11institutetext:  Friedrich–Alexander University Erlangen–Nürnberg, DE
11email<EMAIL_ADDRESS>22institutetext: Imperial College
London, SW7 2AZ, London, UK 33institutetext: ETH Zurich, CH

# Confidence-Aware and Self-Supervised
Image Anomaly Localisation

Johanna P. Müller 11 0000-0001-8636-7986 Matthew Baugh 22 0000-0001-6252-7658
Jeremy Tan 33 0000-0002-9769-068X Mischa Dombrowski 11 0000-0003-1061-8990
Bernhard Kainz 1122 0000-0002-7813-5023

###### Abstract

Universal anomaly detection still remains a challenging problem in machine
learning and medical image analysis. It is possible to learn an expected
distribution from a single class of _normative samples_ , _e.g._ , through
epistemic uncertainty estimates, auto-encoding models, or from synthetic
anomalies in a self-supervised way. The performance of self-supervised anomaly
detection approaches is still inferior compared to methods that use examples
from _known unknown_ classes to shape the decision boundary. However, outlier
exposure methods often do not identify _unknown unknowns_. Here we discuss an
improved self-supervised single-class training strategy that supports the
approximation of probabilistic inference with loosen feature locality
constraints. We show that up-scaling of gradients with histogram-equalised
images is beneficial for recently proposed self-supervision tasks. Our method
is integrated into several out-of-distribution (OOD) detection models and we
show evidence that our method outperforms the state-of-the-art on various
benchmark datasets.

###### Keywords:

Anomaly detection Out-of-distribution detectionPoisson image interpolation
Self-supervision.

##  1 Introduction

Out-of-distribution (OOD) detection builds upon the assumption that the
division into normal and abnormal data is distinct, however, OOD data can
overlap in-distribution (ID) data and may exhibit an infinite number of
descriptive features. We assume for medical imaging data a finite ID
("healthy") distribution space and an infinite OOD ("anomalous") distribution
space. Furthermore, we assume ID consistency for healthy medical images such
that the compatibility condition holds, based on the impossibility theorems
for OOD detection by [8]. As a result, OOD detection algorithms can be capable
of learning the finite ID space and also a finite but sufficient number of ODD
features for inference. We can approximate density-based spaces based on drawn
samples from real unknown (conditioned) probability distributions for covering
uncertainty in the annotation of data, and, therefore, assume the
Realisability assumption [8] for learnable OOD detection referring to the
proposed problem formulation.

The OOD problem for medical imaging can also be seen from a practical,
intuitive point of view. To reflect that multiple human medical experts can
come by different diagnoses given the same image of a patient, we integrate
uncertainty estimates for both ID and OOD data in the form of probability
distributions. Intuitively, we tend to imagine a finite ID space, since we
observe a consistency between ID features which are exhibited by healthy human
individuals from an anatomical point of view. By assuming that, we postulate
that we can present learnable OOD detection through training different types
of algorithms on normal data with synthetically generated anomalies.

Learning from synthetically generated anomalies became a research focus in
medical image analysis research recently [11]. In a medical context, labelling
requires medical expertise and, hence, human resources for generating reliable
ground truth masks for anomaly detection algorithms. Self-supervised tasks
that base on synthetically generated anomalies are considered convenient
mitigation for limited robustness and generalisation abilities that result
from small datasets. An extension of this idea is to leverage the natural
variations in normal anatomy to create a range of synthetic abnormalities. For
example, image patch regions can be extracted from two independent samples and
replaced with an interpolation between both patches [25, 15]. The
interpolation factor, patch size, and patch location can be randomly sampled
from uniform distributions. Any encoder-decoder architecture can be trained to
give a pixel-wise prediction of the patch and its interpolation factor. This
encourages a deep network to learn what features to expect normally and to
identify where foreign patterns have been introduced. The estimate of the
interpolation factor lends itself nicely to the derivation of an outlier
score. Meanwhile, the pixel-wise output allows for pixel- and subject-level
predictions using the same model. However, such synthesis strategies feature
obvious discontinuities. [26, 22] solve the discontinuity problem by using
Poisson image editing, but the resulting anomalies can be so subtle that they
may represent variations of the normal class rather than true anomalies and
these approaches do not provide prediction confidence estimates. Therefore we
propose a new approach to model the ID space and make the following
contributions:

  1. 1.

We propose a revised Poisson Image-interpolation framework for the generation
of salient but still smoothly interpolated anomalies for self-supervision in
unsupervised image anomaly localisation.

  2. 2.

We propose self-supervision with a probabilistic feature extractor –
Probabilistic PII (P-PII) – which allows the generation of stochastic
anomalies with which we are able to simulate multiple annotators.

  3. 3.

We evaluate P-PII on 2D chest radiograph images and 3D CT scans and show that
our method outperforms recently proposed self-supervised anomaly localisation
approaches.

  4. 4.

We show that it is possible to learn feature distributions for ’normal’ tissue
in a self-supervised way from databases that exclusively contain patients with
the disease.

Related Work. The most prominent direction for unsupervised medical anomaly
localisation [27] is dominated by reconstruction-based methods like VAEs [28,
33, 16, 10] as well as other generative models like GANs [31, 1, 21] ,
especially, for image synthesis and data augmentation [7, 11, 9]. New advances
are expected by Diffusion models, which shine with detailed reconstructions
and anomaly maps for detection [29] but they are computationally very
challenging and have not been evaluated in detail yet. Other commonly used
methods include one-class Support Vector Machines, k-Nearest Neighbors and
extensions of these approaches for dimensionality-reduced feature spaces [17,
6]. Probabilistic methods have not been researched in detail for OOD detection
yet. However, they are known for probabilistic segmentation approaches. For
example, the Probabilistic Hierarchical Segmentation (PHISeg) combines a
conditional variational autoencoder (cVAE) with a U-NET setup proposed by [4],
Bayesian U-Nets [23] can model epistemic uncertainty with weak labels and
Monte Carlo estimates [20, 5, 19].

In a medical context, labelling requires medical expertise and, hence, human
resources for generating reliable ground truth masks for anomaly detection
algorithms. Self-supervised tasks are considered as convenient extensions for
improving robustness, uncertainty and generalisation abilities of models and
replace expensive labelling [13, 12, 18, 32]. We modify our backbone models to
allow for OOD detection. To do this, we form a self-supervised task which is
easily interchangeable. The self-supervised principle relies on patch
interpolation from the same or a different source image into a target image.
Since more research work focuses on alleviating the labelling effort by
experts for image data, different generation methods for anomalies emerged.
For Foreign patch interpolation (FPI) [25], two patches of the same location
are extracted from two independent samples and replaced with an interpolation
between both patches. CutPaste [15] updates the original method by translating
patches within an image and allows the effective detection of anomalies in
industrial datasets. Poisson Image Interpolation (PII) [26] overcomes sharp
discontinuities with Poisson editing as an interpolation strategy and
generates more organic and subtle outliers. Natural Synthetic Anomalies (NSA)
[22] are introduced by rescaling, shifting and a new Gamma-distribution-based
patch shape sampling without the use of interpolation factors for an end-to-
end model for anomaly detection.

##  2 Method

Figure 1: Probabilistic PII takes patches from a source image of a given size.
A second mask of circular size, drawn from two normal distributions for radius
and location inside the source patches, allows aggregated anomalies with
smoothly interpolated boundaries. We obtain probabilistic and salient
anomalies.

Self-supervised tasks were considered convenient extensions for improving the
robustness, uncertainty and generalisation abilities of models [13, 12, 18].
Our proposed Probabilistic PII self-supervision task is based on [25] and
builds upon the Poisson image editing implementation by [3]. PII relies on the
relative changes of the source image, the image gradient field
$\mathbf{v_{pq}}$, in the patch region and the patch boundary of the target
image $\delta h$, see Eq. 1. The solution of the underlying mathematical
problem represents the discretised Poisson equation with Dirichlet boundary
conditions, see Eq. 2 and Eq. 4. The intensity values within the patch $h$ are
given by the scalar function $f_{in}$ and $\langle p,q\rangle$ are denoted as
a pixel pair such that $q\in N_{p}$ denote the four directly adjacent
neighbour pixel of $p$. For PII, $\alpha$ determines the influence of the
individual image gradients on the interpolation task.

$v_{pq}=\left\\{\begin{array}[]{ c l
}(1-\alpha)(x_{i_{p}}-x_{i_{q}}),&{\quad\textrm{if
}\left|(1-\alpha)(x_{i_{p}}-x_{i_{q}})\right|>\left|\alpha(x_{i_{p}}-x_{i_{q}})\right|}\\\
\alpha(x_{i_{p}}-x_{i_{q}}),&\quad\textrm{otherwise.}\end{array}\right.$ (1)

$\mathcal{r}f_{in}=\textrm{div}\mathbf{v}\textrm{ over }h$ (2)

The PII task can be reformulated to the following minimisation problem (Eq.
3), given the projection of $\mathbf{v}(\frac{p+q}{2})$ onto the oriented edge
(Eq. 1) [26] and the field of intensity image gradients (Eq. 2). The problem
formulation can be solved via a discrete linear system solver.

$\min_{f_{in}}\iint\displaylimits_{h}\left|{\mathcal{r}f_{in}-\mathbf{v}}\right|^{2},\textrm{
with }f_{in}\Big{|}_{\delta h}=f_{out}\Big{|}_{\delta h}$ (3)

$\min_{f_{in}\big{|}_{h}}\sum_{\langle p,q\rangle\cap h\neq
0}(f_{in,p}-f_{in,q}-v_{pq})^{2},\textrm{ with }f_{in}\Big{|}_{\delta
h}=f_{out}\Big{|}_{\delta h},\forall p\in\delta h,q\in N_{p}$ (4)

Our proposed Probabilistic PII (P-PII) builds upon these mathematical
foundations but incorporates new features and approaches for addressing
current limitations and rethinking its application.

First, we apply P-PII pairwise on allegedly non-anomalous training data
samples but those pairs can be also, _e.g._ , easily reduced to one and single
non-anomalous image sample for reduced memory and time consumption. If
pairwise applied, the allocation of image pairs is randomly drawn from the
image batch. Second, we take patches from different locations of the source
image and interpolate them into different locations inside the target image,
hence, we latch on the patch drawing by NSA [22]. Third, we overcome the
current limitation of PII and PII-based anomaly generation methods regarding
the grade of abnormality of the interpolated patches. If both, source and
target images, are normalised, these anomalous regions are very subtle and
difficult to recognise - compared to real lesions as well. For intensifying
these abnormal features, we introduce an amplification of gradients, through a
scaling factor, during the interpolation into the source patch. This approach
generates less subtle, salient anomalies which are still smoothly interpolated
into the target image. Fourth, we mitigate class imbalance of normal and
anomalous pixels through the generation of $k>1$ anomalies per image with
which we speed up learning to differentiate both classes. Fifth, we introduce
the Probabilistic feature into PII. For simulating the variance of annotations
by multiple raters, as _e.g._ , annotation of lesions by multiple medical
experts, we generate circular anomalies, inside each extracted patch from the
source image. Therefore, we draw anomaly masks whose parameters, radius r and
location (x,y) (Eq. 5), we sample from normal distributions. We ensure with
fixed boundaries of location and radius that the generated anomaly only
touches the boundaries.

