Modeling of fractional order DPG model insight global warming and pollution effect on desertification for control mechanism

Abstract

This study presents a novel fractional-order mathematical model that investigates the dynamic interplay between dust pollutants, plant biomass, and global warming, referred to as the DPG system. This research introduces a fractional-order formulation that incorporates ecological memory and long-range interactions, providing a more realistic representation of desertification dynamics than classical integer-order models. It also establishes a comprehensive analytical numerical framework designed to capture system feedback, assess instability patterns, and evaluate the effectiveness of chaos control mechanisms. The model utilizes Caputo derivatives to capture the memory effects inherent in ecological and atmospheric processes. Key parameters such as dust emission, plant decay, and global warming feedback mechanisms are integrated into a nonlinear system of differential equations. Analytical evaluations ensure the existence, uniqueness, and generalized Hyers-Ulam-Rassias stability of the proposed system. Sensitivity analysis identifies the parameters that have the most significant influence on desertification risk. Furthermore, a Newton polynomial-based numerical scheme is constructed to efficiently simulate system behavior under varying fractional orders. Chaos control strategies are implemented to stabilize the system near critical equilibrium points. Numerical simulations reveal that lower fractional orders dampen dust accumulation, slow plant biomass regeneration, and delay global warming trends, highlighting the efficacy of fractional modeling in capturing real-world environmental inertia and feedback. This research provides a robust analytical and computational foundation for understanding ecosystem resilience in the face of both anthropogenic and climatic stressors.

Introduction

Mathematical models can be used to better understand and describe disease transmission estimates, future population forecasts, and other physical and scientific phenomena^1,2. Global warming has a significant impact on plant biomass because it alters the environment that plants need to grow and thrive. These variables include rising temperatures, rising atmospheric levels, and changing precipitation patterns. Plant biomass, or the total mass of living plant matter in an ecosystem, is crucial for carbon sequestration, soil health, and biodiversity. The ability of plants to survive, grow, and produce may be impacted by stress brought on by global warming³. Long-term disruption of plant communities due to global warming could result in certain species thriving while others suffer. Ecosystems may be significantly impacted by changes in the distribution and abundance of plant biomass, which can lead to alterations in carbon storage, biodiversity, and the availability of food for herbivores^4,5. One of the main causes of desertification is acknowledged to be global warming, as the continuous increase in temperature leads to altered weather patterns and reduced precipitation, both of which negatively impact the vegetation cover in green regions. Plants are increasingly vulnerable to drought and stress as temperatures rise and arid periods lengthen, which slows growth, degrades soil, and reduces fertility. Additionally, as dry spells persist and intensify in many historically green areas, changing weather patterns significantly impact the distribution of surface and groundwater. Land degradation and the eventual conversion of these regions into deserts are the outcomes of this⁶. The health and growth of plants are compromised by dust particles and other airborne contaminants, which have a negative effect on vegetation. One of the most severe impacts on plants is also a barrier to photosynthesis: the accumulation of dust on plant leaves impairs the plant’s capacity to produce the nutrients and energy required for growth by blocking sunlight, which is essential for photosynthesis. In addition, the polluted dust may clog the stomata on the leaves, blocking them and limiting gas exchange. This puts stress on plants and makes them unable to uptake important gases such as carbon dioxide and discharge oxygen⁷. Additionally, dust pollution may contain hazardous materials, such as industrial pollutants and heavy metals, which can alter the growth of plants and damage the structure of leaves. The dust pollutants in the soil can alter its physical and chemical properties, rendering the roots incapable of absorbing nutrients and water, and thereby affecting plant growth as the dust accumulates on the soil. The results of these impacts are reduced health and quality of the vegetation that can be harmful to natural ecosystems and agriculture, a decline in biodiversity, and a risk of desertification as a result of these impacts on the environment and humans⁸. Desertification can be defined as the degradation of land in arid and semi-arid regions due to both natural (such as climate change) and human (such as deforestation and unsustainable farming practices) factors. Desertification reduces agricultural productivity, leading to food shortages and higher rates of poverty, particularly in rural areas where agriculture is the primary industry. It also exacerbates large-scale migration to cities, putting pressure on services and infrastructure.

