Exploring the Evolution of the Food Chain under Environmental Pollution with Mathematical Modeling and Numerical Simulation

Abstract

Environmental pollution has led to many ecological issues, including air, water, and soil contamination. Developing appropriate pollution control measures to mitigate these hazards and protect our environment is critical. In that respect, we developed a mathematical model to study the evolution of ecosystems containing food chains under environmental pollution. We integrate environmental pollution into a three-species food chain model, which includes a prey population, an intermediate predator population, and an apex predator population. The equilibrium points of the model are obtained and we analyze their stability. Numerical simulations are carried out to explore the dynamics of the model. The simulation results show that the model presents complex, chaotic, dynamic behaviors. Our study demonstrates that the interactions of individual populations in the food chain and the effects of environmental pollution can result in complex dynamics. The investigation provides insights into the evolution of the food chain in a polluted environment. Our research shows that pollution can disturb the equilibrium in nature, leading to complex and chaotic effects. Reducing environmental pollution can restore the food chain to an orderly state. Environmental pollution will harm the healthy development of each species in the ecosystem. Reducing pollution and restoring each species’ habitats are effective strategies for restoring a healthy ecosystem. Natural ecosystems are often polluted by domestic and industrial sources. The environmental protection department should allocate more resources to address domestic pollution and enhance domestic wastewater treatment methods. Industrial pollution can be reduced by encouraging companies to invest in treating wastewater and waste gases. It is also vital to prevent the establishment of highly polluting industries in environmentally sensitive environments.

1. Introduction

As an important method to investigate ecological problems, models based on differential equations are often used in ecology to analyze and understand the complex interactions between various components of the system [1,2]. Such models provide insight for understanding and predicting ecosystem behavior. In the literature, various forms of ecological models, including differential equations, networks, and statistical models, are used to study a wide range of ecological processes, such as population dynamics, community interactions, and ecosystems [1,3,4]. By studying these models, researchers can reveal relationships among species and explore species interactions. Mathematical models can also be used to simulate the effects of various scenarios, such as the evolution of environmental conditions or the introduction of new species. These models are beneficial for testing assumptions about the underlying mechanisms in ecosystems and predicting how a given system will respond to different variables or conditions. In addition to their applications in fundamental research, mathematical models are applied in practical settings such as conservation and resource management to assist in decision-making and guide management strategies.

Recently, there has been a growing concern about the increasing severity of pollution and its impact on the natural environment. Environmental and biological statistics indicate that pollutants released into the air, water, and soil can accumulate in living organisms, posing risks to the entire biological chain. This disruption of ecosystems’ natural balance reduces biodiversity and causes the loss of vital natural interactions between species. Statistical studies have shown that different kinds of pollution have consistently affected the health of wildlife populations in the oceans and on land. Accessible data indicated that pollution had a notable influence on the populations of marine mammals and turtles. Research showed that between 2000 and 2015, plastic pollution was responsible for 60% of Atlantic spotted dolphin deaths [5]. In the Greek Seas, plastic pollution caused 28.6% of Risso’s dolphin deaths, and 60% of sperm whales in the same area were found to have macroplastics in their stomachs [5,6,7]. Additionally, microplastic particles were responsible for 40% of Striped dolphin deaths [8,9]. Furthermore, plastic pollution has also affected the Green sea turtle population, with plastic products contributing to 15.9% of annual turtle deaths [10,11]. These statistics highlight the detrimental impact of plastic pollution on marine ecosystems. Statistical studies show that polluted water sources significantly impact the health and mortality of the West Tatunga antelope (Tragelaphus spekii) [12]. After conducting a six-month comprehensive assessment involving data collection and analysis, the authors concluded that most of the region’s open water sources suffer from severe pollution. The water’s acidity levels ranged between 5.98 and 7.74, resulting in a below-average Water Quality Index (WQI) value of 54%. Additionally, the turbidity levels of the water sources ranged from 9.0 to 171.5 NTU, significantly influencing the overall WQI value at each sampling point. The detrimental impacts of air pollution on wildlife health are also documented. For example, Ref. [13] examined the respiratory health and DNA methylation patterns of eastern gray squirrel (Sciurus carolinensis) populations at three locations in London and two additional rural sites in Sussex and North Wales. The surveys revealed that 28% of squirrels in urban areas and 13% in rural areas suffered from lung and airway diseases. Notably, individuals inhabiting high-pollution sites exhibited an immune response to air pollution, significantly increasing the number of alveolar macrophages in their systems. When pollution reduces one or more populations within an ecosystem, it disrupts the delicate balance of the entire ecosystem. This disruption can have disastrous consequences for the overall health and functioning of the ecosystem.

