## Introduction

Since the pioneering work by the Australian statistician P.A.P. Moran in the early 1950s (e.g. Moran 1953), the spatial dimension of population dynamics and especially large-scale synchrony in population fluctuations has received a lot of attention from ecologists (Royama 1992; Ranta *et al*. 1995; Bjørnstad, Ims & Lambin 1999; Koenig 1999). There is a growing interest in extending the analysis of ecological dynamics to include the spatial dimension (e.g. Bascompte & Solé 1997; Tilman & Kareiva 1997; Dieckmann, Metz & Law 2000).

The spatial distribution of individuals can be important when the primary interest is to understand temporal dynamics. For instance, populations at different spatial locations may differ in terms of demography (LaMontagne, Irvine & Crone 2002) and how they respond to environmental fluctuations, i.e. averaging across space can be misleading. The driving environmental variables may also vary across space. Hence, to be able to analyse space–time series data in a rigorous way, population models should allow for spatial variation in growth-rate parameters and environmental variability, as well as spatially correlated error terms (Dennis, Kemp & Taper 1998). In some cases, the populations may also be connected by dispersal, which has turned out to be challenging to estimate, but possible, from space–time series data (Lele, Taper & Gage 1998). Failure to account for spatial patterns may sometimes lead to very different conclusions about an organism's ecology, such as when estimating environment–abundance relationships (Keitt *et al*. 2002).

In this paper, we use a multivariate maximum likelihood framework to study spatio-temporal population dynamics in the red kangaroo (*Macropus rufus*, Desmarest) inhabiting the pastoral zone of South Australia. We are primarily interested in finding out to what extent kangaroo dynamics are affected by rainfall, intraspecific competition, interspecific competition with sheep and cattle, and whether there are any spatial differences in these interactions between different management areas. Previous kangaroo studies have highlighted the importance of the interactions listed above, but no attempt has been made to put them all together in a unified statistical framework.

Because kangaroos interact with sheep via resource competition (Caughley 1987; Edwards 1989) and are harvested for meat and skins (Ramsay 1994; Pople & Grigg 1998; Grigg & Pople 2001), there is a clear need for population models that can be used to evaluate alternative decisions under uncertainty. There is also a need to sort out how the different processes identified above translate into observable kangaroo dynamics and that is our major goal here. Such knowledge is important to be able to make reasonable management decisions, such as optimal harvesting in response to predicted grazing pressure and rainfall. It is also valuable on a more general level because different population processes can give rise to the same patterns in time-series data (Jonzén *et al*. 2002a; Jonzén, Ripa & Lundberg 2002b). However, the system dealt with in this paper is relatively well studied, which minimizes the risk of ignoring key processes governing a population's dynamics (Jonzén *et al*. 2002a).

It has previously been assumed that the red kangaroos inhabiting the study area make up a single large and uniform population (McCarthy 1996), but there are other studies suggesting that there may be differences between at least the western and the central and eastern regions in terms of the numerical response to rainfall (Cairns & Grigg 1993). Furthermore, the study area has been divided by the government management agency into different management regions, each with its own annual harvest quota (SADEH 2002). Hence, there are ecological as well as management reasons to increase the spatial resolution of current understanding and to study the temporal dynamics of the red kangaroo in the different management areas rather than across South Australia as a whole.

### study area and data

The pastoral zone of South Australia covers approximately 282 000 km^{2} in area and comprises a range of different landforms and vegetation types (Laut *et al*. 1977). The aerial survey of kangaroos in the pastoral zone of South Australia was initiated in 1978 and has been conducted annually each winter. Survey methods are described elsewhere (Caughley & Grigg 1981; Cairns & Grigg 1993).

Data on red kangaroo densities (individuals km^{−2}) in six management regions of South Australia (Fig. 1) were collected by winter aerial survey (Caughley & Grigg 1981; Grigg *et al*. 1999) between 1978 and 2002. To reduce the number of parameters that we had to estimate, we amalgamated data from pairs of similar regions, resulting in the following four major areas: Eastern Districts, Gawler, Kingoonya/Maree, and North Flinders Ranges/North-east Pastoral, referred to as areas 1, 2, 3 and 4, respectively. The pooled regions border each other and have similar rainfall statistics.