In order to solve many difficulties in life, mathematical modeling is essential^9,10. Mathematical model of atmospheric pollutant dispersion analysis^11,12. It provides a reflection on how pollutants are transported and the effects they have on the environment. In the study of air pollution, the dynamics of the air pollution process are modeled by the authors of the research, who analyze them using mathematical methods¹. It also discusses possible ways to mitigate it, ensuring that the level of pollution is minimized. It was suggested that a mathematical model can be used to investigate the depletion of plant resource biomass caused by industrialization and environmental pollution. It was found that in the case of small periodic inflows of pollutants, the biomass exhibits periodic variations at low values of the depletion coefficient of the pollutant and approaches equilibrium when the coefficient exceeds a critical value of the depletion coefficient of the pollutant¹³. The other study also investigated the effects of particulate contaminants on plant biomass using a model that involved three state variables: the density of plant biomass, the concentration of dust, and the concentration of water droplets. The finding showed that there is no need to introduce water particles into the lower atmosphere at a time when the level of dust concentration does not reach the threshold limit¹⁴. Moreover, mathematical models have been established to predict the trend of desertification, which is fueled by global warming, as well as the dynamics of dust pollutants¹⁵.

It has been discovered that one of the most effective models for dealing with the complex and memory-dependent dynamics of ecological and environmental systems is fractional-order modeling, especially in cases such as desertification and global warming^16–18. The models incorporate long-term effects and system inertia, which provide a more realistic representation of the processes of dust pollution accumulation and plant biomass decay^19–21. For instance, dust storms, a significant driver of desertification, exacerbate land degradation by reducing vegetation cover and altering soil properties, further intensifying the impacts of climate change²². Studies have shown that fractional derivatives effectively model the cumulative effects of environmental stressors, such as pollution and temperature changes, on ecosystems^23,24. Moreover, the integration of fractional calculus in ecological models provides insights into stabilizing chaotic environmental dynamics, which is critical for developing strategies to mitigate desertification and its socio-economic consequences^25,26. The study²⁷ reveals that the fractional-order emission model exhibits both stable and chaotic behaviors under varying parameter conditions. Effective chaos control strategies are proposed to stabilize environmental dynamics and reduce long-term emission impacts. Kinetic model for sugar oxidation by permanganate in aqueous media²⁸, incorporating a sustainable fractional-order approach to capture memory effects in the reaction dynamics. A fractional-order forestry resource conservation model was developed, incorporating chaos control to mitigate environmental disturbances, with a modified ABC operator employed to simulate the effects of toxin activity and human-caused fires²⁹.

Recent advances in fractional-order epidemic and ecological modeling have underscored the crucial role of memory effects in accurately capturing the complexity of biological and environmental processes. Memory-dependent fractional dynamics formulated through the Caputo derivative have been shown to strongly influence the spread and persistence of Coffee Berry Disease³⁰, underscoring the ability of fractional operators to represent realistic plant-pathogen interactions. Similar advantages are also observed in infectious disease studies: the fractional TB model in³¹ demonstrates that incorporating memory terms yields a more accurate representation of Mycobacterium tuberculosis transmission dynamics. In ecological systems, the fractional prey-predator framework in³² reveals how memory effects enhance the modeling of biological pest-control interactions, while³⁴shows that combining memory with prey refuge and fear-induced responses significantly shapes predator-prey dynamics. Fractional approaches have also proven effective in veterinary epidemiology; for example³³, captures how quarantine and vaccination measures with memory effects more realistically reflect Lumpy Skin Disease transmission. Building on these developments, in-host TB modeling in³⁵ highlights the interplay between hereditary influences and antimicrobial responses in shaping bacterial dynamics, and³⁶ incorporates booster vaccination through the Atangana-Baleanu-Caputo operator to address emerging COVID-19 variants. Additional advancements include the integration of awareness and vaccination strategies into a fractional TB framework in³⁷, and the incorporation of consciousness effects into a Caputo TB formulation in³⁸, both offering strengthened analytical understanding through Laplace-Adomian techniques. Moreover, the analysis in³⁹ employs Caputo fractional-order modeling to evaluate the impact of diagnosis and treatment interventions on tuberculosis transmission, resulting in a more comprehensive and realistic representation of disease progression.

