## Introduction

Mathematical models of the dynamics of density feedback based on abundance time series, including the Ricker, Gompertz and *θ*-logistic, give a functional form to the relationship between sequential population size estimates under the constraints of maximum growth rate and environmental carrying capacity. The estimated strength and type of density feedback arising from these phenomenological models influence projections of population viability and sustainable yield targets in planned conservation and management interventions, such as the proportional annual harvest that would lead to extinction (Holmes *et al*. 2007; Hone, Duncan & Forsyth 2010). Estimates of the growth response of individual populations are obtained from fitting stochastic models of density feedback. However, these models fitted to abundance data often give biased (Lande *et al*. 2002; Freckleton *et al*. 2006), imprecise and ecologically unrealistic (Polansky *et al*. 2009; Clark *et al*. 2010) estimates of growth response parameters.

Bayesian statistical methods provide an explicit means of using prior knowledge to inform the estimation of the parameters of ecological models (Clark 2007; McCarthy 2007). Such information is incorporated into the estimation process by specifying prior distributions for model parameters which weight the likelihood function to generate posterior distributions and can improve the precision of the estimates (McCarthy & Masters 2005). As an example, allometric and demographic predictions of ecological rates can be used within a Bayesian setting as independent, *a priori* distributions. McCarthy, Citroen & McCall (2008) showed that estimated allometric scaling exponents for birds and mammals conform to theoretical predictions and used the corresponding estimates as Bayesian priors to refine estimates of survival rate.

In population dynamics, the stochastic Ricker model (Ricker 1975) predicts the population size *N* (or an index thereof) at time *t* as a function of the population size at the preceding time, the maximum rate of population increase *r*_{m} and the population carrying capacity *K*. Taking *Y*_{t} = log(*N*_{t}), the model is given as follows:

where *ε*_{t−1} is the process (or environmental) variance (a Gaussian-distributed random variable with mean 0 and variance *σ*^{2}). This model can be generalised to account for a nonlinear growth response to changes in population size by adding a shape parameter *θ*:

where the growth response, or return tendency, is then represented jointly by the product of *r*_{m} and *θ* (Saether *et al*. 2008; Clark *et al*. 2010); setting *θ* = 1 gives the Ricker model (see Fig. 1a in Clark *et al*. 2010).

These models can be further generalised to account for sampling errors in the measurement of population size, which is likely to exist in most real-world monitoring situations and can bias parameter estimates (Shenk, White & Burnham 1998; Freckleton *et al*. 2006). State-space time-series models can incorporate both observation and process error (de Valpine & Hastings 2002; de Valpine 2003; Clark & Bjørnstad 2004). Here, recorded population sizes are assumed to represent an unobserved ‘true population state’ plus random errors. Process error represents the stochastic model of temporal variation in the unobserved state variable. de Valpine & Hastings (2002) showed that state-space estimation in linear, Gaussian-distributed models accounting for process and observation error performed better in recovering the simulated parameters than models that ignored either process or observation error.

The state-space formulation of the Ricker model for log-transformed observed population size at time *t*, log(*N*_{t}) = *Y*_{t}, can be given as:

where *Y*_{t} is equal to the unknown ‘true’ population state *X*_{t} plus random observation (i.e. measurement) error *v*_{t}*,* which can be, for example, Gaussian- or Poisson-distributed. The unknown population state *X*_{t} propagates through time via a Ricker model that depends on *r*_{m}, *K* and random process (e.g. environmental) error *ε*_{t-1}, which is usually assumed to follow a Gaussian distribution with mean 0 and process variance *σ*^{2}. Calder *et al*. (2003) used Gibbs sampling to fit a Ricker state-space model (as a linear model on the raw abundance scale); however, the Bayesian framework also allows nonlinear functional forms of density feedback (Wang 2007) and non-Gaussian probability density functions. State-space approaches have been used previously to model the dynamics of mammal populations (Zeng *et al*. 1998; Wang *et al*. 2006), including using Bayesian estimation (Clark & Bjørnstad 2004). Bayesian fitting of these models requires specification of prior distributions for all model parameters, including the initial population state *X*_{0} (Calder *et al*. 2003).

