A model for decoupling emigration from mortality in recapture surveys of slow-moving benthic marine gastropods.

ABSTRACT Decoupling emigration from mortality is critical for accurately estimating mortality in recapture surveys. However, open-population models often suffer from their inability to distinguish between these two parameters. A model is presented here that separates emigration from mortality based on movements of a tagged population of 42 hatchery-reared queen conch, Strombus gigas, released into a 20 m x 20 m plot in the nearshore waters of the Florida Keys. The model was constructed in a 4-step process. Recapture sampling of uniquely tagged individuals was used to derive a frequency distribution based on the movements of these individuals over a given time period. A probability density function was then fit to the frequency distribution. A probability of emigration was assigned to each cell in the plot. This value represented the probability of an individual located in that cell emigrating from the plot. Missing tagged individuals were assigned emigration probabilities (survival) based on the distance between their last observed location and the distance to the periphery of the plot. The overall survival of the population was estimated by constructing a survivorship table to estimate survival within and external to the plot for each sampling interval. After 3 mo, 14 conch were recaptured alive and another eight survivors were estimated to be present outside the plot representing an overall survival of 52.7%.

KEY WORDS: capture-recapture, dispersal, emigration, mortality, queen conch, Strombus gigas

INTRODUCTION

Estimating abundance and mortality in animal populations is essential for accurate evaluations of management strategies. In many cases, recurring recapture surveys provide estimates of these parameters (Seber 1982). However, many open-population mark-recapture models suffer from their inability to distinguish between emigration and mortality (Pollock 1991). Underestimating emigration, then, has the consequence of overestimating mortality. Despite this problem, many investigators continue to use open-population models and incorrectly assume that missing individuals died. One widely used model, the Jolly-Seber model for open populations (Seber 1982), is widely used to estimate mortality in populations of animals; however, this model is unable to distinguish between emigration and mortality without additional information (Pollock 1991, Peter 2001).

For slow-moving benthic gastropods, the estimation of abundance and/or mortality can be especially confounding because of incomplete recovery of tagged individuals due to complex survival strategies such as burial and dispersal (sensu Stoner & Davis 1994). Our studies have shown that recaptures from visual surveys of juvenile queen conch (Strombus gigas, Linaeus) may substantially under-represent those conch actually present (Glazer et al. 1997). Even with relatively slow-moving benthic species, dispersal may be significant (Hesse 1979, Blecha et al. 1992, Matsuda & Akamine 1994, Norkko et al. 2001) making recapture surveys inefficient and, as time progresses, less reliable (Sandland & Kirkwood 1981).

Several researchers have developed models to decouple emigration from mortality based on the movements of individuals in the population. Jackson (1939) proposed a method that estimated emigration based on movements of butterflies between contiguous quadrants in a plot, and he determined mortality by subtracting emigrants. This approach was extended for use in terrestrial snails (Cameron & Williamson 1977). More recently, Zeng and Brown (1987) presented a model that uses dispersal distances to estimate population death rate by using traps. This method was adapted to decouple emigration from mortality in a population of frogs in adjacent ponds (Peter 2001). Beinssen and Powell (1979) presented a method to estimate mortality in a population of abalone by examining movements within a grid system and estimating the proportion of animals not leaving the grid.

The model presented in this study extends that of Beinssen and Powell's (1979) by using movements of a population of tagged individuals within a grid to develop a probability density function from which a probability of emigration may be calculated for each missing individual. This model provides high spatial resolution and provides estimates of emigration probabilities from cells within the plot within which no animals were released, but where some individuals may have been observed and subsequently not recaptured. A 4-step procedure is used which ultimately results in the estimation of survivorship of the population. The model is applied to a hatchery-reared population of a slow-moving benthic gastropod, the queen conch (Strombus gigas Linnaeus, 1758), released into a nearshore marine environment.

