In recent years, researchers have paid more attention to integrated pest management, and its use in the crop field has increased. This strategy emphasizes the implication of biological control factors to minimize the credence of pesticides. In agriculture, forest management, and population health, microbiological pesticides play a significant role in incorporated pest management. In the case of crops, biopesticides provide noticeable pest management dependability as part of incorporated pest management [1]. In North America and Europe, viruses are used as pest control agents against insect pests [2–4]. Agricultural-related awareness programs on radio, TV, mobile and other media might aid in disseminating agricultural knowledge among farmers and ranchers about the hazards of pesticide use on human health as well as the other linked environmental concerns [5–9]. Pesticide overuse is significantly associated with farmers' lack of pesticide knowledge, the impact of pesticide retail outlets, and inaccessibility to non-synthetic pest control methods, while the tendency to overuse reduces higher levels of the learning process in Integrated Pest Management [10]. As a result, farming awareness is essential to prevent crop losses having the least amount of detrimental side effects [11, 12]. Pesticide communication campaigns made it easier for farmers to understand the substantial risks pesticides pose to public health and the ecosystem, and to limit harmful consequences. Farmers primarily learned about pesticide use and hazards through oral communication [13].

Time delay is a key feature in both natural and manmade systems. Kuang provided an example to demonstrate the significance of time delay [14]. He claimed that animals require time to digest their food before moving on to other activities and reflexes. As a result, any species model with no temporal delay is at best an approximation [15]. Many systems as well as industrial plants, including biological systems, machining, metal forming, thermal acoustic systems and many others experience time delays [16–19]. Furthermore, dynamical systems including time delay exhibit far more complex behaviors than those without delay in time [20]. There are two key reasons for the presence of temporal delays in prey-predator systems [21, 22]. The first is the gestation period, and the second is the maturation period. As a result, incorporating delays into predator-prey model is essential for ensuring the realistic nature of these models and demonstrating how well the population dynamics of such models are influenced by previous relevant information. In fact, time delays have a significant impact on the overall characteristics of dynamic systems. Many publications in the literature have described the theory-based analysis of the prey-predator model involving time delay, such as hunting delay [23], dispersal delay [24], predator gestation period [25], as well as intra-specific competitive pressure generated feedback delay [26].

