$ is an eigenvalue of infinite multiplicity.

###### Proof 7

For $\lambda\in\mathbb{C}$ and $(\xi,\eta)\in Y$, we consider the resolvent
equation of $\mathcal{L_{*}}(\alpha)$, which is

$\begin{cases}d\phi_{xx}+f_{u*}\phi+\alpha
u_{*}(G\ast\psi)_{xx}=\lambda\phi+\xi,\\\
h_{u*}\phi+h_{k*}\psi=\lambda\psi+\eta,\\\
\phi(-L)=\phi(L),\;\phi^{\prime}(-L)=\phi^{\prime}(L).\end{cases}$ (3.22)

If $\lambda\neq h_{k*}$, from the second equation of (3.22), we have

$\psi=\dfrac{\eta-h_{u*}\phi}{h_{k*}-\lambda},$ (3.23)

Substituting (3.23) into the first equation of (3.22) and combining with
(3.17) and (3.18), we get

$(f_{u*}-\lambda)\phi+(d\phi_{xx}-\alpha
u_{*}\dfrac{h_{u*}}{h_{k*}-\lambda}(G\ast\phi^{\prime\prime}))=\xi.$ (3.24)

Equation (3.24) has non-zero solutions if and only if

$(f_{u*}-\lambda)\notin\sigma(\mathcal{A})$ (3.25)

holds, where $\mathcal{A}$ is the operator defined in Lemma 3.5 with $a=d$ and
$b=-\alpha u_{*}\dfrac{h_{u*}}{h_{k*}-\lambda}$. Then from Lemma 3.5, (3.25)
is equivalent to

$\dfrac{(f_{u*}-\lambda)(h_{k*}-\lambda)}{d(h_{k*}-\lambda)-\alpha
u_{*}h_{u*}C_{n}(G)}\notin\\{l_{n}\\}_{n\in\mathbb{N}_{0}}.$ (3.26)

It follows that $\mathcal{L_{*}}(\alpha)-\lambda I$ has a bounded inverse
$(\mathcal{L_{*}}(\alpha)-\lambda I)^{-1}$ when (3.26) is satisfied.
Otherwise, $\lambda$ satisfies the following characteristic equation:

$\lambda^{2}-(Tr(J_{*})-dl_{n})\lambda+Det(J_{*})+(\alpha
u_{*}h_{u*}C_{n}(G)-dh_{k*})l_{n}=0.$ (3.27)

Therefore, $\lambda_{\pm n}^{\pm}$ in (3.20) are the roots of (3.27) with
$\mathcal{R}e\lambda_{\pm n}^{-}\leq\mathcal{R}e\lambda_{\pm n}^{+}$, and
(3.21) are eigenfunctions corresponding to $\lambda_{\pm n}^{\pm}$.

If $\lambda=h_{k*}$, we consider

$(\mathcal{L_{*}}(\alpha)-h_{k*}I)\left(\begin{array}[]{c}\phi\\\
\psi\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\end{array}\right),$
(3.28)

that is

$\begin{cases}d\phi_{xx}+f_{u*}\phi+\alpha
u_{*}(G\ast\psi)_{xx}=h_{k*}\phi,\\\ h_{u*}\phi=0,\\\
\phi(-L)=\phi(L),\phi^{\prime}(-L)=\phi^{\prime}(L).\\\ \end{cases}$ (3.29)

Clearly, $\phi=0$ and $(G\ast\psi)_{xx}=0$, which imply that there exist non-
zero solutions to (3.28). Then $h_{k*}$ is also an eigenvalue of
$\mathcal{L_{*}}(\alpha)$, and
$\operatorname{dim}ker(\mathcal{L_{*}}(\alpha)-h_{k*}I)=\infty$.

Note that $Tr(J_{*})<0$, $Det(J_{*})>0$ from the stability of $U_{*}$, and
$l_{n}\geq 0$, which yields $\mathcal{R}e\lambda_{\pm n}^{-}<0$. In other
words, eigenvalues $w_{k*}$ and $\\{\lambda_{\pm n}^{-}\\}_{n\in\mathbb{N}}$
of $\mathcal{L_{*}}$ all have negative real parts, and for $n=0$, we also have

$\lambda_{0}=\dfrac{Tr(J_{*})+\sqrt{Tr(J_{*})^{2}-4Det(J_{*})}}{2}<0.$

On the other hand, the sign of $\lambda_{\pm n}^{+}$ depends on the magnitude
of $\alpha$. Recall $\alpha_{n}$ defined in (1.13), we immediately have the
following proposition.

###### Proposition 3.7

Assume that assumptions (H0)-(H2) are satisfied, and $n\in{\mathbb{N}}$ such
that $C_{n}(G)\neq 0$. Let $\mathcal{L_{*}}(\alpha)$ and $\alpha_{n}$ be
defined in (3.16) and (1.13), respectively. Then $0$ is an eigenvalue of
$\mathcal{L_{*}}(\alpha)$ when $\alpha=\alpha_{n}$ with multiplicity of two
for $n\in\mathbb{N}$, while other eigenvalues have non-zero real parts.
Furthermore,

$\mathcal{N}(\mathcal{L}_{*}(\alpha_{n}))=\operatorname{span}\left\\{\left(1,-\dfrac{h_{u*}}{h_{k*}}\right)\phi_{n}(x),\
\left(1,-\dfrac{h_{u*}}{h_{k*}}\right)\phi_{-n}(x)\right\\}.$ (3.30)

###### Proof 8

From Lemma 3.4, $C_{n}(G)$ is real-valued. Substituting (1.13) into (3.20)
yields $\lambda_{\pm n}^{+}=0$ for $n\in\mathbb{N}$. The conclusion of
eigenfunctions follow from Lemma 3.5.

We can now prove Theorems 1.3 and 1.4.

######  Proof 9 (Proof of Theorem 1.3)

From the assumptions, $h_{k*}<0$. We also know that $\mathcal{R}e\lambda_{\pm
n}^{-}<0$. Finally for $\alpha_{l}<\alpha<\alpha_{r}$, all $\lambda_{\pm
n}^{+}$ are negative. Hence $U_{*}$ is is locally asymptotically stable with
respect to (1.8.a). And when $\alpha<\alpha_{l}$ or $\alpha>\alpha_{r}$, at
least one of $\lambda_{\pm n}^{+}$ is negative, thus $U_{*}$ is unstable.

The proof of Theorem 1.4 is similar by repeating the same analysis. Equation
(1.8.b) has two constant steady states $\widehat{U}_{0}=(0,0)$ and
$\widehat{U}_{*}=(\widehat{u}_{*},\widehat{k}_{*})=(1,\frac{\rho\kappa}{\mu+\beta+\rho})$.
Let $\widehat{h}(u,k):{=}g(u)(\kappa-k)-(\mu+\beta u)k$, we have

$\widehat{h}_{u_{*}}=\frac{\kappa[g^{\prime}(1)(\mu+\beta)-\beta\rho]}{\mu+\beta+\rho},\
\widehat{h}_{k_{*}}=-(\mu+\beta+\rho).$ (3.31)

Other parts are similar to the ones for the proof of Theorem 1.3.

##  4 Bifurcation analysis

In this section, we prove the existence of non-constant steady state solutions
of (1.8.a) through the Bifurcation from Simple Eigenvalue Theorem Crandall and
Rabinowitz (1971). From (3.30), the multiplicity of zero eigenvalue is two, so
we will restrict the solutions to be even functions only to apply the
bifurcation theorem.

For completeness, we first recall the abstract bifurcation theorem, the
definition of $K$-simple eigenvalue and a perturbation result. Consider an
abstract equation $F(\lambda,u)=0$, where $F:\mathbb{R}\times X\rightarrow Y$
is a nonlinear differentiable mapping, and $X,Y$ are Banach spaces. Crandall
and Rabinowitz Crandall and Rabinowitz (1971) obtained the following classical
Bifurcation from Simple Eigenvalue Theorem.

###### Theorem 4.1

(Crandall and Rabinowitz, 1971, Theorem 1.17) Suppose that
$\lambda_{0}\in\mathbb{R}$ and $F:\mathbb{R}\times X\rightarrow Y$ is a twice
continuously differentiable mapping and that

  1. (i)

$F(\lambda,0)=0$ for $\lambda\in\mathbb{R}$,

  2. (ii)

$\operatorname{dim}(\mathcal{N}(F_{u}(\lambda_{0},0)))=\operatorname{codim}(\mathcal{R}(F_{u}(\lambda_{0},0)))=1$,

  3. (iii)

$F_{\lambda
u}(\lambda_{0},0)[\phi_{0}]\notin\mathcal{R}(F_{u}(\lambda_{0},0))$ where
$\mathcal{N}(F_{u}(\lambda_{0},0))=\operatorname{span}\\{\phi_{0}\\}\in X$.