Predicting *r*_{m} and *K* from the *θ*-logistic model, however, gives biased and imprecise estimates because of the inherent trade-off between *r*_{m} and *θ* (Polansky *et al*. 2009; Clark *et al*. 2010), particularly where population abundance fluctuates around *K* and thus contains little information about the true value of *r*_{m}. Hence, one option is to ‘fix’ the *r*_{m} parameter (Saether, Engen & Matthysen 2002) to reduce the negative correlation between *r*_{m} and *θ* which, without some independent knowledge of at least one, tends to make estimates of both parameters ecologically meaningless (Clark *et al*. 2010). The *r*_{m} parameter represents the capacity of a population to grow under conditions where resources are not limiting and competition is negligible (Sibly & Hone 2002; Savage *et al*. 2004a), such that survival and reproduction are maximal. Thus, it can be estimated directly from demographic data using population matrix models parameterised with maximum vital rates (Caswell 2001), or observed abundance counts from populations at low abundance and where survival and reproduction are maximal using the Lotka–Euler equation (Kot 2001). When detailed demographic data are unavailable, a simplified, two-stage version of Cole's equation (Cole 1954) using annual fecundity, age at first reproduction and reproductive life span (and assuming survival probability = 1) can also be used to estimate *r*_{m}, although this approach can overestimate it (Fagan, Lynch & Noon 2010).

Theoretical relationships between a species' life history and rates of population growth can also be used to estimate *r*_{m}. There are several examples where allometric relationships to predict *r*_{m} have been fitted to data sets on mammals and other taxa (Fenchel 1974; Caughley & Krebs 1983; Hennemann 1983; Thompson 1987; Sinclair 1996; McCallum, Kikkawa & Catterall 2000; Duncan, Forsyth & Hone 2007). These theoretical relationships provide the basis for the allometry of interspecific variation in *r*_{m} (Savage *et al*. 2004b), although phylogenetic differences should ideally be incorporated into predictive models (Duncan, Forsyth & Hone 2007). Hone, Duncan & Forsyth (2010) showed for mammals that Cole's equation accurately predicts field-based estimates of *r*_{m}; estimated from age at first reproduction (*α*), *r*_{m} estimates match theoretical predictions (slope = −1) of the (log_{10}–log_{10}) *r*_{m} − *α* relationship. The theoretical slope of −1 is derived from a rearrangement of the simplified Cole's equation where survival in all age classes is fixed at 1 (Duncan, Forsyth & Hone 2007). Hone, Duncan & Forsyth (2010) recommend using predictions of *r*_{m} from *α* over those estimated from allometric relationships based on body mass because the former is independent of phylogeny (Duncan, Forsyth & Hone 2007).

We used two data sets of population growth rates with associated allometry (body mass; 44 species) and demography (age at first reproduction; 64 species) variables to develop predictive models of maximum population growth. To assess the usefulness of this prior information, we used these independent prior distributions for *r*_{m} in a Bayesian framework to estimate posterior distributions of the parameters in the Ricker and *θ-*logistic models fitted to population time series for 36 mammal species. While Bayesian state-space models that account for observation error in the abundance counts have been used previously to investigate the dynamics of mammal populations, ours is the first to incorporate independent prior information about maximum population growth (including uncertainty) into the estimation of density-feedback model parameters. We hypothesised that accurate prior information would give rise to more biologically realistic estimates of maximum population growth for both models, so we compared posterior estimates from informative prior models with those from uninformative (vague) prior models. Additionally, we simulated abundance series from Ricker and *θ-*logistic models with known parameter values and refitted models to these data to assess the characteristics of time series that resulted in biased estimates of *r*_{m}.

We also hypothesised that posterior estimates of *r*_{m} would be sensitive to incorrect (i.e. unrealistic) prior information and were interested in how much the priors affected posteriors for each model. For example, we expected the prior to dominate the posterior for the *θ-*logistic model based on previous research showing high correlation between the *r*_{m} and *θ* estimates and the generally flat likelihood for this model (Polansky *et al*. 2009; Clark *et al*. 2010). Therefore, we assessed relative changes in posterior means from specifying incorrect relative to correct priors as a measure of how much (or little) information was contained in the data about the *r*_{m} parameter.