MATERIALS AND METHODS

Field Study

To examine survival of hatchery-reared queen conch (Strombus gigas) after release into the wild, our laboratory conducted a mark-recapture experiment in October 1996 in the Florida Keys, USA. A total of forty-two 9-cm conch were marked with uniquely-numbered aluminum tags and released uniformly into an existing juvenile conch population within a sea-grass meadow in a back-reef. All conch were released within a 20 m x 20 m plot, further subdivided into 1 m x 1 m cells. Recapture surveys were conducted over a 3-mo period with most surveys conducted weekly; in some cases, poor weather resulted in surveys occurring biweekly. Recaptures were aided by using underwater metal detectors, which our laboratory has demonstrated is a highly efficient method of tag recovery (Glazer et al. 1997). At the time of each recapture survey, the Cartesian coordinates (i.e., cell) of each individual were noted to the nearest meter. I used these data to estimate the survival of the population at each recapture interval by coupling those conch observed alive with an estimation of the proportion of conch emigrating from the plot using a generalized emigration-mortality model (GEMM).

Model Development

Step 1--Development of the Probability Density Function

In the first step, the linear distance traversed for each individual for the recapture period (Fig. 1) was calculated using the Pythagorean Theorem (Eq. 1):

d = [[[([x.sub.t] - [x.sub.(t-1])).sup.2] + [([y.sub.t] - [y.sub.t-1]).sup.2]].sup.0.5] (1)

where d is the distance traveled, [x.sub.t] is the east-west coordinate at time t, and [y.sub.t] is the north-south coordinate at time t.

Based on the simple function:

f(x)|[lambda], (2)

where [lambda] = observed movements of the population for the given time interval, I then derived probability density functions that were based on the frequency distribution of the observed movements of the conch (e.g., Fig. 2) for each time interval collected over the entire study. I developed separate frequency distributions of conch movement for the first week after release (Fig. 2), all other 1-wk time periods combined, and all other 2-wk time periods combined. These three functions were derived because the movements of recaptured conch for the first week after release were significantly different from movements of recaptured conch during all other 1-wk periods and, in the latter case, because in some instances surveys only occurred after 2 wk. In all cases, the frequencies best fit the negative binomial distribution.

[FIGURE 2 OMITTED]

Using the data from the probability density functions, a separate table of movement probabilities was developed for each time period (e.g., Table 1). Each table represents the probabilities of individuals within the population traversing a given distance, p, for that time period and was derived from the expected frequencies from the best-fit probability density function. However, the model requires estimations of the probability of an animal traversing a distance to an edge of the plot or beyond the edge; therefore, an additional probability was derived that represents the sum total of the probability that a conch will move a given distance plus the probabilities that it will move greater distances (Table 1: [p.sub.e]).

Step 2--Constructing a Matrix of Emigration Probabilities

I estimated the probability of emigration from each cell by constructing a matrix (Table 2) that represented the 20 m x 20 m plot. Each cell represented a 1 m x 1 m sampling unit in the plot. Emigration probabilities were assigned to each cell of the matrix and these represented the probability that an individual originating in that cell and not subsequently found alive emigrated from the plot during that time interval (Table 2). The emigration probability is based on the distance from the cell to the perimeter of the plot and was derived from the table of movement probabilities (Table 1). Because an individual can move in any of 4 directions (designated arbitrarily as north [N], east [E], south IS], or west [W]; Fig. 3), the probability assigned to each cell is the mean of the probabilities for movement to the periphery in all 4 directions and is given by:

[[PI].sub.x,y] = ([P.sub.e(N)] + [P.sub.e(E)] + [P.sub.e(S)] + [P.sub.e(W)])/4, (3)

where, [[PI].sub.x,y] = the probability of emigration of an individual found in cell x,y emigrating from the plot, and [P.sub.e(n.w)] = the probability that an individual will move a distance equal to or greater than the distance to the edge of the plot. The N, E, S and W subscripts indicate a directional component to movement and represent movements to the top, right, bottom and left sides of the plot, respectively (Fig. 3).

[FIGURE 3 OMITTED]

Step 3--Assigning Emigration Probabilities to Missing Individuals

I assigned an emigration probability to each missing individual (Eq. 3: [[PI].sub.x,y], e.g., Table 3) after each survey based on the cell where it was last observed alive within the plot (e.g., Table 2). Conversely, mortality for any individual is simply defined as 1-[[PI].sub.x,y]. Live recaptures (i.e., survivors) were assigned a value of [[PI].sub.x,y] = 1 (i.e., there was no emigration or mortality associated with that individual.).