Mathematical models excel at capturing the intricate relationships among elements within complex systems. Through mathematical analysis and computer simulations, researchers and practitioners gain valuable insights into future predictions, optimized outcomes, and informed decision-making. Similarly, statistical models play a vital role across industries by extracting meaningful information from vast datasets. These models allow us to forecast system development trends and provide a reliable foundation for making optimal decisions across diverse scenarios. In recent years, mathematical and statistical models have been particularly instrumental in addressing challenges in supply chain management and the broader field of management science. For example, Ref. [14] introduced a novel mixed-integer mathematical formulation for optimizing cold chain CDT truck schedules. This formulation captures the perishability of goods throughout the entire service process and highlights the presence of temperature-controlled storage areas designated explicitly for perishable products. Ref. [15] proposed an innovative mixed-integer nonlinear mathematical model for ship scheduling problems on liner routes with perishable assets, aiming to minimize the overall route service cost, including asset decay cost. Their model effectively represents the deterioration of perishable assets on board. Due to external factors, perishable products often undergo decay during transportation, leading to significant waste. In order to tackle this challenge, Ref. [16] introduced a novel optimization model for designing intermodal freight networks that facilitate the local and long-distance transportation of perishable products. The primary objective of this model is to minimize the overall costs linked to the transportation and decay of perishable goods. Ref. [17] thoroughly examined path planning in agricultural ground robots, presenting a comprehensive review. The study by the authors extensively analyzes various path-routing methods employed by agricultural ground robots. The research delved into the application of robots in agriculture while also exploring the constraints associated with robot configuration and terrain types within the agricultural domain. Ref. [18] analyzed the attributes of multimodal transport and cold chain distribution, focusing on optimizing logistics routes to enhance the efficiency of cold chain food (CCF) distribution. The authors’ routing model considered customer satisfaction by considering factors such as the punctuality of deliveries and the quality of CCF.

One of the main interests of ecological research is to explore the interactions among different species [19,20]. Such interactions are responsible for the complex behaviors in population dynamics. Interactions in ecosystems can be considered food chains and webs and can be characterized by mathematical models based on differential equations. A food chain or food web consists of several species that are connected through dynamic interactions. As a basis for ecological interaction networks, the study of the food chain was of interest to researchers. The literature suggests that interactions between species result in complex, chaotic impacts. Researchers identified chaotic attractors in predator-prey models [21], food chain models [1,22], and interactive population models [23,24]. Lately, researchers worldwide have investigated ecological models under different circumstances using mathematical models, resulting in significant findings. For instance, Ref. [25] conducted a study on a complete cross-diffusion predator-prey model and investigated their global existence and asymptotic behavior. By developing a fully cross-diffusive predator-prey model and imposing a specific condition on the initial data, the authors established the global existence and asymptotic behavior of classical solutions for the model. Ref. [26] examined the effects of prey sanctuary and fear of predatory risk in the predator-prey model by incorporating these factors into their ecological model. They investigated the dual parameter space of the model to study the transition and chaos mechanisms and analyze the presence of periodic structures. The dynamics of a two-species predator-prey biological system with rate-dependent functional responses and predation capabilities of both juvenile and adult stages of the predator were investigated by [27]. Their findings revealed that the instability of the biological system resulting from supercritical Hopf bifurcations occurs due to the overpredation of juveniles. Moreover, the study showed that juveniles low efficiency and habitat complexity can cause fluctuations and extinctions for each species. In contrast, excessive numbers of juveniles can lead to a drastic decline in predator populations. Ref. [28] examined how enrichment and fishing activities interact in a predator-prey fishery model with Michaelis–Menten-type predatory fish harvesting. They conducted an analytical investigation into the presence and stability of homeostatic states within this system. Additionally, the authors employed numerical simulation analysis to obtain bifurcation results that confirmed their theoretical analysis findings. Ref. [29] developed a predator-prey model that integrates impulses in which predator populations undertake large-scale seasonal migrations. They conducted a global asymptotic stability analysis of this model using impulsive differential equation theory. The analysis yielded a sufficient condition for the persistent existence of the prey extinction boundary cycle solution. These findings contribute to a theoretical understanding of population ecology and management. Ref. [30] created a predator-prey model to examine the natural occurrence of predator-prey interactions. They integrated the cost of perceived fear into a prey species model, which had a Holling type II functional response. Moreover, the team introduced intraspecific competition and pregnancy delay within the predator species. The study suggests that greater levels of fear or increased rates of intraspecific competition contribute to the coexistence and survival of populations. The Allee effect is present in various ecosystems and represents a significant biological phenomenon. It highlights the relationship between population size or density and the average individual fitness of a species or population. The Allee effect has been extensively studied in the literature. For example, Ref. [31] investigated the relevance of the Allee effect predator model in an ecological setting. They explored the effect of the Allee effect on a prey-predator interaction model, specifically on the predator population. The authors revealed the complex and diverse dynamics arising from the Allee effect by studying the model’s dynamic behavior, including bistability, tristable states, and local and global bifurcations. Through numerical bifurcation analysis, Ref. [32] examined ecosystems involving the Allee effect and predator mutual interference. Their study revealed the presence of multiple hydra effects in this type of ecosystem. Furthermore, they observed a direct correlation between predator and prey density when the predator’s density-independent per capita mortality rate increased. Ref. [33] examined the impact of varying levels of soil tillage between vineyard rows on the activity density and diversity of spiders (Araneae) and springtails (Collembola), as well as their potential prey. The study was conducted across 16 vineyards in Austria, analyzing whether non-crop features in the surrounding landscape modified these effects. The authors employed a generalized linear mixed model (GLMM) and conducted data analysis to build their model. The findings highlight the influence of soil tillage intensity on spiders and springtails, as well as the interaction of semi-natural elements (SNE) in the surrounding landscape. Ref. [34] used a Generalized Additive Model (GAM) to investigate the link between the management of top predators in fisheries and their presumed prey. To examine the relationship between fish production and the stocking intensity of European catfish and non-native fish species, the researchers examined mandatory fishing logs from 38,000 recreational anglers gathered from 176 fishing locations in central Bohemia and Prague (Czech Republic) between 2005 and 2017. By utilizing the principles of foraging theory, Ref. [35] made predictions about overfishing linked to fishing locations farther away. The results suggest that habitat changes due to the warm medieval period may result in a substantial decrease in total fish catches.

