## Introduction

Historically, the need for quantitative fisheries management led Ricker (1954, 1958) to develop his highly influential phenomenological model of population dynamics (Geritz & Kisdi 2004), encapsulating the key idea of density dependence as the mechanism preventing unbounded growth. We set out the Ricker model first, and then introduce the shape parameter θ to arrive at the theta-logistic model. Following Lotka (1925) and Volterra (1926) who had formalized and explored simple and specific models of predator–prey interactions, Ricker also built on the work of Verhulst (1838) and Pearl (1828) in generalizing the idea of a population tending to fluctuate around an equilibrium abundance.

What has come to be known as Ricker’s model can be written formally (Cook 1965; Royama 1992) as a difference equation:

that is, the abundance at the next time point *N*_{t+1} is modelled as being equal to the current abundance *N*_{t} multiplied by a value *e*^{r}, where is the growth rate over the time step. The deterministic component of the growth rate declines linearly from a maximum (intrinsic) growth rate *r*_{m} at low abundance, to zero when *N*_{t} = *K* (i.e. *K* quantifies the carrying capacity), and thence negative for *N*_{t}* > K* (identical to the θ* *= 1 trace in Fig. 1a). The *process error* (also called the *environmental variation*) is used in model fitting to capture the real-world differences between population census data and the idealization of the model (see Methods; also Appendix S1, Sections 2 and 6, Supporting information).

Although it can be mathematically ‘derived’ from first principles in various ways (Royama 1992), the Ricker model and its variants are most usefully understood as a mathematical formulation of the ideas they represent (Fig. 1). It is important to appreciate that the Ricker family of models are phenomenological, as opposed to physical or mechanistic. One particularly desirable feature of Eqn 1 is the maintenance of non-negative abundance values (i.e. *e*^{r} > 0, for all *r*). Another is that simple linear regression of (sometimes called pgr for per-capita growth rate – e.g. Sibly *et al.* 2005) against *N*_{t} allows estimates of *K* and *r*_{m} to be extracted from abundance time series: the intercept of the regression (i.e. the *N *=* *0 intercept) equates to *r*_{m}; the slope equates to −*r*_{m}/*K*. Note that the fitting approach we demonstrate here can allow *K* to take any form that the modeller chooses to apply.

Ricker’s phenomenological model is one of a family that includes the Gompertz (Pollard, Lakhani & Rothery 1987), Beverton–Holt (Beverton & Holt 1957) and theta-logistic models, which collectively are used often in applied ecology to estimate maximum sustainable yield targets (Cameron & Benton 2004), temporal abundance patterns (Sæther, Engen & Matthysen 2002), the most effective wildlife management interventions (Caughley & Sinclair 1994), extinction risk (Philippi *et al.* 1987) and epidemiological patterns (Anderson & May 1991). This family of models encapsulates three principal abstractions: that (i) carrying capacity *K* is constant, (ii) factors driving population dynamics can be split into intrinsic (density-dependent) and extrinsic (environmental variation) categories and (iii) the population growth rate *r* can be expressed as a simple function of abundance *N*. It is this last abstraction, being the form of the *r*–*N* relationship (what we call the ‘growth response’ in ecological context) that differentiates the various models; in the Ricker model, this is simply a linear relationship.

The extrinsic factors (e.g. environmental variation in food abundance, predation, disease, breeding conditions) overlay, or can even dominate, the deterministic density dependence. Early debate centred on whether (Andrewartha & Birch 1954) or not (Nicholson 1957) extrinsic forcings overwhelmed intrinsic ones, with modern work establishing that density dependence often plays an important role in species’ dynamics (Turchin & Taylor 1992; Woiwod & Hanski 1992; Zeng *et al.* 1998; Lande *et al.* 2002; Brook & Bradshaw 2006).

While the Ricker model assumes a linear decline in the growth rate with abundance, both theoretical and empirical work (Fowler 1981; Johst, Berryman & Lima 2008) have shown that some populations have concave (as viewed from above) *r*–*N* curves. Concave *r*–*N* curves are typical of so-called ‘*r*-selected’ organisms where density dependence acts strongly at lower densities, whereas convex curves arise from ‘*K*-selected’ species where density dependence acts to reduce growth only at higher densities (although these terms have fallen out of favour). Studies of laboratory fruit fly populations by Gilpin & Ayala (1973) led them heuristically to add a shape parameter θ to the growth term in Ricker’s model, such that;

