## 1 Introduction
The work proves FA feasibility and shows that modified FA outperforms other competing algorithms (GA, ACO, TS, etc.) in terms of results and competence. Parameter estimation of a proposed hyperbolastic type-I diffusion process applying FA is presented in [37]. The low computational cost of FA makes it especially useful for addressing maximum likelihood estimation in diffusion processes. An interesting performance analysis of distance metrics on the exploitation properties and convergence behaviour of FA is presented in [38]. The optimal algorithm tuning based on unique distance metrics shows a new research area for solving large-scale optimization problems. In [39], the enhanced FA is hybridized with the CLT-based K-means algorithm to achieve optimal global convergence. The results show that the hybrid FA-K-means clustering method demonstrates statistically significant superiority compared to other advanced hybrid search variants--GA, DE, PSO, and Invasive Weed Optimization. The CS algorithm attracts attention due to its simplicity and efficiency over GA because of using the Levy flights instead of isotropic random walks [40]. Recently proposed CS algorithm [41] for an energy-efficient robotic mixed-model assembly line balancing problem outperforms GA in terms of obtained objective values. In [42], the CS algorithm employs theoretical anomaly generated by a single structure and different field data sets to estimate the model parameters. As a result, a rapid convergence of the objective function and the model parameters is observed. In [43], a technique combining a CS algorithm and a support vector machine is applied in the prognostic staging of oesophageal cancer based on the prognosis index. The proposed algorithm has the highest prediction accuracy compared to six swarm intelligence algorithms, including the ABC algorithm, FA, Gravitational search algorithm, etc., combined with support vector machine learning techniques. A newly introduced metaheuristic algorithm, Coyote Optimization Algorithm (COA) [44], adopts an interesting technique to achieve a balance between exploration and exploitation [45], as well as the robustness and stability of the algorithm [46]. Binary COA with a hyperbolic transfer function in a wrapper model is applied to a feature selection problem [47]. The algorithm presents low standard deviations when compared to other metaheuristic algorithms and a great convergence curve. An accurate and stable ultrashort-term wind speed prediction method is achieved using chaotic COA [48]. The authors of [49] propose a novel fault diagnosis method in chemical processes based on the Bernoulli shift coyote optimization algorithm. The results demonstrate that the proposed method outperforms other methods in terms of classification accuracy. In [45], COA is proposed for optimal parameter estimation of a proton exchange membrane fuel cell model. The presented results show the superiority of COA over the other compared methods. The known results indicate that metaheuristic techniques, such as GA, FA, CS, and COA, are particularly relevant nowadays, frequently preferred and efficiently used for many optimization problems, especially for model parameters optimization [29; 31; 37; 42; 45; 50]. Moreover, they have not been employed in parameter estimation of TSAD of corn steep liquor model until now which is the motivation to adapt and apply them to the considered optimization problem. The presented study focuses on the mathematical modelling of anaerobic biohydrogen production from corn steep liquor and the following anaerobic processing of the methane production under mesophilic conditions in a two-stage process. Model parameters identification is performed based on GA, FA, CS, and COA metaheuristic algorithms. The specific contributions and innovations of this study are as follows:  
1. For the first time, a structure of nonlinear differential equations of two-stage anaerobic digestion of corn steep liquor is evaluated based on real experimental data. 2.