The chaotic dynamics and behavior of prey-predator interaction: Insights into complex ecological systems and epidemic modeling

Abstract

This study and the literature have shown that the emergence of chaotic behavior has been attributed mostly to predator-prey and competitive dynamics. This is also observed in pandemics, as well as in cancer models, where deterministic chaos or chaotic dynamics can lead to complex oscillations and nonlinear interactions between cell populations. It is important to note that COVID-19 displays the key characteristics of a chaotic system and is one of the deadliest pandemics in recent history.

Key words
chaotic dynamics; prey-predator interaction; time series; phase portrait diagrams; bifurcation diagrams; Lyapunov exponents

INTRODUCTION

In a comprehensive report, Borah et al. 2022 delves deeply into the complex and often chaotic behavior of pandemics in low-to-middle-income countries. Notable examples include the Plague epidemic in Bombay, India; chaotic epidemic crisis management in Mexico; the Ebola Virus epidemic in Guinea, Liberia, and Sierra Leone; and Dengue in Pakistan. Furthermore, the authors observed that specific cancer models demonstrate deterministic chaos or chaotic dynamics, resulting in intricate oscillations. As posited by Jones & Stigul 2021, the current pandemic exhibits the defining characteristics of chaos and has become one of the deadliest in recent history, due in part to its exponential growth. The Coronavirus disease, which emerged in 2019, has a high mortality rate Borah et al. 2022.

The emergence of chaotic behavior can be attributed primarily to predator-prey and competition dynamics, as postulated by Diekmann & Kretzschmar 1991. Nonlinear interactions between cell populations, as observed in cancer models and Parkinson’s disease, also contribute to chaotic behavior. This is supported by the findings of Gross et al. 2004, Stiefs et al. 2009, who have also observed chaotic long-term dynamics in models where the prey is infected.

Hesketh et al. 2020 proposed that ecological models should consider the significance of human-environment interactions in comprehending and modifying human behavior. These models have been integrated at various levels of influence on behavior, including the policy, community, organizational, social, and individual levels. However, fewer studies have explored correlates at the social, physical, and policy levels. In contrast to the behavior of oscillations, the effect of chaos on the stability of ecological models has been a topic of debate for a considerable period of time Stiefs et al. 2009.

In their study, Debbouche et al. (2022) present a conceptual model of the ongoing pandemic that demonstrates chaotic behavior. The system dynamics are investigated via a range of analytical techniques, including bifurcation diagrams, Lyapunov exponents, time series, and phase portraits. Furthermore, deterministic chaos is a common behavior in continuous-time dynamical systems of differential equations with nonlinear terms. These systems exhibit aperiodicity, ergodicity, and sensitivity to initial conditions, as argued by Allen et al. (1993). This underscores the significance of mathematical models as a potential instrument for formulating strategies to anticipate and respond to an impending epidemic or pandemic, as well as to address a disease outbreak in real time. Brauer et al. 2008, Mangiarotti et al. (2020) have highlighted the pivotal role of mathematical modeling in the field of epidemiology. They developed a number of schemes for mathematically modeling infectious epidemics, with compartment models being the most commonly used. These models are divided into classes and determine interactions between them using mathematical formulations. In general, it is challenging to construct a comprehensive model of an epidemic disease within the confines of these formalisms, largely due to the inherent unpredictability of the disease and its rapid evolution in terms of its shape and behavior. The research conducted by Volos et al. (2017) indicated that the propagation of a pandemic is highly sensitive to slight variations in the initial conditions of physical factors, including the number of asymptomatic carriers, infected cases, and undetected cases. In the context of an ongoing pandemic, chaos theory can be a valuable tool for defining, modeling, and analyzing the dynamics of a system. This approach involves taking into account relevant variables, equations that govern these variables, parameter values, constraints of the model, and reformulation of the equations based on existing observations. This methodology has been previously outlined by Mangiarotti et al. (2020).