The novelty of this model lies in combining desertification dynamics with global warming and pollution effects under a unified fractional-order framework, which has not been previously explored. Fractional-order dynamics enable the model to capture memory effects and long-term interactions that classical models cannot represent. We use the Caputo fractional derivative because it supports physically meaningful initial conditions, provides smooth compatibility with integer-order systems, and offers better analytical tractability compared to other operators. Its widespread use in environmental and ecological modeling further supports its suitability for accurately describing desertification and processes driven by pollutants. This study incorporates a fractional-order derivative into the DPG system, allowing it to capture memory and hereditary effects that classical integer-order models cannot. By doing so, the model provides a more accurate representation of system dynamics, offering deeper insights into the long-term behavior and interactions within the system. The contribution lies in combining fractional calculus with stability and chaos analysis to better understand and potentially control the system. The Caputo fractional derivative was selected because it allows for the use of standard initial conditions expressed in integer-order derivatives, making the model more physically interpretable. It effectively captures memory and hereditary effects inherent in the system. Compared to other definitions, it ensures consistency with classical calculus while accommodating non-local dynamics. Its mathematical formulation is well-suited for numerical simulations and stability analysis. Overall, it strikes a balance between accuracy, interpretability, and computational convenience. The fractional-order derivative is necessary because it accounts for non-local interactions and memory effects inherent in the DPG system. These derivatives can model processes where the current state depends not only on the present but also on the past states, which is critical for capturing realistic dynamics that standard integer-order derivatives fail to represent.

The manuscript is structured as follows: Section 2 presents a comprehensive overview of the fractional operator utilized in formulating the proposed model. In Section 3, the fractional-order model is introduced. Section 4 contains a detailed analytical investigation of the model’s dynamics. Section 5 focuses on the stability analysis using the Hyers-Ulam-Rassias stability framework. Chaos control analysis is conducted in Section 6. Numerical simulations demonstrating the model’s behavior are provided in Section 7. Section 8 offers a critical discussion of the results. Finally, Section 9 summarizes the key findings and provides concluding remarks.

Basic concepts

The following are some crucial definitions that could be helpful for system analysis:

Definition 2.1

⁴⁰ Considering a continuous function , the definition of a Caputo operator of order , where , is

1

Definition 2.2

⁴⁰ The Riemann Liouville integration with respect to t is defined as follows when is recognized as a continuous function:

2

with a converging integral.

Formulation of model

The DPG model¹⁴, which includes dust pollutants D(t), plant biomass P(t), and global warming phenomena G(t), is examined in this section. Dust particle emissions into the atmosphere are occurring at a rate of A. The natural rate of depletion of the dust contaminants is . The biomass from plants acts as a dust scavenger by lowering the quantity of dust particles in the atmosphere. The number of dust particles and the density of plant biomass (i.e., ), where is a dust particle depletion rate coefficient, are closely correlated with the decreasing concentration of dust particles. In the absence of the effects of dust pollution and global warming, it is predicted that plant biomass will increase at the intrinsic growth rate (r) and carrying capacity (k). Desertification brought on by climate change is thought to be a factor lowering carrying capacity by cG, where c is the rate at which carrying capacity is reduced as a result of global warming phenomena. It is assumed that the plant biomass is depleted due to the growing concentration of dust particles, where is a plant biomass depletion rate coefficient. Urbanization, industry, contemporary living, and other human activities (i.e., Q) all contribute to the ongoing escalation of the effects of global warming on the earth’s surface and atmosphere. Numerous studies show that reducing fuel usage, plantations, and other factors are just a few of the control techniques that can be used to mitigate or lessen global warming. Thus, let be the rate of global warming depletion due to plantations, and be the rate of global warming depletion due to human control tactics. According to a number of studies, dust particles reduce the amount of heat the planet receives and provide a cooling impact by deflecting some of the sun’s rays before they reach the Earth’s surface. This influence could mitigate some of the effects of global warming. The rate of global warming depletion caused by the cooling effect of dust mass on the climate is . The interaction terms and are empirically justified because dust concentration and plant biomass both intensify temperature-related global warming effects through reduced vegetation cover and enhanced heat absorption. These coupled interactions capture measurable feedbacks observed in environmental studies, where increased dust or decreased plant growth directly amplifies global warming dynamics. Table 1 describes the model parameters, and Figure 1 shows the schematic diagram of the framed model.

Table 1.

Parameters and their values are described.

