2023

[1,2]Marcelo Archanjo JoseThese authors contributed equally to this work.

These authors contributed equally to this work.

[1]Institute of Advanced Studies, University of São Paulo, R. do Anfiteatro,
513, São Paulo, 05508-060, São Paulo, Brazil

2]Center for Artificial Intelligence, C4AI, Av. Prof. Lúcio Martins Rodrigues,
370, São Paulo, 05508-010, São Paulo, Brazil

3]Escola Politécnica, University of São Paulo, Av. Professor Luciano Gualberto
2231, São Paulo, 05508-010, São Paulo, Brazil

# A Multilingual Translator to SQL with Database Schema Pruning to Improve
Self-Attention

<EMAIL_ADDRESS>Fabio Gagliardi Cozman<EMAIL_ADDRESS>*  [  [

###### Abstract

Long sequences of text are challenging in the context of transformers, due to
quadratic memory increase in the self-attention mechanism. As this issue
directly affects the translation from natural language to SQL queries (as
techniques usually take as input a concatenated text with the question and the
database schema), we present techniques that allow long text sequences to be
handled by transformers with up to 512 input tokens. We propose a training
process with database schema pruning (removal of tables and columns names that
are useless for the query of interest). In addition, we used a multilingual
approach with the mT5-large model fine-tuned with a data augmented Spider
dataset in four languages simultaneously: English, Portuguese, Spanish and
French. Our proposed technique used the Spider dataset and increased the exact
set match accuracy results from 0.718 to 0.736 in a validation dataset (Dev).
Source code, evaluations, and checkpoints are available at:
https://github.com/C4AI/gap-text2sql.

###### keywords:

Semantic parsing, SQL generation, deep learning, neural network, natural
language process, text-to-SQL, databases, transformers self-attention,
transformers, Spider dataset

##  1 Introduction

Transformers with the attention mechanism have led to great leaps in natural
language processing (NLP) Attention2017 . However, they do have limitations.
An example is the 512 tokens input limit, as this can be a drawback when
dealing with long text sequences. The number of tokens is not really a
limitation as it can be increased; however, expanding this number increases
memory consumption quadratically and may disperse attention through many
tokens. Different proposals, such as Big Bird BigBird2020 , Longformer
Longformer2020 , Poolingformer Poolingformer2021 , ETC ETC2020 , Linformer
Linformer2020 , Reformer Reformer 2020 , among others, process long text
sequences and address the challenge of memory consumption by letting it grow
near linearly, while keeping good performance.

In this paper we explore techniques that enhance transformers in the context
of natural language to SQL (NL2SQL) translation. Existing NL2SQL parsers
encode the combined text composed of the question and database schema
information, especially the table names, column names, and their relations.
More information about NL2SQL can be found in these surveys: SurveyYu2010
SurveyKim2020  SurveyOzcan2020 .

NL2SQL parsers based on transformers have greatly evolved in the last few
years. The Spider dataset111Spider dataset: https://yale-
lily.github.io/spider. Yu2018b  has had a key role in that progress due to its
features, such as the number of databases, query complexity, etc. The current
leaderboard (measuring exact set match without values) is presented in Table
1.

Table 1: Spider Leaderboard - Exact Set Match without Values in September 2022

Rank | Model | Test | Dev | Reference
---|---|---|---|---
1 | Graphix-3B + PICARD | 0.740 | 0.771 | Anonymous
1 | CatSQL + GraPPa | 0.739 | 0.786 | Anonymous
3 | SHiP + PICARD | 0.731 | 0.772 | Anonymous
4 | G³R + LGESQL + ELECTRA | 0.726 | 0.772 | Anonymous
6 | RESDSQL+T5-1.1-lm100k-xl | 0.724 | 0.781 | Anonymous
6 | T5-SR | 0.724 | 0.772 | Anonymous
7 | S²SQL + ELECTRA | 0.721 | 0.764 | SSSQL2022
8 | LGESQL + ELECTRA | 0.720 | 0.751 | LGESQL2021
9 | T5-3B+PICARD | 0.719 | 0.755 | PICARD2021
|  |  |  |

* Techniques that currently do not have a paper associated are presented as anonymous.

Currently, the works in the first positions that have a paper explaining their
approach are:

\- S²SQL SSSQL2022  is a technique that injects syntactic information of the
question in the encoder, rather than just the question text. They also
introduce a decoupling constraint in order to induce diverse edge embedding
learning.

\- The idea behind LGESQL (Line Graph Enhanced Text-to-SQL) LGESQL2021  is to
use line graphs to include local (1-hop relation) and non-local (extracted
from parameter matrix) features to compute. The line graph relates question
nodes, table nodes, and column nodes. A graph pruning process helps indicate
the relevant graph schema to the related question. The pretrained language
model (PLM) ELECTRA-large-discriminator has achieved the best result.

\- The PICARD PICARD2021  (Parsing Incrementally for Constrained Auto-
Regressive Decoding) approach constrains the decoded tokens during the
inference process to find valid SQL queries through four levels in the parsing
process. The best result within this approach has been achieved with the T5-3b
model.

A technique that has been a reference for many other techniques with good
results in the Spider leaderboard is the RAT-SQL (Relation-Aware Transformer
SQL) RAT-SQL2019  that explored a database schema link with the natural
language questions words, with important results when launched. RAT-SQL+GAP
scheme RAT-SQL+GAP2020  (0.697 Test and 0.718 Dev) is a variant that is used
in this paper as a baseline. GAP means Generation-Augmented Pre-Training; it
employs a custom pre-training in the BART model with learning objectives
related to the NL2SQL task. Such training increases performance when this
model is plugged into the RAT-SQL parser.

When using transformers, the limitation over long input text sequences
strikes. The natural language question is not a problem as it is usually
short; however, the database schema may be large depending on the number of
tables and columns.

We here use the RAT-SQL+GAP when the model is BART-large (the pretrained model
version was downloaded from Github222RAT-SQL+GAP
github:https://github.com/awslabs/gap-text2sql. ) and our multilingual mRAT-
SQL version, but without GAP, when the model is mT5-large, which means the
model is the original form of Hugging Face333Google’s
mT5:https://huggingface.co/google/mt5-large..

The proposal for this paper is to present to the scientific community the
improvement obtained with schema pruning in a multilingual approach. The
motivation was the benefits of schema pruning because in NL2SQL with
transformers it is an open problem to handle databases with big schemas that
produce long input sequences that exceed 512 tokens. The multilingual approach
producing better results was a good side effect, that we notice due to our
previous work mRATSQLGAP2020  using a combination of English and Portuguese
languages, which was expanded with Spanish and French languages here. It is
important to report and present the results of these finds to allow other
researchers to analyze them as a possible choice to be applied in their
context.

The main contributions of this paper are the schema pruning to reduce the
number of tokens to fit in the 512, preserving the relevant tables and column
names used in the queries for the corresponding database, and a multilingual
data augmentation process with four languages: English, Portuguese, Spanish,
and French. Both contributions can be easily incorporated into other NL2SQL
approaches thus making them a viable path to increase benchmark results.

##  2 Multilingual Data Augmentation

Natural language processing now a days as great advances, but mainly in
English Language. Operate with different languages could be problematic due to
the limitation of language models pretrained in those languages. Multilingual
language models are a good option mRATSQLGAP2020  MultiSpider2021  MUCE2022 .

In previous work on NL2SQL in Portuguese mRATSQLGAP2020 , we have found that
it is better to work with multilingual models than with a model for a specific
language (mostly because SQL queries naturally contain many English words).
This was shown in our previous work mRATSQLGAP2020  with the multilingual
model mBART-50. Multilingual models allow training in English and Portuguese
separately, and also the two languages together. We produced better results
when training the model with multiple languages than with a single one even
working with the English language. It is possible to deduce this is an effect
of data augmentation because we double the dataset.

In the current work, we chose mT5 mT52021  multilingual because achieves
better results than mBART-50. Specifically, the mT5 large multilingual model
with 1.2 billion parameters pre-trained with 101 languages, including the
languages we are currently working on English, Portuguese, Spanish, and
French.

The Spider dataset consists of 3 files: train_spider.json (7,000 questions),
train_others (1,659 questions) (both train dataset), and dev.json (1,034
questions) (validation dataset).

We translated natural language questions from the Spider Dataset into
Portuguese, Spanish and French and created versions of the four languages,
with the same corresponding original query. We choose not to translate any
information about the database schema to make the results compatible and
comparable, which means we can make inferences with any of the four languages,
and the resultant query can be evaluated with the Spider test suite
SpiderTestSuite2021 444Spider test suite: https://github.com/taoyds/test-
suite-sql-eval.. In Table 2 the question “What are the maximum and minimum
budgets of the departments?” are presented in four languages; all are related
to the same query: “SELECT max(budget_in_billions), min(budget_in_billions)
FROM department”. The translations were made using the Google Translate
service.

We also created a dataset version that joins the four languages together. The
original Spider has 8659 train and 1034 validation examples. This quad dataset
has 34636 train and 4136 validation examples. Ours is a data augmentation
approach that works with multilingual models.

Table 2: Question sample in English, Portuguese, Spanish and French, related to the same query: “SELECT max(budget_in_billions) , min(budget_in_billions) FROM department” Language | Question
---|---
English | What are the maximum and minimum budgets of the departments?
Portuguese | Quais são os orçamentos máximo e mínimo dos departamentos?
Spanish | ¿Cuáles son los valores máximo y mínimo presupuesto de los departamentos?
French | Quels sont le budget maximum et minimum des départements?
|

##  3 Schema pruning

The transformer self-attention mechanism size limitation also applies to
NL2SQL.

The figure 1 presents a graphical representation of the problem. The figure 1a
shows the ideal situation where the junction of the question text and the
serialized text of the schema (tables names, columns names, and their
relations) fits under the 512 tokens. The figure 1b shows a real example case
where the text that represents the database schema overcomes the limit of 512
tokens. One possible solution for that situation was to expand the limit to
2048 tokens to fit all necessary text, the figure 1c illustrates this
solution.

The problem is not the natural language part (the question) but the database
schema. It is not usual for a question to have too many words (more than 512
tokens), but databases can have many tables and columns that lead to a schema
with more than 512 tokens (when serialized as text). However, one question
will typically not require information of the entire database schema to
generate the expected SQL query. Considering the training data set, even a
group of questions may not require the entire database schema. It is possible
to analyze all questions in the training dataset related to the same database
and see which tables and columns are used.

This idea allows pruning table and column names that are not used for that
pack of questions, reducing in that way the size of the database schema. With
this reduced version of this database schema, It is possible to fit the
natural language question and the database schema under 512 tokens, respecting
the self-attention mechanism limitation. The figure 1d presents the effect of
the schema pruning.

Figure 1: Situations and possibles solutions; a) Ideal situation; b) Example
that overcomes the 512 tokens; c) Possible solution, expand the token limit to
2048 tokens; d) Proposed solution, prune the database schema to fit in 512
token limit.

Currently, the Spider dataset is composed of 166 databases 146 for training
and 20 for validation. The schemas are organized in the tables.json file. RAT-
SQL-GAP has a pre-process step that prepares the information for training and
inference steps.

We noticed that the number of training questions in fact used during the RAT-
SQL-GAP pre-processing are always smaller than the number of questions in the
training dataset. This is due to RAT-SQL+GAP code dropping examples that have
more than 512 tokens.

The original English training dataset has 8659 examples, but just 8558 were
really used, using BartTokenizer, which will interfere with the number of
tokens. 101 examples were rejected due to the combination of the question and
the database schema (table names and column names) being greater than 512
tokens. The quad (English, Portuguese, Spanish, and French together) training
dataset has 34363 examples, but just 33927 were really used, using
MT5Tokenizer. 709 examples were rejected. The number of rejected examples did
not increase by four times, although the quad dataset was four times larger,
because different languages produce questions with a different number of words
and use a different tokenizer (MT5Tokenizer).

We analyzed the rejected examples and organized them by the database in Table
3.