Let $Z$ be any complement of $\operatorname{span}\\{\phi_{0}\\}$ in $X$, then
there exist an open interval $\widehat{I}$ containing $0$ and continuous
functions $\lambda:\widehat{I}\rightarrow\mathbb{R},$
$z:\widehat{I}\rightarrow Z,$ such that $\lambda(0)=\lambda_{0},$ $z(0)=0,$
and $u(s)=s\phi_{0}+sz(s)$ satisfies $F(\lambda(s),u(s))=0.$ Moreover,
$F^{-1}(\\{0\\})$ near $(\lambda_{0},0)$ consists precisely of the curves
$u=0$ and the curves $\\{(\lambda(s),u(s)):s\in\widehat{I}\\}.$

###### Definition 4.2

(Crandall and Rabinowitz, 1973, Definition 1.2) Let $B(X,Y)$ denote the set of
bounded linear maps of $X$ into $Y$, and let $T,K\in B(X,Y)$. Then
$\mu\in\mathbb{R}$ is a $K$-simple eigenvalue of $T$ if

$\operatorname{dim}\mathcal{N}(T-\mu K)=\operatorname{codim}\mathcal{R}(T-\mu
K)=1,$

and if $\mathcal{N}(T-\mu K)=\operatorname{span}\\{\phi_{0}\\}$,
$Kx_{0}\notin\mathcal{R}(T-\mu K)$.

###### Theorem 4.3

(Crandall and Rabinowitz, 1973, Theorem 1.16) Let
$\\{(\lambda(s),u(s)):s\in\widehat{I}\\}$ be the curve of nontrivial solutions
in Theorem 4.1. Then there exist continuously differentiable functions
$r:(\lambda_{0}-\varepsilon,\lambda_{0}+\varepsilon)\rightarrow\mathbb{R}$,
$z:(\lambda_{0}-\varepsilon,\lambda_{0}+\varepsilon)\rightarrow X$,
$\mu:(-\delta,\delta)\rightarrow\mathbb{R}$, $w:(-\delta,\delta)\rightarrow
X$, such that

$\begin{split}F_{u}(\lambda,0)z(\lambda)=r(\lambda)Kz(\lambda),\ \
\lambda\in(\lambda_{0}-\varepsilon,\lambda_{0}+\varepsilon),\\\
F_{u}(\lambda(s),u(s,\cdot))w(s)=\mu(s)Kw(s),\ \
s\in(-\delta,\delta),\end{split}$ (4.1)

where $r(\lambda_{0})=\mu(0)=0$, $z(\lambda_{0})=w(0)=(\phi_{0})$,
$K:X\rightarrow Y$ is the inclusion map with $K(u)=u.$ Moreover, near $s=0$
the functions $\mu(s)$ and $-s\lambda^{\prime}(s)r^{\prime}(\lambda_{0})$ have
the same zeroes and, whenever $\mu(s)\neq 0$ the same sign and satisfy

$\lim\limits_{s\rightarrow
0}\dfrac{-s\lambda^{\prime}(s)r^{\prime}(\lambda_{0})}{\mu(s)}=1.$ (4.2)

To apply the bifurcation theorems, we define a nonlinear mapping
$F:\mathbb{R}\times X\rightarrow Y$ by

$\displaystyle F(\alpha,U)=\left(\begin{array}[]{c}du_{xx}+\alpha(u(G\ast
k)_{x})_{x}+f(u)\\\ g(u)-(\mu+\beta u)k\end{array}\right),$

where $U=(u,k)$, then the Frechét derivative of $F$ at
$(\alpha,U)=(\alpha_{n},U_{*})$ is

$\partial_{U}F(\alpha_{n},U_{*})\left(\begin{array}[]{c}\phi\\\
\psi\end{array}\right)=\left(\begin{array}[]{cc}d\phi_{xx}+f_{u*}\phi+\alpha_{n}u_{*}(G\ast\psi)_{xx}\\\
h_{u*}\phi+h_{k*}\psi\end{array}\right)=\mathcal{L}_{*}(\alpha_{n})\left(\begin{array}[]{c}\phi\\\
\psi\end{array}\right).$ (4.3)

For simplicity, we denote $\mathcal{L}_{*}(\alpha_{n})$ as $\mathcal{L}_{n}$
in the following. Let

$\displaystyle X^{s}=\\{h\in X:h(-x)=h(x),x\in(-L,L)\\},$ $\displaystyle
Y^{s}=\\{h\in Y:h(-x)=h(x),x\in(-L,L)\\}.$

We consider the restriction of $F:{\mathbb{R}}\times X^{s}\to Y^{s}$, and the
restriction of $\mathcal{L}_{n}:X^{s}\to Y^{s}$. Denote
$\mathcal{N}^{s}(\mathcal{L}_{n})$ to be the kernel space of the operator
$\mathcal{L}_{n}$ in $X^{s}$, and $\mathcal{N}^{s}(\mathcal{L}_{n}^{*})$ to be
the kernel space of the adjoint operator $\mathcal{L}_{n}$ in $X^{s}$. Then
$X^{s}$ and $Y^{s}$ have the following decompositions:

$\displaystyle X^{s}=\mathcal{N}^{s}(\mathcal{L}_{n})\oplus X^{s}_{1},\
Y^{s}=\mathcal{N}^{s}(\mathcal{L}_{n})\oplus Y^{s}_{1},$

where

$\displaystyle\mathcal{N}^{s}(\mathcal{L}_{n})=\operatorname{span}\left\\{\left(1,-\dfrac{h_{u*}}{h_{k*}}\right)\cos\left(\dfrac{n\pi}{L}x\right)\right\\},$
$\displaystyle\mathcal{N}^{s}(\mathcal{L}_{n}^{*})=\operatorname{span}\left\\{(1,r_{n})\cos\left(\dfrac{n\pi}{L}x\right)\right\\}\
\text{with}\ r_{n}=\dfrac{dl_{n}-f_{u*}}{h_{u*}},$ $\displaystyle
X^{s}_{1}=\left\\{(h_{1},h_{2})\in
X^{s}:\int_{-L}^{L}\left(h_{1}-\dfrac{h_{u*}}{h_{k*}}h_{2}\right)\cos\left(\dfrac{n\pi}{L}\right)dx=0\right\\},$
$\displaystyle
Y^{s}_{1}=\mathcal{R}^{s}(\mathcal{L}_{n})=\left\\{(h_{1},h_{2})\in
Y^{s}:\int_{-L}^{L}(h_{1}+r_{n}h_{2})\cos\left(\dfrac{n\pi}{L}x\right)dx=0\right\\}.$

Hence,
$\operatorname{dim}(\mathcal{N}^{s}(\mathcal{L}_{n}))=\operatorname{codim}(\mathcal{R}^{s}(\mathcal{L}_{n}))=1.$
We also have

$\partial_{\alpha U}F(\alpha_{n},U_{*})\left(\begin{array}[]{c}\phi\\\
\psi\end{array}\right)=\left(\begin{array}[]{cc}(G\ast\psi)_{xx}\\\
0\end{array}\right),$

thus

$\displaystyle\partial_{\alpha
U}F(\alpha_{n},U_{*})\left[\left(1,-\frac{h_{u*}}{h_{k*}}\right)\cos\left(\dfrac{n\pi}{L}x\right)\right]^{T}=\left(-\frac{h_{u*}}{h_{k*}}\frac{\partial^{2}}{\partial
x^{2}}\left(G\ast\cos\left(\dfrac{n\pi}{L}x\right)\right),0\right)^{T}\notin\mathcal{R}^{s}\left(\mathcal{L}_{n}\right)$

as

$\frac{h_{u*}}{h_{k*}}l_{n}C_{n}(G)\int_{-L}^{L}\cos^{2}\left(\dfrac{n\pi}{L}x\right)dx\neq
0,$

as long as $C_{n}(G)\neq 0$. Now by applying Theorem 4.1, we obtain the
existence of non-constant steady-state solutions of (1.8.a) (Theorem 1.5) and
(1.8.b) (Theorem 1.6).

