Environmental and Ecological Statistics

, Volume 21, Issue 1, pp 41–59

A robust design capture-recapture model with multiple age classes augmented with population assignment data

  • Zhi Wen
  • James D. Nichols
  • Kenneth H. Pollock
  • Peter M. Waser
Article

DOI: 10.1007/s10651-013-0243-6

Cite this article as:
Wen, Z., Nichols, J.D., Pollock, K.H. et al. Environ Ecol Stat (2014) 21: 41. doi:10.1007/s10651-013-0243-6
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Abstract

The relative contribution of in situ reproduction versus immigration to the recruitment process is important to ecologists. Here we consider a robust design superpopulation capture-recapture model for a population with two age classes augmented with population assignment data. We first use age information to estimate the entry probabilities of new animals originating via in situ reproduction and immigration separately for all except the first period. Then we combine age and population assignment information with the capture-recapture model, which enables us to estimate the entry probability of in situ births and the entry probability of immigrants separately for all sampling periods. Further, this augmentation of age specific capture-recapture data with population assignment data greatly improves the estimators’ precision. We apply our new model to a capture-recapture data set with genetic information for banner-tailed kangaroo rats in Southern Arizona. We find that many more individuals are born in situ than are immigrants for all time periods. Young animals have lower survival probabilities than adults born in situ. Adult animals born in situ have higher survival probabilities than adults that were immigrants.

Keywords

Capture-recapture Genetic assignment procedures  Kangaroo rat Robust design Single site Superpopulation Two age groups 

1 Introduction

Understanding the relative contributions of in situ reproduction and immigration to the recruitment process is important to ecologists. Nichols and Pollock (1990) used a two age class capture-recapture model with the robust design to estimate the recruitment from immigration versus in situ reproduction. Wen et al. (2011) proposed a totally different approach to estimate the relative contributions of in situ births and immigrants to the growth of a single site open population without age structure. Here we extend Wen et al.’s approach to a open population with two age classes. We also show that the age information itself can be used to help discriminate between immigrants and local births. All our new models are developed on the basis of the superpopulation capture-recapture model incorporating the robust design.

The robust design (Pollock 1982) combines the properties of open and closed population models. In the robust design, the sampling periods occur at two time scales, with \(\ge \)2 secondary sampling periods within each primary sampling period. The population is assumed to be closed within each primary period (Pollock 1982; Kendall and Nichols 1995; Kendall et al. 1997) and open between primary periods. That is, there can be losses (deaths, emigration) or gains (births, immigration) between primary periods. With the robust design, we can estimate some parameters (such as the capture probabilities and population sizes of the first and last period ) that can not be estimated under the general Jolly–Seber model (Pollock 1982; Pollock et al. 1990; Kendall and Pollock 1992).

The superpopulation model (Crosbie and Manly 1985; Schwarz and Arnason 1996) is a representation of the open population, Jolly–Seber model (Jolly 1965; Seber 1982). In this model, the superpopulation is composed of all the recruits from all periods that are exposed to sampling at any time during the study. The animals of the superpopulation are assumed to enter it via certain entry probabilities between sampling occasions.

Considering a single site open population with two age classes, the animals of the population can only be born on site or originate from outside the site. To analyze the components of recruitment during each period, we assume that we have a “population assignment procedure”, some source of information that allows us to assess whether each newly captured animal was born within the sampled population or immigrated from outside. Correspondingly, we split the entry probability into a new, on site recruitment entry probability and a new, immigrant entry probability. After including age information in the superpopulation capture-recapture model, we prove that it is possible to estimate these two entry probabilities separately for all except the first period. We then combine age information and population assignment information with the standard superpopulation capture-recapture model in order to estimate the entry probability and survival probability for in situ births and immigrants for all sampling periods.

A newly-captured animal may carry both phenotypic and genotypic clues to its population of origin, and both types of information have been used for this purpose (e.g., feather isotope analyses, Hobson 1999; genetic parentage analysis, Waser and Hadfield 2010). A common approach to population assignment is based on the relative probabilities of an individual’s genotype in its source population versus other, nearby populations (Piry et al. 2004). Individuals are genotyped at multiple neutral loci, and the likelihood of their genotypes in potential source populations is estimated assuming that those populations are in Hardy–Weinberg equilibrium (Waser and Strobeck 1998). Paetkau et al. (1995) and Favre et al. (1997) first suggested that individuals with unusual genotypes might come from elsewhere, and also that some individuals whose genotypes were rarer where they were captured than in a neighboring population were probably immigrants from that neighboring population. Since then, statistical methodologies that assign a population of origin to an individual based on its genotype have multiplied (Guinand et al. 2002; Manel et al. 2005). Here we assume there is no uncertainty associated with the laboratory analyses on which the population assignment procedure is based. For example, for genetic assignment procedures we assume there are no errors in the molecular genotyping of the animals.

A “perfect” assignment procedure means that each captured animal can be assigned without error either as an animal born in the population or as an immigrant from outside the population. That is, a “birth state” (immigrant or in situ birth) is assigned to any captured animal where this state is otherwise unknown. An imperfect assignment procedure is more likely in practice. In an imperfect population assignment procedure, the animal can only be assigned to one origin with high probability instead of with probability 1.

We first develop a likelihood framework for the ad hoc approach of Nichols and Pollock (1990), exploiting the fact that young animals are known to be the result of in situ recruitment. We then consider additional population assignment data with the assumption that the population assignment procedure is perfect (i.e., we can accurately assign the animal to be a local birth or an immigrant) and later generalize to a model for imperfect population assignment procedures. The fusion of population assignment data and capture-recapture data will allow us to determine the separate contributions of immigration and in situ reproduction to the growth of the population.

2 Model structure

We consider a standard superpopulation capture-recapture model for a single site population with two age classes, young (0–1 year) and adult (\(\ge \)1 year). Young animals naturally become adults one year after birth. We assume that the time interval between successive primary periods is one year, the same time required to make a transition from young to adult. There are \(K\) primary periods and each primary period has \(l_i \ (l_i \ge 2)\) secondary periods. A new recruit can only come from being born on site or as an immigrant from outside the site.

2.1 Notation

For all the following notation, the first superscript \(r\) of parameters represents the age of the animal with \(r=Y\) (Young) or \(O\) (adult). The second superscript \(h\) of parameters represents the population origin of the animal. If \(h=*\), the population origin of the animal is not considered. If \(h=B\) or \(I\), the animal is either a local birth or an immigrant, respectively.
\(N^{rh}(i)\)

The sampled population, corresponding to all members of the superpopulation that have entered the studied or sampled population at age \(r\) prior to primary period \(i\) and have not yet died or emigrated given that the animals are of population origin \(h, h=*, B, I\). We have \(N^{rh}(i)=N^{Yh}(i)+N^{Oh}(i)\).

\(B_i^{rh}\)

The new recruits (in situ births or immigrants) of age \(r\) to the sampled population between primary period \(i\) and primary period \(i+1\), and still alive and in the population at period \(i+1\) given that the animals are of population origination \(h\). \(B_0^{rh}=N^{rh}(1)\) is the number of animals in the sampled population of age \(r\) at the initial sampling period given that the animal is of population origin \(h\).

