Energy Balance: Molting

Molting requires that a crab expend a great deal of energy generating a new exoskeleton - peak energy expenditure occurs immediately after shedding (deFur 1990) at a rate comparable in magnitude to that of exercising crabs in high salinity water (Booth and McMahon 1992). Shedding of the old exoskeleton for 10.1 to 12.7 cm CW crabs is done within two to three hours (van Engle 1958); it takes approximately 9-12 hours for the shell to acquire a papery or leathery texture; and another 12-24 hours for the shell to fully harden (van Engle 1958; Freeman et al. 1987). During molting, CW increases on average by 15 to 35% over temperatures between 14$^\circ$ to 32$^\circ$ C and salinity of 7.5 to 26 psu (Tagatz 1968). Crabs molt more frequently at higher temperatures (leading to greater daily rates of CW increase) but the percentage increase in CW per molt is smaller (Leffler 1972). Crabs 1.3 to 2.5 cm CW molt every 10 to 15 days, while $> 10$ cm CW crabs molt every 20 to 50 days (Ryer et al. 1990; van Engle 1958). Molting does not occur during November through April. The average CW of sexually mature female crabs ranges between 7.5 and 20 cm with a mean of 14.4 cm (Fisher 1999) or $\approx$ 200 g and males are larger than females (e.g., Olmi III and Bishop 1983).

In the model, molting is triggered when a crab's mass, $G$, becomes greater than the mass when molting is triggered causing molt flag to be set to preparing to molt. The duration of molting depends on the amount of energy (g) that the crab must expend to molt and the rate at which it can expend this energy. When molting starts, the crab cannot eat (eating status = not eating) nor does it move. The total energy a crab must expend to molt depends on its size:

$$= \alpha_{\text{molt}} CW^{\beta_{\text{molt}}}.$$ (A.49)

For $\alpha = 0.02$ g/cm $^{\beta_{\text{molt}}}$ with $\beta_{\text{molt}} = 2$, a 15 cm CW crab will use $\approx 4.5$ g to molt.

The rate at which the crab molts depends on the crab's metabolic rate which governs the rate of mass use due to energy expenditure during molting, $G_$molt (g/hr):

$$= \alpha_{\text{max met}}\,G^{\beta_{\text{met}}} \,\exp\left(-\f... ...\sigma_{\text{molt}}\,G_\text{total to molt}}\right)^2\right) \, f_\text{Temp}.$$ (A.50)

$\alpha_$max met=0.00678 (hr$^{-1}$.g $^{\beta_{\text{met}}-1}$)  (Booth and McMahon 1992, Table 1) and  (Houlihan et al. 1985, Fig 1). Based on the equation, larger crabs expend energy at a faster rate. $G_$total to molt is given by Eqn (A.46), $G_$total molted is the number or grams of energy that the crab has already expended molting, $\sigma_{\text{molt}}=0.25$ governs the rate of mass use from energy expenditure as molting progresses. $f_$Temp is the same as the term in Eqn (A.44) and accounts for a crab's metabolic rate at different temperatures. $G_$molt is shown in Fig. A11 at a temperature of $T_$opt met$=26\ensuremath{^\circ\text{C }}$ as a function of CW and $G_{\text{total molted}}$. A crab with a 15 cm CW will take $\approx 1.25$ days to fully molt.

As a crab molts, $G_$total molted is incremented and the crab stops molting when $G_$total molted$= G_$total to molt. The molt flag is then set to not molting and the crab's state variables are updated, including the crab's current CW, mass next molt triggered, mating status and stomach volume.

The proportion increase of the crab's CW at molting depends on the temperature of the crab's environment and is based on model proposed by Smith (1997). The further the temperature is away from the crab's optimal metabolic temperature, $T_$opt met, the smaller the average percent increase. The proportion increase in CW ranges from $p_{\text{cw l}}= 0.24$ to $p_{\text{cw u}} = 0.32$. The actual proportion increase is generated randomly. Let $u$ be a realization of a uniform RV on $[0, 1]$,

$$\gamma = \begin{cases}\frac{(T - T_{\text{min met}})}{(T_{\text{o... ...et}})} & \text{ if $T_\text{opt met} < T \leq T_{\text{max met}}$}, \end{cases}$$ (A.51)

and $x = u \gamma$. Let $y = (1 - CW/19)^{0.45}$ if CW $\leq$ 19 cm and 0 otherwise. $y$ is used to decrease the changes in CW per molt as the crab's CW gets closer to 19 cm. The random increase in CW is given by:

$$CW_{\text{new}} = (1 + p_{\text{cw,l}} + (p_{\text{cw,u}} - p_{\text{cw,l}})\,x\,y)\, CW_{\text{old}}.$$ (A.52)

where $p_$cw,l is the lower proportion of CW increase and $p_$cw,u is the upper proportion increase.

The new mass when molting triggered is given by applying Eqn (A.36) to $CW_{\text{new}}$ (Eqn A.49) and the crab's new stomach volume is given by Eqn (A.37). In the model, male crabs cease molting once they reach the 20th instar. If a crab is female and is reaching its 18th molt, it mates with a male provided a male with an instar greater than 18 is present within 5 times the maximum interaction distance (See Appendix A.5.3). If the female mates, its mating status is set to mated. If a suitable male is not present, the female will not mate, but will attempt to mate again the next time she molts. Following molting, the crab's eating status is set to foraging and the crab is able to move again.