Roy et al. (2024) developed a predator–prey model that distinguished between immature and mature prey, emphasizing group defense strategies within the mature prey. The system displayed a multitude of bifurcations, including Hopf, saddle-node, transcritical, generalized Hopf, cusp, and Bogdanov–Taken bifurcations. The results demonstrated that elevated levels of fear resulted in a reduction in mature prey density, which could potentially lead to the extinction of the prey population beyond a certain threshold. An increase in maturation rates resulted in the coexistence of immature and mature prey populations, as well as an elevated predator density. The model exhibited a range of intriguing and diverse dynamical phenomena, including various forms of bistability across distinct bi-parameter planes. Moreover, Din 2018 examined the qualitative behavior of a modified prey–predator model, incorporating density-dependent per capita growth rates and a Holling type II functional response. The study examined the positivity of solutions, boundedness, and local asymptotic stability of equilibria for continuous and discrete-time prey–predator systems. Additionally, the study presented two novel generalized hybrid feedback control methods for chaos control under the influence of period-doubling and Neimark–Sacker bifurcations. Furthermore, In Gilpin & Feldman (2017), an eco-evolutionary predator-prey model was discussed in which the population growth of the prey was influenced by a density-dependent fitness landscape. The model demonstrated the potential for chaotic dynamics even in the absence of external environmental variation, due to the alternating influence of stabilizing and disruptive selection on the fitness landscape. The authors drew parallels between the fitness function and free energy in statistical mechanics, employing the theory of first-order phase transitions to elucidate the rapid cycling observed in the chaotic dynamics. The model proposed that chaotic dynamics driven by evolution could provide stability in ecosystems and create opportunities for speciation during disruptive selection periods. This could serve as an observational indicator. Nevertheless, Alexi et al. (2023) presented a study examining the impact of pandemics on ecosystems. The researchers developed an eco-epidemiological model that considers species interactions and multiple pathogen strains. The study revealed that pandemics result in substantial ecological alterations and underscored the necessity for efficacious management of such occurrences. Furthermore, the research deepened our understanding of the relationship between ecology and epidemiology, thereby aiding in the development of strategies to mitigate the ecological impacts of multi-species pandemics. Ghosh et al. (2024) investigated the effect of attractive and repulsive higher-order interactions in globally and non-locally coupled prey–predator Rosenzweig–MacArthur systems. These interactions led to the emergence of complex spatiotemporal chimera states, which were otherwise unobserved in the model system with only pairwise interactions.

The study is based on the classical Lotka-Volterra model, and the results demonstrate that the interaction between the predator and the prey species affects the reproduction rate of the healthy prey species. This work illustrates the potential utility of mathematical models in enabling the formulation of strategies for anticipating and responding to an impending epidemic or pandemic, as well as for managing a disease outbreak in real time.

MATERIALS AND METHODS

Mathematical model

Eilersen et al. (2020) analyzed a dynamical model of an ecosystem consisting of one predator and three prey species, one of which carries a disease while the others are immune. The authors assumed that healthy and infected animals are equally difficult to catch and equally nutritious for the predator. They found that the system exhibits chaotic behavior for a wide range of parameters. Gakkhar & Naji (2003) pointed out that the emergence of chaotic behavior is mainly due to predator-prey and competition dynamics.

The authors proposed the following model of the classical Lotka-Volterra equations Eilersen et al. (2020), which is represented by the system of equations (1).

$$\begin{array}{rl}\frac{d{x}_s}{dt}& =a{x}_s-R{x}_s{x}_i-a{x}_{s}z\\ \frac{d{x}_i}{dt}& =R{x}_s{x}_i-a{x}_{i}z-w\\ \frac{dy}{dt}& =by-acyz\\ \frac{dz}{dt}& =d(x_s+x_i+y)z-dz\end{array}$$ (1)

The given equation describes the populations of x_s and x_i as healthy and infected populations of the susceptible prey species, respectively. The population of the immune prey species is represented by y, and z represents the population of predators.

a= the reproduction rates of the healthy prey.

R= the disease basic reproduction number or disease infectivity. In real epidemics R varies from around 1.

b= the immune growth rates.

c= the rate of the prey of species x and y eaten by predator.

d= the rate at that predators starve in the absence of prey.

The Jacobian matrix for the system of equations (1) is presented in the following equation:

$$A=\left[\begin{array}{cccc}a-R{x}_i-az& -R{x}_s& 0& a{x}_s\\ R{x}_i& R{x}_s-az-1& 0& -a{x}_i\\ 0& 0& b-acz& -acy\\ dz& dz& dz& d(x_s+x_i+y)-d\end{array}\right]$$ (2)

Numerical methods

The Scilab Software 2023 was used to obtain an approximate solution for system (1), which is given by the following recurrence formula:

$$xp=x-inv(JF(x))\ast (F(x))$$ (3)

The solution, xp, is an approximation. F(x) represents the matrix function of the system functions, and JF(x) represents the Jacobian of the matrix function.

