# On a hypercycle equation with infinitely many members

Alexander S. Bratus1,2,111e-mail<EMAIL_ADDRESS>, Olga S.
Chmereva3,222e-mail<EMAIL_ADDRESS>, Ivan Yegorov4,333Also known as Ivan
Egorov; e-mail<EMAIL_ADDRESS>, Artem S.
Novozhilov5,444Corresponding author; e-mail<EMAIL_ADDRESS>
${}^{\textrm{\emph{1}}}$Russian University of Transport, Obraztsova 15, Moscow
127994, Russia
${}^{\textrm{\emph{2}}}$Moscow Center of Fundamental and Applied Mathematics,
Lomonosov Moscow State University, Moscow 119992, Russia
${}^{\textrm{\emph{3}}}$Lomonosov Moscow State University, Leninskie Gory,
MSU,
2nd educational building, Moscow, 119991, Russia
${}^{\textrm{\emph{4}}}$KLA Corporation, 3 Technology Dr, Milpitas, CA 95035,
USA
${}^{\textrm{\emph{5}}}$Department of Mathematics, North Dakota State
University, Fargo, ND, 58108, USA

###### Abstract

A hypercycle equation with infinitely many types of macromolecules is
formulated and studied both analytically and numerically. The resulting model
is given by an integro-differential equation of the mixed type. Sufficient
conditions for the existence, uniqueness, and non-negativity of solutions are
formulated and proved. Analytical evidence is provided for the existence of
non-uniform (with respect to the second variable) steady states. Finally,
numerical simulations strongly indicate the existence of a stable nonlinear
wave in the form of the wave train.

#### Keywords:

Hypercycle, mixed functional differential equations, integro-differentia
equations.

#### AMS Subject Classification:

92D10, 92D15

##  1 Introduction

An important class of replicator models involves systems of nonlinear ordinary
differential equations with dynamics restrained by the standard simplex in the
state space and describes macromolecular interactions in various problems of
population genetics and evolutionary game theory [21, 13, 14], as well as in
theories of the origin of life [22, 11].

Of special interest is the hypercycle model that was proposed by M. Eigen and
P. Schuster [11]. It was related to the prebiotic evolution hypothesis stating
that self-replicating molecules are predecessors of RNA, DNA and eventually of
cells. The hypercycle model has been thoroughly studied from the points of
view of both population genetics [13] and mathematical frameworks for systems
of nonlinear ordinary differential equations [21, 16].

A classical hypercycle is a finite closed network of self-replicating
macromolecules (species) which are connected so that each of them catalyzes
the replication of the successor, with the last molecule reinforcing the first
one; see Fig. 1. From the sociological perspective, the catalytic support for
the replication of other molecules resembles altruistic behavior, in contrast
to conventional autocatalysis [15, 13].

Figure 1: A hypercyclic network.

Let us briefly recall the mathematical formulation of the classical hypercycle
model together with its key properties.

Consider a hypercycle with $n$ macromolecules (species) labeled by
$\>i\,=\,1,2,\ldots,n\>$ (see Fig. 1), and denote the corresponding relative
(normalized) time-varying frequencies by $u_{i}(t)$. One has

$u_{i}(t)\geqslant 0,\quad i\,=\,1,2,\ldots,n,$ (1)
$\sum_{i=1}^{n}u_{i}(t)\>=\>1,$ (2)

i. e., $\>u\,=\,(u_{1},u_{2},\ldots,u_{n})\>$ belongs to the standard simplex
in $\mathbb{R}^{n}$, and the hypercyclic dynamics is described by

$\left\\{\begin{aligned}
&\frac{\mathrm{d}u_{i}(t)}{\mathrm{d}t}\>\>=\>\>u_{i}(t)\,(k_{i}\,u_{i-1}(t)\>-\>f(t)),\quad
i\,=\,1,2,\ldots,n,\\\
&f(t)\>=\>\sum_{i=1}^{n}k_{i}\,u_{i}(t)\,u_{i-1}(t),\quad
u_{0}(t)\>=\>u_{n}(t),\\\ &t\geqslant 0,\\\ &u_{i}(0)\>=\>u_{i}^{0},\quad
i\,=\,1,2,\ldots,n.\end{aligned}\right.$ (3)

Here $k_{i}\,u_{i-1}(t)$ is the fitness of the $i$-th species, and $f(t)$ is
the mean fitness of the entire system (the representation for $f(t)$ can be
obtained by adding up all of the $n$ dynamical equations and using the
relation (2)). Note also that the standard simplex is invariant with respect
to this dynamical system.

The following properties hold:

  * •

the hypercyclic system is permanent [13, §13], i. e., there exists $\delta>0$
(independent from the initial coordinates
$\,u_{1}^{0},u_{2}^{0},\ldots,u_{n}^{0}$) such that $\>u_{i}^{0}>0$,
$\,i\,=\,1,2,\ldots,n,\>$ imply

$\liminf\limits_{t\,\to\,+\infty}\,u_{i}(t)\>>\>\delta,\quad
i\,=\,1,2,\ldots,n;$

  * •

for $n=2,3,4$, the hypercycle admits a globally stable steady state with
positive components [13, §12], while there appears a stable limit cycle if
$n\geqslant 5$ [16];

  * •

when $m$ disjoint hypercycles compete with each other in the same environment,
a unique hypercycle is finally established with the other $m-1$ going to
extinction for almost all initial conditions [12];

  * •

evolutionary adaptation can help hypercycles be resistant to parasites but
also allows for a phase transition phenomenon similar to the error threshold
in the quasispecies models (which divides the selective phase of evolution
with a clear dominance from the random phase with a markedly more uniform
distribution), as was numerically investigated in [2].

Moreover, the paper [4] studied an extension of the hypercycle model with each
species distributed on a line segment and influenced by a homogeneous
diffusion, which led to a system of $n$ partial differential equations instead
of the ordinary differential equations (3). This modeling approach was further
developed in [5, 3, 6].

However, the actual number of macromolecules in a hypercycle may be huge, and
this may significantly complicate the numerical analysis of the associated
dynamical system (3). It may therefore be reasonable to represent the
macromolecules as points in some line segment (of cardinality continuum) and
to construct an appropriate distributed model of hypercyclic replication. Such
a methodology was previously implemented for Crow–Kimura and Eigen
quasispecies models, with a single integro-differential equation replacing a
large number of ordinary differential equations [10, 8, 7]. A crucial step in
the construction of a hypercycle model with a continuum of species is to
incorporate the catalyzing effects (along a continuous loop in contrast to the
finite closed chain in Fig. 1). Possible diffusive behavior is also worth
taking into account. Besides, the cyclic structure has to be ensured by
stating appropriate boundary conditions at the endpoints of the line segment
describing the species. For these purposes, the current work formulates a new
distributed hypercycle model based on a second-order partial integro-
differential equation with spatial delay and mixed boundary conditions. It
should be emphasized that the general theory of mixed functional differential
equations including in particular partial or integro-differential equations
with spatial delays is still at an early stage of development. An introduction
to this promising area of mathematical research can be found in [20], and some
related applications are presented in [18, 1, 9].

This paper is organized as follows. The problem is stated in Section 2. The
existence, uniqueness, and nonnegativity of the solution are discussed in
Section 3, with the proofs moved to Appendix. Section 4 provides steady-state
analysis. Section 5 presents numerical simulation results for the dynamic
model. Finally, concluding remarks are given in Section 6.

##  2 Problem statement

Let macromolecules (species) in a hypercycle be represented as points of the
interval $0\leqslant x<2\pi$, and denote the relative frequency of
macromolecule $x$ at time $t$ by $u(x,t)$. The normalization condition (2)
from the classical model transforms to

$\int_{0}^{2\pi}u(x,t)\,\mathrm{d}x\>=\>1\quad\forall t\geqslant 0,$ (4)

and it is also convenient to incorporate periodicity with respect to $x$:

$u(x+2\pi,\,t)\>=\>u(x,t)\quad\forall x\in\mathbb{R}\quad\forall t\geqslant
0.$ (5)

Hence, species $x$ and $x+2n\pi$ are considered to be equivalent for any
integer $n$. Furthermore, assume the existence of a constant parameter
$h\in(0,2\pi)$ such that replication of species $x$ is catalyzed by species
$x-h$. Next, let $k(x)$ and $\varphi(x)$ be functions that describe the
replication rates and initial distribution, respectively, and introduce a
diffusion coefficient $\alpha>0$.

