Integrated population models poorly estimate the demographic contribution of immigration

Estimating the contribution of demographic parameters to changes in population growth is essential for understanding why populations fluctuate. Integrated population models (IPMs) offer a possibility to estimate the contributions of additional demographic parameters, for which no data have been explicitly collected—typically immigration. Such parameters are often subsequently highlighted as important drivers of population growth. Yet, accuracy in estimating their temporal variation, and consequently their contribution to changes in population growth rate, has not been investigated. To quantify the magnitude and cause of potential biases when estimating the contribution of immigration using IPMs, we simulated data (using northern wheatear Oenanthe oenanthe population estimates) from controlled scenarios to examine potential biases and how they depend on IPM parameterization, formulation of priors, the level of temporal variation in immigration and sample size. We also used empirical data on populations with known rates of immigration: Soay sheep Ovis aries and Mauritius kestrel Falco punctatus with zero immigration and grey wolf Canis lupus in Scandinavia with near‐zero immigration. IPMs strongly overestimated the contribution of immigration to changes in population growth in scenarios when immigration was simulated with zero temporal variation (proportion of variance attributed to immigration = 63% for the more constrained formulation and real sample size) and in the wild populations, where the true number of immigrants was zero or near‐zero (kestrel 19.1%–98.2%, sheep 4.2%–36.1% and wolf 84.0%–99.2%). Although the estimation of the contribution of immigration in the simulation study became more accurate with increasing temporal variation and sample size, it was often not possible to distinguish between an accurate estimation from data with high temporal variation versus an overestimation from data with low temporal variation. Unrealistically, large sample sizes may be required to estimate the contribution of immigration well. To minimize the risk of overestimating the contribution of immigration (or any additional parameter) in IPMs, we recommend to: (a) look for evidence of variation in immigration before investigating its contribution to population growth, (b) simulate and model data for comparison to the real data and (c) use explicit data on immigration when possible.


| INTRODUC TI ON
Quantifying the relative contribution of demographic parameters to population growth is essential for understanding the processes influencing population dynamics (Caswell, 2000;Coulson et al., 2005;Koons et al., 2016). This is important in population research, where identifying these relationships not only can help to predict the effectiveness of targeted conservation measures on population growth but also to provide information about the spatial scale at which conservation management should be taken . For example, if local reproduction is a strong contributor to population growth, this suggests that measures supporting reproduction at the local scale may be an effective strategy at managing the population, while a strong contribution of immigration would instead suggest that supportive measures should be undertaken at a larger spatial scale. However, acquiring data on all demographic parameters and their temporal variation is often challenging.
In particular, the measures of immigration are often limited or absent from datasets (Abadi et al., 2010). For this reason, modelling approaches such as integrated population models (IPMs) have been developed that use other demographic data to estimate the missing demographic parameters and their contribution to population growth rates (Abadi et al., 2010;Schaub & Fletcher, 2015;Schaub et al., 2013).
IPMs combine data on demographic rates with data on population size to allow: (a) an estimate of changes in both demographic rates and population growth rate in a joint analysis, and in some cases (b) to estimate additional demographic parameters (Riecke et al., 2019) for which no data have been explicitly collected by integrating information from available data on other parameters and population growth (Kéry & Schaub, 2011;Schaub & Abadi, 2011). Therefore, IPMs offer the exciting possibility to investigate how changes in a demographic parameter are associated with changes in population growth rate, even in cases where no explicit data on this parameter are available (Millon et al., 2019). Hence, a rapidly increasing number of recent studies have used IPMs to estimate the contribution of such additional parameters, typically immigration (e.g. Schaub et al., 2012;Taylor et al., 2018;Weegman et al., 2017), but also productivity or breeding success when populations are assumed to be closed to immigration (Baillie et al., 2009;e.g. Besbeas et al., 2002;Nuijten et al., 2020).
However, since no explicit data are used, estimating the contribution of an additional demographic parameter such as immigration rate must be based on particular modelling assumptions. It has been shown that estimation of the mean immigration rate can be sensitive to the parametrization and priors chosen (Schaub & Fletcher, 2015), and that systematic bias in the estimation of other parameters results in biased estimation of immigration (Riecke et al., 2019). Similarly, the estimation of the temporal variation of immigration (and consequently its contribution to temporal variation in population growth) could also depend on how this variation is parameterized and on the presence of any bias in the temporal variation of the other parameters. Indeed, any residual temporal variation of the other demographic parameters (e.g. temporal random noise in detection probability, density dependence, temporal trends), if not explicitly considered in the models, will likely result in bias of the contribution of immigration to changes in population growth. Given this, caution is needed when interpreting the findings of the many studies that show immigration has the strongest contribution to changes in population growth rate (70% of 44 immigration parameters estimated from 23 studies compiled in Table 1).
Because it is the residual variation that is used to estimate 'missing' parameters like immigration (together with the variance of the observation model), the model parameterization is likely to influence these estimates (Paquet et al., 2019;Saunders et al., 2018;Schaub et al., 2013). Despite this, the vast majority of IPM studies interpret 4.2%-36.1% and wolf 84.0%-99.2%). Although the estimation of the contribution of immigration in the simulation study became more accurate with increasing temporal variation and sample size, it was often not possible to distinguish between an accurate estimation from data with high temporal variation versus an overestimation from data with low temporal variation. Unrealistically, large sample sizes may be required to estimate the contribution of immigration well. 4. To minimize the risk of overestimating the contribution of immigration (or any additional parameter) in IPMs, we recommend to: (a) look for evidence of variation in immigration before investigating its contribution to population growth, (b) simulate and model data for comparison to the real data and (c) use explicit data on immigration when possible.