The parameters of the system (Eq. 1) have been proposed by Eilersen et al. (2020).

a = 7/400, b = 0.0208, c = 2, d = 0.3098, R = 1.025 and R=1.5.

The parameters a, d, and R were varied between 0 and 2, and 1 and 5, respectively. The most relevant results were obtained for these parameter variations. Other ranges and parameters were tested, but the results were not satisfactory. The initial conditions for x_s, x_i, y, and z were 2.5, 0.001, 0.2, and 0.4, respectively. The time variation ranged from 0 to 450 days with a step of 0.04.

In this work, time series and phase portraits are presented in both 2D and 3D. The phase portraits display several stabilizations and/or critical points, each of which was analyzed to determine its stability. The eigenvalues of each point were calculated using the Jacobian matrix (2), and the Lyapunov exponents were also calculated to determine the behavior of the curves.

RESULTS

A numerical simulation of system (1) was conducted using the previously specified parameter values. The numerical values of the parameters and the initial values of the variables mentioned in the previous section were used to obtain the graphical results. The results of varying time up to 40 days with a step of 0.1 are shown in the following time series figures. Figure 1 illustrates the variables Xs, Xi, Y, and Z as a function of time, while varying the parameter a, which represents the reproduction rates of the prey’s healthy population.

Figure 1
Time series showing the variation in reproductive rates of healthy prey (a) varies.

Figure 1 illustrates the time series of healthy prey, depicting the interaction between prey and predator at the initial instant for parameter a (approximately zero) in a dotted line. While the health of the prey (represented by Xs) exhibits a sharp rise and fall, the predator (represented by Z) demonstrates a relatively minor increase. Following an interval of 11 days, the value of Xs begins to increase once more, although the growth curve displays oscillatory characteristics and exhibits damped behavior. Similarly, the same damped oscillatory behavior is observed in Z. Additionally, Eilersen et al. (2020) observed a dampening of oscillations and a transient nature of chaotic dynamics in their results. However, after 30 days, the Xs and Z variables, along with other parameters, exhibit chaotic characteristics, while the other variables (Xi and Y) show no effect. For parameter a (approximately 2), illustrated with dashed lines, the variable Xs undergoes a decline after approximately five days. Nevertheless, this has no further impact on the predator. Meanwhile, the predator Z exhibits growth until approximately seven days, after which it declines due to a lack of food. The variable Xi, which represents infected prey, exhibits slight growth and decline without any significant impact on the predator. Therefore, the variable Y, which represents immune prey, does not exert a discernible influence.

Figure 2 demonstrates the time series in the absence of prey. It illustrates the variables Xs, Xi, Y, and Z as a function of time when the parameter d is varied from 0 to 2 and the time is increased up to 10 days. The variables Xs and Z, represented by dotted lines when d is approximately zero, exhibit an equilibrium interaction up to three days, after which they decay almost simultaneously. However, the Xi variable demonstrates a slight increase until approximately four days, followed by a decline, while the Y variable exerts no influence. For values of d proximate to 2, as illustrated by the dashed lines, the Xs variable exhibits a decline analogous to that observed in the dotted lines. However, the Xi variable demonstrates an initial increase until approximately four days, after which it declines. Consequently, the Z variable increases, indicating that the predator is feeding on infected prey. For extended periods, Xs, Xi, Y, and Z attain equilibrium, except for d values approaching 2.

Figure 2
Shows a time series in which the starvation rate of the predator in the absence of prey (d) varies.

Figure 3 displays the time series of the disease infectivity, wherein the R parameter (which varies between 1 and 5) exhibits a rapid decline within a few days. However, the variable Xi exerts a considerable influence during the initial stages, exhibiting growth and subsequent decay in a damped oscillatory manner. Concurrently, Z demonstrates a gradual growth and decay as R increases, exhibiting an increasingly oscillatory pattern. This suggests that Z may be influenced by Xs and Xi. Over extended periods, all variables tend to decline and reach a state of equilibrium.

Figure 3
Illustrates the temporal variation in disease infectivity (R).