Near a bifurcation point $\alpha=\alpha_{n}$, it follows from Shi (1999) that
the sign of $\alpha_{n}^{\prime}(0)$ or the one of
$\alpha_{n}^{\prime\prime}(0)$ when $\alpha_{n}^{\prime}(0)=0$ determine the
bifurcation direction. If $\alpha_{n}^{\prime}(0)\neq 0$, then a transcritical
bifurcation occur and a non-trivial solution exists when
$\alpha(\neq\alpha_{n})$ is close to the bifurcation point $\alpha_{n}$. If
$\alpha_{n}^{\prime}(0)=0$ and $\alpha_{n}^{\prime\prime}(0)\neq 0$, then a
pitchfork bifurcation occurs at $\alpha=\alpha_{n}$. The pitchfork bifurcation
is forward if $\alpha_{n}^{\prime\prime}(0)>0$ and there are two (zero) non-
trivial solutions for $\alpha>\alpha_{n}$ ($\alpha<\alpha_{n}$), and it is
backward if $\alpha_{n}^{\prime\prime}(0)<0$. Since

$\displaystyle\langle\zeta,F_{UU}(\alpha_{n},U_{*})\left[\left(1,-\dfrac{h_{u*}}{h_{k*}}\right)\cos\left(\dfrac{n\pi}{L}x\right)\right]^{2}\rangle$
$\displaystyle=$
$\displaystyle\int_{-L}^{L}\Biggl{[}\left(f_{uu*}+r_{n}\left(h_{uu*}-2h_{uk*}\dfrac{h_{u*}}{h_{k*}}\right)\right)\dfrac{1+\cos\left(\dfrac{2n\pi}{L}x\right)}{2}+2\alpha_{n}l_{n}C_{n}(G)\dfrac{h_{u*}}{h_{k*}}\cos\left(\dfrac{2n\pi}{L}x\right)\Biggr{]}\cos\left(\dfrac{n\pi}{L}x\right)dx=0,$

where $f_{uu*}\stackrel{{\scriptstyle\bigtriangleup}}{{=}}f_{uu}(U_{*}),\
h_{uu*}\stackrel{{\scriptstyle\bigtriangleup}}{{=}}h_{uu}(U_{*}),\
h_{uk*}\stackrel{{\scriptstyle\bigtriangleup}}{{=}}h_{uk}(U_{*})=-\beta$, and
$\zeta\in(Y^{s})^{*}$ satisfying $N(\zeta)=\mathcal{R}(\mathcal{L}_{n})$. Then

$\begin{split}\alpha_{n}^{\prime}(0)=&-\dfrac{\left\langle\zeta,F_{UU}(\alpha_{n},U_{*})\left[\left(1,-\dfrac{h_{u*}}{h_{k*}}\right)\cos\left(\dfrac{n\pi}{L}x\right)\right]^{2}\right\rangle}{2\left\langle\zeta,F_{\alpha
U}(\alpha_{n},U_{*})\left[\left(1,-\dfrac{h_{u*}}{h_{k*}}\right)\cos\left(\dfrac{n\pi}{L}x\right)\right]\right\rangle}=0.\end{split}$
(4.4)

We further calculate $\alpha_{n}^{\prime\prime}(0)$ as in Shi (1999),

$\displaystyle\alpha_{n}^{\prime\prime}(0)=-\dfrac{\left\langle\zeta,F_{UUU}\left(\alpha_{n},U_{*}\right)\left[\left(1,-\dfrac{h_{u*}}{h_{k*}}\right)\cos\left(\dfrac{n\pi}{L}x\right)\right]^{3}\right\rangle}{3\left\langle\zeta,F_{\alpha
U}\left(\alpha_{n},U_{*}\right)\left[\left(1,-\dfrac{h_{u*}}{h_{k*}}\right)\cos\left(\dfrac{n\pi}{L}x\right)\right]\right\rangle}-\dfrac{\left\langle\zeta,F_{UU}(\alpha_{n},U_{*})\left[\left(1,-\dfrac{h_{u*}}{h_{k*}}\right)\cos\left(\dfrac{n\pi}{L}x\right)\right]\left[\Theta\right]\right\rangle}{\left\langle\zeta,F_{\alpha
U}\left(\alpha_{n},U_{*}\right)\left[\left(1,-\dfrac{h_{u*}}{h_{k*}}\right)\cos\left(\dfrac{n\pi}{L}x\right)\right]\right\rangle},$
(4.5)

where $\Theta=\left(\Theta_{1},\Theta_{2}\right)$ is the unique solution of

$F_{UU}\left(\alpha_{n},U_{*}\right)\left[(1,-\dfrac{h_{u*}}{h_{k*}})\cos\left(\dfrac{n\pi}{L}x\right)\right]^{2}+F_{U}\left(\alpha_{n},U_{*}\right)\left[\Theta\right]=0.$
(4.6)

From Shi et al. (2021) and (4), we assume
$\Theta=\left(\Theta_{1},\Theta_{2}\right)$ has the following form

$\Theta_{1}=\Theta_{1}^{1}+\Theta_{1}^{2}\cos\left(\dfrac{2n\pi}{L}x\right),\
\ \Theta_{2}=\Theta_{2}^{1}+\Theta_{2}^{2}\cos\left(\dfrac{2n\pi}{L}x\right).$
(4.7)

Combining (4.6) and (4.7), after calculation, we have

$\displaystyle\Theta_{1}^{1}$ $\displaystyle=-\dfrac{f_{uu*}}{2f_{u*}},\ \ \ \
\ \ \ \ \ \ \ \
\Theta_{2}^{1}=\dfrac{h_{u*}f_{uu*}-f_{u*}\left(h_{uu*}-2h_{uk*}\dfrac{h_{u*}}{h_{k*}}\right)}{2f_{u*}h_{k*}},$
$\displaystyle\Theta_{1}^{2}$
$\displaystyle=-\dfrac{h_{k*}\left(\dfrac{f_{uu*}}{2}+2\alpha_{n}l_{n}C_{n}(G)\dfrac{h_{u*}}{h_{k*}}\right)+2\alpha_{n}u_{*}l_{n}C_{n}(G)\left(h_{uu*}-2h_{uk*}\dfrac{h_{u*}}{h_{k*}}\right)}{h_{k*}(f_{u*}-4dl_{n})+4\alpha_{n}u_{*}h_{u*}l_{n}C_{n}(G)},$
$\displaystyle\Theta_{2}^{2}$
$\displaystyle=\dfrac{h_{u*}\left(\dfrac{f_{uu*}}{2}+2\alpha_{n}l_{n}C_{n}(G)\dfrac{h_{u*}}{h_{k*}}\right)-\dfrac{1}{2}\left(f_{u*}-4dl_{n}\right)\left(h_{uu*}-2h_{uk*}\dfrac{h_{u*}}{h_{k*}}\right)}{h_{k*}\left(f_{u*}-4dl_{n}\right)+4\alpha_{n}u_{*}h_{u*}l_{n}C_{n}(G)}.$

Hence, by (4.5), $\alpha^{\prime\prime}(0)$ can be calculated as follows:

$\displaystyle\alpha_{n}^{\prime\prime}(0)$
$\displaystyle=\dfrac{\dfrac{f_{uuu*}+r_{n}g_{uuu*}}{2}+f_{uu*}\left(\Theta_{1}^{1}+\dfrac{\Theta_{1}^{2}}{2}\right)+\alpha_{n}\dfrac{h_{u*}}{h_{k*}}l_{n}C_{n}(G)\left(\Theta_{1}^{1}-\dfrac{\Theta_{1}^{2}}{2}\right)}{\dfrac{h_{u*}}{h_{k*}}l_{n}C_{n}(G)}$
$\displaystyle\ \
+\dfrac{-\alpha_{n}l_{n}C_{n}(G)\Theta_{2}^{2}+r_{n}\left[\left(h_{uu*}-h_{uk*}\dfrac{h_{u*}}{h_{k*}}\right)\left(\Theta_{1}^{1}+\dfrac{\Theta_{1}^{2}}{2}\right)+h_{uk*}\left(\Theta_{2}^{1}+\dfrac{\Theta_{2}^{2}}{2}\right)\right]}{\dfrac{h_{u*}}{h_{k*}}l_{n}C_{n}(G)},$

where $f_{uuu*}\stackrel{{\scriptstyle\bigtriangleup}}{{=}}f_{uuu}(U_{*}),\
g_{uuu*}\stackrel{{\scriptstyle\bigtriangleup}}{{=}}g_{uuu}(U_{*})$. If
$\alpha_{n}^{\prime\prime}(0)>0$, then a forward pitchfork bifurcation occurs;
and If $\alpha_{n}^{\prime\prime}(0)<0$, then a backward pitchfork bifurcation
occurs.

From Theorem 4.3, we obtain the stability of the nonconstant steady-state
solution of problem (1.8.a) and (1.8.b) obtained in Theorems 1.5 and 1.6,
respectively.

###### Theorem 4.4

Suppose the conditions of Theorem 1.5 are satisfied, and let
$(\alpha_{n}(s),U(s,\cdot))$ ($|s|<\delta$) be the non-constant steady state
solutions bifurcating from the constant ones at $\alpha=\alpha_{n}$. Then a
pitchfork bifurcation occurs at $\alpha=\alpha_{n}$ if
$\alpha_{n}^{\prime\prime}(0)\neq 0$.

