Using encounter data to improve capture-recapture abundance estimates

Capture-recapture studies are widely used in ecology to estimate population sizes and demographic rates. However, when individuals are hard to capture, encounters may result in failed captures where the identity of an individual is unknown. Such encounters are ignored in traditional capture-recapture analyses. Here we show that failed captures provide lower bounds on population size that can increase the precision of abundance estimates. Analytical results and simulations indicate that encounter data improve abundance estimates when capture probabilities are low, and when there are few repeat surveys. We present a hierarchical Bayesian approach for integrating failed captures and auxiliary encounter data in statistical capture-recapture models. This approach can be integrated with existing capture-recapture models, and may prove particularly useful for hard to capture species in data-limited settings.


Introduction
whether individual i was captured on survey k: y i,k , so that y i,k = I(y * i,k = 3), where I is 48 an indicator function that is equal to one if the condition inside the parentheses is satisfied, 49 otherwise it equals zero. In other words, we only observe y * i,k when y * i,k = 3. Additionally, with σ set to some small fixed value.

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The parameters consist of states z 1 , ..., z M , outcomes y * 1,1 , ..., y * M,K , and the probabilities of 57 inclusion (ω), encounter (η), and capture (κ). The data consist of the capture histories . CC-BY 4.0 International license It is made available under a perpetuity. preprint (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this . http://dx.doi.org/10.1101/2020.01.22.915314 doi: bioRxiv preprint first posted online Jan. 23, 2020; Abundance lower bounds from encounter and capture data 78 The total population size is bounded from below by the number animals encountered on 79 any one survey, assuming each animal can be encountered once at most (i.e., individuals 80 are not double-counted). If n k is the number of unique animals encountered on survey k, 81 the lower bound on abundance from encounter data n min = max(n 1 , ..., n K ) is the maximum 82 of K independent binomial random variables with sample size N and probability η. The 83 probability mass function of this lower bound is thus given by: where F (n) is the cumulative distribution function of a binomial random variable.

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Population size must also be greater than or equal to the number of unique captured 86 individuals. A probability mass function for the lower bound on abundance from capture data 87 (c min : the number of unique captured individuals) can be derived with a binomial distribution.

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The binomial sample size is the true population size (N ), and the probability of success is 89 the probability of being captured one or more times: 1 − (1 − ηκ) K . The probability mass 90 function for c min , the abundance lower bound derived from the capture data, is: When the expected lower bound from encounter data exceeds the expected lower bound from 92 6 . CC-BY 4.0 International license It is made available under a perpetuity. preprint (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in simulated capture-recapture data were in agreement with the theoretical expectations derived 126 from the probability mass functions of n min and c min ( Figure 2).

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When the lower bound on abundance is greater for encounter data than capture data, the joint CC-BY 4.0 International license It is made available under a perpetuity. preprint (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this . http://dx.doi.org/10.1101/2020.01.22.915314 doi: bioRxiv preprint first posted online Jan. 23, 2020; encounter data increase the precision of population abundance estimates by increasing the 140 lower bound on abundance. Such data are essentially "free" in encounter-capture-recapture 141 study designs, and can be included by modifying the likelihood function (and not the 142 underlying state model) of capture-recapture models.

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In addition to increasing the precision of abundance estimates, encounter data can increase here, we assume that whether an animal is marked or not is unknown until it is captured (as 156 would be the case for subdermal PIT tags in an amphibian). CC-BY 4.0 International license It is made available under a perpetuity. preprint (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this . http://dx.doi.org/10.1101/2020.01.22.915314 doi: bioRxiv preprint first posted online Jan. 23, 2020; CC-BY 4.0 International license It is made available under a perpetuity. preprint (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this . http://dx.doi.org/10.1101/2020.01.22.915314 doi: bioRxiv preprint first posted online Jan. 23, 2020;

Figure legends
JAGS code for this model with uniform priors over ω and p is: 17 . CC-BY 4.0 International license It is made available under a perpetuity. preprint (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this . http://dx.doi.org/10.1101/2020.01.22.915314 doi: bioRxiv preprint first posted online Jan. 23, 2020; } } } Baseline model with failed captures 256 We add failed captures to the baseline model by introducing a categorical parameter y * i,k , 257 which represents "not encountered" (y * i,k = 1), "failed capture" (y * i,k = 2), or "capture" 258 (y * i,k = 3). If η is the encounter probability, κ is the probability of capture conditional on 259 encounter, and f k is the number of failed captures, then as described in the main text the 260 posterior is: JAGS code for this model with uniform priors over ω, η, and κ is: 18 . CC-BY 4.0 International license It is made available under a perpetuity. preprint (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in  . CC-BY 4.0 International license It is made available under a perpetuity. preprint (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this . http: //dx.doi.org/10.1101/2020.01.22.915314 doi: bioRxiv preprint first posted online Jan. 23, 2020 a k ∼ Binomial(η, N ), where N = i z i . Note that there need not be an equal number (K) capture-recapture 272 surveys and auxiliary encounter surveys, but here we assume this is the case to simplify 273 notation. This model permits decomposing the parameter p, which represents the marginal 274 probability of capture, into the product of the encounter probability η and the probability of 275 capture conditional on encounter κ. 276 Then, the posterior is: JAGS code for this model with uniform priors over ω, η, and κ is: CC-BY 4.0 International license It is made available under a perpetuity. preprint (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in JAGS code for this model with uniform priors over ω, η, and κ is: . CC-BY 4.0 International license It is made available under a perpetuity. preprint (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in

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. CC-BY 4.0 International license It is made available under a perpetuity. preprint (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for this . http://dx.doi.org/10.1101/2020.01.22.915314 doi: bioRxiv preprint first posted online Jan. 23, 2020;