######  Assumption 2.1.

$h=\mathrm{const}\in(0,2\pi)$, $\>\alpha=\mathrm{const}>0$,
$\>k\colon\,[0,2\pi]\to(0,+\infty)\>$ is a twice continuously differentiable
positive function satisfying $\,k(0)=k(2\pi)$,
$\,k^{\prime}(0)=k^{\prime}(2\pi),\,$ and
$\>\varphi\colon\,[-h,2\pi]\to[0,+\infty)\>$ is a twice continuously
differentiable nonnegative function satisfying

$\int_{0}^{2\pi}\varphi(x)\,\mathrm{d}x\>=\>1,\quad\>\varphi(x)\,=\,\varphi(x+2\pi)\quad\forall
x\in[-h,0].$

A hypercycle model can then be stated in the form of the following initial-
boundary value problem for a second-order partial integro-differential
equation with a nonlinear source term involving a spatial delay:

$\left\\{\begin{aligned} &\frac{\partial u(x,t)}{\partial
t}\>\>=\>\>u(x,t)\>(k(x)\,u(x-h,\,t)\>-\>f[u(\cdot,t)])\>\,+\>\,\alpha\,\frac{\partial^{2}u(x,t)}{\partial
x^{2}}\,,\\\ &u(x-h,\,t)\>=\>u(x-h+2\pi,\,t)\quad\mathrm{if}\quad-h\leqslant
x-h<0,\\\ &u(x,0)\,=\,\varphi(x)\quad\mbox{(initial condition)},\\\
&u(0,t)\,=\,u(2\pi,t),\quad\frac{\partial u}{\partial
x}\,(0,t)\,=\,\frac{\partial u}{\partial x}\,(2\pi,t)\quad\mbox{(boundary
conditions)},\\\ &0\leqslant x\leqslant 2\pi,\quad t\geqslant
0.\end{aligned}\right.$ (6)

Here

$f[u(\cdot,t)]\>\,=\>\,\int_{0}^{2\pi}k(x)\,u(x,t)\,u(x-h,\,t)\,\mathrm{d}x\quad\forall
t\geqslant 0$ (7)

is the mean fitness of the system. This representation can be obtained by
integrating the dynamic equation in (6) over $0\leqslant x\leqslant 2\pi$ and
using the normalization property (4) as well as the boundary condition for
$\partial u/\partial x$. Similar arguments can also help us verify that, under
Assumption 2.1, a solution of (6) should satisfy

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\int_{0}^{2\pi}u(x,t)\,\mathrm{d}x\right)\>\,=\>\,0\quad\forall
t\geqslant 0,$

which ensures (4). Moreover, the mixed boundary conditions come from the
aforementioned periodicity (cyclic structure) with respect to $x$. Thus, the
relations (6) and (7) can serve as a distributed modification of the classical
hypercycle model (3).

Note that the catalytic structure leading to a spatial delay is an important
feature of our distributed hypercycle model in comparison with distributed
quasispecies models [10, 8, 7].

##  3 Existence, uniqueness, and nonnegativity of the solution

This section addresses the existence, uniqueness, and nonnegativity of the
classical solution to the initial-boundary value problem (6), with detailed
proofs moved to Appendix.

Introduce an arbitrary time horizon $T>0$ and the Banakh space
$\,\mathbf{C}^{2,1}([0,2\pi]\times[0,T])\,$ consisting of continuous functions
$\>[0,2\pi]\times[0,T]\,\ni\,(x,t)\>\longmapsto\>w(x,t)\,\in\,\mathbb{R}\>$
such that $\>\partial w/\partial x$, $\partial^{2}w/\partial x^{2}$, $\partial
w/\partial t\>$ are continuous on $\,[0,2\pi]\times[0,T]$. The continuity on
the boundary of the rectangle means that the corresponding bounded limits from
the interior exist and are taken as the values on the boundary. The norm in
$\,\mathbf{C}^{2,1}([0,2\pi]\times[0,T])\,$ is defined as

$\displaystyle\|w\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\>=\>\>\max_{\begin{subarray}{c}0\leqslant
x\leqslant 2\pi,\\\ 0\leqslant t\leqslant
T\end{subarray}}\>\left\\{|w(x,t)|\>+\>\left|\frac{\partial w(x,t)}{\partial
x}\right|\right.$ (8)
$\displaystyle\left.+\>\left|\frac{\partial^{2}w(x,t)}{\partial
x^{2}}\right|\>+\>\left|\frac{\partial w(x,t)}{\partial t}\right|\right\\}.$

The classical solution is searched for in this space.

The space $\mathbf{C}^{2}([0,2\pi])$ is similarly defined (but with excluded
dependence on $t$).

Consider the auxiliary linear problem

$\left\\{\begin{aligned} &\mathcal{A}v(x,t)\>\>=\>\>\frac{\partial
v(x,t)}{\partial t}\>-\>\alpha\,\frac{\partial^{2}v(x,t)}{\partial
x^{2}}\>\>=\>\>\psi(x,t),\\\ &v(x,0)\,=\,\varphi(x)\quad\mbox{(initial
condition)},\\\ &v(0,t)\,=\,v(2\pi,t),\quad\frac{\partial v}{\partial
x}\,(0,t)\,=\,\frac{\partial v}{\partial x}\,(2\pi,t)\quad\mbox{(boundary
conditions)},\\\ &0\leqslant x\leqslant 2\pi,\quad 0\leqslant t\leqslant
T,\end{aligned}\right.$ (9)

with a right-hand side $\psi$ that lies in
$\,\mathbf{C}^{2,1}([0,2\pi]\times[0,T])\,$ and satisfies the boundary
conditions

$\psi(0,t)\,=\,\psi(2\pi,t),\quad\frac{\partial\psi}{\partial
x}\,(0,t)\,=\,\frac{\partial\psi}{\partial x}\,(2\pi,t)\quad\>\forall
t\in[0,T].$ (10)

The corresponding solution is obtained in Appendix A.1, and, with the help of
Assumption 2.1, one can directly verify that it also belongs to
$\,\mathbf{C}^{2,1}([0,2\pi]\times[0,T])$. The relations (A.1), (A.4), (A.5)
give rise to the operators $\mathcal{E},\mathcal{E}_{1},\mathcal{E}_{2}$ such
that

$v(x,t)\>=\>\mathcal{E}[\varphi,\psi](x,t)\>=\>\mathcal{E}_{1}\varphi(x,t)\,+\,\mathcal{E}_{2}\psi(x,t).$
(11)

Note that $\mathcal{E}_{1},\mathcal{E}_{2}$ are linear operators.
Straightforward calculations based on (A.1), (A.4), (A.5) lead to the
estimates

$\displaystyle\|\mathcal{E}_{1}\varphi\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\>\leqslant\>\>c_{1}\>\|\varphi\|_{\mathbf{C}^{2}([0,2\pi])}\,,$
(12)
$\displaystyle\|\mathcal{E}_{2}\psi\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\>\leqslant\>\>c_{2}\>\|\psi\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\,,$
$\displaystyle\|\mathcal{E}[\varphi,\psi]\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\>\leqslant\>\>c_{1}\>\|\varphi\|_{\mathbf{C}^{2}([0,2\pi])}\>\,+\>\,c_{2}\>\|\psi\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}$

with some positive constants $c_{1},c_{2}$.