\(u_i^{rh}\)

The number of unmarked animals of age \(r\) caught on at least one secondary occasion within primary period \(i\) given that the animals are of population origin \(h\). The newly captured unmarked young animals \(u_i^{Yh}\) can only be the animals entering the superpopulation at primary period \(i\) at age \(Y\). The newly captured unmarked adults \(u_i^{Oh}\) can be

  • Animals that entered the sampled population before period \(i-1\) as either young or adult, survived to period \(i\), and were never captured before period \(i\).

  • Animals that entered the sampled population after period \(i-1\) and before or during period \(i\) at adult age.

\(R_i^{rh}\)

The number of marked animals of age \(r\) that are released back into the population following primary period \(i\) given that the animals are of population origin \(h\).

\(m_{ij}^{rh}\)

The number of animals caught in primary period \(j\) that were last caught in primary period \(i\) at age \(r\) given that the animals are of population origin \(h (1\le i<j \le K)\).

\(X_{i}^{ rh \omega }\)

The number of animals of age \(r\) having capture history \(\omega \) over the \(l_i\) secondary periods of primary period \(i\) given that the animals are part of \(R_i^{rh}\) and are of population origin \(h\).

\(\beta _i^{rh}\)

The probability that an animal of the superpopulation enters the sampled population at age \(r\) between primary periods \(i\) and \(i+1\), and survives until sampling period \(i+1\) given that the animals are of population origin \(h\). \(\beta _0^{rh}\) is the probability that a member of the superpopulation came into the sampled population before the study started and survived to primary period 1 at age \(r\) given population origin \(h\).

\(\varphi _i^{rh}\)

The probability that an animal of the sampled population survives from primary period \(i\) to primary period \(i+1\) given that the animal is of age \(r\) at primary period \(i\) and are of population origin \(h\).

\(p_{ij}^{rh}\)

The secondary period capture probability, i.e., the probability that an animal of the sampled population of age \(r\) is captured on secondary period \(j\) of primary period \(i\) given population origin \(h, q_{ij}^{rh}=1-p_{ij}^{rh}\).

\(\breve{p}_i^{rh}\)

The primary period capture probability, i.e., the probability that an animal of the sampled population of age \(r\) and population origin \(h\) is caught at least once in the \(l_i\) secondary periods of primary period \(i, \breve{p}_i^{rh}=1-\prod _{j=1}^{l_i} (1-p_{ij}^{rh})\).

\(\psi _i^{rh}\)

The probability that a member of the superpopulation is alive and present in the sampled population at age \(r\) and is unmarked at primary period \(i\) given that the animals is of population origin \(h\) (Schwarz and Arnason 1996; Appendix B).

\(\tau _{i,j}^{rh}\)

The probability that an animal is alive and not captured by primary period \(j\) given that its last capture is in primary period \(i\) at age \(r\) and of population origin \(h, i<j \le K\) (Refer to Appendix B and C for calculation details).

\(\pi _{i,t}\)

The probability that the \(t\)th animal of \(u_i^{O*}\) is assigned as a local birth based on the population assignment procedures.

Assuming no tag mortality and all captured animals released back to the population, we have \(R_i^{Yh}=u_i^{Yh}\), and \(R_i^{Oh}=u_i^{Oh}+ \sum \nolimits _{r=Y,O}\sum \nolimits _{j=1}^{i-1}m_{j,i}^{rh}\).

2.2 Standard superpopulation capture-recapture model with two age classes

In the traditional superpopulation capture-recapture model for a single site population with two age classes, the likelihood can be written as
$$\begin{aligned} L=L_1*L_2*L_3 \end{aligned}$$
(1)
where \(L_1\) deals with the capture of unmarked animals of the primary periods, \(L_2\) is the probability distribution of the recapture data of primary periods conditional on the releases of each period, and \(L_3\) is the likelihood corresponding to the captures and recaptures of the secondary sampling periods conditional on capture at least once within the primary period.

Now we show how the age information of animals can be used to estimate the contributions of in situ births and immigrants to the population. We assume that surveys are timed so that young animals have not yet had an opportunity to disperse out of their birth population. If so, newly captured young animals will never be immigrants, and young animals enter the superpopulation as soon as they are born. Newly captured adults, on the other hand, are composed of two groups. One group consists of animals born in situ, entering into the superpopulation as young and not captured until later, when they have become adults. The other group consists of immigrants entering the superpopulation at an adult age.

We first split the entry probability \(\beta _i\) into \(\beta _i^{Y*}\) and \(\beta _i^{O*}\) according to the age of the animals when they enter the superpopulation. Then considering the population origin and the assumption that young animals are caught at their natal population, we have \(\beta _i^{Y*}=\beta _i^{YB}, i=0, 1, \ldots , K-1\) and \(\beta _i^{O*}=\beta _i^{OI}, i=1, \ldots , K-1\). However, for the first sampling period of adults, we have a more complex expression of \(\beta _0^{O*}=\beta _0^{OB}+\beta _0^{OI}\). This is because unmarked adults at any time will be composed of animals that were born on site and animals that were immigrants, and at sampling period 1 there is no prior information that can be used to decompose this mixture.

