1. Introduction

This paper presents a simple review of the role and common methods of chaos theory and fractal theory in current research on population dynamics.

The study of population dynamics is crucial for understanding ecosystem stability and biodiversity. It not only helps to reveal the characteristics of populations and the factors influencing their changes but also plays an important role in predicting population development, making management decisions, and determining conservation measures. The patterns of population dynamics can reflect the health of biological communities and their ability to adapt to environmental changes, making it a core issue in ecological research.

Chaos theory can help us understand certain population dynamics. It emphasizes the nonlinear characteristics of population dynamics and the high sensitivity to initial conditions, known as the "butterfly effect," where small changes can lead to long-term and fundamental impacts [5]. Chaos theory is useful for analyzing ecosystems with nonlinear characteristics, as it helps uncover the complexity and unpredictability in population dynamics.

On the other hand, fractal theory provides more specific tools for studying spatial distribution patterns and scale effects in populations. Fractal models can effectively describe and explain the spatial complexity and patterns in nature. Applying fractal theory can help researchers gain a deeper understanding of the spatial structure and distribution patterns of populations, providing a scientific basis for biodiversity conservation and ecosystem management.

This paper organizes and analyzes several specific models and research methods from chaos theory and fractal theory, demonstrating their role in studying changes in population dynamics.

2. Theories

Chaos theory studies seemingly random states that arise in deterministic nonlinear dynamical systems. Its core research topics include nonlinear dynamics, attractors, and sensitivity to initial conditions. The application of chaos theory in the analysis of population dynamics mainly lies in understanding the complexity and unpredictability of population changes. For example, population growth can be influenced by various factors, such as environmental changes and resource competition, and the interaction of these factors may cause population dynamics to exhibit chaotic characteristics [2].

Fractal theory is a mathematical tool used to describe complex shapes and patterns in nature. In modeling population distribution, fractal theory provides a method to describe and analyze spatial distribution patterns of populations. For instance, the spatial distribution of populations may exhibit fractal characteristics, where the distribution of population density follows a self-similar pattern observable at different scales. Through fractal analysis, researchers can better understand the spatial structure and dynamic changes of populations [2].

3. Literature review

In existing research, chaos and fractal theories are widely applied to analyze population dynamics, particularly in understanding Species Abundance Distribution (SAD).

The application of fractal theory in SAD research mainly involves using fractal dimensions to describe species distribution patterns within communities. For example, the fractal model proposed by Su et al. (referred to as the _p_ model) assumes a power-law relationship between the descending order of species abundance and their rank. In this model, SAD (Nr/N1) approximates 1: 1/2: 1/3: 1/4: 1/5, and so on. This model has relatively low data requirements, strong general applicability, and fits actual community samples well. The model uses the fractal parameter _p_ to characterize the basic features of community SAD, where a higher _p_ value indicates lower species diversity in the community. The introduction of this model provides a new perspective on the general rules of SAD, and it has been validated across multiple biological community databases [2].

Researchers have conducted detailed validation of the fractal model and the general rules of SAD using global biological community databases, such as Brown University's Foraminifera Database (BFD). These databases contain extensive species and community samples, providing an opportunity to test the model's universality. Through statistical analysis, it was found that the median and mean values of the fractal parameter _p_ in different databases are close to 1, supporting the general rule of SAD. This finding suggests that, despite differences in species diversity among communities, their abundance distribution follows similar mathematical laws.

Chaos theory studies seemingly disordered complex dynamic behaviors that arise in deterministic systems. In population dynamics, chaotic phenomena may be related to the unpredictability of population numbers and their sensitivity to initial conditions. The application of chaos theory helps in understanding how population numbers exhibit complex dynamic changes under specific environmental conditions, which is of great significance for population prediction and management.

In the future, research may focus on how to apply these theories to practical ecological conservation and management, as well as how to integrate modern technologies (such as remote sensing, GIS, etc.) to enhance the predictive capability of the models.

When analyzing the research progress in animal population dynamic models, we can explore it from both quantitative models and spatial models [3].