The idea for the proof of an existence and uniqueness result for (6) is to
apply the Banach fixed point theorem (see, e. g., [19, Theorem 4.16]) to the
iterative process

$u^{0}(x,t)\>=\>\varphi(x),\quad
u^{m+1}\>=\>\mathcal{E}[\varphi,\,\mathcal{F}u^{m}],\quad m\,=\,0,1,2,\ldots,$
(13)

where

$\displaystyle\mathcal{F}u^{m}(x,t)\>\,=\>\,u^{m}(x,t)\>(k(x)\,u^{m}(x-h,\,t)\>-\>f[u^{m}(\cdot,t)]),$
(14) $\displaystyle
f[u^{m}(\cdot,t)]\>\,=\>\,\int_{0}^{2\pi}k(x)\,u^{m}(x,t)\,u^{m}(x-h,\,t)\,\mathrm{d}x$

(see the right-hand side of the dynamic equation in (6), as well as the mean
fitness definition (7)). Because of the nonlinearity in (14), this proof
approach works only under the quite restrictive technical condition that the
quantity

$\bar{k}\>\>=\>\>\max_{0\leqslant x\leqslant
2\pi}\>\max\,\\{|k(x)|,\,|k^{\prime}(x)|,\,|k^{\prime\prime}(x)|\\}$ (15)

is sufficiently small. The proof of an extended existence and uniqueness
result remains an open task. One may in general expect the unique solution to
exist under weaker conditions.

Due to the boundary conditions for $\varphi,\varphi^{\prime},k,k^{\prime}$ at
$x=0$ and $x=2\pi$ (recall Assumption 2.1), the relations (10) should hold
with $\psi=\mathcal{F}u^{m}$ for all $m=0,1,2,\ldots$ Using also the fact that
$\>\int_{0}^{2\pi}u^{0}(x,t)\,\mathrm{d}x\>=\>\int_{0}^{2\pi}\varphi(x)\,\mathrm{d}x\>=\>1,\>$
one arrives at

$\int_{0}^{2\pi}u^{m}(x,t)\,\mathrm{d}x\>\,=\>\,\int_{0}^{2\pi}\mathcal{F}u^{m}(x,t)\,\mathrm{d}x\>\,=\>\,1,\quad
m\,=\,0,1,2,\ldots$ (16)

The following lemma is established in Appendix A.2.

######  Lemma 3.1.

Under Assumption 2.1, the sequence of the norms
$\>\|u^{m}\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}$, $\,m=0,1,2,\ldots,\>$
is uniformly bounded for sufficiently small values of (15).

The proof of the existence and uniqueness result (Theorem 3.2) is given in
Appendix A.3.

######  Theorem 3.2.

Let Assumption 2.1 hold. Then for for sufficiently small values of (15), there
exists a unique classical solution
$\>u\,\in\,\mathbf{C}^{2,1}([0,2\pi]\times[0,T])\>$ to the distributed
hypercycle problem (6).

The solution nonnegativity result (Theorem 3.3) is proved in Appendix A.4
under the condition that the catalytic shift (delay) parameter $h$ is
sufficiently small. This condition seems reasonable when comparing the
distributed hypercycle model (6),(7) to the classical model (3) with a large
number of species $n$. Indeed, the relation between $h$ and $n$ can be set up
to $\,h=2\pi/n,\,$ so that $h\to 0$ as $n\to\infty$.

######  Theorem 3.3.

Let Assumption 2.1 hold, and let a solution
$\>u_{h}\,\in\,\mathbf{C}^{2,1}([0,2\pi]\times[0,T])\>$ to the problem (6)
exist for all $\,0<h\leqslant h_{0},\,$ where
$\>h_{0}=\mathrm{const}\in(0,2\pi)$. Assume also

$\sup_{0<h\leqslant
h_{0}}\,\|u_{h}\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\,<\>\,\infty.$
(17)

Then $u_{h}$ is a nonnegative function for sufficiently small $h$.

######  Remark 3.4.

Let Assumption 2.1 hold, and let $\bar{k}$ be small enough to apply Lemma 3.1
and Theorem 3.2. First, note that $c_{1},c_{2}$ in the estimates (12) do not
depend on $h$, since the auxiliary problem (9) does not contain $h$. From the
proofs of Lemma 3.1 and Theorem 3.2 in Appendix, one can see that the
parameters in the corresponding estimates also do not depend on $h$. Hence,
the condition (17) is fulfilled in this case.

##  4 Steady-state analysis

System (6), (7) has the following equilibrium solution $v(x)=(2\pi)^{-1}$ for
$k(x)=k=\text{const}>0$. First we show that this equilibrium is unstable.

Let $\psi\in\mathbf{C}^{2,1}$. Consider function

$v_{\varepsilon}(x,t)=\frac{1}{2\pi}+\varepsilon\psi(x,t)$ (18)

that satisfies condition (4). Then

$\int_{0}^{2\pi}\psi(x,t)\mathrm{d}x=0.$ (19)

Plugging $v_{\varepsilon}$ into (6) and keeping only the linear with respect
to $\varepsilon$ terms we obtain

$\displaystyle\frac{\partial\psi}{\partial t}(x,t)=$ $\displaystyle
k\bigl{(}\psi(x,t)+\psi(x-h,t)\bigr{)}-k\psi(x,t)$
$\displaystyle-\frac{k}{2\pi}\int_{0}^{2\pi}\bigl{(}\psi(x,t)-\psi(x-h,t)\bigr{)}\mathrm{d}x+\alpha\frac{\partial^{2}\psi}{\partial
x^{2}}(x,t),$ $\displaystyle\psi(x,0)=$ $\displaystyle\psi_{0}(x),\quad
x\in[0,2\pi],$

which, after simplification and taking into account (19), leads to

$\begin{split}\frac{\partial\psi}{\partial
t}(x,t)=&k\psi(x-h,t)+\alpha\frac{\partial^{2}\psi}{\partial x^{2}}(x,t),\quad
t>0,\\\ \psi(x,0)=&\psi_{0}(x),x\in[0,2\pi].\end{split}$ (20)

It is natural to look for a solution to this problem in the form of a Fourier
series

$\psi(x,t)=\sum_{n=-\infty}^{+\infty}c_{n}(t)e^{-inx},$

which yields

$\frac{\mathrm{d}c_{n}}{\mathrm{d}t}(t)=c_{n}(t)\left(ke^{inh}-\alpha
n^{2}\right),\quad n=0,\pm 1,\pm 2,\ldots.$ (21)

For $n=0$ $c_{0}(t)\to\infty$ for $t\to\infty$ and hence we proved that the
equilibrium $v(x)=(2\pi)^{-1}$ is unstable.

Now we look into possible spatially non-homogeneous equilibria, which must
satisfy the following problem

$\begin{split}\alpha\frac{\mathrm{d}^{2}v}{\mathrm{d}x^{2}}(x)&+v(x)\bigl{(}kv(x-h)-\bar{f})\bigr{)}=0,\quad
x\in(0,2\pi),\\\
v^{\prime}(0)&=v^{\prime}(2\pi),\quad\bar{f}=k\int_{0}^{2\pi}v(x)v(x-h)\mathrm{d}x,\\\
v(x-h)&=\begin{cases}v(x-h),&\text{if }x-h\geqslant 0,\\\
v(2\pi+x-h),&\text{if }x-h<0,\end{cases}\\\
\int_{0}^{2\pi}v(x)\mathrm{d}x&=\int_{0}^{2\pi}v(x-h)\mathrm{d}x=1.\end{split}$
(22)

As before, here we assume that $k(x)=k=\text{const}>0$.

If we assume that $v\in\mathbf{C}^{2}[0,2\pi]$ then

$v(x-h)=v(x)-hv_{x}(x)+\frac{h^{2}}{2}v_{xx}+o(h^{2}).$

Using this representation and keeping only the terms up to $o(h^{2})$ the
first equation in (22) becomes

$v_{xx}(x)\left(\alpha+\frac{kh^{2}}{2}v(x)\right)=\bar{f}v(x)-kv(x)\bigl{(}v(x)-hv_{x}(x)\bigr{)}.$

We rewrite this second order equation as the following system:

$\begin{split}\frac{\mathrm{d}v}{\mathrm{d}x}(x)&=w(x),\\\
\frac{\mathrm{d}w}{\mathrm{d}x}(x)&=\frac{\bigl{(}\bar{f}-kv(x)+khw(x)\bigr{)}v(x)}{\alpha+2^{-1}kh^{2}v(x)}\,,\end{split}$
(23)

with the boundary conditions $w(0)=w(2\pi)$.