There is a lack of mathematical modeling on agricultural awareness to limit plant pests as well as diseases. Daudi et al. [27] proposed a dynamic model using the fractional derivative operator for maize growth as well as interactions with fall armyworms. They described the basic reproductive number, which was the average amount of newborns generated by a single female moth over the course of a lifespan. The resilience of the trivial equilibria, as well as the positive equilibria of the dynamical system, were investigated by Li et al. [28] and the threshold requirements for pest destruction and system permanence were determined. Xiang et al. [29] explored the influence of MD controls on the dynamical behavior of the pest systems by adding the gestational delay and sex pheromones. First, the system's bounds, stability, as well as bifurcation were discussed. Second, by integrating the constraint violating function, an optimized control problem depending on sex pheromone and pesticides was reduced into an analogous optimized parameter decision issue. The bifurcation control of mosaic viruses fractional order infection models for Jatropha curcas with agricultural awareness and an executing delay was examined by Liu et al. [30] Hopf bifurcation generated by executing delay was explored for the unregulated system by examining the corresponding characteristic equation. They found that changing the fractional order had a considerable impact on bifurcation dynamics. Kumari et al. [31] employed the Integrated Pest Management (IPM) technique to create a mathematical model that used a combination of chemical and biological management. The feasibility of pest eradication and non-trivial equilibria state were examined, and the local stability of the pest eradication equilibria state was investigated further. Shi et al. [32] presented a unique population Smith framework with continual delay as well as impulsive phase adaptive control and examined how it may be used in pest management. The model's singularity was first qualitatively examined, and then the presence and uniqueness of order one periodical orbits were considered in order to calculate the frequency of chemical control implementation. A Filippov prey predator model incorporating time delay was introduced by Arafa et al. [33], where the delay time indicated the changes in the natural enemy's growth rate before discharging it to fatten up pests. The bifurcation parameter time delay was used to derive the threshold constraints for the stability of the equilibrium. Utilizing Hopf bifurcation, it was proven that whenever the time delay parameter crosses through specific critical levels, a periodical oscillation phenomenon arises. They also established the equation of slipping motion and addressed the sliding phase dynamics using the Filippov convex approach. Al Basir et al. [34] presented prey predator framework for assessing the impact of delay time in crop pest management utilizing agricultural awareness-based treatments. The authors indicated that the application of biological insecticides is proportionate to the pest population density on the plantation. The presence of steady states, as well as their stability, had been examined. Allen-Perkins and Estrada [35] constructed an epidemic model to explore disease transmission and control in planted agricultural farms as a long-term pest management strategy. In an epidemiological susceptible, infectious and removed model, the model includes the mobility of aphids carrying a virus in an agricultural farm, the spatial dispersion of plants in a planted field, and the existence of “trapped crops.” Abraha et al. [36] studied a mathematical model for crop pest control that took into account plant biomass, pests, and the impact of farmer awareness. The basic reproductive number and delays in time were used to determine the presence as well as stability of the equilibria. Whenever time delays approach critical values, stability transitions happen due to Hopf-bifurcation. The delayed system's cost-effectiveness was assessed using optimal control-theory. Rossini et al. [37] presented a mathematical framework for calculating the analytical solutions to the second variant of the distributed delay model. The researchers also investigated how the model behaved when it came to representing the population of insect pests in various environmental factors, particularly with respect to temperature. Al Basir et al. [38] proposed a mathematical model including delay to investigate the impact of public awareness on agricultural pest management using crop biomass, and pests. The basic reproductive number was used to determine the presence and stability conditions of the equilibria. The Hoph bifurcation analysis was performed at the epidemic equilibria with time delay as the bifurcation parameter.

Numerical approaches are frequently employed in science and engineering to solve mathematical problems for which exact solutions are difficult or impossible to grab. Only a limited number of differential equations can be solved analytically. There are several analytical methods to solve ordinary differential equations (ODEs). Although several ODEs have closed form solutions that can be obtained using renowned analytical methods, numerical methods must be evolved and applied to obtain numerical solutions of a differential equation under a predefined initial history. Many researchers used a variety of numerical methods to simulate the solution of mathematical models, acquiring results that were more accurate than those found in the literature, such as [39–43]. Researchers have recently focused their efforts on the numerical solutions of numerous mathematical models in the realm of epidemiology, such as the HIV model [44], COVID-19 [45], plant disease model [46], tuberculosis propagation model [47], computer virus transmission model [48]. Although the above mentioned techniques have high precision and consistency but they require considerable memory and long computational cost. Consequently, the procedures for this technique present noteworthy challenges that may be resolved in order to ensure that the solution is precise and consistent. Therefore, several efforts have been made by researchers to develop efficient techniques for solving linear and non-linear ODE systems [49–51]. The importance of numerical solutions is emphasized in the literature listed above. As a result of these considerations, the authors have decided to use the ADS (Adams) and ERK (explicit Runge–Kutta) numerical solvers to solve the delay differential system [52–56]. The Adams predictor-corrector approach [46, 57–60] is also a more efficient and straightforward numerical tool for solving delay differential systems.