In nature, pollution from industry can have many adverse effects on populations of different species. Various forms of pollution, such as chemicals and heavy metals, can be toxic to some species, especially at high concentrations, causing illness or death in individuals, which affects the overall population size. Many species rely on food sources from plants or other organisms to survive. If these food sources are contaminated with pollutants, diseases and deaths may occur. Habitat is critical for a population. Pollution can lead to the destruction of habitats vital to the survival of some species. Once a habitat is contaminated, it will be harder for a population to find food, shelter, and other resources. Over time, this may result in a massive decrease in the number of animals in the population. In recent years, water pollution has damaged aquatic ecosystems and resulted in the loss of habitat for fish and other marine species, leading to the elimination of many fish populations. Some pollutants can disrupt reproductive processes in some species, resulting in lower fertility rates and smaller population sizes. Pollution can also directly increase mortality in some species. On the one hand, a polluted environment increases the vulnerability of animals to disease. On the other hand, it is difficult for species to survive and reproduce in a contaminated environment. Pollution, especially carbon dioxide emissions, is becoming a significant contributor to climate change, causing changes in temperature and weather patterns that can affect the health of various species. In recent years, the ecological environment has faced significant challenges due to the prevalence of pollution. Consequently, controlling pollution has become an indispensable necessity for the sustainable development of humanity. As a result, extensive attention has been devoted to studying pollution and its management in the literature. For example, Ref. [36] developed a forest model based on the DPSIR framework to investigate the significant differences in forest presence and sulfur dioxide emissions (P2) across various regions. Ref. [37] designed and evaluated a multiscale air pollution modeling system with high-resolution capabilities. This system can generate detailed air pollutant concentration data in urban areas and facilitate the identification of distinct atmospheric processes in support of air quality management strategies. The research also enables the estimation of PM pollution exposure for large populations. Ref. [38] integrated the pollution load module with the established Hydro-Informatic Modeling System (HIMS) model to investigate the potentially harmful impacts of pollutants on the environment. The researchers conducted simulations on generating and transporting pollution in semiarid and sub-humid regions, aiming to quantify the pollution load. Ref. [39] conducted a study on assessing heavy metal pollution using reactive heavy metals (RHMs). Through their revised geoaccumulation and Hakanson index, they determined that heavy metal pollution exists in the farmland of the study area, but the ecological risk is considered low. Their findings highlight the need to address industrial activities, traffic emissions, and soil erosion to reduce pollution levels. Ref. [40] has tackled the issue of limited access to hydrological data by devising a valuable model for predicting and addressing metal pollution incidents in drinking water sources, even when parameters are incomplete or uncertain. The authors presented a diffusion model for cadmium pollution in drinking water sources, considering the spatial distribution of pollution incidents, the dispersion of pollution sources, and the concentration diffusion status in drinking water sources across varying hydrological periods and initial pollutant qualities. Ref. [41] proposed a fresh approach for anticipating sediment organic pollution indicators and a productive technique for evaluating and managing sediment pollution, which has posed a significant challenge. They constructed a predictive model based on machine learning for organic pollution indicators and have presented a plan for assessing organic pollution in sediments. Ref. [42] used machine learning techniques, including Bayesian optimization, to build a prediction model. The authors presented a new framework for anticipating long-term water quality that can provide technical assistance for emergency pollution management.