Kangaroos are shot throughout the year and the carcasses are brought to nearby refrigeration units or dealer sites throughout the pastoral zone. The density (individuals km^{−2}) of kangaroos harvested on properties in South Australia each year were determined from shooter and dealer returns collated by the South Australian government conservation agency (currently the Department for Environment and Heritage). Records of harvested kangaroos were available only for the State as a whole in 1978 and 1979. Therefore, the proportion of the State total that was shot in each region in 1980 was used to apportion the State total in 1978 and 1979. The number of red kangaroos harvested in each region between consecutive aerial surveys was used in the analysis.

We used rainfall data from each of the management areas collected during the 12-month period prior to the start of the winter census. Rainfall during this period has been found to have the best positive correlation with the population rate of change of red kangaroos between two censuses following this 12-month period of rain, at least on the broad spatial scale of the whole pastoral zone in South Australia (McCarthy 1996).

Sheep are the predominant domestic stock in the South Australian pastoral zone, with cattle run mostly on properties in the north of the zone. Mean annual numbers of sheep and cattle on properties in the pastoral zone were determined from graziers’ stock returns collated by the South Australian government department responsible for primary industries (currently Department of Water, Land and Biodiversity Conservation). Cattle numbers were converted to dry sheep equivalents by multiplying by eight, the conversion factor recommended by the Department of Water, Land and Biodiversity Conservation in South Australia (Reid 1990). Henceforth, we refer to the combined mean as Dry Sheep Equivalents (DSE), which will be expressed as a density (km^{−2}).

### methods of analysis

#### (a) Background to kangaroo modelling research

Several different models of macropod populations have been developed (reviewed by Cairns 1989) and the focus of most studies has been to document a general impact of (time-lagged) rainfall on population rate of change in the red kangaroo (e.g. Caughley, Bayliss & Giles 1984; Bayliss 1985a,b; Cairns & Grigg 1993; McCarthy 1996). Rainfall is a proxy for pasture growth and biomass and is important for predicting fluctuations in kangaroo populations. In the pastoral zone of South Australia, initial data analysis (Cairns & Grigg 1993) found that red kangaroo populations respond to rainfall at short time-lags on the scale of single management areas of 20 000–40 000 km^{−2}. However, a longer lagged effect of rainfall was found using a longer time series on a broad scale across the entire South Australian pastoral zone (McCarthy 1996).

Statistical density dependence (i.e. a relationship between population rate of change and density) was detected on a broad scale covering the entire pastoral zone of South Australia (McCarthy 1996), but the processes underlying that pattern are not fully understood. Whereas intraspecific competition for food may seem to be a logical explanation, one must also consider that kangaroos compete with sheep and cattle (Edwards 1989). The conventional wisdom is that domestic livestock density has a marked influence on the long-term density of kangaroos, but only a negligible effect on their short-term dynamics (Caughley 1987). In other words, domestic herbivore density should not affect the rate of change of kangaroos. However that has not been shown statistically. In this paper we explore different models for the impact of domestic herbivore density on kangaroo population dynamics at the regional scale.

#### (b) Models

Let *N*_{i,(t)} be the kangaroo population density (individuals km^{−2}) in area *i* at time *t* for *i* = {1,2,3,4} corresponding to Eastern Districts, Gawler, Kingoonya/Maree, and North Flinders Range/North-east Pastoral, respectively. We assume the mapping of density from one year to the next is described by a multivariate stochastic Ricker model (Dennis *et al*. 1998) including harvesting (*H*), rainfall (*R*) and the DSE density (*S*) as covariates. Hence, the full model can be written

where *a*_{i} and *b*_{i} are constants for each area *i*, often referred to as a drift or location parameter and statistical density dependence, respectively. We assume the dynamics are influenced by a regionally and time-dependent environmental random variable *E*_{i,(t)} that is drawn from a multivariate normal distribution with mean zero and variance-covariance matrix Σ. The parameters *c*_{i} and *d*_{i} capture the local response to rainfall and DSE in each area. To be able to compare the response to kangaroo density, rainfall and DSE across the areas, we standardized these time series to zero mean and unit variance.

