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The authors have declared that no competing interests exist.

Experiments involving mosquito mark-release-recapture (MRR) design are helpful to determine abundance, survival and even recruitment of mosquito populations in the field. Obstacles in mosquito MRR protocols include marking limitations due to small individual size, short lifespan, low efficiency in capturing devices such as traps, and individual removal upon capture. These limitations usually make MRR analysis restricted to only abundance estimation or a combination of abundance and survivorship, and often generate a great degree of uncertainty about the estimations.

We present a set of Bayesian biodemographic models designed to fit data from most common mosquito recapture experiments. Using both field data and simulations, we consider model features such as capture efficiency, survival rates, removal of individuals due to capturing, and collection of pupae. These models permit estimation of abundance, survivorship of both marked and unmarked mosquitoes, if different, and recruitment rate. We analyze the accuracy of estimates by varying the number of released individuals, abundance, survivorship, and capture efficiency in multiple simulations. These methods can stand capture efficiencies as low as usually reported but their accuracy depends on the number of released mosquitoes, abundance and survivorship. We also show that gathering pupal counts allows estimating differences in survivorship between released mosquitoes and the unmarked population.

These models are important both to reduce uncertainty in evaluating MMR experiments and also to help planning future MRR studies.

Mosquito-borne diseases such as dengue and malaria impose a global burden with recurrent outbreaks. Recently, emergence of arboviral diseases caused by Zika and chikungunya viruses has also become a global concern. Knowledge about the ecology of mosquito populations under natural conditions may provide significant aid to help designing more effective vector control strategies. Quantitative metrics such as the abundance of mosquito populations are difficult to be measured in the field without resorting to experiments with markers. There are, however, limitations to these kinds of experiments such as short mosquito lifespan, marking limitations due to small body size, low efficiency in capturing devices such as traps, and once-only individual capture. Due to these limitations most methods estimate either only abundance or a combination of abundance and survivorship. In this work, we present statistical methods designed to estimate abundance, survivorship and recruitment using inference models and information such as counts of pupae. Results indicate that having low capture efficiencies as often observed in field assays still permits good estimation. Also, low number of released mosquitoes compromise density and survival estimations. We expect these methods to be helpful to people collecting mosquito field data and for health analysts to evaluate possible outcomes of control interventions.

Mark-release-recapture (MRR) methods applied to study mosquito populations permit analysis of vector survival, dispersal, and abundance in natural environment. Various mosquito species, in particular of the

By their nature, mosquito MRR experiments have important design restrictions that hinder the application of more sophisticated capture-recapture models such as the commonly known Jolly-Seber method [

Here we build Bayesian models that leverage the concepts behind the Fisher-Ford model [

We used the capture counts of adult females obtained from trap collections in the Z-10 neighborhood located at the city of Rio de Janeiro, Brazil, during an MRR experiment described by Villela

Traps used in MRR experiments capture mosquitoes possibly using substances to attract them. Capturing, however, is not perfect as only a subset of all released mosquitoes are collected at traps due to not fully covering the experiment area. Each single trap covers a limited area over which a mosquito can be attracted to it and trapped. Both this capability of being attracted and the probability of being captured, once attracted, are together described quantitatively here as trap capture efficiency _{0}.

We consider survivorship only during the adult stage. There is evidence of senescence in _{u} for unmarked mosquitoes).

Recruitment rate

Here abundance

Collection of immature individuals may happen before MRR experiment. Since searches are typically imperfect, we describe the efficiency of pupal search by parameter

Several designs are used for mosquito MRR trials. Guerra

We simulated multiple scenarios numerically (for instance, varying number of releases, capture probabilities and survival probabilities). Each simulation requires initial conditions and parameters such as abundance _{u} of survival for both marked and unmarked individuals, the daily recruitment rate _{0}, the number of released mosquitoes _{pupae}. If the pupal search is imperfect, the number of pupae collected is given by _{pupae}, where _{i} and _{i} of individuals captured at traps, both marked and unmarked ones, respectively.

List of variables used in the models and their respective descriptions.

