# Essential concepts of digital topology
(digital $k$-connectivity and $k$-adjacencis for digital products)

Sang-Eon Han  Department of Mathematics Education, Institute of Pure and
Applied Mathematics
Jeonbuk National University, Jeonju-City Jeonbuk, 54896, Republic of Korea
e-mail<EMAIL_ADDRESS>Tel: 82-63-270-4449.

###### Abstract.

The paper refers to several concepts which are essential to studying digital
objects from the viewpoint of digital topology: digital $k$-connectivity or
digital $k$-adjacency, $C$-compatible and normal $k$-adjacency for a digital
product. Since L. Boxer has often mentioned the origins of these concepts in
an inaccurate way, we discuss something incorrectly cited or mentioned in
Boxer’s papers according to the facts.

AMS Classification: 68U05
Keywords: digital topology, digital image, digital $k$-connectivity,
$k$-adjacency, $C$-compatible $k$-adjacency, normal $k$-adjacency.

##  1\. Terminology for studying digital images

The present paper will often follow the notation. ${\mathbb{N}}$ indicates the
set of natural numbers (or positive integers), ${\mathbb{Z}}$ means the set of
integers, $``$$:=$” is used for introducing a new term and $``$$\subset$”
stands for a subset as usual. Besides, for $a,b\in{\mathbb{Z}}$: the set of
integers, $[a,b]_{\mathbb{Z}}:=\\{t\in{\mathbb{Z}}\,|\,a\leq t\leq b\\}$. We
also abbreviate the term $``$digital topological” as $DT$. In addition,
$X^{\sharp}$ indicates the cardinality of the given set $X$.

##  2\. Digital $k$-connectivities, digital $k$-neighborhood, a length between
two points, and simple closed $k$-curves in ${\mathbb{Z}}^{n},n\geq 2$

In order to study digital images (or digital objects) from the viewpoint of
digital topology, we had better consider digital $k$-connectivities (or
digital $k$-adjacencies) for the $n$-dimensional lattice set
${\mathbb{Z}}^{n},n\in{\mathbb{N}}$. However, before 2004 (see [6, 7, 8]),
focusing on the $3$-dimensional world, all papers in digital topology dealt
with only low dimensional digital image $(X,k)$, i.e.,
$X\subset{\mathbb{Z}}^{3}$ with one of the $k$-adjacency of ${\mathbb{Z}}^{3}$
[3, 28, 31, 32]. For instance, the $2$-adjacency for ${\mathbb{Z}}$; the $4$\-
and $8$-adjacency for ${\mathbb{Z}}^{2}$; and the $6$-, $18$-, and
$26$-adjacency for ${\mathbb{Z}}^{3}$ [31, 32].
After generalizing these $k$-connectivities into those for $n$-dimensional
cases, $n\in{\mathbb{N}}$, since 2004 ([6, 7, 8]), a digital image $(X,k)$ has
often assumed to be a set $X$ in ${\mathbb{Z}}^{n}$ with one of the
$k$-adjacency relations of ${\mathbb{Z}}^{n}$ from (2.2) below. Indeed, the
$k$-adjacency relations for $X\subset{\mathbb{Z}}^{n},n\in{\mathbb{N}}$, were
initially established in [8] (see also [6, 7, 11, 16]). In detail, for a
natural number $t$, $1\leq t\leq n$, the distinct points
$p=(p_{i})_{i\in[1,n]_{\mathbb{Z}}}$ and
$q=(q_{i})_{i\in[1,n]_{\mathbb{Z}}}\in{\mathbb{Z}}^{n}$ are $k(t,n)$-adjacent
(see the page 74 of [8]) if

$None$ $\text{at most}\,\,t\,\,\text{of their coordinates differ by}\,\,\pm
1\,\,\text{and the others~{}coincide.}$

According to this criterion, the digital $k(t,n)$-connectivities of
${\mathbb{Z}}^{n},n\in{\mathbb{N}}$, are formulated [8] (see also [16]) as
follows:

$None$
$k:=k(t,n)=\sum_{i=1}^{t}2^{i}C_{i}^{n},\text{where}\,\,C_{i}^{n}:={n!\over(n-i)!\
i!}.$

For instance, in ${\mathbb{Z}}^{4}$, $8$-, $32$-, $64$-, and $80$-adjacency;
in ${\mathbb{Z}}^{5}$, $10$-, $50$-, $130$-, $210$-, and $242$-adjacency; and
in ${\mathbb{Z}}^{6}$, $k(1,6)=12$, $k(2,6)=72$, $k(3,6)=232$, $k(4,6)=472$,
$k(5,6)=664$, $k(6,6)=728$-adjacency are considered [6, 16].
The first formula for the $k$-adjacency relations of
${\mathbb{Z}}^{n},n\in{\mathbb{N}}$, is shown in [6, 7, 8] and the second
version of them was suggested in [11] and the process of formulating them was
mentioned in [13] (see Appendix of [13]). Eventually, the final version of
them was established in [16] as in (2.2).
Besides, the notation $``$$k(t,n)$” of (2.2) indeed gives us much information
such as the dimension $n$ and the number $t$ associated with the statement of
(2.1). For instance, we may consider an $8$-adjacency in both
${\mathbb{Z}}^{2}$ and ${\mathbb{Z}}^{4}$. Then, in the case that we follow
the notation $k:=k(t,n)$, we can easily make a distinction between them by
using $k(2,2)=8=k(1,4)$ (see (2.1) and (2.2)). Meanwhile, in Boxer and et
al.’s paper [5], the notation $c_{2}=8$ in ${\mathbb{Z}}^{2}$ and $c_{1}=8$ in
${\mathbb{Z}}^{4}$ are shown. Indeed, the notation of
$c_{u},u\in[1,n]_{\mathbb{Z}}$, is exactly based on the criterion of (2.1).
However, Boxer did not refer to this origin. Boxer indeed took just only
another notation with the same meaning as the criterion referred to in (2.1).
Even though the usage can be just a matter of preference, Han’s approach using
$k:=k(t,n)$ can be helpful to study more complicated cases such as digital
products with a $C$-compatible, a normal, and a pseudo-normal $k$-adjacency
[8, 12, 15, 26] and their applications.
Hereinafter, a relation set $(X,k)$ is assumed in ${\mathbb{Z}}^{n}$ with a
$k$-adjacency of (2.2), called a digital image. Namely, we now abbreviate
$``$digital image $(X,k)$” as $``$$(X,k)$” if there is no confusion.

Let us now recall a $``$digital $k$-neighborhood of the point $x_{0}$” in
$(X,k)$. For $(X,k)$ and a point $x_{0}\in X$, we have often used the notation

$None$ $N_{k}(x_{0},1):=\\{x\in X\,|\,\,x\,\,\text{is}\,\,k\text{-adjacent
to}\,\,x_{0}\\}\cup\\{x_{0}\\},$

which is called a digital $k$-neighborhood of the point $x_{0}$ in $(X,k)$.
The notion of (2.3) indeed facilitates some studies of digital images, e.g.,
digital covering spaces, digital homotopy, digital-topological ($DT$-, for
brevity) $k$-group [20, 21], and so on. According to the literature on the
digital $k$-neighborhood of (2.3), Kong and Rosenfeld [28] and Bertrand [3]
used the following notation (see the lines 14-16 from the bottom on the page
359 of [28]), as follows:
In ${\mathbb{Z}}^{2}$, for $p\in{\mathbb{Z}}^{2}$,
$N(p):=\\{x\,|\,\,x\,\,\text{is}\,\,8\text{-adjacent to}\,\,p\\}\cup\\{p\\}$
and
in ${\mathbb{Z}}^{3}$, for $p\in{\mathbb{Z}}^{3}$,
$N(p):=\\{x\,|\,\,x\,\,\text{is}\,\,26\text{-adjacent to}\,\,p\\}\cup\\{p\\}$.
In ${\mathbb{Z}}^{3}$, for $p\in{\mathbb{Z}}^{3}$,
$N^{\ast}(p):=\\{x\,|\,\,x\,\,\text{is}\,\,26\text{-adjacent to}\,\,p\\}$ [2].
To make a distinction from these notations, Han [6, 7, 8] used the notation of
(2.3) because it has some advantages. Meanwhile, based on the notation of
(2.3), we can represent it by using another term such as $N_{k}(x_{0})$,
$D_{k}(x_{0})$, or $E_{k}(x_{0})$ (see the page 3 of [17]), and so forth.
However, the notation of (2.3) has the following history. The notation of
(2.3) indeed comes from the $``$digital $k$-neighborhood of $x_{0}\in X$ with
radius $\varepsilon\in{\mathbb{N}}$” in [6, 7, 8] (see (2.4) below). More
precisely, for $(X,k),X\subset{\mathbb{Z}}^{n}$ and two distinct points $x$
and $x^{\prime}$ in $X$, a simple $k$-path from $x$ to $x^{\prime}$ with $l$
elements is said to be a sequence $(x_{i})_{i\in[0,l]_{\mathbb{Z}}}$ in $X$
such that $x_{0}=x$ and $x_{l}=x^{\prime}$ and further, $x_{i}$ and $x_{j}$
are $k$-adjacent if and only if $|i-j|=1$ [28]. We say that a length of the
simple $k$-path, denoted by $l_{k}(x,y)$, is the number $l$ [8]. For instance,
let $(X=\\{x_{0}=(0,0),x_{1}=(1,-1),x_{2}=(2,0),x_{3}=(1,1)\\},8)$. Then we
observe that $l_{8}(x_{0},x_{1})=1,l_{8}(x_{0},x_{2})=2$, and
$l_{8}(x_{0},x_{3})=1.$

