Before proceeding, note that the transition matrix for the elemental distance process is and, from (30), we can write (32).
This is possible because, as we will show, the elemental distance process is itself a Markov chain when and the transition matrix is symmetric.
Even for the case, we note that the elemental distance process is well approximated by a Markov chain, and then the expression obtained for the probability of collision can be interpreted as a good approximation to the true collision probability.
On the other hand, even when the elemental distances do not follow a Markov chain, since, the equilibrium probability, the elemental distance process is well approximated by the Markov chain with transition matrix obtained by replacing in (23) and (24) with, such that.
This is of course contingent on the severity of the attack; I am always grateful for good days when the distancing process can work its spell.
If a range expansion was in fact the primary determinant of the directional trends observed in S. o. caurina and P. monticola, we suggest that this process may have disrupted a previously-existing isolation-by-distance process and lead to the inability of these two procedures to detect underlying patterns.
Scolopale rods grow from assembled actin and may also play a role in this distancing process and cap elongation (in fact, we do not clearly observe scolopale rods using actin labeling until approximately 24 hrs APF, although defects in GFP-NompA distribution are already detected at 18 hrs APF (Fig. 2)).
Continue with this distancing process until you are certain that you have allowed enough time for a crush to be out of the question.
Collectively, these results suggest that some IMMADs might work preferentially for long distance coordination processes, while others might work preferentially over shorter distances.
The occurrence of spatial correlation over short distances can be interpreted as the result of current low distance migration/drift processes.