The total number of emigrants at time t([E.sub.t]) in the population was defined as the sum of the individual emigration probabilities (Eq. 4). This represented the estimated number of conch that had emigrated during that time interval.

[E.sub.t] = [m.summation over i=0] [PI.sub.m], (4)

where m is the number of missing individuals. Therefore, the estimated number of survivors in the population is defined as the sum total of the estimated number of emigrants [E.sub.t] plus the number of recaptured survivors at time t ([R.sub.t]):

[S.sub.t] = [E.sub.t] + [R.sub.t] (5)

Step 4--Constructing the Cumulative Survivorship Table

In the fourth step, I determined the mortality of the entire population by constructing a cumulative survivorship table that detailed estimated survival within and external to the plot for each sampling interval (Table 4). Survival within the plot was estimated for each time period by using the GEMM, with consideration given to the running total of the estimated number of survivors. The total number of animals emigrating (i.e, emigrants) for each sampling period was estimated from the sum of the emigration probabilities for all conch that were not observed during that survey and each animal's last known location. A mortality rate for each sampling period was calculated based on the estimated number of total survivors for the previous time period, the number of live recaptures for the current time period, and the estimated number of emigrants during the current time period. This mortality rate was applied to the estimated number of surviving emigrants (i.e., estimated number of live animals outside the plot) for the previous time period (Table 4: emigrants/cumulative survivors) to estimate the number of surviving emigrants from the previous time period. For each time period, the total number of surviving individuals equaled the total number of individuals recovered alive plus the estimated total number of surviving emigrants from that time period. Cumulative survivorship for the entire population was calculated by dividing the survivors at time t (i.e., sum of the recovered individuals plus the cumulative survivors) by the number originally tagged and released.

Several other parameters for the entire population were derived from recaptures and emigrants. The number of deaths equaled the number of missing animals less the number that were believed to have emigrated during the current time period added to the estimated number of emigrants from previous time periods that died during the current time period. Cumulative deaths are simply the running total of the deaths during time t. The mortality coefficient, Z, is estimated from the standard equation (sensu Gulland 1983):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

where [N.sub.2] = survivors at time [t.sub.2], [N.sub.1] = survivors at time [t.sub.1], and ([t.sub.2] - [t.sub.1]) = time between recapture expressed in as a fraction of a year.

RESULTS

After approximately 3 months, 14 conch were recovered alive and 28 were missing (Table 4). Of these 28, I estimated that 11 had emigrated, 8 of which had survived. Furthermore, I estimated that 17 had died within the plot and by attaching mortality rates within the plots to the emigrants at each time interval, three of the estimated eight emigrants had died. Thus, a total of 22 conch had survived and 20 had died resulting in a 52.7% survival of the population over the time of the study. The overall mortality coefficient over the 13 wk experiment was Z = 2.57.

DISCUSSION

Dispersal is a parameter that can significantly affect estimates of survival in a variety of populations and is often estimated using mark-recapture experiments (sensu Schwarz et al. 1993). In open-population models, mortality may be overestimated because of the inability of these models to distinguish between emigration and mortality (Barndorff-Nielsen 1972, Cameron & Williamson 1977). In this study, 29% (8 of 28) of missing individuals were estimated to have emigrated and survived after emigration. Clearly, if these emigrants had been quantified as deaths, the overall mortality of the population would have been greatly exaggerated--67% instead of 47%. Therefore, by examining dispersal rates of the population, it is possible to quantitatively estimate the probability of mortality of each missing individual.

There are four possible outcomes for a tagged individual in a recapture survey and probabilities of survival may be attached to each outcome. In the first two cases, direct observations allow the researcher to assign a survival probability to the individual. First, the animal may be recaptured alive and so the probability of survival would be set to 1 for mathematical purposes. In the second case, the individual may have been recovered dead and so the assigned survival probability would be 0.

On the other hand, there are instances where the animal was not recovered. In these cases, the individual may have emigrated, or the individual may have been overlooked in the surveys. In both instances, survival would range from 0 to 1 and would be based on the best information available to the researcher. Quantifying either of these parameters will greatly facilitate making informed decisions about the fate of a missing individual. The model described herein presented a method of integrating direct observations of survival with indirect estimations of emigration to develop a comprehensive model for estimating mortality in an open population.