The parameter a was varied by setting R at 1.0 and 1.5, as suggested by Eilersen et al. (2020). In each step of the process with a value of 0.2, the time period was varied from 0 to 450 days. The data supporting this is shown in Figures 4, 5, 6, and 7. As a consequence of this variation, the figures exhibited disparate points of stability and non-stability, as illustrated in Table I. At each critical point, we ascertained whether convergence was occurring by calculating the eigenvalues using the Jacobian matrix (2) and identifying the characteristics of each point. Figures 1-3 (time series), which were also analyzed by Borah et al. (2022), demonstrate the absence of a notable influence of Xi and Y variables on Z.

Figure 4
(a) shows the phase portraits of healthy prey (Xs), infected prey (Xi), and predator (Z). (b) presents the projections in the Xs-Z planes and (c) projections in plane Xs-Xi, varying a when R=1.5.

Figure 5
(a) shows the phase portraits of three distinct entities: health prey (Xs), immune prey (Y), and predator (Z). (b) presents a bifurcation diagram of the Xs-Y projection, wherein the variable a is varied for R=1.5.

Figure 6
Shows the phase portraits of healthy prey (Xs), infected prey (Xi), and predators (Z) in (a), and the projection in the Xs-Z plane with varying a for R=1.025 in (b).

Figure 7
(a) shows the phase portraits of healthy prey (Xs), immune prey (Y), and predator (Z). In (b), the projection in the Xs-Y plane displays a bifurcation diagram. The parameter a was varied for R=1.025.

Table I
Illustrates the critical points and their corresponding eigenvalues as depicted in the phase portrait diagrams.

	Variables Xs-Xi-Z, varying a for R=1.5, Figure 4a
Point		Eigenvalues: λ 1 , : λ2: λ3: λ4	Lyapunov exponents	Point characteristic
1.1;0.01;0.2;1.5, for a=0.8 converge		-2.132, -0.210+0.619i, -0.210 - 0.619i, -0.508		Stable point
1.6;0.2;0.2;0.75, for a=1.0 No converge		0.174+1.010i, 0.174-1.01i, -1.241, +0.512	0.0248310 0.0248310 -0.669730	Unstable point
3.5;0.2;0.2;1.52, for a=1.7 no converge		-4.911, -0.133+1.805i, -0.133-1.805i, +1.597		Saddle point
	Variables Xs-Y-Z, varying a for R=1.5, Figure 5a
0.01; 0.5; 0.2; 2.202, for a=0.5 converge		-0.166 -1.367 -2.069 -1.918		Stable point
1.442; 0.1 ;0.3; 1.242, for a=0.8 no converge. At the beginning of the branch		-1.698, -0.123+0.866i, -0.123-0.866i, 0.251		Unstable point
2.522 0.2 0.5 1.793, for a=1.6 no converge		-0.739+1.624i, -0.739- 1.624i, +0.366, -5.476		Saddle point
	Variables Xs-Xi-Z, varying a for R=1.025, Figure 6a
1.057;0.1;0.2;2.447, for a=0.7 Converge		-0.642+ 0.380i, -0.642-0.380i, -1.411, -3.1580		Stable point
3.21;0.2;0.2;2.771, for a=1.7 converge		-0.671 -2.11 + 1.295i -2.105 - 1.295i -9.159		Stable point
	Variables Xs-Y-Z, varying a for R=1.025, Figure 7a
0.361;0.2;0.1;1.079, for a=0.4 converge		-0.2485+0.196i, -0.248-0.196i, -0.948, -0.613		Stable point
2.572;0.2;0.1;1.597, for a=1.4 not converge		-0.527+1.319i, -0.527-1.319i, -0.033, -4.238		Saddle point

Figure 4 presents the phase portraits of the prey and the projections in the planes. Figure 4a in the phase portraits diagram illustrates the interaction of the variables Xs, Xi, and Z in three-dimensional space, with the parameter a varying and R=1.5 held constant. This reveals the existence of three points of stability and non-stability (see Table I), namely a stable point with a chaotic attractor characteristic, an unstable point with a chaotic characteristic, and a saddle point. Additionally, the interaction of the Xs-Z and Xs-Y variables in two dimensions is illustrated. Figure 4b represents the stable point with chaotic attractor characteristics, the unstable point with chaotic characteristics, and the other with a saddle point. Figure 4c displays the interaction between Xs and Xi, which also demonstrates the presence of a stable point with a chaos attractor characteristic and a saddle point. In their study, Borah et al. (2022) observed that the gradual evolution of chaos is distinctly visible through a period-doubling pathway.