  1. (i)

At $\alpha=\alpha_{r}>0$, suppose that $\alpha_{r}=\alpha_{N}$ for
$N\in{\mathbb{N}}$, then the pitchfork bifurcation is forward and the
bifurcating solutions are locally asymptotically stable with respect to
(1.8.a) if $\alpha_{N}^{\prime\prime}(0)>0$, and it is backward and the
bifurcating solutions are unstable if $\alpha_{N}^{\prime\prime}(0)<0$. The
other bifurcating solutions near $\alpha_{n}>0$ for $n\neq N$ are all
unstable.

  2. (ii)

At $\alpha=\alpha_{l}<0$, suppose that $\alpha_{l}=\alpha_{M}$ for
$M\in{\mathbb{N}}$, then the pitchfork bifurcation is backward and the
bifurcating solutions are locally asymptotically stable with respect to
(1.8.a) if $\alpha_{M}^{\prime\prime}(0)<0$, and it is foward and the
bifurcating solutions are unstable if $\alpha_{M}^{\prime\prime}(0)>0$. The
other bifurcating solutions near $\alpha_{n}<0$ for $n\neq M$ are all
unstable.

###### Proof 10

Let $r(\alpha)$ be the eigenvalue of the corresponding linearization operator
for the constant solution $U_{*}$ such that $r(\alpha_{n})=0$. From (4.1),

$F_{U}\left(\alpha,U_{*}\right)\left[\phi(\alpha),\psi(\alpha)\right]^{T}=r(\alpha)K\left[\phi(\alpha),\psi(\alpha)\right]^{T},\
\alpha\in(\alpha_{n}-\epsilon,\alpha_{n}+\epsilon).$ (4.8)

Here $K:X\to Y$ is the inclusion map $i(U)=U$. From Theorem 3.6, we thus get

$r(\alpha)=\dfrac{B_{n}+\sqrt{B_{n}^{2}-4C_{n}}}{2},$

where $B_{n}$ and $C_{n}$ are defined in (3.20). Moreover, from the definition
of $B_{n}$ and $C_{n}$, we obtain

$r^{\prime}(\alpha_{n})=\dfrac{u_{*}h_{u*}C_{n}(G)l_{n}}{B_{n}}.$ (4.9)

In particular, we have
$\operatorname{sign}(r^{\prime}(\alpha_{n}))=\operatorname{sign}(\alpha_{n})$
from (4.9) and (1.13). From (4.4), we have $\alpha_{n}^{\prime}(0)=0$ for any
$n\in{\mathbb{N}}$, and $\alpha^{\prime\prime}(0)$ can be calculated as in
(4.5). Thus a pitchfork bifurcation occurs at $\alpha=\alpha_{n}$ if
$\alpha_{n}^{\prime\prime}(0)\neq 0$.

Suppose that $\alpha_{r}=\alpha_{N}>0$ for some $N\in{\mathbb{N}}$, then
$r^{\prime}(\alpha_{N})>0$. If $\alpha_{N}^{\prime\prime}(0)>0$, then
$\alpha_{N}^{\prime}(s)>0$ for $s\in(0,\delta)$, and
$\alpha_{N}^{\prime}(s)<0$ for $s\in(-\delta,0)$. From Theorem 4.3,
$Sign(-s\alpha_{N}^{\prime}(s)r^{\prime}(\alpha_{N}))=Sign(\mu(s))$, where
$\mu(s)$ is the eigenvalue of the corresponding linearization operator for the
bifurcating solution at $\alpha=\alpha(s)$. Thus $\mu(s)<0$ for
$0<|s|<\delta$. Since all other eigenvalues of linearized equation at
$(\alpha_{n}(s),U(s,\cdot))$ are negative near $\alpha=\alpha_{r}$, then
$(\alpha_{n}(s),U(s,\cdot))$ is locally asymptotically stable. Similarly if
$\alpha_{N}^{\prime\prime}(0)<0$, then $\mu(s)>0$ for $0<|s|<\delta$ hence
$(\alpha_{n}(s),U(s,\cdot))$ is unstable. The cases for bifurcation point
$\alpha=\alpha_{l}<0$ or $\alpha\neq\alpha_{l},\alpha_{r}$ can be proved in a
similar way.

###### Theorem 4.5

Suppose the conditions of Theorem 1.6 are satisfied. Then the same results as
in Theorem 4.4 hold for equation (1.8.b), with $U_{*}$ and $\alpha_{n}$
replaced by $\widehat{U}_{*}$ and $\widehat{\alpha}_{n}$.

When the bifurcating solutions near $\alpha=\alpha_{l}$ or $\alpha_{r}$ are
locally asymptotically stable, one can observe a small amplitude non-constant
steady state solution of (1.8.a) with a prescribed wave pattern (corresponding
to $N$ or $M$ in Theorem 4.4). When the bifurcating solutions near
$\alpha=\alpha_{l}$ or $\alpha_{r}$ are unstable, the corresponding
bifurcating branch will bend back through a saddle-node bifurcation and likely
to a large amplitude non-constant steady state solution of (1.8.a) with the
same prescribed wave pattern.

##  5 Analysis of the model with a top-hat detection function

In this section, we study some specific cases to demonstrate some of our
analytical results corresponding to different growth functions $g(\cdot)$
which exhibit different rates of growth for large arguments. Depending on the
functional form of memory uptake $g(\cdot)$, we establish a number of
monotone/nonmonotone properties of the bifurcation values $alpha_{n}(R)$. In
each subsection we plot the relevant bifurcation curves $\alpha_{l}(R)$,
$\alpha_{r}(R)$, along with a depiction of the corresponding steady state
profiles near and far away from these critical values. For the numerical
simulations we use a pseudo-spectral method with a forward-Euler time-stepping
scheme. Trajectories are run until subsequent time steps are within a
tolerance of $10^{-6}$. Note that the spatial domain chosen is $(0,2\pi)$,
equivalent to choosing $\Omega=(-\pi,\pi)$.

To this end, let $L=\pi$ and $f(u)=u(1-u)$. In all cases, we choose
$d=\mu=\beta=1.0$ and $\rho=5$. We then choose the following three cases of
$g(u)$ to analyze equation (1.8.a) and equation (1.8.b) with the top-hat
detection function defined in (1.7):

(i) $g(u)=\dfrac{2\rho u^{2}}{1+u^{2}}$; (ii) $g(u)=\dfrac{2\rho u^{2}}{1+u}$;
(iii) $g(u)=\rho u^{2}$.

As previously noted in Remark 1.2, cases (i) and (ii) have a global weak
solution for either problem (1.8.a) or (1.8.b). In case (iii), a global weak
solution is only guaranteed by Theorem 1.1 for problem (1.8.b).

###  5.1 Case (i) in equation (1.8.a)

In this case, $g(u)=\dfrac{2\rho u^{2}}{1+u^{2}}$, obviously, $f$ and $g$
satisfy assumptions (H1)-(H2). One can calculate that
$(u_{*},k_{*})=\left(1,\dfrac{\rho}{\mu+\beta}\right)$ and

$\displaystyle f_{u*}=-1,\ f_{uu*}=-2,\ f_{uuu*}=0,\ g_{u*}=\rho,\
h_{u*}=\dfrac{\rho\mu}{\mu+\beta},$ (5.1) $\displaystyle h_{k*}=-(\mu+\beta),\
h_{uu*}=g_{uu*}=-\rho,\ h_{uk*}=-\beta,\ h_{kk*}=0,$ $\displaystyle
h_{uuu*}=g_{uuu*}=h_{uuk*}=h_{ukk*}=h_{kkk*}=0.$

We consider the following model with the top-hat detection function:

$\begin{cases}u_{t}=du_{xx}+\alpha(u\overline{k}_{x})_{x}+u(1-u),&x\in(-\pi,\pi),\
t>0,\\\ k_{t}=\dfrac{2\rho u^{2}}{1+u^{2}}-(\mu+\beta u)k,&x\in(-\pi,\pi),\
t>0,\end{cases}$ (5.2)

subject to periodic boundary conditions, where $G(x)$ is defined as in (1.7)
such that $0<R<\pi$. From (1.4), we have

$\overline{k}(x)=\dfrac{1}{2\pi}\int_{x-R}^{x+R}\frac{1}{2R}k(y)dy,\
-\pi\leqslant x\leqslant\pi,$ (5.3)

and for $G$ in (1.7), we have

$C_{n}(G)=\frac{\sin(nR)}{2\pi nR}.$ (5.4)