$\textbf{r}\sim\mathcal{N}_{Radius}(\mu,\sigma)\quad\quad(\textbf{x},\textbf{y})\sim\mathcal{N}_{Location}(M=\langle\mu_{\textbf{x}},\mu_{\textbf{y}}\rangle,\Sigma=\langle\sigma_{\textbf{x}},\sigma_{\textbf{y}}\rangle)$
(5)

For using P-PII as the self-supervised task for OOD detection, we decided on
intensity-based label generation. Based on the mean of all anomalies of each
patch, we use the absolute difference between the original target image and
the mean final image as the label. Additionally, we have a variance map of all
anomalies which can be used for further statistical evaluation or integration
into the optimisation problem.

##  3 Evaluation and Results

Data. We use the http://db.jsrt.or.jp/eng.phpJSRT database [24] as an
exemplary smaller medical imaging dataset which includes 154 conventional
chest radiographs with lung nodules and 93 radiographs without a nodule. For
each patient, only one image is attributed. We re-scaled all images from
$2048\times 2048$ matrix size to $512\times 512$ in order to hold the
conditions for all datasets equal. The subset without pathological findings
serves as our training dataset.
https://wiki.cancerimagingarchive.net/pages/viewpage.action?pageId=1966254
LIDC-IDRI [2] covers 1018 cases in the form of CT scans with 7371 lesions,
which were marked by at least one radiologist. We also divide the dataset into
lesion slices and anomaly-free slices by extracting the context slices from
each volume with a margin of about 5 slices on either side of the lesion,
which approximates the maximum possible margin of lesions given slice
thickness and lesion diameter. We use the first 800 cases as a training
dataset, and the rest for validation and testing. The large-scale dataset
https://nihcc.app.box.com/v/DeepLesion DeepLesion [30] contains 32,735 lesions
in 32,120 computed tomography (CT) slices from 10,594 studies of 4,427 unique
patients. Since the image size varies throughout the dataset, we resize each
image to the smallest occurring size, $512\times 512$. Each lesion slice is
provided as part of an imaging volume which provides the 3D context of the
lesion. We divide the dataset into lesion slices and anomaly-free slices by
extracting the context slices from each volume with a margin of about 10 mm on
either side of the lesion. As a result, we have 386,587 anomaly-free slices
and 4831 annotated anomalous slices. We test the quality of performance for
all models on ID and OOD data samples, which were not seen during training.
For JSRT, the test set consists of $19$ ID samples and $154$ OOD samples. For
the large datasets, we drew a test cohort of $500$ ID and $500$ ($478$ for
LIDC-IDRI) OOD samples. For LIDC-IDRI and DeepLesion, both ID and OOD samples
are from patients not occurring in the training dataset. Note that the models
are trained on healthy tissue of ill patients for the datasets LIDC-IDRI and
DeepLesion, which is different to the dataset JSRT for which we only
differentiate between ill and healthy patients/samples.

Pre-processing and Training. We use histogram equalisation to the normalised
images for contrast enhancement, adopted from MIMIC-CXR-JPG [14]. We apply
this type of equalisation to all datasets. We train all models for a fixed
number of $100,000$ steps with a fixed batch size of $16$. We used PNY NVIDIA
A100s with at least 18 GB of memory per job. The training runtime was approx.
4 days. The backbone models and P-PII were implemented in Python and
TensorFlow.

Metrics. Choosing suitable metrics for evaluating OOD detection methods is
important to effectively evaluate the performance of a method and make valid
comparisons with other approaches. We chose the Area under the receiver
operating characteristic (AUROC) for sample- and pixel-wise binary
classification between OOD and ID samples/pixels as a threshold-less metric.
We refer with _OOD_ to anomalous samples/pixels and with _ID_ to normal
(’healthy’) input samples/pixels. Average Precision (AP) takes both precision
and recall into account and is considered a sample-based evaluation metric
here. In medical imaging analysis, false negatives are more critical than
false positives, especially, in lesion detection. Therefore, we include the
Free-response receiver operating characteristic (FROC) score as an evaluation
measure.

Sensitivity analysis. We perform an ablation study to investigate the impact
of revised PII as a self-supervision task for various backbone models (U-Net,
Monte-Carlo Dropout (rate=0.1) U-Net, PHiSeg). All backbone models have the
same depth of five levels, PHiSeg includes two additional resolution levels.
We examine the influence of selected augmentation functions for small-scale
datasets or datasets suffering from class imbalance for improving the
performance of self-supervised training.

Figure 2: Exemplary anomaly prediction on test data with U-net, input image in
grey, heatmap of prediction in red, ground truth bounding box in blue.

Results. We evaluated all models with the training checkpoint for best dice.
We show quantitative results in Tab. 1 for all backbone models. We observed an
increase of pixel-wise AUROC of up to $13\%$ for U-net and PHiSeg and $18\%$
for Dropout U-net, for the JSRT dataset. For LIDC-IDRI, we achieve values
improved by up to $53\%$ for PHiSeg. For DeepLesion, we determined an increase
of $34\%$ with PHiSeg and $9\%$ with U-net for pixel-wise AUROC. Emphasising
the sensitivity level of $0.27$ for 10 avg. FPS, we increased the performance
of the U-net, trained with PII, threefold with our proposed self-supervision
task. Sample-wise AUROC was improved the most for the JSRT dataset with
$45\%$, whereas we observed AUROC values $<0.5$ for LIDC-IDRI and, partially,
for DeepLesion and JSRT. An increased amount of false positives in predicting
anomalous samples results for sample-wise AP for the large datasets. We show
qualitative results for the prediction of U-net as a backbone model in Fig. 2.
The prediction on JSRT is quantitatively better, but there are still false
positive pixels in all examples, especially, for the larger datasets. We
compare augmentation functions for further enhancing the performance of P-PII,
see Tab. 2. We compare both best-performing models and obtain an increase of
$1\%$ with scaling of the input image and combining scaling, random rotation
in between $\pm 10^{\circ}$ and elastic deformation. Further improvement was
achieved by scaling the input for the Dropout U-net which resulted in
enhancing image-wise AUROC about $3\%$. The highest improvement can be
achieved through the use of augmentation functions yielding a sensitivity of
$11\%$ for U-net with combined augmentation, and $19\%$ for Dropout U-net with
scaling.

Table 1: Results; for PHiSeg, mean of $50$ drawn samples from likelihood network; AUC - Area under the Receiver operating characteristic (AUROC), FC - Free-response Receiver operating characteristic (FROC) for $10$ average FPs |  |  | JSRT [24] | DeepLesion [30] | LIDC-IDRI [2]
---|---|---|---|---|---
|  |  | Pixel | Sample | Pixel | Sample | Pixel | Sample
(lr)4-7(lr)8-11(lr)12-15  PII [25] | Model |  | AUC | FC | AUC | AP | AUC | FC | AUC | AP | AUC | FC | AUC | AP
U-Net |  | $0.80$ | $0.08$ | $0.44$ | $0.87$ | $0.68$ | $0.00$ | $0.50$ | $0.49$ | $0.50$ | $0.00$ | $0.36$ | $0.39$
MC U-Net |  | $0.76$ | $0.01$ | $0.55$ | $0.90$ | $\mathbf{0.74}$ | $0.00$ | $0.53$ | $\mathbf{0.55}$ | $0.59$ | $\mathbf{0.01}$ | $0.40$ | $0.43$
PHiSeg |  | $0.67$ | $0.00$ | $0.51$ | $0.90$ | $0.41$ | $\mathbf{0.01}$ | $0.47$ | $0.48$ | $0.43$ | $0.00$ | $\mathbf{0.52}$ | $\mathbf{0.50}$
Ours P-PII | U-Net |  | $\mathbf{0.90}$ | $\mathbf{0.27}$ | $\mathbf{0.64}$ | $\mathbf{0.94}$ | $\mathbf{0.74}$ | $\mathbf{0.01}$ | $\mathbf{0.56}$ | $0.52$ | $\mathbf{0.69}$ | $\mathbf{0.01}$ | $0.33$ | $0.38$
MC U-Net |  | $\mathbf{0.90}$ | $0.26$ | $\mathbf{0.64}$ | $0.93$ | $0.72$ | $\mathbf{0.01}$ | $0.47$ | $0.49$ | $0.67$ | $\mathbf{0.01}$ | $0.38$ | $0.41$
PHiSeg |  | $0.76$ | $0.06$ | $0.63$ | $0.93$ | $0.55$ | $\mathbf{0.01}$ | $0.55$ | $0.51$ | $0.66$ | $\mathbf{0.01}$ | $0.41$ | $0.44$
|  |  |  |  |  |  |  |  |  |  |  |  |  |
Table 2: Sensitivity analysis of augmentation functions for small-scale datasets on P-PII for JSRT [24]; scaling, combined (rotation $\pm 10^{\circ}$, elastic deformation, scaling). |  |  |  | AUROC |  | AP |  | FROC
---|---|---|---|---|---|---|---|---
| Model | Augmentation |  | Pixel | Image |  | Image |  | 10FPs
(lr)5-6(lr)8-8(lr)10-10  Ours P-PII | U-Net | scaling |  | $\mathbf{0.91}$ | $0.60$ |  | $0.93$ |  | $0.27$
MC U-Net | scaling |  | $\mathbf{0.91}$ | $\mathbf{0.66}$ |  | $\mathbf{0.94}$ |  | $\mathbf{0.31}$
(lr)2-10 | U-Net | combined |  | $\mathbf{0.91}$ | $0.60$ |  | $0.92$ |  | $0.30$
| MC U-Net | combined |  | $\mathbf{0.91}$ | $0.59$ |  | $0.93$ |  | $0.25$
|  |  |  |  |  |  |  |  |

Discussion. Self-supervision with P-PII enables all models to detect also very
small lesions, see Fig. 2, which is still a major challenge for other anomaly
localisation models, in both, a supervised and self-supervised context. We
improve upon the issue of decreasing sensitivity for increasing average FPs in
FROC, which we observe for the baseline method. With augmentation functions
the performance of models trained with PII increases the sensitivity
significantly by up to $19\%$. The limited quantitative performance on
DeepLesion and LIDC-IDRI is likely due to the fixed training steps which could
be insufficient for large datasets and also the foreground-background class
imbalance could influence the results for large datasets. These issues need to
be approached in further studies. Considering the number of false positive
predicted regions, we would require expert analysis if those regions are
correlated with real aberrations in the input images. For now, we can only
interpret them as visually perceived abnormal regions in the input images,
_e.g._ , dense regions in the lung hilum. Compared to the original PII
implementation we achieved a shortening of at least half of the training time
through the usage of Poisson image editing through discrete sine
transformation [3]. This allows us to sample from different source images
multiple times for probabilistic representations of anomalies while still
being faster than the baseline.