As with any model, recognition of the assumptions is critical and should be emphasized. In general, the model assumes that the animals move uniformly and that there are no subsets within the population that behave differently. More specifically, the model assumes that the rates of movement do not change over the recapture time period. If they do differ, the researcher must be able to test for this and to conduct separate runs of the GEMM for each instance in which movements are significantly different. Additionally, the movements of the population must not differ during that time period immediately after release and before the first recapture occurs from the movements that occur between subsequent recapture intervals during the rest of the study, because there are no recapture data from which a probability density function may be derived. Therefore, the derived function may underestimate emigration and, therefore, overestimate mortality.

The model also assumes that each movement of each individual used in developing the probability density function is representative of the population as a whole. If the movement data of an individual that was recaptured frequently are pooled with the movement data of individuals recaptured less frequently and these pooled data are used to develop the probability density function, effects associated with lack of independence may arise.

Furthermore, the movement of the population during any sampling interval must be either random or homogeneous. If subsets of the population move differently than other individuals or subsets, the estimation of the emigration probabilities assigned to each cell may be erroneous. However, the model is robust to directional movements (i.e., nonrandom) of the population because the increase in emigration probability associated with movement in one direction is proportionately offset by the decrease in probability of emigration to the opposite direction. For example, if the entire population moves to the north, the probability of emigration to the north will increase proportionate to the decrease in the probability of emigration to the south.

Another assumption of the model is that mortality rates outside the study area are identical to those inside. If this assumption is violated, the number of surviving emigrants may be over- or underestimated. This will result in inaccurate estimates of the total number of survivors and, ultimately, of the survivorship of the population.

Additionally, landscape features must not affect the movements of a subset of the population. Although it is not necessary that the landscape be homogeneous, heterogenous or discontinuous landscapes may affect the ability of each individual in the population to behave similarly. In some cases, this bias may be accounted for if the sample size is large enough and those movements are treated as outliers when developing the probability density distribution.

Finally, there must be a sufficient number of recaptures to fit the movement frequencies to a distribution. With too few recaptures, the probability density function may not accurately reflect the movement of the population. Ultimately, the confidence in the survival probability assigned to a missing individual will be based on the goodness of fit of the data to the derived probability density function.

Anderson et al. (1993) observed that relatively few biologists conduct treatment-control field experiments using tagged individuals, and they recommended that more emphasis be placed on this approach. Furthermore, given the increased interest in releasing hatchery-reared animals to supplement existing or restore depleted populations (e.g., Harrison 1986, Stoner 1997), effective methods must be developed to evaluate the efficacy of a release program (Stoner & Glazer 1998).

Although this approach is probably best suited to relatively slow-moving, benthic invertebrates such as bivalves (e.g., scallops), other gastropods (e.g., abalone), and perhaps echinoderms (e.g., urchins), it is possible to extend this method to estimate emigration probabilities for other species as long as one is able to estimate with reliability the movements of a population and the assumptions are met. For example, as telemetry equipment becomes more sophisticated and less expensive, in situ studies are measuring, with a high degree of temporal sensitivity, the movements of populations of tagged crustaceans (Panuluris argus: R. Bertelsen, Florida Fish and Wildlife Conservation Commission, pers. comm.) and fishes including the temperate snapper, Pagrus auratus (Hartill et al. 2003) and the tropical grouper, Epinephelus striatus (Bolden 2001). These data may then be used to develop emigration probabilities from predefined areas thereby facilitating tests of ecologic theory (e.g., the examination of "spillover" from marine reserves.) Thus, this model provides a relatively simple method of allowing researchers to evaluate programs using hatchery-released animals and to allow ecologists to conduct treatment-control field experiments using tagged individuals.

ACKNOWLEDGMENTS

The author thanks Robert Muller and Rodney Bertelsen, who provided constructive comments to early drafts of this manuscript. Richard Jones and Gabriel Delgado assisted in field surveys. Judy Leiby and Jim Quinn of Florida Fish and Wildlife Conservation Commission provided editorial comments. Mary Enstrom and Sherry Dawson of The Nature Conservancy (TNC) organized the numerous volunteers who participated in the field surveys. This work was supported, in part, under funding from the Department of the Interior, U.S. Fish and Wildlife Service, Partnerships for Wildlife Grant Number P-1. The Florida Fish and Wildlife Conservation Commission supplied additional funding.