In order to manage crop pests, insecticides or other preventative measures are frequently used. The emphasis is shifted to educating farmers about alternative techniques including rotation of crops, biological insect control, and cultural practices by incorporating agricultural knowledge into the pest management approach. Through integration, agricultural pest control may be approached holistically and sustainably [61]. To model and simulate the dynamics of agricultural pest populations while taking into account a variety of elements such as environmental conditions, insect life cycles, and farming practices, numerical analytic techniques such as the use of differential equations and optimization methods can be used. The model can offer insights to the population dynamic of pests, the effects of various management techniques, and the ideal time for putting control measures into place by using numerical analysis. By offering data-driven advice to farmers and decision-makers, this quantitative method improves the decision-making process [62]. In general, the notion is innovative since it addresses agricultural pest control by combining principles of farming awareness with numerical analytic methods. This multidisciplinary approach emphasizes eco-friendly and sustainable practices while also offering a quantitative foundation for analyzing and improving pest management tactics. Combining these factors helps researchers create agricultural pest management strategies that are both more practical and ecologically responsible. This research may help with the creation of efficient and long-lasting farming awareness campaigns, the adoption of integrated pest management techniques, and the alleviation of farmer difficulties brought on by crop pests. Time delay models really have the potential to increase complexity because of the intrinsic properties of temporal latencies and their impact on system dynamics. The implementation of numerical solutions for non-linear delayed systems may be challenging and computationally expensive. Researchers may require sophisticated software, outstanding durability computing devices, as well as expertise in both computational and mathematical modeling strategies. In this study, we used state-of-the-art numerical techniques like Adam (ADS) and explicit Runge–Kutta (ERK) to find the numerical solution of the non-linear delayed CPM model. The presented study has the following salient features:

The rest of the article's layout is as follows: In the second section, the non-linear delayed CPM model is formulated. The third section provides a detailed overview of the methodology. The fourth section provides the analysis, discussion, and graphical interpretation of approximate solutions. The fifth section presents the analysis-based conclusion.

The model [38] integrated agricultural biomass, pests, and the population's awareness. Crop biomass density, susceptible pest density, infected pest density, and aware people density are the four compartments incorporated in the model. Logistical evolution for the densities of crop biomass is assumed, since crop fields have a finite size (though it might be large), with a net growth rate a and N is carrying capacity. Pests that are susceptible to the crop are attacked, significantly reducing the crop. Let λ represent the pest attack rate on crops [38].

To take into consideration their interests as well, aware individuals may keep the crops under observation and, if properly trained, will squirt biopesticides or integrate them into fertilizer applications to manage the insect invasion. The massive term lA(t)S(t) can be used to introduce the awareness action rate l, which results from deliberate human activities and control actions like the application of biopesticides [38].

Pests that are infected can also harm the crop, ϕδ, although at a much lesser rate ϕ < 1. Here, c represents the pests' natural death rate, and the infection mortality rate β is a result of knowing human behavior, like the application of insecticides [38].

μ and μ₁ represent the “conversion efficacy” of susceptible as well as infected pests, or how well the pests can use plant components. Since pests influenced by pesticides are less effective, μ > μ₁. Because of media initiatives and increased public awareness, farmers now have a higher level of awareness, which is denoted by A. Additionally, it is expected that the exposure of the resilient pests influences the rate at which local information is increasing at a rate d. The loss of memory causes farmers' levels of consciousness to decline at a rate υ [38].

A delay in observing the number of pests or their activity might occur in a field. Typically, this prediction is produced by studying past incidences of pest prevalence. As a consequence, there are differences in the degree of awareness and the application of preventive countermeasures. Enforcement of such remedies is anticipated to be delayed. The number of pests present at time (t-τ) (or time τ > 0 in some cases) will determine how intense the awareness campaigns are at time t.

The following modified mathematical model results from the abovementioned assumptions [38].

and initial conditions are as:

Each parameter used in the mathematical model is described in the nomenclature. The parameters' descriptions and default values are listed in Table 1 as per in Al Basir et al. [38]. These default parameter values are used to generate each scenario. The non-linear delayed CPM model by using numerical values can be mathematically defined for one of the cases as:

This section includes a detailed presentation of the learning methodologies that are used to determine the approximate solutions of the non-linear delayed CPM model.