As a general case, we assume \(\{u_i^{r*}, N-\sum \nolimits _{i,r} u_i^{r*}\}\) forms a multinomial distribution conditional on the superpopulation size \(N\). The superpopulation and the new estimable entry parameters are \(\{N, \beta _i^{YB}, \beta _0^{O*}, \beta _{i}^{OI}\}\). Notice that we do not decompose \(\beta _0^{O*}\) because the components turn out not to be estimable. We get the first likelihood component,
$$\begin{aligned} L_1 \left( N, \beta _i^{YB}, \beta _0^{O*}, \beta _{i}^{OI}, \varphi _i^r, p_{ij}^r |u_i^{r*}\right)&= \frac{N!}{u.!(N-u.!)} p.^{u.} (1-p.)^{N-u.} \nonumber \\&\quad \times \frac{u.!}{\displaystyle \prod \nolimits _{i=1}^K \prod \nolimits _{r=Y,O} u_i^{r*}!} \prod _{i=1}^K \prod _{r=Y,O} \left( \frac{\psi _i^{r*}\breve{p}_i^{r*}}{p.}\right) ^{u_i^{r*}}\nonumber \\ \end{aligned}$$
(2)
where \(p.= \sum \nolimits _{i,r}(\psi _i^{r*}\breve{p}_i^{r*})\) and \(u.= \sum \nolimits _{i,r}(u_i^{r*})\).
Similarly, for recaptured animals of primary periods (\(L_2\)), let \(m_{i.}^{r*}=\displaystyle \sum \nolimits _{j=i+1}^K m_{ij}^{r*}\) and \(\tau _{i.}^{r*}=1-\displaystyle \sum \nolimits _{j=i+1}^K \tau _{i,j}^{r*}\breve{p}_j^{r*}\). Conditional on the releases of primary period \(i\) at different age classes, (\(R_i^{r*}\)), \(\{m_{i,i+1}^{r*}, m_{i,i+2}^{r*}, \ldots , m_{i,K}^{r*}, R_i^{r*}-m_{i.}^{r*}\}\) of the subsequent sampling periods are assumed to follow a multinomial distribution. We also assume the animals released in one primary period are independent of the animals released in another primary period or released at a different age. The likelihood component \(L_2\) will be the product of the multinomial distributions conditional on \(R_i^r, i=1, \ldots , K-1\). Then we have the likelihood for recaptures across the primary periods,
$$\begin{aligned} L_2\left( \varphi _i^{r*}, \breve{p}_i^{r*}\big | R_i^{r*}, m_{ij}^{r*}\right)&= \prod _{r=Y,O}\prod _{i=1}^{K-1} \frac{R_i^{r*}!}{\prod _{j=i+1}^K m_{ij}^{r*}! (R_i^{r*}-m_{i.}^{r*})!} \nonumber \\&\quad \times \prod _{j=i+1}^{K}\left( \tau _{i,j}^{r*} \breve{p}_{j}^{O*}\right) ^{m_{ij}^{r*}}\left( \tau _{i.}^{r*}\right) ^{R^{r*}_i-m_{i.}^{r*}} \end{aligned}$$
(3)
Within every primary period, the population is closed. The animals can be captured at any of the secondary periods. Therefore the conditional likelihood for animals captured in the secondary periods, conditional on being captured at least once within a primary period, is a multinomial distribution. Here for ease of presentation, we assume that there are two secondary periods within each primary period, but other numbers of secondary periods are also easily modeled.
$$\begin{aligned} L_3\left( p_{ij}^{r*} | X_{i}^{r* \omega }, R_i^{r*} \right)&= \prod _{r=Y,O}\prod _{i=1}^K \frac{R_i^{r*}}{X_i^{r*10}!X_i^{r*01}!X_i^{r*11}!} \left( \frac{p_{i1}^{r*} q_{i2}^{r*}}{\breve{p}_i^{r*}}\right) ^{X_i^{r*10}}\nonumber \\&\quad \times \left( \frac{q_{i1}^{r*} p_{i2}^{r*}}{\breve{p}_i^{r*}}\right) ^{X_i^{r*01}} \left( \frac{p_{i1}^{r*} p_{i2}^{r*}}{\breve{p}_i^{r*}}\right) ^{X_i^{r*11}} \end{aligned}$$
(4)
From the fact that \(\beta _i^{Y*}=\beta _i^{YB} (i=0,\ldots , K-1)\) and \(\beta _i^{O*}=\beta _i^{OI} \ (i \ne 0)\), we conclude that for a single site population with two age classes, after including age information, we can estimate all \(\beta _i^{YB}\) and nearly all \(\beta _i^{OI}\) except \(\beta _0^{OI}\). For the adult animals of the initial sampling period, we can only estimate \(\beta _0^{O*}\) and can not estimate \(\beta _0^{OB}\) and \(\beta _0^{OI}\) separately. This is because \(\beta _0^{OB}\) and \(\beta _0^{OI}\) always appear together in the term \(\psi _0^O \ (\psi _0^O=\beta _0^{OB}+\beta _0^{OI}\) ) in the likelihood. Further for \(u_0^{O*}\), we cannot distinguish \(u_0^{OB}\) from \(u_0^{OI}\) based on our classical capture-recapture history. We also cannot estimate the survival probability of in situ births and immigrants, \(\varphi _i^{OB}\) and \(\varphi _i^{OI}\), separately because we cannot split the newly captured unmarked adults into in situ births and immigrants. However, the approach of the next section will solve these problems.

2.3 Augmenting the model with perfect population assignment information

Now we consider incorporating additional population assignment information into our superpopulation model with one site and two age classes to aid in the estimation of the separate contributions of in situ reproduction and immigration to the growth of the population. Assuming perfect population assignment information, we can split newly captured unmarked adults \(u_i^{O*}=u_i^{OB}+u_i^{OI}\) according to their assigned populations of origin. Correspondingly, we can decompose the entry probabilities according to population of origin as \(\beta _i^{O*}=\beta _i^{OB}+\beta _i^{OI}\). We assume that young animals are captured at their natal place with \(u_i^{Y*}=u_i^{YB}\) and hence \(\beta _i^{Y*}=\beta _i^{YB}\). We also can allow the in situ births and immigrants to have different survival probabilities. The likelihood of the superpopulation capture-recapture model integrated with perfect population assignment information now becomes
$$\begin{aligned} L=L_1^{g}*L_2^{g}*L_3 \end{aligned}$$
(5)
where \(L_1^{g}\) is the likelihood for the newly captured in situ births and immigrants incorporating information on population of origin, \(L_2^g\) is the modified likelihood for the recapture component over primary periods conditional on the releases \(R_i^{YB}, R_i^{OB}\), and \(R_i^{OI}\), and \(L_3\) is the likelihood for the secondary period component. \(L_3\) is the same as in the standard superpopulation capture-recapture model if we assume the capture probability is the same for immigrants and in situ births which is often a reasonable assumption in practice.
Conditional on the superpopulation \(N\), the newly captured unmarked in situ births and immigrants (\( u_i^{YB}, u_i^{Oh}, N-\sum \nolimits _i (u_i^{YB}+\sum _h u_i^{Oh}) \)) are assumed to follow a multinomial distribution. Similar to the standard superpopulation capture-recapture model, we have
$$\begin{aligned}&L_1^{g} \left( N, \beta _i^{YB}, \beta _i^{Oh}, \breve{p}_i^{rh}, \varphi _i^{YB}, \varphi _i^{Oh}, | u_i^{YB}, u_i^{Oh} \right) \nonumber \\&\quad = \frac{N!}{u.! (N-u.)!} (p.)^{u.} (1-p.)^{N-u.} \frac{u.!}{\prod _{i=1}^K u_i^{YB}! \prod _{h=B,I}{u_i^{Oh}!}} \nonumber \\&\quad \quad \times \prod _{i=1}^K \left( \frac{\psi _i^{YB}\breve{p}_i^{YB}}{p.}\right) ^{u_i^{YB}} \prod _{h=B,I}\left( \frac{\psi _i^{Oh}\breve{p}_i^{Oh}}{p.}\right) ^{u_i^{Oh}} \end{aligned}$$
(6)
where \(u.=\displaystyle \sum \nolimits _{i=1}^K (u_i^{YB}+\sum \nolimits _{h=B,I}u_i^{Oh})\), and \(p.=\displaystyle \sum \nolimits _{i=1}^K (\psi _i^{YB}\breve{p}_i^{YB }+\sum \nolimits _{h=B,I}\psi _i^{Oh}\check{p}_i^{Oh})\).
For the recaptures of the primary periods, conditional on \(R_i^{rh}\), all newly recaptured animals \(\{m_{i,i+1}^{rh}, m_{i,i+2}^{rh}, \ldots , m_{i,K}^{rh}, R_i^{rh}-\sum \nolimits _{j=i+1}^K m_{ij}^{rh}\}\) from subsequent primary sampling periods are assumed to follow a multinomial distribution. Also the animals are assumed to be independent given that they are released in different primary periods at different ages and are of different population origin. We have,
$$\begin{aligned}&L_2^{g}\left( \varphi _i^{YB},\varphi _i^{Oh}, \breve{p}_i^{Oh} | R_i^{YB}, R_i^{Oh}, m_{ij}^{YB}, m_{ij}^{Oh},\right) \nonumber \\&\quad = \prod _{i=1}^{K-1} \left\{ \prod _{r=Y,O} \frac{R_i^{rB}!}{\prod _{j=i+1}^K m_{ij}^{rB}! (R_i^{rB}-m_{i.}^{rB})!} \prod _{j=i+1}^{K}(\tau _{i,j}^{rB}\breve{p}_{j}^{ OB})^{m_{ij}^{rB}}(\tau _{i.}^{rB})^{R_i^{rB}-m_{i.}^{rB}} \right. \nonumber \\&\quad \quad \left. \times \frac{R_i^{OI}!}{\prod _{j=i+1}^K m_{ij}^{OI}! (R_i^{OI}-m_{i.}^{OI})!} \prod _{j=i+1}^{K}(\tau _{i,j}^{OI}\breve{p}_{j}^{OI})^{m_{ij}^{OI}}(\tau _{i.}^{OI})^{R_i^{OI}-m_{i.}^{OI}} \right\} \end{aligned}$$
(7)
where \(m_{i.}^{rB}=\displaystyle \sum \nolimits _{j=i+1}^K m_{ij}^{rB}, m_{i.}^{OI}=\displaystyle \sum \nolimits _{j=i+1}^K m_{ij}^{OI}, \tau _{i.}^{rB}=1-\displaystyle \sum \nolimits _{j=i+1}^K \tau _{i,j}^{rB}\breve{p}_j^{OB}\), and \(\tau _{i.}^{OI}=1-\displaystyle \sum \nolimits _{j=i+1}^K \tau _{i,j}^{OI}\breve{p}_j^{OI}\).