Environmental pollution is a primary concern that poses a significant threat to ecosystems globally. Pollution can arise from various sources, including human activities, natural disasters, and industrial processes, and can adversely impact ecosystems’ physical and biological aspects. Therefore, it is essential to study the impacts of environmental pollution on ecosystems and develop strategies to mitigate those impacts. Research on environmental pollution can provide valuable information on the causes and consequences of pollution, informing decision-makers and stakeholders of effective actions to prevent or reduce pollution. Moreover, investigating the impact of pollution on ecosystems can help identify vulnerable areas and species that require protection. Mathematical modeling is a valuable tool for research in this area as it provides a simplified representation of complex environmental systems, allowing researchers to study the effects of pollution in a controlled and systematic manner. In addition, mathematical models can be employed to simulate the effectiveness of various pollution control measures and policies, aiding policymakers in making informed decisions.

Although the use of numerical simulation to study ecological problems is widely described in the literature, numerical simulation for studying ecological development under the influence of pollution has been less mentioned. Research that employs modeling to examine the development of ecosystems under environmental pollution can shed light on the intricate interactions between population species within the food chain and pollution, revealing ecological consequences and offering a valuable foundation for pollution reduction strategies. Mathematical analysis and numerical simulation offer intuitive and comprehensive means to depict the interconnections among various ecosystem elements and assess the influence of environmental pollution on them. Hence, this paper employs modeling and computer simulation techniques to address this issue.

By studying the dynamic behavior of the food chain in the presence of pollution, this paper aims to investigate effective measures for reducing pollution, drawing on the above reasons. We integrate pollution factors into an ecological model based on differential equations to examine the impacts of pollution on the evolution of a three-species food chain. The rest of the paper is organized as follows: The model formulation will be introduced in Section 2. Then, in Section 3, we obtain the equilibrium points of the model and analyze their stability. Numerical simulation is presented in Section 4. We conclude the paper with a discussion in Section 5.

2. Model Formulation

The interactions between predators and prey in ecosystems have captured the attention of researchers. Unraveling the underlying mechanisms driving predator-prey systems enhances our understanding and offers valuable insights into conserving a sustainable ecosystem. Extensive literature has explored prey-predator and food chain models, employing various functional responses to characterize population interactions [43,44,45,46,47].

For instance, Type I functional responses assume a linear increase in intake rates with food density. However, these models often overlook the time predators spend processing food and the subsequent decline in their predation ability. In reality, food processing requires a certain amount of time and affects the predator’s capture efficiency. To address this, Type II functional responses were developed, incorporating a rectangular hyperbola that considers the predator’s food processing capabilities [44]. Moreover, Type III functional responses assume saturation at high prey densities [46], while Holling Type IV functional responses introduce a swarming effect resulting in dome-shaped functional responses [43,45,47]. Each type of functional response offers distinct perspectives on population interactions. In this work, we employed Type II and Type IV functional responses. Then, by integrating pollution into ecosystem models, we investigate the intricate interplay between pollution and population dynamics, aiming to understand the long-term consequences of these interactions on ecosystem and population health.

In this section, we develop an ecological model to investigate the evolution of a three-species food chain in a polluted location. We consider an inhabited habitat to be composed of three species that interact with each other. The 3D ecosystem we study consists of a prey population (x), an intermediate predator population (y), and an apex predator population (z).

Our model depicts the ecosystem, as shown in Figure 1, where the interplay between prey, intermediate predators, and superior predators is represented by the arrows. Notably, the entire ecosystem is subject to the pervasive effects of environmental pollution.

