From NOREMARK to MARK: software for estimating demographic parameters using mark–resight methodology

Abstract

Mark–resight methods can be a less expensive and less invasive alternative to traditional mark–recapture for estimating abundance and related demographic parameters. This is because only a single marking event is required, and subsequent sighting data from both marked and unmarked individuals are used for estimation. Mark–resight is therefore particularly appealing when working with limited budgets or sensitive species. These methods have been applied to many taxa, including avians such as Bald Eagles (Haliaeetus leucocephalu), Sage Grouse (Centrocercus urophasianus), and New Zealand Robins (Petroica australis). However, previous model and software development overwhelmingly focused on abundance estimation and did not extend much beyond classic Lincoln–Petersen methods. For this purpose, Program NOREMARK provided a convenient, but limited, means for analysis. To address these limitations, a new suite of likelihood-based mark–resight estimators has been developed and implemented within Program MARK. Unlike the estimators in NOREMARK, these new models provide a theoretical basis for model selection and multimodel inference, the incorporation of spatio-temporal and individual covariate information, full utilization of the robust design (including the estimation of survival and state transition probabilities), and all the other analysis tools that MARK provides. In addition, MARK no longer requires that the exact number of marked individuals available for sighting be known, thereby greatly extending the potential use of mark–resight methodology for population monitoring. Here, we provide a review of mark–resight methodology and present the various mark–resight estimators that are implemented in Program MARK.

Appendix

For LNE, the expectation and variance formulae for $T_{u_j}$ are:

$$ E(T_{u_j})=(N_j-n_j)\sum_{i=1}^{k_j}\mu_{ij} $$

and

$$ \hbox{var}(T_{u_j})\quad=(N_j-n_j)\left[\sum_{i=1}^{k_j}\mu_{ij}(1-\mu_{ij})+\sum\sum_{l \ne i} (\gamma_{lij}-\mu_{lj}\mu_{ij})\right], $$

where

$$ \gamma_{lij} = \int \left[ {\frac{\exp(\sigma_jz_j+\beta_{lj})} {1+\exp(\sigma_jz_j+\beta_{lj})}}+{\frac{\epsilon_{lj}} {n_j}}\right] \left[{\frac{\exp(\sigma_jz_j+\beta_{ij})} {1+\exp(\sigma_jz_j+\beta_{ij})}}+{\frac{\epsilon_{ij}} {n_j}}\right] \phi(z_j) dz_j. $$

For IELNE, the expectation and variance formulae for $T_{u_{ij}}$ are:

$$ {E}(T_{u_{ij}})=\left(\bar{N}_j+\alpha_{ij}-M_{ij}\right) \mu_{ij} $$

and

$$ \hbox{var}(T_{u_{ij}})=(\bar{N}_j+\alpha_{ij}-M_{ij}) \mu_{ij}\left(1-\mu_{ij}\right). $$

For (Z)PNE, the expectation and variance formulae for $T_{u_j}$ are:

$$ \hbox{E}(T_j)= U_j \left[\exp\left({\frac{\sigma_j^{2}} {2}}+\alpha_j\right)+{\frac{\epsilon_j}{n_jI_j+{\frac{n_j^{*}(1-I_j)} {1-\exp(-\lambda_{j|s})}}}}\right] $$

and

$$ \hbox{var}(T_j)= U_j \left\{ \exp\left({\frac{\sigma_j^{2}} {2}}+\alpha_j\right) +\exp(2\alpha_j)\left[\exp(2\sigma_j^{2})-\exp(\sigma_j^{2})\right]+{\frac{\epsilon_j} {n_jI_j+{\frac{n_j^{*}(1-I_j)}{1-\exp(-\lambda_{j|s})}}}} \right\}, $$

where λ_j|s = exp(σ ²_j /2 + α_j).