As in (3.16), the linearized operator at $(\alpha,U_{*})$ is

$\displaystyle\mathcal{L}_{*}(\alpha)\left(\begin{array}[]{c}\phi\\\
\psi\end{array}\right)=\partial_{U}F(\alpha,U_{*})\left(\begin{array}[]{c}\phi\\\
\psi\end{array}\right)=\left(\begin{array}[]{cc}d\phi_{xx}-\phi+\frac{\alpha}{4RL}\left(\psi_{x}(x+R)-\psi_{x}(x-R)\right)\\\
\frac{\rho\mu}{\mu+\beta}\phi-(\mu+\beta)\psi\end{array}\right).$ (5.11)

From Theorem 3.6, the spectrum of $\mathcal{L}_{*}(\alpha)$ is consisted of
$h_{k*}=-(\mu+\beta)<0$ and eigenvalues $\lambda_{n}^{\pm}$ which satisfy the
characteristic equation

$\lambda^{2}+(1+\mu+\beta+dn^{2})\lambda+(\mu+\beta)+\left(\alpha\dfrac{\rho\mu}{\mu+\beta}\dfrac{\sin(nR)}{2\pi
nR}+d(\mu+\beta)\right)n^{2}=0,$ (5.12)

where $l_{n}$ is defined in (1.9), $n\in\mathbb{Z}$. When $n=0,$ from (3.3),
all eigenvalues of (5.12) have negative real parts, hence the constant
solution $U_{*}$ is locally asymptotically stable with respect to non-spatial
dynamics. Note that (5.12) is an even function of $n$, so we consider
$n\in\mathbb{N}$ below.

From (1.13), (1.14) and (5.4), we obtain

$\displaystyle\alpha_{n}^{R}$
$\displaystyle=\frac{Det(J_{*})-dh_{k*}n^{2}}{u_{*}h_{u*}n^{2}\frac{\sin(nR)}{2\pi
nR}}=\dfrac{-2\pi
nR(\mu+\beta)^{2}}{\rho\mu\sin(nR)}\left(d+\dfrac{1}{n^{2}}\right),$
$\displaystyle\Sigma^{+}$
$\displaystyle=\left\\{n\in\mathbb{N}:nR\in\cup_{j=0}^{\infty}(2j\pi,(2j+1)\pi)\right\\},$
$\displaystyle\Sigma^{-}$
$\displaystyle=\left\\{n\in\mathbb{N}:nR\in\cup_{j=0}^{\infty}((2j+1)\pi,(2j+2)\pi)\right\\},$
(5.13) $\displaystyle\alpha_{l}$
$\displaystyle=-\frac{2\pi(\mu+\beta)^{2}}{\rho\mu}\min_{n\in\Sigma^{+}}\dfrac{nR}{\sin(nR)}\left(d+\dfrac{1}{n^{2}}\right),$
$\displaystyle\alpha_{r}$
$\displaystyle=-\frac{2\pi(\mu+\beta)^{2}}{\rho\mu}\max_{n\in\Sigma^{-}}\dfrac{nR}{\sin(nR)}\left(d+\dfrac{1}{n^{2}}\right),$

In Figure 1 we numerically compute the values $\left|\alpha_{l}\right|$ and
$\alpha_{r}$ and plot them with respect to the perceptual radius $R$. Now we
can apply Theorems 1.3 and 1.5 to (5.2) to have the following results.

###### Theorem 5.1

Let $\alpha_{n}^{R},\Sigma^{+},\Sigma^{-},\alpha_{l},\alpha_{r}$ be defined in
(5.13). Then the constant steady state solution $U_{*}=(1,\rho/(\mu+\beta))$
is locally asymptotically stable with respect to (5.2) when
$\alpha_{l}<\alpha<\alpha_{r}$ and is unstable when $\alpha<\alpha_{l}$ or
$\alpha>\alpha_{r}$. Moreover non-constant steady state solutions of (5.2)
bifurcate from the the branch of constant solutions
$\Gamma_{0}=\\{(\alpha,U_{*}):\alpha\in{\mathbb{R}}\\}$ near
$\alpha=\alpha_{n}^{R}$, and these solutions are on a curve
$\Gamma_{n}=\\{(\alpha_{n}(s),u_{n}(s,\cdot),k_{n}(s,\cdot)):|s|<\delta\\}$
such that $\alpha_{n}(0)=\alpha_{n}^{R}$ and $\alpha_{n}^{\prime}(0)=0$.
Moreover, the following monotonicity properties hold.

  1. (i)

Suppose $n\in\Sigma^{+}$ so that $\dfrac{\sin(nR)}{nR}>0$. Then
$\alpha_{n}(R)<0$, and $\alpha_{n}(R)$ (in particular, $\alpha_{l}$) is
monotonically increasing with respect to $\rho$, is monotonically decreasing
with respect to $d$ and $\beta$, and is not monotone with respect to $\mu$.

  2. (ii)

Suppose $n\in\Sigma^{-}$ so that $\dfrac{\sin(nR)}{nR}<0$. Then
$\alpha_{n}(R)>0$, and $\alpha_{n}(R)$ (in particular, $\alpha_{r}$) is
monotonically increasing with respect to $d$ and $\beta$, is monotonically
decreasing with respect to $\rho$, and is not monotone with respect to $\mu$.

After calculation,

$\alpha_{n}^{\prime\prime}(0)=\frac{-2n^{4}-\frac{143}{24}n^{2}-\frac{5}{24}}{-\frac{5}{4}n^{2}\frac{\sin(nR)}{2\pi
nR}}.$

Therefore, at $\alpha=\alpha_{r}$, the pitchfork bifurcation is forward and
the bifurcating solutions are locally asymptotically stable with respect to
(5.2), at $\alpha=\alpha_{l}$, the pitchfork bifurcation is backward and the
bifurcating solutions are locally asymptotically stable with respect to (5.2).

The monotonicity properties described above can be understood intuitively. In
case (i), we consider cases of aggregation ($\alpha<0$), and so a decreasing
behaviour requires higher rates of advection to destabalize the constant
steady state, while an increasing behaviour allows for destabalization of the
constant steady state at lower advection rates. As is generally understood for
diffusion-advection models, diffusion has a stabalizing effect, and higher
rates of diffusion therefore require comparably high magnitudes of advection
to destabalize the constant steady state. Similarly, an increased value of
$\beta$, a ‘rate of safe return’, also requires an increased magnitude of
advection to destabalize the constant steady state. This suggests that the
population cannot return to previously visited locations too quickly if
patterns are to persist. In case (ii), we flip the sign of the advection rate
and consider the segregataion case ($\alpha>0$), in which case the direction
of the monotonicities also switch, but the understanding of this behaviour is
identical to case (i). Most interestingly, perhaps, is the nonmonotone
behaviour with respect to the memory decay rate $\mu$. In fact, in this case
$\alpha_{l}$ ($\alpha_{r}$) is concave down (up), and so there is a critical
value $\mu^{*}>0$ so that the rate of advection required to destabalize the
constant steady state is minimal. This is in contrast to Theorem 5.2 in case
(ii), where monotonicity with respect to $\beta$ is lost.

Figure 1 demonstrates the more complex relationship between the perceptual
radius $R$ and the sizes of $\alpha_{r}(R)$ and $\alpha_{l}(R)$. Of note is
the nonmonotone behaviour, particularly for smaller perceptual radii. This
wavelike behaviour is most pronounced for $\alpha_{r}(R)$. It is also easy to
see that $\lvert\alpha_{l}(R)\rvert<\alpha_{r}(R)$ when $0<R<\pi/2$ (indeed
this holds for $R<2.2$ from Figure 1). But when $R$ is larger than $2.2$,
either $\lvert\alpha_{l}(R)\rvert$ or $\alpha_{r}(R)$ could be the larger one.
This in general shows that the advection rate needed to destabilize the
positive equilibrium is larger when the perceptual radius is larger. When the
perceptual radius is less than half of domain size, the attractive advection
rate needed to destabilize the positive equilibrium is larger than the
repulsive one. In Figures 2-3, we plot the solution profiles just before, just
after, and far beyond the analytically calculated bifurcation points
$\alpha_{l}^{*}$ and $\alpha_{r}^{*}$, respectively.

Figure 1: The bifurcation curves for Section 5.1 for the perceptual radius $R$
versus the advection rate $\alpha$. In the subsequent Figures 2-3, we fix
$R=2.5$ and use the bifurcation curves to test values near and far from the
critical values $\alpha_{r}^{*}$ and $\alpha_{\ell}^{*}$ where the constant
steady state is expected to be destabalized.  Figure 2:  The steady state
solutions corresponding to Section 5.1 for the negative $\alpha$ (aggregation)
case. We test $\sim 0.1$ before, $\sim 0.1$ after, and twice the value of the
critical value $\alpha_{\ell}^{*}$ given in Figure 1.  Figure 3:  The steady
state solutions corresponding to Section 5.1 for the positive $\alpha$
(segregation) case. We test $\sim 0.1$ before, $\sim 0.1$ after, and twice the
value of the critical value $\alpha_{r}^{*}$ given in Figure 1.

