Single species population fluctuations in patchy environments

Gurney, William and Nisbet, R.M. (1978) Single species population fluctuations in patchy environments. American Naturalist, 12 (988). pp. 1075-1090. ISSN 0003-0147

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In this paper we have assessed the utility of the "zero-correlation" approximation in modeling population fluctuations in a patchy environment. We used a stochastic extension of the original zero-correlation technique to calculate the spectrum and total intensity of fluctuations arising from the stochasticity of colonization and extinction. Comparison of the results of these calculations with the results of realizations of distributed versions of otherwise identical models reveals a systematic discrepancy between the relationship of behavior to microscopic parameters predicted by our zero-correlation analysis and that observed in our realizations. The severity of this discrepancy rises rapidly as the average occupancy of the universe (H0/N) falls, demonstrating clearly that it is primarily a product of the spatial correlation effects neglected by the zero-correlation analysis. Despite its restricted success in predicting the relationship between microscopic and macroscopic parameters observed in our simulations, the zero-correlation analysis predicts with considerable accuracy the interrelations among the various macroscopic aspects of the behavior of our distributed systems. In particular, the predicted relationship between mean patch occupancy (H0/N) and relative fluctuation size (σH/H0), σH/H0 = N-(1/2)[(N/H0)2 - N/H0]1/2, is in very close agreement with that observed in our simulations. This relationship, whose crucial importance is that it enables us to make estimates of the mean time to global extinction from τE ∼ exp [1/2(H0/σH)2], appears to be particularly robust. It is altered neither by a very considerable change in our picture of the mechanism of extinction, nor by the presence of a period at the beginning of the occupied phase of the patch life cycle during which emigration does not occur. It is only very weakly affected by an "unavailable" period following the extinction of the population of a particular patch, but even here the discrepancy is much smaller than could normally be observed in the field. We finally conclude with considerable confidence that a locally unstable population in a patchy environment containing N sites can persist for an appreciable time by balancing extinction and recolonization, only if the average proportion of occupied patches is greater than a limiting value of order of magnitude 3N-(1/2).