The addition of perfect population assignment data enables the \(\beta _0^{OB}, \beta _0^{OI}, \varphi _i^{OB}\) and \(\varphi _i^{OI}\) to be estimated separately, and this cannot be achieved using the capture-recapture data alone. In other words, perfect population assignment data allow us to distinguish in situ reproduction and immigration. We could also construct a more general model by assuming that the immigrants have a different capture probability from that of in situ birth animals, but we do not develop that generalization here.

2.4 Augmenting the model with imperfect population assignment information

As in the case where we had perfect population assignment information, any animal captured for which the birth state is unknown has a population assignment procedure carried out so that a state of origin is assigned. However, now this assignment information is subject to uncertainty, which is what occurs in practice. Consider a new adult genotyped at capture in primary period two. The assignment procedure assigns the animal as an in situ birth with probability \(\pi \) (\(0\le \pi \le 1\) ). We view \(\pi \) as the probability that the animal is an in situ birth from the site, and \(1-\pi \) as the probability that it is an immigrant originating outside the site. If the assignment procedure is a good one, then when the animal is truly an in situ birth, \(\pi \) will be close to one and \((1-\pi )\) will be close to zero; whereas when the animal is truly an immigrant, \(\pi \) will be close to zero and (\(1-\pi \)) will be close to one. The perfect assignment procedure is a special case of the imperfect assignment procedure when \(\pi =1\) if the animal is truly an in situ birth and \(\pi =0\) if the animal is truly an immigrant.

To generalize our model to the more realistic case of imperfect population assignment procedure, we take a resampling approach similar to multiple imputation (Rubin 1987). We assume the young animals are captured at their natal locations, so only adults need to have their origination assigned. Consider that each newly captured adult animal of primary period \(i, t \, (t=1, \ldots , u_i^{O*})\), has a population assignment probability \(\pi _{i,t}, 0\le \pi _{i,t}\le 1\). We impute a pseudo perfect data set \(q\) times using the following procedures. First, the origin of each newly captured adult animal of primary period \(i\) is randomly allocated by a uniform random variable \(z_{i,t} \in [0,1]\), (\(t=1, \ldots , u_i^{O*}\)). If \(z_{i,t}\in [0, \pi _{i,t}]\), we assign this animal as an in situ birth, whereas if \(z_{i,t}\in (\pi _{i,t}, 1]\), we assign it as an immigrant. We apply this assignment procedure to all newly captured adults of all primary sampling periods. Then we split \(u_i^{O*}\) into \(u_i^{OB}\) and \(u_i^{OI}\) as in the perfect assignment procedure and get a “pseudo perfect” data set. We repeat this imputation \(q\) times and employ the superpopulation capture-recapture model based on perfect assignment to estimate the parameters. More specifically, for the \(n\)th imputation, let \(z_{i,t}^{(n)}\in [0,1]\) be the random variable corresponding to the \(t\)th animal of \(u_i^{O*}\) which has assignment probability of \(\pi _{i,t}, u_{i}^{(n)OB}=\sum \nolimits _{t=1}^{u_i^{O*}}I(z_{i,t}^{(n)}\le \pi _{i,t})\) and \(u_{i}^{(n)OI}=u_i^{O*}-u_{i}^{(n)OB}\). The likelihood for the data under the \(n\)th imputation is
$$\begin{aligned} L^{(n)}=L_{1}^{(n)g}\times L_{2}^{(n)g} \times L_{3}^{(n)} \end{aligned}$$
(8)
with
$$\begin{aligned}&L_1^{(n)g} \left( N, \beta _i^{YB}, \beta _i^{Oh}, \breve{p}_i^{rh}, \varphi _i^{YB}, \varphi _i^{Oh}, | u_i^{YB}, u_i^{^{(n)}Oh}, \pi _{i,t}, z_{i,t}^{^{(n)}} \right) \nonumber \\&\quad = \frac{N!}{u.! (N-u.)!} (p.)^{u.} (1-p.)^{N-u.} \frac{u.!}{\prod _{i=1}^K u_i^{YB}! \prod _{h=B,I}{u_i^{(n)Oh}!}} \nonumber \\&\quad \times \prod _{i=1}^K \left( \frac{\psi _i^{YB}\breve{p}_i^{Yh}}{p.}\right) ^{u_i^{YB}}\prod _{h=B,I} \left( \frac{\psi _i^{Oh}\breve{p}_i^{Oh}}{p.}\right) ^{u_i^{(n)Oh}} \end{aligned}$$
(9)
\(L_2^{(n)g}\) and \(L_3^{(n)}\) are defined in Eq. 5 with \(u_i^{Oh}=u_i^{(n)Oh}\). For each of the \(q\) sets of “pseudo perfect” data, we estimate the parameters under \(L^{(n)}\). Then we can take various distributional statistics (means, medians, standard deviations, and percentiles) of the \(q\) estimates. By so doing, we obtain estimators that reflect the inherent uncertainty due to the assignment information being imperfect, in addition to the usual uncertainty associated with capture-recapture inference. This resampling algorithm can be summarized as follows:
  • Step 1: \(q=100, n=1, \theta =( \beta _i^{YB}, \beta _i^{Oh}, \varphi _i^{YB}, \varphi _i^{Oh}, \breve{p}_i^{rh}, N )\).

  • Step 2: \(i=1,\ldots , K, t=1, \ldots , u_i^{O*}\), \(z_{i,t}^{(n)} \sim [0,1], u_{i}^{(n)OB}=0, u_{i}^{(n)OI}=0\).