Variables | Description |
---|---|

_{i} |
Number of unmarked individuals at time |

Number of marked individuals released in the field. | |

Probability of daily survival for marked individuals. | |

_{u} |
Probability of daily survival for unmarked individuals. |

Daily recruitment rate. | |

Efficiency of pupal search. | |

Probability of capture at traps at a day period. | |

_{pupae} |
Number of pupae collected before the experiment. |

_{i}, |
Number of marked individuals captured at day _{i} |

_{i}, |
Number of unmarked individuals captured at day _{i}. |

_{0} |
Daily capture efficiency at mosquito traps. |

Pupal maturation time | |

D | Number of collection days in the MRR experiment. |

Total number of marked individuals recaptured in the experiment. | |

Total number of unmarked individuals captured in the experiment. |

The inference models describe relationships between the known values, such as number _{i} and _{i} of marked and unmarked mosquitos collected at traps at day

Let _{i} captured at each day _{c}).

For a first naïve model M_{0}, we consider the probability _{i} of capture to be only dependent on the trap capture efficiency _{0}. For a second model M_{S}, we describe the capturing probability by a product of capture efficiency _{0} and time effects, to be estimated. Therefore, log (_{i}) = _{0} + _{1}, where the estimated capture efficiency _{0} = exp(_{0}) and estimated survival probability _{1}). Such a model is more general than model M_{0}, since the basic assumption in model M_{0} is equivalent to assume simply that _{1} = 0, which corresponds to no mortality effects at any time _{0} and M_{S} are Bayesian counterparts to multiple values of abundance obtained by Lincoln-Petersen and Fisher-Ford estimators, respectively. For a third model M_{B}, we allow for removal of individuals given the daily captures at traps in a Bayesian counterpart to the model proposed by Buonaccorsi _{i} = _{0}(1 − _{0})^{i−1}^{i}, for marked individuals and _{i} = _{0}(1 − _{0})^{i−1}, for unmarked individuals. For unmarked individuals, this model does not permit estimation of probability of daily survival _{u}. In this case, the underlying assumption is that over a short period of time, typically few days, recruitment is equal to mortality.

The observed number _{i} of unmarked individuals collected at traps is modeled as _{i} ~ Poisson(_{i}), for models M_{0}, M_{S}, and M_{B}, where the abundance number _{0} ~

We build two other models that include a recruitment component, including one considering the number of pupae collected from experiments before releasing mosquitoes. We build these models using relationships also described for model M_{B}, _{i} captured at each day _{c}).

We define model M_{RSU} for which we assume survival of unmarked individuals equal to the one of marked individuals, i.e., essentially _{u} = _{i} of surviving individuals from start of the experiment and the total number _{i} of recruited individuals. In the model a number of mosquitoes given by a recruitment rate _{i} = b_{0}(1 − _{0})^{i−1}^{i}. We have the same vector of probability capture described for model M_{B}, _{i} = _{0}(1 − _{0})^{i−1}^{i}. In model M_{RSU}, the sum of remaining individuals is a latent variable given by _{i} ~ _{0})^{i−1}^{i}) and the number of recruited individuals is another latent variable given by

We define model M_{RP} distinguishing the probabilities of daily survival of unmarked and marked mosquitoes, in order to estimate parameter _{u}. We describe the number of immature collected before the experiment to be _{pupae} ~ _{a}(1 − _{u}),_{a} is a factor that describes how extensive is the immature search, _{a} represents an adjustment since the immature search typically covers only a fraction of the area surveyed, or alternatively a fraction of the number of premises. We have the remaining and recruited individuals assessed in the same way, but survivorship for unmarked individuals is given by _{u}: capture counts of surviving individuals

For both models M_{RSU} and M_{RP}, the observed number of individuals is given by _{i} ~ _{0}, _{i} + _{i}). We use a prior distribution for abundance _{0} ~ _{u} =

As a reference,

Models are built using observed data and different assumptions. Depending on observed data, each model permits distinct parameters to be estimated. Some of these models are closely related to other methods proposed in the literature as shown in the counterpart model column.

