Recently, the mathematical models become one of the core attention for the researchers. The mathematical models perform an emerging fields of interest with various new potentials in the food chain supply models. The growing interests in the mathematical form of the food supply chain models is provided in number of applications. The food chain models arise in various modifications, like as environmental models performs the status of food chain-supply, ecological consequences of global bifurcations in some food chain models, food chain length on trophic efficiencies in simple stoichiometric aquatic food chain models, productivity controls food-chain properties in microbial communities, predicting terrestrial food chain behavior of xenobiotics, oil spill–food chain interaction model for coastal waters, three-species food chain with a lower bound on the bottom population, chance and chaos in population biologymodels of recurrent epidemics and food chain dynamics, etc.

The food chain models mostly appear in the form of prey-preadtor models and there are various mathematical models that designate the natural phenomena based on the prey-predator investigations along with the collaborations of different species. The functional response term in the prey-predator modelling has an important role to present that most of the prey affects the predators with the use of time. There are numerous functional responses species that have been reported in the literature, such as a ratio-dependent, Beddington–DeAngelis and the Holling phase I to III . One of the important models is food supply, which is applied in the association of multiple prey or predators. The updated form of the food supply system together with common qualitative investigations and numerous communications have been presented in related scientific papers. The mathematical models are famous to present the system of nonlinear differential models, e.g., SITR based coronavirus model, dengue virus, nervous stomach, vector disease model, mosquito dispersal and economical-environmental system.

The extensive investigations based on the industrial implimentations using the food packaging has been perform to exploit the chemical pollution because of the food distribution and its storage, infective microbes and antagonistic environmental circumstances gas, dust, light, and wetness. The supply storage and chain of food pakages is considered complex areas in the industry of the food that is required to be addressed precisely to confirm the food safety as well as enhance the shelf-life. Low cost of the production, easy disposal, inertness, and material based on the polymeric reusability are the main targets in the industry of the food. The bad health issues in these days are getting high due to the ignorance of these indicated factors.

The “Allee effects” defined in 1930 and named by the famous scientist Allee also play a vital role in the food chain models by using the addictive form along with weak and strong behaviour representations of the Allee effects.. These effects allocate the progress to reduce the growing rate by using the small quantity of public. The Allee effects appear in the fishery, vertebrates, invertebrates and plants. The Allee effects occasionally indicate the negative influences in the dispensation of population dynamics based on the fishery. The “Allee effects” have been divided into multiplication and addition. Initially, Singh et al. described the double shape of “Allee effects” with the improved Gower-Leslie system based on the prey predator, in which prey population shows the various junction associated with the suitable parameters. Vinoth et al formulated a mathematical model to investigate the dynamical food supply system using the “Allee effect” based on the addition. The communication food supply models have been designed based on the two or more prey and predators. A differential form of the food supply chain system use the analysis of mutual qualitative along with the multiple relationships. Few researchers presented the multiple trophic-level of food supply systems through the structure of logistic prey, Holling type or Lotka–Volterra predator and top-predator.

A recently accepted research paper in Fractal and Fractional Journal performed by our research group where we provided the numerical performances of the fractional kind of food supply model to get the accurate and realistic solutions by using the stochastic procedures of the scaled conjugate gradient neural networks. The stochastic procedures can play an important role in the food chain supply models by providing an accurate data. The stochastic solvers with the global and local search schemes have been used to exploit the variety of applications in recent years. Few well-known applications of the stochastic solvers are heat conduction in human head, Thomas fermi nonlinear singular model, nonlinear nervous stomach model, nonlinear dynamics of the coronavirus models, functional form of the singular models, biological nonlinear Leptospirosis system, the infectious-based HIV models, functional form of the delay differential system, theory of thermal explosion, periodic boundary value problems, nonlinear influenza disease model, higher order singular systems, waste plastic management in the ocean, nonlinear model of the smoking, mathematical model underlying immune-chemotherapeutic treatment for breast vancer using the neural networks, vaccination and Wolbachia on dengue transmission dynamics in the nonlinear model, SIDARTHE COVID-19 pandemic mathematical model, dynamical nonlinear susceptible infected and quarantine differential model.

The significant procedures regarding to the generalization have been provided in the research paper by using the Adam scheme, while the numerical procedures are implemented with the default parameter setting to generate the model dataset. The hidden neurons have been selected 15 in the study along with the data selection for the FKFS model is chosen as 82%, for training and 9% for both testing and authorization. The artificial intelligence abilities based supervised learning SCGNNs have been performed with best cooperation in the indices, including complexity, premature convergence, overfitting and underfitting cases. Additionally, these parameters of the networks are set after exhaustive simulation studies, experience, knowledge and care and small variations in these setting results in degraded performance of the networks.

The second phase of the stochastic scaled conjugate gradient is expressed by using the generic per-ception based on the solo neuron model. The single layered neural network structure, while the designed layer construction, a single input layer vector having 15 hidden numbers of neurons in the hidden layer along with the three outcomes in the outer layer as described in subfigure for solving the mathematical food chain suppy model. The stochastic based scaled conjugate gradient are applied by using the ‘Matlab’ software (nftool command) for the appropriate sections of hidden neurons, testing statistics, learning methods and verification statics. Whereas the implementation performances of the SCGNNs scheme to solve the mathematical food chain suppy model along with the parameter setting is provided. The networks training is performed using the proposed stochastic scaled conjugate gradient scheme.

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 CHAIR37.