LITERATURE CITED

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Cameron, R. A. D. & P. Williamson. 1977. Estimating migration and the effects of disturbance in mark-recapture studies on the snail Cepaea nemoralis L. J. Anita. Ecol. 46:173-179.

Glazer, R. A., R. J. Jones, L. A. Anderson & K. J. McCarthy. 1997. The use of underwater metal detectors to locate outplants of the mobile marine gastropod, Strombus gigas. Proc. Gulf Caribb. Fish. Inst. 49: 503-509.

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Peter, A. H. 2001. Survival in adults of the water frog Rana lessonae and its hybridogenetic associate Rana esculenta. Can. J. Zool. 79:652-661.

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ROBERT GLAZER

Florida Fish and Wildlife Conservation Commission, Fish and Wildlife Research Institute, 2796 Overseas Highway, Ste. 119, Marathon, Florida 33050

E-mail: [e-mail]

TABLE 1.

The expected probability, p, of a 9-cm hatchery-reared queen conch
moving a given distance in meters (the number in parentheses) the
first week after release for a study conducted in October 1996 in the
Florida Keys, USA. The probabilities are based upon the observed
movements of queen conch recaptured one week after release and
are derived from a negative binomial distribution. The value for
p(0) represents the probability of an individual not moving from the
cell where it was released during the time interval. The column
headed with [p.sub.e] represents the probability of an individual
moving at least the indicated distance and is the sum of the
probabilities for movements equal to or greater than that distance.

        P         [P.sub.e]

  P(0) = 0.0624
  P(1) = 0.1016     0.9376
  P(2) = 0.1180     0.8359
  P(3) = 0.1185     0.7180
  P(4) = 0.1098     0.5995
  P(5) = 0.0966     0.4897
  P(6) = 0.0820     0.3931
  P(7) = 0.0678     0.3111
  P(8) = 0.0549     0.2433
  P(9) = 0.0437     0.1884
P(10+) = 0.1447     0.1447

TABLE 2.
Emigration probabilities expressed in percentages
for each cell in one 10 m x 10 m quadrant of the
GEMM matrix for the first week after release of
9-cm hatchery-reared queen conch. Each quadrant is
a mirror image of the adjacent quadrant. Each number
in the paired numbers along the axes represents the
Cartesian coordinate of each cell for each quadrant
of the matrix. The percentages represent the
probability of an individual emigrating from the
plot from the cell in the matrix. For example, a
missing individual previously captured in cell 0,19;
0,0; 19,0; or 19,19 has a probability of 0.5411 of
having emigrated. The probability assigned to each
cell is based on the mean probability of an individual
traversing the distance to the edge of the plot in
either of four directions arbitrarily designated as
N, S, E, or W.

9,10   34.29   31.75   28.80   25.84   23.10
8,11   35.39   32.85   29.90   26.93   24.19
7,12   36.76   34.22   31.27   28.31   25.56
6,13   38.45   35.91   32.96   30.00   27.25
5,14   40.50   37.96   35.01   32.05   29.30
4,15   42.92   40.38   37.43   34.46   31.72
3,16   45.66   43.12   40.17   37.21   34.46
2,17   48.62   46.08   43.13   40.17   37.43
1,18   51.57   49.03   46.08   43.12   40.38
0,19   54.11   51.57   48.62   45.66   42.92
       0,19    1,18    2,17    3,16    4,15

9,10   20.68   18.63   16.94   15.57   14.47
8,11   21.77   19.72   18.03   16.66   15.57
7,12   23.15   21.10   19.40   18.03   16.94
6,13   24.84   22.79   21.10   19.72   18.63
5,14   26.89   24.84   23.15   21.77   20.68
4,15   29.30   27.25   25.56   24.19   23.10
3,16   32.05   30.00   28.31   26.93   25.84
2,17   35.01   32.96   31.27   29.90   28.80
1,18   37.96   35.91   34.22   32.85   31.75
0,19   40.50   38.45   36.76   35.39   34.29
        5,14    6,13    7,12    8,11    9,10

TABLE 3.
Queen conch released 21 October 1996 and not recovered on
28 October 1996. "Tag" represents the unique identifier of
the outplant. The X and Y columns represent the Cartesian
coordinate of the cell into which the conch was released.
The 11 represents the emigration probability from the plot
for that individual (i.e., survival) and is based on the
coordinate where the individual was last observed. The total
is the sum of the emigration probabilities (i.e., estimated
number of survivors) and is calculated using equation 4 (see
text). The total for 11 is slightly lower than that reported
in Table 3 because of rounding errors. The column 1-11
represents the estimated mortality attached to that individual.