Furthermore, Figure 5 shows the phase portraits of three distinct entities and a bifurcation diagram, where Figure 5a exhibits the iterative behavior of the Xs-, Y-, and Z-variables, wherein the parameter a is varied when R=1.5 is set. Three critical points are observed, each exhibiting distinct characteristics. One is a stable point with a chaotic attractor, another displays a bifurcation diagram, and the third is a saddle point. Figure 5b provides a more detailed representation of the bifurcation diagram on the projection Xs-Y. As stated by Elnawawy et al. (2021), a bifurcation diagram reveals periodic windows and examines the robustness of chaotic behavior in response to parameter variations. Consequently, chaotic dynamics represent an effective approach for regulating gene expression, which is essential for complex cellular processes involved in maintaining cellular homeostasis (Uthamacumaran 2021).

Next, Figure 6 illustrates the phase portraits of three distinct entities and a bifurcation diagram. The interaction of the Xs-Xi-Z variables varying the parameter a, with R=1.25 is set, it is shown in Figure 6a, wherein the parameter a is varied. This figure illustrates two critical points: a stable point exhibiting attracting chaos behavior and a saddle point. Additionally, Figure 6b displays the projection in the Xs-Z, illustrating a stable point and a saddle point. Elnawawy et al. (2021) propose that chaotic attractors may be indicative of therapy resistance, tumor recurrence, and cancer stemness. Despite mathematical cancer models indicating that the emergence of chaotic attractors may indicate aggressive (adaptive) cancer states, their detection from empirical datasets remains underexplored.

Figure 7 presents the phase portraits and the projection in the plane. Figure 7a illustrates the interaction of the Xs, Ys, and Zs variables as the parameter a is varied while R is held constant at 1.025. The diagram demonstrates the existence of two critical points: one exhibiting stable chaos attractor behavior and the other manifesting as a saddle point. The bifurcation diagram behavior is depicted in the Xs-Y projection, as illustrated in Figure 7b. The mathematical models employed for the description of intricate dynamical processes in cancer have been demonstrated to be a valuable instrument for the examination of chaotic dynamics in cancerous phenomena, as evidenced by Elnawawy et al. (2021)

In their study, Hat et al. (2016) analyzed the bifurcation diagram in the context of cancer research. This diagram is a visual representation utilized in mathematical and computational models to elucidate the manner in which alterations in system parameters can result in disparate states or behaviors of a tumor. It facilitates the identification of critical points at which a minor alteration in parameters can precipitate a substantial shift in the system’s dynamics, such as a transition from a stable state to uncontrolled growth (tumor development) or vice versa. In technical terms, a bifurcation diagram displays the potential steady states or equilibria of a system as a function of a parameter. The diagram illustrates how the number and stability of these steady states change as the parameter varies. This tool is of particular utility in the field of cancer research, facilitating a comprehensive understanding of the manner in which diverse genetic or environmental factors may influence the progression or treatment of the disease. Bifurcation diagrams can elucidate the phenomenon of hysteresis in cancer, whereby the tumor’s growth trajectory influences its present behavior and potential treatment responses. This information is of paramount importance for the advancement of personalized medicine approaches, as a tumor may exhibit disparate responses to the same treatment depending on its developmental trajectory.

DISCUSSION

Chaotic and ecosystems

Strogatz (2018) argued that argument that deterministic chaos is a common phenomenon observed in continuous-time dynamical systems of differential equations with nonlinear terms. Such systems are characterized by aperiodicity, ergodicity and sensitivity to initial conditions. In their study Momani et al. (2021) presented numerical results and graphs demonstrating the existence of chaos in the arbitrary order SIR epidemic system through the lens of a fractional order nonlinear system. Furthermore, it is proposed that this technique has significant potential for comprehending the intricate behaviors of diverse biological systems with chaotic characteristics. The time series outcomes of this study are highly consistent with those of the authors, even though the fractional-order nonlinear system technique was not utilized. Mangiarotti et al. (2020) have employed time series plots and phase portraits to demonstrate that their proposed model for the novel coronavirus disease COVID-19 displays chaotic behavior. The authors have identified the existence of chaos in their model by comparing their results with observed data. The chaotic behavior observed in these systems is primarily attributed to predator-prey and competition dynamics, as previously documented by Gakkhar & Naji (2003). Deterministic chaos is a common behavior in continuous-time dynamical systems of differential equations with nonlinear terms that exhibit aperiodicity, ergodicity, and sensitivity to initial conditions. This was confirmed by Strogatz (2018).