Table 3: Number of examples dropped from the training dataset during pre-processing. Database | Original dataset | Quad Dataset
---|---|---
| BartTokenizer | MT5Tokenizer
Baseball_1 | 82 | 328
cre_Drama_Workshop_Groups | 19 | 328
Soccer_1 | 0 | 53
Total rejected | 101 | 709

The questions related to these three databases are all in the training dataset
file train_spider.json.

In order to understand the reasons why just these three databases are related
to the problem of exceeding the limit of 512 tokens, the number of original
tables and columns was analyzed, see Table 3. These databases have a large
number of tables and columns, for example, the Baseball_1 DB has 353 columns;
if each column name uses two unique words, the 512 tokens limit is exceeded
merely by the names of the columns.

We created a code to analyze all queries related to these databases and
present the tables that are really used, allowing the deletion of tables not
used. To validate the process of pruning, we did it manually using the DB
Browser for SQLite. First, we deleted tables not used in the queries indicated
by the code; later the not-used columns when the deletion of tables was not
enough. For column deletion, we did not develop a specific code, the deletion
was made based on the name of the column and on a visual inspection of the
queries that used the table related.

After the pruning, we updated the tables.json file with a new section that
reflects the modified database. The dataset file train_spider.json has indexes
related to the original tables and columns. We create a code to update these
indexes with the modified pruned version. Table 4 shows the new numbers of
tables and columns in the pruned version.

This approach was made to evaluate the effects of using the entire Spider
dataset (without drop examples); it can be applied to other training datasets,
despite the manual effort, because it is one-time work. Creating an algorithm
that makes the schema pruning automatically for each drop example candidate
(more than 512 tokens) is an option, but it is worth considering that the
schema pruning for each example drop candidate will create a new schema per
example, not per database. To make the pruning schema per database, it is
necessary to aggregate all the drop example candidates (more than 512 tokens)
and relate the database candidates to make the pruning considering all the
queries (with the tables and columns used).

Table 4: Tables and columns sizes for rejected databases before and after schema pruning. Database | Tables | Columns | Tables | Columns
---|---|---|---|---
| Original | Original | after pruning | after pruning
baseball_1 | 26 | 353 | 13 | 87
cre_Drama_Workshop_Groups | 18 | 100 | 15 | 80
soccer_1 | 7 | 87 | 5 | 57
|  |  |  |

##  4 Experiments and Analysis

###  4.1 Multilingual Data Augmentation

The experiments were performed on the following equipment: AMD Ryzen 9 3950X
16-Core Processor, 64GB RAM, 2 GPUs NVidia GeForce RTX 3090 24GB running
Ubuntu 20.04.2 LTS.

First, we reproduced the results of RAT-SQL+GAP RAT-SQL+GAP2020  in our
environment to use it as a baseline (fine tuning with 41000 steps and Batch
Size=12); since it is BART-large, the GAP is active (pre-trained model by the
RAT-SQL+GAP group), the train and validation datasets are in English. Figure
2a shows the ”Exact Set Match without Values” accuracy result 0.718, which is
the same as the RAT-SQL+GAP RAT-SQL+GAP2020  paper. This metric considers the
inferred query, but not the values. To validate the multilingual approach, we
fine-tuned the mT5 model just with the original English Spider train dataset,
and later with the quad (English, Portuguese, Spanish and French) Spider train
dataset. Both with 51,000 steps, Batch Size=4 (this value was chosen to fit
the model in our GPU memory). The validation dataset is in English for the
three cases. A diagram with the three combinations is presented in Figure 2 on
the left side. Figure 2b shows an inference result of 0.864 for the model
trained just in English and Figure 2c shows a result of 0.715 for the model
trained with the quad dataset. The three tests presented in Figure 2 were made
with the standard self-attention of 512 tokens.

Figure 2: Exact Set Match without Values, the diagram on the left, and the
results on the right; a) BART-large trained in English, infer in
English(baseline); b) mT5-large model trained in English, infer in English; c)
mT5-large model trained in English, Portuguese, Spanish and French, infer in
English.

It is possible to conclude that the multilingual model mT5 produces better
results when trained with more languages. The results in Figure 2b and c show
the increase from 0.684 to 0.715 for the same mT5-large model first trained in
English and after with the quad train dataset (English, Portuguese, Spanish
and French). This increase can be credited to a data augmentation effect that
was enough to make mRAT-SQL (without GAP) achieve the value of 0.715, near to
the BART-large baseline of 0.718 with RAT-SQL+GAP. This makes the training
process simpler because a pre-training with the model is not necessary before
the final NL2SQL training.

###  4.2 Schema pruning

To understand the influence of the schema pruning we fine-tune the mT5 model
with the same quad dataset (without pruning), hereinafter called ”standard
quad” (English, Portuguese, Spanish, and French) Spider train dataset and
after with the quad dataset with schema pruning, hereinafter called ”FIT quad”
(English, Portuguese, Spanish, and French) Spider train dataset. Both with
120,000 steps, Batch Size=4 and the standard self-attention of 512 tokens. We
increased the number of steps to analyze as the model will converge with more
steps than the 51,000 used in the prior tests, mainly because the training
with mt5-large and the quad dataset achieved the best checkpoint in the last
step (on an average rising slope). The validation dataset is in English for
both cases. Figure 3c shows the inference result of 0.718 for the model
trained with the standard quad dataset and Figure 3d shows results of 0.736
for the model trained with FIT quad database, whereby the schema was pruned.

The increase in fine-tuning steps indicates to be adequate; the best
checkpoints were 77,500 for the standard quad dataset and 105,100 for the FIT
quad train dataset.

Another possible approach to include all text sequences during the fine-tuning
process is to increase the max number of tokens in the transformer self-
attention mechanism. In our case, for using the whole standard quad train
dataset, it was necessary to increase from 512 to 2048 tokens. Due to the
memory consumption, we had to reduce the Batch Size to just 1, but it was
necessary to increase the number of steps to 480000 to get a good convergence
in the model training. Figure 3b shows the inference result of 0.697.

The use of the FIT quad Spider train dataset had a huge influence on the
results raising from 0.718 Figure 3c to 0.736 Figure 3d. It can be deduced
that the integral use of the training dataset, without the exclusions caused
by exceeding 512 tokens, provided the best training samples. The attempt to
increase the limit of 512 tokens to 2048 does not produce good results of
0.697 Figure 3b. In fact, it was worst than 0.718. Figure 3c achieved by the
mt5-large fine-tuned with the standard quad train dataset. The possible cause
is that the attention mechanism became too sparse. A diagram with the four
combinations is presented in Figure 3 on the left side.

Table 5: Difficult levels for the exact set match without values. | Max |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
Model | Token | Train | Eval | Easy | Medium | Hard | Extra | All
| Limit |  |  |  |  |  |  |
Bart-large | 512 | en | en | 0.899 | 0.744 | 0.667 | 0.428 | 0.718
mT5-large | 2048 | en-pt-es-fr | en | 0.855 | 0.753 | 0.540 | 0.476 | 0.697
mT5-large | 512 | en-pt-es-fr | en | 0.879 | 0.756 | 0.580 | 0.518 | 0.718
mT5-large | 512 | FIT-en-pt-es-fr | en | 0.895 | 0.776 | 0.603 | 0.530 | 0.736
|  |  |  |  |  |  |  |

Table 5 shows the question/query example difficulty level (easy, medium, hard,
and extra hard) for the exact set match without values for the four cases. The
improvement of the mT5 large fine-tuned with the FIT quad train dataset can be
noticed in all levels if compared with mT5 large fine-tuned with the standard
quad train dataset. The specific value of the mT5 large fine-tuned with the
FIT quad train dataset for extra hard examples: 0.530 is the best of all the
fine-tuning we performed, yet not reported here.

The schema pruning that produced the FIT datasets shows important results, but
it was just used in the training dataset because the validation dataset does
not have examples requiring more than 512 tokens. It is possible to apply the
same schema pruning approach to the validation dataset because we have the
query related to the question to select unused tables and columns. In future
real cases of NL2SQL where only the question and the database schema are
available, it will be difficult to perform the schema pruning in the inference
time. One option is to analyze the need of the complete schema in an inference
endpoint and create a short schema version compatible with the limit of 512
tokens to get good inferences.

Figure 3: Exact Set Match without Values, the diagram on the left, and the
results on the right; a) BART-large trained in English, inferred in English
(baseline); b) mT5-large model trained with the max number of token 2048 in
English, Portuguese, Spanish and French, inferred in the English dataset
standard; c) mT5-large model trained in English, Portuguese, Spanish and
French dataset standard, inferred in English; d) mT5-large model trained in
English, Portuguese, Spanish and French Dataset FIT, inferred in English.

###  4.3 Multilingual inference

The mT5-large fine-tuned with the quad dataset can infer questions in each of
the four languages trained. Figure 4 shows the results for the exact set match
without values, for inferences with the validation dataset in English (Figure
4b and 4c), translated into Portuguese (Figure 4d and 4e), Spanish (Figure 4f
and 4g) and French (Figure 4h and 4i). These results were produced with
mT5-large fine-tuned with the standard quad (English, Portuguese, Spanish and
French) Spider train dataset Figure 4b, 4d, 4f and 4h. The mT5-large fine-
tuned with FIT quad (English, Portuguese, Spanish and French) Spider train
dataset Figure 4c, 4e, 4g and 4i. For all the results, the checkpoint that
produced the best result for each language was selected.

Languages different from English have lower results, which can be attributed
to the pre-training of the model mT5 mT52021 . It used much more tokens in
English (2,733 B) than in Portuguese(146 B), Spanish(433 B), and French(318
B). Another aspect, English words in the question represent tables and columns
names with the same words; for other languages, the words in the question
represent tables and columns names in English, once we preserved the English
language in the Database Schema. The mT5 model fine-tuned with the FIT quad
training dataset shows better inference for all 4 languages if compared to the
mT5 model fine-tuned with the standard FIT quad training dataset. This
reinforces the importance of schema pruning.

Figure 4: Exact Set Match without Values for multilingual inferences:
a) BART-large trained in English dataset standard, inferred in English
(baseline);
b) mT5-large model trained in English, Portuguese, Spanish and French standard
quad dataset, inferred in English;
c) mT5-large model trained in English, Portuguese, Spanish and French FIT quad
dataset, inferred in English;
d) mT5-large model trained in English, Portuguese, Spanish and French standard
quad dataset, inferred in Portuguese;
e) mT5-large model trained in English, Portuguese, Spanish and French FIT quad
dataset, inferred in Portuguese;
f) mT5-large model trained in English, Portuguese, Spanish and French standard
quad dataset, inferred in Spanish;
g) mT5-large model trained in English, Portuguese, Spanish and French FIT quad
dataset, inferred in Spanish;
h) mT5-large model trained in English, Portuguese, Spanish and French standard
quad dataset, inferred in French;
i) mT5-large model trained in English, Portuguese, Spanish and French FIT quad
dataset, inferred in French..

##  5 Conclusion and Future Work

This work introduced new ideas to NL2SQL, particularly for multilingual
settings. For exact set match accuracy, proposed techniques increased standard
metrics from 0.718 to 0.736 with the Dev validation dataset. Note that the
original Spider is not used entirely, because mRAT-SQL, which borrows the same
code from RAT-SQL+GAP, drops examples that exceed 512 tokens. To prove the
hypotheses that pruning the schema tables and columns names will help the
training processes with more examples, we manually performed this pruning and
created a FIT version of the Spider dataset that does not have any examples
excluded, this allowed the self-attention transformer mechanism to treat the
entire training dataset. This Spider FIT dataset version can easily plug in
other techniques that use the Spider dataset. The next step is to plug the
Spider FIT dataset in another technique to evaluate the results.

## Abbreviations

DEV Validation dataset
ETC Extended transformer construction
GAP Generation-augmented pre-training
mT5 Multilingual text-to-text transfer transformer model
NLP Natural language processing
NL2SQL Natural language to SQL
LGESQL Line graph enhanced text-to-SQL
PICARD Parsing incrementally for constrained auto-regressive decoding
RAT-SQL Relation-aware transformer SQL
SQL Structured query language
S²SQL Syntax to question-schema graph encoder for text-to-SQL
T5 Text-to-text transfer transformer model

## Declarations

  * •

Funding This work was carried out at the Center for Artificial Intelligence
(C4AI-USP), with support by the Sao Paulo Research Foundation (FAPESP grant
#2019/07665-4) and by the IBM Corporation. The second author is partially
supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnologico
(CNPq), grant 312180/2018-7.