Symbol	Description14	Value	Units/Physical Interpretation
A	Amount of dust particles released into the atmosphere from various sources	10	mg m year (constant atmospheric dust emission rate)
	Dust pollution coefficient of natural depletion	0.1	year (natural deposition and removal rate of dust)
	Depletion coefficient of dust contaminants due to plant biomass	0.01	m mg year (dust scavenging/capture efficiency per unit biomass)
r	Growth rate of plant biomass	0.22	year (intrinsic logistic growth rate of vegetation)
k	Plant biomass carrying capacity	30	g m (maximum sustainable aboveground dry biomass density)
c	Effects of global warming on desertification (reduction of carrying capacity)	0.01	dimensionless (proportional reduction of k per unit G)
	Depletion coefficient of plant biomass due to dust pollutants	0.001	m mg year (biomass loss rate per unit dust concentration)
Q	Causes of increase in global warming (anthropogenic forcing)	0.821	arbitrary units year (external warming input)
	Contribution rate of dust pollution to global warming reduction (cooling effect)	0.001	m mg year (short-wave reflection/cooling by dust aerosols)
	Contribution rate of plant biomass to global warming reduction	0.003	m g year (cooling via carbon sequestration and evapotranspiration)
	Reduction rate of global warming from human intervention/mitigation	0.001	year (rate of warming mitigation through policy/technological measures)

Fig. 1.

Schematic representation of the proposed model.

To incorporate memory and hereditary effects inherent in ecological and environmental processes, the present model employs the Caputo fractional derivative of order . The Caputo operator is well-suited for modeling systems where initial conditions are defined in terms of integer-order derivatives. The following system of fractional differential equations describes the interactions among DPG under the influence of fractional-order dynamics:

3

with the initial conditions

Positively invariant region

In this section, we show that the system (3) positively invariant feasibility region is provided by the closed set

Lemma 3.1

If they begin in , the solutions of system (3) are restricted and non-negative.

Proof

According to the model (3), we discover that

4

the system (4) states that the vector field lies in the region on each hyperplane covering the non-negative orthant with .

Summing the equations gives:

Using the fact that all parameters and states are positive, there exists a constant such that

Applying the fractional comparison principle, we get

where is function.

Therefore, N(t) is bounded, and consequently, D(t), P(t), G(t) are all bounded.

Equilibrium point analysis

We examine the model’s equilibrium states to identify the desertification point and the non-desertification point (3). The points of desertification are

The non-desertification points are

In the proposed D–P–G fractional-order model, the reproduction number represents the threshold quantity that determines whether the plant population P(t) can successfully establish and persist in an environment dominated by desertification pressure D(t). To compute , we evaluate the dynamics of the plant equation near the plant-free equilibrium. Setting , the equilibrium level of D is obtained from the first equation as

Linearizing the plant equation around yields a per-capita growth rate

where r represents intrinsic plant growth, and represents the loss of plants due to interaction with the prevailing desertification level. The reproduction number is then defined as the ratio of the plant growth force to the effective loss term, giving

Thus, implies that plant growth dominates desertification pressure, enabling plant invasion and persistence, while indicates that desertification suppresses vegetation growth, driving the system toward a plant-free state.

Figure 2 displays the impact of parameters on the number of reproductions. The basic reproduction number does not directly depend on the fractional derivative parameter. It is typically derived from the system’s equilibrium and transmission rates, reflecting the average number of secondary effects produced by one unit. The fractional derivative influences the system’s dynamics and memory effects over time, but itself remains a threshold parameter determined by the underlying rates and interactions in the model. However, the fractional order can affect the speed at which the system approaches or departs from the equilibrium associated with .

Fig. 2.

Impact of parameter variations on the basic reproduction number .

Sensitivity analysis

Sensitivity analysis of this model helps identify which parameters most influence desertification dynamics driven by global warming and dust pollutants. It reveals how small changes in factors such as emission rates, plant decay, or pollution impact the survival of vegetation. This insight supports the development of targeted environmental policies to prevent or reverse land degradation. Figure 3 shows the impact of key parameters on Partial derivatives of the reproductive number, given the pertinent parameters, can be used to examine the sensitivity of .

5

Fig. 3.

PRCC results showing the influence of key variables on .

Existence and uniqueness

To confirm the reliability of the (DGP) model governed by Caputo fractional derivatives, it is essential to establish the existence and uniqueness of its solutions. This section provides the theoretical foundation ensuring that the model yields well-defined and consistent dynamics over time. To maintain clarity and simplicity in the analysis, we begin by considering the following three kernel functions:

6

Equation (6) can be analyzed and solved within the framework of the Caputo fractional-order derivative operator

7

and are positive real constants. We now proceed to verify that the Caputo fractional-order system defined in equation (3) satisfies the Lipschitz continuity condition.

