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Non-linearities are commonly observed in the responses of organisms to environment. They potentially modify the qualitative and quantitative properties of population dynamics. We studied how non-linear responses to environment, or "noise filters", influence population variability and extinction risk by introducing coloured noise to the growth rate in the Hassell single-species model. The consequences of noise filtering were analysed by comparing the model dynamics with linearly filtered and non-linearly filtered noise that have the same mean. Three biologically plausible filters we used: saturating, unimodal optimum type, and sigmoid responses. Filtering can either decrease or increase population variability when compared to linear noise response. The effect of noise filtering on variability is most pronounced with stable population dynamics and the outcome depends on the filter type, population growth rate, and noise colour. Non-linear noise filtering predominantly increases extinction risks when population growth rate is low (R < 5). As an exception, saturating filter has a window of decreased risk at very low growth rate and reddened environment. In the unstable range of the dynamics (15 < R < 25), the optimum and saturating type responses can decrease the extinction risk whereas the sigmoid response increases extinction risk. Noise filtering can also alter the route to extinction in slowly growing populations from slow to rapid decline towards extinction in optimum and saturating responses. These results suggest that accounting for the non-linear responses to environment should be considered when estimating extinction risks and population variability. Moreover, the non-linear responses make noise colour a more important factor in these analyses.
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