  * •

Conflict of interest/Competing interests The authors have no relevant
financial or non-financial interests to disclose.

  * •

Ethics approval and consent to participate Not applicable.

  * •

Consent for publication Not applicable.

  * •

Availability of data and materials https://github.com/C4AI/gap-text2sql

  * •

Code availability https://github.com/C4AI/gap-text2sql

  * •

Authors’ contributions The two authors had an equivalent contribution to the
paper write.

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# Extinction, persistence and growing in a degenerate logistic model with
impulses ††thanks: The second author is supported by the National Natural
Science Foundation of China (No. 12271470) and Carlos Alberto Santos
acknowledges the support of CNPq/Brazil Proc. $N^{o}$ $311562/2020-5$ and
FAPDF grant 00193.00001133/2021-80.

Willian Cintra1, Zhigui Lin2,†, Carlos Alberto Santos1 and Phyu Phyu Win2
1 Department of Mathematics, University of Brasilia, BR-70910900 Brasilia, DF,
Brazil
2 School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

> Abstract. This paper deals with an impulsive degenerate logistic model,
> where pulses are introduced for modeling interventions or disturbances, and
> degenerate logistic term may describe refugees or protections zones for the
> species. Firstly, the principal eigenvalue depending on impulse rate, which
> is regarded as a threshold value, is introduced and characterized. Secondly,
> the asymptotic behavior of the population is fully investigated and
> sufficient conditions for the species to be extinct, persist or grow
> unlimitedly are given. Our results extend those of well-understood logistic
> and Malthusian models. Finally, numerical simulations emphanzise our
> theoretical results highlighting that medium impulse rate is more favorable
> for species to persist, small rate results in extinction and large rate
> leads the species to an unlimited growth.
>
> MSC: primary: 35K57; 35R35; secondary: 92D25.
>
> Keywords: degenerate logistic model, impulsive problem, threshold value,
> unlimited growth, persistence or extinction

##  1 Introduction

This paper characterizes the dynamics of the impulsive and diffusive logistic-
model possibly in the presence of a refugee or protection zones

$\displaystyle\left\\{\begin{aligned} &u_{t}=d\Delta
u+a(t,x)u-b(t,x)u^{p},&&t\in((n\tau)^{+},(n+1)\tau],\,\,x\in\Omega,\\\
&u(t,x)=0,&&t>0,\,\,x\in\partial\Omega,\\\
&u(0,x)=u_{0}(x),&&x\in\overline{\Omega},\\\
&u((n\tau)^{+},x)=cu(n\tau,x),&&x\in\overline{\Omega},\
n=0,1,2,\cdots,\end{aligned}\right.$ (1.1)

where $\Omega$ is a connected bounded domain of $\mathbb{R}^{N}$ ($N\geq 1$)
with smooth boundary $\partial\Omega$, $p>1$ and $\tau>0$ are constants,
$t\in((n\tau)^{+},(n+1)\tau]$ expresses that the equation holds for
$t\in((n\tau),(n+1)\tau]$, and the unknown $u$ takes its value
$u((n\tau)^{+},x)$ instead of $u((n\tau),x)$ at the initial time of the time
interval $((n\tau),(n+1)\tau]$ for $n=0,1,2,\cdots$.

All meanings of variables and parameters in model (1.1) are given below:

$\bullet$ $u(t,x)$ expresses the density of species at time $t\geq 0$ and in
space $x\in\Omega$;

$\bullet$ $d>0$ is the diffusion rate;

$\bullet$ $a(t,x)(\in C^{\theta/2,\theta}([0,\tau]\times\overline{\Omega})$
for some $0<\theta\leq 1$) is a $\tau-$ periodic function, and denotes the
intrinsic growth rate of the population. It can be negative, which means that
the resources on position $x$ at time $t$ are not conducive to survival;

$\bullet$ $b(t,x)(\in C^{\theta/2,\theta}([0,\tau]\times\overline{\Omega}))$
is a nonnegative and $\tau$-periodic function that can vanish on a smooth
subdomain $\Omega_{0}$ of $\Omega$ (under additional assumptions on $b$, the
subset $b^{-1}(0)$ may behave as periodic-parabolic protection or refugee
zone, see below);

$\bullet$ $cu$ is the impulsive function, and an impulse occurs at every time
$t=n\tau$ $(n=0,1,2,\cdots)$ within the successive stage of growth and
disperse. Usually, the linear impulse function is used to represent birth
pulse if $c>1$ and harvesting phenomenon if $0<c<1$.

On the condition $c=1$, which means there is no impulse, problem (1.1) has
attracted much attention recently, see [1, 3, 6, 11, 16] and references
therein. Especially, a degenerate diffusive logistic equation was analyzed in
[9] from a numerical point of view and revealed that the number of nodal
solutions and the number of components in the bifurcation diagrams strongly
depends on the number and position of the components where the weight function
in front of the nonlinearity vanishes.

Recently, the single species model has also been extended to two species
model, see for example [8], that analyzed a spatially heterogeneous diffusive
predator-prey model constructing an $S$-shaped component of coexistence
states, which causes the existence of, at least, three coexistence states.

Corresponding elliptic problems to (1.1), without impulse and its variants,
were well studied, see [4, 5] and references therein, while corresponding
periodic-parabolic problems has also attracted much attention, see for
instance [7, 16] and references therein. Recently, it was proved in [10] the
existence of positive periodic solutions for a periodic-parabolic problem
revealing that the linear part of reaction term and the nonlinear part make
quite different effects on the dynamical behavior of positive periodic
solutions.

Motivated by the above results, besides growth, death, and disperse [20], we
are more interested in the distribution and dynamics of species influenced by
impulsive perturbation [2, 18, 21, 22] on the presence of a degenerate
diffusive logistic term.

This paper is organized as follows. Section 2 contains some necessary
theoretical results about the principal eigenvalue for a periodic eigenvalue
problem with impulse. In Section 3, we establish a global existence result in
the parameter $c>0$ for an impulsive periodic problem associated to problem
(1.1). Sections 4 is devoted to studying dynamics of the unique solution of
problem (1.1). Finally, numerical simulations and a ecological explanations
are given in Section 5.

##  2 An eigenvalue problem with impulse condition

Our main result about the distribution and the dynamic of the population
(Theorem 4.1 below) strongly depends on the behavior of the principal
eigenvalue of the periodic $\mu$-eigenvalue problem

$\displaystyle\left\\{\begin{aligned}
&\phi_{t}-d\Delta\phi-a(t,x)\phi+yb(t,x)\phi=\mu\phi,&&t\in(0^{+},\tau],\,\,x\in\Omega,\\\
&\phi(t,x)=0,&&t\in[0,\tau],\,\,x\in\partial\Omega,\\\
&\phi(0,x)=\phi(\tau,x),&&x\in\overline{\Omega},\\\
&\phi(0^{+},x)=z\phi(0,x),&&x\in\overline{\Omega},\end{aligned}\right.$ (2.1)

for any $z>0$ given.

The existence of the principal eigenvalue

$\mu_{1}=\mu_{1}(d,a(t,x),yb(t,x),z),$

for any $z>0$ and $y\geq 0$ given, and fixed $d,a(t,x)$ and $b(t,x)$, can be
guaranteed by using Poincaré map of a linear operator [14, 15] or by using
Krein-Rutman theorem [11, 12] on a Banach space involving impulse [19]. For
the convenience of the reader, we will give a sketch of the proof below.

To overcome the difficulties introduced by the impulse condition, we consider
the following equivalent eigenvalue problem

$\displaystyle\left\\{\begin{array}[]{lll}\xi_{t}-d\Delta\xi-a(t,x)\xi+yb(t,x)\xi=\mu\xi,&t\in(0,\tau],\,\,x\in\Omega,\\\\[2.84526pt]
\xi(t,x)=0,&t\in[0,\tau],\,\,x\in\partial\Omega,\\\\[2.84526pt]
\xi(0,x)=z\xi(\tau,x),&x\in\overline{\Omega},\end{array}\right.$ (2.5)

under the understanding that $\phi(t,x)=\xi(t,x)$ for
$t\in(0,\tau],x\in\overline{\Omega}$, $\phi(0^{+},x)=\xi(0,x)$ for
$x\in\overline{\Omega}$, and $\phi(0,x)=\xi(\tau,x)$ for
$x\in\overline{\Omega}$.

Now, let $W$ be a Banach space

$\displaystyle\begin{array}[]{rll}W=&{D}^{0,1}_{0}([0,\tau]\times\overline{\Omega}):=\\{\xi(t,x)\in{C}^{0,1}([0,\tau]\times\overline{\Omega}):\
\xi(t,x)=0\ \forall(t,x)\in[0,\tau]\times\partial\Omega,\\\
&\xi(0,x)=c\xi(\tau,x),\ \forall x\in\overline{\Omega}\\}\end{array}$

with its positive cone

${W^{+}}:=\textrm{closure}\\{\xi\in W:\,\xi(t,x)\gg 0\
\forall(t,x)\in[0,\tau]\times\partial\Omega\\},$

and its interior

$\textrm{Int}(W^{+})=\\{\xi\in W:\ \xi(t,y)\gg 0\
\forall(t,x)\in[0,\tau]\times\partial\Omega\\}$

being nonempty, where $\nu$ is the outward unit normal vector on
$\partial\Omega$, and $\xi\gg 0$ means that $\xi(t,x)>0$ for all
$(t,x)\in[0,\tau]\times\Omega$ and $\frac{\partial\xi}{\partial\nu}(t,x)<0$
for all $(t,x)\in[0,\tau]\times\partial\Omega$, normally called as a strongly-
positive function in $[0,\tau]\times\Omega$.

Let
$M^{*}=1+\max_{[0,\tau]\times\overline{\Omega}}|a(t,x)|+\ln(\max\\{1/z,1\\})$.
For any $\xi(t,x)\in W$ given, we first conclude that the linear problem

$\displaystyle\left\\{\begin{array}[]{lll}\phi_{t}-d\Delta\phi=(-M^{*}+a(t,x)-yb(t,x))\phi+\xi,&t\in(0,\tau],\,\,x\in\Omega,\\\\[2.84526pt]
\phi(t,x)=0,&t\in[0,\tau],\,\,x\in\partial\Omega,\\\\[2.84526pt]
\phi(0,x)=z\phi(\tau,x),&x\in\overline{\Omega}\end{array}\right.$ (2.10)

admits a unique solution
$\phi\in{C}^{(1+\alpha)/2,1+\alpha}([0,\tau]\times\overline{\Omega})\bigcap W$
for some $0<\alpha<1$.

Next, define $\mathcal{A}\xi=\phi$. Owing to the embedding
${C}^{(1+\alpha)/2,1+\alpha}\hookrightarrow{C}^{0,1}$ be compact,
$\mathcal{A}$ is a linear compact operator. Moreover, $\mathcal{A}$ is
strongly positive with respect to $W$ according to the strong maximum
principle and Hopf’s boundary lemma. Thus, by the use of Krein-Rutman theorem,
we obtain that there exist a unique $\sigma_{1}:=r(\mathcal{A})>0$ and a
function $\phi\in\textrm{Int}(W^{+})$ such that
$\mathcal{A}\phi=\sigma_{1}\phi$, and therefore

$\mu_{1}:=1/\sigma_{1}-M^{*}$

is the principal eigenvalue of (2.5) and its corresponding eigenfunction
$\phi$ satisfies $\phi(t,x)>0$ in $[0,\tau]\times\Omega$ and
$\frac{\partial\xi}{\partial\nu}(t,x)<0$ for
$(t,x)\in[0,\tau]\times\partial\Omega$, that is, $\phi\gg 0$.