##  4 Conclusion

We analyse the proposed self-supervised learning method, P-PPI, on multiple
three backbone models and three small- and large-scale datasets from the
medical imaging domain. We exploit the influence of augmentation functions for
the self-supervision task and present probabilistic anomalies, which are
described for the first time for applications in OOD detection. Our
investigations highlight previous limitations when using Poisson image
interpolation for the generation of synthetic anomalies. We improve pixel-wise
AUROC by up to $18\%$ and sample-wise AUROC by up to $45\%$ in comparison to
baseline methods. Additionally, we enhanced the pixel-wise sensitivity to $10$
avg. FPs up to $38\%$. We also show that it is possible to learn feature
distributions for normal tissue in a self-supervised way from databases that
exclusively contain patients with the disease (DeepLesion and LIDC-IDRI). In
future work, the integration of the generated variance maps into the loss
function has a high potential for pushing unsupervised probabilistic learning
further towards integration into clinical workflows.

_Acknowledgements_ : The authors gratefully acknowledge the scientific support
and HPC resources provided by the Erlangen National High Performance Computing
Center (NHR@FAU) of the Friedrich-Alexander-Universität Erlangen-Nürnberg
(FAU) under the NHR projects b143dc and b180dc. NHR funding is provided by
federal and Bavarian state authorities. NHR@FAU hardware is partially funded
by the German Research Foundation (DFG) – 440719683. Additional support was
also received by the ERC - project MIA-NORMAL 101083647, DFG KA 5801/2-1, INST
90/1351-1 and by the state of Bavaria.

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H. Huynh and A. Kalkan

Pullback and forward attractors of contractive difference equations

Huy Huynh and Abdullah Kalkan*

Universität Klagenfurt,
Institut für Mathematik,
9020 Klagenfurt am Wörthersee, Austria
E-mail<EMAIL_ADDRESS>
E-mail<EMAIL_ADDRESS>
*Corresponding author

# Pullback and forward attractors of contractive difference equations

###### Abstract

The construction of attractors of a dissipative difference equation is usually
based on compactness assumptions. In this paper, we replace them with
contractivity assumptions under which the pullback and forward attractors are
identical. As a consequence, attractors degenerate to unique bounded entire
solutions. As an application, we investigate attractors of integrodifference
equations which are popular models in theoretical ecology.

Pullback attractor, Forward attractor, Contractive mapping, Dissipative
difference equation, Semilinear difference equation, Contractive difference
equation, Integrodifference equation

Huy Huynh received his MSc in Applied Mathematics at Université de Rennes 1,
France in 2018. His research interests include partial difference equations,
dynamical systems, fluid dynamics, numerical analysis and their applications
in modelling and discretisation. At present, he is a PhD student in Applied
Mathematics at Universität Klagenfurt, Austria.

After studying mathematics at the Ludwig-Maximilians-University Munich,
Germany, Abdullah Kalkan received his MSc in Mathematics at University of
Augsburg, Germany and PhD in Applied Mathematics at Universität Klagenfurt,
Austria. His research interests include dynamical systems, numerical analysis,
stochastic processes and stability theory of stochastic differential
equations.

This research of the authors has been supported by the Austrian Science Fund
(FWF) under Grant Number P 30874-N35.

##  1 Introduction

Different attractor notions for nonautonomous difference equations has
received a large amount of attention over recent years as they reflect the
long-term behaviour of a process and consist of bounded entire solutions.
Whilst the theory of attraction to autonomous systems is well established
where attractors are given by invariant $\omega$-limit sets, the
generalization to nonautonomous systems is not always appropriate. This leads
us to considering nonautonomous sets that are not necessarily invariant. There
are two ways to describe attractors, namely forward attractors and pullback
attractors (Huynh et al. (2020); Kloeden et al. (2000, 2016)). These two types
are independent concepts, whereby in the autonomous case they are equivalent
(Kloeden et al. (2011)).

In order to construct an attractor, one is in need of dissipativity and
compactness properties. In general, nevertheless, the latter may not be easy
to verify and thus can be replaced by alternative assumptions such as
contractivity. Indeed, an example of a contractive difference equation which
is not compact will be shown in Subsection 2.3. For this reason, our goal here
is to investigate the existence and structure of pullback and forward
attractors under the assumption of contraction. Furthermore, based on abstract
fixed-point theorems, error estimates are provided to numerically approximate
attractors. In particular, we discuss two classes of dissipative difference
equations and show that forward and pullback attractors degenerate to unique
bounded entire solutions: first, semilinear difference equations in Banach
spaces and then contractive difference equations in general complete metric
spaces.

Our abstract theory can be applied to integrodifference equations in Banach
spaces of continuous functions over a compact domain. The right-hand side of
such equations consists of nonlinear integral operators of Hammerstein type,
which in theoretical ecology usually describe the spatial dispersal and
evolution of species with nonoverlapping generations (Jacobsen et al. (2015);
Kot et al. (1986); Lutscher (2019)). On the other hand, as integrodifference
equations are infinite-dimensional dynamical systems in the sense that one
cannot implement the solutions, we are interested in their spatial
discretisation to simulate the dynamical behaviours obtained in finite-
dimensional state spaces. Numerical simulations require discretisation. Thus,
appropriate discretisation to replace our integral equations by some
corresponding methods are given by common techniques in numerical analysis,
e.g., Nyström method (Atkinson et al. (2009)).

As an example, we consider the following realistic problem in which a species
lives in a habitat and an integrodifference equation is used to model it. For
simplicity, the habitat here can be considered as the compact interval
$\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$ in one dimension and the initial
condition can be chosen as any function in
$C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$. Now we iterate our equation and
consider the growth and distribution of the species in each generation. Due to
sufficient or insufficient natural conditions, the growth or decay of the
total population can be observed in different seasons. Furthermore, ecologists
take certain measures to protect and promote their growth and distribution in
order to prevent the species from dying out. Therefore, we use and compare
various supportive measures for the total population, i.e., inhomogeneities in
our equation in the last subsection, in order to determine the best possible
influence on the population.

The contents of this paper are as follows: In Section 2, we first establish
the necessary terminology and provide the results for the unique existence and
construction of attractors of two classes of dissipative difference equations:
semilinear difference equations in Banach spaces and contractive difference
equations in complete metric spaces. In addition, results for the latter are
based on a version of contraction principle for composite mappings. Followed
by that, Section 3 addresses the related notions for integrodifference
equations of Hammerstein type and presents some applications of our abstract
theory.

#### Notation

Let ${\mathbb{Z}}$ be the set of all integers and
${\mathbb{N}}:=\\{1,2,...\\}$ the positive integers. On a metric space
$(U,d)$, $I_{U}$ is the identity map, $B_{r}(x)$ and $\bar{B}_{r}(x)$ the open
and closed balls respectively, with center $x\in U$ and radius $r>0$. We write
$\operatorname{dist}(x,A):=\inf_{a\in A}d(x,a)$ for the distance of $x$ from a
set $A\subseteq U$ and $B_{r}(A):=\\{x\in X:\,\operatorname{dist}(x,A)<r\\}$
for its $r$-neighborhood. The Hausdorff semidistance of bounded and closed
sets $A,B\subseteq U$ is then defined as

$\displaystyle\operatorname{dist}(A,B):=\sup_{a\in A}\inf_{b\in B}d(a,b).$

A subset ${\mathcal{A}}\subseteq{\mathbb{Z}}\times U$ with $t$-fibers defined
by ${\mathcal{A}}(t):=\\{u\in U:\,(t,u)\in{\mathcal{A}}\\}$ is called a
nonautonomous set.

##  2 Nonautonomous difference equations

Let $(U,d)$ be a complete metric space. The paper deals with nonautonomous
difference equations of the form

$\displaystyle u_{t+1}=H_{t}(u_{t})$ ($\Delta$)

with right-hand side $H_{t}:U\to U$. For an initial time
$\tau\in{\mathbb{Z}}$, a forward solution to ($\Delta$) is a sequence
$(u_{t})_{\tau\leq t}$ in $U$ satisfying

$\displaystyle u_{t+1}\equiv H_{t}(u_{t})$ (1)

for all $\tau\leq t\in{\mathbb{Z}}$ while an entire solution
$(u_{t})_{t\in{\mathbb{Z}}}$ satisfies (1) on ${\mathbb{Z}}$. We define the
general solution $\varphi:\\{(t,\tau,u)\in{\mathbb{Z}}^{2}\times U:\,\tau\leq
t\\}\to U$ to ($\Delta$) by

$\displaystyle\varphi(t,\tau,u):=\begin{cases}u,&\tau=t,\\\
H_{t-1}\circ\ldots\circ H_{\tau}(u),&\tau<t.\end{cases}$

By the definition of $\varphi$, the process property

$\displaystyle\varphi(t,s,\varphi(s,\tau,u))=\varphi(t,\tau,u)$ (2)

holds for all $\tau\leq s\leq t\in{\mathbb{Z}}$ and $u\in U$. A solution
$u^{\ast}=(u^{\ast}_{t})_{t\in{\mathbb{Z}}}$ to ($\Delta$) is called globally
attractive, if

$\displaystyle\lim_{t\rightarrow\infty}d(u^{\ast}_{t},\varphi(t,\tau,u_{\tau}))=0$

holds for all $\tau\in{\mathbb{Z}}$ and $u_{\tau}\in U$.

A nonautonomous set ${\mathcal{A}}$ is called

  * •

positively invariant or invariant, if

$\displaystyle H_{t}({\mathcal{A}}(t))\subseteq{\mathcal{A}}(t+1)\textrm{ or
}H_{t}({\mathcal{A}}(t))={\mathcal{A}}(t+1)$

holds for all $t\in{\mathbb{Z}}$, respectively,

  * •

compact, if every $t$-fiber ${\mathcal{A}}(t)$ is compact.

Moreover, a nonautonomous set ${\mathcal{A}}$ with the two properties of
invariance and compactness is called a forward attractor of ($\Delta$), if

$\displaystyle\lim_{t\to\infty}\operatorname{dist}(\varphi(t,\tau,B),{\mathcal{A}}(t))=0\quad\text{for
all bounded }B\subseteq U$

and a pullback attractor of ($\Delta$), if

$\displaystyle\lim_{\tau\to-\infty}\operatorname{dist}(\varphi(t,\tau,B),{\mathcal{A}}(t))=0\quad\text{for
all bounded }B\subseteq U.$

We denote equations ($\Delta$) as $\theta$-periodic for some
$\theta\in{\mathbb{N}}$, if

$\displaystyle H_{t+\theta}=H_{t}$ (3)

holds for all $t\in{\mathbb{Z}}$ and $\theta$ is consequently denoted the
period. In case of $\theta=1$, i.e., $H_{t+1}=H_{t}=:H$ for each
$t\in{\mathbb{Z}}$, we say ($\Delta$) is autonomous.