Theorem 4.1

The previously defined kernels, , , and , each satisfy the Lipschitz continuity condition, ensuring the well-posedness of the corresponding fractional-order system.

Proof

Firstly, the kernel atisfies the Lipschitz condition. The corresponding norm, evaluated for two functions D and , is given by:

8

let , where both , and are assumed to be bounded functions. The corresponding norms for the remaining model equations can be derived analogously, following the same procedure.

We aim to demonstrate that the Caputo fractional-order model (3) possesses at least one solution. Accordingly, the equivalent integral formulation of equation (7) is given as:

9

The initial conditions, which are assumed to be positive, serve as the starting values for the first iteration. The following expressions represent the differences between successive iterative terms.

10

11

12

It’s essential

13

We proceed to analyze the following results using the system of equations defined in (10)-(12).

The above equation is simplified by applying the triangle inequality, resulting in the following form:

The kernel is verified to satisfy the Lipschitz condition as demonstrated in equation (7). Therefore, we can express it as:

Utilizing the information provided in equation (13), the previously derived inequality can be simplified as follows.

14

Similarly, we derive the following results based on the corresponding expressions.

15

Theorem 4.2

The fractional-order Caputo model (3) admits an analytical solution at

Proof

The boundedness of D(t), P(t), andG(t) along with the satisfaction of the Lipschitz condition, follows directly from the recursive formulation and the results established in Equation (14).

As a result, the previously obtained solutions remain valid. However, to demonstrate that the above functions indeed represent the solution of the proposed model, we proceed by considering the following approach

As a result, we obtain

By applying the Lipschitz condition,

This yields,

Evaluating at , we obtain

When approaches n, we get

In a similar manner, we conclude that

This shows the existence of a solution.

We now aim to determine whether the solution is unique. Suppose, therefore, that another solution to the proposed model exists, denoted by D(t).

16

By applying the norm to Equation (16), we obtain

17

Theorem 4.3

In this case, the analytical solution of the Caputo fractional model is proven to be unique.

18

Proof

Assume that (18) is equivalent to (17), such that

Therefore, we deduce that

which implies that the solutions are identical. This implies that

confirming the uniqueness of the solution. Applying the same procedure to another function yields the following results.

Thus, the proof confirms that, with respect to the Caputo operator, the proposed model (3) possesses a unique solution.

Hyers-Ulam-Rassias stability

To assess the robustness of the proposed Caputo fractional-order DPG model, we investigate its Hyers-Ulam-Rassias stability. This analysis determines whether small perturbations in initial conditions lead to bounded deviations in system behavior. Establishing such stability ensures the model’s reliability for long-term ecological predictions in the face of environmental fluctuations. This subsection presents the stability analysis of the fractional model (3) based on the concepts of Hyers-Ulam-Rassias stability. To proceed, it is necessary to rewrite the model (3) in the following form.

19

where, the vector and denotes a continuous vector-valued function.

Definition 5.1

Let the fractional order satisfy and be a continuous mapping. Then, the model (19) is said to be Hyers-Ulam stable if there exist and , such that, for every solution , the following inequality holds:

solution of model (19), such as

Definition 5.2

Assume that the fractional order satisfies . Let and be continuous mappings. Then, the model (19) is said to exhibit generalized Hyers-Ulam-Rassias stability with respect to if there exists a , such that, for every solution the following inequality holds:

20

solution of model (19), such as

To establish the Hyers-Ulam-Rassias stability of model (19), we proceed under the following assumptions:

• is a continuous mapping.

• such that for each solution

  
• Let be an increasing mapping, and let such that

  

Theorem 5.1

Suppose that assumptions hold. Then, the generalized model (19) is Hyers-Ulam-Rassias stable with respect to on the given interval, provided that

Proof

Let be a solution of the model. Then, the unique solution of model (19) is obtained from the following expression:

Based on the results established in equation (20), it can be concluded that

So,

Now, , so

Applying Gronwall’s inequality yields the following result

By setting , we obtain

21

Inequality (21) establishes the generalized Hyers-Ulam-Rassias stability of model (19) with respect to .