  • Step 3: if \(z_{i,t}^{(n)}\in [0, \pi _{i,t}]\), then \(u_{i}^{(n)OB}=u_{i}^{(n)OB}+1\), Otherwise, \(u_{i}^{(n)OI}=u_{i}^{(n)OI}+1\).

  • Step 4: \(\hat{\theta }_n \,\)=\(\, \mathrm Argmax _{\theta }\left( L^{(n)}(\theta ) | u_i^{YB}, u_{i}^{(n)Oh}, m_{ij}^{YB}, m_{ij}^{(n)Oh}, X_{i}^{(n)r \omega }, z_{i,t}^{(n)}, \pi _{i,t}\right) \) where \(L^{(n)}\) is defined in equation 2.9.

  • Step 5: if \(n<q, n=n+1 \), go to step 2. Otherwise,

  • Parameter estimate: \(\hat{\theta }=1/q \displaystyle \sum \nolimits _{n=1}^q{\hat{\theta }_n}\).

  • Variance estimate: \(\hat{var\hat{(\theta )} }= \frac{\sum \nolimits _{n=1}^q \sigma _n^2}{q}+(1+\frac{1}{q})\frac{1}{q-1} \sum \nolimits _{n=1}^q(\hat{\theta }_n-\bar{\hat{\theta }})^2\).

The first term of the variance estimate is the natural variability inherent to the data and the model and is calculated by the information matrix. \(\sigma _n^2\) is the variance of \(\hat{\theta }\) if the generated data were the perfect case (no imputation needed). The second term is the between imputation variance associated with the uncertainty of the assignment process. Please refer to Rubin (1987) for more detail.

2.5 Assumptions

The superpopulation capture-recapture model with the robust design and population assignment data is a generalization of the standard superpopulation model, and therefore the assumptions required are similar.
  1. 1.

    All members of the superpopulation, \(N\), from the same population of origin have the same time-specific entry probabilities (homogeneous entry probabilities).

     
  2. 2.

    Local born animals and immigrants have homogeneous survival rates.

     
  3. 3.

    Animals of the same age present during sampling period \(i\) have the same time-specific capture probabilities (homogeneous capture probabilities).

     
  4. 4.

    Marks do not affect the survival or behavior of the animal, are not lost, and are recorded correctly.

     
  5. 5.

    Each animal behaves independently with respect to survival, entry, and detection probabilities.

     
  6. 6.

    There is no uncertainty associated with the laboratory analysis on which the population assignment procedure is based. For example, if the assignment procedure is a genetic one, there are no errors in the molecular genotyping of the animals.

     

3 Simulation

In the simulation, the study is assumed to last five years with three secondary periods within each primary period. The capture probabilities are the same for the three secondary periods within each year. However, the primary capture probabilities and the survival probabilities are assumed to vary across the primary sampling periods. The parameters take different values for different sets of simulations. Secondary capture probabilities take values 0.5 and 0.2, survival probabilities take values 0.3 and 0.15 for young animals and 0.35 and 0.55 for adult animals, (reflecting the fact that young animals usually have lower survival probabilities), and superpopulation size takes values 1,000 and 10,000. For the entry probabilities, the first sampling period almost always has more new recruits than later sampling periods, and there are typically more in situ births than immigrants. This scenario is reasonable in practice because most animals of many species stay in the same population throughout their lives. Following a complete factorial design, our simulations were done under 16 different scenarios. Here we only report a representative subset of the simulation results.

First, we test the performance of the standard superpopulation capture-recapture model with age information but without population assignment information (“Stand model” columns). Table 1 reports part of the simulation results from 500 repetitions under four different situations. From the lower two panels of Table 1, we can see that when the superpopulation size is \(N=\) 10,000, the standard model has really good estimates, whose maximum relative root mean square error (RRMSE \(\%\)) is \(9.71\,\%\) for entry probability estimates and \(16.26\,\%\) for survival probability estimates. Even in the worst case, when the superpopulation size is \(N=\) 1,000, \(\varphi _i^{YB}=0.15 \), and \(\varphi _i^{O*}=0.55, 62\,\%\) of the estimates having RRMSE lower than \(23\,\%\).
Table 1

Simulation results of the superpopulation capture-recapture model augmented with assignment information (“Aug model” columns) and the standard superpopulation capture-recapture model (“Stand model” columns)

Par.

RRMSE

Par.

RRMSE

Par.

RRMSE

Par.

RRMSE

Aug model

Stand model

Aug model

Stand model

Aug model

Stand model

Aug model

Stand model

\(p_{ij}^Y=0.5, p_{ij}^O=0.2, \varphi _i^{YB}=0.3, \varphi _i^{OB}=0.35, \varphi _i^{OI}=0.35, N=\) 1,000

\(\beta _0^{YB}\)

14.65

15.10

\(\beta _0^{OB}\)

9.93

13.03

\(\varphi _1^{YB}\)

34.12

32.91

\(\varphi _2^{OB}\)

20.11

20.09

\(\beta _1^{YB}\)

14.13

14.95

\(\beta _0^{OI}\)

14.46

 

\(\varphi _2^{YB}\)

32.49

33.82

\(\varphi _2^{OI}\)

21.41

 

\(\beta _2^{YB}\)

12.89

15.57

\(\beta _1^{OI}\)

18.37

27.73

\(\varphi _3^{YB}\)

37.76

33.07

\(\varphi _3^{OB}\)

20.34

22.13

\(\beta _3^{YB}\)

14.12

15.00

\(\beta _2^{OI}\)

18.29

21.84

\(\varphi _4^{YB}\)

34.61

41.67

\(\varphi _3^{OI}\)

22.88

 

\(\beta _4^{YB}\)

14.36

16.29

\(\beta _3^{OI}\)

20.73

21.34

\(\varphi _1^{OB}\)

21.66

19.27

\(\varphi _4^{OB}\)

18.70

27.84

\(N\)

4.28

4.77

\(\beta _4^{OI}\)

18.33

23.55

\(\varphi _1^{OI}\)

27.58

-

\(\varphi _4^{OI}\)

26.03

 

\(p_{ij}^Y=0.5, p_{ij}^O=0.2, \varphi _i^{YB}=0.15, \varphi _i^{OB}=0.55, \varphi _i^{OI}=0.55, N=\) 1,000

\(\beta _0^{YB}\)

13.51

13.77

\(\beta _0^{OB}\)

9.30

12.94

\(\varphi _1^{YB}\)

37.94

42.81

\(\varphi _2^{OB}\)

12.51

12.31

\(\beta _1^{YB}\)

14.91

15.43

\(\beta _0^{OI}\)

14.63

 

\(\varphi _2^{YB}\)

43.68

46.57

\(\varphi _2^{OI}\)

14.70

 

\(\beta _2^{YB}\)

14.54

14.41

\(\beta _1^{OI}\)

20.00

34.13

\(\varphi _3^{YB}\)

43.37

48.37

\(\varphi _3^{OB}\)

11.84

14.35

\(\beta _3^{YB}\)

15.86

15.46

\(\beta _2^{OI}\)

18.30

22.76

\(\varphi _4^{YB}\)

49.96

51.03

\(\varphi _3^{OI}\)

