Spatial synchrony and harvesting in fluctuating populations:Relaxing the small noise assumption.

In spatio-temporal population dynamic models, the most important concept, in addition to mean and variance of local density fluctuations, is the spatial scale of fluctuations in density expressed by studying the spatial autocovariance function. Analytical formulas for this scale in models with local density regulation, dispersal and spatially autocorrelated noise, are rather simple when based on asymptotic theory giving linear models in the limit as the environmental variance approaches zero. The accuracy of these analytical small noise approximations has, however, not been investigated theoretically. Here, we work out improved approximations for the scale as well the spatial autocorrelation function using non-linear logistic local dynamics and going to the next order of approximation with respect to the environmental variance. Generally, it turns out that the asymptotic results are remarkably accurate under moderate fluctuations in density but may be inaccurate for very large fluctuations. For populations with small dispersal capacity, the main error comes from the fact that the logistic dynamics is non-linear, and this error is partly wiped out as dispersal increases. Proportional harvesting has a large effect on the dynamics in spatial as well as non-spatial models, increasing population fluctuations and their spatial scale. The optimal harvesting rate with respect to expected yield per time unit, however, is only to a small extent affected by the magnitude of population fluctuations unless these are very large, so that asymptotic results are applicable over a large range of population fluctuations.