A new quantitative model of multiple transitions between discrete stages, applied to the development of crustacean larvae

Many biological processes are described in terms of transitions between discrete stages. For example, crustacean larvae generally pass through a. number of stages that are punctuated by transitional molting events. On the other hand, some continuous processes, such as embryo development, are frequently described in terms of discrete stages. Despite the widespread use of such conceptual models, a mathematical model that quantitatively describes the transitions between multiple stages has not been developed for crustacean larvae. I describe a model of multiple transitions between stages that can be fitted to such data and that holistically describes the processes and allows explicit, quantitative comparisons among treatments or studies. The base of the model is the logistic equation that is frequently used to model a transition between 2 stages. By summing together multiple logistic equations, one for each transition between stages, the model can accommodate multiple stages. Variance is modeled by treating each transition as a binomial distribution and summing the variance from each transition. To demonstrate, I fitted the model to data on larval development of red and blue king crabs (Paralithodes camtschaticus and P. platypus). The model provides an excellent fit for these data and quantitatively describes the process of larval development for 'these crab species.