######  Definition 1.

[6] (see also [8]) For a digital image $(X,k)$ on ${\mathbb{Z}}^{n}$, the
digital $k$-neighborhood of $x_{0}\in X$ with radius $\varepsilon$ is defined
in $X$ to be the following subset of $X$

$None$ $N_{k}(x_{0},\varepsilon)=\\{x\in
X\,|\,l_{k}(x_{0},x)\leq\varepsilon\\}\cup\\{x_{0}\\},$

where $l_{k}(x_{0},x)$ is the length of a shortest simple $k$-path from
$x_{0}$ to $x$ and $\varepsilon\in{\mathbb{N}}$.

For instance, let
$Y=\\{y_{0}=(0,0),y_{1}=(1,0),y_{2}=(1,1),y_{3}=(0,1),y_{4}=(-1,2)\\}$ and
$(Y,8)$. Then $N_{8}(y_{0},1)=Y\setminus\\{y_{4}\\}$ and consider
$N_{8}(y_{0},2)=Y$.

Owing to Definition 1, it turns out that $N_{k}(x_{0},1)$ of (2.3) is a
special case of $N_{k}(x_{0},\varepsilon)$ of (2.4). Note that the
neighborhood $N_{k}(x_{0},2)$ plays an important role in establishing the
notions of a radius $2$-local $(k_{1},k_{2})$-isomorphism and a radius
$2$-$(k_{1},k_{2})$-covering map which is strongly associated with the
homotopy lifting theorem [7] and is essential for calculating digital
fundamental groups of digital images $(X,k)$ [5, 8].

We say that $(Y,k)$ is $k$-connected [28] if for arbitrary distinct points
$x,y\in Y$ there is a finite sequence $(y_{i})_{i\in[0,l]_{\mathbb{Z}}}$ in
$Y$ such that $y_{0}=x$, $y_{l}=y$ and $y_{i}$ and $y_{j}$ are $k$-adjacent if
$|\,i-j\,|=1$ [28]. Besides, a singleton is assumed to be $k$-connected.

A simple closed $k$-curve (or $k$-cycle) with $l$ elements in
${\mathbb{Z}}^{n},n\geq 2$, denoted by $SC_{k}^{n,l}$ [8, 28], $l(\geq
4)\in{\mathbb{N}}$, is defined as the sequence
$(y_{i})_{i\in[0,l-1]_{\mathbb{Z}}}\subset{\mathbb{Z}}^{n}$ such that $y_{i}$
and $y_{j}$ are $k$-adjacent if and only if $|\,i-j\,|=\pm 1(mod\,l)$. Indeed,
the number $l(\geq 4)$ is very important in studying $k$-connected digital
images from the viewpoint of digital homotopy theory. The number $l$ of
$SC_{k}^{n,l}$ can be an even or an odd number. Namely, the number $l$ depends
on the dimension $n$ and the $k$-adjacencies, as follows:

$None$ $\left\\{\begin{aligned} &(1)\,\text{in the case of}\,\,k=2n(n\neq
2),\,\,\text{we have}\,\,l\in{\mathbb{N}}_{0}\setminus\\{2\\};\\\
&(2)\,\text{in the case of}\,\,k=4(n=2),\,\,\text{we
obtain}\,\,l\in{\mathbb{N}}_{0}\setminus\\{2,6\\};\\\ &(3)\,\text{in the case
of}\,\,k=8(n=2),\,\,\text{we
have}\,\,l\in{\mathbb{N}}\setminus\\{1,2,3,5\\};\\\ &(4)\,\text{in the case
of}\,\,k=18(n=3),\,\,\text{we
obtain}\,\,l\in{\mathbb{N}}\setminus\\{1,2,3,5\\};\,\text{and}\\\
&(5)\,\text{in the case of}\,\,k:=k(t,n)\,\,\text{such that}\,\,3\leq t\leq
n,\\\ &\,\text{we
have}\,\,l\in{\mathbb{N}}\setminus\\{1,2,3\\}.\end{aligned}\right\\}$

######  Remark 2.1.

(see (5) of [18]) The number $l$ of $SC_{k}^{n,l}$ depends on the dimension
$n\in{\mathbb{N}}\setminus\\{1\\}$ and the number
$l\in\\{2a,2a+1\\},a\in{\mathbb{N}}\setminus\\{1\\}$.

##  3\. $DT$-versions of the adjacencies of graph products established by
Berge [1] and Harray [25]; a $C$-compatible and normal $k$-adjacency of a
digital product established by Han [8, 12, 15]

Based on the classical adjacencies of a Cartesian product of two graphs in [1]
and [25], the papers [8, 12, 15] introduced digital topological ($DT$-, for
brevity) versions of them. More precisely, assume two digital images
$(X,k_{1})$ on ${\mathbb{Z}}^{n_{1}}$ and $(Y,k_{2})$ on
${\mathbb{Z}}^{n_{2}}$ and further, consider a Cartesian product $X\times
Y\subset{\mathbb{Z}}^{n_{1}+n_{2}}$. Then we need some meaningful
$k$-adjacencies of ${\mathbb{Z}}^{n_{1}+n_{2}}$ for $X\times Y$ to be a
digital image $(X\times Y,k)$, where the $k$-adjacency of
${\mathbb{Z}}^{n_{1}+n_{2}}$ should be considered according to (2.2). Indeed,
the adjacencies for a Cartesian product graph in [25] and a strong product
graph in [1] facilitated the establishment of some $k$-adjacencis for a
digital product $X\times Y\subset{\mathbb{Z}}^{n_{1}+n_{2}}$. The former was
helpful to develop the so-called $C$-compatible $k$-adjacency [15] (or the
$L_{C}$-property of a Cartesian product in [12]) and the latter was motivated
to develop a normal $k$-adjacency of $X\times Y$ [8]. However, note that a
$C$-compatible $k$-adjacency and a normal $k$-adjacency of $X\times Y$ need
not be equal to the classical adjacencies of a Cartesian product graph in [25]
and a strong product of graphs in [1], respectively (see Example 3.5 and
Remark 3.6). More precisely, given two classical graphs, an adjacency of a
Cartesian product graph in [25] always exists. However, a $C$-compatible
$k$-adjacency has its own features (see Example 3.5 and Remark 3.6).
Similarly, comparing with the adjacency of strong product graph in [1], we
clearly observe that a normal $k$-adjacency has its own features (see Example
3.5 and Remark 3.6).

Motivated by the adjacency of a Cartesian product graph in [25], a
$C$-compatible $k$-adjacency of a digital product was defined as follows:

######  Definition 2.

[8] ($DT$-version of the adjacency of a product graph in [25]) For two digital
images $(X,k_{1})$ on ${\mathbb{Z}}^{n_{1}}$ and $(Y,k_{2})$ on
${\mathbb{Z}}^{n_{2}}$, consider a Cartesian product $X\times
Y\subset{\mathbb{Z}}^{n_{1}+n_{2}}$. We say that distinct two points
$p:=(x,y)$ and $q:=(x^{\prime},y^{\prime})$ is $C$-compatible $k$-adjacent if
and only if
(1) $x=x^{\prime}$, $y$ is $k_{2}$-adjacent to $y^{\prime}$ or
(2) $y=y^{\prime}$, $x$ is $k_{1}$-adjacent to $x^{\prime}$.
Then we say that the $k$-adjacency of $(X\times Y,k)$ is $C$-compatible with
the given two $(X,k_{1})$ and $(Y,k_{2})$.