Quantitative models primarily focus on the changes in population size over time, with classic models including the "J"-shaped growth model, "S"-shaped growth model, and the Lotka-Volterra model.

a. The "J"-shaped growth model assumes unlimited population growth under ideal conditions, and is used to describe the rapid initial growth of populations [4].

b. The "S"-shaped growth model, on the other hand, takes environmental carrying capacity into account, suggesting that population growth slows as it approaches environmental limits, eventually reaching a stable state [4].

c. The Lotka-Volterra model is a predator-prey model that describes the dynamic relationship between predator and prey populations, demonstrating periodic fluctuations [5].

Spatial models consider the spatial distribution and interactions of populations, including heterogeneous population models, spatially explicit population models, and individual-based population models (IBMs).

a. The heterogeneous population model emphasizes the impact of environmental heterogeneity on population dynamics, such as the effects of different habitats on population growth and dispersion.

b. The spatially explicit population model provides a more detailed description of population distribution in space, including spatial diffusion and migration of populations.

c. The individual-based population model (IBM) focuses on individual behaviors and interactions, using simulations of individual decision-making to predict population dynamics.

In recent years, researchers have improved the predictive capabilities of models for actual population dynamics by integrating Geographic Information Systems (GIS) and remote sensing technologies. The parameter estimation and validation methods for these models are also continuously being refined, using statistical techniques such as maximum likelihood estimation and Bayesian methods to enhance accuracy. In the future, related research may place greater emphasis on the mechanistic and predictive aspects of models, as well as on how to apply model results to practical species conservation and management. With advancements in computational power and the development of big data technologies, future models are likely to become more complex, capable of handling more variables and larger datasets. Interdisciplinary research approaches, such as integrating ecology, mathematics, statistics, and computer science, will contribute to the development of more precise and practical population dynamics models.

4. Methodology

In studying population dynamics, a rich variety of mathematical models and computational methods provide tools for quantifying and predicting population changes [6]

4.1. Time sequence

Time sequence analysis identifies patterns of population dynamics by analyzing data sequences of population numbers over time. This method is suitable for time series data of different populations, such as predators and prey, and can help researchers understand fluctuations and trends in population numbers.

A typical example is the NLAR model (Nonlinear Autoregressive Model), which is particularly suitable for handling non-stationary time series data. The NLAR model captures complex patterns of population number changes by considering the nonlinear relationships between historical data points. This model can accommodate nonlinear trends in population dynamics, providing effective predictions, especially when population numbers exhibit significant nonlinear characteristics.

In one piece of literature, the author discusses the application of nonlinear autoregressive models in time series modeling and forecasting in detail [7]. The article presents a method for constructing optimal point predictors based on simulation and/or bootstrapping algorithms to address challenges in multi-step forecasting. Additionally, the literature develops bootstrap prediction intervals to enhance the accuracy and reliability of predictions.

There is also the GM(1,1) model (Grey Prediction Model), which is used for forecasting time series with significant random fluctuations. This model generates a trend-obvious sequence by cumulatively summing the original data, then establishes a prediction model, and finally obtains the predicted values by performing a cumulative reduction. The GM(1,1) model can provide high prediction accuracy in population dynamics forecasting, especially when the data series exhibits random fluctuations.

Other literature has proposed an improved grey NGM(1,1) prediction model and applied it to the forecasting of emerging infectious diseases [8]. In this study, the author significantly improved the model's prediction accuracy by optimizing the selection of initial values, constructing background values, and parameter estimation methods. The literature discusses in detail how to use particle swarm optimization algorithms to find optimal initial values and how to estimate the grey parameters of the model using weighted least squares, thereby achieving a combined optimization of the model.

4.2. Identifying and quantifying chaotic phenomena in population dynamics:

Chaotic phenomena in population dynamics manifest as sensitivity to initial conditions and uncertainty in long-term predictions. To identify and quantify this chaotic phenomenon, researchers can employ various methods. Power spectrum analysis can be used to identify whether chaotic behavior exists by analyzing the spectral characteristics of time series data. The Lyapunov exponent can be calculated to quantify the level of chaos in the system. Phase space reconstruction allows researchers to observe whether the trajectories of population dynamics exhibit the characteristics of chaotic attractors. Finally, nonlinear prediction can be utilized to forecast population dynamics and analyze prediction errors and the degree of model fit.