Remark

The generalized Hyers-Ulam-Rassias stability allows for bounded deviations that can vary over time, making it more flexible than classical stability theorems. This is particularly relevant for the current fractional-order model, as exact solutions may not be feasible due to the system’s complexity. Unlike the original Hyers-Ulam approach, which assumes fixed bounds on deviations, the Rassias extension accommodates variable bounds, providing a more realistic measure of system robustness under uncertainty.

Chaos control

In this section, the linear feedback control method is employed to stabilize system (3) at its equilibrium points. The chaos control analysis is included because the interactions among dust pollutants, plant biomass, and global warming naturally produce nonlinear feedbacks that can lead to chaotic fluctuations. Such instability corresponds to real phenomena observed in desertification processes, such as sudden declines in vegetation or irregular cycles of dust storms. Introducing control terms represents environmental actions like dust reduction, vegetation enhancement, and climate-related mitigation. Stabilizing the system around equilibrium helps prevent abrupt ecological shifts that accelerate land degradation. Therefore, chaos control is ecologically relevant, as it reflects how targeted interventions can maintain system stability and slow the progression of desertification. The controlled form of the fractional-order system (3) is analyzed as follows:

22

where , , and are control variables and denotes the equilibrium point of the system. The Jacobian matrix of system (22) evaluated at is given by

with the selected control values and based on the given parameter values, the characteristic roots of the polynomial corresponding to the equilibrium point are determined as follows.

23

Furthermore, the characteristic roots of equation (23) are calculated as , , and are computed as the characteristic roots of Equation (23). Since all eigenvalues have negative real parts, the equilibrium point is asymptotically stable. The Figs. 4, 5, 6 shows the system behavior before and after applying control. The graphs clearly illustrate that uncontrolled dynamics exhibit irregular, chaotic fluctuations, while the controlled system stabilizes over time. Figs. 7, 8, 9 show the chaotic surface phase pattern of the DPG under varying parameter values. The system can exhibit bifurcation for specific parameter values, where small changes in parameters lead to qualitative shifts in dynamics.

Fig. 4.

D(t) behavior before and after control.

Fig. 5.

P(t) behavior before and after control.

Fig. 6.

G(t) behavior before and after control.

Fig. 7.

Chaotic surface phase pattern of the D(t) under varying parameter values.

Fig. 8.

Chaotic surface phase pattern of the P(t) under varying parameter values.

Fig. 9.

Chaotic surface phase pattern of the G(t) under varying parameter values.

Numerical scheme

In this section, a numerical method is developed to solve the proposed model (3). To obtain the numerical solution of the fractional-order system, a novel numerical scheme based on Newton polynomials is constructed.

24

The detailed derivation of the numerical scheme will be moved to an appendix.