In Definition 2, even though the logic for establishing a $C$-compatible
$k$-adjacency of $X\times Y$ seems to be similar to that for the adjacency of
a Cartesian product graph in [25], we need to note that the $C$-compatible
$k$-adjacency of $X\times Y$ of Definition 2 is different from the Cartesian
adjacency of [12] (see Example 3.5 and Remarks 3.6).

Owing to the digital $k$-neighborhood of (2.3), Definition 2 can be
represented as follows:

######  Definition 3.

(A $C$-compatible $k$-adjacency of a digital product [15] and the
$L_{C}$-property of a Cartesian product in [12])
Assume two digital images $(X_{i},k_{i})$ on
${\mathbb{Z}}^{n_{i}},k_{i}:=k(t_{i},n_{i}),i\in\\{1,2\\}$. Then we say that a
$k$-adjacency of $X_{1}\times X_{2}\subset{\mathbb{Z}}^{n_{1}+n_{2}}$ is
$C$-compatible with the given $k_{i}$-adjacency of $(X_{i},k_{i})$,
$i\in\\{1,2\\}$ if every point $p:=(x_{1},x_{2})$ in $X_{1}\times X_{2}$
satisfies the following property:

$None$ $N_{k}(p,1)=(N_{k_{1}}(x_{1},1)\times\\{x_{2}\\})\cup(\\{x_{1}\\}\times
N_{k_{2}}(x_{2},1)),$

where the $k$-adjacency is one of the typical $k$-adjacencies of
${\mathbb{Z}}^{n_{1}+n_{2}}$ stated in (2.2).

After comparing Definitions 2 and 3, the following is obtained.

######  Proposition 3.1.

[14] A $k$-adjacency of $X_{1}\times X_{2}\subset{\mathbb{Z}}^{n_{1}+n_{2}}$
is $C$-compatible with the given $k_{i}$-adjacency of $(X_{i},k_{i})$,
$i\in\\{1,2\\}$, if and only if every point $p:=(x_{1},x_{2})$ in $X_{1}\times
X_{2}$ satisfies the following property:

$None$ $N_{k}(p,1)=(N_{k_{1}}(x_{1},1)\times\\{x_{2}\\})\cup(\\{x_{1}\\}\times
N_{k_{2}}(x_{2},1)),$

where the $k$-adjacency is one of the typical $k$-adjacencies of
${\mathbb{Z}}^{n_{1}+n_{2}}$ stated in (2.2).

In Definitions 2 and 3, we strongly need to focus on the $k$-adjacency of
$X\times Y\subset{\mathbb{Z}}^{n_{1}+n_{2}}$ as in (2.2). We can easily
observe some difference between a $C$-compatible $k$-adjacency and the
adjacency of a graph product of [25], as follows:

######  Remark 3.2.

Assume two digital images $(X,k_{1})$ on ${\mathbb{Z}}^{n_{1}}$ and
$(Y,k_{2})$ on ${\mathbb{Z}}^{n_{2}}$. Not every $X\times Y$ alway has a
$C$-compatible $k$-adjacency in ${\mathbb{Z}}^{n_{1}+n_{2}}$ and further, the
number $k$ of a $C$-compatible $k$-adjacency need not be unique (see Example
3.5 and Remark 3.6).

Note that the notion of a $C$-compatible $k$-adjacency is essential to
developing a $DT$-$k$-group (see [20, 21]).

Unlike the $C$-compatible $k$-adjacency of Definitions 2 and 3, motivated by
the strong product graph in [1], a normal $k$-adjacency of digital product was
defined as follows:

######  Definition 4.

[8] ($DT$-version of the adjacency of a strong product graph in [1]) For two
digital images $(X,k_{1})$ on ${\mathbb{Z}}^{n_{1}}$ and $(Y,k_{2})$ on
${\mathbb{Z}}^{n_{2}}$, consider a Cartesian product $X\times
Y\subset{\mathbb{Z}}^{n_{1}+n_{2}}$. We say that distinct two points
$p:=(x,y)$ and $q:=(x^{\prime},y^{\prime})$ is normally $k$-adjacent if and
only if
(1) $x=x^{\prime}$, $y$ is $k_{2}$-adjacent to $y^{\prime}$ or
(2) $y=y^{\prime}$, $x$ is $k_{1}$-adjacent to $x^{\prime}$ or
(3) $x$ is $k_{1}$-adjacent to $x^{\prime}$ and $y$ is $k_{2}$-adjacent to
$y^{\prime}$.
Then we say that the $k$-adjacency of $(X\times Y,k)$ is normally $k$-adjacent
with the given two $(X,k_{1})$ and $(Y,k_{2})$

In Definition 4, we strongly focus on the $k$-adjacency of
${\mathbb{Z}}^{n_{1}+n_{2}}$ for $X\times Y$ according to (2.2). In Definition
4, even though the logic for establishing a normal $k$-adjacency of $X\times
Y$ seems to be similar to that for the adjacency of a strong product graph in
[1], it is clear that the normal $k$-adjacency of $X\times Y$ of Definition 4
is different from the adjacency of [1] (see Example 3.5 and Remarks 3.6).

Using the digital $k$-neighborhood of (2.3), Definition 4 can be represented
as follows:

######  Definition 5.

(A normal $k$-adjacency of a digital product [8, 15]) Assume two digital
images $(X_{i},k_{i})$ on
${\mathbb{Z}}^{n_{i}},k_{i}:=k(t_{i},n_{i}),i\in\\{1,2\\}$. Then we say that a
$k$-adjacency of $X_{1}\times X_{2}\subset{\mathbb{Z}}^{n_{1}+n_{2}}$ is
normal if every point $p=(x_{1},x_{2})$ in $X_{1}\times X_{2}$ satisfies the
following property:

$None$ $N_{k}(p,1)=N_{k_{1}}(x_{1},1)\times N_{k_{2}}(x_{2},1),$

where the $k$-adjacency is one of the $k$-adjacencies of
${\mathbb{Z}}^{n_{1}+n_{2}}$ stated in (2.2).

Based on Definitions 4 and 5, the following is obtained.

######  Proposition 3.3.

[15] A $k$-adjacency of $X_{1}\times X_{2}\subset{\mathbb{Z}}^{n_{1}+n_{2}}$
is normal with the given $k_{i}$-adjacency of $(X_{i},k_{i})$, $i\in\\{1,2\\}$
if and only if every point $p=(x_{1},x_{2})$ in $X_{1}\times X_{2}$ satisfies
the following property:

$None$ $N_{k}(p,1)=N_{k_{1}}(x_{1},1)\times N_{k_{2}}(x_{2},1),$

where the $k$-adjacency is one of the $k$-adjacencies of
${\mathbb{Z}}^{n_{1}+n_{2}}$ stated in (2.2).

######  Remark 3.4.

Assume two digital images $(X,k_{1})$ on ${\mathbb{Z}}^{n_{1}}$ and
$(Y,k_{2})$ on ${\mathbb{Z}}^{n_{2}}$. Not every $X\times Y$ alway has a
normal $k$-adjacency in ${\mathbb{Z}}^{n_{1}+n_{2}}$ and further, a normal
$k$-adjacency need not be unique (see Example 3.5 and Remark 3.6).

To confirm some difference between the adjacency of a Cartesian product graph
in [25] and the $C$-compatible $k$-adjacency of a digital product and further,
the adjacency of a strong product graph in [1] and the normal $k$-adjacency of
a digital product of Definitions 4 and 5, we observe the following:

######  Example 3.5.

(1) $SC_{4}^{2,4}\times SC_{4}^{2,4}$ has a $C$-compatible $8$-adjacency.
(2) $SC_{26}^{3,5}\times SC_{26}^{3,5}$ has a $C$-compatible
$k(3,6)$-adjacency and a normal $k(6,6)$-adjacency.
(3) Each of $SC_{8}^{2,6}\times SC_{8}^{2,6}$ and $SC_{8}^{2,4}\times
SC_{8}^{2,6}$ has a normal $80$-adjacency.
(4) $SC_{8}^{2,4}\times SC_{8}^{2,4}$ has a $C$-compatible $k$-adjacency,
$k\in\\{32=k(2,4),64=k(3,4)\\}$.
(5) Let
$X=\\{x_{0}=(0,0,0),x_{1}=(1,1,0),x_{2}=(1,2,1),x_{3}=(0,3,1),x_{4}=(-1,2,1),x_{5}=(-1,1,0)\\}$
in ${\mathbb{Z}}^{3}$ with an $18$-adjacency. Indeed, $(X,18)$ is an example
for $SC_{18}^{3,6}$. Then $X^{2}$ has a normal $k$-adjacency,
$k\in\\{k(4,6)=472,k(5,6)=664,k(6,6)=728\\}$.

