Algorithm: Movement

Many crustaceans are tolerant of hypoxic exposure and can regulate oxygen consumption at low oxygen concentrations (McMahon 2001). However, both the impacts of hypoxia and a crab's reaction to it are stage specific. Adult crabs avoid hypoxic water (Das and Stickle 1994; Lowery and Tate 1986; Pihl et al. 1991). Juvenile crabs (20-25 mm carapace width, CW) prefer oxygen concentrations of 4.5 to 5.8 mg/L, but have a reduced ability to detect low oxygen and to find and remain in optimal oxygen conditions (Das and Stickle 1994). Crabs exposed to normoxia grow faster, have a higher feeding rate, molt more and have a shorter intermolt interval than those exposed to low oxygen conditions (Das and Stickle 1993). Adult males tend to spend their lives in lower salinity water located in the upper part of the estuary. Prior to mating, females are distributed throughout the estuary, but following mating remain in higher salinity waters near the mouth of the estuary.

Individual crabs in this model are assumed to be moving continuously with a rate, $v\geq 0$ (m/hr), at an angle $\theta \in [0, 2\pi)$ relative to the horizontal axis of the estuary. Movement is determined by the crab's local environment or the finest level triangle containing the crab. Since crabs move continuously in space, determining this fine level triangle is accomplished using the nested conforming triangulation of the estuary (Fig. A1). After determining the coarsest-level triangle containing the crab with an iterative search, one then searches within the triangles that each coarser level triangle is divided into (Fig. A1) and one repeats this process down the nested triangulation levels. The average computational cost of finding the finest-level triangle containing the crab using such an algorithm is much less than if we were to exhaustively search over all finest-level triangles. Such nested triangulations also play a central role in adaptive multigrid algorithms used to numerically solve partial differential equations (e.g., Braess 1997) and will be useful if more detailed and mathematically complex models (e.g., a partial differential equation model for DO) are ever required.

Unlike the above algorithm, movement of individuals in IBMs is typically done by dividing the spatial domain into small cells with individuals moving between neighboring cells when updated. Such movement algorithms, however, have limitations relative to the one outlined above (Tischendorf 1997). First, the size of the grid is limited by memory capacity and computational time required to iterate over the entire grid. Second, the fixed spatial structure implies equal resolution for both landscape features and individual movements. Finally, individuals can only move in complete cell units and are limited in movement angles. For such a movement algorithm to work, the distance between cells must be small relative to the individual's actual movement rates and a balance must be found between computational tractability (number of cells that can be stored in memory) and the time resolution of the model. The smaller each individual cell, the smaller the time resolution over which the model can be advanced, but the greater the model's computational demands.

The crab's new direction of movement, $\theta_{\text{new}}$, is arrived at by generating a new random change in direction based on a variety of factors, including oxygen concentration, salinity, depth, whether the crab is fleeing as a result of an interaction or if the crab has hit the boundary of the estuary. $\theta_{\text{new}}$ is kept with $[0, 2\pi)$.

If the crab is not fleeing and it has not hit the estuary boundary, then $\theta_{\text{new}}$ is generated randomly based on trying to maximize or minimize environment variables. The environment variables are not all of equal importance in determining a crab's direction of motion. In the list below, conditions further down in the list over-ride earlier ones. Let $u(\psi)$ denote a realization of a uniform random variable generated over the interval $[\psi - \pi/4, \psi + \pi/4]$ and let $\theta_c$ denote the current direction of the crab. Let $\theta_p$ denote the direction in which the particular environment variable is maximum/minimum (depending on the context) and is computed based on the values of the variable at the three nodes defining the triangle of the crab's current triangle in the case of salinity and depth and using the neighboring set of triangles for DO.