which resulted in the Theta-Ricker or theta-logistic model (Gilpin & Ayala 1973; Thomas, Pomerantz & Gilpin 1980). Thus, the growth response is parameterized jointly by *r*_{m} and θ (Fig. 1). For a population growing from low abundance, the parameter θ reflects those aspects of a species’ evolved life history (demographic rates) that determine how abruptly growth slows as abundance interacts with resource availability (Freckleton *et al.* 2003; Sibly *et al.* 2005) and type of competition (Johst, Berryman & Lima 2008). When concave (θ* *< 1), the growth response (i.e. the ‘*r*–*N* curve’) ideally characterizes a population unable to recover quickly from extrinsic perturbations, whereas a convex growth response (θ* *> 1) implies that density feedback occurs mainly above some (relatively large) threshold abundance (Fowler 1981; Owen-Smith 2006), typical, for instance, of the population dynamics in large mammals (McCullough 1999; Owen-Smith 2006). For θ* = *0·1, the growth response at *N = K*/2 is 6·7% of *r*_{m} compared with 50% for the Ricker (θ* = *1), and 99·9% when θ* *=* *10; in the latter case, growth drops to 90% of *r*_{m} at *N *∼* *0·8 *K* and to 50% at ∼0·93 *K*. We henceforth use this range (0·1 ≤ θ* *≤ 10) as sufficiently inclusive for most real populations (Fig. 1).

Census data for real populations usually show a fluctuating population rather than one growing steadily from low abundance. Thus, when fitting models to such data, what is actually being measured is only that part of the growth response for *N* ∼ *K*. Also, as these phenomenological models describe only a population’s tendency to return to carrying capacity, the actual dynamics of a real population over each time step will often differ greatly from the tendency suggested by the model.

Although mechanistic models of population dynamics are typically more desirable to investigate the role of density feedback, for most populations census data are more readily available than detailed life-history data, especially those of sufficient duration to estimate process variability in vital rates (Brook & Bradshaw 2006). As such, a primary applied use of the theta-logistic is to describe growth dynamics at population sizes distant from *K*: for instance, to determine viability of small populations or sustainable yields of harvested populations (Philippi *et al.* 1987; Cameron & Benton 2004). Because maximum sustainable yield occurs where the product of abundance and growth rate are maximal, the use of an incorrect growth response can lead to a calculated yield that is sub-optimal or unsustainable (i.e. if curvature is less or more concave than supposed).

Sibly *et al.* (2005) analysed abundance time series for mammals, birds, fish and insects and controversially (Doncaster 2006; Getz & Lloyd-Smith 2006; Ross 2006; Sibly *et al.* 2006) found that concave *r*–*N* curves dominated (i.e. concave growth responses if taken as ecologically real). This substantive work has been important in bringing attention to the problems we examine. Despite differences in methodology between Sibly *et al.* (2005) and that used here (examined and compared explicitly in the Appendix S1, Section 4, Supporting information), we also observe a dominance of concave *r*–*N* curves when fitting real abundance data; however, we show that these cannot be taken as ecologically realistic and thus reject the conclusion that concave growth responses are shown to dominate in nature.

In this work, we demonstrate and clearly explain our observation that estimating the curvature of the growth response from abundance data, in particular using the theta-logistic model, is highly problematic. The specific technical point at the core of this explanation has been independently shown by Polansky *et al.* (2009), and we thus focus on providing a complementary and less technical explanation of the specific problem and its consequences, while also exploring some general problems in obtaining meaningful results when applying phenomenological models to census data. While we include some analysis and discussion of real data, the foundational demonstrations are based on simulated data; the logic being that simulated data have known dynamics and thus allow the results of model fitting to be interpreted against a known reality. We use simulated data to examine: (i) a population fluctuating around a constant carrying capacity (i.e. stationary time series), (ii) recovery of a population from 10% of carrying capacity, and (iii) a population subject to sinusoidal variation in *K*. Further, we: (i) demonstrate the usefulness of confidence intervals from resampling, (ii) contrast the simulation results with those from real abundance data, and (iii) provide an example that demonstrates the implications of poorly defined θ for estimating extinction risk. Throughout we stress the relationship between the parameters θ and *r*_{m} in jointly parameterizing the *r–N* curve for the theta-logistic model. We also (iv) examine what can happen when the assumption of a constant *K* is not met.

The overall approach – back-fitting simulated data to examine how readily generating parameters can be recaptured – is a standard technique that has broad applicability in ecological modelling. The contrast between the analysis of simulated and real data illustrates this point, and also the need for careful consideration of model assumptions, measurement error and confidence intervals when using models to estimate real ecological parameters. An extensive Appendix S1 (Supporting information) contains (i) further details and explanations in relation to the practical aspects of model fitting that will be useful for those not well-versed in fitting mathematical models to ecological data, and (ii) some more technical aspects, in particular, the geometry of the objective landscape (likelihood profile) which is characterized by *r*_{m} − θ gullies containing an arc of parameter fits of similar likelihood (equivalent to the ‘ridges’ described in Polansky *et al.* 2009).