######  Remark 3.6.

When comparing the adjacency of a Cartesian product graph in [25] and a strong
product graph in [1] and the $C$-compatible and the normal $k$-adjacency of a
digital product, respectively, we observe some difference among them. Given
two graphs, even though there are adjacencies for a Cartesian product graph in
[25] and a strong product graph in [1], the $DT$-versions of them have some
their own features, as follows:
Let $MSC_{18}:=(x_{i})_{i\in[0,5]_{\mathbb{Z}}}$ in ${\mathbb{Z}}^{3}$, where
$x_{0}=(0,0,0),x_{1}=(1,-1,0),x_{2}=(1,-1,1),x_{3}=(2,0,1),x_{4}=(1,11),x_{5}=(1,1,0)$
as a special kind of an example for $SC_{18}^{3,6}$. Then the following is
obtained.
(1) $MSC_{18}\times MSC_{18}\subset{\mathbb{Z}}^{6}$ has neither a
$C$-compatible $k$-adjacency nor a normal $k$-adjacency in ${\mathbb{Z}}^{6}$.
(2) $SC_{4}^{2,4}\times SC_{8}^{2,6}$ does not have a normal $k$-adjacency in
${\mathbb{Z}}^{4}$, $k\in\\{8,32,64,80\\}$.

As mentioned in Example 3.5 and Remark 3.6, even though there are some big
differences between the $C$-compatible $k$-adjacency of Han and the adjacency
of Harray in [22], and further, between the normal $k$-adjacency of Han and
the adjacency of Berge in [1], Boxer often does not make a distinction between
them in his papers.

######  Remark 3.7.

(Utilities of the $C$-compatible and normal $k$-adjacency)
(1) Calculation of digital fundamental groups of digital products [12, 15].
(2) The $C$-compatible $k$-adjacency is essential for developing
$DT$-$k$-groups [20, 21].

##  4\. Further works

We have discussed some concepts of digital topology which have been used to
study digital topology. Besides, we explained the $DT$-version of the
$k$-adjacencies of digital products which incorrectly mentioned in Boxer’s
works. Furthemore, as a further work, we will discuss some history of the
notions of digital $k$-continuity, digital $k$-isomorphism, local
$k$-isomorphism, and radius $2$-local $k$-isomorphism. In addition, based on
the works [3, 4, 5, 6, 7, 8, 10, 11, 19, 22, 24, 27, 29, 30], we will remark
on Boxer’s misconception on Han’s papers [14, 19] and a revision of a
pseudo-$(k_{1},k_{2})$-covering space.

Conflicts of Interest: The author declares no conflict of interest.

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# Periodic Traveling Waves in an Integro-Difference Equation With Non-
Monotonic Growth and Strong Allee Effect

Michael Nestor and Bingtuan Li M. Nestor’s email is<EMAIL_ADDRESS>B. Li
was partially supported by the National Science Foundation under Grant
DMS-1515875 and Grant DMS-1951482. Department of Mathematics, University of
LouisvilleLouisville, KY 40292

###### Abstract

We derive sufficient conditions for the existence of a periodic traveling wave
solution to an integro-difference equation with a piecewise constant growth
function exhibiting a stable period-2 cycle and strong Allee effect. The mean
traveling wave speed is shown to be the asymptotic spreading speed of
solutions with compactly supported initial data under appropriate conditions.
We then conduct case studies for the Laplace kernel and uniform kernel.

Key words: Integro-difference equation, period two cycle, Allee effect,
periodic traveling wave.

AMS Subject Classification: 92D40, 92D25.

##  1 Introduction

Integro-difference equations in the form

$\displaystyle
u_{n+1}(x)\;=\;Q[u_{n}](x)\;:=\,\int^{\infty}_{-\infty}k(x-y)\,g\left(u_{n}(y)\right)\,\mathrm{d}y$
(1.1)

are of great interest in the studies of invasions of populations with discrete
generations and separate growth and dispersal stages. They have been used to
predict changes in gene frequency [8, 9, 10, 14, 17], and applied to
ecological problems [2, 3, 4, 5, 7, 11, 12, 13]. Previous rigorous studies on
integro-difference equations have assumed that the growth function is
nondecreasing [17, 18], or is non-monotone without Allee effect [10, 16]. The
results show existence of constant spreading speeds and travelng waves with
fixed shapes and speeds. Sullivan et al. [15] demonstrated numerically that an
integro-difference equation with a non-monotone growth function exhibiting a
strong Allee effect can generate traveling waves with fluctuating speeds, and
Otto [13] showed that such an equation can have robust non-spreading
solutions. In this paper we give conditions for the existence of a periodic
traveling wave with two intermediate speeds for an integro-difference equation
with a piecewise constant growth function that exhibits a strong Allee effect
and a period-2 cycle.

Piecewise constant growth functions have been used in the studies of integro-
difference equations; see for example [6, 11, 13, 15]. Such an equation is
analytically tractable and it can provide specific insights into the dynamics
of solutions. For the piecewise constant growth function

$g(u)=\begin{cases}0,&\text{if }u<a,\\\ 1,&\text{if }u\geq a,\end{cases}$
(1.2)

with $0<a<1$, which exhibits a strong Allee effect and monotonicity, Kot et
al. [6] and Lutscher [11] (Section 6.1) investigated the range expansion of
the population and the spreading speed, and Sullivan et al. [15] studied
oscillations in spreading speeds when the dispersal kernel is density
dependent. More discussions about integro-difference equations with piecewise
growth functions can be found in Lutscher [11] (Chapter 15).

In this paper, we consider the integro-difference equation (1.1) with

$g(u)=\begin{cases}0,&\text{if }u<a,\\\ 1,&\text{if }a\leq u\leq b,\\\
m,&\text{if }u>b,\end{cases}$ (1.3)

with $0<a<m<b<1$. $g(u)$ is a piecewise constant non-monotone growth function
exhibiting a strong Allee effect [1]. Specifically, it has a stable fixed
point at zero and a stable period two cycle $(1,m)$ with $a$ the Allee
threshold value. This is the function considered in Otto [13] where non-
spreading solitions are studied. It may be viewed as an extension of (1.2). A
graphical demonstration of $g$ defined by (1.3) is given by Figure 1.

Figure 1: The growth of $g(u)$ defined by (1.3) with $0<a<m<b<1$.

We rigorously construct periodic traveling waves with two intermediate speed
for (1.1) with $g$ given by (1.3) and a proper dispersal kernel $k$. To the
best of our knowledge, this is the first time that traveling waves with
oscillating speeds have been analytically established for scalar
spatiotemporal equations with constant parameters. We also show that the mean
traveling wave speed is the spreading speed of solutions with compactly
supported initial data under appropriate conditions. We finally conduct case
studies for the Laplace kernel and uniform kernel.

##  2 Results

In this section, we present our main results about our model. Throughout, we
assume all population density functions are bounded elements of
$C(\mathbb{R})$, the space of real-valued continuous functions. It is
straightforward to verify this space is closed under iteration of $Q$. We
denote the norm of a function $u\in C(\mathbb{R})$ by $||u||_{\infty}$.

We make the following assumptions about the dispersal kernel.

######  Hypothesis 2.1.

$k(x)$ is a piecewise differentiable function satisfying

  1. (H1)

$k(x)\geq 0$ for all $x$, and $\int_{-\infty}^{\infty}k(x)\,\mathrm{d}x=1$,

  2. (H2)

$k(x)=k(-x)$ for all $x$,

  3. (H3)

$k(x)>0$ for all $x\in(-\sigma,\sigma)$ for some $0<\sigma\leq\infty$,

  4. (H4)

for all $\lambda\in(0,1)$, for all $A,B\in\mathbb{R}$ with $A>B$, the
expression

$k(x-A)-\lambda k(x-B)$

has $<2$ sign changes, and

  5. (H5)

for all $\lambda\in(0,1)$, for all $A,B\in\mathbb{R}$ with $A>B$, the
expression

$k(x-r-A)-\lambda k(x-r-B)-k(x+r+A)+\lambda k(x+r+B)$

has $<4$ sign changes for sufficiently large $r>0$.

