The intricate dynamics of plant-insect interactions have long captivated ecologists and biologists due to their profound implications for ecosystem stability and agricultural productivity. A notable example of such interaction is the relationship between plants and herbivorous insects, such as aphids, which are significant contributors to crop damage and yield loss globally [1]. Plants have evolved sophisticated defense mechanisms to counteract these insect attacks, initiating a complex interplay of molecular signals and physiological responses aimed at mitigating damage and ensuring survival [2, 3]. In this context, the role of Botrytis Induced Kinase-1 ( B I K 1 ) has emerged as a critical component of the plant defense arsenal. BIK1 is involved in the phosphorylation of the flagellin receptor FLS2 and BAK1 proteins, initiating a cascade of defense responses that include the production of phytoalexins and other defensive compounds [4]. These responses are modulated by signaling pathways mediated by jasmonic acid (JA) and salicylic acid (SA), which are crucial for the activation of plant immune responses [5, 6]. Recent research has highlighted the interaction between BIK1 and Phytoalexin Deficient-4 (PAD4), another key player in the plant defense response. PAD4 is known to enhance the plant’s resistance to aphids by promoting the synthesis of antixenotic compounds that deter insect feeding [7]. However, B I K 1 has been shown to suppress PAD4 expression, thereby modulating the plant’s defensive capabilities [8]. Given the pivotal roles of B I K 1 and P A D 4 in plant-insect interactions, understanding the molecular underpinnings of their interaction and its impact on plant health is of paramount importance. We further enhance the deterministic model [9] by incorporating MEMS-based feedback mechanisms, allowing for real-time dynamic adjustments in plant biomass and molecular signaling based on environmental conditions and sensor data. Our study addresses critical gaps in existing research by incorporating stochastic elements to capture the inherent variability and extreme events in these systems, offering novel insights into how molecular signaling pathways influence plant defense mechanisms and herbivore population dynamics. The model simulates the dynamics of plant biomass (P), insect herbivore density (I), P A D 4 protein, and B I K 1 protein under various conditions. Mathematical modeling has evolved from deterministic approaches, such as those by [10, 11], to stochastic models that account for variability and randomness, as highlighted by [12, 13], and [14]. These stochastic models provide a more realistic depiction of ecological systems by incorporating environmental noise, as seen in works like [15, 16].

The stochastic processes, particularly Lévy noise, enhances the model’s ability to capture extreme events, such as insect outbreaks, which are not well represented by Gaussian noise alone. Stochastic modeling techniques, as demonstrated by [17, 18], offer valuable insights for predicting population dynamics, evaluating control strategies, and understanding the influence of environmental variability. The use of Lévy noise in our model allows for the simulation of significant, abrupt changes in plant-insect interactions, providing a comprehensive representation of random phenomena. This approach underscores the importance of stochastic models in understanding complex biological systems and informing agricultural and environmental management practices. The characteristic function of a Lévy process is given by, E e i u L t = exp t i u γ − 1 2 σ 2 u 2 + ∫ R \ 0 e i u x − 1 − i u x 1 | x | < 1 ν d x , where γ represents the drift coefficient, σ 2 is the variance of the Gaussian part, and ν is the Lévy measure that describes the jump intensity and distribution.

The plant-insect interaction system is a well-studied model in ecology, evolving from predator-prey analogies to more sophisticated mathematical models incorporating plant immunity concepts. This study extends these models by including molecular interactions in the plant defense system, inspired by [9]. The primary objective of this research is to develop a stochastic mathematical model to analyze plant-insect interaction dynamics, focusing on the molecular interplay between P A D 4 and B I K 1 proteins. It explores the impact of noise types, deterministic, Gaussian, and Lévy noise on the system’s variability and stability. This research contributes to understanding plant-insect interactions at the molecular level, with potential applications in agriculture. It highlights the benefits of adaptive feedback control for plant protection, dynamically adjusting to changing conditions. By incorporating noise, especially Lévy noise, the model captures extreme fluctuations, providing a realistic depiction of variability in plant-insect interactions. The study aims to answer key questions about the effects of control strategies, noise types, and the broader ecological implications of these findings.

The system of differential equations governing the deterministic model is [9], d y 1 d t = a 1 y 1 1 − y 1 K − a 2 y 1 y 2 , d y 2 d t = a 3 y 1 y 2 − a 4 y 2 − a 5 y 3 y 2 , d y 3 d t = a 6 y 1 y 2 − a 7 y 3 − a 8 y 3 y 4 , d y 4 d t = a 9 y 1 y 2 − a 10 y 4 − a 11 ⋅ I n BIK1 . (1)

The stochastic differential equations is, d y 1 = a 1 y 1 1 − y 1 K − a 2 y 1 y 2 d t + σ 1 y 1 d W 1 , d y 2 = a 3 y 1 y 2 − a 4 y 2 − a 5 y 3 y 2 d t + σ 2 y 2 d W 2 , d y 3 = a 6 y 1 y 2 − a 7 y 3 − a 8 y 3 y 4 d t + σ 3 y 3 d W 3 , d y 4 = a 9 y 1 y 2 − a 10 y 4 − a 11 ⋅ I n BIK1 d t + σ 4 y 4 d W 4 , (2)