Uthamacumaran’s (2021) research showed that cancers are complex cybernetic systems that exhibit strange signaling attractors and a pattern gene expression like reaction-diffusion systems. The author also argues that chaos, despite appearing random, may serve as a robust biomarker for tumor complexity and is bound to well-defined pattern structures in state space, known as strange attractors. The concept of “strange signaling attractors” refers to the idea that cancer cells can enter into stable yet abnormal states due to the altered dynamics of their signaling networks. These attractors can cause the cells to behave in ways that promote tumor growth and survival. This analogy is used to explain how cancer cells can create complex spatial patterns of growth and invasion, much like the patterns seen in certain chemical reactions.

The study of chaotic dynamics in gene expression has revealed its potential role in regulating and controlling genes. The extant research indicates that chaotic behavior in transcription factors has the capacity to modulate gene expression, up-regulating specific gene families even in the presence of extrinsic and intrinsic noise. This modulation can result in increased production of protein complexes and enhanced efficiency of their assembly, as proposed by Heltberg et al. (2019). Moreover, chaotic dynamics have been linked to pluripotency in cells. As stated by Furusawa & Kaneko (2009), cells exhibiting irregular oscillations in gene expression demonstrate the capacity to differentiate into multiple cell types. Nevertheless, as differentiation progresses, these irregular oscillations dissipate, leading to a decline in pluripotency. These results suggest that chaotic dynamics have functional implications in cellular processes, including gene expression control, rather than being mere random fluctuations. Such dynamics can contribute to the heterogeneity of cell states, which is beneficial in various biological contexts, such as multi-toxic environments Heltberg et al. (2019). It is, however, important to note that chaotic dynamics can be efficient under certain conditions. They are integral components of a larger and intricate regulatory network that orchestrates gene expression.

In accordance with the research conducted by Gilpin & Feldman (2017), the phenomenon of chaotic behavior within a predator-prey ecosystem was discussed. This could entail investigating how interactions between predator and prey populations can result in complex, unpredictable, and non-linear dynamics. The chaotic behavior observed in such ecosystems may manifest as the emergence of irregular population cycles, sensitive dependence on initial conditions, and the potential for sudden, unpredictable shifts in population dynamics. Furthermore, the paper may examine the implications of chaotic behavior for ecosystem stability, resilience, and management strategies.

Borah et al. (2022) observed that certain cancer models exhibit deterministic chaos or chaotic dynamics, resulting in complex oscillations. As noted by Jones & Stigul 2021, the current pandemic exhibits the defining characteristics of chaos and has become one of the deadliest in recent history. Diekmann & Kretzschmar (1991), observed in cancer models and Parkinson’s disease, also contribute to chaotic behavior. Gross et al. (2004), Stiefs et al. (2009), who also observed chaotic long-term dynamics. Debbouche et al. (2022) presented a conceptual model of the ongoing pandemic that showed chaotic behavior. This was demonstrated through the presentation of a variety of mathematical results, including bifurcation diagrams, Lyapunov exponents, time series, and phase portraits, contextualized within the framework of nonlinear dynamical systems of differential equations. As noted by Allen et al. (1993), nonlinear dynamical differential equations manifest properties such as aperiodicity, ergodicity, and sensitivity to initial conditions. This highlights the importance of mathematical models as a potential tool for developing strategies to anticipate and respond to an impending epidemic or pandemic, as well as to manage a disease outbreak in real time. Sajan et al. (2024) developed a predator-prey model. The system exhibited a variety of bifurcations, including Hopf, saddle-node, transcritical, generalized Hopf, cusp, and Bogdanov-Taken bifurcations. The chaotic dynamics observed in the study can be applied to the modeling of the spread of epidemics and diseases. An understanding of the chaotic behavior of infectious diseases can assist in the prediction of their spread, the design of control measures, and the development of effective treatment strategies.