The approximate numerical solutions for compartments B(t), S(t), I(t), and A(t) of the non-linear delayed CPM model are presented here in this section. The dynamics of the non-linear delayed CPM model are investigated for sundry scenarios each comprising of 1–5 cases by means of ADS and ERK numerical solvers with input points from 0 to 300 and step size 0.5. The approximate solutions for the sundry scenarios with 1–5 cases of the non-linear delayed CPM model are computed by varying the crop biomass growth rate, rate of pest attacks, pest natural death, disease associated mortality rate and memory loss of aware people.as listed in Table 2. Figure 1 presented the flowchart of the designed methodology.

The dynamics of crop biomass density are shown in Figures 2A, B, respectively, using the ADS and ERK numerical solvers for the variation in crop biomass growth rate, i.e., a for the non-linear delayed CPM model. The crop biomass density has been found to increase as the value of a increases. Figures 2C, D for various values of a illustrate the effects of susceptible pest density. The graph shows that as the value of a increases, so does the density of pests that are susceptible. Figures 3A, B show how infected pests' behavior varies as the value of a changes. There is an increase in the density of infected pests for larger values of a. The effects of people's level of awareness for various values of a are depicted in Figures 3C, D. The graph illustrates how increasing the value of a raises the level of awareness. Table 3 presents the numerical results for the classes B(t), S(t), I(t), and A(t) for scenario 1, case-1 of the non-linear delayed CPM model. Using the strength of ADS and ERK numerical solvers for cases 1 to 5 of scenario 2, the dynamics of the non-linear delayed CPM model for the rate of pest attacks, i.e., δ, is investigated for all four classes B(t), S(t), I(t), and A(t) and graphically shown in Figures 4, 5 respectively. The numerical outcomes for the classes B(t), S(t), I(t), and A(t) for scenario 2, case-1 of the non-linear delayed CPM model are shown in Table 3. Raising the value of δ increases the density of crop biomass, as presented in Figures 4A, B. For case-1 of pest attacks rate, the maximum value of B(t) is approximately between 5 to 45, oscillates from 0 to 150 days, and then maintains steady state behavior. The maximum value for case-2 is between 5–30 and it initially exhibits oscillating behavior in the range of 0 to 250 days before becoming stable in the range of 250 to 300 days. Cases 3 to 5, as depicted in Figures 4A, B, exhibit oscillations with varying amplitudes across the time interval. As the value of pest attacks i.e., δ expanded, the density of susceptible pests also increased, as seen in Figures 4C, D.

Susceptible pest density demonstrated oscillatory behavior from 0 to 150 days before returning to steady state behavior, whereas cases 2 to 5 exhibit oscillatory behavior with varying amplitudes from 0 to 300 days, as shown in Figures 4A, B. For compartment I(t) of the non-linear delayed CPM model, Figures 5A, B depict the effects of pest attacks rate. The graphs show that increasing the value of δ will result in decreasing the infected pest density. The impact of pest attacking rate, i.e., δ is also determined for compartment A(t) of the non-linear delayed CPM model. As seen in Figures 5C, D, the awareness level decreases as the value of pest attacks grow.

Similarly, for scenario 3 of non-linear delayed CPM model, the dynamics of the four compartments B(t), S(t), I(t), and A(t) are explored by varying the value of pest natural death rate i.e., c, which is represented by c and graphically portrayed in Figures 6, 7 respectively. The numerical solutions for compartments B(t), S(t), I(t), and A(t) for scenario 3, case-1 of the non-linear delayed model are computed and listed in Table 3. The influence of the pest's natural death rate on crop biomass density using the ADS and ERK numerical solvers respectively, is shown in Figures 6A, B. The effects of the pest natural death rate on the density of susceptible pests are shown in Figures 6C, D. It is noticed that the number of susceptible pests reduced as c increased. Figures 7A, B demonstrated how the density of infected pests decreases as the value of c rises. In Figures 7C, D, the level of people's awareness can be analyzed. It is worth noting that the larger value of the natural pest's death rate i.e., c, causes a decrease in people's awareness.

