A discrete model of ontogenetic growth
Introduction
Many generally accepted classical growth curves (i.e., the logistic, Richards and Gompertz equations) can be considered special or extended cases of the von Bertalanffy paradigm (Tjørve and Tjørve, 2010; 2017). Some subsequent growth equations developed from the allometric scaling laws of metabolism, such as the ontogenetic growth model (OGM) (West et al., 2001). Other modifications to these models (Hou et al., 2008; Zou et al., 2011) remain part of the von Bertalanffy family of equations. From experimental results, we know that the von Bertalanffy paradigm (Von Bertalanffy, 1957) is a reasonable mathematical framework for ontogenetic growth. Under this framework, the growth rate of organisms (dm/dt) can be expressed as:where m represents the body weight at time t; k and λ are constants of assimilation and dissimilation, respectively; θ and δ are scaling exponents (see Table 1 for a list of symbols), where θ ≤ δ and δ = 1. The first and second terms of the right-hand side of Eq. (1) represent the rates of synthesis and decomposition of organic matter, respectively. Initially, Eq.(1) did not have any particular physical or biological argumentation. It considered only θ and δ as fitting parameters and could be justified according to goodness-of-fit (Makarieva et al., 2004). However, this flexibility also means different combinations of parameter values can produce very similar growth curves (Paine et al., 2012).
Some theoretical explanations based on energy assimilation and expenditure or metabolic use (including growth) and maintenance (e.g. Hou et al., 2008; West et al., 2001) may be closely related to the establishment of Eq.1, and have attracted much attention amongst described growth models (e.g. Cao et al., 2019, Hou et al., 2008, Martyushev and Terentiev 2015, West et al., 2001, Zou et al., 2011). Although these equations can possibly clarify the physical or biological significance of the parameters in Eq. (1), such as treating θ as a fixed metabolic exponent, they still have limitations as a mechanistic model with stable state variables (Marshall and White, 2018). A central issue is that an organism's resources and energy cannot always be represented by a stable state (Sousa et al., 2009). For example, there may be conditions where an organism may only maintain metabolism (Lescourret and Génard, 2005). The metabolism exponent of plants can decrease from 1 to 3/4 with growth (Enquist et al., 2007; Mori et al., 2010), and there is enormous variation in animals between different taxa and mass ranges (Glazier, 2005; Moses et al. 2008). The incorporation of these state changes is necessary for more accurate growth equations.
Another neglected but important detail is the change in the growth rate. Size, playing a central role in organism design and functions, is closely related to resource uptake and trade-off (Gibert et al., 2016; Mencuccini et al., 2005; Peter, 2003). For example, clownfish (Amphiprion percula) can adjust their size and growth rate to prevent intraspecific conflict usually related to resource competition (Peter, 2003). In some experiments, size even has a greater effect on age-related declines in relative growth and net assimilation rates than cellular senescence (Mencuccini et al., 2005). Thus organism growth rate is typically considered to change instantaneously with its size. However, this treatment, based on mathematical change may not be accurate. This is because the formation of cells or tissues is primarily controlled genetically and by physiological activities, with the intrinsic or developmental growth rate independent of size. Numerous developing cells or unit tissues together determine the growth of an organism. Although size affects the number of cells or unit tissue, the growth rate during this period, i.e. the number of all developing unit tissue multiplied by their average intrinsic growth rate, does not change immediately with the formation of new tissue or slight size increases. Consequently, we consider that the effect of an organism's size on its growth rate is discontinuous.
Under the Bertalanffy paradigm, we derived a discrete growth model from the perspective of resource allocation and discrete growth. Compared with our previous discrete growth model (Shu et al., 2016), this new model is universal and has a more complete mathematical and biological foundation.
Section snippets
Model basic form
We first introduced time parameter T to refer to the formation time of a unit of tissue, which is controlled by genetic and physiological activity, and then considered that organism tissue is composed of numerous unit tissues. The growth rate of an organism within the tissue formation period can be expressed as f(m)/T, where f(m) is the total biomass of new tissue created during this period. Meanwhile, T can also be infinitely small, so that f(m)/T → dm/dt. Mathematically, this treatment
Model performance
Hou et al. (2008) first assumed b = 3/4, and then derived a general growth equation for endothermic animal, i.e., ontogenetic growth model (OGM). On this basis, Zou et al. (2011) expanded OGM to explain the effects of temperature on ectotherm ontogenetic growth and development. Since the equations of Hou et al. (2008) and Zou et al. (2011) can be supported by a series of growth and/or assimilation data for ectothermic and endothermic animals with diverse body sizes and taxa, the IGM and its
Discussion
Like most growth models, the IGM also considered the energy or resource allocation dependent on size as a key factor driving growth. The difference is that the IGM cancels the constraint of resting metabolism exponent (b) and highlights the formation time of unit tissue (T). These improvements casuse the IGM can directly reflect changes and connections in the transport system, some physiological activities, and the resource intake, thereby effectively revealing some underlying growth details.
Data accessibility statement
All data generated or analyzed during this study are included in this article.
CRediT authorship contribution statement
Shu-miao Shu: Conceptualization, Methodology, Data curation, Writing – original draft. Wan-ze Zhu: Funding acquisition, Project administration, Writing – review & editing. George Kontsevich: Formal analysis, Writing – review & editing. Yang-yi Zhao: Software, Validation. Wen-zhi Wang: Software, Data curation. Xiao-xiang Zhao: Formal analysis. Xiao-dan Wang: Funding acquisition, Project administration.
Declaration of Competing interest
We would like to submit the enclosed manuscript entitled “A discrete model of ontogenetic growth”, which we wish to be considered for publication in Ecological Modelling. No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole
Acknowledgements
We thank Mr. Zhi-qiang Xiao for important advice concerning the science and the manuscript. This work was supported by the Second Tibetan Plateau Scientific Exploration (2019QZKK0404), the Strategic Priority Research Program of Chinese Academy of Sciences (XDA20020401), the National Key Research and Development Program of China (2017YFC0505004), and the National Natural Science Foundation of China (41977396).
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