Extinction risk and the 1/f family of noise models

In order to predict extinction risk in the presence of reddened, or correlated, environmental variability, fluctuating parameters may be represented by the family of 1/f noises, a series of stochastic models with different levels of variation acting on different timescales. We compare the process of parameter estimation for three 1/f models (white, pink and brown noise) with each other, and with autoregressive noise models (which are not 1/f noises), using data from a model time-series (length, T) of population. We then calculate the expected increase in variance and the expected extinction risk for each model, and we use these to explore the implication of assuming an incorrect noise model. When parameterising these models, it is necessary to do so in terms of the measured ("sample") parameters rather than fundamental ("population") parameters. This is because these models are non-stationary: their parameters need not stabilize on measurement over long periods of time and are uniquely defined only over a specified "window" of timescales defined by a measurement process. We find that extinction forecasts can differ greatly between models, depending on the length, T, and the coefficient of variability, CV, of the time series used to parameterise the models, and on the length of time into the future which is to be projected. For the simplest possible models, ones with population itself the 1/f noise process, it is possible to predict the extinction risk based on CV of the observed time series. Our predictions, based on explicit formulae and on simulations, indicate that (a) for very short projection times relative to T, brown and pink noise models are usually optimistic relative to equivalent white noise model; (b) for projection timescales equal to and substantially greater than T, an equivalent brown or pink noise model usually predicts a greater extinction risk, unless CV is very large; and (c) except for very small values of CV, for timescales very much greater than T, the brown and pink models present a more optimistic picture than the white noise model. In most cases, a pink noise is intermediate between white and brown models. Thus, while reddening of environmental noise may increase the long-term extinction probability for stationary processes, this is not generally true for non-stationary processes, such as pink or brown noises.