The dynamical behavior of the four compartments B(t), S(t), I(t), and A(t) for cases 1 to 5 of scenario 4 with the variability in disease associated death rate, i.e., β is analyzed and graphically portrayed in Figures 8, 9 respectively. The numerical solutions for scenario 4, case-1 of the non-linear delayed CPM model are calculated for all four classes B(t), S(t), I(t), and A(t) and presented in Table 4. The dynamics of crop biomass density are portrayed in Figures 8A, B exploiting the potential of the ADS and ERK numerical solvers for the variability in the disease associated death rate, i.e., β. It has been found that the crop biomass density falls as the value of β increases. Figures 8C, D show the effect of disease associated death rate on the density of susceptible pests. It is evident from Figure that raising the value of β would lead to a rise in the density of susceptible pests. The behavior of infected pest density for the variation in disease associated death rate is shown in Figures 9A, B. As the value of β is raised, it may be observed that the density of infected pests will decrease. Figures 9C, D show the impact of disease associated death rate against awareness level in people. The graphical representation presented that increasing the value of β causes the awareness level to decrease.

Using the strength of ADS and ERK numerical solvers, the dynamics for memory loss of aware people, i.e., υ , are investigated for all four compartments B(t), S(t), I(t), and A(t) for scenario 5, cases 1 to 5 of the non-linear delayed CPM model. The numerical outcomes of all four compartments B(t), S(t), I(t), and A(t) for case-1 of scenario 5 are provided in Table 4. The behavior of crop biomass density for the varying values of υ is depicted in Figures 10A, B, and it can be observed that crop biomass density decreases for higher values of υ. Figures 10C, D illustrated how the density of susceptible pests increases as the value of υ increases. Figures 11A, B show the dynamics of infected pest density for the variation in memory loss of aware people, i.e., υ. One may witness that the density of infected pests increases continuously in the first three cases, for υ = 0.01, 0.02, and 0.03, and then decreases again in the subsequent two cases, for υ = 0.04, and 0.05, in the interval of 0 to 300 days. Consequently, the density of infected pests shows varied behavior for different values of υ. Figures 11C, D portrayed the effect of memory loss in aware people, i.e., υ on awareness level compartment A(t). It is clearly noticed from Figures 11C, D that the awareness level is decreased as the value υis increased.

In this research, the numerical approximate solution of the non-linear delayed CPM system supervised by ODEs is investigated effectively to portray the dynamic impacts of unforeseen interactions between crops and pests, rehabilitation strategies, and environmental factors across time. Based on the presented model, the dynamic nature of crop biomass density [B(t)], susceptible pest density [S(t)], infected pest density [I(t)] and awareness level of the population [A(t)] may be forecasted effectively. Analysis based on the approximate numerical outcomes as well as graphic interpretations of the non-linear delayed CPM model is carried out by means of sundry scenarios by varying the different parameters utilized in the model. The approximate numerical solution of the non-linear delayed CPM model is computed by exploiting the state-of-the-art Adams (ADS) and explicit Runge–Kutta (ERK) numerical techniques. Compared with real-time models, delayed models exhibit greater realism because they take into account the interval between contact and infection. As delay affects processes along with dynamics, mathematically it impacts stability. This analysis can help to create predictive models for upcoming outbreaks and shed light on the efficacy of various pest management techniques. The numerical analysis that is being given makes it possible to optimize pest control tactics, analyze risks, educate people, and pursue continual improvement. It is essential for improving agricultural methods, reducing crop losses, and advancing environmentally friendly pest control strategies.

In the future, soft computing approaches based on artificial intelligence algorithms may be used to study the dynamics of epidemic models and other non-linear systems [68–73].

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Conceptualization, methodology, and writing—review and editing: NA and MR. Software: AK. Validation: MR. Formal analysis: MS and MR. Investigation: IA. Writing—original draft preparation: NA. Visualization: NA and MS. Project administration: AK and IA. All authors have read and agreed to the published version of the manuscript.