###  2.1 Semilinear difference equations

We consider nonautonomous difference equations ($\Delta$) in Banach spaces
$(\tilde{U},\|\cdot\|)$. Let $L_{t}\in L(\tilde{U})$, $t\in{\mathbb{Z}}$, be a
sequence of bounded linear operators and $K_{t}:\tilde{U}\to\tilde{U}$,
$t\in{\mathbb{Z}}$, be mappings. We consider ($\Delta$) of the semilinear form

$\displaystyle H_{t}(u):=L_{t}u+K_{t}(u).$ (4)

The general solution to ($\Delta$) of the form (4) by the variation of
constants formula (Pötzsche, 2010, Theorem 3.1.16, p. 100) is of the form

$\displaystyle\varphi(t,\tau,u_{\tau})=\Phi(t,\tau)u_{\tau}+\sum_{s=\tau}^{t-1}\Phi(t,s+1)K_{s}\left(\varphi(s,\tau,u_{\tau})\right)\quad\text{for
all }\tau\leq t,$ (5)

where the transition operator $\Phi:\\{(t,\tau)\in{\mathbb{Z}}^{2}:\tau\leq
t\\}\to L(\tilde{U})$ is defined by

$\displaystyle\Phi(t,\tau):=\begin{cases}L_{t-1}\ldots L_{\tau},&\tau<t,\\\
I_{\tilde{U}},&\tau=t.\end{cases}$

The next lemma is helpful for proving the main result in this subsection:

###### Lemma 2.1

Let $H_{t}:\tilde{U}\to\tilde{U}$ be of the semilinear form (4). Suppose there
exist reals $\kappa_{t}\geq 0,\alpha_{t}>0$ for all $t\in{\mathbb{Z}}$ and
$\gamma\geq 1$ such that

$\displaystyle\|K_{t}(u)-K_{t}(\bar{u})\|\leq\kappa_{t}\|u-\bar{u}\|$ (6)

and

$\displaystyle\|\Phi(t,\tau)\|\leq\gamma\prod_{r=\tau}^{t-1}\alpha_{r}$ (7)

hold for all $\tau\leq t\in{\mathbb{Z}}$ and $u,\bar{u}\in\tilde{U}$, then the
general solution of (4) satisfies the estimate

$\displaystyle\|\varphi(t,\tau,u)-\varphi(t,\tau,\bar{u})\|\leq\gamma\|u-\bar{u}\|\prod_{r=\tau}^{t-1}(\alpha_{r}+\gamma\kappa_{r})$
(8)

for all $\tau\leq t\in{\mathbb{Z}}$ and $u,\bar{u}\in\tilde{U}$.

###### Proof.

Proof Let $\tau\leq t\in{\mathbb{Z}}$ and $u,\bar{u}\in\tilde{U}$. The
variation of constants formula (5) and the assumptions (6)-(7) then imply

$\displaystyle\|\varphi(t,\tau,u)-\varphi(t,\tau,\bar{u})\|$
$\displaystyle\leq\gamma\|u-\bar{u}\|\prod_{r=\tau}^{t-1}\alpha_{r}+\gamma\sum_{s=\tau}^{t-1}\kappa_{s}\|\varphi(s,\tau,u)-\varphi(s,\tau,\bar{u})\|\prod_{r=s+1}^{t-1}\alpha_{r}.$

This leads to the estimate

$\displaystyle\|\varphi(t,\tau,u)-\varphi(t,\tau,\bar{u})\|\dfrac{1}{\prod_{r=\tau}^{t-1}\alpha_{r}}$
$\displaystyle\leq\gamma\|u-\bar{u}\|$
$\displaystyle\qquad+\gamma\sum_{s=\tau}^{t-1}\dfrac{\kappa_{s}}{\alpha_{s}}\|\varphi(s,\tau,u)-\varphi(s,\tau,\bar{u})\|\dfrac{1}{\prod_{r=\tau}^{s-1}\alpha_{r}}.$

Then the discrete Grönwall inequality from (Pötzsche, 2010, Proposition
A.2.1(a), p. 348) yields the claimed estimate (8). ∎

In order to construct pullback and forward attractors of the equation
($\Delta$) in the semilinear form (4), we assume ($\Delta$) is pullback
absorbing, i.e., there is a nonautonomous set
${\mathcal{B}}\subseteq{\mathbb{Z}}\times\tilde{U}$ satisfying

  * •

there exists a real $\rho>0$ such that
${\mathcal{B}}(t)\subseteq\bar{B}_{\rho}(0)$ for all $t\in{\mathbb{Z}}$,

  * •

for all bounded nonautonomous subsets $C\subseteq{\mathbb{Z}}\times\tilde{U}$,
there exists a real $S\in{\mathbb{N}}$ such that

$\displaystyle\varphi(t,t-s,C(t-s))\subseteq{\mathcal{B}}(t)\quad\text{for all
}s\geq S,$

  * •

${\mathcal{B}}$ is positively invariant.

###### Theorem 2.2

Let ($\Delta$) of the form (4) satisfy the assumptions of Lemma 2.1. If
($\Delta$) has a pullback absorbing set ${\mathcal{B}}$ and

$\displaystyle\lim_{\tau\to-\infty}\prod_{r=\tau}^{t-1}(\alpha_{r}+\gamma\kappa_{r})=0$
(9)

for a fixed $t\in{\mathbb{Z}}$, then ($\Delta$) has a unique bounded entire
solution $u^{\ast}=(u^{\ast}_{t})_{t\in{\mathbb{Z}}}$ and a unique pullback
attractor
${\mathcal{A}}^{\ast}:=\\{(t,u^{\ast}_{t})\in{\mathbb{Z}}\times\tilde{U}:t\in{\mathbb{Z}}\\}$.
If, in addition,

$\displaystyle\lim_{t\to\infty}\prod_{r=\tau}^{t-1}(\alpha_{r}+\gamma\kappa_{r})=0$
(10)

holds for a fixed $\tau\in{\mathbb{Z}}$, then ${\mathcal{A}}^{\ast}$ is also
the forward attractor of ($\Delta$).

The following proof is in fact the discrete time version of the proof of
(Kloeden:2020, Theorem 5.4, p. 35).

###### Proof.

Proof Let ${\mathcal{B}}$ be a pullback absorbing set of ($\Delta$),
$\tau\in{\mathbb{Z}}$ and $t_{n}\leq\tau$ a monotone decreasing sequence with
$t_{n}\to-\infty$ as $n\to\infty$ and $b_{n}\in{\mathcal{B}}(t_{n})$ for all
$n\in{\mathbb{N}}$.

  * (I)

Define the sequence

$\displaystyle\phi_{n}:=\varphi(\tau,t_{n},b_{n})$

for all $n\in{\mathbb{N}}$. It is easy to see that
$\phi_{n}\in{\mathcal{B}}(\tau)$ due to the positive invariance of
${\mathcal{B}}$. Moreover, $(\phi_{n})_{n\in{\mathbb{N}}}$ is Cauchy, i.e.,
for all $\epsilon>0$, there exists a real $N\in{\mathbb{N}}$ such that

$\displaystyle\|\phi_{n}-\phi_{m}\|<\epsilon\quad\text{for all }n,m\geq N.$

Indeed, w.l.o.g. for $m\geq n$, the process property (2) yields

$\displaystyle\phi_{m}=\varphi(\tau,t_{m},b_{m})=\varphi(\tau,t_{n},\varphi(t_{n},t_{m},b_{m}))=\varphi(\tau,t_{n},\bar{x}_{n,m})$

where $\bar{x}_{n,m}:=\varphi(t_{n},t_{m},b_{m})\in{\mathcal{B}}(t_{n})$ due
to the positive invariance of ${\mathcal{B}}$. Then by the definition of
$\phi_{n}$, the estimate (8) and the triangle inequality, one obtains

$\displaystyle\|\phi_{n}-\phi_{m}\|$
$\displaystyle=\|\varphi(\tau,t_{n},b_{n})-\varphi(\tau,t_{n},\bar{x}_{n,m})\|$
$\displaystyle\leq\gamma\|b_{n}-\bar{x}_{n,m}\|\prod_{r=t_{n}}^{\tau-1}(\alpha_{r}+\gamma\kappa_{r})$
$\displaystyle\leq
2\gamma\rho\prod_{r=t_{n}}^{\tau-1}(\alpha_{r}+\gamma\kappa_{r}).$

Combining this with the assumption (9) yields $\|\phi_{n}-\phi_{m}\|<\epsilon$
for some sufficiently large $n$. Therefore, $(\phi_{n})_{n\in{\mathbb{N}}}$ is
Cauchy and consequently has a unique limit $u^{\ast}_{\tau}$ due to the
completeness of $\tilde{U}$.

  * (II)

The sequence $(u^{\ast}_{\tau})_{\tau\in{\mathbb{Z}}}$ constructed in step (I)
satisfies $u^{\ast}_{\tau}\in{\mathcal{B}}(\tau)$ and
$u^{\ast}_{\tau+1}=H_{\tau}(u^{\ast}_{\tau})$. Hence,
$(u^{\ast}_{t})_{t\in{\mathbb{Z}}}$ is an entire solution to ($\Delta$) and
bounded due to $u^{\ast}_{t}\in{\mathcal{B}}(t)\subseteq\bar{B}_{\rho}(0)$ for
all $t\in{\mathbb{Z}}$.

  * (III)

The aim of this step is to show the uniqueness of a bounded entire solution
$(u^{\ast}_{t})_{t\in{\mathbb{Z}}}$ to ($\Delta$) in ${\mathcal{B}}$.

Let $(v^{\ast}_{t})_{t\in{\mathbb{Z}}}$ be another bounded entire solution to
($\Delta$) in ${\mathcal{B}}$. By the definition of an entire solution to
($\Delta$), the estimate (8), the triangle inequality and the assumption (9),
we obtain

$\displaystyle\|u^{\ast}_{t}-v^{\ast}_{t}\|$
$\displaystyle=\|\varphi(t,\tau,u^{\ast}_{\tau})-\varphi(t,\tau,v^{\ast}_{\tau})\|$
$\displaystyle\leq\gamma\|u^{\ast}_{\tau}-v^{\ast}_{\tau}\|\prod_{r=\tau}^{t-1}(\alpha_{r}+\gamma\kappa_{r})$
$\displaystyle\leq
2\gamma\rho\prod_{r=\tau}^{t-1}(\alpha_{r}+\gamma\kappa_{r})\xrightarrow[\tau\to-\infty]{}0.$

Hence, $u_{t}^{\ast}=v_{t}^{\ast}$ for all $t\in{\mathbb{Z}}$, implying the
uniqueness of the bounded entire solution $(u^{\ast}_{t})_{t\in{\mathbb{Z}}}$
to ($\Delta$) in ${\mathcal{B}}$. Moreover, since a pullback attractor
consists of all bounded entire solutions (cf. (Pötzsche, 2010, Corollary
1.3.4, p. 17)), the nonautonomous set
${\mathcal{A}}^{\ast}=\\{(t,u^{\ast}_{t}):t\in{\mathbb{Z}}\\}$ is the pullback
attractor.

