PREDATOR-PREY MODEL WITH FUZZY PARAMETERS AND FUZZY INITIAL CONDITIONS: A SYSTEMATIC LITERATURE REVIEW

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INTRODUCTION
The predator-prey model represents a dynamic system of differential equations that characterizes the interaction between two species: predator and prey.The foundational work on  Terry [11]; Xie [5]; Ye et al. [12]; and Zu & Mimura [6].In existing literature, all parameters and initial values in the predator-prey model are typically assumed to be certain.However, in practice, uncertainty often arises due to unclear, inadequate, or incomplete information regarding variables and parameters resulting from errors in observations, measurements, experiments, and so on.To address this issue, various approaches have been employed, including stochastic, fuzzy, and fuzzy stochastic approaches, to construct models that account for uncertainty.
In recent decades, the application of fuzzy theory has proven to be a valuable tool for mathematical modeling of real-world phenomena.This approach enables the representation of uncertain variables and parameters using intervals and fuzzy numbers.In the study of differential equations in a fuzzy environment, the term fuzzy differential equation is used as a reference to differential equations with fuzzy coefficients, differential equations with fuzzy initial values or fuzzy limit values, or even differential equations that relate to functions in the space of fuzzy intervals [15]- [18].The stability of fuzzy dynamical systems in population dynamics has also been discussed through fuzzy differential equations and the problem of fuzzy initial values [18].Various methods are used in solving fuzzy differential equations including through the Hukuhara derivative approach (H-derivative) [15], [16] The advancement of fuzzy differential equations has led to significant advancements in the field of the fuzzy predator-prey model, as evidenced by studies conducted by [38]- [40].In this particular investigation, the classical deterministic predator-prey model Lotka-Volterra, was reformulated by incorporating fuzzy initial conditions.Moreover, the Lotka-Volterra predatorprey model has undergone further development to incorporate additional biological processes.As a result, research on the fuzzy predator-prey model continues to expand and flourish.This review employs a systematic literature review (SLR) as an approach to provide a overview of existing studies and emerging trends in the field of the fuzzy predator-prey model.
To support the SLR process, we use the Preferred Reporting Items for Systematic reviews and Meta-Analyses (PRISMA) method.The PRISMA method requires material from database mining articles sourced from various digital libraries.The study incorporates a bibliometric analysis to establish connections between the data presented in the reviewed articles.Through data visualization techniques, the analysis aims to examine the content, patterns, and trends within the collection of documents by measuring the significance of terms and calculating the frequency of keywords or topics that appear simultaneously in the article under review.Additionally, an evolutionary analysis was conducted to identify temporal changes in the topic covered.
In essence, this study offers a survey report that presents a systematic examination of the fuzzy predator-prey model.It encompasses various aspects, including the development and modification of the model, the methodologies employed in studying and analyzing the model, as well as an evaluation of the strengths and weaknesses associated with each approach.Finally, it will offer the development of models and approaches that can be used in the studying and analyzing models as novelty for further research.

Scientific Article Data
The research process for this study involved searching and collecting relevant article data.
The initial step was to conduct a systematic search of indexed publications in three databases: Google Scholar, Dimensions, and Scopus.The search was performed using specific keywords to ensure comprehensive coverage of relevant literature.The keywords used in the search included: ("Predator-Prey Model" OR "Predator Prey Model" OR " Prey Predator Model" OR "Prey-Predator Model") AND ("Fuzzy Set" OR "Fuzzy Numbers") AND ("Fuzzy Differential Equations" OR "Fuzzy Initial Value" OR "Fuzzy Initial Condition" OR " Fuzzy Initial Population" OR "Fuzzy Parameters").For more details, refer to Table 1 for the specific keywords combinations employed during each stage of the search.

KW-III KW-I AND KW-II
The article search process was facilitated by utilizing the Publish or Perish software.This software enabled the creation of a database containing articles relevant to the specified keywords.
The information within the database encompassed various aspect, including the author's name, article title, publication date, publisher, number of citations, and abstract text.The search was limited to articles written in English and published in peer-reviewed journals between the years 2014 and 2023.By utilizing Publish or Perish, a comprehensive collection of articles meeting the specified criteria was obtained, forming the basis for further analysis and examination in this study.
The initial step involved searching for articles related to the first keyword.The search yielded 313 articles from Google Scholar, 296 articles from Dimension, and 21 articles from Scopus.
Furthermore, by using the second keyword results are obtained 245 articles from Google Scholar, 152 articles from Dimension, and 10 articles from Scopus.Finally, by combining the first keyword with the second keyword using the "AND" operator, the search resulted in 183 articles from Google Scholar, 112 articles from Dimension, and 5 articles from Scopus.To facilitate the selection process, the articles obtained from the search results in the three databases were stored in BibTex format.A summary of the search results from the three databases, based on the aforementioned keywords, is provided in Table 2.The search results for the next stage comprised at total of 300 articles obtained from the three databases using the third keyword.These articles will be further examined and analyzed in subsequent phase of the research.