The standard analysis of (23) yields that there are two equilibria $O(0,0)$
and $A(\bar{f}k^{-1},0)$. The former one is a saddle point for any parameter
values, and the latter one is a stable focus for $h<0$, unstable focus for
$h>0$; if $h=0$ then the eigenvalues of the Jacobi matrix are pure imaginary
complex conjugate.

We note that for $h=0$ system (23) becomes Hamiltonian, with the Hamiltonian

$H(v,w)=\frac{w^{2}}{2}+U(v),\quad
U(v)=\frac{v^{2}}{\alpha}\left(\frac{kv}{2}-\frac{\bar{f}}{2}\right).$

The graph of potential $U$ immediately implies that equilibrium $A$ in the
case $h=0$ is Lyapunov stable, with a family of closed orbits surrounding it,
which implies that there are infinitely many orbits that satisfy the condition
$w(x_{1})=w(x_{2})=0$ for two points $x_{2}>x_{1}>0$ (see Fig. 2).

Figure 2: Graph of the potential for system (23) with $h=0$ (left) and the
corresponding phase portrait (right).

The conditions identified above are sufficient to invoke the Hopf bifurcation
theorem (in the form given, e.g., in [17], Section 3C) to conclude that for
sufficiently small $|h|$ there must be nonconstant periodic solutions that
collapse at a point when $|h|\to 0$. Among all such periodic solutions we are
interested in one that satisfies $w(0)=w(2\pi)=0$. We claim that such solution
can always be found if one is allowed to consider sufficiently small values of
$\alpha$ (see Fig. 3).

Figure 3: Periodic solution of (23) for sufficiently small $\alpha$ and $h>0$.

Indeed, from equations (23) we see that decreasing $\alpha=o(h)$ leads to
increase of velocity of movements along the phase portrait, and therefore
there will always be sufficiently small values of $\alpha$ for which the
required condition is satisfied; moreover, it is clear that for some choice of
$\alpha$ the orbit will travel around the equilibrium point only once, for
some $\alpha$ will travel twice, etc. As a result of these reasoning we put
forward the following conjecture.

######  Conjecture 4.1.

For sufficiently small values of $h$ and diffusion coefficient $\alpha=o(h)$
there exist periodic solutions to (22), with are $o(h^{2})$-close to the
periodic solutions to (23).

An additional support to this conjecture can be seen from the following
argument.

If we assume that $v$ is sufficiently smooth in (22), i.e.,

$v(x-h)=\sum_{j=0}^{N}\frac{(-1)^{j}}{j!}h^{j}v^{(j)}(x)+o(h^{N}),$

then problem (22) can be rewritten as a system of $N$ (dropping the terms of
the smaller order) equations of the first order

$\begin{split}\frac{\mathrm{d}v_{1}}{\mathrm{d}x}(x)&=v_{2}(x),\quad\frac{\mathrm{d}v_{2}}{\mathrm{d}x}(x)=v_{3}(x),\quad\ldots,\quad\frac{\mathrm{d}v_{N-1}}{\mathrm{d}x}(x)=v_{N}(x),\\\
\frac{\mathrm{d}v_{N}}{\mathrm{d}x}(x)&=\frac{(-1)^{N+1}N!}{kh^{N}}\left(k\sum_{j=0}^{N-1}\frac{(-1)^{j}}{j!}h^{j}v_{j+1}+\alpha\frac{v_{3}}{v_{1}}-\bar{f}\right),\end{split}$
(24)

which has the equilibrium $A(\bar{f}k^{-1},0,\ldots,0)$. Linearizing around
this equilibrium yield the Jacobi matrix with the characteristic polynomial

$(-1)^{N+1}\frac{kh^{N}}{N!}\lambda^{N}+(-1)^{N-1}\frac{h^{N-1}}{(N-1)!}\lambda^{N-1}+\ldots-\frac{h^{3}}{3!}\lambda^{3}+\left(\frac{h^{2}}{2!}+\frac{\alpha}{\bar{f}}\right)\lambda^{2}-h\lambda+1.$

When $h\to 0$ we again have two purely imaginary complex root, and the same
Hopf theorem [17] implies the existence of periodic solutions that collapse
into the point when $h\to 0$.

Therefore, it is highly probably that problem (22) admits solutions different
from a constant, which may be non-unique; and therefore the original system
(6), (7) admits spatially non-uniform equilibria.

##  5 Numerical solutions

In the previous section we found a family of spatially-nonhomogeneous
stationary solution. We did not study their stability analytically because it
represents an independent and complex problem on its own. Here we present some
numerical evidence that these solutions are most probably unstable because in
numerical experiments spatial and temporal oscillations are usually observed.

To conduct the numerical experiments we use an explicit finite difference
scheme and numerical approximation of the integral on the given interval.
Choosing the parameters as follows:
$x\in[0,2\pi],\,\alpha=0.05,t\in[0,300],h=0.4,k(x)=1,\varphi(x)=(2\pi)^{-1}(\sin(5x+\pi/4)+1)$
we can see that the numerically obtained solution $u$ represents a non-linear
wave which oscillates with respect to both spatial and temporal variables (see
Fig. 4).

Figure 4: Numerical solution of problem (6) with
$x\in[0,2\pi],\,\alpha=0.05,t\in[0,300],h=0.4,k(x)=1,\varphi(x)=(2\pi)^{-1}(\sin(5x+\pi/4)+1)$.

Choosing a different initial condition does not change the result
qualitatively (see Fig. 5).

Figure 5: Numerical solution of problem (6) with
$x\in[0,2\pi],\,\alpha=0.05,t\in[0,500],h=0.4,k(x)=1,\varphi(x)=(2\pi)^{-1}(\sin(x+\pi/4)+1)$.

Among other things we can see that the numerical simulations strongly indicate
that the spatially continuous hypercyclic system (6) is permanent similar to
its discrete counterpart. It is an interesting open mathematical problem to
analyze the observed oscillations analytically.

##  6 Conclusion

In this paper we introduced a continuous analogue of the classical discrete
hypercycle (3). The dynamics of (3) is well understood; it is, however, an
open challenge to fully investigate the properties of the solutions of the
introduced partial integro-differential equation with spatial delay (6). Here
we started such investigation by proving the existence and uniqueness theorem
for this problem and showing, according to biological interpretation of the
model, non-negativity of its solutions. We also presented some analytical
evidence that there are spatially nonuniform stationary solutions and
numerical evidence that the actual asymptotic behavior of the system tends to
both spatial and temporal oscillations. It is a challenging open problem to
provide an analytical support for our observations.

We would like to conclude this paper with a short historic anecdote. Famous
Soviet mathematician Anatolii Myshkis (1920–2009) is mostly known as one of
the founders of the field of delay differential equations. Less known is the
fact that the last significant mathematical object of his study was a class of
differential equations with delay with respect to other than time variable
[20]. Two of the authors of the present paper were fortunate to attend a talk
by prof. Myshkis in early 2000’s devoted to this topic. As it was usual in
mathematical presentations in Soviet Union and later in Russia, the talk
concluded with a somewhat long discussion of the results. In particular prof.
Myshkis mentioned that the class of problem he had been studying did not get
much attention in the mathematical community in part because of the lack of
applications and mathematical models of natural phenomena described by such
equations. Several ideas were suggested in which direction one may look for
applications, but none of them was very convincing. For us it is important to
state that the mathematical model (6) we consider in this text is very similar
to the mixed functional differential equations considered in [20] and it is
our hope that our model will attract more attention to this interesting class
of mathematical problems.

## Acknowledgements

The work of ASB was supported by grant $075-15-219-1621$ from Ministry of
science and higher education of the Russian Federation.