16.68

-

\(\beta _4^{YB}\)

12.51

15.66

\(\beta _3^{OI}\)

19.99

23.06

\(\varphi _1^{OB}\)

12.85

11.65

\(\varphi _4^{OB}\)

13.93

17.95

\(N\)

3.23

3.81

\(\beta _4^{OI}\)

18.68

21.86

\(\varphi _1^{OI}\)

18.88

 

\(\varphi _4^{OI}\)

16.87

 

\(p_{ij}^Y=0.5, p_{ij}^O=0.2, \varphi _i^{YB}=0.3, \varphi _i^{OB}=0.35, \varphi _i^{OI}=0.35, N=\) 10,000

\(\beta _0^{YB}\)

4.31

4.35

\(\beta _0^{OB}\)

3.19

3.85

\(\varphi _1^{YB}\)

9.76

9.90

\(\varphi _2^{OB}\)

6.17

6.17

\(\beta _1^{YB}\)

4.35

4.42

\(\beta _0^{OI}\)

4.52

 

\(\varphi _2^{YB}\)

10.21

10.30

\(\varphi _2^{OI}\)

6.71

 

\(\beta _2^{YB}\)

4.72

4.89

\(\beta _1^{OI}\)

5.96

8.10

\(\varphi _3^{YB}\)

9.20

10.06

\(\varphi _3^{OB}\)

6.39

7.32

\(\beta _3^{YB}\)

4.46

4.61

\(\beta _2^{OI}\)

5.73

6.52

\(\varphi _4^{YB}\)

11.65

12.63

\(\varphi _3^{OI}\)

7.41

 

\(\beta _4^{YB}\)

4.86

5.22

\(\beta _3^{OI}\)

5.83

7.03

\(\varphi _1^{OB}\)

5.48

5.69

\(\varphi _4^{OB}\)

5.84

7.79

\(N\)

0.45

1.33

\(\beta _4^{OI}\)

5.76

6.82

\(\varphi _1^{OI}\)

7.39

 

\(\varphi _4^{OI}\)

1.23

 

\(p_{ij}^Y=0.5, p_{ij}^O=0.2, \varphi _i^{YB}=0.15, \varphi _i^{OB}=0.55, \varphi _i^{OI}=0.55, N=\) 10,000

\(\beta _0^{YB}\)

4.68

4.74

\(\beta _0^{OB}\)

2.87

3.87

\(\varphi _1^{YB}\)

12.84

13.63

\(\varphi _2^{OB}\)

4.03

4.17

\(\beta _1^{YB}\)

4.67

4.89

\(\beta _0^{OI}\)

4.70

 

\(\varphi _2^{YB}\)

14.41

13.29

\(\varphi _2^{OI}\)

4.38

 

\(\beta _2^{YB}\)

4.97

4.97

\(\beta _1^{OI}\)

6.12

9.71

\(\varphi _3^{YB}\)

14.49

14.83

\(\varphi _3^{OB}\)

3.73

3.98

\(\beta _3^{YB}\)

4.87

4.92

\(\beta _2^{OI}\)

6.08

7.39

\(\varphi _4^{YB}\)

15.90

16.26

\(\varphi _3^{OI}\)

4.94

 

\(\beta _4^{YB}\)

4.81

4.96

\(\beta _3^{OI}\)

5.76

6.97

\(\varphi _1^{OB}\)

4.25

4.04

\(\varphi _4^{OB}\)

4.41

5.59

\(N\)

3.23

3.81

\(\beta _4^{OI}\)

5.89

6.84

\(\varphi _1^{OI}\)

4.70

 

\(\varphi _4^{OI}\)

6.87

 

The RRMSE is the relative root of mean squared errors of the estimates. The standard model estimates \(\beta _0^{O*}\) instead of \(\beta _0^{OB}, \beta _0^{OI}\) and \(\varphi _i^{O*}\) instead of \(\varphi _i^{OB}, \varphi _i^{OI}\). The simulation results are obtained for \(\beta _0^{YB}=0.06, \beta _0^{OB}=0.2, \beta _0^{OI}=0.1, \beta _i^{YB}=0.05, \beta _i^{OI}=0.11\), and other parameter values are listed in the table

Next we want to assess the consequences of adding the population assignment data to the superpopulation capture-recapture model with age information. We compare the simulation results for the superpopulation capture-recapture model with perfect population assignment data (“Aug model” columns) and the superpopulation capture-recapture model that incorporates only age information (“Stand model” column). When we compare the second column to the third column and the fifth column to the sixth column of Table 1, we can see the augmented model has lower RRMSE than the standard model for all of the entry probability estimates, no matter what true values the parameters take.

We cannot compare the entry probability of adults at the first primary period between the augmented and standard models, because in the augmented model we estimate \(\beta _0^{OB}\) and \(\beta _0^{OI}\), while in the standard model, we estimate only their sum \(\beta _0^{O*}\). We also cannot directly compare the estimates of adult survival probabilities between the augmented and standard models, because in the augmented model we estimate \(\varphi _i^{OI}\) and \(\varphi _i^{OB}\), while in the standard model, we can only estimate \(\varphi _i^{O*}\). Nevertheless, from Table 1 we can see that the augmented model and the standard model have comparable survival estimates, which is unlike the estimates of entry probabilities. This is because the estimate of survival probability mainly depends on the capture-recapture data (\(L_2\) component of the likelihood) which are almost the same for both the augmented and standard models. For both models, the higher the parameter values, the better the performance of the estimators. For example, comparing the first panel of Table 1 to the second panel, when entry and capture probabilities remain constant, the relative bias and relative standard errors for \(\hat{\varphi _i}^{YB}\) increase for both augmented and standard models as \(\varphi _i^{YB}\) decreases from 0.3 to 0.15 and the relative bias and the relative standard errors for \(\hat{\varphi _i}^{O*}\) (\(\hat{\varphi _i}^{OB}\) and \(\hat{\varphi _i}^{OI}\) for augmented model) decrease when \(\varphi _i^{O*}\) (\(\varphi _i^{OB}\) and \(\varphi _i^{OI}\) for augmented model) increases from 0.35 to 0.55. Similarly, when we compare the second panel to the fourth panel of Table 1 as the superpopulation size \(N\) increases from 1,000 to 10,000 while other parameter values remain unchanged, the relative bias and relative standard errors of all estimates decrease dramatically. We have similar results for the other different scenarios where parameters take different values.

Now we investigate how the uncertainty of the assignment procedure affects the estimators of the model. In our simulations, we explored imperfect population assignment procedures with four different levels of uncertainty. The first level is the perfect case and assumes the assignment probability to be one (zero) given the animal is an in situ birth (immigrant); the second assumes the assignment probability is uniform between (0.95, 1) given the adult animal is an in situ birth and uniform between (0, 0.05) if it is an immigrant; the third assumes the assignment probability is uniform between (0.9, 1) given the adult animal is an in situ birth and uniform between (0, 0.1) if it is an immigrant; and the fourth assumes the assignment probability is uniform between (0.8, 1) given the adult animal is an in situ birth and uniform between (0, 0.2) otherwise.