In this food chain, the prey population of size x at the lowest end is the only food source for the intermediate predatory population of size y. Intermediate predators of size y survive by capturing prey while, at the same time, serving as a food source for apex predator species (population density z). We assume that the prey population follows logistical growth. Suppose r is the intrinsic growth rate of the prey species, and K is its carrying capacity. Here we use a modified Holling-IV functional response to model the interaction between prey and intermediate predators. In this model, we use the Holling-II functional response to characterize the interaction between the intermediate predator species and the top predator species. Here, $c_x$ is the rate at which the intermediate predator species attack the prey species, $c_y$ is the net gain of the intermediate predator species over the prey species, $c_z$ is the rate at which the top predator species attack the intermediate predator species, and $c_u$ is the net gain of the top predator species over the intermediate predator species. In our model, $d_y$ and $d_z$ are the natural death rates of the intermediate predator species and the apex predator species. We assume that these three species live in chronically contaminated habitats. Environmental pollution can result in animal mortality. The pollution-related death rates for prey, intermediate predators, and top predators are, respectively, $p_x$, $p_y$ and $p_z$. Based on the above discussion, we derive the following model:

$$\begin{array}{c}\dot{x}=rx\left(1-\frac{x}{K}\right)-\frac{c_{x}xy}{a{x}^2+s}-p_{x}x,\\ \dot{y}=\frac{c_{y}xy}{a{x}^2+s}·\frac{1+cy}{1+cy+wz}-\frac{c_{z}yz}{y+t}-d_{y}y-p_{y}y,\\ \dot{z}=\frac{c_{u}yz}{y+t}-d_{z}z-p_{z}z.\end{array}$$

(1)

In model (1), the initial conditions are $x\left(0\right)\ge 0$, $y\left(0\right)\ge 0$ and $z\left(0\right)\ge 0$.

3. Dynamics of the Model

In this section, we consider the equilibrium points of the model and their stability. In order to obtain the equilibrium of the model, we let $\dot{x}=0$, $\dot{y}=0$, and $\dot{z}=$0. Then, we can obtain the equilibrium points of the model by solving the system of linear equations. Here, we are particularly interested in the stability of the biologically feasible equilibrium points.

System (1) admits the trivial equilibrium $E_1=\left(\mathrm{0,0},0\right)$ and the predator-free equilibrium $E_2=\left(\frac{K(r-p_x)}{r},\mathrm{0,0}\right)$. Now we consider their stability. The Jacobian matrix at the equilibrium point $E_1$ is

$$J(E_1)=\left(\begin{array}{ccc}r-p_x& 0& 0\\ 0& -d_y-p_y& 0\\ 0& 0& -d_z-p_z\end{array}\right)$$

We notice that the three eigenvalues of $J\left(E_1\right)$ are ${\lambda }_1=r-p_x$, ${\lambda }_2=-d_y-p_y$ and ${\lambda }_3=-d_z-p_z.$ Obviously, ${\lambda }_2<0$ and ${\lambda }_3<0$. Thus, if $r-p_x<0$, i.e., $p_x>r$, the trivial equilibrium $E_1=\left(\mathrm{0,0},0\right)$ is asymptotically stable. We notice that $p_x$ is the pollution related death rate of the prey species, indicating that severe environmental pollution leads to the extinction of the prey population. It is easy to see that $E_2$ is biologically feasible if and only if $r\ge p_x$, i.e., $\frac{K(r-p_x)}{r}\ge 0$. We then consider the Jacobian matrix of the system at the equilibrium point $E_2$ given by:

$$J(E_2)=\left(\begin{array}{ccc}{p}_x-r& \frac{c_{x}K(p_x-r)}{r\left(\frac{a{K}^2{(p_x-r)}^2}{r^2}+s\right)}& 0\\ 0& \frac{c_{y}K\left({r-p}_x\right)}{r\left(\frac{a{K}^2{(r-p_x)}^2}{r^2}+s\right)}-d_y-p_y& 0\\ 0& 0& -d_z-p_z\end{array}\right)$$

The three eigenvalues of $J\left(E_2\right)$ are ${\lambda }_1=p_x-r$, ${\lambda }_2=-\frac{c_{y}K(p_x-r)}{r\left(\frac{a{K}^2{(p_x-r)}^2}{r^2}+s\right)}-d_y-p_y$ and ${\lambda }_3=-d_z-p_z.$ It is easy to see that for the feasible equilibrium point $E_2$, ${\lambda }_1<$0, and ${\lambda }_3=-d_z-p_z<0.$ Thus, when $\frac{c_{y}K\left({r-p}_x\right)}{r\left(\frac{a{K}^2{(r-p_x)}^2}{r^2}+s\right)}<d_y+p_y$, equilibrium point $E_2$ is asymptotically stable.