###  2.1 Periodic traveling waves

Let $w_{1},w_{2}\in C(\mathbb{R})$ be bounded, continuous functions defined by

$w_{1}(x):=\int_{x}^{\infty}k(y)\,\mathrm{d}y$ (2.1)

and

$w_{2}(x):=Q[w_{1}](x+c^{*})=\int_{-\infty}^{\infty}k(x+c^{*}-y)g(w_{1}(y))\,\mathrm{d}y$
(2.2)

where $c^{*}$ is a constant defined by

$c^{*}:=\frac{1}{2}\sup\\{x\in\mathbb{R}\mid Q[w_{1}](x)=a\\}.$ (2.3)

The following theorem shows that $w_{1}$ and $w_{2}$ generate a traveling
wave-like solution to (1.1). We term this solution a periodic traveling wave
since it alternates between two distinct wave profiles (with different limits
at $-\infty$) while traveling with mean speed $c^{*}$.

######  Theorem 2.2.

Let $(W_{n})_{n=0}^{\infty}$ be a sequence of bounded, continuous functions
defined by

$W_{n}(x):=\begin{cases}w_{1}(x-nc^{*}),&n\text{ even},\\\
w_{2}(x-nc^{*}),&n\text{ odd}.\end{cases}$ (2.4)

If $||w_{2}||_{\infty}<b$, then $W_{n+1}=Q[W_{n}]$ for all $n\geq 0$.

###### Proof.

For $i=1,2$, define the right inverse of $w_{i}$ by

$\Phi_{i}(\ell):=\sup\\{x\in\mathbb{R}\mid w_{i}(x)=\ell\\},\quad i=1,2.$
(2.5)

Note that $\Phi_{i}(\ell)$ is finite for $0<\ell<||w_{i}||_{\infty}$, where
$||w_{1}||_{\infty}=1$, and $a<||w_{2}||_{\infty}<b$ by the assumption of this
theorem.

By the translation invariance property of $Q$, it suffices to show

$Q[w_{1}](x)=w_{2}(x-c^{*})$ (2.6)

and

$Q[w_{2}](x)=w_{1}(x-c^{*})$ (2.7)

for all $x\in\mathbb{R}$. The first equation follows immediately from
definition (2.2). To prove the second, observe that $w_{1}$ is monotonically
decreasing and satisfies $w_{1}(\infty)=0$ and $w_{1}(-\infty)=1$ (see for
example Figure 2). Thus,

$g(w_{1}(x))=\begin{cases}m,&x<\Phi_{1}(b),\\\ 1,&\Phi_{1}(b)\leq
x\leq\Phi_{1}(a),\\\ 0,&x>\Phi_{1}(a).\end{cases}$

Applying the integro-difference operator yields

$\displaystyle w_{2}(x-c^{*})$ $\displaystyle=Q[w_{1}](x)$
$\displaystyle=m\int_{-\infty}^{\Phi_{1}(b)}k(x-y)\,\mathrm{d}y+\int_{\Phi_{1}(b)}^{\Phi_{1}(a)}k(x-y)\,\mathrm{d}y$
$\displaystyle=w_{1}(x-\Phi_{1}(a))-(1-m)\,w_{1}(x-\Phi_{1}(b)).$

We can then differentiate with respect to $x$:

$\frac{\,\mathrm{d}w_{2}}{\,\mathrm{d}x}\Big{|}_{x-c^{*}}=-k(x-\Phi_{1}(a))+(1-m)\,k(x-\Phi_{1}(b)).$

From (H4), $\frac{\,\mathrm{d}w_{2}}{\,\mathrm{d}x}$ has at most 1 sign
change; thus, $w_{2}$ has at most 1 turning point. We also have $w_{2}(x)<b$
for all $x$, hence $g(w_{2}(x))\in\\{0,1\\}$.

We claim

$g(w_{2}(x))=\begin{cases}1,&x\leq c^{*},\\\ 0,&x>c^{*},\end{cases}$ (2.8)

where $c^{*}$ is defined by (2.3). This can be shown by arguing in cases
depending on the number of turning points of $w_{2}$.

  1. Case 1.

If $w_{2}$ has no turning points, then $w_{2}$ is monotone decreasing with
$0\leq w_{2}(x)\leq m$ for all $x$. It follows by the intermediate value
theorem (IVT) that $c^{*}=\Phi_{2}(a)$ satisfies (2.8).

  2. Case 2.

If $w_{2}$ has one turning point, say $x_{0}\in\mathbb{R}$, then $w_{2}$ must
be increasing for $x<x_{0}$ and decreasing for $x>x_{0}$. It follows that
$x_{0}$ is a global maximum, and $w_{2}$ is increasing on $(-\infty,x_{0})$
and decreasing on $(x_{0},\infty)$. We have $g(w_{2}(x))=1$ for $x<x_{0}$. By
the IVT, $c^{*}=\Phi_{2}(a)$ is the unique solution to (2.8).

Taking the convolution of equation (2.8) with $k(x)$ yields

$Q[w_{2}](x)\,=\,\int_{-\infty}^{c^{*}}k(x-y)\,\mathrm{d}y\,=\,w_{1}(x-c^{*}).$

This completes the proof. ∎

(a) $w_{1}(x)$

(b) $w_{2}(x)$

Figure 2: Profiles of the intermediate traveling waves $w_{1}$ and $w_{2}$,
with $k(x)$ equal to the Laplace kernel (see Section 3). Blue points mark the
intersection of the curves with the level sets $\ell=a$ and $\ell=b$.

Theorem 2.2 describes a rightward periodic traveling wave, but we can define
the corresponding leftward periodic traveling wave via $x\mapsto W_{n}(-x)$.
This is due to the fact that $Q$ is symmetric and translation invariant.

We investigated the intermediate wave speed generated by the periodic
traveling wave solution (2.4). To do this, we define a sequence
$(\xi_{n})_{n=0}^{\infty}$ of continuous functions $(0,m)\to\mathbb{R}$ by

$\xi_{n}(\ell):=\sup\\{x\in\mathbb{R}:W_{n}(x)=\ell\\},\quad\ell\in(0,m).$
(2.9)

This expression yields the rightmost intersection of the curve $u=W_{n}(x)$
with the horizontal line $u=\ell$. By computing the first difference, we
obtain a new function sequence $(c_{n})_{n=0}^{\infty}$ (with the same domain
and range) defined by

$c_{n}(\ell):=\xi_{n+1}(\ell)-\xi_{n}(\ell),\quad\ell\in(0,m).$ (2.10)

From Theorem 2.2, we have that $c_{n}(\ell)$ converges to a period-$2$ cycle
for every $\ell\in(0,m)$.

Our results about the intermediate wave speed sequence are summarized in the
following corollary.

######  Corollary 2.3.

Assume the hypothesis of Theorem 2.2 holds. Then for all $\ell\in(0,m)$, the
sequence $c_{n}(\ell)$ satisfies

$c_{n}(\ell)=c^{*}+\begin{cases}\Phi_{1}(\ell)-\Phi_{2}(\ell),&n\text{
even},\\\ \Phi_{2}(\ell)-\Phi_{1}(\ell),&n\text{ odd}.\end{cases}$ (2.11)

###### Proof.

By Theorem 2.2, we have

$\displaystyle\xi_{n}(\ell)$
$\displaystyle=\begin{cases}\sup\\{x\in\mathbb{R}\mid
w_{1}(x-nc^{*})=\ell\\},&n\text{ even},\\\ \sup\\{x\in\mathbb{R}\mid
w_{2}(x-nc^{*})=\ell\\},&n\text{ odd},\end{cases}$
$\displaystyle=nc^{*}+\begin{cases}\Phi_{1}(\ell),&n\text{ even},\\\
\Phi_{2}(\ell),&n\text{ odd}.\end{cases}$

for every $\ell\in(0,m)$. Taking the first difference yields

$\displaystyle c_{n}(\ell)$ $\displaystyle=\xi_{n}(\ell)-\xi_{n-1}(\ell)$
$\displaystyle=nc^{*}+\begin{cases}\Phi_{1}(\ell),&n\text{ even},\\\
\Phi_{2}(\ell),&n\text{
odd},\end{cases}-(n-1)\,c^{*}+\begin{cases}\Phi_{2}(\ell),&n\text{ even},\\\
\Phi_{1}(\ell),&n\text{ odd},\end{cases}$
$\displaystyle=c^{*}+\begin{cases}\Phi_{1}(\ell)-\Phi_{2}(\ell),&n\text{
even},\\\ \Phi_{2}(\ell)-\Phi_{1}(\ell),&n\text{ odd}.\end{cases}$

∎

Thus, for each fixed $\ell\in(0,m)$, we can define intermediate traveling wave
speeds $c_{1}^{*}$ and $c_{2}^{*}$ by

$c_{1}^{*}(\ell):=c^{*}+\Phi_{1}(\ell)-\Phi_{2}(\ell)$ (2.12)

and

$c_{2}^{*}(\ell):=c^{*}+\Phi_{2}(\ell)-\Phi_{1}(\ell).$ (2.13)

It follows that $\eqref{q}$ has a periodic traveling wave with wave profiles
$w_{1}(x)$ and $w_{2}(x)$, intermediate wave speeds $c_{1}^{*}(\ell)$ and
$c_{2}^{*}(\ell)$, and mean wave speed
$\frac{1}{2}\left(c_{1}^{*}(\ell)+c_{2}^{*}(\ell)\right)=c^{*}$. Note that the
intermediate wave speeds depend on $\ell$, but their sum is identically equal
to $2c^{*}$.

