key: cord-0993928-oztavbhm
authors: Sabir, Zulqurnain; Ali, Mohamed R.; Sadat, R.
title: Gudermannian neural networks using the optimization procedures of genetic algorithm and active set approach for the three-species food chain nonlinear model
date: 2022-01-18
journal: J Ambient Intell Humaniz Comput
DOI: 10.1007/s12652-021-03638-3
sha: 77564caf6ce24cd775920e90db387832e7e4656e
doc_id: 993928
cord_uid: oztavbhm

The present study is to investigate the Gudermannian neural networks (GNNs) using the optimization procedures of genetic algorithm and active-set approach (GA-ASA) to solve the three-species food chain nonlinear model. The three-species food chain nonlinear model is dependent upon the prey populations, top-predator, and specialist predator. The design of an error-based fitness function is presented using the sense of the three-species food chain nonlinear model and its initial conditions. The numerical results of the model have been obtained by exploiting the GNN-GA-ASA. The obtained results through the GNN-GA-ASA have been compared with the Runge–Kutta method to substantiate the correctness of the designed approach. The reliability, efficacy and authenticity of the proposed GNN-GA-ASA are examined through different statistical measures based on single and multiple neurons for solving the three-species food chain nonlinear model.

The study of two and three trophic-level based on food chain systems using the structure of logistic prey X, specialist Lotka-Volterra predator Y and top-predator Z (Freedman and Waltman 1977; Freedman and So 1985; Muratori and Rinaldi 1992; Kuznetsov and Rinaldi 1996; Rinaldi et al. 1996; El-Owaidy et al. 2001; Umar et al. 2019) . The general form of the state system based on the three species food chain nonlinear model is written as (Aziz-Alaoui 2002):

The above system represents the three-dimensional food chain nonlinear model that has been investigated analytically/ numerically with the prey population X, which implemented as a single food predator Y together with prey of a top-predator Z. The features of prey X along with the species Y present the modeling of Volterra scheme, which indicates the predator population reduces exponentially in the absence of prey. The association of species Z together with its prey Y is formed based on the Leslie-Gower scheme (Leslie and Gower 1960) , which indicates the predator population reduces to the reciprocal of per capita availability of its most special food (Upadhyay et al. 1998) . l 1 , l 2 and l 3 are the positive initial conditions (Ics). The model parameter's detail of the three-dimensional food chain nonlinear model is expressed in the Table 1. (1)

The stochastic computing processes have been executed to solve a large variety of nonlinear systems, few of them are fractional singular systems (Sabir et al. 2021c, d, e) , like higher order singular systems (Ayub et al. 2021; Sabir et al. 2021a ), dengue fever system (Umar et al. 2020b, c, d, e) , SITR based COVID-19 models (Umar et al. 2020b (Umar et al. , c, d, e, 2021a , delay singular function system Sabir et al. 2021b) , SIR system for spreading infection and treatment (Umar et al. 2021a, b) , mosquito release system in the heterogeneous environment (Umar et al. 2020a) , doubly singular nonlinear models Sabir et al. 2020a, b, c) , rank-constrained spectral clustering (Li et al. 2018a, b) , zero-shot event detection system (Li et al. 2019) , fuzzy K-means clustering associated discriminative embedding scheme (Li et al. 2018a, b) , multiclass classification systems (Yan et al. 2020) . dynamic affinity graph construction strategy for spectral clustering , enhanced multilayer piezoelectric transducer design (Naz et al. 2021) , performance investigation of the heat sink of functionally graded material of the porous fin (Ahmad et al. 2021) , impact of heat transfer in a Bodewadt flow model (Awais et al. 2021) , thin film flow model over a stretched surface (Uddin et al. 2021) and state estimation problems arising in underwater Markov chain maneuvering targets (Ali et al. 2021) . All these utmost applications inspired the authors to explore/exploit/investigate artificial intelligencebased computational solver to solve the governing model of three-species food chain nonlinear model as presented in a set of Eq. (1). A brief summary of innovative insights and contributions of the presented study is listed in terms of salient features as follows:

• A novel application of artificial intelligent knacks via Gudermannian neural networks (GNNs) models optimized with genetic algorithm and active-set approach (GA-ASA), i.e., GNNs-GA-ASA is introduced to solve a mathematical model of the three-species food chain nonlinear systems (TS-FCNS). • The design of an error-based fitness function is effectively portrayed for TS-FCNS for the dynamics of the prey populations, top-predator and specialist predator. 

The mathematical formulations of the three-dimensional food chain nonlinear model together with derivatives are derived as:

where, W indicates an unidentified weight vector, given as: 

Rate at Y will decrease in the omission of X d 3

Surplus loss in the species of Z due to severe insufficiency of its selected food Y c 3 Development rate of Z A merit function, i.e., Gudermannian function

The merit function is provided as:

indicate the proposed results of the system (1). Likewise, the Eqs. (5)-(7) represent an error function of the three-dimensional food chain nonlinear model and its ICs.