Bayesian models | Description | Estimation | ||||
---|---|---|---|---|---|---|

Number of recaptures | Survivorship | Removal of individuals | Number of pupae | Counterpart model | ||

M_{0} |
Yes | - | - | - | Lincoln-Petersen estimator | Abundance, |

M_{S} |
Yes | Yes | - | - | Fisher- Ford estimator [ |
Abundance, survival |

M_{B} |
Yes | Yes | Yes | - | Buonaccorsi et al. [ |
Abundance, survivorship |

M_{RSU} |
Yes | Yes | Yes | - | - | Abundance, survivorship, recruitment |

M_{RP} |
Yes | Yes | Yes | Yes | - | Abundance, survivorship (marked and unmarked), recruitment |

We implemented the simulation tool using the R platform [_{0}, M_{S}, M_{B}, M_{RSU}, M_{RP}). We analyze the simulation data via Monte-Carlo Markov chain simulations (MCMC), by running 3 separate chains, 360,000 iterations during each of the chains, with a 320,000 burn-in period. These numbers sufficed for good convergence except otherwise noted within our results. We use R to load the simulation data and streamline pre-processed data via package R2JAGS [

We estimated abundance of _{0}, M_{S}, M_{B}, M_{RSU}, and M_{RP}. Results from using model M_{0} reveal a much larger abundance (_{S}, M_{B}, M_{RSU}, M_{RP}, respectively. Probability of daily survival from posterior distributions obtained from analyses of models M_{B}, M_{RSU}, and M_{RP} were very similar (_{RP} the mean probability of daily survival was 0.77 (95% CI: 0.72–0.83). The mean recruiting rate was estimated at 530 mosquitoes per day (95% CI: 383–701) for model M_{RP}. Since the method is sensitive to the number of pupae collected in the field, we estimate abundance using model M_{RP} considering various alternative possibilities such as a twofold, half and a quarter of the collected number of pupae (

(A) Abundance of mosquitoes (number of females in the Z-10 area). (B) Probability of daily survival. (C) Recruiting rate, where _{pupae} is the number of pupae collected. Outlier values are shown by points.

_{0}, M_{S}, M_{B}, M_{RSU}, M_{RP}. Model M_{RP} included the assumed input values of abundance within credibility intervals in 16 simulation studies (indicated by the number of asterisks in the M_{RP} column). For assumed values of abundance of 8,000 mosquitoes and above, model M_{RP} underestimated the abundance. For values of probability of daily survival of unmarked individuals less than 0.8, model M_{RP} resulted in either overestimation (study 19) or underestimation (studies 23 and 25).

The simulation study number refers to the study identifier (

Simulation Study | Abundance (input value) | Abundance mean value estimates at thousands (cred. intervals) | ||||
---|---|---|---|---|---|---|

M_{RP} |
M_{RSU} |
M_{B} |
M_{S} |
M_{0} |
||

1 | 4,000 | 4.2 (3.6–5.1)* | 3.6 (3.0–4.2)* | 3.4 (2.8–4.1)* | 2.5 (2.0–2.9) | 8.4 (7.6–9.3) |

2 | 4,000 | 3.3 (2.5–4.3)* | 2.4 (1.9–3.1) | 2.7 (1.9–3.7) | 1.5 (1.1–2.0) | 5.8 (5.0–6.7) |

3 | 4,000 | 3.9 (3.2–4.9)* | 3.1 (2.6–3.7) | 3.1 (2.4–3.7) | 2.1 (1.7–2.6) | 7.0 (6.2–7.9) |

4 | 4,000 | 4.4 (3.5–5.5)* | 3.2 (2.6–3.9) | 3.9 (3.0–4.8)* | 2.6 (2.1–3.1) | 7.8 (6.9–8.8) |

5 | 4,000 | 3.8 (3.3–4.3)* | 3.5 (3.0–4.1)* | 3.0 (2.5–3.4) | 2.3 (2.0–2.7) | 8.7 (7.9–9.5) |

6 | 4,000 | 4.2 (3.6–4.8)* | 3.7 (3.2–4.3)* | 3.5 (3.0–4.0)* | 2.8 (2.4–3.2) | 9.1 (8.4–9.9) |

7 | 2,000 | 2.3 (2.0–2.7)* | 1.9 (1.6–2.3)* | 2.2 (1.9–2.7)* | 1.7 (1.4–2.1)* | 6.9 (6.2–7.7) |

8 | 8,000 | 6.5 (5.5–7.7) | 6.1 (5.3–7.0) | 4.6 (3.8–5.5) | 3.3 (2.8–3.9) | 11.7 (10.7–12.8) |

9 | 6,000 | 5.3 (4.5–6.2)* | 4.7 (4.0–5.4) | 4.1 (3.4–4.9) | 3.0 (2.5–3.5) | 10.1 (9.2–11.1) |