###  2.2 Periodic spreading solutions

By applying the results of Theorem 2.2, we were able to prove the asymptotic
spreading speed of solutions with compact initial data. In particular, we
showed that $c^{*}$ is the asymptotic spreading speed of compactly supported
initial data with sufficient weight above the Allee threshold but below the
overcompensation threshold.

######  Theorem 2.4.

Assume the hypothesis of Theorem 2.2 holds. Suppose $u\in C(\mathbb{R})$ is
bounded, non-negative, and has compact support. If

  1. i.

$||u||_{\infty}<b$,

  2. ii.

the set $A:=\\{x\in\mathbb{R}:u(x)\geq a\\}$ is connected and has sufficiently
large length, and

  3. iii.

$c^{*}>0$,

then the sequence $(u_{n})_{n=0}^{\infty}$ defined by $u_{0}=u$ and
$u_{n+1}=Q[u_{n}]$ for $n\geq 0$ spreads asymptotically with mean speed
$c^{*}$.

###### Proof.

We define two continuous functions
$\tilde{w_{1}},\tilde{w_{2}}:\mathbb{R}\times[0,\infty)\to\mathbb{R}$ by

$\tilde{w_{1}}(x,r):=w_{1}(x)-w_{1}(x+2r)=\int_{-2r}^{0}k(x-y)\,\mathrm{d}y$
(2.14)

and

$\tilde{w_{2}}(x,r):=Q[\tilde{w_{1}}(\,\cdot\,,r)](x+c^{*}).$ (2.15)

Observe that $\tilde{w_{1}}$ is symmetric with respect to $x=-r$, and
$\tilde{w_{2}}$ is symmetric with respect to $x=-r-c^{*}$. This notation can
be justified by observing that $\tilde{w_{1}}(x,r)\to w_{1}(x)$ and
$\tilde{w_{2}}(x,r)\to w_{2}(x)$ as $r\to\infty$ for all $x$. We also set

$\phi_{i}(r,\ell):=\sup\\{x\in\mathbb{R}\mid\tilde{w}_{i}(x,r)=\ell\\},\quad
i=1,2.$ (2.16)

$\phi_{1}$ and $\phi_{2}$ are the right-inverse of $\tilde{w_{1}}$ and
$\tilde{w_{2}}$, respectively. For a fixed $r>0$, the value of
$\phi_{i}(r,\ell)$ is finite if
$0<\ell<||\tilde{w}_{i}(\,\cdot\,,r)||_{\infty}$, with the latter bound
converging to $||w_{i}||_{\infty}$ as $r\to\infty$. They satisfy

$\lim_{r\to\infty}\phi_{i}(r,\ell)=\Phi_{i}(\ell),\quad
0<\ell<||w_{i}||_{\infty},\quad i=1,2.$

We now apply the growth function to $\tilde{w_{1}}$, assuming $r$ is
sufficiently large enough so that
$a<||\tilde{w_{1}}(\,\cdot\,,r)||_{\infty}<b$; this is guaranteed by taking
$r>\Phi_{1}\left(\frac{1-a}{2}\right)$. We have

$g\left(\tilde{w_{1}}(x,r)\right)=\begin{cases}0,&x<-2r-\phi_{1}(r,a),\\\
1,&-2r-\phi_{1}(r,a)<x<-2r-\phi_{1}(r,b),\\\
m,&-2r-\phi_{1}(r,b)<x<\phi_{1}(r,b),\\\ 1,&\phi_{1}(r,b)<x<\phi_{1}(r,a),\\\
0,&\phi_{1}(r,a)<x.\end{cases}$

Note that the above expression converges to $g(w_{1}(x))$ as $r\to\infty$ for
all $x$. We then apply the convolution operator:

$\displaystyle\tilde{w_{2}}(x-c^{*},r)=Q[\tilde{w_{1}}(\,\cdot\,,r)](x)$
$\displaystyle=\int_{-2r-\phi_{1}(r,a)}^{\phi_{1}(r,a)}k(x-y)\,\mathrm{d}y-(1-m)\,\int_{-2r-\phi_{1}(r,b)}^{\phi_{1}(r,b)}k(x-y)\,\mathrm{d}y$
$\displaystyle=w_{1}(x-\phi_{1}(r,a))-(1-m)\,w_{1}(x-\phi_{1}(r,b))$
$\displaystyle+(1-m)\,w_{1}(x+2r+\phi_{1}(r,b))-w_{1}(x+2r+\phi_{1}(r,a)).$

Computing the derivative of this expression yields

$\displaystyle\frac{\,\mathrm{d}\tilde{w_{2}}}{\,\mathrm{d}x}\Big{|}_{x-c^{*}}$
$\displaystyle=-k(x-\phi_{1}(r,a))+(1-m)\,k(x-\phi_{1}(r,b))$
$\displaystyle-(1-m)\,k(x+2r+\phi_{1}(r,b))+k(x+2r+\phi_{1}(r,a)).$

By (H5), the number of turning points of $\tilde{w_{2}}$ is at most 3
(assuming $r$ is sufficiently large). Furthermore, $\tilde{w_{2}}$ is
symmetric around $x=-r-c^{*}$; thus the sign changes must come in pairs or
occur exactly at $x=-r-c^{*}$. We can immediately exclude the cases of 0 and 2
turning points because $\tilde{w_{2}}$ is non-negative, vanishes at
$\pm\infty$, and is not identically zero. As in the proof of Theorem 2.2, this
leaves two cases:

  1. Case 1.

If $\tilde{w_{2}}$ has 1 turning point, then it must occur at $x=-r-c^{*}$. We
have $\tilde{w_{2}}(-r-c^{*},r)=w_{2}(-r-c^{*})-w_{2}(r+c^{*})\to m$ as
$r\to\infty$; hence, it is sufficient to take $r$ large enough so that
$|\tilde{w_{2}}(-r-c^{*},r)-m|<\min\\{|a-m|,|b-m|\\}$. It follows by the
triangle inequality that $\tilde{w_{2}}(x,r)\leq\tilde{w_{2}}(-r-c^{*},r)<b$
for all $x$.

  2. Case 2.

If $\tilde{w_{2}}$ has 3 turning points, then again by symmetry, they must be
given by $\\{-r-c^{*}-t,-r-c^{*},-r-c^{*}+t\\}$, for some $t>0$. We can label
them in increasing order by $t_{1}<t_{2}<t_{3}$. It follows that $t_{1}$ and
$t_{3}$ are maxima with
$\tilde{w_{2}}(t_{1},r)=\tilde{w_{2}}(t_{3},r)\to||w_{2}||_{\infty}$ as
$r\to\infty$, and $t_{2}$ is a local minima with $\tilde{w_{2}}(t_{2},r)\to m$
as $r\to\infty$. Thus $\tilde{w_{2}}$ is monotone increasing on
$(-\infty,t_{1})\cup(t_{2},t_{3})$ and monotone decreasing on
$(t_{1},t_{2})\cup(t_{3},\infty)$. By taking $r$ sufficiently large, we can
make
$\left|\tilde{w_{2}}(t_{1},r)-||w_{2}||_{\infty}\right|=\left|\tilde{w_{2}}(t_{3},r)-||w_{2}||_{\infty}\right|<\left|b-||w_{2}||_{\infty}\right|$
and $\left|\tilde{w_{2}}(t_{2},r)-m\right|<\left|a-m\right|$.