###  5.2 Case (ii) in equation (1.8.a)

Let $g(u)=\dfrac{2\rho u^{2}}{1+u}$ in equation (1.8.a). Similar to case (i),
we have

$\displaystyle(u_{*}$
$\displaystyle,k_{*})=\left(1,\frac{\rho}{\mu+\beta}\right),\ f_{u*}=-1,\
f_{uu*}=-2,\ f_{uuu*}=0,\ g_{u*}=\dfrac{3\rho}{2},$ (5.14) $\displaystyle
h_{u*}=\dfrac{3\rho\mu+\rho\beta}{2(\mu+\beta)},\
h_{k*}=-(\mu+\beta),h_{uu*}=g_{uu*}=\dfrac{\rho}{2},\ h_{uk*}=-\beta,$
$\displaystyle h_{kk*}=0,\
h_{uuu*}=g_{uuu*}=-\dfrac{3\rho}{4},h_{uuk*}=h_{ukk*}=h_{kkk*}=0.$

and

$\alpha_{n}(R)=\frac{Det(J_{*})-dh_{k*}n^{2}}{u_{*}h_{u*}n^{2}\frac{\sin(nR)}{2\pi
nR}}=\dfrac{-2(\mu+\beta)^{2}}{\rho(3\mu+\beta)\frac{\sin(nR)}{2\pi
nR}}\left(d+\dfrac{1}{n^{2}}\right).$ (5.15)

In Figure 4, we again numerically compute $\left|\alpha_{l}\right|$ and
$\alpha_{r}$ and plot them with respect to the perceptual radius $R$. In
Figures 5-6, we again plot the solution profiles just before, just after, and
far beyond the analytically calculated bifurcation points $\alpha_{l}^{*}$ and
$\alpha_{r}^{*}$. In this case the stability of the constant steady-state is
the same as the result of Theorem 5.1 (which we omit here), while the
monotonicity of $\alpha_{n}(R)$ with respect to parameters is slightly
different, we state the theorem below.

###### Theorem 5.2

Let $\alpha_{n}(R)$ be defined in (5.15).

  1. (i)

Suppose $n\in\Sigma^{+}$ so that $\dfrac{\sin(nR)}{nR}>0$. Then
$\alpha_{n}(R)<0$, and $\alpha_{n}(R)$ (in particular, $\alpha_{l}$) is
monotonically increasing with respect to $\rho$, monotonically decreasing with
respect to $d$, and is not monotone with with respect to either $\mu$ or
$\beta$.

  2. (ii)

Suppose $n\in\Sigma^{-}$ so that $\dfrac{\sin(nR)}{nR}<0$. Then
$\alpha_{n}(R)>0$, and $\alpha_{n}(R)$ (in particular, $\alpha_{r}$) is
monotonically increasing with respect to $d$, monotonically decreasing with
respect to $\rho$, and is not monotone with respect to either $\mu$ or
$\beta$.

This Theorem, similar to Theorem 5.1 for case (i), retains the expected
monotonicity properties with respect to $\rho$ and $d$, while we lose
monotonicity with respect to $\beta$. Indeed, the curves $\alpha_{l}$ and
$\alpha_{r}$ are concave with respect to each parameter, suggesting the
existence of critical values $\mu^{*}>0$ and $\beta^{*}>0$ so that the
magnitude of advection required to destabalize the constant steady state is
minimal. This highlights a key difference caused by the choice of memory
uptake $g(\cdot)$.

Figure 4: The bifurcation curves for Section 5.2 for the perceptual radius $R$
versus the advection rate $\alpha$. In the subsequent Figures 5-6, we fix
$R=2.5$ and use the bifurcation curves to test values near and far from the
critical values $\alpha_{r}^{*}$ and $\alpha_{\ell}^{*}$ where the constant
steady state is expected to be destabalized.  Figure 5:  The steady state
solutions corresponding to Section 5.2 for the negative $\alpha$ (aggregation)
case. We test $\sim 0.1$ before, $\sim 0.1$ after, and twice the value of the
critical value $\alpha_{\ell}^{*}$ given in Figure 4.  Figure 6:  The steady
state solutions corresponding to Section 5.2 for the positive $\alpha$
(segregation) case. We test $\sim 0.1$ before, $\sim 0.1$ after, and twice the
value of the critical value $\alpha_{r}^{*}$.

###  5.3 Case (iii) in equation (1.8.b)

In this case, we choose $g(u)=\rho u^{2}$ in equation (1.8.b). From (3.31) and
(1.15), bifurcation point $\widehat{\alpha}_{n}(R)$ has the following
expression

$\widehat{\alpha}_{n}(R)=\frac{-(\rho+\mu+\beta)^{2}}{\kappa\rho(2\mu+\beta)\frac{\sin(nR)}{2\pi
nR}}\left(d-\frac{f^{\prime}(1)}{l_{n}}\right).$ (5.16)

Similar to Theorem 5.1, we have the following.

###### Theorem 5.3

Let $\alpha_{n}(R),\Sigma^{+},\Sigma^{-},\alpha_{l},\alpha_{r}$ be defined in
(5.16) and (1.16). Then there are non-constant steady-state solutions that
bifurcation from the constant solution $(1,\frac{\rho}{\rho+\mu+\beta})$ near
$\widehat{\alpha}_{n}(R)$ of system (1.8.b). Moreover, the constant solution
$(1,\frac{\rho}{\rho+\mu+\beta})$ is locally asymptotically stable when
$\widehat{\alpha}_{l}<\widehat{\alpha}<\widehat{\alpha}_{r}$ and unstable when
$\widehat{\alpha}<\widehat{\alpha}_{l}$ or
$\widehat{\alpha}>\widehat{\alpha}_{r}$.

###### Theorem 5.4

Let $\widehat{\alpha}_{n}(R)$ be defined in (5.16).

  1. (i)

Suppose $n\in\Sigma^{+}$ so that $\dfrac{\sin(nR)}{nR}>0$. Then
$\widehat{\alpha}_{n}(R)<0$, and $\widehat{\alpha}_{n}(R)$ (in particular,
$\widehat{\alpha}_{l}$) is monotonically increasing with respect to $\kappa$,
monotonically decreasing with respect to $d$, and is not monotone with respect
to any of $\rho$, $\mu$ or $\beta$;

  2. (ii)

Suppose $n\in\Sigma^{-}$ so that $\dfrac{\sin(nR)}{nR}<0$. Then
$\widehat{\alpha}_{n}(R)>0$, and $\widehat{\alpha}_{n}(R)$ (in particular,
$\widehat{\alpha}_{r}$) is monotonically increasing with respect to $d$,
monotonically decreasing with respect to $\kappa$, and is not monotone with
respect to any of $\rho$, $\mu$ or $\beta$.

We again compare to Theorem’s 5.1-5.2: the monotonicity with respect to $d$
remains, while a quadratic growth for the memory uptake function causes all
other previously monotone cases to be nonmonotone! In case (iii), we also have
a new parameter $\kappa$, which is the theoretical maximal memory capacity of
the organism. It is biologically reasonable, therefore, for an increase in
this memory capacity to decrease the magnitude of advection required to
destabalize the constant steady state.

In Figure 7, we plot the bifurcation curves $\alpha_{l}(R)$ and
$\alpha_{r}(R)$. Figures 8-9 again show the solution profiles as done in cases
(i) and (ii).

Figure 7: The bifurcation curves for Section 5.2 for the perceptual radius $R$
versus the advection rate $\alpha$. In the subsequent Figures 8-9, we fix
$R=2.5$ and use the bifurcation curves to test values near and far from the
critical values $\alpha_{r}^{*}$ and $\alpha_{\ell}^{*}$ where the constant
steady state is expected to be destabalized.  Figure 8:  The steady state
solutions corresponding to Section 5.3 for the negative $\alpha$ (aggregation)
case. We test $\sim 0.1$ before, $\sim 0.1$ after, and twice the value of the
critical value $\alpha_{\ell}^{*}$.  Figure 9:  The steady state solutions
corresponding to Section 5.3 for the positive $\alpha$ (segregation) case. We
test $\sim 0.1$ before, $\sim 0.1$ after, and twice the value of the critical
value $\alpha_{r}^{*}$.