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## Appendix

###  A.1 Solution of the auxiliary linear problem (9)

The boundary conditions are obviously satisfied if one represents the solution
as the Fourier series

$v(x,t)\>\>=\>\>\frac{a_{0}(t)}{2}\>+\>\sum_{n=1}^{\infty}(a_{n}(t)\cos(nx)\,+\,b_{n}(t)\sin(nx)).$
(A.1)

Similarly, the initial profile and the right-hand side of the equation are
written as

$\varphi(x)\>\>=\>\>\frac{\varphi_{10}}{2}\>+\>\sum_{n=1}^{\infty}(\varphi_{1n}\cos(nx)\,+\,\varphi_{2n}\sin(nx)),$
(A.2)
$\psi(x,t)\>\>=\>\>\frac{\psi_{10}(t)}{2}\>+\>\sum_{n=1}^{\infty}(\psi_{1n}(t)\cos(nx)\,+\,\psi_{2n}(t)\sin(nx)).$
(A.3)

Here we use the fact that the functional system $\>1/\sqrt{2\pi}$,
$\>\left(1/\sqrt{\pi}\right)\cos(nx)$, $\>\left(1/\sqrt{\pi}\right)\sin(nx)$,
$\>n=1,2,3,\ldots,\>$ is orthonormal in the Hilbert space
$\,\mathcal{L}^{2}([0,2\pi];\,\mathbb{R})\,$ of square Lebesgue-integrable
real-valued functions on $[0,2\pi]$. The Fourier coefficients in (A.2) and
(A.3) are determined by

$\displaystyle\varphi_{10}\>\,=\>\,\frac{1}{\pi}\,\int_{0}^{2\pi}\varphi(x)\,\mathrm{d}x,\quad\psi_{10}(t)\>\,=\>\,\frac{1}{\pi}\,\int_{0}^{2\pi}\psi(x,t)\,\mathrm{d}x,$
$\displaystyle\varphi_{1n}\>\,=\>\,\frac{1}{\pi}\,\int_{0}^{2\pi}\varphi(x)\cos(nx)\,\mathrm{d}x,$
$\displaystyle\varphi_{2n}\>\,=\>\,\frac{1}{\pi}\,\int_{0}^{2\pi}\varphi(x)\sin(nx)\,\mathrm{d}x,$
$\displaystyle\psi_{1n}(t)\>\,=\>\,\frac{1}{\pi}\,\int_{0}^{2\pi}\psi(x,t)\cos(nx)\,\mathrm{d}x,$
$\displaystyle\psi_{2n}(t)\>\,=\>\,\frac{1}{\pi}\,\int_{0}^{2\pi}\psi(x,t)\sin(nx)\,\mathrm{d}x,$
$\displaystyle n\,=\,1,2,3,\ldots$

Using integration by parts together with the conditions
$\,\varphi(0)=\varphi(2\pi)$, $\,\varphi^{\prime}(0)=\varphi^{\prime}(2\pi)\,$
(recall Assumption 2.1), and (10), one arrives at the following
representations:

$\displaystyle\varphi_{1n}\>\,=\>\,\frac{1}{n\pi}\,\int_{0}^{2\pi}\varphi(x)\,\mathrm{d}(\sin(nx))\>\,=\>\,-\frac{1}{n\pi}\,\int_{0}^{2\pi}\varphi^{\prime}(x)\sin(nx)\,\mathrm{d}x$
(A.4)
$\displaystyle\qquad=\>\,\frac{1}{n^{2}\pi}\,\int_{0}^{2\pi}\varphi^{\prime}(x)\,\mathrm{d}(\cos(nx))\>\,=\>\,-\frac{1}{n^{2}\pi}\,\int_{0}^{2\pi}\varphi^{\prime\prime}(x)\cos(nx)\,\mathrm{d}x,$
$\displaystyle\varphi_{2n}\>\,=\>\,-\frac{1}{n\pi}\,\int_{0}^{2\pi}\varphi(x)\,\mathrm{d}(\cos(nx))\>\,=\>\,\frac{1}{n\pi}\,\int_{0}^{2\pi}\varphi^{\prime}(x)\cos(nx)\,\mathrm{d}x$
$\displaystyle\qquad=\>\,\frac{1}{n^{2}\pi}\,\int_{0}^{2\pi}\varphi^{\prime}(x)\,\mathrm{d}(\sin(nx))\>\,=\>\,-\frac{1}{n^{2}\pi}\,\int_{0}^{2\pi}\varphi^{\prime\prime}(x)\sin(nx)\,\mathrm{d}x,$
$\displaystyle\psi_{1n}(t)\>\,=\>\,-\frac{1}{n^{2}\pi}\,\int_{0}^{2\pi}\frac{\partial^{2}\psi(x,t)}{\partial
x^{2}}\,\cos(nx)\,\mathrm{d}x,$
$\displaystyle\psi_{2n}(t)\>\,=\>\,-\frac{1}{n^{2}\pi}\,\int_{0}^{2\pi}\frac{\partial^{2}\psi(x,t)}{\partial
x^{2}}\,\sin(nx)\,\mathrm{d}x,$ $\displaystyle n\,=\,1,2,3,\ldots$

Plugging the relations (A.1), (A.3) into the equation of (9) yields

$\dot{a}_{0}(t)\>=\>\psi_{10}(t),\quad\dot{a}_{n}(t)\,+\,\alpha
n^{2}a_{n}(t)\>=\>\psi_{1n}(t),\quad\dot{b}_{n}(t)\,+\,\alpha
n^{2}b_{n}(t)\>=\>\psi_{2n}(t),$

while the initial condition with (A.1), (A.2) implies

$a_{0}(0)\>=\>\varphi_{10},\quad a_{n}(0)\>=\>\varphi_{1n},\quad
b_{n}(0)\>=\>\varphi_{2n},$

and one consequently obtains the sought-after Fourier coefficients:

$\displaystyle
a_{0}(t)\>\>=\>\>\varphi_{10}\>+\>\int_{0}^{t}\psi_{10}(s)\,\mathrm{d}s,$
(A.5) $\displaystyle a_{n}(t)\>\>=\>\>e^{-\alpha
n^{2}t}\,\varphi_{1n}\>+\>\int_{0}^{t}e^{-\alpha
n^{2}(t-s)}\,\psi_{1n}(s)\,\mathrm{d}s,$ $\displaystyle
b_{n}(t)\>\>=\>\>e^{-\alpha n^{2}t}\,\varphi_{2n}\>+\>\int_{0}^{t}e^{-\alpha
n^{2}(t-s)}\,\psi_{2n}(s)\,\mathrm{d}s,$ $\displaystyle n\,=\,1,2,3,\ldots$

###  A.2 Proof of Lemma 3.1

From (14)–(16), one derives

$\mathcal{F}u^{m}(x,t)\>\,=\>\,u^{m}(x,t)\>\int_{0}^{2\pi}(k(x)\,u^{m}(x-h,\,t)\>-\>k(\xi)\,u^{m}(\xi-h,\,t))\>u^{m}(\xi,t)\>\mathrm{d}\xi$

and, therefore,

$\|\mathcal{F}u^{m}\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\,\leqslant\>\,c_{3}\bar{k}\,\|u^{m}\|^{3}_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\,,\quad
m\,=\,0,1,2,\ldots,$

where $\,c_{3}=\mathrm{const}>0$. Together with (12) and (13), this yields

$\displaystyle\left\|u^{m+1}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\,\leqslant\>\,c_{4}\>+\>c_{2}c_{3}\bar{k}\,\|u^{m}\|^{3}_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\,,$
(A.6) $\displaystyle m\,=\,0,1,2,\ldots,\quad
c_{4}\>=\>c_{1}\,\|\varphi\|_{\mathbf{C}^{2}([0,2\pi])}\,,\quad
u^{0}(x,t)\>=\>\varphi(x).$

We need the following auxiliary algebraic properties for nonnegative
parameters:

$\displaystyle(p_{1}+p_{2})^{3}\>\>=\>\>p_{1}^{3}\>+\>3p_{1}^{2}p_{2}\>+\>3p_{1}p_{2}^{2}\>+\>p_{2}^{3}$
(A.7)
$\displaystyle\qquad\qquad\>\>\>\>\leqslant\>\>p_{1}^{3}\>+\>6\,(\max\,\\{p_{1},p_{2}\\})^{3}\>+\>p_{2}^{3}$
$\displaystyle\qquad\qquad\>\>\>\>\leqslant\>\>p_{1}^{3}\>+\>6\left(p_{1}^{3}+p_{2}^{3}\right)\>+\>p_{2}^{3}\>\>\leqslant\>\>7\left(p_{1}^{3}+p_{2}^{3}\right),$
$\displaystyle(p_{1}+p_{2}+p_{3})^{3}\>\>\leqslant\>\>7p_{1}^{3}\>+\>7(p_{2}+p_{3})^{3}\>\>\leqslant\>\>7p_{1}^{3}\>+\>7^{2}p_{2}^{3}\>+\>7^{2}p_{3}^{3},$
$\displaystyle(p_{1}+p_{2}+\ldots+p_{l})^{3}\>\>\leqslant\>\>7p_{1}^{3}\>+\>7^{2}p_{2}^{3}\>+\>\ldots+\>7^{l-2}p_{l-2}^{3}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\>\>\>+\>7^{l-1}p_{l-1}^{3}\>+\>7^{l-1}p_{l}^{3},$
$\displaystyle p_{1}\geqslant 0,\>\>\>p_{2}\geqslant
0,\>\>\>\ldots,\>\>\>p_{l}\geqslant 0,\>\>\>l\,=\,2,3,4,\ldots$

Let $\bar{k}$ be small enough to satisfy

$\displaystyle c_{2}c_{3}\bar{k}\,\leqslant\,\beta,\quad
c_{2}c_{3}\bar{k}\,\|\varphi\|^{3}_{\mathbf{C}^{2}([0,2\pi])}\>\leqslant\>\beta,\quad
0<\beta<1,$ (A.8) $\displaystyle 7^{3}\beta^{2}c_{4}^{4}\,\leqslant\,1,\quad
7^{3}\beta^{5}\,\leqslant\,1.$

Then

$\displaystyle\left\|u^{1}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\,\leqslant\>\,c_{4}\>+\>c_{2}c_{3}\bar{k}\,\left\|u^{0}\right\|^{3}_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}$
(A.9)
$\displaystyle\qquad\qquad\qquad\quad\>\>\>=\>\,c_{4}\>+\>c_{2}c_{3}\bar{k}\,\|\varphi\|^{3}_{\mathbf{C}^{2}([0,2\pi])}\>\,\leqslant\>\,c_{4}+\beta,$
$\displaystyle\left\|u^{2}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\,\leqslant\>\,c_{4}\>+\>c_{2}c_{3}\bar{k}\,\left\|u^{1}\right\|^{3}_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}$
$\displaystyle\qquad\qquad\qquad\>\leqslant\>\,c_{4}\,+\,\beta(c_{4}+\beta)^{3}\>\,\leqslant\>\,c_{4}\,+\,7\beta\left(c_{4}^{3}+\beta^{3}\right)$
$\displaystyle\qquad\qquad\qquad\>\leqslant\>\,c_{4}\,+\,7\beta
c_{4}^{3}\,+\,7\beta^{4}.$

Let us now verify the general relation

$\displaystyle\left\|u^{m}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\,\leqslant\>\,c_{4}\,\sum_{i=0}^{m-1}\left(7\beta
c_{4}^{2}\right)^{i}\>+\>\left(7\beta^{4}\right)^{m-1},$ (A.10) $\displaystyle
m\,=\,2,3,4,\ldots$

by induction. Since (A.9) serves as the basis of induction ($m=2$), it remains
to show that (A.10) implies

$\left\|u^{m+1}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\,\leqslant\>\,c_{4}\,\sum_{i=0}^{m}\left(7\beta
c_{4}^{2}\right)^{i}\>+\>\left(7\beta^{4}\right)^{m}.$ (A.11)

Using (A.6)–(A.8) and (A.10), one obtains

$\displaystyle\left\|u^{m+1}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\>\leqslant\>\>c_{4}\>+\>c_{2}c_{3}\bar{k}\,\|u^{m}\|^{3}_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}$
$\displaystyle\quad\leqslant\>\>c_{4}\>\,+\>\,\beta\,\left(\sum_{i=0}^{m-1}(7\beta)^{i}\,c_{4}^{2i+1}\>+\>\left(7\beta^{4}\right)^{m-1}\right)^{3}$
$\displaystyle\quad\leqslant\>\>c_{4}\>\,+\>\,\beta\,\left(\sum_{i=0}^{m-1}7^{i+1}\,(7\beta)^{3i}\,c_{4}^{3(2i+1)}\>+\>7^{m}\left(7\beta^{4}\right)^{3(m-1)}\right)$
$\displaystyle\quad=\>\>c_{4}\>\,+\>\,c_{4}\,\sum_{i=1}^{m}\beta\,7^{i}\,(7\beta)^{3i-3}\,c_{4}^{6i-4}\>\,+\>\,\beta\,7^{m}\left(7\beta^{4}\right)^{3m-3}$
$\displaystyle\quad\leqslant\>\>c_{4}\>\,+\>\,c_{4}\,\sum_{i=1}^{m}\left(7\beta
c_{4}^{2}\right)^{i}\,\left(7^{3i-3}\,\beta^{2i-2}\,c_{4}^{4i-4}\right)\>\,+\>\,\left(7\beta^{4}\right)^{m}\,\left(7^{3m-3}\,\beta^{8m-11}\right)$
$\displaystyle\quad\leqslant\>\>c_{4}\>\,+\>\,c_{4}\,\sum_{i=1}^{m}\left(7\beta
c_{4}^{2}\right)^{i}\,\left(7^{3}\beta^{2}c_{4}^{4}\right)^{i-1}\>\,+\>\,\left(7\beta^{4}\right)^{m}\,\left(7^{3}\beta^{5}\right)^{m-1}\,\beta^{3m-6}.$

This leads to (A.11), because $\,7^{3}\beta^{2}c_{4}^{4}\leqslant 1$,
$\,7^{3}\beta^{5}\leqslant 1,\,$ and $\,\beta^{3m-6}\leqslant\beta^{0}=1\,$
($m\geqslant 2$) in line with (A.8). Thus, (A.10) holds, and, consequently,

$\displaystyle\left\|u^{m}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\,\leqslant\>\,c_{4}\,\sum_{i=0}^{\infty}\left(7\beta
c_{4}^{2}\right)^{i}\>+\>1\>\,<\>\,\infty,$ $\displaystyle m\,=\,2,3,4,\ldots$

(one has $\>7\beta
c_{4}^{2}\,=\,7^{-\frac{1}{2}}\,(7^{3}\beta^{2}c_{4}^{4})^{\frac{1}{2}}\,<\,1\>$
and
$\,7\beta^{4}\,=\,\left(7^{3}\beta^{5}\right)^{\frac{1}{3}}\,\beta^{\frac{7}{3}}\,<\,1\,$
according to (A.8)), which completes the proof.

###  A.3 Proof of Theorem 3.2

By virtue of (11)–(13), one has

$\displaystyle\left\|u^{m+1}-u^{m}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\,=\>\,\left\|\mathcal{E}_{2}\left(\mathcal{F}u^{m}-\mathcal{F}u^{m-1}\right)\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}$
$\displaystyle\leqslant\>\,c_{2}\,\left\|\mathcal{F}u^{m}-\mathcal{F}u^{m-1}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\,,\quad
m\,=\,1,2,3,\ldots$

If one obtains the estimate

$\displaystyle\left\|\mathcal{F}u^{m}-\mathcal{F}u^{m-1}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}$
(A.12)
$\displaystyle\qquad\qquad\leqslant\>\,c_{5}\bar{k}\,\left\|u^{m}-u^{m-1}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\,,$
$\displaystyle m\,=\,1,2,3,\ldots,\quad c_{5}=\mathrm{const}>0,$

then

$\displaystyle\left\|u^{m+1}-u^{m}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\>\,\leqslant\>\,c_{2}c_{5}\bar{k}\,\left\|u^{m}-u^{m-1}\right\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\,,$
$\displaystyle m\,=\,1,2,3,\ldots,$

and the Banach fixed point theorem (see, e. g., [19, Theorem 4.16]) implies
the sought-after result (note that $c_{2}c_{5}\bar{k}<1$ for sufficiently
small $\bar{k}$). But (A.12) follows from the representations