In their study, the authors, Santra et al. (2024), examined the fascinating phenomenon of cooperative hunting strategies employed by predators in their natural habitats. They use mathematical modeling and statistical analysis to understand how factors like prey density, predator speed, and communication affect hunting success. Key points include a) Spatio-temporal Modeling: The researchers use a model that considers both space and time to simulate predator-prey interactions realistically. This helps understand the impact of dynamic factors on hunting outcomes. (b) PRCC Analysis: Partial rank correlation coefficient (PRCC) analysis is used to identify the parameters that most influence hunting success. This helps direct further research and understand ecological factors driving cooperative hunting. (c) Key Insights: The study provides insights into optimal group sizes for hunting, the relationship between predator numbers and hunting success, and the influence of prey density. (d) Broader Implications: The findings contribute to understanding predator-prey relationships, inform conservation strategies for endangered predators, and may have applications in fields like robotics and swarm intelligence. The research combines a robust mathematical framework with insightful statistical analysis, offering valuable insights into cooperative hunting dynamics and their broader implications.

According to the literature, chaotic phenomena occur naturally in, for example, covid-19, cancer, Parkinson’s disease, biological systems, prey-predator, competition dynamics and all can be simulated by differential equation with nonlinear term. According to our results in this behavior of prey-predator interaction work, the following results have been presented, such as bifurcation diagrams, Lyapunov exponents, time series, and phase portraits. Thus, we can safely say that our work is consistent with the literature and can be used to study chaotic and ecosystems.

Fields outside of ecology

In a comprehensive examination of the intricate dynamics of host-parasitoid systems, Kumar et al. (2020) emphasized the necessity of considering both spatial structures and chaotic behaviors to achieve a comprehensive understanding and effective management of these systems. The role of parasitoids as significant natural enemies in various ecological and evolutionary processes is highlighted. The study discusses the potential for parasitoids to be utilized in the biological control of insect pests, emphasizing their importance as more than mere participants in chaotic population dynamics. The authors Khan et al. (2019) investigated the chaotic behavior of the model using numerical simulations and find that the non-singular derivative model exhibits chaotic behavior, which is characterized by complex and unpredictable patterns of disease spread. Furthermore, the study investigates the potential of non-singular derivatives to model fields beyond ecology, including physics and engineering. The authors illustrate the potential of non-singular derivatives to model intricate systems in these disciplines, including chaotic behavior. For instance, they discuss the application of non-singular derivatives to model the propagation of disease in social networks, which can manifest chaotic behavior due to the intricate interactions between individuals. The article underscores the potential of non-singular derivatives to model complex systems across diverse fields, including ecology, physics, and engineering, and demonstrates its capacity to capture chaotic behavior in these systems.

In their analysis, Gilpin & Feldman (2017) discussed of chaotic behavior in predator-prey ecosystems may have implications for various fields outside of ecology. For example, insights derived from this research may prove relevant to mathematical modeling and dynamical systems theory, particularly in the context of non-linear systems and complex dynamics. Furthermore, an understanding of chaotic behavior in ecological systems may have applications in fields such as environmental science, conservation biology, and even certain areas of economics and sociology where complex, interconnected systems play a role. In a study published by Alexi et al. (2012), the impact of pandemics on ecosystems was examined. The researchers developed an eco-epidemiological model that considers species interactions and multiple pathogen strains. The study demonstrated that pandemics result in significant ecological alterations, underscoring the necessity for effective management of such occurrences. Furthermore, the research facilitated a deeper understanding of the interconnection between ecology and epidemiology, thereby enabling the development of effective strategies to mitigate the ecological consequences of multi-species pandemics.

Hesketh et al. (2020) suggested that ecological models should consider the importance of human-environment interactions in understanding and modifying human behavior. The effect of chaos on the stability of ecological models has been debated for quite some time Stiefs et al. (2009). Gilpin & Feldman (2017) presented an eco-evolutionary predator-prey model in which prey population growth was influenced by a density-dependent fitness landscape. The model demonstrated the potential for chaotic dynamics even in the absence of external environmental variation, exhibited rapid cycling in the chaotic dynamics, and suggested that evolution-derived chaotic dynamics could provide stability in ecosystems.