10 | 4,000 | 3.5 (2.9–4.1)* | 3.0 (2.5–3.6) | 2.6 (2.1–3.1) | 2.0 (1.7–2.3) | 7.1 (6.4–7.9) |

11 | 4,000 | 2.7 (1.9–4.1)* | 2.1 (1.5–2.8) | 2.2 (1.3–3.3) | 1.0 (0.7–1.5) | 4.6 (3.8–5.5)* |

12 | 4,000 | 3.9 (2.8–5.3)* | 2.4 (1.8–3.1) | 3.2 (2.2–4.8)* | 1.6 (1.2–2.2) | 4.8 (4.1–5.7) |

13 | 4,000 | 3.2 (2.3–4.4)* | 2.5 (1.9–3.2) | 2.3 (1.6–3.3) | 1.4 (1.0–1.8) | 5.7 (4.9–6.7) |

14 | 10,000 | 8.6 (7.1–10.0) | 7.4 (6.5–8.4) | 5.5 (4.6–6.6) | 4.2 (3.6–4.8) | 12.0 (11.0–13.1) |

15 | 4,000 | 4.0 (3.2–4.9)* | 3.3 (2.7–3.9) | 3.4 (2.7–4.3)* | 2.4 (2.0–2.9) | 8.1 (7.3–9.1) |

16 | 4,000 | 3.7 (3.3–4.3)* | 3.6 (3.2–4.1)* | 2.7 (2.3–3.1) | 2.1 (1.8–2.4) | 8.6 (7.9–9.4) |

17 | 4,000 | 4.6 (4.0–5.2)* | 4.4 (3.8–4.9)* | 3.4 (3.0–3.9) | 2.6 (2.3–3.0) | 10.1 (9.2–10.9) |

18 | 4,000 | 3.6 (3.1–4.3)* | 3.0 (2.5–3.5) | 2.9 (2.4–3.5) | 2.2 (1.8–2.6) | 6.6 (6.0–7.3) |

19 | 4,000 | 4.8 (4.1–5.5) | 3.4 (2.9–4.0) | 3.1 (2.6–3.6) | 2.4 (2.0–2.8) | 5.5 (5.0–6.1) |

20 | 12,000 | 9.4 (8.0–10.7) | 9.0 (8.0–10.1) | 6.2 (5.1–7.3) | 4.6 (4.0–5.4) | 14.6 (13.4–16.0) |

21 | 14,000 | 10.8 (9.5–12.2) | 9.9 (8.8–11.2) | 7.4 (6.1–9.0) | 5.3 (4.6–6.2) | 16.1 (14.8–17.5) |

22 | 4,000 | 3.8 (3.3–4.4)* | 3.3 (2.8–3.8) | 2.8 (2.4–3.3) | 2.3 (2.0–2.6) | 6.9 (6.3–7.5) |

23 | 4,000 | 3.5 (3.0–3.9) | 3.2 (2.8–3.6) | 2.6 (2.3–3.0) | 2.2 (1.9–2.5) | 7.1 (6.6–7.8) |

24 | 4,000 | 4.2 (3.6–4.8)* | 3.5 (3.0–4.0) | 2.9 (2.5–3.4) | 2.4 (2.0–2.7) | 5.8 (5.3–6.4) |

25 | 4,000 | 3.5 (3.0–3.9) | 3.0 (2.6–3.4) | 2.4 (2.1–2.7) | 2.0 (1.8–2.3) | 5.8 (5.3–6.3) |

Comparing assumed input values for study 1 and its estimations in _{RP} results in abundance of 4,220 mosquitoes (95% CI: 3,572–5,067) for an assumed abundance value of 4,000 mosquitoes. We also estimated probability of daily survival (PDS) for unmarked at 0.86 and marked individuals at 0.77 and recruitment rate 624 individuals/day. Analysis from model M_{RSU} reveals an estimation of a 95% credibility interval also containing the abundance value for simulation study 1. The probability of daily survival, however, is wrongly estimated due to the assumption of equal survival rates for all individuals whether marked or not. Model M_{B} permits estimation of abundance, probability of daily survival (marked individuals) and trap capture efficiency. Estimates given by model M_{B} are also close to the assumed values, which are well within the 95% credibility intervals. Model M_{S} does not consider removal of individuals, an assumption that proves costly since it underestimated both the abundance and probability of daily survival. Model M_{0} results greatly overestimate abundance due to not considering the daily survival.

