Empirical distribution of Chapman’s estimator in simulations with the standard sampling scheme.

<p>Study-reach length is 90 (arbitrary units), study-reach abundance <i>n</i> = 300, sample-1 capture probability <i>q</i> = 0.1, 0.3, 0.5, 0.7, sample-2 capture probability <i>q</i>′ = <i>q</i>, and maximum movement distance <i>δ</i> = 0, 30, 60, 90. Dispersal is balanced, so the average change in abundance between samples is approximately zero. Symbols and lines are defined as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0200733#pone.0200733.g003" target="_blank">Fig 3</a>.</p

Effects of expanding the study reach on the empirical distribution of Chapman’s estimator with constant sampling effort.

<p>Top row: standard sampling scheme applied to a study reach of length 30 (arbitrary units) and abundance <i>n</i> = 120. Middle row: standard sampling scheme applied to an expanded study reach of length 90 and abundance <i>n</i> = 360. Bottom row: modified sampling scheme applied to an expanded study reach of length 90 and abundance <i>n</i> = 360, with 120 fish in each of zones U, C, and D. Sampling effort (duration of sampling) is the same in all cases, resulting in higher capture probabilities when sampling is restricted to a single zone. Maximum movement distance <i>δ</i> = 60 in all cases. <i>q</i>: sample-1 capture probability; <i>q</i>′: sample-2 capture probability; : average proportion of individuals in the study reach that exited between samples (standard sampling scheme only); , , : average proportion of individuals in zones U, C, and D of the study reach that exited the study reach between samples (modified sampling scheme only); : average number of immigrants in simulations; : average relative bias with respect to <i>n</i> (percent) of Chapman’s estimator in simulations; IQR: empirical interquartile range. Solid red lines: <i>n</i>; dashed blue lines: empirical mean of . See text for additional explanation.</p

Empirical distribution of Chapman’s estimator in simulations with the modified sampling scheme.

<p>Study-reach length is 90 (arbitrary units), study-reach abundance <i>n</i> = 300, sample-2 capture probability <i>q</i>′ = 0.1, 0.3, 0.5, 0.7, sample-1 capture probability <i>q</i> = 1 − (1 − <i>q</i>′)³, and maximum movement distance <i>δ</i> = 0, 30, 60, 90. Sample-1 capture probabilities ensure sampling effort (duration of sampling) is the same as in sample 2. Dispersal is balanced. Other symbols and lines are defined as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0200733#pone.0200733.g003" target="_blank">Fig 3</a>.</p

Simulation results with unbalanced migration.

<p>Simulation results with unbalanced migration.</p

Comparison of standard and modified sampling schemes applied to a stream reach subject to dispersal.

<p>The study reach is shown in tints of blue, parts of the population domain outside the study reach in yellow. Black dots: fish marked and released in sample 1 (unmarked fish not shown). Top row: Standard scheme applied to a short study reach. Middle row: Standard scheme applied to an expanded study reach. Bottom row: Modified scheme applied to an expanded study reach, with zones U, C, and D indicated (darker blue: samples 1 and 2 are taken; lighter blue: only sample 2 is taken). Left panels show locations of marked fish upon completion of sample 1. Center panels show dispersal of marked fish in progress. Right panels show locations of marked fish when sample 2 is taken.</p

Minimum sufficient abundances and asymptotic relative biases.

<p>Minimum sufficient abundances and asymptotic relative biases.</p

Definition sketch for comparing bias of standard and modified sampling schemes.

<p>Five equal-size zones of a stream reach are shown. Zone C is the original study reach; zones U, C, and D constitute the expanded study reach. Each zone contains <i>n</i>₀ fish when sample 1 is taken. Isotropic dispersal occurs between samples 1 and 2, with probability <i>p</i>₀ of exiting any given zone. See text for additional details.</p

Simulation results with balanced migration.

<p>Simulation results with balanced migration.</p