  * (IV)

In this step, additionally suppose the assumption (10) holds for one
$\tau\in{\mathbb{Z}}$ and consequently for all $\tau\in{\mathbb{Z}}$. We want
to prove that $\mathcal{A}^{\ast}$ is also the forward attractor to (4).
Indeed, for all $u_{\tau}\in\tilde{U}$, by the definition of $u^{\ast}_{t}$,
the estimate (8), the triangle inequality and the assumption (10), one obtains

$\displaystyle\|\varphi(t,\tau,u_{\tau})-u^{\ast}_{t}\|$
$\displaystyle=\|\varphi(t,\tau,u_{\tau})-\varphi(t,\tau,u^{\ast}_{\tau})\|$
$\displaystyle\leq\gamma\|u_{\tau}-u^{\ast}_{\tau}\|\prod_{r=\tau}^{t-1}(\alpha_{r}+\gamma\kappa_{r})$
$\displaystyle\leq
2\gamma\rho\prod_{r=\tau}^{t-1}(\alpha_{r}+\gamma\kappa_{r})\xrightarrow[t\to\infty]{}0.$

Thus, ${\mathcal{A}}^{\ast}$ is also the forward attractor.

The proof is now set and done. ∎

Next we regard a special case of the previous results where ($\Delta$) of the
form (4) is $\theta$-periodic for some $\theta\in{\mathbb{N}}$, i.e.,

$\displaystyle L_{t+\theta}=L_{t}\textrm{ and }K_{t+\theta}=K_{t}$ (11)

hold for all $t\in{\mathbb{Z}}$, as follows.

###### Corollary 2.3

Let ($\Delta$) of the form (4) satisfy the assumptions of Lemma 2.1. If there
exists a $\theta\in{\mathbb{N}}$ such that the assumptions (11) hold for all
$t\in{\mathbb{Z}}$ and

$\displaystyle\prod_{r=0}^{\theta-1}(\alpha_{r}+\gamma\kappa_{r})<1,$

then the unique bounded entire solution $u^{\ast}$ to ($\Delta$) is
$\theta$-periodic and the nonautonomous set
${\mathcal{A}}^{\ast}=\\{(t,u^{\ast}_{t})\in{\mathbb{Z}}\times\tilde{U}:t\in{\mathbb{Z}}\\}$
is the pullback and forward attractor of ($\Delta$).

###### Proof.

Proof The periodicity of $L_{t}$ and $K_{t}$ in the assumptions (11) extends
to the right-hand side $H_{t}$ of ($\Delta$) as well as to the constants
$\kappa_{t}$ and $\alpha_{t}$ in the assumptions (6) and (7), respectively.
This satisfies the assumptions (9) and (10) for a fixed $t\in{\mathbb{Z}}$ and
a fixed $\tau\in{\mathbb{Z}}$, respectively. Hence, Theorem 2.2 implies
($\Delta$) has a unique bounded entire solution $u^{\ast}$ and the pullback
and forward attractor ${\mathcal{A}}^{\ast}$. Moreover, thanks to (Pötzsche,
2010, Proposition 1.4.5, p. 22),
$\\{u^{\ast}_{t+\theta}\\}={\mathcal{A}}^{\ast}(t+\theta)={\mathcal{A}}^{\ast}(t)=\\{u_{t}^{\ast}\\}$
for all $t\in{\mathbb{Z}}$. This proves the periodicity of $u^{\ast}$. ∎

###  2.2 Contractive difference equations

In this subsection, we return to nonautonomous difference equations ($\Delta$)
in general complete metric spaces $(U,d)$ whose right-hand side $H_{t}$
satisfies the following standing assumptions:

  * (i)

there exists a real $\lambda_{t}\geq 0$ such that

$\displaystyle
d(H_{t}(u),H_{t}(\bar{u}))\leq\lambda_{t}d(u,\bar{u})\quad\text{for all
}t\in{\mathbb{Z}},u,\bar{u}\in U,$

  * (ii)

there exists a sequence $(\tilde{u}_{t})_{t\in{\mathbb{Z}}}$ such that
$\left(H_{t}(\tilde{u}_{t})\right)_{t\in{\mathbb{Z}}}$ is bounded for all
$t\in{\mathbb{Z}}$, i.e., there exist a real $R>0$ and a point $\hat{u}\in U$
such that

$\displaystyle H_{t}(\tilde{u}_{t})\subseteq B_{R}(\hat{u})\quad\text{for all
}t\in{\mathbb{Z}}.$

The following lemma enables us to prove our results in this subsection. It is
a variant of the contraction mapping principle (Deimling, 1985, Theorem
17.1(a), p. 187) where the iterates $F^{T}$ of a mapping $F$ are defined
recursively via $F^{0}:=I_{X}$ and $F^{T}:=F\circ F^{T-1}$ for all
$T\in{\mathbb{Z}},T>0$:

###### Lemma 2.4

Let $(X,d_{\infty})$ be a complete metric space and $F:X\to X$. If there exist
a $T\in{\mathbb{N}}$ and a real $\ell\in[0,1)$ such that

$\displaystyle d_{\infty}(F^{T}(x_{1}),F^{T}(x_{2}))\leq\ell
d_{\infty}(x_{1},x_{2})\quad\quad\text{for all }x_{1},x_{2}\in X,$

then $F$ possesses a unique fixed point $x^{\ast}\in X$. Moreover, the error
estimate

$\displaystyle
d_{\infty}(x^{\ast},F^{tT}(x))\leq\dfrac{\ell^{t}}{1-\ell}d_{\infty}(x,F^{T}(x))$
(12)

is satisfied for all $t\in{\mathbb{N}}$ and $x\in X$.

The Lipschitz condition for the composite mappings and the error estimate in
Lemma 2.4 applied to ($\Delta$) can be stated as follows.

###### Theorem 2.5

Let the assumptions $(i)$-$(ii)$ hold. If there exists a $T\in{\mathbb{N}}$
such that $\sup_{s\in{\mathbb{Z}}}d(u,\varphi(s,s-T,u))<\infty$ for all $u\in
U$ and

$\displaystyle\ell:=\sup_{\tau\in{\mathbb{Z}}}\prod_{r=\tau}^{\tau+T-1}\lambda_{r}<1,$
(13)

then the following results hold:

  * (a)

($\Delta$) possesses a unique bounded entire solution
$u^{\ast}=(u_{t}^{\ast})_{t\in{\mathbb{Z}}}$. Moreover, $u^{\ast}$ is globally
attractive and the error estimate

$\displaystyle\sup_{s\in{\mathbb{Z}}}d(u_{s}^{\ast},\varphi(s,s-tT,u))\leq\dfrac{\ell^{t}}{1-\ell}\sup_{s\in{\mathbb{Z}}}d(u,\varphi(s,s-T,u))$
(14)

is satisfied for all $t\in{\mathbb{N}}$ and $u\in U$,

  * (b)

the nonautonomous set
${\mathcal{A}}^{\ast}:=\\{(t,u^{\ast}_{t})\in{\mathbb{Z}}\times
U:t\in{\mathbb{Z}}\\}$ is the pullback and forward attractor of ($\Delta$).

###### Proof.

Proof

  * (a)

In order to apply Lemma 2.4, let us consider the space $X:=\ell^{\infty}(U)$
of all bounded sequences $(u_{t})_{t\in{\mathbb{Z}}}$ in $U$ equipped with the
metric

$\displaystyle d_{\infty}(u,v):=\sup_{t\in{\mathbb{Z}}}d(u_{t},v_{t})$

and define the mapping

$\displaystyle F(u)_{t}:=H_{t-1}(u_{t-1})$

for all $t\in{\mathbb{Z}}$. Obviously, $\ell^{\infty}(U)$ is a complete metric
space and a sequence $u^{\ast}=(u_{t}^{\ast})_{t\in{\mathbb{Z}}}$ is a bounded
entire solution to ($\Delta$) if and only if $u^{\ast}=F(u^{\ast})$.

We first show that $F(u)$ is bounded for each $u\in\ell^{\infty}(U)$. By the
assumptions (i)-(ii) and the triangle inequality, one obtains

$\displaystyle d(H_{t}(u_{t}),\hat{u})\leq
d(H_{t}(u_{t}),H_{t}(\tilde{u}_{t}))+d(H_{t}(\tilde{u}_{t}),\hat{u})\leq\sup_{t\in{\mathbb{Z}}}\lambda_{t}d(u,\tilde{u})+R$

and passing to the supremum over $t$ yields
$d_{\infty}(F(u),\hat{u})\leq\sup_{t\in{\mathbb{Z}}}\lambda_{t}d(u,\tilde{u})+R$.
Thus, $F$ maps the bounded sequences into the bounded sequences, i.e., $F:X\to
X$ is well-defined.

Next, we show by induction

$\displaystyle F^{T}(u)_{\tau+T}=\varphi(\tau+T,\tau,u_{\tau}).$ (15)

Indeed, the initial case for $T=1$ holds by the definition. We then assume the
induction hypothesis (15) holds for a particular $T\in{\mathbb{N}}.$ By the
induction hypothesis, one obtains

$\displaystyle F^{T+1}(u)_{\tau+T+1}$ $\displaystyle=[F\circ
F^{T}(u)]_{\tau+T+1}=F(F^{T}(u))_{\tau+T+1}$
$\displaystyle=H_{\tau+T}(F^{T}(u)_{\tau+T})=\varphi(\tau+T+1,\tau,u_{\tau}),$

which implies the induction step for $T+1\in{\mathbb{N}}$. Hence (15) holds
for all $T\in{\mathbb{N}}$.

Finally, we compute the Lipschitz constant of $F^{T}$ for some
$T\in{\mathbb{N}}$

$\displaystyle d(F^{T}(u)_{\tau+T},F^{T}(\bar{u})_{\tau+T})$
$\displaystyle=d(\varphi(\tau+T,\tau,u),\varphi(\tau+T,\tau,\bar{u}))$
$\displaystyle\leq\lambda_{\tau+T-1}d(\varphi(\tau+T-1,\tau,u),\varphi(\tau+T-1,\tau,\bar{u}))$
$\displaystyle\leq\lambda_{\tau+T-1}\ldots\lambda_{\tau}d(u,\bar{u})=\left(\prod_{r=\tau}^{\tau+T-1}\lambda_{r}\right)d(u,\bar{u})$
$\displaystyle\leq\left(\sup_{\tau\in{\mathbb{Z}}}\prod_{r=\tau}^{\tau+T-1}\lambda_{r}\right)d_{\infty}(u,\bar{u})=\ell
d_{\infty}(u,\bar{u})$

for all $u,\bar{u}\in\ell^{\infty}(U)$ and passing to the supremum over all
$\tau\in{\mathbb{Z}}$ yields

$\displaystyle d_{\infty}(F^{T}(u)_{\tau+T},F^{T}(\bar{u})_{\tau+T})\leq\ell
d_{\infty}(u,\bar{u}).$

Applying Lemma 2.4 with $\ell<1$ then derives that $F$ has a unique fixed
point $u^{\ast}$, which in turn is a unique bounded entire solution of
($\Delta$).