##  6 Discussion

The role of spatial memory in driving the movement of animals has long been of
interest to both empirical ecologists Fagan et al. (2013) and mathematical
modellers Wang and Salmaniw (2023a). In this work, we consider the
incorporation of a nonlocal advection term in the PDE to model movement in
response to remembered space use, where the memory map is described
dynamically by an additional ODE. The nonlocal advection term is crucial from
both biological and mathematical standpoints Painter et al. (2023).
Biologically, it more accurately captures an essence of how organisms sense
their surrounding environment and make movement decisions based on that
information Martinez-Garcia et al. (2020). This is a useful step forward in
our mathematical representation of animal movement ecology, making the
modelling formulation more applicable to what is observed in the natural
world. Mathematically, however, nonlocality introduces technical difficulties
which deserve a careful and robust study.

One of the significant contributions of this paper is the establishment of a
well-posedness result, proving the existence and uniqueness of a global
solution, ruling out the possibility of a finite-time blow up. In particular,
we showed the existence and uniqueness of a solution even when considering the
discontinuous top-hat detection function. This result broadens the existing
literature in a crucial way, providing answers to some open questions found in
Wang and Salmaniw (2023a). Prior to this work, the existence of solutions for
this specific class of models remained an open question. Our result not only
bridges this gap but also opens doors for more complex models incorporating
different types of detection functions.

Another significant contribution lies in our robust bifurcation and spectral
analyses. Previous efforts, motivated more directly by the ecological
application, have often relied solely on a linear stability analyses Briscoe
et al. (2002); Potts and Lewis (2016b), which can be insufficient in scenarios
where the essential spectrum is nonempty. This does not occur in classical
reaction-diffusion systems, but may be the case when nonlocal advection is
introduced. Moreover, the comprehensive approach used here reveals a more
nuanced understanding of the system’s stability, describing more qualitative
features of the solution profile near these critical bifurcation points.

From this bifurcation analysis, we establish a number of monotonicity and non-
monotonicity results for the critical bifurcation parameters, with the
monotonic properties depending on the functional form given chosen for the
memory uptake rate $g(\cdot)$. In the special cases considered here, a bounded
functional form has the most monotonicity properties, while a roughly linear
or quadratic functional form appears to remove most of the monotonicity
properties. This suggests the existence of critical values so that the
magnitude of advection required to destabalize is minimized, an interesting
feature that deserves future study. For example, is the critical value
dependent on whether the population aggregates or segregates?

Our numerical simulations using a pseudo-spectral method further complement
our analytical results. These numerical methods are particularly well-suited
for dealing with nonlocal advection-diffusion problem, demonstrating some of
the interesting dynamics found in these models.

While this work provides new insights and fills existing gaps in the
literature, further studies are needed to explore more general functional
forms, higher-dimensional space, and for domains with a physical boundary.
From a biological perspective, there are various aspects to memory at play
that are not modelled here Fagan et al. (2013). In reality, animals’ advective
tendencies will not simply be towards (or away from) areas they have
previously visited. Rather they will assess the quality of those places – e.g.
whether they contain access to food or shelter, if they have had aggressive or
favourable encounters there – and adjust their advective tendencies
accordingly Lewis et al. (2021). Our work paves the way for analysing these
more detailed and realistic memory effects.

One tricky, yet important, feature will be the inclusion of heterogeneous
landscapes, for example where some areas are better than others for foraging
or hiding from predators Van Moorter et al. (2009); Merkle et al. (2014).
Analysis of nonlinear PDEs often takes place on a homogeneous environment, but
to connect better to the ecological community, theory on pattern formation in
heterogeneous environments will be of fundamental importance Krause et al.
(2020). Additionally, it will be important to account for between-population
interactions via multi-species models with explicit inclusion of memory
processes Potts and Lewis (2019). A possible way in to this would be to
analyse existing models on territory, some which are simply multi-species
extensions of the model analysed here Potts and Lewis (2016a, b), by placing
these on solid mathematical foundations through developing existence theory,
and gaining greater insights into territorial pattern formation through
rigorous spectral and bifurcation analyses.

##  7 Acknowledgments

The authors thank Mark Lewis for his insightful comments on model development.
DL was partially supported by a HIT PhD Scholarship and the University of
Alberta. YS is supported by NSERC grant PDF-578181-2023. JRP acknowledges
support of Engineering and Physical Sciences Research Council (EPSRC) grant
EP/V002988/1. JS is partially supported by the U.S. National Science
Foundation grants OCE-2207343 and DMS-185359. HW gratefully acknowledges
support from the Natural Sciences and Engineering Research Council of Canada
(Discovery Grant RGPIN-2020-03911 and NSERC Accelerator Grant
RGPAS-2020-00090) and the Canada Research Chairs program (Tier 1 Canada
Research Chair Award).

