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首页> 外文期刊>Advances in Pure Mathematics >A Two-Step Growth Curve: Approach to the von Bertalanffy and Gompertz Equations
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A Two-Step Growth Curve: Approach to the von Bertalanffy and Gompertz Equations

机译：两步增长曲线：von Bertalanffy和Gompertz方程的方法

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Many curves have been proposed and debated to model individual growth of marine invertebrates. Broadly, they fall into two classes, first order (e.g. von Bertalanffy) and sigmoidal (e.g. Gompertz). We provide an innovative approach which demonstrates that the growth curves are not mutually exclusive but that either may arise from a simple three-stage growth model with two steps (k1 and k2) depending on the ratio of the growth parameters . The new approach predicts sigmoidal growth when is close to 1, but if either growth from stage A to stage B or B to C is fast relative to the other, the slower of the two steps becomes the growth limiting step and the model reduces to first order growth. The resulting curves indicate that there is a substantial difference in the estimated size at time t during the period of active growth. This novel two-step rate model generates a growth surface that allows for changes in the rate parameters over time as reflected in the new parameter n(t) = k1(t) -?k2(t). The added degree of freedom brings about individual growth trajectories across the growth surface that is not easily mapped using conventional growth modeling techniques. This two (or more) stage growth model yields a growth surface that allows for a wide range of growth trajectories, accommodating staged growth, growth lags, as well as indeterminate growth and can help resolve debates as to which growth curves should be used to model animal growth. This flexibility can improve estimates of growth parameters used in population models influencing model outcomes and ultimately management decisions.=
机译：已经提出和讨论了许多曲线来模拟海洋无脊椎动物的个体生长。大致来说，它们分为两类，一类（例如von​​ Bertalanffy）和S形（例如Gompertz）。我们提供了一种创新的方法，该方法证明了增长曲线不是互斥的，但其中的任何一条都可能取决于具有两个步骤（k1和k2）的简单三阶段增长模型，具体取决于增长参数的比率。新方法可以预测乙状结肠的增长接近于1，但是如果从阶段A到阶段B的增长或从阶段B到阶段C的增长相对于另一阶段快，那么这两个步骤中的较慢者将成为增长限制步骤，并且模型会减少到订单增长。所得曲线表明，在活跃生长期间，在时间t的估计大小存在很大差异。这种新颖的两步速率模型生成了一个增长表面，该速率表面允许速率参数随时间变化，如新参数n（t）= k1（t）-？k2（t）所反映。增加的自由度带来了整个生长表面上的单个生长轨迹，而使用传统的生长建模技术很难轻松地绘制出这些轨迹。这两个（或多个）阶段的增长模型产生的增长表面允许广泛的增长轨迹，适应分阶段的增长，增长滞后以及不确定的增长，并且可以帮助解决有关应使用哪种增长曲线进行建模的争论动物生长。这种灵活性可以改善人口模型中使用的增长参数的估计，这些参数会影响模型结果并最终影响管理决策。

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