The Gompertz model (Gompertz, 1825) was an empirical model originally proposed to represent human mortality rates. Throughout the 20th century, the Gompertz model has undergone modifications and reparameterizations, resulting in the emergence of different forms of the modified Gompertz model (Rietz, 1921; Ludwig, 1929; Winsor, 1932; Laird, 1964; Jefferies et al., 1982; Gibson et al., 1987, Gibson et al., 1988; Rogers et al., 1987; Zwietering et al., 1990). Some of the modified Gompertz models have increased the efficiency in analyzing growth phenomena and obtaining valuable parameters, including the maximum growth rate and amount (Winsor, 1932; Zwietering et al., 1990; Lay et al., 1996).

These Gompertz models have subsequently found applications in various biotechnological areas in the 20th and 21st centuries (Tjørve and Tjørve, 2017). These applications include understanding the microbial growth, cell growth, and animal growth kinetics during biological processes (Ribeiro, 2017; Zardin et al., 2019; Chaturvedi et al., 2023; Czajkowska et al., 2024). Additionally, the Gompertz model has been extensively used in environmental biological processes, contributing significantly to environmental remediation efforts (Wang and Chen, 2006, Wang and Chen, 2009; Wang and Chu, 2016; Wang and Yin, 2022). At the end of the 20th century, the Gompertz model began to be utilized to analyze the kinetics of bioproduction processes (Lay et al., 1996). Towards the end of the 20th century, the Gompertz model began to be employed for analyzing the kinetics of bioproduction processes (Lay et al., 1996). In the 21st century, the Lay-modified Gompertz model has been widely applied to various bioproduction processes, including biohydrogen, biomethane, and caproate production (Wang and Wan, 2009; Zhang et al., 2022; Yin and Wang, 2022a, Yin and Wang, 2022b; Yang and Wang, 2019, Yang and Wang, 2020, Yang and Wang, 2021).

However, there have been some issues and gaps in the utilization of the Gompertz model. Firstly, the origins of the Gompertz model and how this model, initially designed to describe mortality, can be applied in modeling microbial growth kinetics and further to model bioproduction kinetics are essential; however, these aspects have rarely been reviewed. Secondly, various modified Gompertz models have been applied to the growth phenomena; the deduction of the modified Gompertz model and the selection of the proper modified Gompertz model should be thoroughly discussed and reviewed. In addition, citing the original research when referencing different forms of the model is essential, but this consistency is not achieved in many published papers. Thus, it is significant to standardize the citations of the models. Thirdly, many frequently used Gompertz models only consider internal mortality (age-dependent mortality). Since external mortality (age-independent random mortality) can also occur in some growth processes (Makeham, 1860, Makeham, 1867), it is important to modify the Gompertz model to include both internal and external mortality.

Based on the above, we systematically studied the applications of the Gompertz model in biotechnological areas. The objectives of this paper were: (1) to review the original rationale and basic concepts of the Gompertz model; (2) to comprehensively review the modifications/reparameterizations of the Gompertz model; (3) to provide an overview of the applications of this model in growth analysis and bioproduction kinetics; and (4) to identify the limitations of the Gompertz model in current studies and propose prospects for its modification (considering internal and external mortality) and application.