In practical applications, researchers need to choose appropriate models and methods based on the characteristics of population data. For instance, the ARMA model may be more suitable for stationary time series, while the NLAR model or GM(1,1) model may be more effective for non-stationary time series. Through these methods, researchers can gain a better understanding of and make predictions about population dynamics, providing scientific evidence for ecological conservation and resource management.

5. Case study

The Research Progress on Survey Methods for Wild Giant Panda Population Numbers is a great case demonstrating the role of chaos and fractals in the analysis of population dynamics [9]. Chaos theory can help us understand the complexity and unpredictability of changes in wild giant panda population numbers. For example, population numbers may exhibit nonlinear and dynamic changes due to factors such as food resources, habitat changes, diseases, and human activities. In practical research, time series analysis can be used to identify chaotic characteristics in population number fluctuations. If the population data shows sensitivity to initial conditions, this may indicate the presence of chaotic phenomena. For instance, a small environmental change, such as an outbreak of disease, could lead to significant changes in population numbers, which may continue to influence population dynamics in subsequent years.

Fractal theory can be used to analyze the spatial distribution patterns of giant panda populations. For instance, the habitats of giant pandas may exhibit fractal characteristics, where the distribution of populations in different areas may follow a self-similar pattern. By calculating the fractal dimension of habitat distribution, researchers can gain a better understanding of the spatial structure and dynamics of giant panda populations. This analysis can help identify key habitat areas and provide guidance for conservation efforts.

In the study of wild giant panda population dynamics, researchers can analyze multi-year population data. This data can be used to establish population dynamic models and predict future trends. If the data shows periodic fluctuations, it may indicate that population dynamics are influenced by periodic factors, such as cyclical changes in food resources. Conversely, if the data exhibits irregular fluctuations, it may suggest that population dynamics are affected by chaotic factors.

The application of chaos and fractal theory in the conservation and management of wild giant pandas can help researchers identify and predict key factors affecting population dynamics. For example, by understanding the chaotic characteristics of population number changes, better conservation measures can be formulated to address potential declines in population numbers. The application of fractal theory can help researchers identify spatial distribution patterns of panda populations, which is crucial for habitat conservation and planning protected areas.

6. Conclusion

Chaos and fractal theory hold significant potential value in the study of population dynamics, but they also have limitations. By integrating these theories with traditional ecological methods, a more comprehensive understanding and prediction of population changes can be achieved. Future research needs to find a better balance between theory and practice to fully leverage the advantages of these theories.

Chaos theory emphasizes the nonlinear characteristics of ecosystems and population dynamics, revealing that seemingly random complex behaviors can occur even in deterministic systems. Fractal theory, through the concepts of self-similarity and fractal dimension, provides a method to describe and quantify the complexity of spatial distribution in populations. Fundamentally, the "butterfly effect" in chaos theory indicates that population dynamics are extremely sensitive to initial conditions, meaning that even minor changes can lead to significant long-term impacts. Chaos and fractal theory offer new mathematical tools and models, such as time series analysis, attractor analysis, and fractal dimension calculations, which help to gain deeper insights into the underlying mechanisms of population dynamics. Fractal theory particularly emphasizes the spatial distribution patterns of populations, aiding researchers in identifying and predicting spatial distribution and diffusion patterns.

In the future, researchers may continue to deepen the application of chaos and fractal theory in population dynamics, developing new mathematical models and computational methods to better describe and predict these dynamics. By combining knowledge from ecology, physics, mathematics, and computational science, they can explore the complexities of population dynamics and the applicability of these theories across different ecosystems and biological taxa. Conducting more field surveys and experimental studies to validate the predictions of chaos and fractal theory, while collecting data from various ecosystems and biological groups to test and refine theoretical models, will be crucial. Utilizing modern technologies, such as remote sensing, GIS, and ecological sensors, to gather extensive population dynamic data and applying data mining and machine learning techniques will help uncover new patterns in population dynamics. Furthermore, applying chaos and fractal theory to practical ecological conservation and resource management issues, such as endangered species protection, invasive species management, and ecosystem restoration, can enhance the scientific rigor and effectiveness of management strategies. Strengthening education on chaos and fractal theory for students in ecology and related fields will equip them with the skills to apply these theories to solve real-world problems.

Through the exploration of these research directions, we can still anticipate a greater role for chaos and fractal theory in future studies of population dynamics.