This section indicates the optimization procedures to solve the three-dimensional food chain nonlinear model using the stochastic procedures based on GNN-GA-ASA.

Genetic algorithm is known as a famous, optimization global search scheme implemented to solve the linear/nonlinear models. It is performed to tackle both constrained/ unconstrained systems using the typical selection processes.

GA is usually applied to regulate the results of the accurate population to solve the numerous complex/steep models of ideal training. Recently, GA is implemented in the brain tumor images (Simi et al. 2020) , hospitalization expenditure systems (Tao et al. 2019) , Thomas-Fermi model , feature diversity in cancer microarray (Sayed et al. 2019) , radiation protective in the bismuth-borate glasses (Wilson 2019), nonlinear electric circuit models ), heat conduction model ), HIV infection model (Umar et al. 2020b, c, d, e) , wire coating with Oldroyd 8-constant fluid model (Munir et al. 2019 ), prediction differential system (Sabir et al. 2020a, b, c) , periodic differential model (Sabir et al. 2020a, b, c) and cloud service optimization procedures (Yang et al. 2019) . ASA is applied in pricing American better-of option on two assets (Gao et al. 2020) , pressure-dependent models of water distribution systems with flow controls (Piller et al. 2020) , nonlinear optimization with polyhedral constraints (Hager and Tarzanagh 2020) , numerical solution of the optimal control problem governed by partial differential equation (Azizi et al. 2020) , electrodynamic frictional contact problems (Abide et al. 2021 ) and quadratic semidefinite program with general constraints (Shen et al. 2021 ). The optimization process-based GA-ASA is applied to control the slowness of GA.

The performance through statistics based on the semi-interquartile range (S.I.R), mean absolute deviation (MAD), variance account for (VAF) and Theil's inequality coefficient (TIC) along with the global representation are observed to solve the three-dimensional food chain nonlinear model, given as:

where X , Ŷ and Ẑ are the proposed solutions.

(9) S.I.R = −0.5 Q 1 − Q 3 , Q 1 Q 3 are the 1st 3rd quartiles,

The simplified form of the three-dimensional food chain nonlinear model using suitable parameter values is given as:

An objective function using the three-dimensional food chain nonlinear model is written as:

The mathematical results of the stochastic procedures based on GNN-GA-ASA: Figures 1, 2 and 3 illustrates the best weight vectors, result comparisons and the values of AE to solve the threedimensional food chain nonlinear model using the stochastic procedures based on GNNs-GA-ASA. The best weight values are illustrated in the three-dimensional food chain nonlinear model in Fig. 1a -c for 30 variables or 10 neurons. These weight vectors are established in Eqs. (15-17) . The comparative performance of the results for the threedimensional food chain nonlinear model is illustrated in Fig. 1d-f . The plots of the AE have been established in Fig. 2a -c for the three-dimensional food chain nonlinear model. The statistical operator plots along with the performances of the boxplots are illustrated in Fig. 3 to solve the three-dimensional food chain nonlinear model. The convergence measures are plotted using the TIC, MAD and EVAF to solve the three-dimensional food chain nonlinear model. The complexity of GNNs-GA-ASA in terms of execution time consumed for learning of the weights of the networks is calculated and it is found in the close vicinity of 30 ± 10 for the single runs of the algorithm.

This study aims to investigate the Gudermannian neural networks (GNNs) using the optimization procedures of genetic algorithm and active-set approach (GA-ASA) to solve the three-species food chain nonlinear model. An error function is constructed using the three classes of the threespecies food chain nonlinear model names as prey populations, top-predator and specialist predator and its initial conditions. The exactness of the scheme GNN-GA-ASA is observed by comparing the proposed results and the reference Runge-Kutta results to solve the three-dimensional food chain nonlinear model. The AE values are found in good measures to solve the three-dimensional food chain nonlinear model, i.e. around 10 -05 -10 -07 . The performances of the operators TIC, EVAF and MAD proved the good illustrations to solve the three-dimensional food chain nonlinear model. The statistical Mean, S.I.R, Min, Max, MED and STD performances for 30 independent runs validate the correctness of the proposed stochastic procedures based on GNN-GA-ASA. Furthermore, the global performances through statistical trials of MED and S.I.R have been competently applied to solve the three-dimensional food chain nonlinear model.

In the future, the proposed stochastic procedures based on GNN-GA-ASA are accomplished to solve the environmental economic systems (Kiani et al. 2021; Nisar et al. 2021) , information security models (Masood et al.2019 (Masood et al. , 2020 (Masood et al. , 2021 and fluid dynamic models (Awan et al. 2020 Raja et al. 2020; Umar et al. 2020b, c, d, e) .

Funding Not applicable.

No data is used to support this study.

There is no conflict of interest. All authors contributed equally.

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