Concerning the error estimate (14), take a point $\bar{u}\in U$ and consider
the constant sequence $u:=(\bar{u})_{t\in{\mathbb{Z}}}\in\ell^{\infty}(U)$. By
the definition of $d_{\infty}$ and the error estimate (12), one obtains

$\displaystyle d(u_{s}^{\ast},\varphi(s,s-tT,u))$
$\displaystyle=d(u_{s}^{\ast},F^{tT}(u)_{s})\leq
d_{\infty}(u^{\ast},F^{tT}(u))$
$\displaystyle\leq\dfrac{\ell^{t}}{1-\ell}d_{\infty}(u,F^{T}(u))=\dfrac{\ell^{t}}{1-\ell}\sup_{s\in{\mathbb{Z}}}d(u,\varphi(s,s-T,u))$

for all $s\in{\mathbb{Z}}$ and passing to the supremum over all
$s\in{\mathbb{Z}}$ proves (14). Furthermore, it is not difficult to see that
$u^{\ast}$ is also globally attractive.

  * (b)

First, we verify the following properties:

    * $\bullet$

${\mathcal{A}}^{\ast}$ is invariant:
${\mathcal{A}}^{\ast}(t+1)=\\{H_{t}(u^{\ast}_{t})\\}=H_{t}\left(\\{u^{\ast}_{t}\\}\right)=H_{t}\left({\mathcal{A}}^{\ast}(t)\right)$
for all $t\in{\mathbb{Z}}$,

    * $\bullet$

${\mathcal{A}}^{\ast}$ is compact:
${\mathcal{A}}^{\ast}(t)=\\{u^{\ast}_{t}\\}$ is a singleton and thus compact.

Now, let $B\subseteq U$ be bounded and $(\tau,u_{\tau})\in{\mathbb{Z}}\times
B$, i.e., there exist a real $R>0$ and a point $u\in U$ so that
$u_{\tau},u^{\ast}_{\tau}\in B_{R}(u)$. The definition of
${\mathcal{A}}^{\ast}$, the triangle inequality and the assumptions (i)-(ii)
derive

$\displaystyle\inf_{a\in{\mathcal{A}}^{\ast}(t)}d(\varphi(t,\tau,u_{\tau}),a)$
$\displaystyle=d(\varphi(t,\tau,u_{\tau}),u_{t}^{\ast})=d(\varphi(t,\tau,u_{\tau}),\varphi(t,\tau,u_{\tau}^{\ast}))$
$\displaystyle\leq\left(\prod_{r=\tau}^{t-1}\lambda_{s}\right)d(u_{\tau},u_{\tau}^{\ast})\leq
2R\prod_{r=\tau}^{t-1}\lambda_{s}$

for all $u_{\tau}\in B$ and passing to the supremum over all $u_{\tau}\in B$
yields

$\displaystyle\operatorname{dist}(\varphi(t,\tau,B),{\mathcal{A}}^{\ast}(t))\leq
2R\prod_{r=\tau}^{t-1}\lambda_{s}.$

Hence, combining this with the limit relations
$\lim_{t\rightarrow\infty}\prod_{r=\tau}^{t-1}\lambda_{s}=\lim_{\tau\rightarrow-\infty}\prod_{r=\tau}^{t-1}\lambda_{s}=0$
implies ${\mathcal{A}}^{\ast}$ is the pullback and forward attractor of
($\Delta$).

This completes the proof of the theorem. ∎

We next formulate the result of periodic attractors as follows.

###### Corollary 2.6

If there exists a $\theta\in{\mathbb{N}}$ such that (3) holds for all
$t\in{\mathbb{Z}}$ and

$\displaystyle\prod_{r=0}^{\theta-1}\lambda_{r}<1,$ (16)

then the unique bounded entire solution $u^{\ast}$ to ($\Delta$) is
$\theta$-periodic and the nonautonomous set
${\mathcal{A}}^{\ast}=\\{(t,u^{\ast}_{t})\in{\mathbb{Z}}\times
U:t\in{\mathbb{Z}}\\}$ is the pullback and forward attractor of ($\Delta$).

###### Proof.

Proof The assumption (3) yields both ($\Delta$) and $\lambda_{t}$ are also
$\theta$-periodic, implying the assumption (13) is satisfied for $T:=\theta$.
The rest of the proof is verbatim to that in the proof of Corollary 2.3 where
Theorem 2.5 is used instead. ∎

###  2.3 Examples of contractive non-compact difference equations

In this subsection, we provide an example where a difference equation is
contractive but not compact in order to explain the reason behind our
alternative assumption. In Banach spaces
$C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$ with

$\displaystyle\|v\|$
$\displaystyle:=\sup_{|x|\leq\tfrac{L}{2}}|v(x)|\quad\text{for all }v\in
C\left[-\tfrac{L}{2},\tfrac{L}{2}\right],$ (17)

we consider nonautonomous difference equations ($\Delta$) with right-hand side
given by

$\displaystyle H_{t}(u)(x):=\dfrac{b_{t}(x)u(x)}{1+|u(x)|}$

where $b_{t}:\left[-\tfrac{L}{2},\tfrac{L}{2}\right]\to[0,\infty)$ is
continuous and $h_{t}\in C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$. It is
obvious that $H_{t}$ is well-defined in
$C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$, i.e.,
$H_{t}:C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]\to
C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$. However, $H_{t}$ may not be
completely continuous. Indeed, if $H_{t}$ is completely continuous, i.e.,
$H_{t}$ maps any bounded set of $\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$ into
a relatively compact subset of $\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$, then
$\overline{H_{t}(B_{1}(0))}$ is compact. This implies that
$\overline{H_{t}(B_{1}(0))}$ contains a closed ball $\overline{B}_{r}(0)$ with
some $r>0$. Therefore, by a consequence of Riesz’s lemma (Diestel, 1984,
Theorem 4, p. 3), the dimension of the state space $\dim
C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$ is finite, which does not hold.
Hence, the operator $H_{t}$ is not compact.

On the other hand, it can be easily seen that $H_{t}$ is contractive, i.e.,
$H_{t}$ satisfies Assumption (i) in Subsection 2.2. where
$d(u,\bar{u}):=\|u-\bar{u}\|$ for all $u,\bar{u}\in
C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$ and
$\lambda_{t}:=\sup_{|x|\leq\tfrac{L}{2}}b_{t}(x)$ for all $t\in{\mathbb{Z}}$.
Indeed, letting $u,\bar{u}\in C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$, one
obtains

$\displaystyle|H_{t}(u)(x)-H_{t}(\bar{u})(x)|$
$\displaystyle=b_{t}(x)\left|\dfrac{u}{1+|u(x)|}-\dfrac{\bar{u}}{1+|\bar{u}(x)|}\right|$
$\displaystyle\leq b_{t}(x)|u(x)-\bar{u}(x)|\leq\lambda_{t}\|u-\bar{u}\|.$

and passing to the supremum over all
$x\in\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$ then yields the contractivity of
$H_{t}$.

In conclusion, one can construct attractors of nonautonomous difference
equations under the assumption of contractivity instead of compactness.

##  3 Applications

Now, we apply the theory from the previous section to integrodifference
equations (shortly IDEs) as applications. To be more precise, we study a class
of IDEs involving Hammerstein integral operators satisfying a global Lipschitz
condition and being defined on the space of continuous functions over a
compact domain.

###  3.1 Integrodifference equations

In this part, we deal with scalar IDEs of Hammerstein type which are equations
$\eqref{deq}$ with right-hand side given by

$\displaystyle
H_{t}(u):=\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}k_{t}(\cdot,y)g_{t}(y,u(y))\,{\mathrm{d}}y+h_{t}$
(18)

under the following standing assumptions:

  * •

the kernel $k_{t}:\left[-\tfrac{L}{2},\tfrac{L}{2}\right]^{2}\to{\mathbb{R}}$
is continuous and satisfies

$\displaystyle\sup_{|x|\leq\tfrac{L}{2}}\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}|k_{t}(x,y)|\,{\mathrm{d}}y<\infty,$

  * •

the growth function
$g_{t}:\left[-\tfrac{L}{2},\tfrac{L}{2}\right]\times{\mathbb{R}}\to{\mathbb{R}}$
is such that $g_{t}$ is bounded and satisfies a global Lipschitz condition for
all $t\in{\mathbb{Z}}$, i.e., there exists a continuous function
$\hat{g}_{t}:\left[-\tfrac{L}{2},\tfrac{L}{2}\right]\to[0,\infty)$ such that
for all $x\in\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$ and
$z,\bar{z}\in{\mathbb{R}}$, one has

$\displaystyle|g_{t}(x,z)-g_{t}(x,\bar{z})|\leq\hat{g}_{t}(x)|z-\bar{z}|\quad\text{for
all }t\in{\mathbb{Z}};$

additionally,
$g_{t}(\cdot,z):\left[-\tfrac{L}{2},\tfrac{L}{2}\right]\to{\mathbb{R}}$ is
continuous and $g_{t}(x,0)=0$ for all $t\in{\mathbb{Z}}$ and
$x\in\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$,

  * •

the inhomogeneity $h_{t}\in C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$
satisfies $\sup_{t\in{\mathbb{Z}}}\|h_{t}\|<\infty$ where the norm $\|\cdot\|$
is defined in (17).

The right-hand side (18) of ($\Delta$) defines a Hammerstein integral operator
whose properties are summarized in the following theorem (Pötzsche, 2019,
Theorem B.5 and Corollary B.6):

###### Theorem 3.1

Let $t\in{\mathbb{Z}}$. The Hammerstein operator
$H_{t}:C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]\to
C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$ is well-defined and satisfies the
global Lipschitz condition

$\displaystyle\|H_{t}(u)-H_{t}(\bar{u})\|\leq\sup_{|x|\leq\tfrac{L}{2}}\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}|k_{t}(x,y)|\hat{g}_{t}(y)\,{\mathrm{d}}y\|u-\bar{u}\|$
(19)

for all $u,\bar{u}\in C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$; one obtains

$\displaystyle\lambda_{t}:=\sup_{|x|\leq\tfrac{L}{2}}\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}|k_{t}(x,y)|\hat{g}_{t}(y)\,{\mathrm{d}}y$

as the Lipschitz constant of the Hammerstein operator $H_{t}$.

Notice that the Lipschitz constant $\lambda_{t}$ in Theorem 3.1 is the same as
the constant $\lambda_{t}$ in Assumption (i) in Subsection 2.2.