Results of proposed scheme

In this section, we present the numerical results of the DPG model governed by the Caputo fractional operator. The simulations are conducted using data specified in Table 1. The horizontal axis represents time (in years) in each plot. The vertical axes show the physical units as follows: Dust concentration D(t) in , Plant biomass P(t) in , and Global warming index G(t) in arbitrary units (a.u.), consistent with the original integer-order DPG model¹⁴. The figure (10a) illustrates the temporal evolution of dust pollutants under different fractional orders, including the classical integer-order case. It is observed that the dust concentration rises more rapidly when , indicating a more immediate response to external influences. In contrast, for fractional orders, the progression of is more gradual. This behavior highlights the memory-retaining property of the Caputo fractional operator, whereby the influence of past states dampens the immediate growth of dust pollution. The fractional order decreases. In Figure 10b, the growth of dust pollutants becomes progressively slower. The introduction of long-memory effects through lower values effectively captures the inherent delays and inertia in the environmental processes governing dust accumulation. These results confirm the system’s sensitivity to the fractional order and the importance of selecting an appropriate for accurate ecological modeling. In figure (10c) show that higher fractional orders lead to steeper growth in , the memory effect in lower orders significantly suppresses the rate of increase. This suggests that incorporating fractional dynamics provides a nuanced understanding of how pollutant levels evolve in response to both immediate and accumulated environmental stresses. The Fig. 11a presents the decline in plant biomass over time, comparing the integer-order and fractional-order formulations. When , the model predicts a relatively faster depletion of biomass, suggesting an abrupt response to dust-induced stress. Conversely, fractional orders exhibit a more moderate decline. The persistence introduced through fractional calculus captures the delayed degradation mechanisms in ecological systems, where biomass does not degrade instantly but is influenced by cumulative exposure to pollutants. The simulations (11b) reveal a gradual reduction in biomass as decreases. This demonstrates that memory effects inherently slow the degradation process, thereby aligning the model more closely with empirical observations in arid ecosystems. Lower values are shown to confer more resilience to biomass, indicating that fractional-order dynamics are essential for capturing long-term ecological sustainability. The results show that under fractional dynamics in Figure 11c, the degradation of is notably less severe compared to the integer-order case. This suggests that fractional-order models are more capable of representing the delayed responses of plant systems to environmental pressures. In Figure 12a, the evolution of the global warming variable is shown under different values of . For , exhibits a faster and steeper increase, indicative of an aggressive warming response to dust and biomass dynamics. However, fractional orders result in a more attenuated and smoother progression. This demonstrates the critical role of memory in modeling global warming, where feedback mechanisms such as carbon storage and pollutant heat absorption act over extended periods. As the order of differentiation is reduced, the warming trajectory becomes increasingly tempered in Figure 12b. This reflects the fact that climate systems do not respond instantaneously but rather accumulate and manifest impacts over long durations. Such behavior validates the use of fractional calculus in climate modeling, providing a more realistic assessment of future warming trajectories under environmental stressors. The simulation (12c) assesses the contributions of dust pollutants and biomass to global warming. The figure demonstrates that while all fractional orders lead to a gradual rise in , lower values of yield a more delayed response. This aligns with real-world climate systems where the impact of pollutants and deforestation on global temperatures is cumulative and delayed, further supporting the appropriateness of fractional-order formulations. The fractional order model captures memory effects and dynamic behaviors more accurately. It demonstrates improved realism in representing interactions among pollutants, plants, and climate. These differences underscore the significance of using fractional derivatives over classical integer-order models. Fractional orders (−0.99) represent memory effects in the system, capturing how past states influence current dynamics. Higher values indicate stronger memory and slower decay of past effects. This allows modeling real-world processes with long-term dependencies more accurately than integer-order models. Using data from a previous study allows the ecological interpretation to be directly linked to observed environmental processes. This ensures that the modeled dust levels, plant biomass, and warming trends reflect realistic ecosystem behavior, thereby increasing the scientific relevance of the results. The graphical analysis across all three variables, dust pollutants, plant biomass, and global warming, clearly shows that fractional-order models, due to their intrinsic memory effects, provide a more realistic and gradual evolution of system dynamics compared to the classical integer-order models. Lower values of the fractional order systematically introduce delays and smooth transitions in each variable, aligning with observed ecological and environmental behavior. Therefore, incorporating Caputo fractional derivatives enhances the model’s ability to simulate complex, real-world phenomena related to desertification and climate change. We provide a scientific interpretation of the model outcomes using real data obtained from a previously published study. The parameter values and initial conditions employed in our simulations are directly derived from empirically measured field data, ensuring that the model reflects realistic ecological conditions. By grounding the numerical results in observed patterns, the analysis captures how the system behaves under actual environmental scenarios. The model has been validated using real data obtained from a previously published study. The parameter values and initial conditions employed in the simulations are directly sourced from empirical measurements reported in that work, ensuring that the system dynamics reflect realistic environmental conditions. While the present study does not include an independent data-collection process, the use of these verified empirical data sets provides indirect validation by demonstrating that the model reproduces behavior consistent with observed trends.

Fig. 10.

Simulation of dust concentration D(t) under different fractional orders. (a) Comparison with ω = 1. (b) Different fractional order values. (c) β = 0.01,α = 0.003,c = 0.8, γ₁ = 0.1, γ₂ = 0.3.

Fig. 11.

Simulation of plant biomass P(t) under different fractional orders. (a) Comparison with ω = 1. (b) Different fractional order values. (c) β = 0.01,α = 0.003,c = 0.8, γ₁ = 0.1, γ₂ = 0.3.

Fig. 12.

Simulation of the global warming index G(t) under different fractional orders. (a) Comparison with ω = 1. (b) Different fractional order values. (c) β = 0.01,α = 0.003,c = 0.8, γ₁ = 0.1, γ₂ = 0.3.

Tables 2, 3, 4 examine the impact of different fractional orders on the dynamics of D, P, and G.

Table 2.

