Algorithm: Interaction and Aggression Outcomes between Crabs

Crabs are highly cannibalistic with large crabs (14-16 cm carapace width) accounting for very high rates of juvenile mortality (40-90% in Chesapeake Hines and Ruiz (1995), $>$85% in Alabama Spitzer et al. (2003)), with somewhat lower rates farther north (Heck Jr and Coen 1995). Juvenile crabs in turn prey on smaller crabs and megalopae (Moksnes et al. 1997). A variety of fish and birds are also known to consume blue crab, but in insignificant numbers relative to larger crabs (Hines, in press; Hines and Ruiz 1995). Blue crabs engage in aggressive and defensive behaviors to fend off other crabs from food or mates. In tank experiments, such behaviors occurred when two crabs were within 10 cm of each other  (Jachowski 1974). Larger individuals almost always dominate smaller ones (Dittel et al. 1995) and males typically dominate females (Jachowski 1974). Many invertebrate and vertebrate species respond to threat or injury by autotomizing an appendage. Approximately 25% and 18% of the crabs sampled from Chesapeake Bay in 1986 and 1987, respectively, were missing one or more limbs, usually a cheliped (Smith 1990). We do not model autotomy because limited levels of autotomy do not appear to alter crab behaviors (Jachowski 1974) or life history (Smith and Hines 1991; Smith 1995).

In the model, we assume that a potential interacting crab can be chosen randomly based on the distance between the crab being updated and neighboring crabs. The set of neighboring crabs is found by aggregating the crabs located on the neighboring fine-level triangles within $d_$max inter=12 m (the max interaction distance) of the crab being updated. Crabs that are further than $d_{\text{max inter}}$ apart are assumed not to interact. Each crab stores the last crab it interacted with and this crab is excluded from the set of possible interactions. The distance between each neighboring crab, $C$, and the crab being updated, $d(C, C_{\text{update}})$, is computed and $C$ is randomly designated as a potential interacting crab if $u < \sqrt{1 - \frac{d(C, C_{\text{update}})}{d_{\text{max inter}}}}$ where $u$ is a realization of a uniform random variable on $[0, 1]$. The potential interaction time of these two crabs is generated randomly over the smallest time interval until either crab was scheduled to be updated again (Appendix A.5.2). Over all potential interaction times, the pair of potentially interacting crabs with the smallest randomly generated update time is stored in the priority queue and in each crab's interaction set. As a result, crabs that are closer to each other have a greater chance of interacting.

A scheduled interaction will not occur if one of the crabs is updated because of an earlier scheduled update. If the interaction does occur, possible outcomes include that nothing happens to either crab, one crab is killed and nothing happens to the other, or one crab flees while nothing happens to the other. The actual outcome of the interaction depends on a large number of factors: gut fullness, whether one or both crabs are molting, the sexes of the interacting crabs, whether an interacting female is in her final molt, and the size difference between the crabs. The rules governing the outcome of interactions are not complex, but many different cases must be considered. These interaction rules are given below. A killed crab becomes food for the crab that killed it. If a feeding crab flees, its mass of food feeding on (Appendix A.5.6) is passed off to the attacking crab.

The interaction rules between two crabs, A and B are symmetric and so will only be described relative to Crab A. If the crabs are both molting, nothing happens to either of them. If crab A is molting then:

1. If A molting and male and B is not molting.
   • If the CW width of A is less than 1.4 times the CW of B, A dies and nothing happens to B.
   • If the CW width of A is greater than 1.4 times the CW of B. Nothing happens to A and B flees.
2. A molting, female and not on terminal molt, B is not molting.
   • Exactly as in the two cases in 1).

3. A molting, female and on terminal molt, B is not molting and male.
   • Nothing happens to A or B.
4. A molting, female and on terminal molt, B is not molting and female.
   • Exactly as in the two cases in 1).

Finally, we discuss the case when both crabs are not molting. The idea behind this rule is that crabs only attack other crabs and kill/eat them if their gut is almost empty and there are particular size differences between the two crabs. Let $c = 1 - \frac{V_{\text{stom}} - V_f}{V_\text{stom}}$, where $V_f$ is the volume of food in the crab's stomach. $c$ is between zero and 1 and denotes ``gut emptiness''. Let $\mu = 2$, $\sigma = 0.5$ and $u$ be a realization of standard uniform random variable on $[0, 1]$. Let $\alpha_$inter$=0.8$ and

$$\nu = \alpha_ \exp\left(-\left(\frac{CW_A/CW_B - \mu}{\sigma}\right)^2\right).$$ (A.26)

Then,

• If $c_A > 0.95$, then
  • if $u \leq \nu$, Crab B killed by Crab A.
  • else B flees, nothing happens to A.
• If $c_A \leq 0.95$ then B flees.

For $c_A > 0.95$ for Crab A, if both crabs are the same size, then there is only a small chance that B is killed. When A is twice as big as B, B is killed with probability 0.8, and when A is 3 times as big as B, B has a low probability of being killed since it is less likely that A would bother with such small prey.