K E Y W O R D S
immigration, integrated population models, parameter estimation, temporal variation, transient Life  Est mean immigr: estimated mean immigration rate. Estimated contribution: estimated contribution (mean and 95% CIs) of immigration to population growth as value of the correlation, contribution (i.e. % of variation in population growth explained by immigration) or regression coefficient (see column 'Immigr method').
Method: method used to calculate the estimated contribution of immigration: cor = correlation coefficient, contr = LTRE contribution and reg = regression coefficient.
>95% certainty: whether the 95% CIs of the estimated contribution of immigration overlapped zero. Strongest contribution: whether immigration was estimated to be the strongest contributor to variation in growth rate compared with the other demographic parameters.
Biological conclusions: whether biological conclusions were drawn from the contribution of immigration.
Conservation conclusions: whether recommendations or management decisions were suggested based on the contribution of immigration.
Bias discussed: whether potential modelling biases that could explain the contribution of immigration were discussed.
this contribution exclusively as immigration (Table 1). Thus, there is an obvious need to estimate what proportion of the estimated contribution of immigration is truly due to immigration and not to some other component of the residual variation.
Here we use a combination of simulated and empirical data to assess the accuracy of IPMs in estimating immigration as a driver of temporal changes in population growth when immigration is not measured directly. First, we used simulation scenarios to confirm the existence and examine the cause of bias in estimating immigration within the IPM framework. Using 'perfect datasets' simulated with known immigration, we investigate whether IPM parametrization,

| Simulation study
To assess how accurate IPMs are at estimating the temporal variation and contribution of immigration to changes in population growth rate, we simulated data using known parameter values for fecundity, apparent survival (i.e. accounting for both emigration and mortality), population size and immigration. We then applied IPMs to the simulated data and compared modelled parameter estimates to the known values (see Appendix 1 for the scripts used to simulate data and to fit IPMs to simulated data). We simulated a series of datasets (and their underlying time-varying demographic parameters) using the structure of IPMs adapted from the real-data IPM example in Kéry and Schaub (2011) with time varying (random) vital rates, demographic stochasticity accounted for using Poisson and Binomial distributions and a Poisson distribution for the observation model of the count data. We simulated the number of immigrants rather than an immigration rate as it has been suggested to better estimate immigration, particularly in small populations, whereas modelling immigration rate may lead to unrealistically high estimates due to its dependency to population size (Schaub & Fletcher, 2015;. To obtain realistic parameter values for the simulations, we fitted the same IPM structure to a dataset from a northern wheatear population of central Sweden that is open to immigration (Paquet et al., 2020). For all parameters except the number of immigrants (for which further details are given below), we used the posterior medians from this fit in the simulation (see Appendix 1 for parameter values).