*H*_{i,(t)} is the instantaneous harvest mortality in area *i* between winter in year *t* − 1 and winter in year *t* calculated as

where *C*_{i,(t)} is the total number of harvested animals per km^{2} in each area *i* between consecutive aerial surveys. To approximate the annual harvest fraction, we have to divide *C*_{i,(t)} by the geometric mean of population density in year *t* − 1 and *t* because harvesting is not a discrete event and population density is estimated only once every year (see Cairns & Grigg 1993). By not fitting a coefficient to the harvest term, we are assuming that the effect of harvesting is constant for a given population-growth rate, and the population-growth rate was simply adjusted for harvest rate (i.e. we assumed the coefficient was one). Estimating a coefficient (other than 1) would be seeking the level of compensation (< 1) or perhaps superadditivity (> 1) in harvesting in addition to the density dependence that was modelled, and the effect of harvesting is not the focus of this paper.

Finally, we also consider a different model structure motivated by the theory of ratio-dependent consumer–resource interactions (Arditi & Ginzburg 1989) and a previous study on red kangaroo dynamics in South Australia (McCarthy 1996). This second model is

where *b*_{i} is now the regression coefficient with respect to the ratio of kangaroo density and rainfall (i.e. not standardized as above), *d*_{i} is the regression coefficient with respect to the ratio of DSE and rainfall. These ratios were standardized to mean zero and unit variance to facilitate the comparison across areas.

#### (c) Parameter estimation and model selection

The stochastic multivariate Ricker model with covariates (equation 1) can be considered a hybrid between an ecological and a statistical model in the sense that the parameters have an ecological interpretation, but the model can be expressed as a statistical time-series model on a logarithmic scale. If we define ln(*N*) ≡ *X*, we can write equation 1 as a multivariate first-order nonlinear autoregression (NLAR) model (Tong 1990) with linear covariates on the log-scale:

where the boldface indicates that the parameters and data have vector (**a**, **b**, **c**, **d**, **H**_{(t)} and **E**_{(t)} are column vectors) or matrix (**N**_{(t−1)}, **R**_{(t−1)} and **S _{(}**

_{t−1)}are diagonal matrices) structure. The likelihood function for a multivariate NLAR with Gaussian error structure is presented in Dennis

*et al*. (1998) and the log-likelihood, which was used for estimating the unknown parameters, can be written as

where **Z**_{t} is a vector of residuals at time *t*, *m* is the number of areas (*m* = 4) and all sums are from time *t* = 1 to *t* = *q* (= 25). We obtain maximum-likelihood estimates of the unknown parameters by minimizing the negative log-likelihood (–ln *L*(**a**, **b**, **c**, **d**, Σ)) using the Nelder–Mead simplex algorithm (Press *et al*. 1994). For a more detailed explanation of the multivariate normal likelihood function, see Dennis *et al*. (1998).

We calculate the likelihood for each of a set of candidate models where equations 1 and 3 describe the full models assuming either additive (equation 1) or ratio-dependent (equation 3) effects of density and rainfall. We confront models with and without a term for density dependence and/or DSE, but harvesting, rainfall and a location parameter were included in all models. Each model could be further classified as having population-specific or global parameter values and the variance-covariance matrix had either zero or non-zero off-diagonal elements. Model selection was guided by the information-theoretic approach and we used the Akaike Information Criteria corrected for small sample size, AIC_{c} (Burnham & Anderson 1998; p. 51) to rank the alternative models. We ignored observation error because we have no *a priori* reason to assume that the magnitude of the observation error should differ among the alternative models, and relative differences should therefore remain similar (LaMontagne *et al*. 2002).

Finally, we undertook a residual analysis of the best model as determined by the smallest AIC_{c} value to make sure that the residuals were approximately normally distributed and not strongly serially correlated. For this purpose, one can treat the residuals from the four areas as approximately independent (Tong 1990). We performed Lilliefors test for goodness of fit to a normal distribution at the α = 5% level (Conover 1980), and we analysed the residuals for autocorrelation by estimating the partial autocorrelation function. The critical value of the partial autocorrelation coefficient is considered significantly differently from zero at the 5% level if it is greater than | 2/√*n* | = 0·408, where *n* is the length of the residual vector (Chatfield 1999).