$\displaystyle\mathcal{F}u^{m}(x,t)\,-\,\mathcal{F}u^{m-1}(x,t)$
$\displaystyle\qquad=\>\>u^{m}(x,t)\>(k(x)\,u^{m}(x-h,\,t)\>-\>f[u^{m}(\cdot,t)])$
$\displaystyle\qquad\quad\>\>-\>\,u^{m-1}(x,t)\,\left(k(x)\,u^{m-1}(x-h,\,t)\>-\>f\left[u^{m-1}(\cdot,t)\right]\right)$
$\displaystyle\qquad=\>\>k(x)\,u^{m}(x-h,\,t)\,\left(u^{m}(x,t)\,-\,u^{m-1}(x,t)\right)$
$\displaystyle\qquad\quad\>\>+\>\,k(x)\,u^{m-1}(x,t)\,\left(u^{m}(x-h,\,t)\,-\,u^{m-1}(x-h,\,t)\right)$
$\displaystyle\qquad\quad\>\>-\>\,u^{m}(x,t)\,\left(f[u^{m}(\cdot,t)]\,-\,f\left[u^{m-1}(\cdot,t)\right]\right)$
$\displaystyle\qquad\quad\>\>-\>\,f\left[u^{m-1}(\cdot,t)\right]\,\left(u^{m}(x,t)\,-\,u^{m-1}(x,t)\right),$
$\displaystyle
f\left[u^{m-1}(\cdot,t)\right]\>\,=\>\,\int_{0}^{2\pi}k(x)\,u^{m-1}(x,t)\,u^{m-1}(x-h,\,t)\,\mathrm{d}x,$
$\displaystyle
f[u^{m}(\cdot,t)]\>\,=\>\,\int_{0}^{2\pi}k(x)\,u^{m}(x,t)\,u^{m}(x-h,\,t)\,\mathrm{d}x,$
$\displaystyle f[u^{m}(\cdot,t)])\>-\>f\left[u^{m-1}(\cdot,t)\right]$
$\displaystyle\qquad=\>\>\int_{0}^{2\pi}k(x)\,u^{m}(x-h,\,t)\,\left(u^{m}(x,t)\,-\,u^{m-1}(x,t)\right)\,\mathrm{d}x$
$\displaystyle\qquad\quad\>\>+\>\,\int_{0}^{2\pi}k(x)\,u^{m-1}(x,t)\,\left(u^{m}(x-h,\,t)\,-\,u^{m-1}(x-h,\,t)\right)\,\mathrm{d}x,$

$m=1,2,3,\ldots,\,$ and the uniform boundedness of
$\,\left\\{\|u^{m}\|_{\mathbf{C}^{2,1}([0,2\pi]\times[0,T])}\right\\}_{m=0}^{\infty}\,$
(the latter takes place for sufficiently small $\bar{k}$ due to Lemma 3.1).
This completes the proof.

###  A.4 Proof of Theorem 3.3

For $h\in(0,h_{0}]$, let $x^{*}_{h}\in[0,2\pi]$ and $t^{*}_{h}\in[0,T]$
satisfy

$u_{h}(x^{*}_{h},t^{*}_{h})\>=\>\min_{\begin{subarray}{c}0\leqslant x\leqslant
2\pi,\\\ 0\leqslant t\leqslant T\end{subarray}}u_{h}(x,t).$ (A.13)

The first-order necessary minimum conditions imply

$\displaystyle\frac{\partial u_{h}}{\partial
t}\,(x^{*}_{h},t^{*}_{h})\>\geqslant\>0\quad\mathrm{for}\>\>\,t^{*}_{h}=0,$
(A.14) $\displaystyle\frac{\partial u_{h}}{\partial
t}\,(x^{*}_{h},t^{*}_{h})\>=\>0\quad\mathrm{for}\>\>\,0<t^{*}_{h}<T,$
$\displaystyle\frac{\partial u_{h}}{\partial
t}\,(x^{*}_{h},t^{*}_{h})\>\leqslant\>0\quad\mathrm{for}\>\>\,t^{*}_{h}=T.$

If $\,0<x^{*}_{h}<2\pi,\,$ the first- and second-order necessary minimum
conditions yield $\>\dfrac{\partial u_{h}}{\partial
x}\,(x^{*}_{h},t^{*}_{h})\,=\,0\>$ and

$\frac{\partial^{2}u_{h}}{\partial
x^{2}}\,(x^{*}_{h},t^{*}_{h})\>\geqslant\>0.$ (A.15)

If $x^{*}_{h}=0$ or $x^{*}_{h}=2\pi$, then

$\displaystyle
u_{h}(x^{*}_{h},t^{*}_{h})\>=\>u_{h}(0,t^{*}_{h})\>=\>u_{h}(2\pi,t^{*}_{h}),$
$\displaystyle\frac{\partial u_{h}}{\partial
x}\,(x^{*}_{h},t^{*}_{h})\>=\>\frac{\partial u_{h}}{\partial
x}\,(0,t^{*}_{h})\>=\>\frac{\partial u_{h}}{\partial x}\,(2\pi,t^{*}_{h})$

(by virtue of the boundary conditions in (6)), and the minimum can be reached
at both points $(0,t^{*}_{h})$, $(2\pi,t^{*}_{h})$ only for $\>\dfrac{\partial
u_{h}}{\partial x}\,(x^{*}_{h},t^{*}_{h})\,=\,0,\>$ so Taylor’s theorem with
the Lagrange remainder again leads to (A.15).

Since $\>u_{h},\,\partial u_{h}/\partial x,\,\partial^{2}u_{h}/\partial
x^{2}\>$ are continuous on $\,[0,2\pi]\times[0,T]\,$ and
$\>u_{h}(x-h,\,t)\,=\,u_{h}(x-h+2\pi,\,t)\>$ for $\,-h\leqslant x-h\leqslant
0,\,$ Taylor’s theorem with Lagrange remainder and the condition (17) imply

$\displaystyle\max_{\begin{subarray}{c}0\leqslant x\leqslant 2\pi,\\\
0\leqslant t\leqslant
T\end{subarray}}\,\left|u_{h}(x-h,\,t)\>-\>\left(u_{h}(x,t)\>-\>h\,\frac{\partial
u_{h}(x,t)}{\partial x}\right)\right|\>\>\leqslant\>\>c_{6}h^{2}$ (A.16)
$\displaystyle\forall h\in(0,h_{0}],$

where

$\displaystyle c_{6}\>\,=\>\,\sup_{0<h\leqslant
h_{0}}\,\max_{\begin{subarray}{c}0\leqslant x\leqslant 2\pi,\\\ 0\leqslant
t\leqslant T\end{subarray}}\,\left|\frac{\partial^{2}u_{h}(x,t)}{\partial
x^{2}}\right|\>\,<\>\,\infty.$ (A.17)

Hence,

$\displaystyle
f[u_{h}(\cdot,t)])\>\,=\>\,\int_{0}^{2\pi}k(x)\,u_{h}(x,t)\,u_{h}(x-h,\,t)\,\mathrm{d}x$
$\displaystyle\quad\>\,=\>\,\int_{0}^{2\pi}k(x)\,(u_{h}(x,t))^{2}\,\mathrm{d}x\>-\>h\,\int_{0}^{2\pi}k(x)\,u_{h}(x,t)\,\frac{\partial
u_{h}(x,t)}{\partial x}\,\mathrm{d}x\>+\>o(h,t)$ $\displaystyle\forall
h\in(0,h_{0}],$

where

$\lim_{h\,\to\,+0}\,\frac{\max\limits_{0\leqslant t\leqslant
T}\,|o(h,t)|}{h}\>\,=\>\,0.$ (A.18)

Using integration by parts and the boundary conditions for $k$ and $u_{h}$
(see Assumption 2.1 and (6)), one arrives at