The choice of noise addition is grounded in both biological and mathematical reasoning. Biologically, in ecological systems like plant-insect interactions, fluctuations in population densities and molecular levels are typically influenced by current population or protein levels [19–21]. Environmental stresses, such as insect outbreaks or weather conditions, do not affect the system uniformly but have a state-dependent effect: larger populations or protein levels experience more significant impacts. As a result, multiplicative noise (state-dependent noise) is biologically appropriate because it reflects that larger variables are more susceptible to noise. Mathematically, adding noise terms directly allows for stochastic perturbation while preserving the structure of the deterministic model. This approach simplifies the analysis and is commonly used in models involving stochastic dynamics through stochastic differential equations (SDEs), providing a meaningful representation of randomness.

P A D 4 is a crucial gene that helps Arabidopsis plants defend against aphids. When aphids feed on the plant, P A D 4 gets activated, boosting the plant’s defenses. P A D 4 helps in two main ways: by producing substances that deter aphids (antixenosis) and by creating chemicals that can harm aphids (antibiosis). P A D 4 ’s activity is regulated by other genes like TPS11 and LOX5. TPS11 deals with sugar metabolism, while LOX5 is involved in fatty acid metabolism. TPS11’s activity increases in the plant shoots when aphids attack, while LOX5 activity increases in the leaves. Aphid attacks also trigger another gene, MPL1, which helps in defense but works independently of P A D 4 . The B I K 1 gene suppresses P A D 4 . If B I K 1 is less active (as in B I K 1 mutants), P A D 4 levels rise, making the plant more resistant to aphids. The interaction between B I K 1 and P A D 4 involves ethylene (ET), a plant hormone that also plays a role in defense. P A D 4 is essential for a full defense response against aphids, involving the production of deterrents and toxins. The plant’s ability to emit ethylene is crucial for repelling aphids and is dependent on P A D 4 activity [22–27]. The mathematical model formation on this basis is shown in Figure 1A. This underlying process is shown in Figure 1B. The variables of the model are shown in Table 1.

The key assumptions of the model 1,2 are as follow,

i. The model assumes that the plant biomass (P), insect herbivore density (I), PAD4 protein (PAD4), and BIK1 protein (BIK1) are homogeneously mixed within the environment. This means that these components are evenly distributed, and their interactions occur uniformly throughout the system.

In the following subsections, various numerical results are provided. The values of the parameters used in the model are given in Table 2. The Euler-Maruyama method is used to solve the stochastic differential equations iteratively as follow, y 1 t + Δ t = y 1 t + k 1 y 1 t 1 − y 1 t K − k 2 y 1 t y 2 t Δ t + σ Δ t ⋅ η 1 t , y 2 t + Δ t = y 2 t + k 3 y 1 t y 2 t − k 4 y 2 t − k 5 y 3 t y 2 t Δ t + σ Δ t ⋅ η 2 t , y 3 t + Δ t = y 3 t + k 6 y 1 t y 2 t − k 7 y 3 t − k 8 y 3 t y 4 t Δ t + σ Δ t ⋅ η 3 t , y 4 t + Δ t = y 4 t + k 9 y 1 t y 2 t − k 10 y 4 t − k 11 ⋅ 0.1 Δ t + σ Δ t ⋅ η 4 t , where η i ( t ) represents the noise term. For Gaussian noise, η i ( t ) is from a normal distribution N ( 0,1 ) , η i t ∼ N 0,1 .

For Lévy noise, η i ( t ) is from a Lévy distribution. η i t ∼ Lvy α , β . where, Δ W t ∼ N 0 , Δ t . And Δ W t = Δ t ⋅ Z , where Z is a standard normal random variable ( Z ∼ N ( 0,1 ) ) .