| Scenarios
To quantify the bias in the estimation of the temporal variation of immigration and its contribution, we simulated six scenarios that resulted from the combination of using three levels of temporal variation in the number of immigrants (no, moderate or strong) and two levels of sample size. In the no variation in immigration scenario, we kept the number of immigrants fixed at the median of its value as estimated from the wheatear data (N imm,t = e imm = 32, which corresponds to 39% of the initial population size). For the moderate and strong variation in immigration scenarios, we fixed the temporal standard deviation of the number of immigrants (on the log scale), hereafter imm , to either 0.2 (simulating moderate temporal variation) or 0.4 (simulating strong temporal variation).
To understand how sensitive the estimates of imm are to the amount of data available, we simulated 24 years' datasets of 'normal' sample size (the same sample sizes per year as in the real wheatear dataset, see Appendix 1 for sizes of simulated datasets) or 'large' sample sizes (sample sizes per year 10 times larger than the wheatear data; with the initial population size and mean number of immigrants also 10 times bigger).

| Estimating time-varying immigration and its contribution to population growth rate
To estimate temporal variation in the number of immigrants, we fitted the two most typically used formulations on each dataset (Table 1). In the first, most widely used type of IPM (hereafter IPM Pois ), the number of immigrants is strictly positive and allowed to vary around a mean value according to a Poisson log-normal distribution (e.g. Schaub et al., 2012Schaub et al., , 2013Taylor et al., 2018). In the second type of IPM (hereafter IPM NoConst ), the number of immigrants is a fixed parameter estimated independently for each year e.g. Szostek et al., 2014), without constraining the number of immigrants to be positive, nor to vary randomly around a mean.
We estimated the contribution of immigration to changes in population growth rate using the two most common methods. First, we calculated the correlation coefficient between the annual number of immigrants and the annual population growth rates for each posterior sample, as well as the proportion of positive coefficients . Second, we computed recently developed transient Life Table Response Experiment (LTRE) contributions for immigration rate (Koons et al., 2016(Koons et al., , 2017Taylor et al., 2018). This method has the advantage of summing into a meaningful quantity, which should approximate the variation in population growth rate and therefore allows an estimate of the proportion of variation in annual population growth that is explained by the variation in immigration rate (Paquet et al., 2019;Taylor et al., 2018).

| Case studies on three real populations
To assess how accurate IPMs are at estimating the temporal variation of immigration and its contribution in real populations, we compare estimated and true contribution of immigration using longterm data from real populations with known rates of immigration (see Appendix 2 for details on sample sizes, methods and references describing data collection). We built Bayesian IPMs with time varying (random) vital rates. We modelled female breeders only, assuming females are the limiting sex (Rankin & Kokko, 2007). A detailed description of the models is provided in Appendix 3.
For each IPM, we obtained posterior distributions from three independent MCMC chains. Details on prior distribution, initial values, number of iterations, convergence assessment and posterior predictive checks can be found in Appendix 4. All simulations and estimations of posterior distributions were performed using JAGS, version 4.2.0 (Plummer, 2003(Plummer, , 2015 run using the rjags package (Plummer, 2013) code to compute LTRE contributions and correlations is provided in Appendix 5.

| Simulation study
Both IPM parameterizations satisfactorily predicted the mean (i.e. 32 and 320 for both sample sizes) and annual number of immigrants in all scenarios (see Figure S1 for illustration on one of the 100 datasets in each scenario). Nevertheless, IPMs overestimated the LTRE contribution of immigration ( Figure 1; Figure S2, panels A and D) as well as the correlation between the estimated number of immigrants and population growth rate ( Figure S3) for most of the simulations where the number of immigrants did not vary (Table S1; Figure 2). This is because the variation in the number of immigrants was overestimated, due to estimation uncertainty combined with the fact that the standard deviation is constrained to be positive ( Figure 3).
When the simulated number of immigrants varied moderately or

| Case studies
For all case studies, estimated numbers of immigrants were small relative to population size, and 95% credible intervals almost always included zero ( Figure S4). Their temporal variation (estimated with the IPM Pois parameterization) was also low and did not clearly deviate from zero ( Figure S5). The kestrel and the sheep populations are closed to immigration, and for the wolf population, where immigration did occur twice, the posterior contribution of the true immigration rate was close to zero −3.46 × 10 -5 (95% CrI: −0.16.8 × 10 -5 ,