$\displaystyle\int_{0}^{2\pi}k(x)\,u_{h}(x,t)\,\frac{\partial
u_{h}(x,t)}{\partial x}\,\mathrm{d}x$
$\displaystyle\qquad=\>\,\frac{1}{2}\left.\left(k(x)\,(u_{h}(x,t))^{2}\right)\right|_{x=0}^{x=2\pi}\>-\>\frac{1}{2}\,\int_{0}^{2\pi}k^{\prime}(x)\,(u_{h}(x,t))^{2}\,\mathrm{d}x$
$\displaystyle\qquad=\>\,-\frac{1}{2}\,\int_{0}^{2\pi}k^{\prime}(x)\,(u_{h}(x,t))^{2}\,\mathrm{d}x$
$\displaystyle\forall h\in(0,h_{0}],$

and, therefore,

$f[u_{h}(\cdot,t)])\>\,=\>\,\int_{0}^{2\pi}\left(k(x)\,+\,\frac{h}{2}\,k^{\prime}(x)\right)\,(u_{h}(x,t))^{2}\,\mathrm{d}x\>+\>o(h,t)\quad\forall
h\in(0,h_{0}].$

According to Assumption 2.1, $k$ is continuously differentiable and positive
on $[0,2\pi]$, so there exist constants $c_{7}>0$ and
$h^{\prime}_{0}\in(0,h_{0}]$ such that

$\min_{0\leqslant x\leqslant
2\pi}\,\left\\{k(x)\,+\,\frac{h}{2}\,k^{\prime}(x)\right\\}\>\,\geqslant\>\,c_{7}\quad\forall
h\in(0,h^{\prime}_{0}],$

leading to

$f[u_{h}(\cdot,t)])\>\,\geqslant\>\,c_{7}\int_{0}^{2\pi}(u_{h}(x,t))^{2}\,\mathrm{d}x\>+\>o(h,t)\quad\forall
h\in(0,h^{\prime}_{0}].$

Furthermore, the normalization condition (4) and the Cauchy–Schwarz inequality
for the Hilbert space $\,\mathcal{L}^{2}([0,2\pi];\,\mathbb{R})\,$ yield

$1\>\,=\>\,\left(\int_{0}^{2\pi}1\cdot
u_{h}(x,t)\,\mathrm{d}x\right)^{2}\>\,\leqslant\>\,2\pi\int_{0}^{2\pi}(u_{h}(x,t))^{2}\,\mathrm{d}x$

and, consequently,

$f[u_{h}(\cdot,t)])\>\geqslant\>\frac{c_{7}}{2\pi}\,+\,o(h,t)\quad\forall
h\in(0,h^{\prime}_{0}].$

Together with (A.18), this implies the existence of a constant
$h^{\prime\prime}_{0}\in(0,h^{\prime}_{0}]$ such that

$c_{8}\>\,=\>\,\inf_{0<h\leqslant
h^{\prime\prime}_{0}}\,\min\limits_{0\leqslant t\leqslant
T}\,f[u_{h}(\cdot,t)])\>\,>\>\,0.$ (A.19)

Let also

$w_{h}(x,t)\>=\>u_{h}(x,t)\,e^{\Phi_{h}(t)},\quad\Phi_{h}(t)\>=\>\int_{0}^{t}f[u_{h}(\cdot,s)]\,\mathrm{d}s\quad\forall
h\in(0,h_{0}].$ (A.20)

Then

$\displaystyle\frac{\partial w_{h}}{\partial
t}\,(x^{*}_{h},t^{*}_{h})\>\>=\>\>\frac{\partial u_{h}}{\partial
t}\,(x^{*}_{h},t^{*}_{h})\>e^{\Phi_{h}(t^{*}_{h})}$ (A.21)
$\displaystyle\qquad+\>\,u_{h}(x^{*}_{h},t^{*}_{h})\>e^{\Phi_{h}(t^{*}_{h})}\>f[u_{h}(\cdot,t^{*}_{h})]\quad\forall
h\in(0,h_{0}].$

Moreover, (6) and (A.20) lead to

$\displaystyle\frac{\partial w_{h}}{\partial
t}\,(x^{*}_{h},t^{*}_{h})\>\>=\>\>k(x^{*}_{h})\>u_{h}(x^{*}_{h},t^{*}_{h})\>u_{h}(x^{*}_{h}-h,\,t^{*}_{h})\>e^{\Phi_{h}(t^{*}_{h})}$
$\displaystyle+\>\,\alpha\>\frac{\partial^{2}u_{h}}{\partial
x^{2}}\,(x^{*}_{h},t^{*}_{h})\>e^{\Phi_{h}(t^{*}_{h})}\quad\forall
h\in(0,h_{0}],$

and, therefore,

$\displaystyle\frac{\partial w_{h}}{\partial
t}\,(x^{*}_{h},t^{*}_{h})\>\>\geqslant\>\>k(x^{*}_{h})\>u_{h}(x^{*}_{h},t^{*}_{h})\>u_{h}(x^{*}_{h}-h,\,t^{*}_{h})\>e^{\Phi_{h}(t^{*}_{h})}$
$\displaystyle\forall h\in(0,h_{0}]$

due to (A.15) and $\alpha>0$. Together with the relations (17), (A.16) and the
continuity of $k$ on $[0,2\pi]$, this yields the existence of a constant
$c_{9}>0$ such that

$\displaystyle\frac{\partial w_{h}}{\partial
t}\,(x^{*}_{h},t^{*}_{h})\>\>\geqslant\>\>k(x^{*}_{h})\>(u_{h}(x^{*}_{h},t^{*}_{h}))^{2}\>e^{\Phi_{h}(t^{*}_{h})}$
(A.22)
$\displaystyle\qquad\qquad\qquad\quad\>\,-\>\,c_{9}h\>|u_{h}(x^{*}_{h},t^{*}_{h})|\>e^{\Phi_{h}(t^{*}_{h})}\quad\forall
h\in(0,h_{0}].$

Let us now establish the sought-after property by contradiction. Assume the
existence of a sequence
$\,\\{h_{i}\\}_{i=1}^{\infty}\,\subset\,(0,h^{\prime\prime}_{0}]\,$ such that
$\>\lim\limits_{i\to\infty}h_{i}\,=\,0\>$ and
$\>u_{h_{i}}(x^{*}_{h_{i}},t^{*}_{h_{i}})\,<\,0\>$ for all $i\in\mathbb{N}$.
Then $t^{*}_{h_{i}}\neq 0$ (since $\,u_{h_{i}}(x,0)=\varphi(x)\geqslant 0$),
(A.14) reduces to $\>\dfrac{\partial u_{h_{i}}}{\partial
t}\,(x^{*}_{h_{i}},t^{*}_{h_{i}})\,\leqslant\,0,\>$ and (A.21) leads to

$\displaystyle\frac{\partial w_{h_{i}}}{\partial
t}\,(x^{*}_{h_{i}},t^{*}_{h_{i}})\>\>\leqslant\>\>u_{h}(x^{*}_{h_{i}},t^{*}_{h_{i}})\>e^{\Phi_{h_{i}}(t^{*}_{h_{i}})}\>f[u_{h_{i}}(\cdot,t^{*}_{h_{i}})].$
(A.23)

From (A.22) and (A.23), one obtains

$\displaystyle
u_{h_{i}}(x^{*}_{h_{i}},t^{*}_{h_{i}})\>f[u_{h_{i}}(\cdot,t^{*}_{h_{i}})]\>\>\geqslant$
$\displaystyle\>k(x^{*}_{h_{i}})\>(u_{h_{i}}(x^{*}_{h_{i}},t^{*}_{h_{i}}))^{2}$
$\displaystyle-\>\,c_{9}h_{i}\>|u_{h_{i}}(x^{*}_{h_{i}},t^{*}_{h_{i}})|,$

and, consequently,

$f[u_{h_{i}}(\cdot,t^{*}_{h_{i}})]\>\>\leqslant\>\>k(x^{*}_{h_{i}})\>u_{h_{i}}(x^{*}_{h_{i}},t^{*}_{h_{i}})\>\,+\>\,c_{9}h_{i}$

(due to $\>u_{h_{i}}(x^{*}_{h_{i}},t^{*}_{h_{i}})\,<\,0$). Together with
(A.19) and the positivity of $k$ on $[0,2\pi]$, this implies that
$\,0<c_{8}<c_{9}h_{i}\,$ for all $i\in\mathbb{N}$, which contradicts with
$\>\lim\limits_{i\to\infty}h_{i}\,=\,0$. Thus, $u_{h}$ should indeed be a
nonnegative function for sufficiently small $h$.