Coming back to the problem (2.1) with impulse, we now have the existence of
the principal eigenvalue $\mu_{1}$, whose monotonicity with respect to $d$ and
$y$ follows from similar arguments as done in [1, 16].

###### Lemma 2.1

Problem (2.1) admits a principal eigenvalue
$\mu_{1}:=\mu_{1}(d,a(t,x),yb(t,x),z)$ with its corresponding eigenfunction
$\phi(t,x)\gg 0$ in $[0,\tau]\times\Omega$. Moreover,
$\mu_{1}(d,a(t,x),yb(t,x),z)$ is increasing in $d>0$ and $y\geq 0$.

By the periodicity of the terms in problem (2.1), let us use indistinctly
$\mu_{1}$ and $\phi$ as the principal eigenvalue and the corresponding
eigenfunction, respectively, of the problem

$\displaystyle\left\\{\begin{aligned}
&\phi_{t}-d\Delta\phi-a(t,x)\phi+yb(t,x)\phi=\mu\phi,&&t\in((n\tau)^{+},(n+1)\tau],\,\,x\in\Omega,\\\
&\phi(t,x)=0,&&t\in[0,\tau],\,\,x\in\partial\Omega,\\\
&\phi((n\tau),x)=\phi((n+1)\tau,x),&&x\in\overline{\Omega},\
n=0,1,2,\cdots,\\\
&\phi((n\tau)^{+},x)=z\phi(n\tau,x),&&x\in\overline{\Omega},\
n=0,1,2,\cdots,\end{aligned}\right.$ (2.11)

for any $\tau,z>0$ given.

Besides the eigenvalue problem (2.1) (or (2.11)), the next one will play an
important tool in many of our proofs. Let the periodic $\lambda$-eigenvalue
problem

$\displaystyle\left\\{\begin{aligned}
&-\psi_{t}-d\Delta\psi-a(t,x)\psi+yb(t,x)\psi=\lambda\psi,&&t\in(0^{+},\tau],\,\,x\in\Omega,\\\
&\psi(t,x)=0,&&t\in[0,\tau],\,\,x\in\partial\Omega,\\\
&\psi(0,x)=\psi(\tau,x),&&x\in\overline{\Omega},\\\
&\psi(0^{+},x)=\frac{1}{z}\psi(0,x)),&&x\in\overline{\Omega}.\end{aligned}\right.$
(2.12)

So, we have the below result.

###### Lemma 2.2

Problem (2.12) admits a principal eigenvalue
$\lambda_{1}:=\lambda_{1}(d,a(t,x),yb(t,x),z)$. Its corresponding
eigenfunction is $\psi$ satisfying $\psi(t,x)\gg 0$ in $[0,\tau]\times\Omega$.
Moreover, $\lambda_{1}=\mu_{1}$, where $\mu_{1}$ is the principal eigenvalue
of problem (2.1).

Proof: The existence of the principal eigenvalue
$\lambda_{1}:=\lambda_{1}(d,a(t,x),yb(t,x),z)$ follows the same arguments as
done in the proof of Lemma 2.1 to prove the existence of $\mu_{1}$. Below let
us prove that $\lambda_{1}=\mu_{1}$. Let $(\mu_{1},\phi)$ and
$(\lambda_{1},\psi)$ be the pairs of principal eigenvalues and eigenfunctions
of the eigenvalues problems (2.1) and (2.12), respectively.

So, by multiplying the first equation of problem $\eqref{a013}$ by $\psi$, we
obtain

$\psi\phi_{t}-d\psi\Delta\phi-a(t,x)\phi\psi+yb(t,x)\phi\psi=\mu_{1}\phi\psi,$

while the product of $\phi$ by the first equation of the problem $\eqref{e28}$
leads us to the equation

$-\phi\psi_{t}-d\phi\Delta\psi-a(t,x)\phi\psi+yb(t,x)\psi\phi=\lambda_{1}\phi\psi.$

Now, by subtracting the last equation from the former, one yields

$(\psi\phi)_{t}-d(\psi\Delta\phi-\phi\Delta\psi)=(\mu_{1}-\lambda_{1})\psi\phi,$

whence follows by integration in $x\in\Omega$, $t\in(0^{+},\tau]$, and the
other conditions in problems (2.1) and (2.12), that

$(\mu_{1}-\lambda_{1})\int^{\tau}_{0^{+}}\int_{\Omega}\phi\psi=0$

leading to $\mu_{1}=\lambda_{1}$. This ends the proof.  $\Box$

Due to the importance that the variable $y\geq 0$ and $z>0$ will play on our
arguments, let us denote by

$\Sigma(y,z)=\mu_{1}(d,a(t,x),yb(t,x),z)$

the principal eigenvalue of problem (2.1) for fixed $d,a(t,x)$ and $b(t,x)$,
and by

$\Sigma(\infty,z)=\lim_{y\to\infty}\Sigma(y,z)\leq\infty,$

for any $z>0$ fixed. The above limit is well defined due to the fact that
$\Sigma(y,z)$ be increasing with respect to $y\geq 0$ for any $z>0$ fixed. The
following results brings up more properties on $\Sigma(y,z)$.

###### Theorem 2.3

Let $z>0$.

  1. $(i)$

if $a(t,x)=a(t)$ for $t\in[0,\tau]$ and $x\in\overline{\Omega}$, then

$\Sigma(0,z)=d\lambda_{0}-\frac{1}{\tau}\int^{\tau}_{0}a(s)ds-\frac{\ln
z}{\tau},$

where $\lambda_{0}>0$ is the principal eigenvalue of $-\Delta$ in $\Omega$
under Dirichlet boundary condition;

  2. $(ii)$

if $b(t,x)>0$ in $[0,\tau]\times\overline{\Omega}$, then

$\Sigma(\infty,z):=\lim_{y\to\infty}\Sigma(y,z)=+\infty,\ \mbox{for any }z>0\
\mbox{fixed};$

  3. $(iii)$

if $b(t,x)\equiv 0$ in $[0,\tau]\times\overline{\Omega}$, then

$\mu_{1}(d,a_{M}(t),yb(t,x),z)\leq\Sigma(0,z)=\Sigma(\infty,z)\leq\mu_{1}(d,a_{m}(t),yb(t,x),z),$

where

$a_{m}(t)=\min_{x\in\overline{\Omega}}a(t,x))\ \textrm{and
}\,a_{M}(t)=\max_{x\in\overline{\Omega}}a(t,x);$

  4. $(iv)$

if $b(t,x)\not\equiv 0$ in $[0,\tau]\times\overline{\Omega}$, then
$\Sigma(y,z)$ is a continuous and strictly increasing function of $y\geq 0$
for each $z>0$;

  5. $(v)$

Assume that $b(t,x)>0$ on $[0,\tau]\times\partial\Omega$. Then, for each
$z>0$, $\Sigma(\infty,z)<+\infty$ if, and only if, there exists a continuous
map $\kappa:\,[0,\tau]\to\Omega$ such that
$(t,\kappa(t))\in\,\textrm{int}\,b^{-1}(0)$ for all $t\in[0,\tau]$.

Proof: Item $(i)$. Under the assumption $a(t,x)=a(t)$ for $t\in[0,\tau]$ and
$x\in\overline{\Omega}$, the eigenvalue problem (2.1), for $y=0$, becomes

$\displaystyle\left\\{\begin{aligned}
&\phi_{t}-d\Delta\phi-a(t)\phi=\Sigma(0,z)\phi,&&t\in(0^{+},\tau],\,\,x\in\Omega,\\\
&\phi(t,x)=0,&&0<t\leq\tau,\,\,x\in\partial\Omega,\\\
&\phi(0,x)=\phi(\tau,x),&&x\in\overline{\Omega},\\\
&\phi(0^{+},x)=z\phi(0,x),&&x\in\overline{\Omega}.\end{aligned}\right.$ (2.13)

By taking

$\displaystyle f(t)=\left\\{\begin{array}[]{lll}1,&\,t=0,\\\\[2.84526pt]
z,&\,t=0^{+},\\\\[2.84526pt]
ze^{(-d\lambda_{0}+\frac{1}{\tau}\int^{\tau}_{0}a(s)ds+\Sigma(0,z))t},&\,t\in(0^{+},\tau]\end{array}\right.$
(2.17)

and defining $\phi(t,x)=f(t)\psi(x)$, it is straightforward to show that the
principal eigenvalue $\Sigma(0,z)$ is explicitly expressed as

$\ \Sigma(0,z)=d\lambda_{0}-\frac{1}{\tau}\int^{\tau}_{0}a(s)ds-\frac{\ln
z}{\tau},$

where $\lambda_{0}>0$ is the principal eigenvalue of $-\Delta$ in $\Omega$
under Dirichlet boundary condition and $\psi(x)>0$ in $\Omega$ is its
corresponding eigenfunction.

The item $(ii)$ is a consequence of [1, Theorem 6.2 (b)] due to the fact that

$\int^{\tau}_{0}\min_{x\in\overline{\Omega}}b(s,x)ds>0.$

About the item $(iii)$. Since we are assuming that $b(t,x)\equiv 0$ in
$[0,\tau]\times\overline{\Omega}$, problem (2.1) becomes

$\displaystyle\left\\{\begin{aligned}
&\phi_{t}-d\Delta\phi-a(t,x)\phi=\Sigma(y,z)\phi,&&t\in(0^{+},\tau],\,\,x\in\Omega,\\\
&\phi(t,x)=0,&&t\in[0,\tau],\,\,x\in\partial\Omega,\\\
&\phi(0,x)=\phi(\tau,x),&&x\in\overline{\Omega},\\\
&\phi(0^{+},x)=z\phi(0,x),&&x\in\overline{\Omega}\end{aligned}\right.$ (2.18)

so that

$\Sigma(0,z)=\Sigma(\infty,z)=\mu_{1}(d,a(t,x),0,z)\leq\mu_{1}(d,a_{m}(t),yb(t,x),z),$

for each $z>0$, due to the fact that $\mu_{1}$ be increasing with respect to
$a(t,x)$, and the item $(i)$ just proved. Similarly, we prove the estimate
from below.

About the item $(iv)$. Under the assumption $b(t,x)\not\equiv 0$ in
$[0,\tau]\times\overline{\Omega}$, we can take advantage of the ideas of the
proofs of [1, Corollary 2.3 and Lemma 2.2] to show the $\Sigma(y,z)$ is
continuous and the strict increasing with respect to $y\geq 0$.

Finally, by considering the equivalent eigenvalue problem (2.5), instead of
(2.1), the proof of $(v)$ follows, with minors adjustments, from the proof of
the main result in [16], after using a tricky of a novel generalized version
of the parabolic maximum principle on general domains. This ends the proof.
$\Box$

###### Theorem 2.4

For any $y\geq 0$ and $z_{1},z_{2}>0$, we have

$\Sigma(y,z_{1})-\Sigma(y,z_{2})=\frac{\ln(z_{2}/z_{1})}{\tau}.$ (2.19)

In particular, for any $y\geq 0$, one has:

  1. $(i)$

$\Sigma(y,z)$ is continuous and strictly decreasing with respect to $z>0$,

  2. $(ii)$

$\lim_{z\to 0^{+}}\Sigma(y,z)=+\infty$ and
$\lim_{z\to\infty}\Sigma(y,z)=-\infty$,

Proof: First, we point out that the proof of the above items $(i)$ and $(ii)$
are an immediate consequence of the equality (2.19). To prove (2.19), let
$(\Sigma(y,z_{1}),\phi)$ be the eigen-pair of the eigenvalue problem (2.1),
that is,

$\displaystyle\left\\{\begin{aligned}
&\phi_{t}-d\Delta\phi-a(t,x)\phi+yb(t,x)\phi=\Sigma(y,z_{1})\phi,&&t\in(0^{+},\tau],\,\,x\in\Omega,\\\
&\phi(t,x)=0,&&0<t\leq\tau,\,\,x\in\partial\Omega,\\\
&\phi(0,x)=\phi(\tau,x),&&x\in\overline{\Omega},\\\
&\phi(0^{+},x)=z_{1}\phi(0,x),&&x\in\overline{\Omega}.\end{aligned}\right.$

By taking

$\psi(t,x)=e^{\frac{\ln(z_{1}/z_{2})}{\tau}t}\phi(t,x),\ x\in\Omega,\
t\in(0^{+},\tau],\ \mbox{and }\psi(0,x)=\psi(\tau,x)=z_{1}/z_{2}\phi(\tau,x),$

we obtain that $\psi$ satisfies

$\left\\{\begin{aligned}
&\psi_{t}-d\Delta\psi-a(t,x)\psi+yb(t,x)\psi=\left(\Sigma(y,z_{1})+\frac{\ln(z_{1}/z_{2})}{\tau}\right)\psi,&&t\in(0^{+},\tau],\,\,x\in\Omega,\\\
&\psi(t,x)=0,&&0<t\leq\tau,\,\,x\in\partial\Omega,\\\
&\psi(0,x)=\psi(\tau,x),&&x\in\overline{\Omega},\\\
&\psi(0^{+},x)=\phi(0^{+},x)=z_{1}\phi(0,x)=z_{1}\phi(\tau,x)=z_{2}\psi(0,x),&&x\in\overline{\Omega},\end{aligned}\right.$

whence follows by the uniqueness of the principal eigenvalue of the above
eigenvalue problem that the equality (2.19) holds. This ends the proof.
$\Box$

About $\Sigma(\infty,z)$, for $z>0$, we obtain form the above theorem the next
result.