Since the entry probability estimates depend mainly on the population assignment procedure, we analyze the changes of relative bias, relative standard error, and RRMSE for each entry probability parameter in detail (Figs. 12).
Fig. 1

The effects of different levels of uncertainty of population assignment on estimates of entry probabilities for young animals (\(\hat{\beta }_i^{YB}, i=0, \ldots , 4\)). The five panels represent the five simulated primary periods. Each point reflects the results of 1,000 simulations with \(p_{ij}^{Yh}=0.5, P_{ij}^{Oh}=0.2, \varphi _{k}^{Yh}=0.3, \varphi _{k}^{Oh}=0.35, \beta _0^{YB}=0.2, \beta _{k}^{YB}=0.06, \beta _0^{OB}=0.2, \beta _0^{OI}=0.1, \beta _{k}^{OI}=0.11, N=\) 1,000, \(i=1, \ldots , 5, k=1,2,3,4, j=1,2,3\). Relative bias, relative SE, and RRMSE are illustrated separately

Fig. 2

The effects of different levels of uncertainty of population assignment on estimates of entry probabilities for adult animals \(\hat{\beta }_0^{OB}, \hat{\beta }_i^{OI}, i=1,\ldots , 4\). The five panels represent the five simulated primary periods. Each point reflects the results of 1,000 simulations with \(p_{ij}^{Yh}=0.5, p_{ij}^{Oh}=0.2,\ \varphi _{k}^{Yh}=0.3, \varphi _{k}^{Oh}=0.35, \beta _0^{YB}=0.2, \beta _{k}^{YB}=0.06, \beta _0^{OB}=0.2, \beta _0^{OI}=0.1, \beta _{k}^{OI}=0.11, N=\) 1,000, \(i=1,\ldots , 5, k=1,2, 3,4, j=1,2,3\). Relative bias (lines of “o- - -o”), relative SE (lines of “\(\triangle \)- - -\(\triangle \)”), and RRMSE (lines of “+- - - +”) are illustrated separately

From each panel of Figs. 1 and 2, as the reliability of the assignment procedure increases from \(\pi \in (0.8,1)\) to \(\pi =1\), the relative bias and RRMSE of the estimates of the entry probabilities for young animals and adult animals decrease. The relative standard errors of these entry probabilities do not have such an apparent decreasing trend. The estimates of the entry probabilities become better and better as the assignment procedure becomes more reliable. Our simulation results for other parameter values showed similar patterns. So we conclude that the decreases of the RRMSE of the estimates based on better assignment procedures are mainly caused by the decrease of the relative bias of the estimates. The simulation results of capture probabilities and survival probabilities under the scenarios of Figs. 1 and 2 are listed in Appendix A.

4 Example: kangaroo rats in Southern Arizona

As in our previous analysis (Wen et al. 2011), we illustrated this model with capture-recapture and population assignment data from a study of banner-tailed kangaroo rats Dipodomys spectabilis in Southern Arizona, U.S.A. (Skvarla et al. 2004; Waser et al. 2006; Sanderlin et al. 2012). In this study, a set of \(8\) small populations was surveyed using the robust design. Populations were defined spatially; each “population” was a cluster of up to 50 mounds separated from other clusters by several hundred meters of empty habitat. During each survey (primary sampling period), we trapped animals at their dens on an average of three nights (secondary sampling periods). For each trapped animal, we recorded location, sex and age (“young” for animals less than one year old, “adult” for animals older than one year), based on size and reproductive maturity, i.e., descended testes or elongated nipples indicating lactation. We gave each newly-captured animal an individually-numbered pair of ear tags and took a small tissue sample for genetic analyses. For the one-site, two-age-class model that we describe here, we considered capture histories from a single, central population during one survey each year (July/August) between 1994 and 2001.

We were able to genotype 90 % of all individuals at an average of 7.7 microsatellite loci (for the remaining 10 %, sample loss or degradation prevented us from obtaining reliable genotypes). As in Wen et al. (2011), we used genotypes from our central population to estimate the allele frequencies expected in animals born in situ.

We used GENECLASS2 (Piry et al. 2004) to assign a population of origin to each adult. As in Wen et al. (2011) we used GENECLASS2 to derive an assignment probability vector for each animal that was first captured as an adult, based on the likelihood of its genotype in the central population and its likelihood in surrounding populations. The population assignment procedure implemented by GENECLASS2 is imperfect, but we also handled the data as though they were “perfect”.

In our real data analysis, we considered 12 different models with different constraints for the capture probability, survival probability, and entry probability parameters. To choose the “best” model for the kangaroo rat data set, we use \(BIC\) and \(AIC\) as the model selection statistics. The \(BIC\) and \(AIC\) values for all 12 models are listed in Table 2.
Table 2

Kangaroo rat data set probability models, model with lowest \(AIC\) or \(BIC\) is preferred

Model name

\(BIC\)

\( AIC\)

Model name

\(BIC\)

\( AIC\)

\(p(c,r,c)\beta (c,r,h)\varphi (c,r,h)\)

2,578.69

2,583.02

\(p(c,c,c)\beta (c,r,h)\varphi (c,r,h)\)

2,587.96

2,591.81

\(p(c,r,c)\beta (t,r,h)\varphi (c,c,c)\)

2,744.89

2,754.52

\(p(c,c,c)\beta (t,r,h)\varphi (c,r,h)\)

2,746.58

2,756.69

\(p(c,r,c)\beta (t,r,h)\varphi (c,r,h)\)

2,748.08

2,758.67

\(p(c,c,c)\beta (t,r,h)\varphi (c,c,c)\)

2,757.39

2,766.54

\(p(c,c,c)\beta (c,r,h)\varphi (c,c,c)\)

2,770.11

2,773.00

\(p(c,r,c)\beta (c,r,h)\varphi (c,c,c)\)

2,771.37

2,774.74

\(p(c,r,c)\beta (t,r,h)\varphi (c,r,c)\)

2,819.31

2,828.99

\(p(c,r,c)\beta (c,r,h)\varphi (c,r,c)\)

2,929.77

2,933.14

\(p(c,c,c)\beta (t,r,h)\varphi (c,r,c)\)

2,830.89

2,840.04

\(p(c,c,c)\beta (c,r,h)\varphi (c,r,c)\)

2,931.72

2,934.61

The first letter within parentheses is “c” if the parameter is constant over time, otherwise, it would be “t”. The second letter within parentheses is “c” if the parameter is constant for local born young animals and local born old animals, otherwise, the the second letter should be “r”. The third letter within parenthesis is “c” if the parameter is the same for in situ births and immigrants, otherwise, the third letter should be “h”

In Table 2 we can see that the model \(p(c,r,c)\beta (c,r,h)\varphi (c,r,h)\) is the most suitable model for the kangaroo rat data set; it has the smallest \(AIC\) and \(BIC\) values. This model assumes that capture probability is the same for all animals and all sampling periods, that entry probability is dependent on population of origin and time, and that survival probability depends on both age and population of origin. Parameter estimates from the kangaroo rat data set under the “best” model are presented in Table 3.
Table 3

Parameter estimates for the kangaroo rat data under the “best” superpopulation capture-recapture model augmented with assignment data (model specified in paragraph 5 of Sect. 4)

Par.

Est.

