Models for Jointly Estimating Abundances of Two Unmarked Site-Associated Species Subject to Imperfect Detection

  • Rafael A. Moral
  • John Hinde
  • Clarice G. B. Demétrio
  • Carolina Reigada
  • Wesley A. C. Godoy


In ecological field surveys, it is often of interest to estimate the abundance of species. It is frequently the case that unmarked animals are counted on different sites over several time occasions. A natural starting point to model these data, while accounting for imperfect detection, is by using Royle’s N-mixture model (Biometrics 60:108–115, 2004). Subsequently, many multivariate extensions have been proposed to model communities as a whole. However, these approaches are used to study species richness and other community-level variables and do not focus on the relationship between two site-associated species. Here, we extend the N-mixture modelling framework to model two site-associated species abundances jointly and propose to measure the influence of one species’ abundance on the populations of the other and study how this changes over time and space. By including a new parameter in the abundance distribution of one of the species, linking it to abundance of the other, our proposed model treats extra variability as an effect induced by an associated species’ abundance and allows one to study how environmental covariates may affect this. Using results from simulation studies, we show that the model is able to recover true parameter estimates. We illustrate our approach using data from bald eagles and mallards obtained in the 2015 survey of the North American Breeding Bird Survey. By using the joint model, we were able to separate overdispersion from mallard-induced variability and hence what would be accounted for with a dispersion parameter in the univariate framework for the eagles was explained by covariates related to mallard abundance in the joint model. Our approach represents an attractive, yet simple, way of modelling site-associated species populations jointly. Conservation ecologists can use the approach to devise management strategies based on the strength of association between species, which may be due to direct interactions and/or environmental effects affecting both species’ populations. Also, mathematical ecologists can use this framework to develop tools for studying population dynamics under different scenarios. Supplementary materials accompanying this paper appear on-line.


Bald eagle Binomial–Poisson mixture Bivariate models Mallard N-mixture models Replicated counts Site association 



We thank the Associate Editor and two anonymous referees for their constructive comments which helped improve the manuscript.


RAM was funded by FAPESP (Grant No. 2014/12903-8). Exchange visits between CGBD and JH were partially supported by CNPq, FAPESP and SFI. CR and WACG were funded by CNPq.

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Copyright information

© International Biometric Society 2017

Authors and Affiliations

  1. 1.Departamento de Ciências Exatas, Escola Superior de Agricultura Luiz de QueirozUniversidade de São PauloPiracicabaBrazil
  2. 2.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland
  3. 3.Departamento de Ecologia e Biologica EvolutivaUniversidade Federal de São CarlosSão CarlosBrazil
  4. 4.Departamento Entomologia e Acarologia, Escola Superior de Agricultura Luiz de QueirozUniversidade de São PauloPiracicabaBrazil

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