The integration of Micro-Electromechanical Systems (MEMS) in plant systems has revolutionized the field of precision agriculture by providing real-time monitoring and adaptive control capabilities. MEMS sensors are widely used for tracking environmental parameters such as soil moisture, temperature, humidity, and nutrient levels, enabling more efficient and precise irrigation and fertilization strategies. For example, The increasing demand for the miniaturization of biosensors has driven growing interest in microelectromechanical systems (MEMS) [44, 45], along with nanoelectromechanical systems (NEMS) and microfluidic or lab-on-a-chip based biosensors [35, 46]. These compact systems provide enhanced accuracy, sensitivity, specificity, and cost-efficiency, while also offering high-performance biosensing capabilities. MEMS-based biosensors leverage a range of detection methods, including optical, mechanical, magnetic, and electrochemical approaches. For optical detection, probes like organic dyes, semiconductor quantum dots, and other fluorescence markers are commonly employed. In magnetic MEMS biosensors, nanoparticles such as magnetic, paramagnetic, or ferromagnetic particles are utilized. Mechanical MEMS biosensors operate based on changes in surface stress or mass [47], where biochemical reactions or analyte adsorption on the cantilever induce surface stress changes. Electrochemical MEMS biosensors, on the other hand, rely on amperometric, potentiometric, or conductometric detection methods [48–51]. For each variable, sensor-based feedback terms can be introduced to adjust the differential equations based on real-time data gathered by MEMS. This can include intervention strategies or feedback loops that modify plant biomass growth, herbivore density, and molecular signals. We extend the stochastic Equation 2 to include MEMS input, denoted by a new variable M ( t ) , representing MEMS sensor feedback, d y 1 = a 1 y 1 1 − y 1 K − a 2 y 1 y 2 + M 1 t d t + σ 1 y 1 d W 1 , d y 2 = a 3 y 1 y 2 − a 4 y 2 − a 5 y 3 y 2 + M 2 t d t + σ 2 y 2 d W 2 , d y 3 = a 6 y 1 y 2 − a 7 y 3 − a 8 y 3 y 4 + M 3 t d t + σ 3 y 3 d W 3 , d y 4 = a 9 y 1 y 2 − a 10 y 4 − a 11 ⋅ I n BIK1 + M 4 t d t + σ 4 y 4 d W 4 .

For each variable y i ( t ) , the MEMS feedback terms M i ( t ) are defined as follows, M 1 t = β 1 ⋅ Control Function for  y 1 t , M 2 t = β 2 ⋅ Control Function for  y 2 t , M 3 t = β 3 ⋅ Control Function for  y 3 t , M 4 t = β 4 ⋅ Control Function for  y 4 t .

The parameters β 1 , β 2 , β 3 , and β 4 represent scaling factors that control the strength of the feedback applied by the MEMS sensors for each variable. Figure 9 presents the results incorporating MEMS feedback into the system. In 5, the plant biomass y 1 shows a significant oscillatory behavior in the stochastic case, with a generally increasing trend after an initial decline. The MEMS feedback helps stabilize biomass growth, resulting in quicker recovery compared to the deterministic solution, which shows a smoother, slower increase. In 5, the insect herbivore density y 2 demonstrates a steady decline, with MEMS further suppressing insect growth. The stochastic model exhibits more variability and faster suppression of herbivores, while the deterministic model shows a smoother, more gradual reduction. The PAD4 protein dynamics y 3 in five show an initial peak followed by a decline and eventual stabilization, with MEMS feedback leading to more active and pronounced fluctuations in the stochastic case compared to the smoother deterministic curve. In 5, the BIK1 protein y 4 follows a similar trend to the other variables, with MEMS inducing stronger oscillations in the stochastic model, particularly towards the later stages. The deterministic solution, on the other hand, stabilizes more smoothly without such fluctuations. The implementation of MEMS introduces dynamic feedback into the system, leading to faster recovery, better control of herbivore density, and more pronounced fluctuations in protein levels. This highlights MEMS’ effectiveness in dynamically regulating plant-insect interactions and molecular signals.

Our comparison of noise conditions revealed distinct behaviors in plant biomass and insect herbivore density. Deterministic conditions produced smooth trajectories, while Gaussian noise caused moderate fluctuations, and Lévy noise led to extreme variations, effectively capturing sudden changes in biological systems. High P A D 4 levels and low B I K 1 levels were associated with increased plant biomass and reduced herbivore density, indicating effective plant defense mechanisms. Moderate noise intensity ( σ = 0.05 ) sustained P A D 4 activity, providing a balance between system stability and variability.

Lévy noise had a significant impact on P A D 4 and herbivore density, with moderate noise intensities sustaining P A D 4 activity for optimal plant defense. Queir plots revealed non-linear regulatory interactions between P A D 4 and B I K 1 , emphasizing the role of stochastic dynamics in biological networks. The results suggest that moderate noise intensity is optimal for maintaining system function, with MEMS-based feedback showing potential for adaptability by continuously adjusting key parameters.

This study provides a comprehensive analysis of the dynamic interactions between plants and insect herbivores, focusing on the molecular interplay between P A D 4 and B I K 1 proteins, under various control strategies and noise conditions. The inclusion of different noise types in the model reveals significant insights into the system’s variability. Gaussian noise introduces moderate fluctuations around the deterministic trends, while Lévy noise induces more extreme fluctuations and variability, capturing the sudden changes and extreme events often observed in biological systems. This emphasizes the necessity of incorporating stochastic elements, particularly Lévy noise, to accurately model and understand the complex dynamics of plant-insect interactions. By incorporating MEMS-driven adaptive feedback, we demonstrated how real-time sensor-based interventions can enhance the system’s stability and effectiveness, particularly in maintaining plant health and controlling insect populations under varying noise conditions. Our study advances our understanding of plant-insect interactions and offers valuable insights into the role of noise and control strategies in modulating system behavior. These findings have implications for the development of effective management strategies in agricultural and ecological contexts, paving the way for more robust and sustainable approaches to mitigating the impacts of insect herbivory on plant health and productivity.