In both cases, we have $a\leq\tilde{w_{2}}(x,r)\leq b$ on a closed interval of
radius $r+c^{*}+\phi_{2}(r,a)$ and $\tilde{w_{2}}(x,r)<a$ elsewhere. Taking
composition with $g$ yields

$g(\tilde{w_{2}}(x-c^{*},r))=\begin{cases}0,&x<-2r-c^{*}-\phi_{2}(r,a),\\\
1,&-2r-c^{*}-\phi_{2}(r,a)\leq x\leq c^{*}+\phi_{2}(r,a),\\\
0,&x>c^{*}+\phi_{2}(r,a).\end{cases}$

Hence,

$\displaystyle Q^{2}[\tilde{w_{1}}(\,\cdot\,,r)](x)$
$\displaystyle=w_{1}(x-c^{*}-\phi_{2}(r,a))-w_{1}(x+2r+c^{*}+\phi_{2}(r,a)).$

Shifting the expression left by $c^{*}+\phi_{2}(r,a)$ units yields

$\displaystyle Q^{2}[\tilde{w_{1}}(\,\cdot\,,r)](x+c^{*}+\phi_{2}(r,a))$
$\displaystyle=\tilde{w_{1}}(x,r+c^{*}+\phi_{2}(r,a)).$ (2.17)

We are now prepared to prove the theorem using an inductive argument. Let $A$
be the level set defined in the statement of the theorem. Without loss of
generality, we write $A=[-r_{0},r_{0}]$, for some $r_{0}>0$. We have

$g(u_{0}(x))=\begin{cases}1,&-r_{0}\leq x\leq r_{0},\\\
0,&\text{else}.\end{cases}$

for all $x$. Applying the convolution operator yields

$\displaystyle u_{1}(x)$ $\displaystyle=w_{1}(x-r_{0})-w_{1}(x+r_{0})$
$\displaystyle=\tilde{w_{1}}(x-r_{0},r_{0}).$

Let $(r_{n})_{n=0}^{\infty}$ be a sequence of real numbers satisfying

$r_{n+1}=r_{n}+c^{*}+\phi_{2}(r_{n},a).$ (2.18)

If $r_{0}$ is sufficiently large, we can show that $r_{n}$ is strictly
increasing and unbounded. This is due to our assumption that $c^{*}>0$. Since
$\phi_{2}(r,a)\to c^{*}$, it suffices to assume $|\phi_{2}(r,a)-c^{*}|<c^{*}$,
i.e. $r_{0}$ is sufficieintly large so that $\phi_{2}(r,a)>0$ for all
$r>r_{0}$. Plugging this into the recurrence (2.18) yields
$r_{n+1}-r_{n}=c^{*}+\phi_{2}(r,a)>c^{*}>0$.

This argument shows that, assuming $r_{0}$ is sufficiently large, equation
(2.17) may be applied recursively. From the principle of induction, it follows
that $u_{2n+1}(x)=\tilde{w_{1}}(x-r_{n},r_{n})$ for all $n\geq 0$. Knowing
that $(r_{n})$ is unbounded allows us to use equation (2.18) to compute the
asymptotic spreading speed as the following limit.

$\lim_{n\to\infty}(r_{n+1}-r_{n})=\lim_{n\to\infty}(c^{*}+\phi_{2}(r_{n},a))=c^{*}+\lim_{r\to\infty}\phi_{2}(r,a)=2c^{*}.$

This completes the proof. ∎

##  3 Examples

In this section, we construct the periodic traveling wave solution for uniform
and Laplace kernels and derive formulas for the mean spreading speed $c^{*}$
in terms of the model parameters. We verify our results using numerical
simulation, with source code available at https://github.com/mdnestor/ide-
ptws.

###  3.1 Laplace kernel

We applied our main results to the Laplace dispersal kernel, defined

$k(x)=\frac{1}{2}e^{-|x|}.$ (3.1)

$k(x)$ is symmetric and has connected support, satisfying (H1), (H2), and
(H3). We will now prove that $k(x)$ satisfies (H5). Let $A,B\in\mathbb{R}$
with $A>B$, let $\lambda\in(0,1)$, and define $f(x)=k(x-A)-\lambda k(x-B)$.
For $r>0$ sufficiently large, we have

$f(x-r)-f(-x-r)=k(x-r-A)-\lambda k(x-r-B)-k(x+r+A)+\lambda k(x+r+B)$
$=\begin{cases}\frac{1}{2}e^{x}\left(-e^{r+A}+me^{r+B}-me^{-r-B}+e^{-r-A}\right),&x<-r-A,\\\
\frac{1}{2}e^{-x-r-A}+\frac{1}{2}e^{x}\left(e^{-r-A}-me^{-r-B}+me^{r+B}\right),&-r-A\leq
x\leq-r-B,\\\
\frac{1}{2}e^{-x}\left(-e^{-r-A}+me^{-r-B}\right)+\frac{1}{2}e^{x}\left(-me^{-r-B}+e^{-r-A}\right),&-r-B<x<r+B,\\\
\frac{1}{2}e^{-x}\left(-e^{-r-A}+me^{-r-B}-me^{r+B}\right)+\frac{1}{2}e^{x-r-A},&r+B\leq
x\leq r+A\\\
e^{-x}\left(-e^{-r-A}+me^{-r-B}-me^{r+B}+e^{r+A}\right),&x>r+A.\end{cases}$

$f$ is monotone on $(-\infty,-r-A)\cup(r+A,\infty)$ with $f(\pm\infty)=0$;
thus it cannot have any sign changes there. On each of the intervals
$[-r-A,-r-B]$, $(-r-B,r+B)$, and $[r+B,r+A]$, $f$ has the form
$Ce^{x}+De^{-x}$. Depending on the sign of $C$ and $D$, $f$ can have either
one or zero sign changes on each interval. Summing over each interval, it
follows that $f$ has at most $3$ sign changes. This proves (H5); the proof of
(H4) follows a similar argument.

Assuming $||w_{2}||_{\infty}<b$, the periodic traveling wave profiles are
given by

$w_{1}(x)=\begin{cases}1-\frac{1}{2}e^{x},&\text{if }x\leq 0,\\\
\frac{1}{2}e^{-x},&\text{if }x>0,\end{cases}$ (3.2)

and

$w_{2}(x-c^{*})=\begin{cases}m+\left(\frac{1}{2}(1-m)e^{-\Phi_{1}(b)}-\frac{1}{2}e^{-\Phi_{1}(a)}\right)e^{x},&x<\Phi_{1}(b),\\\
1-\frac{1}{2}e^{-\Phi_{1}(a)}e^{x}-\frac{1}{2}(1-m)e^{\Phi_{1}(b)}e^{-x},&\Phi_{1}(b)\leq
x\leq\Phi_{1}(a),\\\
\left(\frac{1}{2}e^{\Phi_{1}(a)}-\frac{1}{2}(1-m)e^{\Phi_{1}(b)}\right)e^{-x},&\Phi_{1}(a)<x,\end{cases}$
(3.3)

with

$c^{*}=\frac{1}{2}\log\begin{cases}\frac{1}{2a}\left(e^{\Phi_{1}(a)}-(1-m)e^{\Phi_{1}(b)}\right),&a\leq\ell_{a},\\\
e^{\Phi_{1}(a)}\left(1-a+\sqrt{(1-a)^{2}-(1-m)e^{\Phi_{1}(b)-\Phi_{1}(a)}}\right),&\ell_{a}<a<\ell_{b},\\\
2\,(m-a)\left[e^{-\Phi_{1}(a)}-(1-m)e^{-\Phi_{1}(b)}\right]^{-1},&a\geq\ell_{b}.\end{cases}$
(3.4)

$\ell_{a}$ and $\ell_{b}$ are constants given by

$\ell_{a}=\frac{1}{2}\left(1-(1-m)e^{\Phi_{1}(b)-\Phi_{1}(a)}\right),\quad\ell_{b}=\frac{1}{2}\left(1+m-e^{\Phi_{1}(b)-\Phi_{1}(a)}\right).$
(3.5)

The formulas for $\Phi_{1}$ and $\Phi_{2}$ are given by

$\Phi_{1}(\ell)=\begin{cases}-\log(2\ell),&\ell\leq\frac{1}{2},\\\
\log(2-2\ell),&\ell>\frac{1}{2},\end{cases}$ (3.6)

and

$\Phi_{2}(\ell)=-c^{*}+\log\begin{cases}\frac{1}{2\ell}\left(e^{\Phi_{1}(a)}-(1-m)e^{\Phi_{1}(b)}\right),&\ell\leq
w_{2}(\Phi_{1}(a)),\\\
e^{\Phi_{1}(a)}\left(1-\ell+\sqrt{(1-\ell)^{2}-(1-m)e^{\Phi_{1}(b)-\Phi_{1}(a)}}\right),&w_{2}(\Phi_{1}(a))<\ell<w_{2}(\Phi_{1}(b)),\\\
2\,(m-\ell)\left[e^{-\Phi_{1}(a)}-(1-m)e^{-\Phi_{1}(b)}\right]^{-1},&\ell\geq
w_{2}(\Phi_{1}(b)).\end{cases}$ (3.7)

