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

Abstract

Abstract-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.

Effects of ocean acidification on red king crab larval survival and development

“…Similarly, development rate, growth rate, mass, and carbon and nitrogen content differed ontogenetically, but not with treatment pH. The developmental rate was on the slower side, but comparable to other studies (Sato and Tanaka, 1949b; Kurata, 1960; Kovatcheva et al, 2006; Long, 2016). Mass and linear size increased with stage as expected (Sato and Tanaka, 1949a; Nakanishi, 1987).…”

“…Larval rearing followed standard red king crab culturing techniques (Swingle et al, 2013; Long, 2016). In brief, zoea were fed daily to excess on a diet of Artemia nauplii enriched with DC DHA Selco (INVE Aquaculture, Salt Lake City).…”

“…The staging data was fit to a multi-stage transition model as per Long (2016). In brief each molt from one stage to the next is modeled as: , where t is the time in degree days (to account for the effect of temperature on the development of ectotherms), t 50 is the time at which 50% of the individuals have molting to the next stage and s is proportional to the abruptness of the transition.…”

“…We calculated the stage duration for each stage as the time between t 50 estimates for each tank. Further, we calculated the slope at each transition as as a measure of the rapidity of the transition between the stages (Long, 2016). Both the stage duration and the Slope estimates were analyzed with an ANOVA with treatment fully crossed with larval stage and tank (nested within treatment) as factors.…”

“…Water temperatures increased throughout larval rearing from 4.8 °C in early April when the experiment started to 9.4 °C in early July when it finished. The pHs averaged within 0.01 units of the nominal values and the pH 7.8 treatment was undersaturated with respect to aragonite and the pH 7.5 undersaturated with respect to calcite (Table 1) Larval rearing followed standard red king crab culturing techniques (Swingle et al, 2013;Long, 2016). In brief, zoea were fed daily to excess on a diet of Artemia nauplii enriched with DC DHA Selco (INVE Aquaculture, Salt Lake City).…”