###### Theorem 2.5

Let $z_{0}>0$ given. Then one has:

  1. $i)$

either $\Sigma(\infty,z_{0})=\infty$ if, and only if,
$\Sigma(\infty,z)=\infty$ for all $z>0$,

  2. $ii)$

or $\Sigma(\infty,z_{0})<\infty$ if, and only if, $\Sigma(\infty,z)<\infty$
for all $z>0$. In such case, we have:

    1. $(ii_{1})$

$\Sigma(\infty,z)$ is continuous and strictly decreasing with respect to
$z>0$,

    2. $(ii_{2})$

$\lim_{z\to 0^{+}}\Sigma(\infty,z)=+\infty$ and
$\lim_{z\to\infty}\Sigma(\infty,z)=-\infty$.

Proof: It is immediate from the equality (2.19) by passing to the limit as
$y\to\infty$.  $\Box$

###### Remark 2.1

After Theorem 2.3-$(v)$ and Theorem 2.5, we deduce that the subset $b^{-1}(0)$
will work as a periodic-parabolic protection zone for the species if, and only
if, $\Sigma(\infty,c_{0})<\infty$ for some $c_{0}>0$, that is, that size of
the impulse $c>0$ does not change the status of the protection zone if it
occurs for some $c_{0}>0$.

##  3 About a impulsive periodic problem associated to Problem (1.1)

It is well known from [17] that problem (1.1) admits a unique global classical
solution $u(t,x)\in
C^{1+\theta/2,2+\theta}((n\tau,(n+1)\tau]\times\Omega)\bigcup
C((n\tau,(n+1)\tau])\times\overline{\Omega})$ $(n=0,1,2,\cdots)$ for any
nonnegative and continuous initial value $u_{0}$ that vanishes on the boundary
$\partial\Omega$, and, as in [3], the dynamic of problem (1.1) is related to
its corresponding periodic problem

$\displaystyle\left\\{\begin{aligned} &U_{t}=d\Delta
U+a(t,x)U-b(t,x)U^{p},&&t\in(0^{+},\tau],\,\,x\in\Omega,\\\
&U(t,x)=0,&&0<t\leq\tau,\,\,x\in\partial\Omega,\\\
&U(0,x)=U(\tau,x),&&x\in\overline{\Omega},\\\
&U(0^{+},x)=cU(0,x),&&x\in\overline{\Omega}.\end{aligned}\right.$ (3.1)

So, we establish the next theorem.

###### Theorem 3.1

One has.

  1. $(i)$

If either $b(t,x)\not\equiv 0$ in $[0,\tau]\times\overline{\Omega}$ and
$\Sigma(0,c)\geq 0$ or $b(t,x)\equiv 0$ in $[0,\tau]\times\overline{\Omega}$
and $\Sigma(0,c)\neq 0$, then problem (3.1) has no positive solution.

  2. $(ii)$

If either $b(t,x)\not\equiv 0$ in $[0,\tau]\times\overline{\Omega}$ and
$\Sigma(\infty,c)\leq 0$ or $b(t,x)\equiv 0$ in
$[0,\tau]\times\overline{\Omega}$ and $\Sigma(\infty,c)\neq 0$, then problem
(3.1) has no positive solution.

  3. $(iii)$

If $\Sigma(0,c)<0<\Sigma(\infty,c)$ $(b(t,x)\not\equiv 0$ in this case$)$,
then the periodic problem (3.1) admits a unique positive periodic solution,
say $U(t,x)$.

Proof: For the item $(i)$. It follows from Lemma 2.2 that $\Sigma(0,c)$ is the
principal eigenvalue of problem (2.12) with $y=0$, that is,

$\displaystyle\left\\{\begin{aligned}
&-\psi_{t}-d\Delta\psi-a(t,x)\psi=\Sigma(0,c)\psi,&&t\in(0^{+},\tau],\,\,x\in\Omega,\\\
&\psi(t,x)=0,&&0<t\leq\tau,\,\,x\in\partial\Omega,\\\
&\psi(0,x)=\psi(\tau,x),&&x\in\overline{\Omega},\\\
&\psi(0^{+},x)=\frac{1}{c}\psi(0,x)),&&x\in\overline{\Omega},\end{aligned}\right.$
(3.2)

where $\psi\gg 0$ denotes the eingenfunction associate to $\Sigma(0,c)$.

Assume, by contradiction, that problem (3.1) admited a positive solution
$U(t,x)$. Following the same strategy as done in the proof of Lemma 2.2, we
would obtain

$(U\psi)_{t}=d(\Delta U\psi-\Delta\psi U)-\Sigma(0,c)U\psi-b(t,x)U^{p}\psi,$

that leads, after integrating in $x\in\Omega$ and $t$ from $0^{+}$ to $\tau$
together with the use of the other conditions in (3.1) and (3.2), to

$\displaystyle-$ $\displaystyle\Sigma(0,c)\int^{\tau}_{0}\int_{\Omega}U\psi
dxdt-\int^{\tau}_{0}\int_{\Omega}b(t,x)U^{p}\psi dxdt$ $\displaystyle=$
$\displaystyle\int_{\Omega}[U(\tau,x)\psi(\tau,x)-U(0^{+},x)\psi(0^{+},x)]dx$
$\displaystyle=$
$\displaystyle\int_{\Omega}[U(0,x)\psi(0,x)-cU(0,x)\frac{1}{c}\psi(0,x)]dx$
$\displaystyle=$ $\displaystyle\ 0.$

So, we must have $\Sigma(0,c)<0$ when $b(t,x)\not\equiv 0$ in
$[0,\tau]\times\overline{\Omega}$, and $\Sigma(0,c)=0$ when $b(t,x)\equiv 0$
in $[0,\tau]\times\overline{\Omega}$. Both situations contradict our
assumptions.

Proof of item $(ii)$. Let us proceed by contradiction again. Suppose that
problem (3.1) admits a positive solution $U(t,x)$. Then $U(t,x)$ is bounded
from above by some positive constant $M^{1/(p-1)}$.

Since $\Sigma(y,c)$ is the principal eigenvalue of problem (2.12), see Lemma
2.2, that is,

$\displaystyle\left\\{\begin{aligned}
&-\psi_{t}-d\Delta\psi-a(t,x)\psi+yb(t,x)\psi=\Sigma(y,c)\psi,&&t\in(0^{+},\tau],\,\,x\in\Omega,\\\
&\psi(t,x)=0,&&0<t\leq\tau,\,\,x\in\partial\Omega,\\\
&\psi(0,x)=\psi(\tau,x),&&x\in\overline{\Omega},\\\
&\psi(0^{+},x)=\frac{1}{c}\psi(0,x),&&x\in\overline{\Omega},\end{aligned}\right.$

where $\psi\gg 0$ is the first eigenfunction associate to $\Sigma(y,c)$, we
obtain, by following the same arguments as done in the proof of Lemma 2.2,
that

$(U\psi)_{t}-d(\psi\Delta
U-U\Delta\psi)=yb(t,x)U\psi-b(t,x)U^{p}\psi-\Sigma(y,g^{\prime}(0))U\psi$

holds, whence follows by integration in $x\in\Omega$, for $t$ from $0^{+}$ to
$\tau$, that

$\displaystyle-$ $\displaystyle\Sigma(y,c)\int^{\tau}_{0}\int_{\Omega}U\psi
dxdt+\int^{\tau}_{0}\int_{\Omega}b(t,x)U\psi\left(y-U^{p-1}\right)dxdt$
$\displaystyle=$
$\displaystyle\int_{\Omega}[U(\tau,x)\psi(\tau,x)-U(0^{+},x)\psi(0^{+},x)]dx$
$\displaystyle=$
$\displaystyle\int_{\Omega}[U(0,x)\psi(0,x)-cU(0,x)\frac{1}{c}\psi(0,x)]dx$
$\displaystyle=$ $\displaystyle\ 0,$

holds for any $y\geq 0$. So, when $b(t,x)\not\equiv 0$ in
$[0,\tau]\times\overline{\Omega}$, we have

$\Sigma(y,c)>0\ \mbox{for any }y>M,$

whence follows, together with the fact that $\Sigma(y,c)>0$ is strict
increasing with respect to $y\geq 0$, that

$\Sigma(\infty,c)=\lim_{y\to\infty}\Sigma(y,c)>0,$

which is a contradiction to our assumption $\Sigma(\infty,c)\leq 0$. If
$b(t,x)\equiv 0$ in $[0,\tau]\times\overline{\Omega}$, then
$\Sigma(\infty,c)=\Sigma(0,c)\neq 0$, as proved in the item $(i)$.

Proof of item $(iii)$. It suffices to construct an upper solution and a lower
solution for the problem (3.1). Since $\Sigma(0,c)<0<\Sigma(\infty,c)$, we
have $b(t,x)\not\equiv 0$ in $[0,\tau]\times\overline{\Omega}$, and there
exist $\delta>0$ and $M>0$ such that

$\Sigma(0,c)<\Sigma(\delta,c)<0<\Sigma(M,c)<\Sigma(\infty,c).$

Let us first construct an upper solution $\overline{U}$ for the periodic
problem (3.1) inspired on ideas from [3, Proposition 3.2]. Since
$\Sigma(M,c)>0$, we are able to find an $\varepsilon>0$ such that

$0<\Sigma_{\varepsilon}(M,c):=\mu_{1}[d,a(t,x),Mb(t,x)\chi_{\Omega_{\varepsilon}}(x),c],$

where

$\Omega_{\varepsilon}:=\\{x\in\Omega:\,\textrm{dist}\,(x,\partial\Omega)>\varepsilon\\},\
\chi_{\Omega_{\varepsilon}}(x)=1\,\textrm{for}\,x\in\Omega_{\varepsilon}\
\textrm{and}\,\chi_{\Omega_{\varepsilon}}(x)=0\,\textrm{for}\,x\in\Omega/\Omega_{\varepsilon}.$

Let $\varphi\gg 0$ be the eigenfunction corresponding to the principal
eigenvalue $\Sigma_{\varepsilon}(M,c)$ of problem (2.1) with
$\max_{(t,x)\in[0,\tau]\times\overline{\Omega}}\varphi(t,x)=1$ (here, we are
considering problem (2.1) with $b(t,x)\chi_{\Omega_{\varepsilon}}$ in the
place of $b(t,x)$). That is,

$\displaystyle\left\\{\begin{aligned}
&\varphi_{t}-d\Delta\varphi-a(t,x)\varphi+Mb(t,x)\chi_{\Omega_{\varepsilon}}(x)\varphi=\Sigma_{\varepsilon}(M,c)\varphi,&&t\in(0^{+},\tau],\,\,x\in\Omega,\\\
&\varphi(t,x)=0,&&t\in[0,\tau],\,\,x\in\partial\Omega,\\\
&\varphi(0,x)=\varphi(\tau,x),&&x\in\overline{\Omega},\\\
&\varphi(0^{+},x)=c\varphi(0,x)),&&x\in\overline{\Omega}.\end{aligned}\right.$

By defining

$\overline{U}=K\varphi,$ (3.3)

we obtain

$\displaystyle\overline{U}_{t}-d\Delta\overline{U}-a(t,x)\overline{U}+b(t,x)\overline{U}^{p}$
$\displaystyle=$ $\displaystyle\
\overline{U}[\Sigma_{\varepsilon}(M,c)+(-M\chi_{\Omega_{\varepsilon}}(x)+K\varphi)b(t,x)]$
$\displaystyle>$ $\displaystyle\
\overline{U}[-M\chi_{\Omega_{\varepsilon}}(x)+K\varphi]b(t,x)$
$\displaystyle\geq$ $\displaystyle\ 0$

for any $K\geq
M/\min_{[0,\tau]\times\overline{\Omega}_{\varepsilon}}\,[\varphi(t,x)]$ given.

