## Introduction

The annual change in abundance or density of a wildlife population is described as the rate of increase. The annual finite rate of change (λ) is estimated as *N*_{t + 1}/*N*_{t} where *N*_{t + 1} is abundance in year *t* + 1 and *N*_{t} is abundance in year *t*. The annual instantaneous rate of change (*r*) is estimated as ln(*N*_{t + 1}/*N*_{t}), or average annual instantaneous rate of change as the slope of the regression of ln*N*_{t} over time, with λ = *e*^{r} (Caughley 1980). The maximum rate of increase under ideal conditions when no resource is in short supply is the intrinsic rate of increase (*r*_{m}) (Caughley 1980).

Caughley (1980) reduced population management to three problems: conservation, sustained yield and pest control. These can be abbreviated as increasing abundance (*r* > 0), maintaining abundance (*r* = 0) and decreasing abundance (*r* < 0). Pest control for conservation, production or human health usually rests on the assumption that a reduction in pest abundance causes a reduction in pest damage (Braysher 1993; Hone 1994). Evidence for the simultaneous drop in pest abundance and damage is equivocal, especially for vertebrate pests (Hone 1994, 1996). To reduce abundance, population growth must be stopped, and to hold abundance at reduced levels the higher population growth that is generated after pest control must be stopped.

Caughley (1980) stated that ‘most populations, although fluctuating from year to year, have a rate of increase which, averaged over several years, is close to zero’. Caughley & Sinclair (1994) state that the ‘rate of increase of a population of vertebrates usually fluctuates gently for most of the time, around a mean of zero’. The frequency distribution of *r* is expected to be a normal distribution (Harris 1986). Aspects of the estimation of rate of increase and biases have been explored by many authors (Caughley & Birch 1971; Harris 1986; Eberhardt 1987; Gerrodette 1987; Eberhardt & Simmons 1992; Johnson 1994) with the general conclusion that such analysis is quite robust but precise estimates require considerable data. Attempts have been made to understand the determinants of rate of increase, for example by using life tables and demographic parameters (mortality and fecundity) (Caughley 1976; Johnson 1994; Sibly & Smith 1998) or using the numerical response (Caughley 1976, 1980, 1987), although these are not examined here.

This paper describes the expected and observed patterns of variation, and hence the means and frequency distributions of rate of increase (*r*), for three species, the European rabbit *Oryctolagus cuniculus* L., red fox *Vulpes vulpes* L. and house mouse *Mus domesticus* Rutty in Australia, and examines the implications of the patterns for wildlife management. The proportions of pests that must be controlled to stop population increases are estimated for each species and compared with observed levels of control achieved in the field.