## References

  * Fagan et al. [2013] William F Fagan, Mark A Lewis, Marie Auger-Méthé, Tal Avgar, Simon Benhamou, Greg Breed, Lara LaDage, Ulrike E Schlägel, Wen-wu Tang, Yannis P Papastamatiou, et al.  Spatial memory and animal movement.  _Ecology letters_ , 16(10):1316–1329, 2013.
  * Briscoe et al. [2002] Brian K Briscoe, Mark A Lewis, and Stephen E Parrish.  Home range formation in wolves due to scent marking.  _Bulletin of Mathematical Biology_ , 64(2):261–284, 2002.
  * Börger et al. [2008] Luca Börger, Benjamin D Dalziel, and John M Fryxell.  Are there general mechanisms of animal home range behaviour? a review and prospects for future research.  _Ecology Letters_ , 11(6):637–650, 2008.
  * Van Moorter et al. [2009] Bram Van Moorter, Darcy Visscher, Simon Benhamou, Luca Börger, Mark S Boyce, and Jean-Michel Gaillard.  Memory keeps you at home: a mechanistic model for home range emergence.  _Oikos_ , 118(5):641–652, 2009.
  * Bracis and Mueller [2017] Chloe Bracis and Thomas Mueller.  Memory, not just perception, plays an important role in terrestrial mammalian migration.  _Proceedings of the Royal Society B: Biological Sciences_ , 284(1855):20170449, 2017.
  * Abrahms et al. [2019] Briana Abrahms, Elliott L Hazen, Ellen O Aikens, Matthew S Savoca, Jeremy A Goldbogen, Steven J Bograd, Michael G Jacox, Ladd M Irvine, Daniel M Palacios, and Bruce R Mate.  Memory and resource tracking drive blue whale migrations.  _Proceedings of the National Academy of Sciences_ , 116(12):5582–5587, 2019.
  * Potts and Lewis [2016a] Jonathan R Potts and Mark A Lewis.  How memory of direct animal interactions can lead to territorial pattern formation.  _Journal of the Royal Society Interface_ , 13(118):20160059, 2016a.
  * Ellison et al. [2020] Natasha Ellison, Ben J Hatchwell, Sarah J Biddiscombe, Clare J Napper, and Jonathan R Potts.  Mechanistic home range analysis reveals drivers of space use patterns for a non-territorial passerine.  _Journal of Animal Ecology_ , 89(12):2763–2776, 2020.
  * Merkle et al. [2014] JA Merkle, D Fortin, and Juan Manuel Morales.  A memory-based foraging tactic reveals an adaptive mechanism for restricted space use.  _Ecology Letters_ , 17(8):924–931, 2014.
  * Merkle et al. [2019] Jerod A Merkle, Hall Sawyer, Kevin L Monteith, Samantha PH Dwinnell, Gary L Fralick, and Matthew J Kauffman.  Spatial memory shapes migration and its benefits: evidence from a large herbivore.  _Ecology Letters_ , 22(11):1797–1805, 2019.
  * Lewis et al. [2021] Mark A Lewis, William F Fagan, Marie Auger-Méthé, Jacqueline Frair, John M Fryxell, Claudius Gros, Eliezer Gurarie, Susan D Healy, and Jerod A Merkle.  Learning and animal movement.  _Frontiers in Ecology and Evolution_ , 9:681704, 2021.
  * Shi et al. [2021] Qingyan Shi, Junping Shi, and Hao Wang.  Spatial movement with distributed memory.  _J. Math. Biol._ , 82(4):Paper No. 33, 32, 2021\.  ISSN 0303-6812.  doi:10.1007/s00285-021-01588-0.  URL https://doi.org/10.1007/s00285-021-01588-0.
  * Wang and Salmaniw [2023a] Hao Wang and Yurij Salmaniw.  Open problems in pde models for knowledge-based animal movement via nonlocal perception and cognitive mapping.  _J. Math. Biol._ , 86(71), 2023a.
  * Benhamou [2014] Simon Benhamou.  Of scales and stationarity in animal movements.  _Ecology Letters_ , 17(3):261–272, 2014.
  * Martinez-Garcia et al. [2020] Ricardo Martinez-Garcia, Christen H Fleming, Ralf Seppelt, William F Fagan, and Justin M Calabrese.  How range residency and long-range perception change encounter rates.  _Journal of Theoretical Biology_ , 498:110267, 2020.
  * Armstrong et al. [2009] Nicola J Armstrong, Kevin J Painter, and Jonathan A Sherratt.  Adding adhesion to a chemical signaling model for somite formation.  _Bulletin of mathematical biology_ , 71(1):1–24, 2009.
  * Painter et al. [2023] Kevin J Painter, Thomas Hillen, and Jonathan R Potts.  Biological modelling with nonlocal advection diffusion equations.  _arXiv preprint arXiv:2307.14396_ , 2023.
  * Bertozzi and Laurent [2007] Andrea L Bertozzi and Thomas Laurent.  Finite-time blow-up of solutions of an aggregation equation in r n.  _Communications in mathematical physics_ , 274:717–735, 2007.
  * Giunta et al. [2022] Valeria Giunta, Thomas Hillen, Mark A. Lewis, and Jonathan R. Potts.  Detecting minimum energy states and multi-stability in nonlocal advection–diffusion models for interacting species.  _Journal of Mathematical Biology_ , 85(56), 2022.
  * Wang and Salmaniw [2023b] Hao Wang and Yurij Salmaniw.  Open problems in pde models for knowledge-based animal movement via nonlocal perception and cognitive mapping.  _J. Math. Biol._ , 86(71), 2023b.
  * Potts and Lewis [2019] Jonathan R Potts and Mark A Lewis.  Spatial memory and taxis-driven pattern formation in model ecosystems.  _Bulletin of mathematical biology_ , 81:2725–2747, 2019\.
  * Harten et al. [2020] L. Harten, A. Katz, A. Goldshtein, M. Handel, and Y. Yovel.  The ontogeny of a mammalian cognitive map in the real world.  _Science_ , 369(1):194–197, 2020.
  * Peer et al. [2021] M. Peer, I.K. Brunec, N.S. Newcombe, and R.A. Epstein.  Structuring knowledge with cognitive maps and cognitive graphs.  _Trends Cogn Sci._ , 25(1):37–54, 2021.
  * Potts et al. [2022] Jonathan R Potts, Luca Börger, Bronson K Strickland, and Garrett M Street.  Assessing the predictive power of step selection functions: How social and environmental interactions affect animal space use.  _Methods in Ecology and Evolution_ , 13(8):1805–1818, 2022.
  * Lewis and Murray [1993] MA Lewis and JD Murray.  Modelling territoriality and wolf–deer interactions.  _Nature_ , 366(6457):738–740, 1993.
  * Moorcroft et al. [2006] Paul R Moorcroft, Mark A Lewis, and Robert L Crabtree.  Mechanistic home range models capture spatial patterns and dynamics of coyote territories in yellowstone.  _Proceedings of the Royal Society B: Biological Sciences_ , 273(1594):1651–1659, 2006.
  * Holling [1965] Crawford Stanley Holling.  The functional response of predators to prey density and its role in mimicry and population regulation.  _The Memoirs of the Entomological Society of Canada_ , 97(S45):5–60, 1965.
  * Fagan et al. [2017] William F Fagan, Eliezer Gurarie, Sharon Bewick, Allison Howard, Robert Stephen Cantrell, and Chris Cosner.  Perceptual ranges, information gathering, and foraging success in dynamic landscapes.  _The American Naturalist_ , 189(5):474–489, 2017\.
  * Ducrot et al. [2018] Arnaud Ducrot, Xiaoming Fu, and Pierre Magal.  Turing and Turing-Hopf bifurcations for a reaction diffusion equation with nonlocal advection.  _J. Nonlinear Sci._ , 28(5):1959–1997, 2018.  ISSN 0938-8974.  doi:10.1007/s00332-018-9472-z.  URL https://doi.org/10.1007/s00332-018-9472-z.
  * Giunta et al. [2021] Valeria Giunta, Thomas Hillen, Mark A Lewis, and Jonathan R Potts.  Local and global existence for non-local multi-species advection-diffusion models.  _arXiv preprint arXiv:2106.06383_ , 2021.
  * Jüngel et al. [2022] Ansgar Jüngel, Stefan Portisch, and Antoine Zurek.  Nonlocal cross-diffusion systems for multi-species populations and networks.  _Nonlinear Analysis_ , 219, 2022.
  * Shi et al. [2020] Junping Shi, Chuncheng Wang, Hao Wang, and Xiangping Yan.  Diffusive spatial movement with memory.  _J. Dynam. Differential Equations_ , 32(2):979–1002, 2020.  ISSN 1040-7294.  doi:10.1007/s10884-019-09757-y.  URL https://doi.org/10.1007/s10884-019-09757-y.
  * Song et al. [2022] Yongli Song, Junping Shi, and Hao Wang.  Spatiotemporal dynamics of a diffusive consumer-resource model with explicit spatial memory.  _Studies in Applied Mathematics_ , 148(1):373–395, 2022.
  * Marciniak-Czochra et al. [2017] Anna Marciniak-Czochra, Grzegorz Karch, and Kanako Suzuki.  Instability of Turing patterns in reaction-diffusion-ODE systems.  _J. Math. Biol._ , 74(3):583–618, 2017.  ISSN 0303-6812.  doi:10.1007/s00285-016-1035-z.  URL https://doi.org/10.1007/s00285-016-1035-z.
  * Li et al. [2017] Ying Li, Anna Marciniak-Czochra, Izumi Takagi, and Boying Wu.  Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis.  _Hiroshima Math. J._ , 47(2):217–247, 2017.  ISSN 0018-2079.  URL http://projecteuclid.org/euclid.hmj/1499392826.
  * Carrillo et al. [2019] José A Carrillo, Katy Craig, and Yao Yao.  Aggregation-diffusion equations: dynamics, asymptotics, and singular limits.  _Active Particles, Volume 2: Advances in Theory, Models, and Applications_ , pages 65–108, 2019.
  * Ladyzhenskaiâ et al. [1968] O.A. Ladyzhenskaiâ, V.A.S.N.N.U. O. A. Ladyženskaja, V.A. Solonnikov, N.N. Uraltseva, and Uralt́ŝeva N.N.  _Linear and Quasi-linear Equations of Parabolic Type_.  American Mathematical Society, translations of mathematical monographs. American Mathematical Society, 1968.  ISBN 9780821886533.  URL https://books.google.ca/books?id=HsE-AAAAIAAJ.
  * Wang [2003] L.H. Wang.  A geometric approach to the calderón-zygmund estimates.  _Acta Math Sinica_ , 19:381–396, 2003.
  * Wu et al. [2006] Z. Wu, J. Yin, and C. Wang.  _Elliptic & Parabolic Equations_.  Elliptic & Parabolic Equations. World Scientific, 2006.  ISBN 9789812700254.  URL https://books.google.ca/books?id=DnCH1_1YffYC.
  * Magal and Ruan [2018] Pierre Magal and Shigui Ruan.  _Theory and applications of abstract semilinear Cauchy problems_ , volume 201 of _Applied Mathematical Sciences_.  Springer, Cham, 2018.  ISBN 978-3-030-01505-3; 978-3-030-01506-0.  doi:10.1007/978-3-030-01506-0.  URL https://doi.org/10.1007/978-3-030-01506-0.  With a foreword by Glenn Webb.
  * Crandall and Rabinowitz [1971] Michael G. Crandall and Paul H. Rabinowitz.  Bifurcation from simple eigenvalues.  _J. Functional Analysis_ , 8:321–340, 1971.  doi:10.1016/0022-1236(71)90015-2.  URL https://doi.org/10.1016/0022-1236(71)90015-2.
  * Crandall and Rabinowitz [1973] M. G. Crandall and P. H. Rabinowitz.  Bifurcation, perturbation of simple eigenvalues and linearized stability.  _Arch. Rational Mech. Anal._ , 52:161–180, 1973.  ISSN 0003-9527.  doi:10.1007/BF00282325.  URL https://doi.org/10.1007/BF00282325.
  * Shi [1999] J. P. Shi.  Persistence and bifurcation of degenerate solutions.  _J. Funct. Anal._ , 169(2):494–531, 1999.  ISSN 0022-1236.  doi:10.1006/jfan.1999.3483.  URL https://doi.org/10.1006/jfan.1999.3483.
  * Potts and Lewis [2016b] Jonathan R Potts and Mark A Lewis.  Territorial pattern formation in the absence of an attractive potential.  _Journal of mathematical biology_ , 72:25–46, 2016b.
  * Krause et al. [2020] Andrew L Krause, Václav Klika, Thomas E Woolley, and Eamonn A Gaffney.  From one pattern into another: analysis of turing patterns in heterogeneous domains via wkbj.  _Journal of the Royal Society Interface_ , 17(162):20190621, 2020.