Another author who made a significant contribution was Samanta (2021). Their work provides a comprehensive and detailed examination of the stability and dynamics of ecological systems. He elucidates the way diverse regulatory mechanisms, including birth and death, competition, and consumption, give rise to alterations in the stability and dynamics of these systems. Additionally, it offers a distinctive perspective on the mathematical intricacies of fundamental ecological models. The author enlightened several pivotal concepts pertaining to the stability and dynamics of ecological systems, including: (a) Dynamical models of single and predator–prey species. This examines the dynamics of single and predator-prey species, elucidating how their interactions influence the stability of the ecosystem. (b) Dynamical models of single-species systems in a polluted environment: This section examines the impact of pollution on the dynamics and stability of a single-species system. (c) An analysis of nonautonomous two-species systems in a polluted environment is presented in this section, which explores the dynamics and stability of two-species systems in a polluted environment. (d) Dynamical models of single-species systems under the influence of environmental noise. This particular section of the text discusses the impact of environmental noise on the dynamics and stability of a single-species system. (e) Stability Behavior in Randomly Fluctuating versus Deterministic Environments of Two Interacting Species. This last section compares the stability behavior of two interacting species in randomly fluctuating environments versus deterministic environments. It also discusses thermodynamic criteria of stability and stochastic criteria of stability.

The authors Dutta et al. (2022), have provided a comprehensive analysis of this theme, including an examination of the cooperative hunting strategies employed by predators in their natural habitats. Mathematical modeling and statistical analysis are employed to elucidate the influence of factors such as prey density, predator speed, and communication on hunting success. The study’s most significant findings include the following: (a) Spatio-temporal modeling. The researchers employ a model that considers both space and time to simulate predator-prey interactions in a manner that is both realistic and accurate. This allows for an understanding of the impact of dynamic factors on hunting outcomes. (b) PRCC Analysis. The partial rank correlation coefficient (PRCC) analysis is employed to ascertain the parameters that exert the most significant influence on hunting success. This facilitates the direction of subsequent research and the comprehension of ecological factors that drive cooperative hunting. (c) Key Insights. The study provides insights into optimal group sizes for hunting, the relationship between predator numbers and hunting success, and the influence of prey density. (d) Broader Implications. The findings contribute to understanding predator-prey relationships, inform conservation strategies for endangered predators, and may have applications in fields like robotics and swarm intelligence. The research combines a robust mathematical framework with insightful statistical analysis, offering valuable insights into cooperative hunting dynamics and their broader implications.

An understanding of chaotic behavior in ecological systems may have applications in fields such as environmental science, conservation biology, and even certain areas of economics and sociology where complex, interconnected systems play a role, as well as in the interconnection between ecology and epidemiology. This study analyzed the dynamics of a prey-predator system, providing insights into the behavior of interacting populations. This can assist in the understanding of the dynamics of real ecosystems and the design of effective conservation and management strategies when the dynamics of chaos are presented.

CONCLUSIONS

The most relevant results of this work will be presented as follows. According to our results in this behavior of prey-predator interaction work, the following results have been presented, such as bifurcation diagrams, Lyapunov exponents, time series, and phase portraits. The results exhibited evidence of chaotic behavior.

This study and a review of the literature show that the emergence of chaotic behavior has been attributed mostly to predator-prey and competitive dynamics. In fields outside of ecology, this phenomenon has also been observed in models of cancer and pandemics. It displays deterministic chaos or chaotic dynamics, which gives rise to complex oscillations and nonlinear interactions between cell populations. It is also noteworthy that the novel coronavirus disease (Covid-19) exhibits the primary qualitative characteristics of a chaotic system and is the most lethal pandemic in recent history.

The chaotic behavior observed in ecological systems may have applications in several fields, including environmental science, conservation biology, and economics and sociology, where complex interconnected systems play a role, as well as in the interconnection between ecology and epidemiology.

We believe that the results of this research could have implications in chaotic and ecosystems, as well as in fields outside of ecology. For example, they could be useful in modelling epidemics and diseases, such as covid-19, cancer, Parkinson’s disease, biological systems, prey-predator dynamics, and competition dynamics. Additionally, they could be applicable in other areas for practical applications.

Mathematical models can play an important role in developing strategies to plan for and deal with disease outbreaks in real time. Nevertheless, it is worth noting that developing a comprehensive model of an epidemic disease within these formalisms represents a significant challenge, particularly given the evolving nature of such diseases and their rapidly changing characteristics, patterns of behavior, and modes of propagation.

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Publication Dates

• Publication in this collection
  02 Dec 2024
• Date of issue
  2024

History

• Received
  2 May 2024
• Accepted
  30 Aug 2024