• Base Case $\theta_{\text{new}}$ is uniformly distributed over $[0, 2\pi)$.
• Salinity Two salinity thresholds are set based on the mean salinity at $d_s = 0.5$ and $d_s = 0.7$ using Eqn (A.6) (Appendix A.3.3). The first value defines a threshold for the juveniles ($v_{j,sal}$) while the second defines a threshold for sexually mature females ( $v_{f, sal}$).
  • Juvenile Crab's CW $<$ 5 cm and the salinity of its environment is $> v_{j,sal}$, then $\theta_{\text{new}} = u(\theta_p)$ where $\theta_p$ is toward lower salinity water.
  • Non-Juvenile Male Crab's CW $\geq 5$ cm, male and salinity $> v_{f, sal}$, then $\theta_$new$= u(\theta_p)$ where $\theta_p$ is toward lower salinity water.
  • Non-Juvenile Female Crab's CW $\geq 5$ cm, female, and salinity $< v_{f, sal}$, then $\theta_$new$= u(\theta_p)$ where $\theta_p$ is toward higher salinity water.

• Depth
  • Juvenile CW $\leq 5$ cm and crab depth $\geq$ 1.5 m, then $\theta_$new$= u(\theta_p)$ where $\theta_p$ is toward shallower water.
  • Non-Juvenile CW $> 5$ cm and crab depth $<$ 1.5 m, then $\theta_$new$= u(\theta_p)$ where $\theta_p$ is toward deeper water.

• DO
  • Moderate DO Surrounding DO $\in [3, 4)$ and if a uniform random number generated over the range $[0, 1]$ is $> DO - 3$, then $\theta_$new$= u(\theta_p)$ towards higher dissolved oxygen. Thus, the closer the DO is to 3, the greater the chance that the crab's direction of movement will be influenced by its DO environment

  • Low DO If crab's DO $< 3$, then $\theta_{\text{new}} = u(\theta_p)$ towards higher dissolved oxygen.

• Flee or Hit Estuary Boundary $\theta_$new$= \theta_c - \pi + u(\theta_c)$.

A crab's maximum rate of movement, $v_{\text{max}}$, for a 200 g crab is $\approx 720$ (m/hr) (Wolcott and Hines 1990; Houlihan et al. 1985; Clark et al. 1999). In the model, the maximum rate of movement decreases with decreasing crab size. The new rate of crab movement, $v$ (m/hr), depends on whether the crab is eating or molting, the temperature of the crab's environment, whether the crab has lost weight because it is starved, whether it is fleeing from another crab, moving out of an hypoxic patch, or moving away from or into higher or lower salinity water. The factor decrease in the crab's maximum rate of movement, $\gamma_{\text{met}}$, depends on how the temperature of the crab's environment differs from $T_{\text{met opt}}$ and whether the crab lost weight due to starvation:

$$\gamma_{\text{met}} = \left(\frac{G}{G_\text{max}}\right)^2 \, f$$ (A.27)

where $G$ is the current mass of the crab, $G_{\text{max}}$ the maximum mass ever attained and $f=f_{\text{Temp}}(T; Q_\text{met}=2.5, T_\text{max met}=35, T_\text{opt met}=26)$ is given by Eqn (A.39).

Let $\alpha_{\text{max move}} = 300^{1-\beta_{\text{move}}}$ (g $^{1 - \beta_\text{move}}$). The actual rate of crab movement, $v$ (m/hr), depends on whether the crab is eating or molting and environmental conditions:

$$v = \begin{cases}0 : \text{ is \textit{eating} or \textit{molting... ...G^{\beta_{\text{move}}-1} : \text{ all other cases} \end{cases}$$ (A.28)

where $z$ is a realization from Beta $(b_1, b_2=20)$ random variable. The value of $b_1$ is given by:

$$b_1 = \begin{cases}2 & : \quad \text{If Salinity conditions not m... ...text{If DO $< 3$\ mg/L, or Fleeing} \\ 1 & : \quad \text{Otherwise} \end{cases}$$ (A.29)

Fig. A6 shows the density functions for the velocity of crab movement for a 200 g crab when the crab is either fleeing low DO, another crab, trying to find more optimal salinity conditions or simply diffusing about the estuary.