 Tab    X    Y     II    1-II

16303    3   14   0.32   0.68
16319   16    3   0.37   0.63
16321    3    4   0.34   0.66
16324    3   19   0.46   0.54
16336    4   12   0.26   0.74
16343   17    1   0.46   0.54
16363   12   11   0.18   0.82
16375   19    2   0.49   0.51
16381    0   15   0.43   0.57
16383   15    0   0.43   0.57
Total             3.74   6.26

TABLE 4.
Cumulative survivorship table for 9-cm juvenile queen conch
outplants released in the Florida Keys in 1996. Separate GEMM
runs were conducted for the first week after release and the
other 1-wk periods because recaptured conch moved significantly
more during the first week after release than any other 1-wk
period. Additionally, the GEMM was run for two-week periods in
order to estimate emigration during those intervals when sampling
did not occur weekly. "Emigrants" are the number of conch that
were missing and were estimated by the GEMM to have emigrated
from the plot. "Cumulative survivors" is the estimated cumulative
number of emigrants surviving and is calculated by multiplying
the instantaneous mortality rate for that time period by the
cumulative surviving emigrants (cumulative survivors). The total
surviving column represents the sum of the individuals recovered
alive ("recovered") plus the estimated number of surviving
emigrants (cumulative survivors). "Cumulative survivorship" is the
percentage of number of tagged conch at the beginning of the study
divided by the sum of the surviving individuals (i.e., recovered
plus cumulative survivors). The estimation of the overall mortality
coefficient, Z, is based upon No = 42 and N, = 22.1. In some cases,
because of rounding errors, the columns do not add to 100%. Wk 0
represents the time of release.

                       Residents              Emigrants

                                                    Cumulative
Week   Date       Recovered   Missing   Emigrants   Survivors

 0     10/21/96      42          0        0.00
 1     10/29/96      32         10        3.74         3.8
 2
 3     11/08/96      30          2        1.05         4.7
 4
 5     11/22/96      30          0        0.00         4.7
 6
 7     12/07/96      28          2        0.92         5.4
 8     12/15/96      26          2        0.62         5.8
 9
10     12/27/96      22          4        1.45         6.7
11     01/03/97      15          7        2.64         8.0
12
13     01/16/97      14          1        0.44         8.1

    Overall                     14         28           11

                                    Total

                   Number        Deaths           Deaths
Week   Date       Surviving   (inside plot)   (outside plot)

 0     10/21/96      42
 1     10/29/96     35.8           6.2
 2
 3     11/08/96     34.7           0.9             0.1
 4
 5     11/22/96     34.7           0.0             0.0
 6
 7     12/07/96     33.4           1.1             0.2
 8     12/15/96     31.8           1.4             0.3
 9
10     12/27/96     28.7           2.6             0.6
11     01/03/97     23.0           4.4             1.3
12
13     01/16/97     22.1           0.6             0.3

    Overall           8            22               17

                                    Total

                  Cumulative                Cumulative
Week   Date         Deaths     Mortality   Survivorship     Z

 0     10/21/96       0                        100%
 1     10/29/96       6.2        14.9%        85.1%        8.31
 2
 3     11/08/96       7.2         3.0%        82.6%        0.81
 4
 5     11/22/96       7.2         0.0%        82.6%         0
 6
 7     12/07/96       8.3         3.6%        79.6%        0.99
 8     12/15/96       9.7         4.9%        75.5%        2.55
 9
10     12/27/96      12.2         9.8%        68.3%        2.67
11     01/03/97      16.6        19.8%        54.7%       11.51
12
13     01/16/97      17.1         3.9%        52.7%        1.04

    Overall           3           17          52.7%        2.57