S.E.

Par.

Est.

S.E.

\(\beta ^{YB}\)

0.104

0.003

\(\varphi ^{YB}\)

0.400

0.053

\(\beta _0^{OB}\)

0.073

0.019

\(\varphi ^{OB}\)

0.441

0.045

\(\beta _0^{OI}\)

0.039

0.015

\(\varphi ^{OI}\)

0.380

0.088

\(\beta ^{OI}\)

0.008

0.002

\( N \)

231

2.83

\(p^Y \)

0.546

0.039

\(p^O\)

0.438

0.045

From Table 3 we can see that there appear to be many more individuals born in situ than immigrants for all periods. Young animals have lower survival probabilities than adults born locally but have higher survival probabilities than adult immigrants. For adults, animals born in situ have higher survival probabilities than immigrants.

Table 4 compares the advantages and disadvantages of both the standard model without population assignment information and the augmented model with population assignment data.
Table 4

Compare the standard superpopulation capture-recapture model with age information but without population assignment information (“Standard Model”) and with population assignment information (“Augmented Model”)

Parameter

Standard model

Augmented model

\(\beta _i^{YB}\)

Estimable

Estimable

\(\beta _0^{O*}\)

Estimable

Estimable

\(\beta _0^{Oh} \)

Not estimable

Estimable

\(\beta _i^{OI}\)

Estimable

Estimable

\(\varphi _i^{YB}\)

Estimable

Estimable

\(\varphi _i^{O*} \)

Estimable

Estimable

\(\varphi _i^{Oh} \)

Not estimable

Estimable

(h = B, I) in this table

5 Discussion

For an open animal population, new recruits are composed of two components, in situ recruits and immigrants that have moved in from outside the population. Understanding the relative contribution of in situ reproduction versus immigration to the recruitment process is important for reasons ranging from the assessment of source-sink and metapopulation dynamics (Pulliam 1988; Hanski 1999; Runge et al. 2006) to spatial conservation prioritization and design of reserves (Moilanen et al. 2009). This separation of recruitment components has been a difficult problem for studies of single site populations, and is not possible using the standard open models derived by Jolly and Seber (Jolly 1965; Seber 1965, 1982). There are two possible sources of information available for separating recruitment components: age information for animals that do not disperse when very young, and population assignment information, based for example on genetic data.

Nichols and Pollock (1990) and Pollock et al. (1990) presented an ad hoc approach using age information for a single population, two age class, robust design to estimate the in situ birth and immigrant recruitment numbers separately. Their approach does not explicitly incorporate the recruitment process into the likelihood. In the current paper, we first develop formally a new superpopulation capture-recapture model (Schwarz and Arnason 1996) for a single population with two age classes, with data collected using the closed robust design protocol (Pollock 1982; Pollock et al. 1990). This initial model permits maximum likelihood estimation of the entry probabilities associated with in situ reproduction and immigration separately for all except the first sampling period. This model can be viewed as a rigorous likelihood approach to the ad hoc estimation methods developed by Nichols and Pollock (1990).

Wen et al. (2011) considered a single age class model which combined a superpopulation open capture-recapture model augmented with data from both a perfect and an imperfect population genetic assignment procedure. This enabled them to separate in situ reproduction from immigration, which is not possible using the single-age open capture-recapture model alone. In this paper we consider both sources of information about recruitment components and combine an age-based robust design capture-recapture model with population assignment data. This allows us to obtain maximum likelihood estimates of entry probabilities for in situ recruits and immigrants for all sampling periods and generalizes the earlier approaches. Our estimates are more efficient than those obtained using the age data alone for the parameters that can be estimated in common. Further we can estimate the parameters for all sampling periods. In addition, the entry probabilities can be directly modeled as functions of time-specific covariates thought to influence either immigration and/or in situ reproduction, providing direct inference about reproductive and dispersal processes.

This method allows us to obtain meaningful population parameter estimates for the kangaroo rat population. Standard mark-recapture analysis (Skvarla et al. 2004), genetic parentage analysis (Waser and Hadfield 2010), and a novel reverse-time approach (Sanderlin et al. 2012) all indicate that dispersal between populations in this species is a rare event. Animals that do emigrate from their birth population do so when young, but after our summer survey. The entry probabilities estimated are consistent with this result (we note that they are obtained with mark-recapture data from only one population, whereas the earlier analyses required data from 4–8 populations). Previous analyses found that banner-tailed kangaroo rats enter traps readily, and that less than half of juveniles survive to adulthood (Skvarla et al. 2004); again, the estimates of capture and juvenile survival probabilities we estimate here are consistent with published results. Previous approaches did not, however, allow us to compare the survival of immigrant and resident adults; here we find that immigrants survive less well. Banner-tailed kangaroo rats depend on their elaborate dens and attendant seed stores to evade predators and survive long rainless periods in their desert habitat; a lower survival rate for immigrants is not surprising given the difficulty that dispersing animals have in establishing or usurping dens and seed stores (Jones 1986). Lower survival rates for immigrants compared to residents may be common in nature, but published estimates are rare (Doligez and Pärt 2008; Devillard and Bray 2009). The ability to separately estimate survival probabilities of immigrants and residents is another big advantage of this approach. There is substantial interest in “costs of dispersal” among investigators interested in the evolution of dispersal tendencies.

We are thus convinced that “augmenting” traditional mark-recapture methodology with population assignment data is an approach with considerable promise, and the kangaroo rat example suggests several directions in which further development might pay off. First, genetic parentage analysis has shown that a small number of animals first captured when young have already immigrated, violating our assumption that \(\beta ^{YI} = 0\) (Waser et al. 2006; Waser and Hadfield 2010). Use of data from such an analysis could be added to the likelihood and incorporated to estimate the probability that a newly caught young animal is a product of in situ reproduction versus immigration. This uncertainty could then be applied to all newly caught young, further generalizing the approach presented here. Second, techniques for making full use of information from animals with missing population assignment data need to be developed. More importantly, genetic assignment procedures implemented in GENECLASS2 and elsewhere may overestimate \(\beta ^{OI}\), for a subtle reason: an individual’s genotype may have a low likelihood in the population where it is trapped not because it is an immigrant, but because its parents (or other ancestors) were immigrants. If some information (e.g., parentage analysis) were available to estimate misclassification probabilities associated with offspring of immigrant parents, then it could be incorporated into our modeling and potentially improve our methodology. It would also be useful to consider incorporating other types of information about population assignment. Finally, an obvious extension of our work would consider a metapopulation system where new animals are comprised of recruits from patches within the system plus immigrants from outside the system.

Acknowledgments

We thank the NSF, for support (DEB 0816925).

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Zhi Wen
    • 1
  • James D. Nichols
    • 2
  • Kenneth H. Pollock
    • 3
  • Peter M. Waser
    • 4
  1. 1.Office of Biostatistics and Epidemiology, Center for Biologics Evaluation and ResearchFood and Drug AdministrationRockvilleUSA
  2. 2.Patuxent Wildlife Research CenterUSGSLaurelUSA
  3. 3.Department of BiologyNorth Carolina State UniversityRaleighUSA
  4. 4.Department of Biological SciencesPurdue UniversityWest LafayetteUSA

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