Besides this, it is immediate to verify that $\overline{U}(t,x)=0$ for
$t\in[0,\tau],\,\,x\in\partial\Omega$,
$\overline{U}(0,x)=\overline{U}(\tau,x),\ x\in\overline{\Omega}$, and
$\overline{U}(0^{+},x)-c\overline{U}(0,x)=0$ for $x\in\overline{\Omega}$
whence follows that $\overline{U}$ is an upper solution for the problem (3.1).

In what follows, let us build a lower solution, say $\underline{U}$. To do
this, define

$\displaystyle\underline{U}(t,x)=\begin{cases}\displaystyle\varepsilon{\phi}(0,x),&t=0,\,\,x\in\overline{\Omega},\\\\[8.53581pt]
\displaystyle\varepsilon{\frac{\rho}{c}}\phi(0^{+},x),&t=0^{+},\,\,x\in\overline{\Omega},\\\\[8.53581pt]
\displaystyle\varepsilon{\frac{\rho}{c}}e^{-\Sigma(\delta,c)t}\phi(t,x),&t\in(0^{+},\tau],\,\,x\in\overline{\Omega},\end{cases}$
(3.4)

where $\varepsilon>0$ is a sufficiently small constant, and $\phi\gg 0$ is the
eigenfunction associated to the principal eigenvalue $\Sigma(\delta,c)$ of the
problem (2.1) with
$\max_{(t,x)\in[0,\tau]\times\overline{\Omega}}\phi(t,x)=1$, that is,

$\displaystyle\left\\{\begin{aligned} &\phi_{t}-d\Delta\phi=a(t,x)\phi-\delta
b(t,x)\phi+\Sigma(\delta,c)\phi,&&t\in(0^{+},\tau],\,\,x\in\Omega,\\\
&\phi(t,x)=0,&&t\in[0,\tau],\,\,x\in\partial\Omega,\\\
&\phi(0,x)=\phi(\tau,x),&&x\in\overline{\Omega},\\\
&\phi(0^{+},x)=c\phi(0,x)),&&x\in\overline{\Omega}.\end{aligned}\right.$

Let us choose the parameters. First, we point out that
$\underline{U}(0,x)=\underline{U}(\tau,x)$ holds if and only if
$\rho=ce^{\Sigma(\delta,c)\tau}$. By setting such $\rho>0$ and noting that
$\rho=ce^{\Sigma(\delta,c)\tau}<c$, due to the fact that $\Sigma(\delta,c)<0$,
we obtain that

$\displaystyle\underline{U}_{t}-d\Delta\underline{U}-a(t,x)\underline{U}+b(t,x)\underline{U}^{p}$
$\displaystyle=$ $\displaystyle\
\underline{U}[-\delta+\varepsilon\frac{\rho}{c}e^{-\Sigma(\delta,c)t}\phi)b(t,x)]$
$\displaystyle\leq$ $\displaystyle\ \underline{U}(-\delta+\varepsilon
e^{-\Sigma(\delta,c)\tau})b(t,x)$ $\displaystyle\leq$ $\displaystyle\ 0$

for any

$0<\varepsilon\leq\delta e^{\Sigma(\delta,c)\tau}$

given. It is an immediate consequence of the definition of $\underline{U}$
that $\underline{U}(t,x)=0,\ t>0,x\in\partial\Omega$, and
$\underline{U}(0^{+},x)=c\underline{U}(0,x),\ x\in\Omega$ so that
$\underline{U}$ is a lower solution to problem (3.1).

By taking $\varepsilon>0$ small enough, if it necessary, we obtain
$\underline{U}\leq\overline{U}$. Then, based on standard upper and lower
solutions technique [18, Theorem 3.4], we have the existence of a
$\tau-$periodic solution for problem (3.1).

For the uniqueness of $\tau-$periodic solution, suppose that $U_{1}$ and
$U_{2}$ are two solutions of problem (3.1), and define

$S=\\{s\in{[0,1]},sU_{1}\leq{U_{2}},\,t\in{[0,\tau]},\,x\in\overline{\Omega}\\}.$

By using the fact that $f(v)/v$ is non-increasing with respect to $v$ in
$(0,\max\limits_{[0,T]\times{\overline{\Omega}}}U_{2}]$, where
$f(v):=a(t,x)v-b(t,x)v^{p}$, we are able to prove, in a similar way as was
done in [18, Theorem 3.4], that that $1\in{S}$ so that $U_{1}\leq U_{2}$.
Similarly, we have $U_{2}\leq U_{1}$ that leads to the uniqueness of solutions
to the problem (3.1). This ends the proof.  $\Box$

Again, by the periodicity of the terms $a(t,x)$ and $b(t,x)$ and the
eigenvalue properties, we will naturally extend the conclusions of Theorem 3.1
to the periodic-parabolic $\lambda$-problem

$\displaystyle\left\\{\begin{aligned} &U_{t}-d\Delta
U=a(t,x)U-b(t,x)U^{p}+\lambda
U,&&t\in((n\tau)^{+},(n+1)\tau],\,\,x\in\Omega,\\\
&U(t,x)=0,&&t\in[0,\tau],\,\,x\in\partial\Omega,\\\
&U((n\tau),x)=U((n+1)\tau,x),&&x\in\overline{\Omega},\ n=0,1,2,\cdots,\\\
&U((n\tau)^{+},x)=cU(n\tau,x),&&x\in\overline{\Omega},\
n=0,1,2,\cdots.\end{aligned}\right.$ (3.5)

That is.

###### Corollary 3.2

One has.

  1. $(i)$

If either $b(t,x)\not\equiv 0$ in $[0,\tau]\times\overline{\Omega}$ and
$\Sigma(0,c)\geq\lambda$ or $b(t,x)\equiv 0$ in
$[0,\tau]\times\overline{\Omega}$ and $\Sigma(0,c)\neq\lambda$, then problem
(3.5) has no positive solution.

  2. $(ii)$

If either $b(t,x)\not\equiv 0$ in $[0,\tau]\times\overline{\Omega}$ and
$\Sigma(\infty,c)\leq\lambda$ or $b(t,x)\equiv 0$ in
$[0,\tau]\times\overline{\Omega}$ and $\Sigma(\infty,c)\neq\lambda$, then
problem (3.5) has no positive solution.

  3. $(iii)$

If $\Sigma(0,c)<\lambda<\Sigma(\infty,c)$, then the periodic problem (3.5)
admits a unique positive periodic solution, say $U_{\lambda}(t,x)$.

About the behavior at the endpoints of the $\lambda$-interval for the
existence, we have the next result.

###### Theorem 3.3

If $b(t,x)\not\equiv 0$ in $[0,\tau]\times\overline{\Omega}$, then the
solution $U_{\lambda}$ is strictly increasing in
$\lambda\in(\Sigma(0,c),\,\Sigma(\infty,c))$ in the sense that
$U_{\lambda_{2}}\gg U_{\lambda_{1}}$ if $\lambda_{2}>\lambda_{1}$. Moreover,

$\displaystyle\lim_{\lambda\downarrow\Sigma(0,c)}\,\|U_{\lambda}\|_{C([0,\tau]\times\overline{\Omega})}=0$
(3.6)

and if $\Sigma(\infty,c)<\infty$, then

$\displaystyle\lim_{\lambda\uparrow\Sigma(\infty,c)}\,\|U_{\lambda}\|_{C([0,\tau]\times\overline{\Omega})}=\infty.$
(3.7)

Proof: First, we note that is a consequence of Theorem 2.3-$(iv)$ that
$b(t,x)\not\equiv 0$ in $[0,\tau]\times\overline{\Omega}$ leads to
$\Sigma(0,c)<\Sigma(\infty,c)$. Assume $\lambda_{2}>\lambda_{1}$. We first
claim that $U_{\lambda_{2}}\geq U_{\lambda_{1}}$ in
$[0,\tau]\times\overline{\Omega}$, where $U_{\lambda_{i}}$ is the solution of
problem (3.5) with $\lambda$ replaced with $\lambda_{i}$ for $i=1,2$. In fact,
$U_{\lambda_{1}}$ is a lower solution for problem (3.5) with
$\lambda=\lambda_{2}$. As done in the proof of Theorem 3.5-$(iii)$, we are
able to build a solution $\overline{U}$ satisfying
$U_{\lambda_{1}}\leq\overline{U}$. So, the claim is derived by a iteration
process, by using the lower solution as the initial iteration, and the
uniqueness of solution for problem (3.5).

Next let us prove that $U_{\lambda_{2}}\gg U_{\lambda_{1}}$. To circumvent the
difficulties induced by the impulse in the parabolic-periodic problem (3.5),
we consider the following equivalent problem

$\displaystyle\left\\{\begin{array}[]{lll}(V_{\lambda})_{t}=d\Delta
V_{\lambda}+a(t,x)V_{\lambda}-b(t,x)V_{\lambda}^{p}+\lambda
V_{\lambda},&t\in(0,\tau],\,\,x\in\Omega,\\\\[2.84526pt]
V_{\lambda}(t,x)=0,&t\in[0,\tau],\,\,x\in\partial\Omega,\\\\[2.84526pt]
V_{\lambda}(0,x)=cV_{\lambda}(\tau,x),&x\in\overline{\Omega}.\end{array}\right.$
(3.11)

This equivalence is reached by taking

$U_{\lambda}(t,x)=V_{\lambda}(t,x)\ \mbox{for }t\in(0,\tau],\
x\in\overline{\Omega}$ (3.12)

and

$U_{\lambda}(0^{+},x)=V_{\lambda}(0,x)\ \mbox{for }x\in\overline{\Omega},\
\mbox{and }U_{\lambda}(0,x)=V_{\lambda}(\tau,x)\ \mbox{for
}x\in\overline{\Omega}.$ (3.13)

So, the proof that $U_{\lambda_{2}}\gg U_{\lambda_{1}}$ holds is obtained by
proving that $V_{\lambda_{2}}\gg V_{\lambda_{1}}$, where $V_{\lambda_{i}}$ is
the solution of the problem (3.11) with $\lambda$ replaced with $\lambda_{i}$
for $i=1,2$.