| D ISCUSS I ON
Using both simulations and real case studies, we show that IPMs can strongly overestimate the contribution of immigration to changes in population growth rate. This happened when immigration was simulated with zero temporal variation and in our case studies where immigration was known to be zero or negligible. The strength of this overestimation varied with how immigration was formulated and with sample size. The estimation of the contribution of immigration to variation in growth rate was more accurate when the true (simulated) variation in the number of immigrants was substantial (i.e. imm = 0.2 or 0.4 on the log scale). However, despite this, it was still often not distinguishable from what was estimated when immigration did not vary. Below, we discuss the implications of these results and provide guidelines for more informed inference when estimating the importance of immigration (or any demographic parameter informed by little or no data) for population dynamics using IPMs.
Although previous empirical work has acknowledged the possibility for bias when estimating the contribution of immigration, for example due to spatial mismatch, lack of fit or unmodelled temporal variation in other parameters (Paquet et al., 2019;Saunders et al., 2018;Schaub et al., 2013), our simulation study shows that the contribution of immigration can be strongly overestimated, even in absence of any such biases. We found that in absence of variation in the number of immigrants and with a realistic sample size, the estimated variation in the number of immigrants was substantial ( Figure 3; Figure S1a) and immigration rate was the demographic parameter contributing the most to changes in population growth rate (LTRE contribution representing 63% (95% CrI 30-95) of the total variation for the IPM Pois parameterization, Figure S2). We found that such bias is particularly strong when using the least constrained formulation of immigration IPM NoConst for both the simulation study and the case studies. This is likely due to the uniform priors used to model the number of immigrants independently each year, which induce spurious temporal variation because of the high uncertainty in estimating yearly immigration. Although more rarely used, such formulation has been recommended instead of the IPM Pois formulation to estimate the mean number of immigrants in cases where it is expected to be small (as for our case studies), because it allows negative values (Schaub & Fletcher, 2015),  What can be done in order to get better inference on the temporal contribution of immigration, or any other parameter, to population growth when using IPMs? We recommend to first look for evidence of variation before investigating its contribution to population growth. This can be done by evaluating the shape of the posterior distribution of its variance and assessing whether its peak clearly differs from zero ( Figure 3). Although computationally more time-consuming, a second recommendation is to proceed as we did in our simulation study. That is, simulate datasets of the same size as the real datasets based on the estimated demographic parameters but where immigration (or any parameter of interest) is fixed in time. Then run the IPM using these simulated datasets and compare the contribution obtained with the one obtained when using the real dataset of interest. We provide a step-by-step procedure and R script on how to do so in Appendix 6. As an illustration, using data from a northern wheatear population, our procedure highlights that although immigration is estimated to be by far the main contributor of changes in population growth, a highly similar contribution would have been estimated in absence of any variation in the number of immigrants (Appendix 6). Because the inclusion of an additional parameter in IPMs can help prevent bias in the estimation of the other parameters (Riecke et al., 2019), it could be the case likewise regarding its temporal variation. Therefore, comparing estimates obtained with and without temporal variation in this additional parameter may be useful for better-informed inference regarding the other demographic parameters. That being said, we recommend not interpreting the contribution of additional parameters biologically F I G U R E 4 Transient LTRE contributions representing the part of variation in growth rate explained by immigration rate, and explained by the sum of all other demographic rates for the kestrel (a and b)   If the main aim of a study was to investigate the temporal contribution of immigration to changes in population growth rate, it is advisable to empirically collect and use explicit data on immigration.
In rare situations, where all breeders in the population are monitored and all offspring are marked, immigration can be estimated as the number of unmarked animals recruited into the population (Link & Barker, 2005). In the cases where all subpopulations are monitored, immigration and emigration from and towards each subpopulation can be estimated using multi-state models (Seward et al., 2019).
Moreover, additional data on individuals' locations can be used in spatially explicit IPMs to estimate individuals' movements and hence immigration and emigration (Chandler & Clark, 2014;Chandler et al., 2018;Paquet et al., 2020), although extrapolation to movements at a scale larger than the study area may be problematic.
Spatially explicit IPMs also allow accounting for spatial autocorrelation of parameters, accommodate data collected at different scales and hence avoid bias due to scale mismatch when estimating immigration. Because IPMs offer the possibility of using different types of data into a single modelling framework, then other types of available data such as spatial and genetic data (Millon et al., 2019) should be included in IPMs for more informed estimations of immigration and its temporal variation.

ACK N OWLED G EM ENTS
The authors thank Michael Schaub, the Editors and two anony-

PE E R R E V I E W
The peer review history for this article is available at https://publo ns.

DATA AVA I L A B I L I T Y S TAT E M E N T
Data can be accessed on Dryad Digital Repository https://doi.org/ 10.5061/dryad.xd254 7dh0 .