In the fisheries study, fish growth is typically in an indeterminate fashion implying a continuous growth function occurs throughout the life. Most existing studies of fish trajectories are continuous; however, it is not true when modelling growth of the crustacean species with discontinuous growth paths. In this context, it is imperative to insure that a different type of model for describing absolute growth in crustaceans. Crustaceans must moult for them to grow. A moulting process is of periodical shedding of the exoskeleton and thus the crustacean growth is known to be a discontinuous process. The sudden growth of crustaceans through the moulting process makes the growth estimation more complex. To model the discontinuous growth, we consider stochastic approaches where the growth model only considers for a monotonically increasing function. To this end, we introduce a subordinator that is a special case of a Levy process. A subordinator is a non-decreasing Levy process, that enabling the individual variability and environmental perturbation to be included in modelling growth. A dataset in the laboratory setting (e.g. in an aquarium) is developed. The motivational dataset is from the ornate rock lobster, Panulirus ornatus, where the growth parameters can be estimated through two inter-correlated variables, namely the intermoult periods and the moult increments. We propose a joint density function, consisting of the moult increments and the intermoult periods. Both of these variables are assumed to be conditionally independent based on the Markov property. In the four-year studies from 1995 to 1999, the growth rates for females and males are estimated averagely 0.307 mm year-1 and 0.205 mm year-1 , respectively. Therefore, the growth parameters of moult increments and intermoult periods can be quantified individually. The corresponding functions will then be convoluted through a simulation approach to obtain a population mean curve for crustaceans.

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<abstract><p>Crustaceans exhibit discontinuous growth as they shed hard shells periodically. Fundamentally, the growth of crustaceans is typically assessed through two key components, length increase after molting (LI) and time intervals between consecutive molts (TI). In this article, we propose a unified likelihood approach that combines a generalized additive model and a Cox proportional hazard model to estimate the parameters of LI and TI separately in crustaceans. This approach captures the observed discontinuity in individuals, providing a comprehensive understanding of crustacean growth patterns. Our study focuses on 75 ornate rock lobsters (<italic>Panulirus ornatus</italic>) off the Torres Strait in northeastern Australia. Through a simulation study, we demonstrate the effectiveness of the proposed models in characterizing the discontinuity with a continuous growth curve at the population level.</p></abstract>

The contemporary methodology for growth models of organisms is based on continuous trajectories and thus it hinders us from modelling stepwise growth in crustacean populations. Growth models for fish are normally assumed to follow a continuous function, but a different type of model is needed for crustacean growth. Crustaceans must moult in order for them to grow. The growth of crustaceans is a discontinuous process due to the periodical shedding of the exoskeleton in moulting. The stepwise growth of crustaceans through the moulting process makes the growth estimation more complex. Stochastic approaches can be used to model discontinuous growth or what are commonly known as "jumps" (Figure 1). However, in stochastic growth model we need to ensure that the stochastic growth model results in only positive jumps. In view of this, we will introduce a subordinator that is a special case of a Levy process. A subordinator is a non-decreasing Levy process, that will assist in modelling crustacean growth for better understanding of the individual variability and stochasticity in moulting periods and increments. We develop the estimation methods for parameter estimation and illustrate them with the help of a dataset from laboratory experiments. The motivational dataset is from the ornate rock lobster, Panulirus ornatus, which can be found between Australia and Papua New Guinea. Due to the presence of sex effects on the growth (Munday et al., 2004), we estimate the growth parameters separately for each sex. Since all hard parts are shed too often, the exact age determination of a lobster can be challenging. However, the growth parameters for the aforementioned moult processes from tank data being able to estimate through: (i) inter-moult periods, and (ii) moult increment. We will attempt to derive a joint density, which is made up of two functions: one for moult increments and the other for time intervals between moults. We claim these functions are conditionally independent given pre-moult length and the inter-moult periods. The variables moult increments and inter-moult periods are said to be independent because of the Markov property or conditional probability. Hence, the parameters in each function can be estimated separately. Subsequently, we integrate both of the functions through a Monte Carlo method. We can therefore obtain a population mean for crustacean growth (e.g. red curve in Figure 1).