Rules: Larval Development and Recruitment

In addition to computing a crab's energy balance, another important component of the model involves development, recruitment and initialization of crabs into the model estuary. Development of spawned crab eggs occurs in the maturation pot object representing the ocean. All eggs released over a single day by all female crabs in the estuary are stored in a single ``egg pot''. The maturation pot is comprised of a sequential, linked list of pots each containing the number of larvae/juvenile crabs and their collective development stage. Once a day, the larvae/young crabs in this list are updated and growth and mortality applied based on the average temperature for that day. Based on the development information for blue crabs (Etherington and Eggleston 2000; van den Avyle 1984; van Engle 1958; O'Leary Amsler and George 1984), it is assumed that crabs take anywhere from 80 to 145 days to go from spawned egg to 7th instar - the stage when the crabs are moved from the egg pots and instantiated into the estuary. Crabs are instantiated at the 7th instar to avoid the computational costs of modeling large numbers of juvenile crabs which have very short lifespans.

It is assumed that the rate at which the crabs progress through development depends solely on average temperature. The development rate of larvae/crabs in each egg pot is taken to be linear with mean temperature, T (Eqn A.4), according to $aT + b$ where $a = (1 - \gamma_{\text{maturation}})/(T_\text{max spawn} - T_\text{min spawn})$ and $b = 1 - aT_$max spawn. $\gamma_$maturation$= 0.55$ is a constant chosen so that at a temperature of $T_$min spawn$= 19\ensuremath{^\circ\text{C }}$, development to 7th instar takes 145 days and at $T_$max spawn$=29\ensuremath{^\circ\text{C }}$ it takes 80 days.

Larvae/crabs in an egg pot die at a constant rate where the proportion surviving after a time interval $t$ is $\exp(-\lambda_$mat$t)$ and $\lambda_$mat$= 5\times 10^{-3}$ (1/hr). Thus, the longer the development the greater the mortality. Over an 80 d development, the proportion of the larvae/crabs surviving is $1\times10^{-4}$ while over 145 d it is only $5.6\times 10^{-8}$. If the mean temperature given by Eqn (A.4) is outside the range $[T_$min spawn$, T_$max spawn$]$, all the larvae/crabs in the egg pots die.

The degree to which megalopae return to a particular estuary is unknown and dispersal between estuaries is likely. However, the large distance between major estuaries in the Mid Atlantic Bight and the coherence between spawning and recruitment events suggests that larvae often return to their parent estuary (Garvine et al. 1997). North Atlantic estuaries show constant low levels of daily settlement punctuated by significant, episodic peaks that can account for $>50$% of the annual settlement at a site (van Montfrans et al. 1995). Variables such as wind vectors, radiant energy and surface temperature are thought to be important predictors of recruitment (Rugolo et al. 1998; Epifanio 1995; Tang 1985).

Thus, the proportion of 7th instar crabs recruited into the model estuary is generated randomly and is negatively correlated with the current density of crabs in the estuary and positively correlated with the number crabs produced by the egg pots. Let $u$ be a realization of a uniform RV on $[0, 1]$ and let $\vartheta =$   Crab Density$/\rho_$max ((#/m$^2$)/(#/m$^2$)). The maximum density of 7th to 20th instar crabs in the estuary is assumed to be no larger than $\rho_$max$= 0.4$ (#/m$^2$). The proportion of the 7th instar crabs instantiated from the egg pots for a given day is $u^{\vartheta}$. Thus, if the crab density is less than $\rho_$max (#/ m$^2$) a larger proportion are instantiated then when the density is greater than $\rho_$max. This reflects the fact that the higher the crab density in the estuary, the smaller the proportion of recruits surviving past the 7th instar.

Instantiated crabs are randomly placed in the estuary. Let $u_x$ and $u_y$ denote realizations from a uniform random variable over $[0, 1]$ and let $d_{x}$ and $d_{y}$ denote the horizontal and vertical dimensions of the rectangle containing the part of the estuary being modeled. Under a wind driven mechanism of recruitment, it is likely that higher densities of megalopae and juvenile crabs will be found toward the mouth of the estuary. The initial location of an instantiated crab is given by $(d_{x}\sqrt{u_x}, d_{y}u_y)$ provided this location is contained in the estuary.

A crab's sex is specified randomly based on a Bernoulli random variable where the probability of male is dependent on the sex ratio (assumed to be 50%). Both initial CW and mass contain a random component. The initial CW of the instantiated 7th instar crab is uniformly distributed over 1.05 to 1.65 cm while the initial mass of the crab, $G_{\text{init}}$, is given by applying Eqn (A.36) to $CW_{\text{init}}/(1+1/2u)$ where $u$ is a realization from a uniform random variable over $[0, 1]$. Thus, the initial mass of the crab is set below the mass at which molting would be triggered. The crab's initial stomach volume is given by Eqn (A.37).