Analyzing a coupled dynamical system of materials recycling in chemostat systems with artificial deep neural network

This manuscript is devoted to study a mathematical model of materials recycling which is motivated from the chemostat model. For the mentioned study we use fractals fractional derivative with power law kernel. A viable strategy for lake restoration, sediment removal from eutrophicated shallow lakes offers the possibility of recycling sediment nutrients for crop cultivation. But it’s still difficult to come up with a good plan for reusing the silt in an environmentally friendly way. One potential long-term solution to the issue of nutrient losses from agricultural soils is the recycling of sediments and related nutrients. Keeping this importance in mind, we investigate the aforementioned model from some different perspectives. Sufficient conditions are derived for the existence and uniqueness of solution to the mentioned model by using fixed point theory. Also, generalized Ulam–Hyers (U–H) stability is studied for the solution of suggested model. In addition, for numerical simulations to the mentioned model, we extend the Euler numerical method. Various graphical presentation under different fractals fractional orders are displayed. In addition to the aforesaid analysis, artificial intelligence (AI) based deep neural networks (DNNs) tools are used for classification of various terminologies including mean square error (MSE), root mean square error (RMSE), variance and standard deviation, regression coefficient. For the mentioned requirement, we use Levenberg–Marquardt training algorithm. Here we remark that training, learning and prediction accuracies of the DNN are testified and verified by using 50 and 5 neurons and maximum of 1000 epochs to deduce the regression R, mean square error (MSE) and root mean square error (RMSE). All these results are displayed using graphical illustration.