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发挥着重要作用，并且在所有学科中都有巨大的实际应用。捕获物种适应性的最流行的增长指标是该领域的“相对增长率”。不同的增长规律，例如指数、逻辑、Gompertz、幂和广义 Gompertz 或广义逻辑，可以根据相对增长率 (RGR) 对大小或时间的单调行为来表征。因此，在这种情况下，可以通过 RGR 真正确定物种适应度。然而，在自然界中，RGR 通常是非单调的，特别是钟形的，特别是在物种适应新环境的情况下 [1]。在这种情况下，物种可能会在两个不同的时间点经历相同的适应性 (RGR)。无法从此 RGR 函数确定物种精确的生长和成熟状态。[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，公制 IMR 存在严重缺陷；(i) IMR 对于特定参数的所有非整数值是难以处理的。(ii) 该措施取决于模型参数。IMR 的数学表达式具有实验者不知道的术语“承载能力”。(iii) 请注意，为了确定一个物种的精确生长状态，还需要了解当种群偏离其承载能力平衡位置时的反应。这是人口生物学中的一个既定概念，俗称回报率。然而，IMR 不提供有关稳态下物质偏转率的信息。因此，我们提出了一种与物种回报率相关的新增长措施，称为“相对增长率的相对反转”（以下简称 RRRGR），它被视为 IMR 的代理，具有类似的数学特性。最后，我们引入了一个随机 RRRGR 模型来指定精确的物种生长和成熟状态。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，使该物种能够以最佳努力达到稳定状态。称为“相对增长率的相对反转”（以下简称 RRRGR），它被视为 IMR 的代理，具有相似的数学特性。最后，我们引入了一个随机 RRRGR 模型来指定精确的物种生长和成熟状态。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，使该物种能够以最佳努力达到稳定状态。称为“相对增长率的相对反转”（以下简称 RRRGR），它被视为 IMR 的代理，具有相似的数学特性。最后，我们引入了一个随机 RRRGR 模型来指定精确的物种生长和成熟状态。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，使该物种能够以最佳努力达到稳定状态。