Selection of Literature Database
In this section, the data collection and selection process was conducted using the PRISMA flowchart, as presented in Figure 1.Out of the initial pool of 300 articles obtained from the combined search results in the three databases using the JabRef software, a series of step were followed to refine the selection.Duplicate articles (80), books (24), and conference proceedings (11) were exluded, resulting in a remaining set of 185 articles.These articles were then assessed based on their titles and abstracts, leading to the identification of 29 relevant articles.The full texts of these 29 articles were obtained and further examined, resulting in a final set of 20 relevant articles.Lastly, a manual selection process involving a careful reading of the complete articles led to identification of 18 articles that were deemed relevant to the research being conducted.

Bibliometric Analysis
In this section, we employed the ".RIS" format to store the articles selected from our final selection process.Subsequently, we conducted bibliometric analysis using VOSviewer software, a widely-used tool in library science research.Bibliometric analysis allows us to gain valuable insights and a comprehensive overview of scientific publications by examining their citation patterns.The analysis results were visualized, providing a clear depiction of the content, patterns, and trends present within our document collection.This visualization facilitated the measurement of term strength and the identification of keywords or topics that appeared concurrently across the research documents under study.To further organize the emerging topics, we utilized clustering techniques.The size of each cluster directly corresponded to the number of articles addressing specific keywords related to our research theme.Larger clusters indicated a broader coverage of the corresponding keywords throughout the database, while smaller clusters indicated a more limited discussion on those specific keywords within the articles.

RESULTS AND DISCUSSION
The following section describes the results of data analysis obtained from 18 articles.
Explanations include article data visualization, development of fuzzy predator-prey models, model analysis, and methods used in model analysis and numerical simulations.

Article Data Visualization
In this section, the results of the bibliometric analysis conducted using VOSviewer are presented in the form of article data visualization.The analysis focused on identifying topics that appeared at least twice out of the total 827 words analyzed, resulting in 351 mappings that met the threshold.From these mappings, 42 distinct topics were derived and categorized into four clusters, represented by different colors: red (18 topics), green (10 topics), blue (8 topics), and yellow (6 topics).The network visualization analysis of the article data is illustrated in Figure 2.

Figure 2. Article data network visualization
The network visualization results reveal that the topic that occurs most frequently is "stability" and "fuzzy model" which appears 27 times each in the analyzed articles.This is closely followed by the topic of "triangular fuzzy numbers" which appears 22 times, and "predator prey model", "fuzzy parameter", and "fuzzy numbers", which appears 21 times each.The topic of "stability" and "fuzzy model" demonstrates strong connections to several other topics, particularly predator-prey models, fuzzy parameter, fuzzy differential equations, fuzzy initial condition, and fuzzy numbers.
The overlay visualization map, as depicted in Figure 3

Overview of Models from Each Article and Potential to Develop for Future Research
The fuzzy predator-prey model was initially introduced in 2008 by Peixoto et al.Following its introduction, publications related to this research have gradually emerged.Table 3 shows the summary of developments in the fuzzy predator-prey model in 2014 to 2023.  3 it can be seen the limited number of articles published on the fuzzy predator-prey model within the 2014-2023 timeframe.On average, only approximately 1 to 3 articles per year were published, accompanied by a range of 0 to 39 citations per year.Notably, the highest number of articles occurred in 2021 with three articles, receiving a total of 18 citations.This indicates that research on the fuzzy predator-prey model remains relatively scarce, presenting significant opportunities for further exploration and development.From Figure 7 it can be seen that of the 18 articles reviewed, the fuzzy predator-prey model that has been discussed generally has added harvesting (50%) and protection (29%) factors.
Biological factors that are still not widely discussed are disease factors (14%) and toxic factors (7%).This opens up opportunities to develop fuzzy predator-prey models by including disease and poison factors for further research.Another biological factor that has not been discussed in the research that has been done in the predator-prey fuzzy model is the alee effect.Allee effect broadly defined as a decrease in the fitness of individuals at low population sizes or densities, which may result in a critical population threshold below population extinction [58].So for some endangered species, the allee effect is more likely.Allee effect is very important for conservation management of endangered species, population development and utilization of natural resources.Therefore, the allee effect in the predator-prey fuzzy model is very important to study and do further research.
Apart from developments in terms of models, purpose and objectives, as well as a research focus, a review of 18 articles was also carried out on discussion material in terms of the concept of fuzzy theory used, including the uncertainty environment (parameters/initial conditions), the use of fuzzy numbers, the methods used, as well as numerical simulations carried out.Table 4 summarizes of the fuzzy theory concepts used in research related to the fuzz predator-prey model based on the literature review.