Approximate solution of D at different fractional order.

t
20	29.0778	29.135	29.1613	29.2316
40	28.8279	28.9284	28.9746	29.0974
60	28.6617	28.7878	28.8466	29.0044
80	28.5413	28.6831	28.7503	28.9329
100	28.4496	28.601	28.6738	28.8748
120	28.3773	28.5343	28.611	28.826
140	28.3187	28.4789	28.5583	28.7841
160	28.2703	28.4318	28.5131	28.7473
180	28.2295	28.3913	28.4738	28.7147
200	28.1948	28.3561	28.4392	28.6854

Table 3.

Approximate solution of P at different fractional order.

t
20	24.8773	24.7982	24.76	24.6508
40	25.1247	25.0191	24.9673	24.8178
60	25.2666	25.1492	25.0907	24.9193
80	25.3629	25.2396	25.1773	24.9921
100	25.434	25.3078	25.2432	25.0488
120	25.4893	25.3619	25.2959	25.0949
140	25.5339	25.4062	25.3395	25.1337
160	25.5709	25.4435	25.3764	25.167
180	25.6021	25.4755	25.4082	25.1962
200	25.6289	25.5034	25.436	25.2221

Table 4.

Approximate solution of G at different fractional order.

t
20	8.0156	8.0228	8.0261	8.0348
40	7.9811	7.995	8.0012	8.0174
60	7.9551	7.9741	7.9826	8.0047
80	7.9341	7.9571	7.9675	7.9943
100	7.9165	7.9429	7.9548	7.9856
120	7.9014	7.9305	7.9437	7.978
140	7.8883	7.9196	7.9339	7.9712
160	7.8768	7.9099	7.9252	7.9651
180	7.8664	7.9012	7.9173	7.9595
200	7.8571	7.8933	7.91	7.9544

Conclusion

In this work, we formulated and analyzed a fractional-order dynamical system to examine the interlinked effects of dust pollutants, plant biomass fluctuations, and global warming. By leveraging the Caputo derivative, the model successfully captured the memory-driven behavior of environmental variables. We demonstrated the existence and uniqueness of solutions, confirmed generalized Hyers-Ulam-Rassias stability, and developed an accurate Newton polynomial-based numerical algorithm for simulation. Sensitivity analysis identified critical parameters, such as dust emission and plant depletion rates, that most significantly influence ecosystem degradation. The chaos control analysis showed that the system could be stabilized around equilibrium states using linear feedback mechanisms. Numerical simulations under various fractional orders indicated that incorporating memory effects leads to more realistic and moderated system dynamics. These findings underscore the importance of fractional calculus in environmental modeling and provide valuable insights for designing effective mitigation strategies to combat desertification and climate change. The study can be extended to utilize other fractional operators, such as Atangana-Baleanu or Caputo-Fabrizio. These operators offer different kernel functions, such as non-singular or Mittag-Leffler kernels, which provide alternative ways to model memory effects and may improve the model’s flexibility and accuracy when fitting real data. Moreover, future extensions of this model to spatially distributed or stochastic frameworks, integrating remote-sensing data sets, or exploring optimal control strategies could further enhance predictive accuracy and practical applicability in environmental management.

Appendix

For simplicity, we make the following observations about the given system

So,

At the point , the system can be expressed as

As a result,

25

Equation (25) can be employed as a substitute for the Newton polynomial to generate the numerical solution of the system.

26

Accordingly, the preceding equation can be reformulated as follows

27

Consequently,

28

Equation (28) can be employed to evaluate the aforementioned integrals

Substituting these values into equation (28) yields the following computational procedure

29

Similarly, corresponding expressions can be derived for the remaining system equations in (24).

Where

Author contributions

Muhammad Farman: Formal analysis, methodology, writing, reviewing, editing, software, and resources. Khadija Jamil: Data curation, investigation, software, writing, and original draft. Saba Jamil: Formal analysis, software, editing, and resources. Ausif Padder: Conceptualization, formal analysis, software, methodology, writing, reviewing, and editing. Hijaz Ahmadf: Data curation, investigation, software, writing, and original draft. Kaushik Dehingia: Formal analysis, software, editing, and resources. Mustafa Bayram: Formal analysis, software, editing, and resources.

Funding

Open access funding provided by Symbiosis International (Deemed University).

Data availability

All data generated or analyzed during this research work are included in this published article.

Declarations

Consent to participate

Each author has approved of and agreed to submit the article.

Consent to publish

Each author has approved of and agreed to submit and publish the article.

Competing interests

The authors declare no competing interests.