The dynamical models are considered very important for the research community. There are various dynamical models, out of which the biological models have gained huge significance. Some of the important biological models are the heat conduction model of the human head, nonlinear food supply chain model, dynamics of nervous stomach nonlinear model, biological nonlinear Leptospirosis system, a nonlinear corneal system of eye surgery using the capability of Morlet wavelet artificial neural networks, novel prevention category in the HIV system, nonlinear influenza disease model, waste plastic management in the ocean, underlying immune-chemotherapeutic treatment for breast cancer using the neural networks, nonlinear prey-predator system, vaccination and Wolbachia on dengue transmission dynamics in the nonlinear model, SIDARTHE COVID-19 pandemic differential model, dynamical nonlinear susceptible, infected and quarantine differential model, nonlinear dengue fever SIR system, infectious disease model considering its anatomical variables, dynamics of nonlinear SITR COVID-19 and nonlinear HIV infection model with CD4+ T-cells. The interface of the predator-prey model is known as the basic interspecies relationships in ecology and biology as well as it represents the basic block based on the complex food chain, biochemical network, and food web networks. Based on the influential investigations of Freedman and Aiello, the growth species systems along with stage construction have drawn substantial consideration. Integrating stage construction for the predator in the network, Xu constructed a Lotka–Volterra delayed type of prey-predator model. Moreover, the investigations based on the stage constructions for both prey and predator have been defined in the modeling. Few interesting outcomes based on the dynamical performances of prey and predator systems have also been discussed in different studies.
There have been many presentations related to predator-prey and top-predator based on the food-chain systems that have been provided in the literature work, like the behavior of the dynamical models based on the ratio-dependent three-species food-chain systems. Cui et al. reported the bifurcation and stability based on the periodic results using the food-chain three-species model. Pei et al. provided a delayed form of the three-species ecosystem using the Holling type of functional response as well as the dynamical system behaviors. Mbava et al. studied the food-chain dynamics along with the species’ diseases. Considerably, the current literature based on the theoretical investigations using the predator-prey and top-predator model is predominantly concerned with/without stage construction. The stage construction for both prey and predator seems to be rare. Such investigations are, however, significant for the researchers to understand the dynamical features based on the food chain systems. Alternatively, the existence of time delays can happen in various models like neural network systems, economic systems, population systems, epidemic models, etc.
The mathematical form of the systems accomplishes a developing area of interest with numerous new capacities in the models of food chain supply. The researchers are taking a keen interest in the food supply chain mathematical models due to a rising in the number of submissions, e.g., ecological significances based on the global bifurcations using the models of the food chain, environmental models accomplish the food chain supply status, length of food chain based on the trophic competences in modest stoichiometric food aquatic chain systems, expecting terrestrial behavior of the food chain based on the xenobiotics, food chain types of three species using the lower bound based on the bottom population, interaction model based on the food chain oil spill based on the coastal waters, etc. There have been a number of studies that have been presented in order to present the industrial implementations based on the food packaging to explore the chemical pollution due to the food delivery and storage, dust, antagonistic conservation circumstances gas, light, and humidity. The storage of the supply, as well as food chain packages, is known as the complex field in the food industry, which is to be addressed exactly to authorize food security and improve the shelf life. Easy disposal, inertness, Low charge of the manufacture, and polymeric reusability are the major goals in the food industry. The issues of bad health in these decades are increasing because of the unawareness of these specified features.
In the food chain models, the “Allee effects” play a significant role. These effects were introduced by a well-known scientist Alee in 1930. The addictive term, together with strong and weak behavior in the Allied effects, has a huge significance. The Alee effects assign the evolution to lessen the growing ratio through the small public number. These effects have a number of applications in the fishery industry, invertebrates, plants, and vertebrates. Sometimes, the Allee effects perform negative impacts on the exemption of population dynamics through the fishery. The division of the Allee effects has been performed into addition and multiplication. Firstly, Singh et al. discovered the double form of the Allee effects using the improved Gower-Leslie model in the form of prey-predator, where the prey population represents the numerous junction linked to the appropriate parameters. Vinoth et al. presented a mathematical system to examine the dynamical form of the food supply model using the additive state of the Allee effects. The communication state of the food supply systems has been performed by using a couple or more than a couple predators and prey. A differential food supply chain form with the mutual qualitative analysis together with the multiple relationships is also presented in various studies.
Thus, using the Meyer wavelet neural network along with the optimization performances of the local and global search schemes is a tool to present the numerical representations of the food supply system. An error-based cost function is developed by using each dynamics of the food supply chain model and its initial/boundary conditions. This cost function will be optimized through the global and local search methods. The use of the Meyer wavelet neural network has never been done in the literature. Hence, it is the best choice for the researchers to design such a neural network that can be constructed by using the combination of the Meyer wavelet and the hybridization of the local/global search schemes. In recent years, stochastic solvers have been exploited not only in the integer order dynamical models but these solvers have been used to provide the numerical solutions of the fraction order dynamical systems. Moreover, the singular, predictions, delay, nonlinear, pantograph, and fluid dynamical models have also been solved by using stochastic numerical computing schemes.
The significant features of the stochastic algorithm are :
- The stochastic procedure based on the Meyer wavelet neural networks are designed first time to solve the three species food chain model numerically.
- The numerical solutions of the three species food chain model are presented by using the proposed Meyer wavelet neural networks using the optimization-based stochastic procedures.
- Authentic, consistent, and matched representations of the three species food chain model represent the stochastic scheme performances based on the Meyer wavelet neural networks together with the optimization-based stochastic procedures.
- The correctness of the designed stochastic approach is observed through the performance of the absolute error.
- The consistency and reliability of the scheme are authorized by taking a larger number of runs for solving the three species food chain model.
- The authorization of the scheme is acknowledged based on different statistical performances to solve the three species food chain model using forty executions.
Finally, I would like to thank Al Bilad Bank Scholarly Chair for Food Security in Saudi Arabia, the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia, for supporting this project under project grant No CHAIR71.
References:
- Al Bilad Bank Scholarly Chair for Food Security in Saudi Arabia, The Deanship of Scientific Research, The Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, PO Box 400, Al-Ahsa 31982, Saudi Arabia.
- Department of Physics, College of Science, King Faisal University, PO Box 400, Al-Ahsa 31982, Saudi Arabia.
