Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108, India.
Department of Mathematical Sciences, University of Cincinnati, 2600 Clifton Ave, Cincinnati, OH, 45221, USA.
J Biol Phys. 2023 Jun;49(2):195-234. doi: 10.1007/s10867-023-09628-0. Epub 2023 Mar 22.
Growth curve models play an instrumental role in quantifying the growth of biological processes and have immense practical applications across all disciplines. The most popular growth metric to capture the species fitness is the "Relative Growth Rate" in this domain. The different growth laws, such as exponential, logistic, Gompertz, power, and generalized Gompertz or generalized logistic, can be characterized based on the monotonic behavior of the relative growth rate (RGR) to size or time. Thus, in this case, species fitness can be determined truly through RGR. However, in nature, RGR is often non-monotonic and specifically bell-shaped, especially in the situation when a species is adapting to a new environment [1]. In this case, species may experience with the same fitness (RGR) for two different time points. The species precise growth and maturity status cannot be determined from this RGR function. The instantaneous maturity rate (IMR), as proposed by [2], helps to determine the correct maturity status of the species. Nevertheless, the metric IMR suffers from severe drawbacks; (i) IMR is intractable for all non-integer values of a specific parameter. (ii) The measure depends on a model parameter. The mathematical expression of IMR possesses the term "carrying capacity" which is unknown to the experimenter. (iii) Note that for identifying the precise growth status of a species, it is also necessary to understand its response when the populations are deflected from their equilibrium position at carrying capacity. This is an established concept in population biology, popularly known as the return rate. However, IMR does not provide information on the species deflection rate at the steady state. Hence, we propose a new growth measure connected with the species return rate, termed the "reverse of relative of relative growth rate" (henceforth, RRRGR), which is treated as a proxy for the IMR, having similar mathematical properties. Finally, we introduce a stochastic RRRGR model for specifying precise species growth and status of maturity. We illustrate the model through numerical simulations and real fish data. We believe that this study would be helpful for fishery biologists in regulating the favorable conditions of growth so that the species can reach a steady state with optimum effort.
生长曲线模型在量化生物过程的生长方面起着重要作用，在各个学科领域都有广泛的实际应用。在这个领域，捕获物种适应性的最流行的生长度量是“相对生长率”。不同的生长规律，如指数、逻辑斯谛、龚珀兹、幂律和广义龚珀兹或广义逻辑斯谛，可以根据相对生长率（RGR）对大小或时间的单调行为来描述。因此，在这种情况下，真正的物种适应性可以通过 RGR 来确定。然而，在自然界中，RGR 通常是非单调的，特别是呈钟形，尤其是在物种适应新环境的情况下[1]。在这种情况下，两个不同的时间点可能具有相同的物种适应性（RGR）。从这个 RGR 函数中无法确定物种的精确生长和成熟状态。[2]提出的瞬时成熟率（IMR）有助于确定物种的正确成熟状态。然而，该度量存在严重的缺陷：（i）对于特定参数的所有非整数值，IMR 都是不可处理的。（ii）该度量取决于模型参数。IMR 的数学表达式包含实验者未知的“承载能力”一词。（iii）注意，为了确定物种的精确生长状态，还需要了解当种群从承载能力的平衡位置偏离时的响应。这是种群生物学中的一个既定概念，通常称为返回率。然而，IMR 没有提供物种在稳定状态下的偏离率的信息。因此，我们提出了一个与物种返回率相关的新生长度量，称为“相对生长率的倒数”（简称 RRRGR），它被视为 IMR 的代理，具有相似的数学性质。最后，我们引入了一个随机的 RRRGR 模型来确定物种生长和成熟的精确状态。我们通过数值模拟和真实鱼类数据来说明该模型。我们相信，这项研究将有助于渔业生物学家调节有利的生长条件，以便物种能够以最佳的努力达到稳定状态。
