The use of matrix population models (MPMs) to portray the life cycles of organisms is a powerful tool for ecological and evolutionary approaches. However, MPMs have been underutilized, despite their importance in demographic studies. One explanation is that students and young researchers are not encouraged to explore the MPMs’ potential. MPMs are often absent in elementary textbooks, and specialized literature can be mathematically challenging, especially if it lacks basic concepts of matrix algebra. Consequently, MPMs’ bibliography is limited to a few books, their application is concentrated in the United States and Europe, and errors are not rare in the literature. Here, we advocate broadening the use of structured populations, where individuals contribute to the population growth rate according to their age, sex, or other variables. We present the theoretical development of the MPMs, relevant examples of ecological studies, and their mathematical proprieties. Finally, we offer mathematical formulation and practical exercises in R in the appendices. We conceived this manuscript to be the first stop for those students and researchers interested in entering the literature about MPMs.

Representar matematicamente os ciclos de vidas dos organismos a partir de modelos matriciais (MPMs) tem sido uma ferramenta poderosa para as mais diversas questões ecológicas e evolutivas. Apesar de ser uma ferramenta essencial nos estudos demográficos, os MPMs continuam sendo subutilizados. Uma explicação potencial para a subutilização dos modelos matriciais pode ser o pouco encorajamento de estudantes ou jovens pesquisadores para explorar suas potencialidades. MPMs geralmente não estão presentes nos livros textos básicos em ecologia e a literatura especializada pode ser bastante desafiadora, principalmente quando conceitos básicos de álgebra de matrizes não foram previamente ensinados durante a formação desses pesquisadores. O resultado disso é uma bibliografia restrita, aplicações dos MPMs concentradas nos Estados Unidos e Europa, além de uma literatura com muitos erros conceituais. Apresentar uma introdução gentil aos MPMs parte de uma tentativa de popularizar suas aplicações desde os anos iniciais da formação do pesquisador não familiarizado, seja com a língua inglesa ou com os conceitos matemáticos relativos à essa literatura. Dessa forma, o objetivo aqui é apresentar o ganho em considerar uma população estruturada (aquela em que cada indivíduo contribui para a população de acordo com seu peso, idade, sexo ou outra variável), o desenvolvimento teórico dos MPMs, algumas aplicações relevantes para estudos ecológicos, além das propriedades matemáticas essenciais dos MPMs – muitas das vezes difíceis de serem entendidas. Por fim, oferecemos nos apêndices formulações matemáticas e exercícios práticos no software R. Esperamos que esse manuscrito seja a primeira parada daqueles estudantes e pesquisadores interessados a adentrar na literatura dos modelos populacionais matriciais.

ABSTRACT Estimates of local population abundances, which require carefully designed sampling procedures, can provide valuable information on population size and density. Even though small mammals are one of the most widely studied vertebrate groups, many surveys have not recorded basic information to estimate local abundances, for instance catching effort. Here we suggest a simple comparative trapping frequency index that can be used as an alternative to the relative abundance index in data sets that only contain the number of species and individuals collected, thus lacking information on sampling effort. To compare trapping frequency and relative abundances we used capture records from more than four years, from seven species of rodents and two marsupial species collected by the Brazilian Plague Service. We calculated the trapping frequency index of each species as the proportion of trapped individuals per total of all individuals caught. We found that this trapping index was significantly correlated with a relative abundance index (number of captured individuals divided by number of trap nights). Our findings suggest that the proposed index may be useful for comparisons in situations where data on catching effort is lacking. The index may also provide a simple, though approximate quantification of relative local abundances, with possible applications in comparative studies (e.g. meta-analysis). We suggest that this index is used in studies that do not focus on obtaining accurate population parameter estimates, but which nonetheless contain data that can still offer a representative measure to compare local population abundances.

A major difficulty in the application of probabilistic models to estimations of mammal abundance is obtaining a data set that meets all of the assumptions of the model. In this paper, we evaluated the concordance correlation among three population size estimators, the minimum number alive (MNA), jackknife and the model suggested by the selection algorithm in CAPTURE (the best-fit model), using long-term data on three Brazilian small mammal species obtained from three different studies. The concordance correlation coefficients between the abundance estimates indicated that the probabilistic and enumeration estimators were highly correlated, giving concordant population estimates, except for one species in one of the studies. The results indicate the adequacy of using enumeration estimates as indexes for population size when scarce data do not allow for the use of probabilistic methods. Differences observed in the behavior of enumeration and probabilistic methods among species and studies can be related to the exclusive sampling design of each area, species-specific movement characteristics and whether a significant portion of the population could be sampled.

A fundamental step in the emerging Movement Theory is the description of movement paths, and the identification of its proximate and ultimate drivers. The most common characteristic used to describe and analyze movement paths is its tortuosity, and a variety of tortuosity indices have been proposed in different theoretical or empirical contexts. Here we review conceptual differences between five movement indices and their bias due to locations errors, sample sizes and scale-dependency: Intensity of Habitat use (IU), Fractal D, MSD (Mean Squared Distance), Straightness (ST), and Sinuosity (SI). Intensity of Habitat use and ST are straightforward to compute, but ST is actually an unbiased estimator of oriented search and ballistic movements. Fractal D is less straightforward to compute and represents an index of propensity to cover the plane, whereas IU is the only completely empirical of the three. These three indices could be used to identify different phases of path, and their path tortuosity is a dimensionless feature of the path, depending mostly on path shape, not on the unit of measurement. This concept of tortuosity differs from a concept implied in the sinuosity of BENHAMOU (2004), where a specific random walk movement model is assumed, and diffusion distance is a function of path length and turning angles, requiring their inclusion in a measure of sinuosity. MSD should be used as a diagnostic tool of random walk paths rather than an index of tortuosity. Bias due to location errors, sample size and scale, differs between the indices, as well as the concept of tortuosity implied. These differences must be considered when choosing the most appropriate index.

Uma população de gambás, Didelphis aurita Wied-Neuwied 1826, foi estudada de 1997 a 2006 por captura, marcação e recaptura numa área de Mata Atlântica no Estado do Rio de Janeiro. Foram estabelecidas coortes, incluindo apenas fêmeas marcadas ainda no marsúpio, classificando-as em cinco classes etárias. Os parâmetros populacionais foram calculados através de tábuas de vida horizontais. A maior mortalidade ocorreu entre as classes etárias I e II, logo após o desmame, sendo que a sobrevivência apresentou uma curva do tipo III. A taxa de fecundidade foi inversamente correlacionada com a sobrevida, o valor reprodutivo inversamente correlacionado com a mortalidade e o valor reprodutivo residual inversamente correlacionado com a taxa de fecundidade. A taxa intrínseca de crescimento sempre esteve próxima de zero, porem ligeiramente negativa. O gambá mostrou-se um bom modelo de mamífero para este tipo de estudo.

A population of the black-eared opossum, Didelphis aurita Wied-Neuwied 1826, was studied during 1997-2006 by capture-mark-recapture in the Atlantic Forest in the state of Rio de Janeiro. Cohorts were established with females marked still in the pouch, classified into five age classes. Population parameters were estimated using life tables. Survival was type III and mortality was greatest in age classes I and II, just after weaning. Fecundity was inversely correlated with survival, reproductive value was inversely correlated with mortality and residual reproductive value was inversely correlated with fecundity. The intrinsic rate of increase was always close to zero and tended to be slightly negative. The black-eared opossum can be considered a good model species for this type of study.