Figure 1 .
Figure 1.PRISMA flow chart , provides insights into the publication timeline of articles related to the analyzed topics.The map utilizes different shades of color, with lighter shades indicating more recent publications and darker shades representing articles published at a later time.Based on the visualization, it is evident that the majority of the topics covered in the articles were published within the period of 2016 to 2022.

Figure 3 .
Figure 3. Visualization of article data overlay

Figure 5
Figure 5.a.Overlay visualization: The topic of fuzzy initial condition Figure 6 provides an overview of the number of articles published and cited between the years 2014 to 2023.

Figure 6 .
Figure 6.Number of publications and citations for the period 2014 -2023 The most cited articles are articles [44] by title " Stability and bionomic analysis of fuzzy parameters based prey-predator harvesting model using UFM" with 36 citations.The second position is an article [51] with the title "Dynamical analysis of a fuzzy phytoplankton-zooplankton model with refuge, fishery protection and harvesting" with 25 citations.This shows that the topic of this research is considered interesting enough to be discussed and has a major contribution to further research.In Table 3 it can be seen that, of the 10 articles with the top citations, 5 articles namely [44], [45], [47], [48], [51] discusses the fuzzy predator-prey model with harvesting.This indicates that the fuzzy predator-prey model with harvesting has been extensively studied and remains an intriguing topic for future research endeavors.From Table 3 it can also be seen that the fuzzy predator-prey model extends the traditional predator-prey model by incorporating uncertainty in parameters and/or initial population expressed as fuzzy numbers.The initial research in this field focused on developing the Lotka-Volterra predator-prey model with fuzzy initial conditions, as evident from studies conducted in 2014 [41].Subsequently, from 2015 onwards, research on the fuzzy predator-prey model expanded to include various additional biological processes.The development of a fuzzy predator-prey model has also

Figure 7 .
Figure 7. Development of a fuzzy predator-prey model based on biological factors.

Table 4 .
Fuzzy theory concepts used in research related to the fuzzy predator-prey model

Table 3 .
Summary of developments in the fuzzy predator-prey model in 2014 to 2023.
[1]ta PREDATOR-PREY MODEL WITH FUZZY PARAMETERS AND FUZZY INITIAL CONDITIONSFrom Table4it can be seen that the environmental uncertainty discussed in the articles reviewed 1 is mostly only in parameters, namely in[1], [44]-[48], [51], [52], [54], [56] , [57] or in initial 2 conditions only, namely in [41]-[43], [49], [50], [53]-[55], [57].There are only two articles that 3 discuss uncertainty in both (parameters and initial conditions), namely in [54], [57], but fuzzy 4 parameters and fuzzy initial conditions are discussed separately.This is an opportunity for future 5 research by combining fuzzy parameters and fuzzy initial conditions so that they are more realistic 6 and in accordance with the conditions in real problems.Meanwhile, most studies carry out 7 simulations using triangular fuzzy numbers, there is only one article that discusses trapezoidal 8 fuzzy numbers, namely in [43], but the discussion is only briefly discussed in theory, but in 9 carrying out simulations they still use triangular fuzzy numbers.Further research development can 10be carried out using trapezoidal fuzzy numbers or others.toshow the existence and uniqueness of the system solution.Nevertheless, this 14 approach exhibits a limitation whereby the diameter of the fuzzy function under investigation must 15 always remain constant or increase, thus restricting its applicability in cases where the diameter 16 may decrease.As a result of this limitation, the solutions obtained in numerous cases may deviate 17 from those that would be expected intuitively based on the nature of the system.Besides using the 18 H-derivative approach, another approach is gH-derivative[56].The gH-derivative was defined 19 based on generalized Hukuhara difference (gH-difference) which is a more general concept than 20 H-difference.Although the existence of gH-differences comes with more limitations little 21 compared to the H-difference, there is a possibility that the gH-difference of the two fuzzy numbers 22 does not exist.Therefore, the existence of the gH derivative of the fuzzy function cannot be 23 guaranteed.This limitation opens opportunities for further research with other approaches to fuzzy 24 derivatives, such as generalized derivative (g-derivative) [32], [33], or granular derivative (gr- 11The method used in studying the fuzzy predator-prey model generally uses the Hukuhara 12 derivative (H-derivative) [41], [42], [44]-[49], [51], [52], [54].The advantage of the H-derivative 13 approach is able 25 derivative) [59].g-derivative and gr-derivative can overcome the shortcomings of H-derivative 26 and gH-derivative, where these two approaches do not require that the diameter of the fuzzy