The critical spreading speed $c^{*}$ for the Laplace kernel thus falls into
one of three cases:

  1. Case 1.

$a\leq\frac{1}{2}$, $b\leq\frac{1}{2}$:

$c^{*}=\frac{1}{2}\log\begin{cases}\frac{b-a(1-m)}{4a^{2}b},&a<\frac{b-a(1-m)}{2b},\\\
\frac{1-a+\sqrt{b(1-a)^{2}-a(1-m)}}{2a\sqrt{b}},&\frac{b-a(1-m)}{2b}\leq
a\leq\frac{b(1+m)-a}{2b},\\\
\frac{a-m}{b(1-m)-a},&a>\frac{b(1+m)-a}{2b}.\end{cases}$

  2. Case 2.

$a\leq\frac{1}{2}$, $b>\frac{1}{2}$:

$c^{*}=\frac{1}{2}\log\begin{cases}\frac{1-4a(1-m)(1-b)}{4a^{2}},&a<\frac{1-4a(1-m)(1-b)}{2},\\\
\frac{1-a+\sqrt{(1-a)^{2}-4a(1-m)(1-b)}}{2a},&\frac{1-4a(1-m)(1-b)}{2}\leq
a\leq\frac{1+m-4a(1-b)}{2},\\\
\frac{4(a-m)(1-b)}{1-m-4a(1-b)},&a>\frac{1+m-4a(1-b)}{2}.\end{cases}$

  3. Case 3.

$a>\frac{1}{2}$, $b>\frac{1}{2}$:

$c^{*}=\frac{1}{2}\log\begin{cases}\frac{1-a-(1-m)(1-b)}{a},&a<\frac{1-a-(1-m)(1-b)}{2(1-a)},\\\
2\,(1-a)\left(1-a+\sqrt{\frac{(1-a)^{3}-(1-m)(1-b)}{1-a}}\right),&\frac{1-a-(1-m)(1-b)}{2(1-a)}\leq
a\leq\frac{b-1+m(1-a)}{2(1-a)},\\\
\frac{4(m-a)(1-a)(1-b)}{1-b+m(1-a)},&a>\frac{b-1+m(1-a)}{2(1-a)}.\end{cases}$

Figure 3 shows the results of a numerical simulation with growth parameters
were $a=0.3$, $m=0.5$, and $b=0.8$ with the Laplace dispersal kernel, for an
average spreading speed of $c^{*}=\frac{1}{2}\ln\frac{22}{9}$. Figures 3(a)
and 3(b) show heatmaps of the periodic traveling wave solution and a spreading
solution with compactly supported initial condition. Figure 3(c) shows a
numerical simulation of the periodic traveling wave solution, while Figure
3(d) shows the asymptotic convergence of the intermediate wave-speed sequence
$c_{n}(\ell)$, evaluated at the Allee threshold $\ell=a$.

(a) Heatmap of the periodic traveling wave.

(b) Heatmap of a periodic spreading solution.

(c) Population density curve of the periodic traveling wave.

(d) Time series plot of intermediate spreading speed.

Figure 3: Heatmaps of (a) the periodic traveling wave, and (b) the
corresponding spreading solution with compactly supported initial data, and
Laplace dispersal kernel. In figure (d), the horizontal blue lines correspond
to the theoretical values derived in Corollary 2.3.

###  3.2 Uniform kernel

The uniform dispersal kernel is given by

$k(x)=\begin{cases}\frac{1}{2},&\text{if }|x|\leq 1,\\\ 0,&\text{if
}|x|>1.\end{cases}$ (3.8)

Like the previous example, $k(x)$ is symmetric around zero. We can verify (H4)
by letting $A>B$ and $\lambda\in(0,1)$ and defining $f(x)=k(x-A)-\lambda
k(x-B)$. There are two cases:

  1. Case 1.

If $|A-B|<2$, then

$f(x)=\begin{cases}0,&x\leq B-1,\\\ -\frac{\lambda}{2},&B-1<x\leq A-1,\\\
\frac{1-\lambda}{2},&A-1<x<B+1,\\\ \frac{1}{2},&B+1\leq x<A+1\\\ 0,&x\geq
A+1.\end{cases}$

  2. Case 2.

If $|A-B|\geq 2$, then

$f(x)=\begin{cases}0,&x\leq B-1,\\\ -\frac{\lambda}{2},&B-1<x<B+1,\\\
0,&B+1\leq x\leq A-1,\\\ \frac{1}{2},&A-1<x<A+1\\\ 0,&x\geq A+1.\end{cases}$

In both cases, the number of sign changes is exactly 1. To extend this
argument to (H5), observe that since $k(x)$ has compact support, we have

$\displaystyle f(x-r)-f(r-x)$ $\displaystyle=k(x-r-A)-\lambda
k(x-r-B)-k(x+r+A)+\lambda k(x+r+B)$
$\displaystyle=\begin{cases}-f(r-x),&x<-r+1,\\\ 0,&-r+1\leq x\leq r-1,\\\
f(x-r),&x>r-1,\end{cases}$

for sufficiently large $r$.

The alternating wave profiles are given by

$\displaystyle w_{1}(x)=\begin{cases}1,&x\in(-\infty,-1),\\\
\frac{1}{2}-\frac{1}{2}x,&x\in[-1,1],\\\ 0,&x\in(1,\infty),\end{cases}$ (3.9)

and

$\displaystyle w_{2}(x+c^{*})=\begin{cases}m,&x\in(-\infty,-2b),\\\
\frac{1-m}{2}x+m+b-mb,&x\in[-2b,-2a),\\\
-\frac{m}{2}x+m+b-mb-a,&x\in[-2a,2-2b),\\\
-\frac{1}{2}x-a+1,&x\in[2-2b,2-2a],\\\ 0,&x\in(2+2a,\infty),\end{cases}$
(3.10)

with

$c^{*}=\begin{cases}1-2a,&\text{if }a\leq b/2,\\\ 1-b+\frac{b-2a}{m},&\text{if
}a>b/2.\end{cases}$ (3.11)

$w_{2}$ has a global maximum at $x=-2a$ so that
$||w_{2}||_{\infty}=m+(b-a)(1-m)$. Thus, a sufficient condition for the
existence of the periodic traveling wave is $w_{2}(-2a)=m+(b-a)(1-m)<b$.

The intermediate wave speeds are given by

$\displaystyle c_{1}^{*}(\ell)$ $\displaystyle=1-2\ell$ (3.12)

and

$\displaystyle c_{2}^{*}(\ell)$ $\displaystyle=\begin{cases}1-4a+2\ell,&a\leq
b/2,\\\ 1-2b+\frac{2b-4a}{m}+2\ell,&a>b/2,\end{cases}$ (3.13)

for $0<\ell<m+(b-a)(1-m)$. By evaluating the intermediate wave speeds at
$\ell=a$, we obtain $c_{1}^{*}(a)=c_{2}^{*}(a)$ if $a\leq b/2$, and
$|c_{1}^{*}(a)-c_{2}^{*}(a)|=2(2a-b)(\frac{1-m}{m})>0$ if $a>b/2$. So for
$a>b/2$, the traveling wave is periodic with two different intermediate wave
speeds. Furthermore, the difference between these two intermediate speeds is
increasing with respect to $a$, the Allee threshold, and decreasing with
respect to $b$, the overcompensation threshold.

Figure 4(a) depicts the periodic traveling wave solution (2.4) with the
uniform kernel with growth parameters are $a=0.325$, $b=0.6$, and $m=0.45$.
The mean spreading speed can be analytically calculated as
$c^{*}=\frac{13}{45}$. The intermediate speeds are $c_{1}^{*}(a)=\frac{7}{20}$
and $c_{2}^{*}(a)=\frac{41}{180}$. Figure 4(b) demonstrates the spreading
phenomena with initial domain rize $r_{0}=1$. Since our uniform kernel is
compact with support $[-1,1]$, a sufficient lower bound on the initial domain
size is $r_{0}\geq 3$.

(a) Heaviside initial data.

(b) Compact initial data ($r_{0}=1$).

Figure 4: Population density plots of (a) the periodic traveling wave, and (b)
a spreading solution with uniform dispersal.

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