Simulated data were obtained using parameter values in the first line (Input value). Results from analysis running MCMC simulations (3 chains, 360,000 iterations, 320,000 burn-in period) are shown in the subsequent lines (Estimation). Mean values and credibility intervals (95%) are obtained from posterior output samples. An asterisk (*) indicates whether the credibility interval contains the assumed input value in the simulation. Parameters not estimated due to the model limitations are signaled by a single dash (-).

Input values | Abundance | Trap capture efficiency | PDS (marked) | PDS (unmarked) | Recruitment (per day) |
---|---|---|---|---|---|

_{0} = 0.05 |
_{u} = 0.85 |
||||

Estimation | |||||

M_{RP} |
4,220 (3,572–5,067)* | 0.049 (0.041–0.59)* | 0.77 (.75-.81)* | 0.86 (0.83–0.88)* | 624 (503–760)* |

M_{RSU} |
3,573 (3,013–4,218)* | 0.06 (0.05–0.07)* | 0.75 (0.72–0.78)* | 0.75 (0.72–0.78) | 993 (865–1,140) |

M_{B} |
3,394 (2,805–4,086)* | 0.05 (0.04–0.06)* | 0.78 (0.75–0.81)* | - | - |

M_{S} |
2,458 (2,048–2,926) | 0.06 (0.05–0.08)* | 0.72 (.69–0.75) | - | - |

M_{0} |
8,402 (7,599–9,305) | 0.018 (0.016–0.020) | - | - | - |

For simulations with at least 1000 marked mosquitoes, mean estimated abundance values are close to the assumed values, which are within the 95% credibility interval. Values below 1,000 marked mosquitoes were not quite as close to the estimation value. Also, the 95% credibility interval in these cases gets much larger as size of the released cohort decreases. Inspection of results from very low values indicates high uncertainty, as expected (

Results are shown for the posterior distributions (mean and 95% credibility intervals) of abundance (A) and capture effciency (B). Horizontal lines indicate the assumed input value for simulation. Points indicate outliers. Released numbers less than 1,000 reveal either mean not close to the assumed value or large 95% credibility interval/poor convergence.

_{RP} model considering only simulation experiments with same abundance values and release numbers, but varying probability of daily survival of marked individuals and unmarked population. Estimation of PDS for both marked cohort and unmarked cohorts are close to the input values assumed in the simulations, although in some cases for marked population the assumed values are closer to the extremes of the 95% credibility intervals.

All simulation parameters are equal at all experiments, except for varying probability of daily survival. Abundance is 4,000 mosquitoes and 2,000 marked mosquitoes are released. Trap capture efficiency is fixed at 0.05. Results are shown for the posterior distributions (mean and 95% credibility intervals). SurvivalU indicates assumed values used for unmarked PDS, whereas SurvivalM indicates assumed values for marked PDS. Red and blue boxplots represent results for marked and unmarked cohorts, respectively.

If efficiency at collecting pupae is low, results from using model M_{RP} indicate estimations deviating from the assumed input values for abundance, recruitment and probability of daily survival.

Here, efficiency describes how good from 0 to 1 counting the immature individuals (pupae) in the pre-MRR phase. Results are shown for the posterior distributions (mean and 95% credibility intervals). Chart A shows abundance results. Chart B indicates recruitment estimates. Chart C indicates survivorship of unmarked individuals. All estimations are intertwined and lowering efficiency causes all of them to deviate from assumed values (black bars). Colors represent the number of released individuals as shown in legend.

_{RSU} are not as close to the expected values. Since estimation of all parameters is intertwined, abundance estimates (Chart A) also get worse.

Results are shown for the posterior distributions (mean and 95% credibility intervals). Colors represent the number of released individuals. When the absolute value of the difference between survival of unmarked and marked individuals is 0.07, estimates of either abundance (A) or recruitment and (B) are not close to the assumed value.

In

Capture efficiency varies from values 0.03 to 0.1. Results are shown for the posterior distributions (mean and 95% credibility intervals). All other parameters (abundance, released numbers, recrutiment, survival) were equal across simulation experiments. Abundance is 4,000 mosquitoes and 2,000 mosquitoes are released. Recruitment rate is at 600 mosquitoes/day. Probabilities of daily survival is 0.85 for unmarked cohort and 0.78 for marked cohort. Surprisingly, high capture efficiency does not decrease the 95% credibility intervals significantly.