In this article, we study the dynamics of the model using the following set of system parameters. r = 1.5, K = 30, $c_x$ = 0.9, $c_y$ = 0.8, $c_z$ = 0.7, $c_u$ = 0.6, w = 1, s = 10, t = 51, c = 4, a = 0.009, $p_x$ = 0.002, $p_y$ = 0.0001, $p_z$ = 0.0001, $d_y$ = 0.073, and $d_z$ = 0.082.

In this case, the system has equilibrium points:

$$\begin{gathered} E_1=\left(\mathrm{0,0},0\right) \\ {E}_2=\left(\mathrm{29.96,0},0\right) \\ {E}_3=\left(0.91,16.15,0\right) \end{gathered}$$

and

$$E_4=\left(18.97,8.08,38.64\right)$$

The eigenvalues of $E_1$ are 1.50,$-0.07$, and $-0.08$, indicating that $E_1$ is unstable. The eigenvalues of $E_2$ are $-1.50$, 1.25, and $-0.08$, indicating that $E_1$ is unstable. The eigenvalues of $E_3$ are $-0.02+0.32i$, $-0.02-0.32i$, $\mathrm{a}\mathrm{n}\mathrm{d} 0.06$, indicating that $E_3$ is unstable. The eigenvalues of $E_4$ are $-0.52$, 0.09 + 0.24 i, and $0.09-0.24i$, indicating that $E_4$ is unstable.

4. Numerical Simulations

Numerical simulations make it possible to study the behavior of a model under different conditions and with different variables. In this section, we use the numerical simulation method to investigate the dynamic behavior of the model (1). Our simulation results show that model (1) displays complicated dynamical behaviors. As shown in Figure 2, choosing r = 1.5, K = 30, $c_x$ = 0.9, $c_y$ = 0.8, $c_z$ = 0.7, $c_u$ = 0.6, w = 1, s = 10, t = 51, c = 4, a = 0.009, $p_x$ = 0.002, $p_y$ = 0.0001, $p_z$ = 0.0001, $d_y$ = 0.073, and $d_z$ = 0.082, the behavior of the system is unpredictable, i.e., it displays a chaotic attractor. A straightforward and intuitive approach to ascertaining the presence of chaos in a system is to examine the phase portrait of its attractors. The numerical simulation of a chaotic attractor displays a non-periodic nature and complex, seemingly random behavior. Additionally, the chaotic attractor’s trajectory remains bound within a specific region of the phase space. Upon observing Figure 2, it becomes evident that the depicted attractor is a chaotic system.

A chaotic system is a type of dynamic system that is highly responsive to initial conditions. Thus, its behavior is unpredictable [48]. To illustrate the system’s sensitivity to the initial condition, we conducted numerical simulations to reveal that the system displays distinct motion trajectories depending on the initial values used. This phenomenon is highlighted in Figure 3. The trajectory of system (1) with an initial value of (4, 10, 3.2) is depicted by the blue curve in Figure 3. The red curve in Figure 3 represents the trajectory of system (1) with an initial value of (0.6, 3.1, 5). It is evident from the simulation results that the dynamic behavior of the same system varies significantly under different initial values. In a chaotic system, the behavior of its trajectories is highly sensitive to minor differences in initial conditions. This sensitivity, known as “sensitive dependence on initial conditions” in chaos theory, can amplify those differences over time, resulting in dramatically different dynamical outcomes. Consequently, attempting long-term predictions for chaotic systems becomes exceedingly challenging. In the context of an ecosystem, the presence of chaos further complicates the task of predicting and controlling the system. Therefore, the intricate nature of chaos in ecosystems warrants a thorough investigation, recognizing the significance of studying this phenomenon in depth.

The bifurcation diagram is a significant tool to show the parameter values at which the unpredictable behavior of the system occurs, as it showcases the evolution of the system’s dynamic behavior with respect to a specific parameter of the system. The bifurcation representation exhibits period doubling, wherein an N-point attractor transitions to a 2N-point attractor, signifying the changes in the system due to variations in the control parameters. In this regard, we utilize bifurcation diagrams to demonstrate the progressive period-doubling evolution that occurs as the system parameter increases.