Let $W=V_{\lambda_{2}}-V_{\lambda_{1}}$. Then due to $U_{\lambda_{2}}\geq
U_{\lambda_{1}}$, we have that $W\geq 0$ and satifies

$\displaystyle\left\\{\begin{array}[]{lll}W_{t}=d\Delta
W+aW+b\xi(t,x)W+(\lambda_{2}-\lambda_{1})V_{\lambda_{2}}+\lambda_{1}W,&t\in(0,\tau],\,\,x\in\Omega,\\\\[2.84526pt]
W(t,x)=0,&t\in[0,\tau],\,\,x\in\partial\Omega,\\\\[2.84526pt]
W(0,x)=cV_{\lambda_{2}}(\tau,x)-cV_{\lambda_{1}}(\tau,x)=cW(\tau,x),&x\in\overline{\Omega},\end{array}\right.$
(3.17)

where

$\xi(t,x)=p\int_{0}^{1}\left((1-s)V_{\lambda_{2}}(t,x)+sV_{\lambda_{1}}(t,x)\right)^{p-1}ds.$

If there was a $(t_{0},x_{0})\in[0,\tau]\times\Omega$ such that
$W(t_{0},x_{0})=0=\min_{[0,\tau]\times\overline{\Omega}}W$, then we could
choose $t_{0}\in(0,\tau]$, and $W\equiv 0$ in
$[0,t_{0}]\times\overline{\Omega}$ by the strong maximum principle. Hence
$W(\tau,x)\equiv W(0,x)\equiv 0$ in $\overline{\Omega}$ due to
$W(0,x)=cW(\tau,x)$ in $\Omega$, which turns $W\equiv 0$ in
$[0,\tau]\times\overline{\Omega}$. This indicates that $W_{t}=d\Delta
W+aW-b\xi(t,x)W+\lambda_{1}W\equiv 0$ in $(0,\tau]\times\Omega$, that leads to
a contradiction with the first equation of (3.17), since
$(\lambda_{2}-\lambda_{1})V_{\lambda_{2}}>0$ in $(0,\tau]\times\Omega$.

For the remaining two limits, we only prove that
$\lim_{\lambda\uparrow\Sigma(\infty,c)}\,\|U_{\lambda}\|_{C([0,\tau]\times\overline{\Omega})}=\infty$,
because the proof of $\lim_{\lambda\downarrow\Sigma(0,c)}=0$ is similar.
Equivalently, it suffices to prove
$\lim_{\lambda\uparrow\Sigma(\infty,c)}\,\|V_{\lambda}\|_{C([0,\tau]\times\overline{\Omega})}=\infty$,
where $V_{\lambda}$ satisfies (3.11). As in [3, Theorem 1.1], if we assumed
$\lim_{\lambda\uparrow\Sigma(\infty,c)}\,\|V_{\lambda}\|_{C([0,\tau]\times\overline{\Omega})}<\infty$,
there would exist an increasing sequence $\\{\lambda_{n}\\}^{\infty}_{n=1}$
with $\lambda_{n}>\Sigma(0,c)$ and $M>0$ such that

$\lim_{n\to\infty}\,\lambda_{n}=\Sigma(\infty,c)<\infty\ \textrm{and}\
V_{\lambda_{n}}<V_{\lambda_{n+1}}\leq
M\,\textrm{in}\,[0,\tau]\times\overline{\Omega}\,\textrm{ for all}\,\ n\geq
1,$

whence follows that the pointwise limit

$\displaystyle
V_{\infty}(t,x):=\lim_{n\to\infty}V_{\lambda_{n}}(t,x)\,\textrm{in}\,[0,\tau]\times\overline{\Omega}\,$
(3.18)

exists and is bounded from below by $V_{\lambda_{1}}$.

Since

$\displaystyle 0<V_{\lambda_{n}}=(\frac{\partial}{\partial
t}-d\Delta)^{-1}[(\lambda_{n}+a(t,x))V_{\lambda_{n}}-b(t,x)V^{p}_{\lambda_{n}}],\
\mbox{for every }n\geq 1,$ (3.19)

is uniformly bounded from above by $M$, and $(\frac{\partial}{\partial
t}-d\Delta)^{-1}$ can be regarded as a compact operator in the space

$\displaystyle\begin{array}[]{ll}{D}_{0}([0,\tau]\times\overline{\Omega})=&:=\\{\xi(t,x)\in{C}([0,\tau]\times\overline{\Omega}):\
\xi(t,x)=0\ \forall(t,x)\in[0,\tau]\times\partial\Omega,\\\
&\xi(0,x)=c\xi(\tau,x),\ \forall x\in\overline{\Omega}\\},\end{array}$

endowed with the norm $\|\cdot\|_{C([0,\tau]\times\overline{\Omega})}$, we can
infer that the pointwise limit in (3.18) is a uniform limit.

So, by letting $n\to\infty$ in (3.19) yields that $V_{\infty}$ is a positive
solution for problem (3.11) with $\lambda=\Sigma(\infty,c)$. Equivalently,
$U_{\infty}$, given by (3.12) and (3.13) is a positive solution of (3.5) with
$\lambda=\Sigma(\infty,c)$, which leads a contradiction to the item $(iii)$ of
Corollary 3.2. This ends the proof.  $\Box$

##  4 The dynamical behavior of the solution

Now, we are in position to prove the dynamical behavior of the unique global
classical solution $u(t,x)\in
C^{1+\theta/2,2+\theta}((n\tau,(n+1)\tau]\times\Omega)\bigcup
C((n\tau,(n+1)\tau])\times\overline{\Omega})$ $(n=0,1,2,\cdots)$. As we
already know, it was established in [17] for any nonnegative and continuous
initial value $u_{0}$ that vanishes on the boundary $\partial\Omega$.

###### Theorem 4.1

For any nonnegative and nontrivial function $u_{0}\in C(\overline{\Omega})$
given, the solution $u(t,x)$ of problem (1.1) behaves in the following way:

  1. $i)$

If either $b(t,x)\not\equiv 0$ in $[0,\tau]\times\overline{\Omega}$ and
$\Sigma(0,c)\geq 0$ or $b(t,x)\equiv 0$ in $[0,\tau]\times\overline{\Omega}$
and $\Sigma(0,c)>0$, then $\lim_{t\to\infty}u(t,x)=0$ uniformly for
$x\in\overline{\Omega}$;

  2. $ii)$

if $\Sigma(0,c)<0<\Sigma(\infty,c)$, then
$\lim_{m\to\infty}u(t+m\tau,x)=U(t,x)$ for any $t\geq 0$ and uniformly for
$x\in\overline{\Omega}$, where $U(t,x)$ is the unique solution of the problem
(3.1);

  3. $iii)$

If either $b(t,x)\not\equiv 0$ in $[0,\tau]\times\overline{\Omega}$ and
$\Sigma(\infty,c)\geq 0$ or $b(t,x)\equiv 0$ in
$[0,\tau]\times\overline{\Omega}$ and $\Sigma(\infty,c)<0$, then
$\lim_{t\to\infty}||u(t,\cdot)||_{C(\overline{\Omega})}=+\infty$.

Proof: We first proved the part $(ii)$. Without loss of generality, let us
assume that $u_{0}(x)>0$ for $x\in\Omega$; otherwise, we may use the initial
time $t_{0}>0$ replaced with $t=0$. Since we are assuming
$\Sigma(0,c)<0<\Sigma(\infty,c)<\infty$, we can choose the upper solution
$\overline{U}$, defined in (3.3), and the lower solution $\underline{U}$,
defined in (3.4). Owing to their definitions and the fact that the eigenvalue
functions $\phi$ and $\varphi$ are strongly positive, a sufficiently small
$\varepsilon>0$ and a large $K>1$ can be selected to lead to the below
inequalities

$\underline{U}(0,x)=\varepsilon\phi(0,x)\leq{u}(0,x)\leq
K\varphi(0,x)=\overline{U}(0,x),\ \mbox{for }x\in\overline{\Omega},$

and

$\underline{U}(0^{+},x)={c\underline{U}(0,x)}\leq c{u}(0,x)={u}(0^{+},x)\leq
K\varphi(0^{+},x)=\overline{U}(0^{+},x)\quad\mbox{for }x\in\overline{\Omega}.$

So, it follows from the comparison principle that
$\underline{U}(t,x)\leq{u}(t,x)\leq\overline{U}(t,x)$ for $t\in(0^{+},\tau]$
and $x\in\overline{\Omega}$. By using a similar process for
$t\in(\tau^{+},2\tau],((2\tau)^{+},3\tau],\cdots$, we infer that

$\underline{U}(t,x)\leq{u}(t,x)\leq\overline{U}(t,x),\,\,t\geq
0,\,\,x\in\overline{\Omega}.$ (4.1)

Below, let us construct two iteration sequences, say $\\{\bar{u}^{(n)}\\}$ and
$\\{\underline{u}^{(n)}\\}$, starting from
$\underline{u}^{(0)}:=\underline{U}$ and $\bar{u}^{(0)}:=\overline{U}$,
respectively, defined by

$\displaystyle\left\\{\begin{array}[]{lll}\underline{u}_{t}^{(n)}-d\Delta\underline{u}^{(n)}+b^{M}\underline{u}^{(n)}=b^{M}\underline{u}^{(n-1)}+a(t,x)\underline{u}^{(n-1)}&\\\
-b(t,x)(\underline{u}^{(n-1)})^{p},&t\in(0^{+},\tau],\,\,x\in\Omega,\\\\[2.84526pt]
\underline{u}^{(n)}(t,x)=0,&t\in[0,\tau],\,\,x\in\partial\Omega\\\
\underline{u}^{(n)}(0,x)=\underline{u}^{(n-1)}(\tau,x),&x\in\overline{\Omega},\\\
\underline{u}^{(n)}(0^{+},x)=c\underline{u}^{(n-1)}(0,x),&\
x\in\Omega,\end{array}\right.$ (4.7)

and

$\displaystyle\left\\{\begin{array}[]{lll}\bar{u}_{t}^{(n)}-d\Delta\bar{u}^{(n)}+b^{M}\bar{u}^{(n)}=b^{M}\bar{u}^{(n-1)}+a(t,x)\bar{u}^{(n-1)}&\\\
-b(t,x)(\bar{u}^{(n-1)})^{p},&t\in(0^{+},\tau],\,\,x\in\Omega,\\\\[2.84526pt]
\underline{u}^{(n)}(t,x)=0,&t\in[0,\tau],\ \ x\in\partial\Omega\\\
\bar{u}^{(n)}(0,x)=\bar{u}^{(n-1)}(\tau,x),&\ x\in\overline{\Omega},\\\
\bar{u}^{(n)}(0^{+},x)=c\bar{u}^{(n-1)}(0,x),&x\in\Omega,\end{array}\right.$
(4.13)

where $b^{M}>0$ is taken as
$b^{M}=\max_{[0,\tau]\times\overline{\Omega}}[-a(t,x)+pb(t,x)(\overline{U})^{p-1}]$
to ensure the monotonicity of the function
$h(x,s):=b^{M}s+a(t,x)s-b(t,x)s^{p}$ with respect to $s>0$.

From (4.1), (4.7) and (4.13), we have

$\underline{u}^{(1)}(0,x)=\underline{u}^{(0)}(\tau,x)\leq{u(\tau,x)}\leq\bar{u}^{(0)}(\tau,x)=\bar{u}^{(1)}(0,x),\\\
$

for $x\in\overline{\Omega}$, and

$\underline{u}^{(1)}(0^{+},x)=c\underline{u}^{(0)}(\tau,x)\leq{cu(\tau,x)}=u(\tau^{+},x)\leq{c\bar{u}^{(0)}(\tau,x)}=\bar{u}^{(1)}(0^{+},x),\\\
$

for $x\in\overline{\Omega}$. Thus, by comparison argument, we have

$\underline{u}^{(1)}(t,x)\leq{u(t+\tau,x)}\leq{\bar{u}^{(1)}(t,x)},\
t\in(0^{+},\tau]\ \mbox{and }x\in\overline{\Omega},$

hold so that a induction process leads to

$\underline{u}^{(1)}(t,x)\leq{u(t+\tau,x)}\leq{\bar{u}^{(1)}(t,x)}\ \mbox{for
}t\geq 0\ \mbox{and }x\in\overline{\Omega}.$

Similarly, we can conclude by iteration that

$\underline{u}^{(m)}(t,x)\leq{u(t+m\tau,x)}\leq{\bar{u}^{(m)}(t,x)},\,t\geq{0},\,x\in\overline{\Omega}$
(4.14)

hold for any $m\in\mathbb{N}$.