Under the assumptions of Theorem 3.1, we obtain from Theorem 2.5:

###### Theorem 3.2

Let ($\Delta$) have the right-hand side (18). If there exists a
$T\in{\mathbb{N}}$ such that (13) holds, then the following results hold:

  * (a)

($\Delta$) possesses a unique bounded entire solution
$u^{\ast}=(u_{t}^{\ast})_{t\in{\mathbb{Z}}}$ in
$C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$. Moreover, $u^{\ast}$ is globally
attractive and the following error estimate

$\displaystyle\sup_{s\in{\mathbb{Z}}}\|u_{s}^{\ast}-\varphi(s,s-tT,u)\|\leq\dfrac{\ell^{t}}{1-\ell}\sup_{s\in{\mathbb{Z}}}\|u-\varphi(s,s-T,u)\|$
(20)

is satisfied for all $t\in{\mathbb{N}}$ and $u\in
C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$,

  * (b)

the nonautonomous set
${\mathcal{A}}^{\ast}:=\left\\{(t,u^{\ast}_{t})\in{\mathbb{Z}}\times
C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]:t\in{\mathbb{Z}}\right\\}$ is the
pullback and forward attractor of ($\Delta$) with right-hand side (18).

###### Proof.

Proof First, notice that the well-definedness and Lipschitz properties of the
Hammerstein type mapping $H_{t}$ as in (18) are stated in Theorem 3.1. Then
applying Theorem 2.5 with $U:=C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$ and
the metric $d$ as the norm $\|\cdot\|$ proves both results of the theorem. ∎

Next we deal with IDEs being $\theta$-periodic for some
$\theta\in{\mathbb{N}}$, i.e.,

$\displaystyle k_{t+\theta}=k_{t},g_{t+\theta}=g_{t}\textrm{ and
}h_{t+\theta}=h_{t}$ (21)

hold for all $t\in{\mathbb{Z}}$. The periodicity of the pullback and forward
attractor of a $\theta$-periodic IDE is stated as follows.

###### Corollary 3.3

Let ($\Delta$) have the right-hand side (18). If there exists a
$\theta\in{\mathbb{N}}$ such that the assumptions (16) and (21) hold for all
$t\in{\mathbb{Z}}$, then the unique bounded entire solution $u^{\ast}$ to
($\Delta$) is $\theta$-periodic and the nonautonomous set
${\mathcal{A}}^{\ast}=\\{(t,u^{\ast}_{t})\in{\mathbb{Z}}\times
C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]:t\in{\mathbb{Z}}\\}$ is the pullback
and forward attractor of ($\Delta$).

###### Proof.

Proof The argument is verbatim to the one given in the proof of Corollary 2.6
where the assumption (21) and Theorem 3.2 are used instead. ∎

###  3.2 Examples

In this subsection, we apply our theoretical results above to IDEs. IDEs are
investigated under commonly used kernels $k_{t}$ and growth functions $g_{t}$
listed in Tables 3.1 and 3.2, respectively (cf. Lutscher (2019)), whose
coefficients $a_{t}$ and
$b_{t}:\left[-\tfrac{L}{2},\tfrac{L}{2}\right]\to[0,\infty)$ satisfy $a_{t}>0$
and $\sup_{|x|\leq\tfrac{L}{2}}|b_{t}(x)|\leq\beta_{t}$, respectively, for all
$t\in{\mathbb{Z}}$ and some real $\beta_{t}$ with
$\sup_{t\in{\mathbb{Z}}}\beta_{t}<\infty$.

| Laplace | Gauß | tent
---|---|---|---
$k_{t}(x,y)$ | $\dfrac{a_{t}}{2}e^{-a_{t}|x-y|}$ | $\dfrac{a_{t}}{\sqrt{\pi}}e^{-a_{t}^{2}(x-y)^{2}}$ | $\max(0,a_{t}-a_{t}^{2}|x-y|)$
$\sup_{|x|\leq\tfrac{L}{2}}\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}|k_{t}(x,y)|\,{\mathrm{d}}y$ | $1-e^{-\tfrac{a_{t}L}{2}}$ | $\operatorname{erf}\left(\dfrac{a_{t}L}{2}\right)$ | $a_{t}L-\dfrac{1}{4}a_{t}^{2}L^{2}$
Table 3.1: Kernels $k_{t}$ and their upper bounds $\sup_{|x|\leq\tfrac{L}{2}}\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}|k_{t}(x,y)|\,{\mathrm{d}}y$ | logistic | Beverton-Holt | Ricker
---|---|---|---
$g_{t}(x,z)$ | $\max(0,b_{t}(x)z(1-z))$ | $\dfrac{b_{t}(x)z}{1+|z|}$ | $ze^{-b_{t}(x)|z|}$
$\sup_{|x|\leq\tfrac{L}{2}}|g_{t}(x,z)|$ | $\dfrac{1}{4}\beta_{t}$ | $\beta_{t}$ | $e^{-1}\beta_{t}$
Table 3.2: Growth functions $g_{t}$ and their upper bounds
$\sup_{|x|\leq\tfrac{L}{2}}|g_{t}(x,z)|$

In order to construct and numerically approximate the pullback and forward
attractor ${\mathcal{A}}^{\ast}=\left\\{(t,u^{\ast}_{t})\in{\mathbb{Z}}\times
C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]:t\in{\mathbb{Z}}\right\\}$ of an IDE
($\Delta$) from Theorem 3.2, the Lipschitz constant $\lambda_{t}$ of the
Hammerstein operator $H_{t}$ from Theorem 3.1 and the error estimate (20) are
required. First, for the Lipschitz function $\lambda_{t}$, one can obtain it
from Table 3.2. Indeed, notice that all the growth functions $g_{t}$ from
Table 3.2 possess the same continuous function $\hat{g}_{t}(x):=\beta_{t}$
(independent of $x$) and thus

$\displaystyle\lambda_{t}=\beta_{t}\sup_{|x|\leq\tfrac{L}{2}}\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}|k_{t}(x,y)|\,{\mathrm{d}}y.$

Then, concerning the error estimate (20), we compute
$\sup_{s\in{\mathbb{Z}}}\|u-\varphi(s,s-T,u)\|$ by

$\displaystyle\|u-\varphi(s,s-T,u)\|$
$\displaystyle\leq\|u\|+\|\varphi(s,s-T,u)\|$
$\displaystyle=\|u\|+\|H_{s-1}\left(\varphi(s-1,s-T,u)\right)\|$
$\displaystyle\leq\|u\|+\sup_{s\in{\mathbb{Z}}}\|H_{s-1}\left(\varphi(s-1,s-T,u)\right)\|$

for all $s\in{\mathbb{Z}}$ by the triangle inequality and the definition of
$\varphi(s,s-T,u)$. Followed by that, we determine the upper bound of
$\sup_{s\in{\mathbb{Z}}}\|H_{s-1}\left(\varphi(s-1,s-T,u)\right)\|$. For each
function $u\in C\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$, one obtains

$\displaystyle|H_{s}(u)(x)|$
$\displaystyle\leq\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}|k_{s}(x,y)||g_{s}(y,u(y))|\,{\mathrm{d}}y+|h_{s}(x)|$
$\displaystyle\leq\sup_{|x|\leq\tfrac{L}{2}}\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}|k_{s}(x,y)|\,{\mathrm{d}}y\sup_{|x|\leq\tfrac{L}{2}}|g_{s}(x,u(x))|+\|h\|_{\infty}$
$\displaystyle=:l_{1}(s,u)+\|h\|_{\infty}$

for all $s\in{\mathbb{Z}}$ and $x\in\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$
where $\|u\|_{\infty}:=\sup_{t\in{\mathbb{Z}}}\|u_{t}\|$ for all
$u\in\ell^{\infty}(C\left[-\tfrac{L}{2},\tfrac{L}{2}\right])$. If
$\sup_{s\in{\mathbb{Z}}}l_{1}(s,u)<\infty$, then subsequent passing to the
supremum over all $x\in\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$ and
$s\in{\mathbb{Z}}$ yield

$\displaystyle\sup_{s\in{\mathbb{Z}}}\|H_{s}(u)\|\leq\sup_{s\in{\mathbb{Z}}}l_{1}(s,u)+\|h\|_{\infty}$

since $l_{1}(s,u)$ is independent of $x$ and $h_{s}$ is bounded above in
$s\in{\mathbb{Z}}$. This implies the upper bounds of
$\sup_{s\in{\mathbb{Z}}}\|H_{s-1}\left(\varphi(s-1,s-T,u)\right)\|$ and
$\|u-\varphi(s,s-T,u)\|$ for all $s\in{\mathbb{Z}}$. Similarly, passing to the
supremum over all $s\in{\mathbb{Z}}$ results in

$\displaystyle\sup_{s\in{\mathbb{Z}}}\|u-\varphi(s,s-T,u)\|\leq\|u\|+\sup_{s\in{\mathbb{Z}}}l_{1}\left(s-1,\varphi(s-1,s-T,u)\right)+\|h\|_{\infty}=:l_{2}$
(22)

and the error estimate (20) is obtained as a result.

Let us now be more precise by illustrating some results using concrete IDEs
and their discretisations. In order to avoid the numerical integration of the
integral operator and obtain a full discretisation, we work with the Nyström
method (cf. Atkinson et al. (2009)) by replacing the integral with a weighted
sum such that the discretisation of ($\Delta$) leads to a sequence of
equations $(\Delta_{n})_{n\in{\mathbb{N}}}$ of the form

$\displaystyle
u_{t+1}(x)=H^{n}_{t}(u_{t})(x)=\sum_{i=0}^{n}\omega_{i}k_{t}(x,\eta_{i})g_{t}(\eta_{i},u_{t}(\eta_{i}))\,{\mathrm{d}}y+h_{t}(x)$
($\Delta_{n}$)

for all $t\in{\mathbb{Z}}$ and $x\in\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$,
assuming given a convergent quadrature rule with nodes $\eta_{i}$ and weights
$\omega_{i}$ for $i=0,1,\ldots,n$. In particular, Example 3.4 below uses the
trapezoidal quadrature rule to approximate integrals with

$\displaystyle h:=\dfrac{L}{n},\eta_{i}:=-\dfrac{L}{2}+ih\textrm{ and
}\omega_{i}:=\begin{cases}\dfrac{h}{2},&i\in\\{0,n\\},\\\
h,&\textrm{otherwise}\end{cases}$

for all $i=0,1,\ldots,n$. Moreover, we can also approximate the total
population $\overline{u}_{t}$ at time $t$ as

$\displaystyle\overline{u}_{t}:=\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}u_{t}(y)\,{\mathrm{d}}y\approx\sum_{i=0}^{n}\omega_{i}u_{t}(\eta_{i}).$

On the other hand, for a discretisation, the error estimate
$\sup_{s\in{\mathbb{Z}}}\|u^{\ast}_{s}-\varphi(s,s-tT,u)\|$ in (20) has to be
less than or equal to some real tolerance $tol>0$ for some sufficiently large
$t\in{\mathbb{Z}}$. Therefore, in order to meet the assumption (13), one can
pick $\ell\leq\dfrac{1}{2}$ and hence by (22)

$\displaystyle\dfrac{\ell^{t}}{1-\ell}\sup_{s\in{\mathbb{Z}}}\|u-\varphi(s,s-T,u)\|\leq\dfrac{\left(\tfrac{1}{2}\right)^{t}}{1-\tfrac{1}{2}}l_{2}=2^{1-t}l_{2}<tol,$

resulting in $t\geq\log_{2}\left(\dfrac{2l_{2}}{tol}\right)$. Thus, we need at
least

$\displaystyle S:=T\log_{2}\left(\dfrac{2l_{2}}{tol}\right)$ (23)

iterations so that the error estimate (20) is satisfied and
${\mathcal{A}}^{\ast}$ is consequently the pullback and forward attractor of
the IDE ($\Delta$).