We are interested in the effects of the intrinsic growth rate of the prey species r on the dynamics of the ecological system. The intrinsic growth rate is a fundamental concept in population biology that plays a crucial role in modeling and predicting changes in population size over time. Biologically speaking, it represents the maximum potential rate at which a population can increase under ideal conditions, such as the absence of limiting factors like limited resources, predation, disease, or environmental constraints. The intrinsic growth rate determines the future development of a population within an ecosystem and is an essential parameter to take into account. Its value not only affects the survival of the population but also influences the development and evolution of other related biological populations within the ecosystem. Therefore, it is essential to evaluate the effect of r on the evolution of the ecological system. As shown in the bifurcation diagram in Figure 4, system (1) displays complicated dynamical behaviors over a wide range of the parameter r. We noticed that for r ∈ [1.50, 1.75] and r ∈ [2.10, 2.20], system (1) has unpredictable behaviors, i.e., it displays chaotic behaviors.

Our focus now turns to examining how the carrying capacity of the prey species, K, affects the evolutionary dynamics of the ecological system. Carrying capacity is a fundamental concept in ecology that refers to the maximum population size an environment or ecosystem can sustainably support based on available resources such as food, water, and shelter. The bifurcation diagram in Figure 5 illustrates the complex dynamic behaviors of system (1) across a wide range of parameter K. Notably, we observed that system (1) displays chaotic behavior for K values within the range of [27.5, 33.5]. That is to say, in this interval, the behavior of the system is unpredictable.

As part of future research, we will develop mathematical models to explore how to design optimal pollution control approaches to promote healthy ecosystem development. We will also establish appropriate control approaches to make the food chain less chaotic or even deterministic. The research is expected to offer practical recommendations for optimal pollution control measures and environmental protection.

Likewise, system (1) exhibits intricate dynamical patterns spanning a diverse range of system parameter s. The bifurcation diagram, Figure 6, depicts the complicated dynamic behaviors of system (1) across a broad parameter spectrum of s.

Next, we examine the impact of parameter w on the dynamic behavior of the system. To this end, we have plotted the simulation results for system (1) on the y-z plane for different values of w, as illustrated in Figure 7. All the other parameter values utilized in Figure 7 are the same as those used in Figure 2. Simulation results indicate that the system behaves chaotically for a wide range of values of w, i.e., the system displays chaotic trajectories for w between 1.0 and 5.0. Within this parameter interval, the behavior of the system is unpredictable.

Now, we demonstrate the presence of chaos. The existence of chaos can be established through various methods. Scholars have formulated diverse approaches to confirm the presence of chaos. For instance, Ref. [49] employed a 0–1 test to showcase it. The 0–1 test approach was further reviewed by [50]. Ref. [51] developed a chaos testing method based on topology. Ref. [52] calculated the largest Lyapunov exponent and conducted tests to detect the occurrence of chaotic dynamics.

In this work, we use the maximum Lyapunov exponent method to confirm the presence of chaos in this system. In the presence of a positive maximum Lyapunov exponent, the system’s orbit becomes unstable and exhibits chaos. Regardless of their initial separation, nearby points along the orbit diverge by arbitrary distances. As the Lyapunov exponent increases in magnitude, the system becomes progressively more unstable. As shown in Figure 8, as the parameter $c_u$ changes, the system will exhibit different dynamic behaviors. Figure 8 shows that when the value of $c_u$ is within a certain range, the system exhibits chaotic behavior.

5. Discussion

Pollution due to industrial production and consumption has continued to harm the natural environment. In order to study the impact of pollution on the ecological environment and to explore strategies to eliminate such impacts, we established a mathematical model of a food chain containing a prey, an intermediate predator, and a top predator species.

We obtained the model’s equilibrium outcomes and their stability using mathematical analysis. To investigate the dynamic behavior of the model, we provide numerical simulations. Our simulation results show that the model presents complex chaotic attractors, i.e., the behavior of the system is unpredictable. Our research shows that the interaction between various populations of food chains and long-term pollution can lead to unpredictable and chaotic phenomena in food chains.

To illustrate the dynamic behavior of the model, a thorough study was conducted utilizing numerical simulation. Our results show that the system’s dynamic behavior varies greatly under different initial values, thus proving the presence of unpredictable behaviors in the system. A more in-depth analysis was conducted by investigating the bifurcation of the system, which revealed that it exhibits complex and diversified dynamic behaviors as the value of the parameters changes. It was found that the system can exhibit unpredictable behavior over a wide range of parameter values, indicating that ecosystem behavior can be unpredictable and complex in the presence of pollution. The findings of this study have far-reaching implications for the management and preservation of ecosystems. The discovery of unpredictable behaviors within ecosystems highlights the need for a more comprehensive approach to environmental management and conservation that considers the unpredictability of ecosystem dynamics. Managing chaos within the ecosystem can be a challenging and complex task, requiring a comprehensive understanding of the underlying mechanisms that govern ecosystem behavior. In light of these findings, it is clear that effective environmental management and conservation efforts must take into account the potential for unpredictable behavior within ecosystems. By better understanding the complex and dynamic nature of ecosystems, we can develop more effective strategies for managing and preserving these vital resources and ensure a sustainable future for generations to come.