We defined Bayesian models to estimate abundance, recruitment and probability of daily survival of mosquito populations in the field from MRR experiment data. Analyses using these models result in posterior distributions for these parameters, hence mean and 95% credibility intervals can be obtained. Moreover, counts from pupal surveys were instrumental to obtain estimated recruitment rates of the wild population. These estimates are particularly interesting since immature counting in breeding sites is one of the most common vector control approaches in countries endemic for arboviruses infections.

Our first set of simple Bayesian inference models is based on estimating the capture efficiency and the probability of daily survival with close relationship to existing methods used for MRR analysis. Since mosquitoes are not often individually captured multiple times (once captured they are effectively removed from the study), a Bayesian model should better describe removal of individuals not only due to mortality but also from the capture process itself. This model has close association to the method proposed by Buonaccorsi _{B} permits inference about the capture efficiency and the probability of daily survival. However, it may not achieve accurate estimations, depending on conditions of large difference between probability survival of marked and unmarked mosquitoes, large abundance or low capture efficiency.

Estimation of recruitment becomes challenging due to the usual mosquito MRR limitations. The concept of using pupal counts for assessment of abundance has been proposed by Focks _{RP} and M_{RSU} also permit to estimate recruitment, either assuming collection of pupae or not, respectively. Depending on this information, we can evaluate any potential difference between daily survival of marked and unmarked mosquitoes.

We estimated abundance, survivorship and recruitment rate of an _{RSU} model) to 4.2 mosquitoes per premise (_{S}). Such twofold increase shows the importance of choosing the appropriate model to describe parameters of _{RP} achieves intervals that include the simulation input value in most of the studied scenarios. Daily recruitment rate in the field was about 0.67 mosquitoes per premise in the analysis from model M_{RP}. In this case, the recruited number would be about a quarter of the total abundance. The effectiveness of vector control approaches such as targeting the most productive container or using chemical compounds (insecticides) might be evaluated based on potential changes on mosquito recruitment rates. For more effective the vector control intervention, greater decrease in recruitment rate would be expected.

Our results from analyses of simulated datasets show that these models can tolerate capture efficiencies as low as the ones observed for mosquito MRR. We also varied the abundance levels, as opposed to the released numbers, and differences in the survivorship between marked, released mosquitoes and the unmarked population. In the case of immature counts (pupae), recruitment rate can also be estimated, but we find it to be highly dependent on extensive pupal collection, which can require extensive resources in the field.

Limitations in the design of mosquito MRR studies expectedly impact estimation of abundance, survivorship and recruitment rates. First, when abundance is large, the number of released mosquitoes is critical, regardless of the method used. Also capture efficiency in regular MRR experiments is usually small, varying in the range of 5–10% [

Collecting pupae in the field can be difficult due to limited accessibility to breeding sites, but we think that results from model M_{RP} should motivate getting such samples to have better estimates. Our results indicate sensitivity of recruiting rate estimates when assuming different number of pupae. Because pupal surveys may have difficult feasibility to be conducted on the routine of vector control programs, our results demonstrate that surveys with varying degrees of imperfection lead to biased estimations of abundance, recruitment and survival rates. Conversely, public health decision makers might adopt models such as M_{RP} and M_{RSU} with attention to these issues. For example, the Brazilian dengue national control program recommends a survey 4–6 times yearly in around 10% of cities of each district of important cities to determine infestation and Breteau Indexes, plus the most productive container type across the country [

Our models estimating recruitment assume that the population stays constant during the short period of experiment time. If such assumption does not hold due to abundance fluctuations occurring as a result of changing environmental conditions, use of insecticides, or any other, we expect difficulties to get accurate estimations applying this modeling, unless the exogenous conditions can be modeled.

Simulated datasets and analyses consider typical designs used for MRR experiments involving mosquito populations of

There is vast literature on MRR experiments to study ecology of wild animal species [

Bayesian models permit us to include all parameters instead of serial parameter estimation and to use prior beliefs, if any, or vague priors in order to obtain not only mean estimations but also credibility intervals. Traditional methods require a sequence of estimations for survival and abundance, and if possible recruitment, from observed field data. Smith and McKenzie [

All experiments used a recruitment rate

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