###### Example 3.4

Suppose in a habitat, a species initially lives mostly on the boundary of the
habitat and its population gradually decreases in the middle areas (cf. a
choice of a constant initial solution $u_{0}$ below). Over four seasons of
each year, it periodically breeds and dies as well as spreads from one place
to another. Not to mention that the total population decays dramatically in
fall and winter before slightly flourishing again in spring and summer every
year resulting from natural conditions and the species may come to extinction
after a period of time. Therefore, ecologists would like to improve the total
population by supporting it in each season with various options and thus
choosing the optimal one. Followed by that, a mathematical model using an IDE
may help to study the system and give the optimal solution in terms of the
total population of the species as a result.

For simplicity, we consider the scalar, $\theta$-periodic and inhomogeneous
IDE

$\displaystyle
u_{t+1}(x)=H_{t}(u_{t})(x):=\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}\dfrac{a}{2}e^{-a|x-y|}\dfrac{b_{t}(y)u_{t}(y)}{1+|u_{t}(y)|}\,{\mathrm{d}}y+h_{t}(x)$
(24)

for all $(x,t)\in\left[-\tfrac{L}{2},\tfrac{L}{2}\right]\times{\mathbb{Z}}$
that fits into the realistic problem and the framework of (18) where

  * •

the domain $\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$ represents the habitat,

  * •

the kernel $k(x,y):=\dfrac{a}{2}e^{-a|x-y|}$ with $a>0$ represents the
dispersal rate of the species from point $x$ to point $y$,

  * •

the periodic growth function $g_{t}(x,z):=\dfrac{b_{t}(x)z}{1+|z|}$ represents
the natural growth rate of the species at point $x$ and day $t$ starting from
the population $z$, while the periodic parameter
$b_{t}(x):=\alpha_{t}\hat{b}(x)$ represents the natural conditions at point
$x$ and day $t$ with

    * $\cdot$

$\alpha_{t}:=C\left(1+\tfrac{1}{2}\sin\left(\tfrac{2\pi
t}{\theta}\right)\right)>0$ for all $t\in{\mathbb{Z}}$,

    * $\cdot$

$\hat{b}(x)\geq 0$ for all $x\in\left[-\tfrac{L}{2},\tfrac{L}{2}\right]$,

    * $\cdot$

$\sup_{|x|\leq\tfrac{L}{2}}|b_{t}(x)|\leq\beta_{t}:=\alpha_{t}\sup_{|x|\leq\tfrac{L}{2}}|\hat{b}(x)|$
for all $t\in{\mathbb{Z}}$,

  * •

the periodic inhomogeneity $h_{t}(x)$ represents the supported population at
point $x$ and day $t$.

With the choice of $C$ as follows

$\displaystyle
C:=\dfrac{1}{1-e^{-\tfrac{aL}{2}}}\sqrt[\theta]{\dfrac{1}{2\prod_{r=0}^{\theta-1}\beta_{r}\left(1+\tfrac{1}{2}\sin\left(\tfrac{2\pi
r}{\theta}\right)\right)}}>0$

for all $t\in{\mathbb{Z}}$, the assumptions (13) and (19) hold where
$\lambda_{t}:=\beta_{t}\left(1-e^{-aL/2}\right)$ and

$\displaystyle\|H_{t}(u)-H_{t}(\bar{u})\|\leq\beta_{t}\left(1-e^{-aL/2}\right)\|u-\bar{u}\|\quad\text{for
all }u,\bar{u}\in C(\left[-\tfrac{L}{2},\tfrac{L}{2}\right]).$

By Theorem 3.2, for $\ell=\prod_{r=0}^{\theta-1}\lambda_{r}\leq\dfrac{1}{2}$,
(24) is contractive and has the $\theta$-periodic pullback and forward
attractor ${\mathcal{A}}^{\ast}=\\{(t,u^{\ast}_{t})\in{\mathbb{Z}}\times
C(\left[-\tfrac{L}{2},\tfrac{L}{2}\right]):t\in{\mathbb{Z}}\\}$ due to
Corollary 3.3. Furthermore, for the initial solution $u_{0}\in
C(\left[-\tfrac{L}{2},\tfrac{L}{2}\right])$ representing the initial
population, the error estimate (20) then becomes

$\displaystyle\sup_{s\in{\mathbb{Z}}}\|u_{s}^{\ast}-\varphi(s,s-t\theta,u_{0})\|\leq
2^{1-t}\left(\|u_{0}\|+(1-e^{-aL/2})\sup_{s\in{\mathbb{Z}}}\beta_{s-1}+\|h\|_{\infty}\right)$

for all $t\in{\mathbb{Z}}$.

By the Nyström method with the trapezoidal quadrature rule, the $t$-fibers
${\mathcal{A}}^{\ast}(t)$ are illustrated in Figure 1 for
$t\in[0,T^{\prime}]$, $L:=6$, $\theta:=365$ (a year), $tol:=10^{-6}$, $a:=10$,
$\hat{b}(x):=2|x|+3$ and

$\displaystyle u_{0}(x):=\begin{cases}2x^{2}+0.5,&x\in[-1,1],\\\
2.5,&\textrm{otherwise}.\end{cases}$

For $T^{\prime}$, in order to observe the periodicity of
${\mathcal{A}}^{\ast}$, we pick $T^{\prime}:=366$ (one year and one day) so
that the same behaviour of the system at days 0 and 365 can be displayed. For
$h_{t}(x)$, we consider four functions $h^{i}_{t}(x),i\in\\{1,2,3,4\\}$, in
the following table:

| $t\in\left(\left.0,\dfrac{\theta}{4}\right.\right]$ | $t\in\left(\left.\dfrac{\theta}{4},\dfrac{\theta}{2}\right.\right]$ | $t\in\left(\left.\dfrac{\theta}{2},\dfrac{3\theta}{4}\right.\right]$ | $t\in\left(\left.\dfrac{3\theta}{4},\theta\right.\right]$
---|---|---|---|---
$h_{t}^{1}(x)$ | $\cos\left(\dfrac{\pi x}{L}\right)$ | $\cos\left(\dfrac{\pi x}{L}\right)$ | $2\cos\left(\dfrac{\pi x}{L}\right)$ | $2\cos\left(\dfrac{\pi x}{L}\right)$
$h_{t}^{2}(x)$ | $\cos\left(\dfrac{\pi x}{L}\right)$ | $2\cos\left(\dfrac{\pi x}{L}\right)$ | $\cos\left(\dfrac{\pi x}{L}\right)$ | $2\cos\left(\dfrac{\pi x}{L}\right)$
$h_{t}^{3}(x)$ | $2\cos\left(\dfrac{\pi x}{L}\right)$ | $2\cos\left(\dfrac{\pi x}{L}\right)$ | $\cos\left(\dfrac{\pi x}{L}\right)$ | $\cos\left(\dfrac{\pi x}{L}\right)$
$h_{t}^{4}(x)$ | $2\cos\left(\dfrac{\pi x}{L}\right)$ | $\cos\left(\dfrac{\pi x}{L}\right)$ | $2\cos\left(\dfrac{\pi x}{L}\right)$ | $\cos\left(\dfrac{\pi x}{L}\right)$
Table 3.3: Inhomogeneities $h_{t}^{i}$, $i\in\\{1,2,3,4\\}$

In Table 3.3, all intervals $\left(\left.0,\tfrac{\theta}{4}\right.\right]$,
$\left(\left.\tfrac{\theta}{4},\tfrac{\theta}{2}\right.\right]$,
$\left(\left.\tfrac{\theta}{2},\tfrac{3\theta}{4}\right.\right]$ and
$\left(\left.\tfrac{3\theta}{4},\theta\right.\right]$ represent four seasons
spring, summer, fall and winter, respectively, and each $h_{t}^{i}$,
$i\in\\{1,2,3,4\\}$, represents different behaviour in each season. For
instance, the populations regarding $h_{t}^{1}$ in fall and winter are twice
those in spring and summer. Last, by the definitions of $u_{0}$, $b_{t}$ and
$h_{t}$, $S=8760$ due to (23).

Finally, in order to meet the optimal choice for the artificial support, we
compute the mean total populations
$\dfrac{1}{\theta}\sum_{t=0}^{\theta-1}\bar{u}_{t}$ over a year for
inhomogeneities $h^{i}_{t}(x),i\in\\{1,2,3,4\\}$, from Table 3.3 as follows.

| $h^{1}_{t}(x)$ | $h^{2}_{t}(x)$ | $h^{3}_{t}(x)$ | $h^{4}_{t}(x)$
---|---|---|---|---
$\dfrac{1}{\theta}\sum_{t=0}^{\theta-1}\bar{u}_{t}$ | $7.9640$ | $5.8614$ | $8.0794$ | $10.1816$
Table 3.4: Mean total populations
$\dfrac{1}{\theta}\sum_{t=0}^{\theta-1}\bar{u}_{t}$ over a year for
inhomogeneities $h^{i}_{t}(x),i\in\\{1,2,3,4\\}$

According to Table 3.4, we can conclude $h^{4}_{t}(x)$ is the best option
among four. To study this behaviour, one can see that if the populations
supported in spring and fall are larger than those in summer and winter to
encounter natural conditions, then species can develop to the highest amount
after some generations. To be more precise, in winter, the total population
witnesses a significant decline and hence requires a larger support in spring.
Meanwhile, despite reaching a peak in summer, the total population is
supported more in fall in order to avoid a dramatic decline due to natural
conditions.

Figure 1: The $t$-fibers ${\mathcal{A}}^{\ast}(t)$ for $t\in[0,365]$ and
$h_{t}(x)=h^{i}_{t}(x)$, $i\in\\{1,2,3,4\\}$ from the top ($i=1$) to the
bottom ($i=4$), respectively.

## Acknowledgments

We are deeply grateful to our supervisor for their support as well as helpful
comments and suggestions on the manuscript. We also thank a colleague of ours
for polishing the language in some parts.

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