Pollution in ecological systems severely affects the environment and the organisms living within it. When pollutants enter an ecosystem, they can affect its normal functioning and disrupt its balance, causing unpredictable behaviors, as envisioned by our model. Since pollution may negatively impact the health and survival of different species in the food chain, it is essential to develop appropriate strategies to control pollutant emissions to improve the overall health and productivity of the ecosystem. Pollution can also indirectly impact humans, as many of the resources we rely on, such as clean air and water, are provided by healthy ecosystems. By eliminating pollution, we can help protect and preserve the ecological systems vital to the health and well-being of our planet and its inhabitants.

Enhancing efforts to mitigate environmental pollution can have a positive impact on ecosystem health. It is essential to reduce activities that contribute to significant environmental degradation. Simultaneously, encouraging enterprises to improve their capacity to reduce emissions can effectively contribute to the restoration of polluted natural environments. Given the factors of cost and efficiency, prioritizing measures such as enhancing sewage treatment capacity and waste gas treatment in enterprises becomes a crucial step to addressing these challenges.

Gaining a deep understanding of ecosystem dynamics is vital for effective management and decision-making processes. This work combines theoretical investigation and numerical analysis to deliver managerial insights into ecosystem management. To restore ecosystems damaged by environmental pollution, it is crucial to enhance ecosystem resilience and improve their structural integrity to withstand disturbances. Measures that can be implemented include conserving biodiversity, safeguarding critical habitats, and reducing pollution sources that can weaken ecosystem resilience. Ecosystems are subject to uncertainties like climate change, invasive species, and human impacts. Policymakers can foster collaboration among sectors such as agriculture, fisheries, and urban development to develop and implement measures for ecosystem improvement. Integrating diverse perspectives and knowledge, encompassing biologists, mathematicians, ecologists, and environmentalists, can lead to comprehensive and effective management strategies. Successful ecosystem management necessitates collaboration and engagement with various stakeholders, including local communities, policymakers, and industry representatives. These managerial insights serve as valuable guidance for policymakers. By fostering resilience, mitigating pollution, and optimizing ecosystem structure, ecological management can be implemented with a long-term perspective, ultimately creating a healthy and sustainable ecological environment for the well-being of future generations.

6. Conclusions

This article presents a mathematical model that examines the ecological impact of environmental pollution. Our approach involves integrating environmental pollution into a three-species food chain model to investigate environmental protection strategies. The model’s equilibrium points are derived, and the stability of these points is analyzed. To study the dynamic behavior of the model, numerical simulations are conducted using computer simulations. Our findings reveal that the model exhibits intricate and chaotic dynamics. We compute the maximum Lyapunov exponent, plot it against the system parameter, and confirm the system’s chaotic behavior. Overall, this research sheds light on improving the natural environment by mitigating pollution from both industrial and residential sources.

To address the persistent pollution challenge and protect the environment, our future research endeavors will focus on designing effective strategies to mitigate pollution and facilitate ecosystem self-recovery. Firstly, we will construct models by leveraging data from long-term monitoring programs to track ecosystems affected by pollution. These models will enable us to predict ecological development trends by incorporating parameters such as water, air, and soil quality, as well as critical species’ health, abundance, and pollution levels. Through this modeling approach, we aim to identify optimal strategies for ecosystem restoration, including thresholds that trigger self-recovery mechanisms within the ecosystems.

Secondly, we will conduct comprehensive assessments and employ modeling techniques based on environmental risk factors. This approach will aid in quantifying the potential ecological risks associated with pollution. For instance, we will utilize reaction-diffusion equations to model and forecast the dispersion of pollutants, allowing us to evaluate their potential for the sustainable development of ecosystems. The outcomes of these research efforts will serve as valuable references for decision-makers, providing insights into effective pollution management strategies.

Furthermore, our models will incorporate socioeconomic and policy measures to establish a holistic approach. We aim to provide valuable insights into promoting sustainable management practices by integrating these aspects into our research. This interdisciplinary approach will enable us to consider pollution mitigation’s social, economic, and policy dimensions, ensuring that our findings contribute to developing comprehensive and effective strategies for protecting the environment and fostering sustainable practices.