Besides this, it is standard to show that

$\underline{u}^{(m)}(t,x)\leq{\underline{u}^{(m+1)}(t,x)},\ \mbox{and
}\overline{u}^{(m+1)}(t,x)\leq{\overline{u}^{m}(t,x)}\ \mbox{for all
}t\geq{0},\,x\in\overline{\Omega},$ (4.15)

hold for any $m\in\mathbb{N}$. So, it follows from (4.14) and (4.15) that the
limits

$\lim\limits_{m\to\infty}{\underline{u}^{(m)}(t,x)}\ \mbox{and
}\lim\limits_{m\to\infty}{\bar{u}^{(m)}(t,x)},\ t\in(0^{+},\tau]\ \mbox{and
}x\in\overline{\Omega}$

exist, and again is standard showing that they are the minimal and the maximal
$\tau$-periodic solutions of problem (3.1), respectively, so that they are
equals due to the uniqueness of solutions of problem (3.1) stated in Theorem
3.1-$(iii)$. This completes the proof of $(ii)$.

Below, let us begin to prove the part $(iii)$. To do this, first we point out
that the arguments just used to prove the item $(ii)$ above, can be repeated
to prove the following Lemma.

###### Lemma 4.2

If $\Sigma(0,c)<\lambda<\Sigma(\infty,c)$, then
$\lim_{m\to\infty}u_{\lambda}(t+m\tau,x)=U_{\lambda}(t,x)$ for any $t\geq 0$
and uniformly for $x\in\overline{\Omega}$, where $U_{\lambda}$ is the unique
solution of the problem (3.5), and $u_{\lambda}$ is a unique positive solution
of

$\displaystyle\left\\{\begin{aligned} &u_{t}=d\Delta
u+a(t,x)u-b(t,x)u^{p}+\lambda
u,&&t\in((n\tau)^{+},(n+1)\tau],\,\,x\in\Omega,\\\
&u(t,x)=0,&&t>0,\,\,x\in\partial\Omega,\\\
&u(0,x)=u_{0}(x),&&x\in\overline{\Omega},\\\
&u((n\tau)^{+},x)=cu(n\tau,x),&&x\in\overline{\Omega},\
n=0,1,2,\cdots.\end{aligned}\right.$ (4.16)

Coming back to the prove of item $(iii)$, we first note that the solution $u$
of problem (1.1) satisfies, by comparison principle, the inequality

$u_{\lambda}(t,x)\leq u(t,x)\ \mbox{in }[0,\infty)\times\overline{\Omega},$

for all $\Sigma(0,c)<\lambda<\Sigma(\infty,c)\leq 0$ given, where
$u_{\lambda}$ is the unique solution of problem (4.16).

So, by Lemma 4.2, we have that

$U_{\lambda}(t,x)=\lim_{m\to\infty}\,u_{\lambda}(t+m\tau,x)\leq\liminf_{t\to\infty}\,u(t+m\tau,x),$

for any $\Sigma(0,c)<\lambda<\Sigma(\infty,c)$, $t\geq 0$ and uniformly for
$x\in\overline{\Omega}$.

So, by letting $\lambda\uparrow\Sigma(\infty,c)$, together with (3.7), yields

$\liminf_{t\to\infty}||u(t,\cdot)||_{C(\overline{\Omega})}=\infty,$

which concludes the proof of $(iii)$.

The proof of the item $(i)$ is similar to that one used to prove the item
$(iii)$, by using (3.6), and we omit the detail with obvious modification.
$\Box$

As a consequence of Theorems 4.1 and 2.5, we have the next result that
highlights the impact of the impulse on the dynamics of the population.

###### Theorem 4.3

Assume that $b(t,x)\not\equiv 0$ in $[0,\tau]\times\Omega$ and $d$, $a(t,x)$
and $b(t,x)$ are fixed. Then there exist $0<c_{*}<\infty$ and
$c_{*}<c^{*}\leq\infty$ such that the unique solution $u(t,x)$ of problem
(1.1) behaves in the following way:

  1. $i)$

if $0<c\leq c_{*}$, then $\lim_{t\to\infty}u(t,x)=0$ uniformly for
$x\in\overline{\Omega}$;

  2. $ii)$

if $c_{*}<c<c^{*}$, then $\lim_{m\to\infty}u(t+m\tau,x)=U(t,x)$ uniformly for
$x\in\overline{\Omega}$, where $U(t,x)$ is the unique solution of the problem
(3.1);

  3. $iii)$

if $c\geq c^{*}$, then
$\lim_{t\to\infty}||u(t,\cdot)||_{C(\overline{\Omega})}=+\infty$.

Proof: It follows from Theorem 2.4 that $\Sigma(0,z)$ is continuous and
strictly decreasing with respect to $z>0$. Moreover, we know that
$\lim_{z\to\infty}\Sigma(0,z)=-\infty$ and $\lim_{z\to
0^{+}}\Sigma(0,z)=+\infty$. Therefore, there exists uniquely $c_{*}>0$ such
that $\Sigma(0,c_{*})=0$, and $\Sigma(0,c)\geq 0$ if, and only if, $c\leq
c_{*}$, which means, by Theorem 4.1-$(i)$, that $\lim_{t\to\infty}u(t,x)=0$
uniformly for $x\in\overline{\Omega}$ if $c\leq c_{*}$, that completes the
proof of $(i)$.

Next, let us set $c^{*}>0$ to prove the items $(ii)$ and $(iii)$, that is, let
us set a $c_{*}<c^{*}\leq\infty$ such that $\Sigma(\infty,c)>0$ for any
$c<c^{*}$. By the assumption $b(t,x)\not\equiv 0$ in $[0,\tau]\times\Omega$,
we have from Theorem 2.3-$(iv)$, that $\Sigma(0,z)<\Sigma(\infty,z)\leq\infty$
for each $z>0$. Besides this, it is a consequence of Theorem 2.5 that just one
situation may occur: either $(a)$ $\Sigma(\infty,z)=\infty$ for any $z>0$ or
$(b)$ $\Sigma(\infty,z)<\infty$ for any $z>0$.

If the possibility $(a)$ occurs, then $\Sigma(\infty,z)=\infty$ for any $z>0$
that naturally leads us to set $c^{*}=\infty$, that is,
$\Sigma(\infty,c)=\infty$ for all $c>0$. This case is similar to the classical
logistic system, that is, unlimited growth does not occur.

Assume that the possibility $(b)$ holds true. Then Theorem 2.5 implies that
there exists a $c^{*}>0$ such that $\Sigma(\infty,c^{*})=0$, and
$\Sigma(\infty,c)\leq 0$ if, and only if, $c\geq c^{*}$. Since
$\Sigma(0,c^{*})<\Sigma(\infty,c^{*})=0$, we have $c_{*}<c^{*}$. So, for all
$c\geq c^{*}$, we have by Theorem 4.1-$(iii)$ that the item $(iii)$ above
occurs so that unlimited growth happens.

To finish the prove, we just note that the above construction of
$0<c_{*}<c^{*}\leq\infty$ implies that $\Sigma(\infty,c)<0<\Sigma(\infty,c)$
holds if, and only if, $c_{*}<c<c^{*}$ so that the claim of the item $(ii)$
above follows from Theorem 4.1-$(ii)$. This ends the proof.  $\Box$

The above analysis can be immediately replicated to the case $b(t,x)\equiv 0$
in $[0,\tau]\times\overline{\Omega}$ by just noting that
$\Sigma(0,z)=\Sigma(\infty,z)$ for any $z>0$ whence follows that
$c_{*}=c^{*}$. So, we have the next result that extends some classical results
for Malthusian model.

###### Theorem 4.4

Assume that $b(t,x)\equiv 0$ in $[0,\tau]\times\overline{\Omega}$, and $d$ and
$a(t,x)$ are fixed. Then there exists $0<c_{*}<\infty$ such that the unique
solution $u(t,x)$ of problem (1.1) satisfies:

  1. $i)$

$\lim_{t\to\infty}u(t,x)=0$ uniformly for $x\in\overline{\Omega}$ if
$c<c_{*}$;

  2. $ii)$

$\lim_{t\to\infty}||u(t,\cdot)||_{C(\overline{\Omega})}=+\infty$ if $c>c_{*}$;

  3. $iii)$

$k\phi(t,x)$ is the positive solution to problem (1.1) for any $k>0$ if
$c=c_{*}$, where $(c_{*},\phi(t,x)$ is the eigen-pair of the eigenvalue
problem (2.1).

Proof: Similarly as in Theorem 4.3, there exists uniquely $c_{*}>0$ such that
$\Sigma(0,c_{*})=0$ since $\Sigma(0,z)$ is continuous and strictly decreasing
with respect to $z>0$, and $\lim_{z\to\infty}\Sigma(0,z)=-\infty$ and
$\lim_{z\to 0^{+}}\Sigma(0,z)=+\infty$. Therefore, if $c<c_{*}$, then
$\lim_{t\to\infty}u(t,x)=0$ uniformly for $x\in\overline{\Omega}$ by Theorem
4.1 $(i)$.

On the other hand, the assumption that $b(t,x)\equiv 0$ in
$[0,\tau]\times\overline{\Omega}$ assures that $\Sigma(0,c)=\Sigma(\infty,c)$
for any $c>0$. Therefore, $\Sigma(0,c_{*})=\Sigma(\infty,c_{*})=0$, which
means that, if $c>c_{*}$, then $\Sigma(\infty,c)<0$ and
$\lim_{t\to\infty}||u(t,\cdot)||_{C(\overline{\Omega})}=+\infty$, unlimited
growing happens.

The result of (iii) is from the definition of the eigenvalue value in the case
that $b(t,x)\equiv 0$ in $[0,\tau]\times\overline{\Omega}$.  $\Box$

##  5 Numerical simulation and biological explanation

In this section, we consider the interval $\Omega=[0,\pi]$, and we will assume
that the impulse occurs at each time $\tau=2$. When $0<c<1$, impulsive
harvesting will take place, while $c>1$, birth pulse may occur.

We now fix some parameters in the model (1.1). Set $d=1$, $a(t,x)=2.0$, the
initial value $u_{0}(x)=0.5\sin{x}+0.2\sin{3x}$, and
$b(t,x):=b(x)=[0.01+0.5\sin(3x)]^{+}$ for $x\in[0,\pi]$ and any $t\geq 0$,
which means that $b(x)\equiv 0$ for
$x\in[\frac{\pi}{3}-\frac{1}{3}\arcsin{0.02},\frac{2\pi}{3}-\frac{1}{3}\arcsin{0.02}]$.

First, by choosing small impulse $c=0.08$, we see in the Fig. 1 that the
species goes to the extinction quickly. This was proved in Theorem 4.3-$i)$.

Figure 1: The dynamics of species $u$ with small pulse $c=0.08$ at every
$\tau=2$. Graph $(a)$ is the right view of the spatiotemporal distribution of
$u$ plotted in graph $(b)$, while in the Graph $(c)$ are plotted the cross
sections view at $t=0,3,5,7,20$. Graphs $(a)-(c)$ imply that the species
$u(t,x)$ gradually tends to zero with time $t$.

Secondly, let us choose $c=0.8$. Figure 2 shows that the solution stabilizes
to a periodic positive state. See Theorem 4.3-$ii)$.

Figure 2: The dynamics of species $u$ with medium pulse $c=0.8$ at every
$\tau=2$. Graph $(a)$ is the right view of the spatiotemporal distribution of
$u$ in the graph $(b)$, and Graph $(c)$ is its cross section views at
$t=0,3,5,7,20$. Graphs $(a)-(c)$ imply that the species $u(t,x)$ gradually
tends to a periodic state.

Lastly, we choose a large impulsive rate $c=7.8$, it follows from Theorems
4.3-$iii)$ that the solution grows quickly, see Figure 3.

Figure 3: The dynamics of species $u$ with large pulse $c=0.08$ at every
$\tau=2$. Graph $(a)$ is the right view of the spatiotemporal distribution of
$u$ in graph $(b)$, and Graph $(c)$ is its cross section views at
$t=0,3,5,7,20$. Graphs $(a)-(c)$ imply that the species $u(t,x)$ increases
quickly.

To sum up, the rate of population extinction will be accelerate with small
impulses, medium-impulse rates are more favorable for species to persist;
however, large impulses make the population increases quickly and eventually
blows up.

It is worth mentioning that all above results hold if $-\Delta$ operator in
model (1.1) is replaced with a second-order uniformly strongly elliptic
operator with obvious modifications.

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